Alternative Investments and Strategies

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World Scientific

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Alternative investments and strategies / edited by Rüdiger Kiesel, Matthias Scherer & Rudi Zagst. p. cm. ISBN-13: 978-9814280105 ISBN-10: 9814280100 1. Investments--Moral and ethical aspects. 2. Portfolio management--Moral and ethical aspects. I. Kiesel, Rüdiger, 1962– II. Scherer, Matthias. III. Zagst, Rudi, 1961– HG4515.13.A498 2010 332.6--dc22 2010013167

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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PREFACE

Asset allocation investigates the optimal division of a portfolio among different asset classes. Standard theory involves the optimal mix of risky stocks, bonds, and cash together with various subdivisions of these asset classes. Underlying this is the insight that diversification allows for achieving a balance between risk and return: by using different types of investment, losses may be limited and returns are made less volatile without losing too much potential gain. These insights are made precise using the benchmark theory of mathematical finance, the Black-Scholes-Merton theory, based on Brownian motion as the driving noise process for risky asset prices. Here, the distributions of financial returns of the risky assets in a portfolio are multivariate normal, thus relating to the standard meanvariance portfolio theory of Markowitz with its risk-return paradigm as above. Recent years have seen many empirical studies shedding doubt on the BlackScholes-Merton model, and motivating various alternative modeling approaches, which were able to reproduce the stylized facts of asset returns (such as heavy tails and volatility clustering) much better. Also, various new asset classes and specific financial tools for achieving better diversification have been created and entered the investment universe. This book combines academic research and practical expertise on these new (often called alternative) assets and trading strategies in a unique way. We include the practitioners’ viewpoint on new asset classes as well as academic research on modeling approaches, for new asset classes. In particular, alternative asset classes such as power forward contracts, forward freight agreements, and investment in photovoltaic facilities are discussed in detail, both on a stand-alone basis and with a view to their effects on diversification in combination with classical asset. We also analyse creditrelated portfolio instruments and their effect in achieving an optimal asset allocation. In this context, we highlight aspects of financial structures which may sometimes be neglected, such as default risk of issuer in case of certificates or the role that model v

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risk plays within asset allocation problems. This leads naturally to the use of robust asset allocation strategies. Extending the classical mean-variance portfolio setting, we include dynamic portfolio strategies and illustrate different portfolio protection strategies. In particular, we compare the benefits of such strategies and investigate conditions under which Constant Proportion Portfolio Insurance (CPPI) may be prefered to Option-Based Portfolio Insurance (OBPI) and vice versa. We also contribute to the understanding of gap risk by analyzing this risk for CPPI and Constant Proportion Debt Obligations (CPDO) in a sophisticated modeling framework. Such analyses are supplemented and extended by an investigation of the optimality of hedging approaches such as variance-optimal hedging and semistatic variants of classical hedging strategies. Many of the articles can serve as guides for the implementation of various models. In addition, we also present state-of-the-art models and explain modern tools from financial mathematics, such as Markov-Switching models, time-changed Lévy models, variants of lognormal approximations, and copula structures. This books combines a unique mix of authors. Also many of our students improved the outcome of the project with critical and insightful comments. Particular thanks goes to Georg Grüll, Peter Hieber, Julia Kraus, Matthias Lutz, Jan-Frederik Mai, Kathrin Maul, Kevin Metka, Daniela Neykova, Johannes Rauch, Andreas Rupp, Daniela Selch, and Christofer Vogt. R. Kiesel, M. Scherer, and R. Zagst

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CONTENTS

Preface

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Part I. Alternative Investments Chapter 1. Socially Responsible Investments

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Sven Hroß, Christofer Vogt and Rudi Zagst 1.1 1.2 1.3

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1.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Recent Research on SRI . . . . . . . . . . . . . . . . . . . How Sustainable is Sustainability? . . . . . . . . . . . . . . 1.3.1 Description of the Dataset . . . . . . . . . . . . . . 1.3.2 Introduction to Markov Transition Matrices . . . . . 1.3.3 Results of Markov Transition Matrices . . . . . . . SRI in Portfolio Context . . . . . . . . . . . . . . . . . . . 1.4.1 Description of the Dataset and Statistical Properties 1.4.2 Markov-Switching Model . . . . . . . . . . . . . . 1.4.3 Fitting the Model Parameters . . . . . . . . . . . . 1.4.4 Simulation of Returns . . . . . . . . . . . . . . . . 1.4.5 Portfolio Optimization Models . . . . . . . . . . . 1.4.6 Definition of Investor Types . . . . . . . . . . . . . 1.4.7 Optimal Portfolios . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 2. Listed Private Equity in a Portfolio Context

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Philipp Aigner, Georg Beyschlag, Tim Friederich, Markus Kalepky and Rudi Zagst 2.1 2.2

Introduction . . . . . . . . . . . . . Defining Private Equity Categories . 2.2.1 Financing Stages . . . . . . 2.2.2 Divestment Strategies . . . vii

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Type of Financing . . . . . . . . . . . . . . . . . . . . . . Classification of Private Equity Fund Investments . . . . . 2.2.4.1 Venture capital funds . . . . . . . . . . . . . . . . 2.2.4.2 Buyout funds . . . . . . . . . . . . . . . . . . . . 2.2.4.3 Leveraged buyouts (LBO) . . . . . . . . . . . . . Investment Possibilities — One Asset, Many Classes . . . . . . . . 2.3.1 Direct Investments . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Private Equity Funds . . . . . . . . . . . . . . . . . . . . . 2.3.2.1 Key players . . . . . . . . . . . . . . . . . . . . 2.3.3 Cash Flow Structure of a Private Equity Fund . . . . . . . . 2.3.4 Fund-of-Funds . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4.1 Structure of a private equity fund-of-funds . . . . 2.3.4.2 Advantages . . . . . . . . . . . . . . . . . . . . . 2.3.4.3 Disadvantages . . . . . . . . . . . . . . . . . . . 2.3.5 Publicly Traded Private Equity . . . . . . . . . . . . . . . 2.3.6 Secondary Transactions . . . . . . . . . . . . . . . . . . . 2.3.6.1 Types of secondary transactions . . . . . . . . . . 2.3.6.2 Buyer’s motivation . . . . . . . . . . . . . . . . . Private Equity as Alternative Asset Class in an Investment Portfolio . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Characteristics of LPE Return Series . . . . . . . . . . . . 2.4.2 Modeling Return Series with Markov-Switching Processes . 2.4.2.1 Markov–Switching models . . . . . . . . . . . . 2.4.2.2 Fitting the parameters . . . . . . . . . . . . . . . 2.4.2.3 Simulation of return paths . . . . . . . . . . . . . 2.4.3 Listed Private Equity in Asset Allocation . . . . . . . . . . 2.4.3.1 Performance measurement . . . . . . . . . . . . . 2.4.3.2 Portfolio optimization frameworks . . . . . . . . 2.4.3.3 Definition of investor types . . . . . . . . . . . . 2.4.3.4 Optimization of portfolios . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 3. Alternative Real Assets in a Portfolio Context

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Wolfgang Mader, Sven Treu and Sebastian Willutzky 3.1 3.2 3.3

Introduction . . . . . . . . . . . . . . . . . Overview on Alternative Real Assets . . . . Modeling Photovoltaic Investments . . . . 3.3.1 General Approach . . . . . . . . . 3.3.2 Definition of the Investment Project

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3.3.3

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Modeling of Risk Factors . . . . . . . . . . . . . . . . . 3.3.3.1 Economic factors . . . . . . . . . . . . . . . . 3.3.3.2 Non-economic factors . . . . . . . . . . . . . . 3.3.3.3 Historical analysis of monthly global irradiance 3.3.3.4 Monte Carlo analysis of yearly global irradiance Photovoltaic Investments in a Portfolio Context . . . . . . . . . . 3.4.1 Setting the Portfolio Context . . . . . . . . . . . . . . . . 3.4.2 Including Photovoltaic Investments in a Portfolio . . . . . 3.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 4. The Freight Market and Its Derivatives

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Rüdiger Kiesel and Patrick Scherer 4.1

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Introduction: the Freight Market . . . . . . . 4.1.1 Vessels . . . . . . . . . . . . . . . . 4.1.2 Cargo . . . . . . . . . . . . . . . . . 4.1.3 Routes . . . . . . . . . . . . . . . . Freight Rates: What Drives the Market? . . . 4.2.1 Demand for Shipping Capacity . . . 4.2.2 Supply of Shipping Capacity . . . . 4.2.3 Costs . . . . . . . . . . . . . . . . . Freight Derivatives: Hedging or Speculating? 4.3.1 Forward Freight Agreement . . . . . 4.3.2 Freight Futures . . . . . . . . . . . . Explanatory Variables . . . . . . . . . . . . 4.4.1 Explanatory Power . . . . . . . . . . 4.4.2 Granger Causality . . . . . . . . . . 4.4.3 Selection Algorithm “Top Five” . . . 4.4.4 Cointegration . . . . . . . . . . . . . Predicting Freight Spot and Futures Rates . . The Backtesting Algorithm . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . .

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Chapter 5. On Forward Price Modeling in Power Markets

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5.3.1

5.4 5.5

A Geometric Brownian Motion Dynamics by Volatility Averaging . . . . . . . . . . . . . . . . . 5.3.2 A Geometric Brownian Motion Dynamics by Moment Matching . . . . . . . . . . . . . . . . . 5.3.3 The Covariance Structure Between Power Forwards . 5.3.4 The Distribution of a Power Forward . . . . . . . . . 5.3.5 Numerical Analysis of the Power Forward Distribution Pricing of Options on Power Forwards . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 6. Pricing Certificates Under Issuer Risk

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Barbara Götz, Rudi Zagst and Marcos Escobar 6.1 6.2 6.3

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Introduction . . . . . . . . . . . . . . . . . The Model . . . . . . . . . . . . . . . . . Pricing of Certificates Under Issuer Risk . . 6.3.1 Building Blocks . . . . . . . . . . 6.3.2 Index Certificates . . . . . . . . . 6.3.3 Participation Guarantee Certificates 6.3.4 Bonus Guarantee Certificates . . . 6.3.5 Discount Certificates . . . . . . . . 6.3.6 Bonus Certificates . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . .

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Chapter 7. Asset Allocation with Credit Instruments

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Barbara Menzinger, Anna Schlösser and Rudi Zagst 7.1 7.2 7.3 7.4

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Introduction . . . . . . . . . . . . . . . . . . . . . Simulation Framework . . . . . . . . . . . . . . . Framework for Total Return Calculation . . . . . . Optimization Framework . . . . . . . . . . . . . . 7.4.1 Mean-Variance Optimization . . . . . . . 7.4.2 CVaR Optimization . . . . . . . . . . . . Model Calibration and Simulation Results . . . . . 7.5.1 Mean-Variance Approach . . . . . . . . . 7.5.2 Conditional Value at Risk . . . . . . . . . 7.5.3 Comparison of Selected Optimal Portfolios Summary and Conclusion . . . . . . . . . . . . .

Chapter 8. Cross Asset Portfolio Derivatives Stephan Höcht, Matthias Scherer and Philip Seegerer

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8.1 8.2 8.3

8.4

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Introduction to Cross Asset Portfolio Derivatives 8.1.1 Definitions and Examples . . . . . . . . Collateralized Obligations . . . . . . . . . . . . A Comparison of CFO with CTSO . . . . . . . . 8.3.1 Structural Features of CFO . . . . . . . 8.3.2 Structural Features of CTSO . . . . . . . 8.3.3 The Different Risks . . . . . . . . . . . 8.3.4 Correlation of Tail Events in CTSO . . . Pricing Cross Asset Portfolio Derivatives . . . . 8.4.1 Pricing Trigger Swaps . . . . . . . . . . 8.4.2 Pricing nth-to-Trigger Baskets . . . . . . 8.4.3 Pricing CTSO . . . . . . . . . . . . . . 8.4.4 Modeling Approaches . . . . . . . . . . 8.4.4.1 The structural approach . . . . 8.4.4.2 The copula approach . . . . . . 8.4.5 An Example for an nth-to Trigger Basket 8.4.5.1 A pricing exercise of Example 3 (structural approach) . . . . . 8.4.5.2 A pricing exercise of Example 3 (copula approach) . . . . . . . 8.4.5.3 Resulting model spreads . . . . Outlook . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . .

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Part II. Alternative Strategies Chapter 9. Dynamic Portfolio Insurance Without Options

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9.3 9.4

Introduction . . . . . . . . . . . Simple Strategies . . . . . . . . 9.2.1 Buy-and-Hold . . . . . 9.2.2 Stop-Loss . . . . . . . 9.2.3 The Bond Floor Strategy 9.2.4 Plain Vanilla CPPI . . . Historical Simulation I . . . . . Advanced Features . . . . . . . 9.4.1 Transaction Costs . . . 9.4.2 Transaction Filter . . . 9.4.3 Lock-in Levels . . . . .

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9.4.4 Leverage and Constrain of Exposure . . . . . 9.4.5 Rebalancing Strategies for the Risky Portfolio 9.4.6 CPPI and Beyond . . . . . . . . . . . . . . . Historical Simulation II . . . . . . . . . . . . . . . . 9.5.1 Transaction Costs and Transaction Filter . . . 9.5.2 Lock-in Levels . . . . . . . . . . . . . . . . . 9.5.3 The Use of Leverage . . . . . . . . . . . . . . 9.5.4 CPPI on a Multi-Asset Risky Portfolio . . . . Implement a Dynamic Protection Strategy with ETF . Closing Remarks . . . . . . . . . . . . . . . . . . . .

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Chapter 10. How Good are Portfolio Insurance Strategies?

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Sven Balder and Antje Mahayni 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Optimal Portfolio Selection with Finite Horizons . . . . . . . . 10.2.1 Problem (A) . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Problem (B) . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Problem (C) . . . . . . . . . . . . . . . . . . . . . . . 10.2.4 Comparison of Optimal Solutions . . . . . . . . . . . . 10.3 Utility Loss Caused by Guarantees . . . . . . . . . . . . . . . . 10.3.1 Justification of Guarantees and Empirical Observations 10.3.2 Utility Loss . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Utility Loss Caused by Trading Restrictions and Transaction Costs . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Discrete-Time CPPI . . . . . . . . . . . . . . . . . . . 10.4.2 Discrete-Time Option-Based Strategy . . . . . . . . . . 10.4.3 Comments on Utility Loss and Shortfall Probability . . 10.5 Utility Loss Caused by Guarantees and Borrowing Constraints . . . . . . . . . . . . . . . . . . . . 10.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 11. Portfolio Insurances, CPPI and CPDO, Truth or Illusion?

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Elisabeth Joossens and Wim Schoutens 11.1 Introduction . . . . . . . . . . . . . . 11.2 Credit Risk and Credit Default Swaps 11.2.1 Credit Risk . . . . . . . . . . 11.2.2 Credit Default Swaps (CDS) . 11.3 Portfolio Insurances . . . . . . . . .

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11.4 Modeling of CPPI Dynamics Using Multivariate Jump-Driven Processes . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Multivariate Variance Gamma Modeling . . . . . . . . . . 11.4.2 Swaptions on Credit Indices . . . . . . . . . . . . . . . . . 11.4.2.1 Black’s model . . . . . . . . . . . . . . . . . . . 11.4.2.2 The variance gamma model . . . . . . . . . . . . 11.4.3 Spread Modeling by Correlated VG Processes . . . . . . . 11.4.3.1 The pricing of CPPIs . . . . . . . . . . . . . . . . 11.4.3.2 Gap risk . . . . . . . . . . . . . . . . . . . . . . 11.5 Recent Developments for CPPI . . . . . . . . . . . . . . . . . . . 11.5.1 Portfolio Insurance: The Extreme Value Approach to the CPPI Method . . . . . . . . . . . . . . . . . . . . . 11.5.2 VaR Approach for Credit CPPI . . . . . . . . . . . . . . . 11.5.3 CPPI with Cushion Insurance . . . . . . . . . . . . . . . . 11.6 A New Financial Instrument: Constant Proportion Debt Obligations 11.6.1 The Structure . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.2 CPDOs in the Spotlight . . . . . . . . . . . . . . . . . . . 11.6.3 Rating CPDOs Under VG Dynamics . . . . . . . . . . . . 11.7 Comparison Between CPPI and CPDO . . . . . . . . . . . . . . . 11.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 12. On the Benefits of Robust Asset Allocation for CPPI Strategies

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Katrin Schöttle and Ralf Werner 12.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Financial Market . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 The Basic Financial Market . . . . . . . . . . . . . . . . . 12.2.2 The Riskless Asset . . . . . . . . . . . . . . . . . . . . . . 12.2.3 The Risky Asset . . . . . . . . . . . . . . . . . . . . . . . 12.2.4 Classical Mean–Variance Analysis . . . . . . . . . . . . . 12.2.5 The Trading Strategy . . . . . . . . . . . . . . . . . . . . . 12.3 The Standard CPPI Strategy . . . . . . . . . . . . . . . . . . . . . 12.3.1 The Simple Case . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 The General Case . . . . . . . . . . . . . . . . . . . . . . 12.3.3 Shortfall Probability of CPPI Strategies . . . . . . . . . . . 12.3.4 Improving CPPI Strategies . . . . . . . . . . . . . . . . . . 12.3.5 CPPI Strategies Under Estimation Risk . . . . . . . . . . . 12.4 Robust Mean–Variance Optimization and Improved CPPI Strategies 12.4.1 Robust Mean–Variance Analysis . . . . . . . . . . . . . . . 12.4.2 Uncertainty Sets Via Expert Opinions or Related Estimators

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12.4.3 Uncertainty Sets Via Confidence Sets . . . 12.4.4 Usage and Implications for CPPI Strategies 12.4.5 CPPIs with Robust Asset Allocations . . . 12.5 Conclusion . . . . . . . . . . . . . . . . . . . . .

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Chapter 13. Robust Asset Allocation Under Model Risk

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Pauline Barrieu and Sandrine Tobelem 13.1 Background . . . . . . . . . . . . . . . . . . . . . . . 13.2 A Robust Approach to Model Risk . . . . . . . . . . . 13.2.1 The Absolute Ambiguity Robust Adjustment . 13.2.2 Relative Ambiguity Robust Adjustment . . . . 13.2.3 ARA Parametrization . . . . . . . . . . . . . 13.3 Some Definitions Relative to the Ambiguity-Adjusted Asset Allocation . . . . . . . . . . . . . . . . . . . . 13.4 Empirical Tests . . . . . . . . . . . . . . . . . . . . . 13.4.1 Portfolios Tested . . . . . . . . . . . . . . . . 13.4.2 Performance Measures . . . . . . . . . . . . . 13.4.3 Results . . . . . . . . . . . . . . . . . . . . . 13.4.3.1 Performances of the different models 13.4.3.2 SEU portfolio . . . . . . . . . . . . 13.4.3.3 Ambiguity robust portfolios . . . . . 13.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 14. Semi-Static Hedging Strategies for Exotic Options

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Hansjörg Albrecher and Philipp Mayer 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Hedging Path-Independent Options . . . . . . . . . . . . . . 14.2.1 Plain Vanilla Options with Arbitrary Strikes are Liquid 14.2.2 Finitely Many Liquid Strikes . . . . . . . . . . . . . 14.3 Hedging Barrier and Other Weakly Path Dependent Options . 14.3.1 Model-Dependent Strategies: Perfect Replication . . . 14.3.2 Model-Dependent Strategies: Approximations . . . . 14.3.3 Model-Independent Strategies: Robust Strategies . . . 14.4 Hedging Strongly Path-Dependent Options . . . . . . . . . . 14.4.1 Lookback Options . . . . . . . . . . . . . . . . . . . 14.4.2 Asian Options . . . . . . . . . . . . . . . . . . . . . 14.5 Case Study: Model-Dependent Hedging of Discretely Sampled Options . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Conclusion and Future Research . . . . . . . . . . . . . . . .

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Chapter 15. Discrete-Time Variance-Optimal Hedging in Affine Stochastic Volatility Models

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Jan Kallsen, Richard Vierthauer, Johannes Muhle-Karbe and Natalia Shenkman 15.1 15.2 15.3 15.4

Introduction . . . . . . . . . . . . . . . . Discrete-Time Variance-Optimal Hedging The Laplace Transform Approach . . . . Application to Affine Stochastic Volatility Models . . . . . . . . . . . . . 15.5 Numerical Illustration . . . . . . . . . . Index

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Part I

Alternative Investments

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1

SOCIALLY RESPONSIBLE INVESTMENTS

SVEN HROß∗ , CHRISTOFER VOGT† and RUDI ZAGST‡ HVB-Stiftungsinstitut für Finanzmathematik, Technische Universität München, Boltzmannstr. 3, 85747 München, Germany ∗ [email protected] † [email protected] ‡ [email protected]

Within the last two decades, the market of socially responsible investing (SRI) has seen unprecedented growth and has become more and more important, not only because of the current financial crisis. This chapter gives a survey of the asset class SRI in general, i.e., market development and investment possibilities. Moreover, the question “How sustainable is sustainability?” is addressed by analyzing SAM Group sustainability rankings of the years 2001–2007. Furthermore, the ability of SRI to contribute to diversification within a portfolio is scrutinized. The analysis is based on simulated returns generated by an autoregressive Markov-Switching model and accounts for different levels of investors’ risk aversion. Optimal portfolios consisting of stocks, bonds, and the respective SRI index show that risk–averse investors mix SRI to an established portfolio consisting of bonds and stocks to reduce the risk and increase the performance. Additionally, the asset class SRI is found to be a substitute for the asset class stocks.

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1.1. INTRODUCTION There are different ways to describe socially responsible investing (SRI). Reference 1 defines SRI as the integration of environmental, social, and corporate governance (ESG) considerations into investment management processes and ownership practices hoping that these factors can have an impact on financial performance. Responsible investment can be practiced across all asset classes. Several reasons can be stated, why the field of SRI has gained great public interest as well as rising economic importance in recent years. Simultaneously to the on-going climate change debate, public scrutiny and political attention have put pressure on businesses to consider both social and environmental issues in their activities. Accompanied by these developments, the SRI market grew strongly during the last decade. SRI does no longer represent a negligible economical niche, but as stated in [2] it might play a crucial financial role in the future. The current size of the worldwide SRI market is according to [3] approximately 5 trillion. With 53% market share, the greatest part of the SRI market is based in Europe followed by the United States with 39%. The rest of the world represents only 8% of the SRI market. According to [4], the size of the SRI market in the United States was $639 billion in 1995 and then grew up to $2159 billion in 1999, which means an average annual growth rate of 36%. From 1999 to 2005, SRI investment volumes only slightly grew up to $2290 billion, but then growth accelerated again resulting in $2711 billion in 2007. The European SRI market experienced an average growth rate of 51% since 2002 from an absolute investment volume of 336 billion in 2002 up to 2665 billion in 2007. Reference 3 estimates that the share of SRI in the total European fund market is about 17.6% in 2008 and largely driven by institutional investors. There are several possibilities to invest into SRI. For example, the SAM Group (www.sam-group.com) offers a wide range of funds covering the total SRI market and also special funds, e.g., on Islamic sustainability. There are also sustainably managed fixed-income funds available. Another possibility is the direct investment into nonlisted companies or projects. In this context, projects like wind farms or solar parks can be mentioned as suitable investment possibilities. Moreover, certificates are available on the market which allow the investor to participate in the SRI market, e.g., index certificates on the European Renewable Energy Index (ERIX Index Certificate, Societe Generale, ISIN: DE000SG1ERX7). The structure of this chapter is as follows. Section 1.2 gives an overview on recent research on SRI. Section 1.3 answers the question “How sustainable is sustainability?” by using Markov transition matrices. Section 1.4 then analyzes SRI in a portfolio context by generating optimal portfolios for different investors using a Markov-Switching model and different optimization frameworks. Finally, Sec. 1.5 concludes.

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1.2. RECENT RESEARCH ON SRI During the last years, several empirical studies analyzed whether SRI produces or destroys shareholder wealth. Many early studies on the performance of SRI use regression models with one or two factors and try to measure Jensen’s alpha. Reference 5 compares 32 SRI funds to 320 non-SRI funds in the United States between 1981 and 1990 and finds no significant average alphas with respect to a value-weighted NYSE index. More advanced studies apply a matching approach to compare SRI and non-SRI funds with similar characteristics, e.g., fund universe and size. Within this approach, management and transaction costs can be included into the analysis, see, e.g., [6] or [7]. As a result, no significant performance differences between SRI and non-SRI could be observed. One problem is that important characteristics might not be taken into consideration. Reference 8 applies a four factor model according to [9] using as regression factors the excess market return, SMB (“Small-minus-Big”: The difference between the return of a small- and of a large-cap portfolio), HML (“High-minus-Low”: The return difference between a value- and a growth-portfolio, i.e., a portfolio containing firms that dispose of a high book-to-market ratio versus firms with a low value relating to this ratio), and MOM (“Momentum”: The return difference betweeen two portfolios, one consisting of last year’s best performers and the other of the worst performers) in order to analyze the performance of United States, German, and British SRI funds. The authors build two portfolios for each country, one containing all SRI funds, the other the conventional funds, and find under — as well as outperformance of SRI, but none of the differences are significant. Furthermore, SRI funds seem to have an investment bias toward growth stocks (low book-to-market value) and small caps (lower market-capitalization). Reference 10 uses eco-efficiency rankings of Innovest to evaluate two equity portfolios that differ in eco-efficiency. The high-ranked portfolio shows significantly higher returns than its low-ranked counterpart over the period 1995–2003. In contrast, [11] finds that SRI investors have to pay for their constrained investment style. Another approach is to look at SRI equity indices to avoid usual problems of mutual funds during a performance analysis, e.g., transaction costs of funds or effects of management skills. Reference 12 analyzes 29 SRI indices and applies different settings to test for differences in risk-adjusted performance compared to a suitable benchmark. The study concludes that SRI screens do not lead to significant performance difference of SRI indices. Yet, no final answer to the question whether SRI produces or destroys shareholder wealth can be given. Independent of these findings, SRI market growth might simply come from the non-financial utility gained by SRI investors. To the authors’ best knowledge, there is yet no such study scrutinizing this effect. Therefore, the focus of Sec. 1.4 lies on the benefits of SRI in a portfolio context. For this, optimal portfolios of bonds, stocks, and SRI will be constructed for different investor types and in different optimization frameworks.

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1.3. HOW SUSTAINABLE IS SUSTAINABILITY? In this section, the endurance of sustainability is analyzed. This is especially important for an SRI investor, who does not want to have too many reallocations in his portfolio. Moreover, sustainability scores should be enduring by the pure definition of the word “sustainability”. For this aim, sustainability scores from SAM Group, one of the world’s most respected companies in the field of SRI assessment, are scrutinized. This study is implemented using Markov transition matrices.

1.3.1. Description of the Dataset The dataset used for the analysis contains the sustainability scores (hereinafter called total score) of 822 companies. The methodology for calculating the total score of a firm is given as follows. A company’s economic, ecologic, and social performance is analyzed, where each of the three dimensions is divided into several criteria. These criteria are weighted with an individual percentage of contribution to derive the final total score. There are general criteria for all industries and specific criteria for companies in a certain sector. The complete dataset consists of 4432 total scores for the different firms and years between 2001 and 2007. However, not every company receives a sustainability score by SAM every year, simply due to the fact that there are firms that are not willing to participate in the assessment process every year. To be more precise, only 185 companies were evaluated by SAM Group in every single of the seven assessment years. The companies in the dataset are a mixture of worldwide well-known multinational companies, such as Adidas AG, Allianz SE, the Coca-Cola Company, and Sony Corporation, as well as rather regional established firms such as Eniro AB from Sweden or the Italian Beni Stabili SpA. It can be seen from Table 1.1 that the total scores over the whole time period range between a rather low rating of 4.97 and a very high score of 92.37, i.e., that the predefined range between 0 and 100 is actually utilized. Interestingly, the median and mean of the overall total scores are slightly above 50, and barely half of the companies received a sustainability score between 43 and 65.

1.3.2. Introduction to Markov Transition Matrices In this section, Markov transition matrices are used to analyze the evolution of the sustainability scores. A high degree of variation within the total scores would be Table 1.1 Minimum 4.97

1st quartile 43.60

Statistics on Total Score. Median 55.48

Mean 53.67

3rd quartile 65.19

Maximum 92.37

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counter-intuitive, due to the fact that sustainability is a long-term affair and thus should not be subject to large-sized jumps, unless extraordinary events occur, e.g., an environmental disaster on an oil producer’s platform. For the following analysis, data of those companies are used for which the sustainability scores are available for two consecutive years. For the entire six-year time period, this leads to a total dataset of 2125 observations. The calculation of the transition matrices is performed as follows: For every single year, companies are ranked by their sustainability score, whereby for every year the 25% best rated companies are assigned to the 1st quartile, the next 25% to the 2nd quartile, and so on. Based on this allocation, empirical transition probabilities from one of the four quartiles to any of the four quartiles after one year can be calculated.

1.3.3. Results of Markov Transition Matrices From the average one-year transition probabilities in Table 1.2, it can be seen that the probability of staying in the current quartile is the highest and ranges from 47.53% for the 2nd quartile to 72.21% for the last quartile.Additionally, the probability decreases in the distance between two quartiles. Furthermore, the probability that a top-ranked firm will end up in the 4th quartile in the following year is only 0.37% and the probability of a “bad” company to be part of the first quartile in the following period is 1.23%. Moreover, Markov transition matrices for every single year 2001–2007 were scrutinized. The results for the single years are quite similar to the average observation in Table 1.2. Finally, a six-year Markov transition matrix was computed. The results are shown in Table 1.3. Nearly half of the companies that were ranked in the first quartile in 2001 were still in the first quartile in 2007. The probability that a highly sustainable company will be part of the worst quartile at the end of the six years is 5.36% and the probability of the opposite case, i.e., a “bad” company ending as a sustainability leader after six years, is 7.02%. Table 1.2 Average One-Year Markov Transition Probabilities (Year 2001–2007). Next year quartile

Last year quartile

1 2 3 4

1 (%)

2 (%)

3 (%)

4 (%)

69.74 23.83 5.15 1.23

25.00 47.53 21.74 5.96

4.90 24.35 49.94 20.60

0.37 4.29 23.17 72.21

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Hroß et al. Table 1.3 Markov Six-Year Transition Probabilities (Year 2001–2007). Next year quartile

Last year quartile

1 2 3 4

1 (%)

2 (%)

3 (%)

4 (%)

46.43 25.00 21.43 7.02

26.79 39.29 14.29 19.30

21.43 32.14 32.14 14.04

5.36 3.57 32.14 59.65

Altogether, the results provide evidence to the assumption that sustainability rankings do not have a high degree of short-term variation.

1.4. SRI IN PORTFOLIO CONTEXT After having analyzed the sustainability of sustainability in the preceding section, this section will scrutinize how SRI can be evaluated with regard to the portfolio context. The main questions to be answered are whether investors shall add SRI investments to their portfolio, and if so, with which weighting. In the conducted portfolio case study, the SRI market is represented by the Advanced Sustainable Performance Index (ASPI). The ASPI is a European index consisting of 120 companies and is published by Vigeo Group, an extra-financial supplier and rating agency in the field of sustainable development and social responsibility (for further information see [13]). In order to include dividend payments to the analysis, total return indices are used, i.e., dividends are reinvested. This approach has two main advantages. First, the index already represents a selected basket of the asset category SRI and the time series are readily available. Second, the predefined index is widespread and thus has the advantage that the companies’ specific risks are already eliminated by diversification. As a result, only the diversification effect of the asset class SRI itself is observed.

1.4.1. Description of the Dataset and Statistical Properties The portfolio analysis is based on daily log-returns of the asset classes bonds (represented by the JP Morgan Global Government Bond Index), stocks (represented by the Dow Jones Total Markets World Index), and SRI (represented, as described above, by the ASPI index) between 1 January 1992 and 30 September 2008. The main empirical statistics are shown in Table 1.4.

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Socially Responsible Investments Table 1.4

Empirical Statistics of Daily Log-Returns.

Empirical statistics Mean Mean (annualized) Standard deviation Standard deviation (annualized) Skewness Excess kurtosis Autocorrelation: lag 1 Autocorrelation: lag 2 Autocorrelation: lag 3 Autocorrelation of squared returns (lag 1) 5% critical value for autocorrelation

Bonds

Stocks

SRI

0.00024 0.05917 0.00389 0.06151 −0.00619 1.44597 0.03266 0.00386 −0.01373 0.03859 0.03026

0.00026 0.06399 0.00803 0.12702 −0.29037 4.11292 0.16873 −0.02818 −0.02108 0.13695 0.03026

0.00038 0.09425 0.01216 0.19230 −0.13149 3.58599 0.00494 −0.02391 −0.06324 0.19301 0.03026

By comparing mean and standard deviation of bonds and stocks, it becomes evident that most of the risk–averse investors would invest the bulk of their wealth in bonds. This is due to the extremely high mean for bonds (5.92% per annum) combined with a low standard deviation. Additionally, bonds display the highest skewness and lowest excess kurtosis, which is generally preferred by risk–averse investors. As it is a debatable point whether past returns indicate the future in a sufficient way, experts’ forecasts about expected returns are often used to solve this shortcoming. By using the Black–Litterman approach to adjust the empirical returns, the empirical mean µemp itself as well as absolute and relative forecasts are taken into account (see, e.g., [14]). This approach can be interpreted as a linear combination of these two components at a given confidence level τ regarding the forecasts. The Black–Litterman expectations µBL can be expressed by (given that L is invertible) µBL = τ · L−1 q + (1 − τ) · µemp ,

(1.1)

where L represents the linear transformation Lµ of the asset classes expected return vector µ and for each forecast, whose actual value is specified in q. The assumptions about the forecasts are taken from [15]. This means that the annual return of bonds is expected to be 3.96% and the equity risk premium amounts to 3.5%. The additional assumption that the difference of the means of stocks and SRI does not change leads to     1 0 0 0.0396/250 L =  −1 1 0  and q =  0.0350/250  . 0 −1 1 0.0303/250 For τ = 0, the Black–Litterman expectations are equal to the empirical means, while τ = 1 leads to expectations which are completely driven by the forecasts. Table 1.5 provides the Black–Litterman expectations for a confidence level of τ = 0.75.

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Hroß et al. Table 1.5 Black–Litterman Expectations for Asset Class Log-Returns. Black–Litterman expectations Daily return Annual return

Bonds

Stocks

SRI

0.00018 0.04449

0.00029 0.07195

0.00041 0.10221

Table 1.6 P-Values of Jarque–Bera and Ljung–Box-Q Tests. Test

Null hypothesis

Jarque–Bera Ljung–Box-Q (Q1) Ljung–Box-Q (Q2) Ljung–Box-Q (Q3) Ljung–Box-Q (QS1)

Bonds

Stocks

Normal distribution <0.001 <0.001 No autocorrelation (up to lag 1) 0.0308 0 No autocorrelation (up to lag 2) 0.0940 0 No autocorrelation (up to lag 3) 0.1354 0 No ac. (squared returns, lag 1) 0.0096 0

SRI <0.001 0.7438 0.2715 0.0002 0

The empirical returns are adjusted for the Black–Litterman expectations by applying a linear shift to the whole dataset. Hence, all other empirical statistics (except for the autocorrelation in squared returns) are unaffected. Skewness and excess kurtosis in Table 1.4 lead to the presumption that the returns of all three asset classes are non-normally distributed. To test for non-normality and autocorrelation, a Jarque–Bera test and a Ljung–Box-Q test (see [16] and [17]) were applied. The p-values of both tests are given in Table 1.6. Italicised values are those smaller than 0.05, for which the null hypothesis can be rejected at a significance level of 5%. As correlations are essential for diversification in a portfolio context, the correlations of the empirical daily log-returns are listed in Table 1.7. The high correlation between stocks and SRI is in line with the findings of [12]. One reason for this fact may be that SRI stocks are simply a subset of the whole stock universe. Nevertheless, each correlation coefficient smaller than one allows to benefit from diversification. Table 1.7 Empirical Log-Returns. Correlation Bonds Stocks SRI

Correlations

of

Daily

Bonds

Stocks

SRI

1 −0.034524 0.044066

−0.034524 1 0.732227

0.044066 0.732227 1

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1.4.2. Markov-Switching Model Due to the described non-normality and autocorrelation of the considered time series of daily log-returns, the standard Black–Scholes model, which implies i.i.d. normally distributed log-returns, is not appropriate to describe the asset returns. As MarkovSwitching models allow for non-normality and autocorrelation at the same time, this model class is applied in this study. To be more precise, a state-independent (first lag) autoregressive Markov-Switching model as introduced in [18] is utilized. Further applications of Markov-Switching models in a portfolio context can, e.g., be found in [15] or [19]. It is assumed throughout that the return of asset class a ∈ {1, 2, 3} at time t is given by rt,a = µst ,a + φa · (rt−1,a − µst−1 ,a ) + t,a ,

(1.2)

where st indicates the state of the markets at time t, µst ,a denotes the mean return of the asset class a in state st . Here, a = 1 corresponds to the asset class bonds, a = 2 to stocks, and a = 3 to the SRI asset class. Furthermore, t = (t,1 , t,2 , t,3 )T represents the innovation at time t with t ∼ N(0, st ) and φa the autocorrelation parameter for asset class a satisfying |φa |<1. The Markov-Switching model allows the market to be in two different regimes (st = 1 for state 1 and st = 2 for state 2). For the reasons adduced in [15], only two meta states are allowed. As a result, all three assets are in the same state at each point of time t. Changes between the two states over time are modeled by a Markov chain with transition probabilities given by the matrix   p11 p12 , P= p21 p22 where P(st = j|st−1 = i) = pij . Therefore, the whole process is parametrized by a vector θ = (µ1 , µ2 , p12 , p21 , φ, 1 , 2 ) with µ1 , µ2 ∈ R3 , φ ∈ [−1, 1]3 , p12 , p21 ∈ [0, 1], and 1 , 2 ∈ R3×3 . The assumption of only two meta states has the great advantage that overfitting problems can be avoided.

1.4.3. Fitting the Model Parameters The model parameters are fitted in a way that the empirical moments equal the moments of the Markov-Switching process best possible. The applied method of moments is described in detail in [18]. With regard to the empirical statistics, the focus of the fitting lies on the first four moments and the autocorrelation of lag 1. The resulting parameters and transition probabilities are displayed in Table 1.8.

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Hroß et al. Table 1.8

Parameter Bonds Stocks SRI

Markov-Switching Model Parameters.

µ1

µ2

σ1

σ2

φ

p12

p21

0.00018 0.00062 0.00064

0.00015 −0.00108 −0.00055

0.00325 0.00519 0.00812

0.00630 0.01473 0.02166

0.03270 0.16550 0.00389

0.0579

0.2425

The parameters allow for the derivation of crucial information about the two possible meta states. State 1 characterizes a bull market for all three assets. The expected return of stocks and SRI is almost the same (15.4% p.a. for stocks and 16% p.a. for SRI) whereas the standard deviation is much higher for SRI. In state 2, the expected return of bonds only suffers a small decline compared to state 1 (from 4.6% p.a. to 3.8% p.a.) but the volatility nearly doubles in this regime. For stocks and SRI state 2 resembles a bearish market with huge losses for both asset classes (−26.9% p.a. for stocks and −13.9% p.a. for SRI) and high standard deviations. The observation of an increasing volatility in falling markets can be found in empirical studies like [20]. The transition probabilities p12 and p21 of the Markov chain imply a realistic stability of the two possible states. If the market is in a bullish or a bearish scenario respectively, the market remains in the current state on the following day with a high probability. As the fitting is based on minimizing the sum of squared deviations between the empirical and theoretical statistics, these are outlined in Table 1.9. The close match is emphasized by the theoretical correlations of the model, which fit the empirical correlations very well (compare Tables 1.7 and 1.10). It is worthwhile to have a look at the correlation structures of the error terms t , which are depicted in Table 1.11. The correlation of the error terms between stocks and SRI is about 0.9 in the “bad” state (state 2) and thus much higher than in the “good” state (state 1) with about 0.47.

Table 1.9

Empirical and Theoretical Statistics of Daily Log-Returns.

Statistics

Mean

Std. dev.

Skewness

Ex. kurt.

Autocorr.

Bonds

Empirical Theoretical

0.0001780 0.0001777

0.003890 0.004020

−0.006192 −0.006200

1.445971 1.517654

0.032663 0.032703

Stocks

Empirical Theoretical

0.0002878 0.0002890

0.008033 0.008109

−0.290370 −0.287749

4.112919 4.122927

0.168726 0.169116

SRI

Empirical Theoretical

0.0004088 0.0004100

0.012162 0.011994

−0.131494 −0.130343

3.585990 3.680593

0.004943 0.004961

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Socially Responsible Investments Table 1.10 Theoretical Correlations of Daily Log-Returns. Correlation Bonds Stocks SRI

Table 1.11 Correlation

Bonds

Stocks

SRI

1 −0.033037 0.043037

−0.033037 1 0.733543

0.043037 0.733543 1

Correlation Structure of Error Terms in States 1 and 2. Bonds

Stocks

SRI

−0.264006 1 0.467326

−0.414615 0.467326 1

0.140301 1 0.896169

0.414109 0.896169 1

State 1 Bonds Stocks SRI

1 −0.264006 −0.414615 State 2

Bonds Stocks SRI

1 0.140301 0.414109

1.4.4. Simulation of Returns The portfolio optimization case study will be based on the distribution of simulated returns according to the model in Sec. 1.4.2. Therefore, a Monte Carlo simulation with 20,000 return paths is conducted. Each path consists of 1250 returns and represents, assuming 250 trading days per year, five years of data for each asset class. The initial state of the underlying Markov chain is drawn from the stationary distribution π = (π1 , 1 − π1 ) with π1 = p21 /(p12 + p21 ). Due to non-normality and the importance of skewness and kurtosis with respect to the risk aversion of an investor (risk–averse investors have a positive preference direction for skewness and a negative one for excess kurtosis), it is necessary to introduce more complex portfolio concepts compared to a pure mean–variance optimization.

1.4.5. Portfolio Optimization Models Having deduced the distributions of the simulated return paths, five different models are applied in order to optimize the investor’s portfolio. Two constraints for the portfolio  weights x ensure a full investment of the available budget ( 3i=1 xi = 1), and avoid short-selling (xi ≥ 0, i = 1, 2, 3).

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Stocks

SRI

Expected return

0.09

Bonds, stocks Bonds, stocks, SRI

0.08

0.07

0.06

0.05

0.04 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

Standard deviation

Figure 1.1 Efficient frontier for one-year time horizon.

The traditional mean–variance framework based on [21] only takes the first two moments of the return distribution into account. It is defined by λ T (1.3) x x, 2 where µ denotes the vector of expected asset returns,  the return covariance–matrix, and λ the risk–aversion parameter of the investor. Figure 1.1 shows the diversification effect which can be obtained by including the asset SRI into an existing portfolio of bonds and stocks. Due to the importance of higher moments within the portfolio context, the powerutility model and frameworks which maximize different performance measures are introduced in the following. The power-utility model is defined by the optimization problem   max E[γ −1 (1 + R(x))γ ], γ < 1 ∧ γ = 0 x , (1.4) max E[U(R(x))] = x  max E[ln(1 + R(x))], γ=0 max xT µ − x

x

where γ describes the risk aversion of the investor and R(x) is the portfolio return depending on the portfolio weights x. The third optimization model maximizes the performance measure introduced in [22]. In this framework, each investor can set a threshold to classify the returns into gains (returns above the threshold) and losses (returns below the threshold). is

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defined as the ratio of probability-weighted gains to losses and thus equal to the upside potential divided by the downside potential. The corresponding optimization problem is given by max τ (R(x)) = max x

x

E[R(x) − τ]+ , E[τ − R(x)]−

(1.5)

where τ is the loss threshold and R(x) the portfolio return for portfolio weights x. The performance measure Score-value considers the difference of upside and downside potential. The risk-free rate r is used as threshold and the downside potential is weighted with a risk–aversion parameter λSc . This leads to an optimization problem defined by max ScoreλSc (R(x)) = max E[R(x) − r]+ − λSc · E[r − R(x)]− . x

x

(1.6)

The Mean-Conditional Value at Risk (MCVaR) is a risk measure referring to the tail of a distribution. It is based on the Conditional Value at Risk (CVaR) defined by CVaR(R(x)) = E[R(x)|R(x) < VaR(R(x))],

(1.7)

where VaR(R(x)) is the Value at Risk (see, e.g., [23]) of the portfolio return R(x) at a given confidence level α (in the case under consideration α = 99.5%). In order to consider both risk and return, the optimization problem is given by max MCVaRλMCVaR (R(x)) = max E[R(x)] − λMCVaR CVaR(R(x)) x

x

(1.8)

with λMCVaR denoting the respective risk–aversion parameter.

1.4.6. Definition of Investor Types All optimization models introduced above take different levels of risk–aversion into account. To consistently define investor types over the different models, the risk– aversion parameters are chosen such that they result in the same optimal asset allocations in a world where only stocks and bonds exist (see, e.g., [19]). In this case study, three investor types with different levels of risk aversion are used. They are represented by their characteristic benchmark portfolios with bonds:stocks equal to 0.7:0.3 (Investor A), 0.5:0.5 (Investor B), and 0.3:0.7 (Investor C). For the one-year time horizon, the respective parameters are given in Table 1.12.

1.4.7. Optimal Portfolios Using the risk–aversion parameters of the three different investor types, optimal portfolios of bonds, stocks, and SRI are constructed. The optimization is conducted for all introduced frameworks with time horizons of one, three, and five years. As the results are quite similar for all three time horizons, only the results of the one-year horizon

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Hroß et al. Table 1.12 Risk–Aversion Parameters for the Three Investor Types (One Year). Risk aversion Investor A Investor B Investor C

λ

γ

τ

λSc

λMCVaR

7.0609 2.8570 1.7908

−6.9009 −2.8353 −1.7820

0.0245 0.0390 0.0427

3.0632 1.8072 1.6328

0.1957 0.0983 0.0768

Table 1.13 Weights and Performance Measures of Optimal Portfolios (One Year). Framework

MV

PU

Omega

Score

MCVaR

Portfolio weights

A

Bonds Stocks SRI

0.6735 0.0759 0.2506

0.6732 0.0722 0.2546

0.6605 0.0624 0.2771

0.6252 0.0360 0.3388

0.6169 0.0376 0.3455

B

Bonds Stocks SRI

0.3968 0.0000 0.6032

0.3955 0.0000 0.6045

0.3832 0.0000 0.6168

0.0000 0.0000 1.0000

0.0000 0.0000 1.0000

C

Bonds Stocks SRI

0.0940 0.0000 0.9060

0.0927 0.0000 0.9073

0.1114 0.0000 0.8886

0.0000 0.0000 1.0000

0.0000 0.0000 1.0000

Risk and performance measures

A

Sharpe Omega Score MCVaR

0.3542 3.5017 −0.0112 0.0329

0.3547 3.5026 −0.0112 0.0330

0.3586 3.5055 −0.0110 0.0331

0.3663 3.4881 −0.0107 0.0333

0.3671 3.4828 −0.0107 0.0333

B

Sharpe Omega Score MCVaR

0.3693 2.3309 0.0208 0.0545

0.3693 2.3309 0.0209 0.0545

0.3687 2.3310 0.0211 0.0547

0.3497 2.2879 0.0285 0.0605

0.3497 2.2879 0.0285 0.0605

C

Sharpe Omega Score MCVaR

0.3540 2.1796 0.0342 0.0673

0.3540 2.1796 0.0342 0.0673

0.3549 2.1796 0.0337 0.0669

0.3497 2.1779 0.0366 0.0695

0.3497 2.1779 0.0366 0.0695

are presented in detail here. Table 1.13 shows the optimal portfolio weights as well as the computed performance measures , Score-value, MCVaR, and Sharpe ratio. The first three were already introduced in Sec. 1.4.5. The well-known Sharpe ratio measures the expected excess return over the riskless investment in units of standard

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deviation (see [24]). The low risk–averse Investor C invests largely into SRI. In the mean-variance (MV), the power-utility (PU), and the Omega framework, he allocates around 90% into SRI and in the Score-value and the MCVaR framework he is completely invested in SRI. As Investor C does not invest in stocks at all, the remainder of his wealth is invested in bonds. The strategy of Investor B is very similar with the only difference being that the portfolio weights of the asset class SRI are smaller in the MV, PU, and Omega framework. The best diversified portfolios are held by the highly risk–averse Investor A. Most of his wealth — between 61% and 68% — is invested into bonds. The majority of the remainder is allocated to SRI, leading to a fraction of 25–35% and only a small amount is invested into stocks (between 3–8%). Overall, the results emphasize that SRI can be interpreted as a substitute for stocks and that only the most risk–averse investor allocates money to the substituted asset class in order to benefit from the diversification effect. Figures 1.2 (a)–(e) illustrate the portfolio allocations in dependence of the respective risk–aversion parameter. The vertical lines indicate the three considered investor types. The solid line displays the risk of the portfolios measured by the standard deviation of the portfolios’ returns. As mentioned above, the proportion invested into stocks is due to the substitution effect always very small or even zero. A further interesting issue is the development of the proportion invested into bonds. In the MV, PU, and Omega framework, the fraction of bonds increases slowly with increasing risk–aversion whereas it ascends late but steeply in the Score-value and the MCVaR framework what results in an enormous reduction of risk from that point on. In order to get more information about the substitution effect, a sensitivity analysis is performed for the mean–variance framework with a one-year time horizon. As only the most risk–averse investor allocates parts of his wealth to stocks, the question arises under which circumstances the less risk–averse Investor C mixes stocks into his portfolio. For this purpose, the Black–Litterman expectation for the annual return of SRI is gradually decreased, starting from a value of 10.22% (see Table 1.5). With the help of the overall probability π1 (see Sec. 1.4.4) the amount of reduction can be split up for the two regimes. For state 1 the reduction is π1 · and for state 2 it equals (1 − π1 ) · . While facing the same risk exposure, the upper hurdle rate for the expected return is determined by a value of 9.27%. This value represents the highest expected return of SRI for which Investor C still invests a small amount into stocks. A further decrease of the expected return leads to a partial replacement of SRI by stocks. The lower hurdle rate is given by an annual SRI return of 7.01%, i.e., if the expected return is below the lower hurdle rate Investor C will not invest in SRI anymore and therefore create the appropriate benchmark portfolio of 70% stocks and 30% bonds.

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Figure 1.2 Optimal portfolio weights in different optimization frameworks.

1.5. CONCLUSION SRI is a growing asset class. By analyzing SAM sustainability scores, it was shown that an SRI portfolio has a high degree of consistency, i.e., sustainable companies are likely to stay sustainable in the future. For the best ranked companies, the probability of being in the 1st quartile again in the next year is 70% on average. Moreover, optimal

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portfolios for different investor types are constructed. The parameter estimates of the underlying Markov-Switching model are based on a time series ranging from 1992 to 2008. The main finding of the conducted case study is that SRI turns out to be a substitute for stocks and that the less risk–averse an investor is the more he invests in SRI, respectively, does a full investment into this asset class. More risk–averse investors use all three asset classes in order to gain from diversification effects. Nevertheless, only a very small fraction (<8%) is invested into stocks.

References [1] Mercer (2008). The language of socially responsible investing, an industry guide to key terms and organizations (January, 2008). [2] Lydenberg, S (2006). Envisioning socially responsible investing — a model for 2006. Journal of Corporate Citizenship, 7, 57–77. [3] European Social Investment Forum. European SRI study (2008). [4] Renneboog, L, J Horst and C Zhang (2008). Socially responsible investments: Institutional aspects, performance, and investor behaviour. Journal of Banking and Finance, 32(9), 1723–1742. [5] Hamilton, S, H Jo and M Statman (1993). Doing well while doing good? The investment performance of socially responsible mutual funds. Financial Anaylsts Journal, 49(6), 62–66. [6] Mallin, C, B Saadouni and R Briston (1995). The financial performance of ethical investment trusts. Journal of Business Finance & Accounting, 22, 483–496. [7] Statman, M (2000). Socially responsible mutual funds. Financial Analysts Journal, 56, 30–39. [8] Bauer, R, K Koedijk and R Otten (2005). International evidence on ethical mutual fund performance and investment style. Journal of Banking and Finance, 29, 1751–1767. [9] Carhart, M (1997). On the persistence in mutual fund performance. Journal of Finance, 52, 57–82. [10] Derwall, J, N Guenster, R Bauer and K Koedijk (2005). The eco-efficiency premium puzzle. Financial Analyst Journal, 61, 51–63. [11] Geczy, C, R Stambaugh and D Levin (2005). Investing in socially responsible mutual funds. Wharton School Working Paper. [12] Schröder, M (2007). Is there a difference? The performance characteristics of SRI equity indexes. Journal of Business Finance and Accounting, 34(1&2), 331–348. [13] Vigeo (2008). Methodology and composition of the ASPI Eurozone. URL http:// www.vigeo.com/csr-rating-agency/en/nos-produits-isr/indice-aspi/indice-aspi.html. [14] Black, F and R Litterman (1990). Asset allocation: Combining investor views with market equilibrium. Goldman Sachs Fixed Income Research. [15] Aigner, P, G Beyschlag, T Friederich, M Kalepky and R Zagst (2008). Optimal risk-return profiles for portfolios including stocks, bonds, and listed private equity. Submitted for publication. [16] Jarque, C and A Bera (1980). Efficient tests for normality, homoskedasticity and serial independence of regression residuals. Economic Letters 6, 6(3), 255–259.

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[17] Ljung, G and G Box (1978). On a measure of lack of fit intime series models. Biometrika, 65(2), 297–303. [18] Timmermann, A (2000). Moments of Markov switching models. [19] Höcht, S, K-H Ng, J Wolf and R Zagst (2008). Optimal portfolio allocation with Asian hedge funds and Asian REITs. International Journal of Services Sciences, 1(1), 36–68. [20] French, K, W Schwert and R Stambaugh (1987). Expected stocks returns and volatility. Journal of Financial Economics, 19, 3–29. [21] Markowitz, H (1952). Portfolio selection. Journal of Finance, 7(1), 77–91. [22] Shadwick, W and C Keating (2002). A universal performance measure. Journal of Performance Measurement, 6(2), 59–84. [23] Jorion, P (2000). Value at Risk: The New Benchmark for Managing Financial Risk, 2nd Ed., McGraw-Hill. [24] Sharpe, W (1964). Capital asset prices, a theory of market equilibrium under conditions of risk. Journal of Finance, 19, 77–91.

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LISTED PRIVATE EQUITY IN A PORTFOLIO CONTEXT

PHILIPP AIGNER∗ , GEORG BEYSCHLAG† , TIM FRIEDERICH‡ , MARKUS KALEPKY§ and RUDI ZAGST¶ ∗

Max-Planck-Str. 1, 81675 München, Germany [email protected] † Brixener Str. 17, 86720 Nördlingen, Germany [email protected] ‡ risklab GmbH, Seidlstraße 24-24a, 80335 München, Germany [email protected] § Ringgasse 24, 55218 Ingelheim, Germany [email protected] ¶ HVB-Stiftungsinstitut für Finanzmathematik, Technische Universität München, Boltzmannstr. 3, 85747 München, Germany [email protected]

This chapter first provides a comprehensive overview of private equity by categorizing the private equity investments into financing stages, divestment strategies, and types of financing. Different ways of investing in the asset class “private equity” are characterized, ranging from direct investments, which are hard to access, to listed private equity (LPE) investments, which provide a liquid means for investors to consider private equity in their portfolios. A Markov–Switching model is presented, which is able to capture the characteristics of the asset class LPE. By applying several risk measures and optimization frameworks, the question of the optimal fraction for an LPE investment in an investor’s portfolio is scrutinized. Depending on the risk aversion of the investor, the optimal fraction of an LPE investment in this study ranges between 0% for a very risk–averse investor, 7.5–11.8% for a moderately risky investor, and 16.9–27.9% for an investor willing to take higher risks. 21

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2.1. INTRODUCTION Private equity investments have probably been ventured since men try to augment their personal wealth. The voyage of Christopher Columbus, who received funding from various noblemen for his supposed expedition to Asia, is often referred to as a well-known example of the historical roots of this asset class. By the middle of the last century, individual financiers, affectionately dubbed business angels, crossed the threshold to the present-day notion of private equity investment: They granted money along with management support to high-tech start-up companies and actively supported young entrepreneurs in order to maximize the value of their own investment. In the 1970s, investment constraints previously imposed on institutional investors were relaxed by providing for vast additional capital channels. Thus, the venture capital market appeared significantly altered by the late 1980s: Not only high-tech start-up companies engrossed the investors’ attention, yet more established enterprises and firms not having technology-related products in their portfolio came into financial support as well, and debt financing grew more and more important. A profuse array of various funding practices such as restructuring, buyout, or mezzanine emerged. Together with the traditional venture capital, they evolved to an asset class known as private equity, which enjoyed enormous growth and returns over the last five years. With the credit crisis followed by an overall economic downturn becoming the prominent topic in 2008, one may raise the question whether private equity is still an asset class worthwhile considering. And indeed, both columnists and scientists forecast great changes which the industry is about to experience: Since the credit markets have basically come to halt, procuring debt for leveraged buyout transactions was virtually impossible, resulting deal volumes to plummet by 75% in 2008. Default probabilities for portfolio companies may be as high as 50% and consequently, most institutional investors have been reducing the exposure to private equity, see [16]. However, private equity still displays characteristics, which make it an interesting part of a well-diversified investment strategy: Since private equity tends to display only a moderate correlation to the public equity market, it may extend the portfolio’s efficient frontier, and thus, yield a better risk–return profile. Furthermore, the persistence in performance of top-quartile private equity firms has also proven valid in past crises and is likely to do so in the future, see [1, 5, 21]. Private equity funds might even play a key role in the economic recovery process. Currently, the private equity industry holds $450 billion waiting to be invested [16]; thus, besides governments and sovereign wealth funds, they are the only source for liquidity to an ailing economy. Since private equity funds tend to generate even greater returns when acquiring portfolio companies during grave economic periods, now might be the right timing to increase one’s exposure to private equity vehicles, see [1].

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The remainder of this article is structured as follows: First the basic ideas and different categories concerning private equity are delineated, then various investment vehicles are expounded, and finally means of determining the optimal fraction of listed private equity entities in a portfolio are developed.

2.2. DEFINING PRIVATE EQUITY CATEGORIES Delineating the notion of “private equity” is certainly not facile. In general, private equity is a broad term that refers to any type of equity investment in an asset in which the equity is not freely tradable on a public stock market. Furthermore, a private equity investment usually entails a rather long engagement period, although it is always temporary with an exit strategy sometimes already existing at the time of the investment being ventured. Before one can probe into the various classes of private equity (see Sec. 2.3) it is pertinent to scrutinize the criteria which distinguish them.

2.2.1. Financing Stages Private equity investments can be conducted in various stages of a company’s financial life-cycle. Roughly speaking, it is differentiated between an early stage, an expansion stage, and a late stage. Early Stage Financing. Early stage financing comprises seed financing, start-up financing, and first stage financing. During the seed phase solely a conception of the business and the product exist, with the founder himself or some acquaintances of his raising the money required. The start-up period is dedicated to the composition of a business plan while the financial burden is still primarily imposed on the previously mentioned persons. With the first stage commencing, the entrepreneur launches production and selling of his product. These first revenues, however, are easily outnumbered by the increasing expenditures. Expansion Stage Financing. During the expansion stage, the business strives for continuous growth and attempts to break even; then, an expansion to new markets and a diversification of the array of products is aimed at, and the sales and distribution process is adapted to this development. Compared with the early stage, the risk involved is lowered significantly. Late Stage Financing. With the late stage commencing, the business is usually fairly well established. According to its situation and its prospects, there are various possibilities to be proceeded with. One might be the alignment of the enterprise to a future

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IPO, i.e., initial public offering of its share at a stock market; doing this, bridge financing provides for an ameliorated equity ratio. A financing aimed at changing the capital structure of the portfolio company is also called “recapitalization” or “consolidation.” Another possible strategy might be a separation of the firm from its former owners. This can be achieved by a management-buyout (MBO) with the former executives becoming the new shareholders or by a management-buyin (MBI), where previously unengaged investors assume both stakes and operational control. With the new owner usually not being able to raise the money required for the purchase on their own, buyout financing, which can be composed of equity, debt, and mezzanine products, is required. If the debt ratio of the deal exceeds 50%, the expression of a leveraged buyout applies, see [13]. Turnaround financing comes in handy if the enterprise struggles and feels inclined toward augmenting the equity ratio to become attractive to debt capital providers again. Funds specializing in this practice usually offer also consultant services to increase the odds of a successful turnaround. This type of investment strategy is sometimes referred to as “distressed financing.”

2.2.2. Divestment Strategies Initial Public Offering (IPO). An IPO is probably the most remunerative exit strategy for a private equity investor, as the sale price achieved is usually quite notable due to a high degree of transparency and decent market liquidity. Furthermore, it provides for a good publicity to both the private equity fund and the portfolio company, which facilitates the alignment of interests. However, the downside of this exit strategy is the great dependence on a favorable stock market environment and significant costs due to regulatory and legal constraints. Trade Sale. Selling the investment to another corporation is called “trade sale.” As the purchaser predominately operates in the same industry like the portfolio company, he may benefit from strategic advantages when incorporating the venture to be sold into his corporate structure and thus pays even a higher price than an IPO might generate. Additionally, transaction fees are usually lower than in an IPO sale. Although the portfolio company mostly thrives upon such an arrangement, its management might be hesitant, as their own position and independence could be affected. Secondary Sale/Financial Sale. An exit is called “secondary sale,” if the purchaser is also a financial investor; thus, a synonym for this way of exit is “financial sale.” This type of divestment is usually chosen if the seller has undergone a change of investment strategy or requires capital, which cannot be procured anywhere else. Consequently,

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due to lofty yield expectations of the purchaser, the capital gains obtainable with a secondary sale are rather limited while transaction costs tend to be fairly moderate. Buy-Back. “Buy-back” refers to a divestment approach, where the target company, especially its management, acquires the shares to be sold by the private equity investor. This strategy is apt for rather less successful companies, which cannot be hived off by a trade sale or IPO, whereas some family-controlled businesses actually insist on a contractual agreement on buy-back conditions prior to letting a private equity fund invest in their business. Total Loss/Liquidation/Distressed. Liquidation of a business is undoubtedly the least pleasant way of disinvestment. It is only adopted if the company is “dead more worth than alive” and will certainly experience grave resistance by the portfolio company’s management and staff. In case of a liquidation, the provider of debt (i.e., the creditor) holds claims, which are senior to those of equity investors; with liabilities usually exceeding the assets of a distressed company, its liquidation mostly results in a total loss of money for the private equity investor. Sometimes, exiting a firm that is financially troubled or even already insolvent is categorized as “distressed.” In that case, private equity companies specialized in turnaround-financing might pose a follow-on investor, see [15].

2.2.3. Type of Financing The type of capital used for a private equity investment also accounts for its classification. The basic categories are equity, debt, and mezzanine. Equity Financing. Equity is, as the expression “private equity” suggests, the most common capital source of the private equity industry. An equity investor enjoys a proportional claim to a company’s assets, which usually depends on his share of the company’s book value. He is mostly entitled to vote at the shareholders’ meeting and thus, wields influence on the company’s business strategy and its management. In case of an insolvency, equity is characterized by its junior treatment in the liquidation process. Equity investors only recover the invested capital after all claims resulting from debt financing are satisfied. Therefore, they face the peril of not only losing the invested principal, yet also might be obliged to inject fresh capital into the firm, if it has a corporate structure rendering the stakeholder personally liable, see [4]. Evidently an equity investment is deemed to be riskier (than debt financing), and thus the expected return is higher due to the risk premium, which an equity investor expects. Owing to these characteristics, equity is the preferred type of financing for rather young companies, as it does not require a collateral which these companies mostly could not

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provide for. Equity is provided to the portfolio company either by purchasing shares from the previous stakeholder, or raising “fresh” equity capital by the private equity investor, or granting equity capital in the form of a dormant partnership. Debt Financing. Opposed to equity, debt is provided to a company only for a predetermined and limited time. The debt investor’s claim is senior to claims of equity providers in case of an insolvency. Therefore, he is not liable, and he only risks the loss of the committed principal in case of a default. The debt investor’s return, i.e., the interest payments, are due at predefined, fixed dates regardless of the portfolio company’s performance, which accounts for the name “fixed income investment.” As the cash flows — apart from default situations — are predictable, the risk and, consequently, also the return associated with debt financing is lower than in the equity case. Debt is predominately a prevalent means of financing late-stage portfolio companies. Furthermore, debt does not affect the ownership situation, which is often desired by family-owned portfolio companies, and might grant tax-related benefits to the invested firms, as it can deduct the interest payments from the taxable revenues. Mezzanine Financing. Mezzanine financing can be deemed a hybrid structure between equity and debt financing. The first mezzanine financing is recorded to have taken place in 1910, when the Computing-Tabulating-Recording Company (today: IBM) issued high yield bonds for the purpose of start-up financing. Sometimes, mezzanine capital is furnished with an “equity-kicker”, which allows for a participation in capital gains in the case of an IPO. Along with the typical interest payment associated with debt, mezzanine capital will often include an equity stake in the form of warrants attached to the debt obligation or a debt conversion feature identical to that of a convertible bond. Mezzanine financing is junior to debt and senior to equity financing in case of a default. It is only secured by the equity of the company, and not the company’s tangible assets (e.g., property, cash, or accounts receivable). Therefore, mezzanine capital is a more expensive financing source for a company than secured debt or senior debt, as its risk of default is greater, which results in the investors demanding a higher interest rate, usually 200–800 basis points above a “regular” loan, see [15].

2.2.4. Classification of Private Equity Fund Investments According to these previously depicted criteria, private equity funds can generally be grouped into two categories: (1) venture capital and (2) buyout funds.

2.2.4.1. Venture capital funds Venture capital funds invest in new (i.e., early or expansion-stage) companies with a promising idea, innovation, or product predominantly to be found in technology,

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telecommunications, or healthcare sectors. They mostly provide equity financing, opt for an IPO as preferred exit route, and therefore are strongly correlated with small cap indices. Extensive market and product due diligence is the major focus during the selection process. The fund has to handle the immeasurable uncertainty associated with its investment and relies on its industry know-how, product development, and commercialization expertise; therefore, providing consultancy services to the portfolio company is crucial to succeed. The investor focuses on few “winners” and accepts many write-offs, see [19].

2.2.4.2. Buyout funds Buyout funds invest in established, i.e., late stage companies that need capital for changing the ownership. Buyout funds are mainly debt financed and tend to perform well during bearish public equity market periods, when debt is available at low costs. Intensive financial due diligence is the major focus during the selection process. Putting emphasis on established industry sectors, the fund has to handle rather measurable risks and relies on its financial engineering and corporate restructuring expertise. The investor focuses on a high percentage of successful investments with only limited number of write-offs. As previously depicted, “buyout fund” is a generic term comprising management buyout (MBO), management buy-in (MBI), and leveraged buyout (LBO) activities.

2.2.4.3. Leveraged buyouts (LBO) As LBOs account for the bulk of private equity deals, it is pertinent to shed more light on this type of buyout transaction. The purpose of financial leverage with regard to investments is fairly straightforward: The private equity fund incurs debt, which is reinvested in portfolio companies along with equity capital. If the overall investment’s return is higher than the cost of interest, the private equity investor yields a greater return on its equity stake than if it does not borrow. The idea of a leveraged buyout hinges on the validity of the Modigliani–Miller Theorem, see [20]. It states that the value of a firm is unaffected by how the firm is financed. It does not matter if the firm’s capital is raised by issuing stock or selling debt. If this was not the case, there would be an arbitrage possibility. Therefore, when a private equity fund conducts a leveraged buyout, the value of the target company — as far as theory goes — ought not to decrease. Hence, it is possible that a large portion of the purchase price is debt financed — sometimes up to 95% of the target company’s total capitalization. Doing this, private equity firms can conduct large acquisitions without having to commit a lot of capital. Typically, the loan is borrowed through a combination of repayable bank facilities and/or public or privately placed bonds, which may be classified as high-yield debt, also called “junk bonds.” The debt will predominantly appear on the acquired

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company’s balance sheet and the acquired company’s free cash flow will be used to repay it, see [4]. Although the sources of value generation for buyout deals are manifold, they can be basically cut down to the following three aspects: Bondholders. Investors holding bonds of the target company are the obvious losers of a leveraged buyout deal: As previously mentioned, the debt of a buyout is mostly collateralized with the target company’s assets; therefore, the existing bond holders would have to “share” the target company with the upcoming bondholders in case of a default. Thus, bond prices tend to plummet, as soon as a leveraged buyout is announced. Tax shields. Debt financing displays the great advantage that the interest the company pays to creditors is a tax-deductible expense, while dividends, the “cost” of equity capital, are not. The amount of tax payments which the company can avoid by incurring debt is called “tax-shield.” Provided that the value of a company is unaffected by the way it is financed (Modigliani–Miller Theorem [20]), the present value of the taxshield equals the increase in value of a firm when a private equity fund raises the amount of debt in its target’s capital structure. Incentives. Greater incentives for the company’s management and its employees can account for yet another source of value generation as people need to increase their efforts to come up with the cash needed for debt service. Furthermore, senior managers are usually obliged to place a considerable amount of their personal wealth in the company they lead.

2.3. INVESTMENT POSSIBILITIES — ONE ASSET, MANY CLASSES When contemplating an investment in private equity, one should bear in mind that there are manifold possibilities of obtaining exposure to this asset class each of which are visualized in Fig. 2.1.

2.3.1. Direct Investments The most genuine way of investing in private equity certainly is the direct purchase of shares of a single company. As this requires superior skills pertaining to the selection and the supervision of the acquired firm, the timing and execution of the divestment, it usually takes a team of professionals to perform these duties and meet these challenges adequately. With 30% of direct investments resulting in a total

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Figure 2.1 Private equity investment possibilities.

loss, with target-companies usually stating a preference for an established private equity investor, and with the fraction of private equity in a standard portfolio being fairly low and consequently, the cost associated with a direct investment disproportionately high, it is usually a prudent choice to opt for different fashions of private equity investments.

2.3.2. Private Equity Funds Instead of directly giving money to the target companies the investor can also pick an investment fund as financial intermediary, which assumes all duties from the selection process to the exit planning, see [13].

2.3.2.1. Key players General Partner. The term “general partner” (GP) is equivalent to the expression “fund manager” or better “fund management team” of a private equity fund, as it is usually operated by more than one person. The general partner contributes about 1% of the invested capital to his private equity fund. Fund Management Company. The management company commonly supervises various general partners. It receives an annual management fee of about 1.5–2.5% of the committed capital. Fund. The private equity fund is established by a limited partnership agreement between the general partner and the limited partners. The general and the limited partners arrange their commitment to the private equity fund with Participation Agreement 1 (see Fig. 2.2).

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Figure 2.2 Structure of a private equity fund.

Limited Partner. The limited partners provide for the bulk of the committed capital (∼99%) and are accordingly called “investor” or “sponsor.” The investment activity of large institutional investors carries today’s industry. Therefore, the main players in the market are pension funds, insurers, funds-of-funds, banks, government agencies, corporate investors, foundations, and family offices. Target Companies. Target companies, also dubbed portfolio or investee companies, are the entities into which a private equity fund directly invests. This investment is often referred to as “deal.” Investment Process. Prior to conducting an investment, the private equity fund’s general partner imposes a rigorous scrutiny on the companies he is interested in. On average, only 25% of the businesses requesting financing from a private equity fund manage to pass this hurdle, see [15]. These enterprises are to experience an in-detail due diligence. Usually, the due diligence process contains three components: The legal due diligence appraises all important contracts and agreements, the financial due diligence assesses the completeness and accuracy of financial statements, and the business due diligence probes into the economic prospects of the enterprise. With the due diligence being completed the parties may agree upon a letter of intent binding them to proceed with the selection process. The transaction process concludes with the signing of the Participation Agreement 2 (see Fig. 2.2) and the closing which comprises the transfer of the target company’s stakes in return to payment of the purchase price. Banks. Banks usually play an important role in leveraged buyout transactions as they provide for the necessary amount of debt capital. The target company predominantly serves as collateral for these loans (see Fig. 2.2).

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2.3.3. Cash Flow Structure of a Private Equity Fund Limited Partners’Contribution. The limited partners usually contribute about 99% of a fund’s volume. It is pertinent to differ between the committed capital, which is the maximum amount the limited partners may be obliged to deposit into the fund and the drawdown. The latter denotes the money actually requested by the general partner which he does whenever there is a worthwhile investment opportunity. On the other hand, cash distributions to the limited partners are made as soon as a deal is realized by the fund manager. From the standpoint of the limited partners, the investment process entails substantial uncertainty: The timing of both drawdowns and distribution is unpredictable and the valuation of the portfolio is sometimes not meaningful, as regular marking to market of the investment is too costly and thus left at the general partner’s discretion. Management Fees. Management fees provide for a base compensation so the fund manager can support ongoing activities of the fund. On the other hand, the management team should not make any profits on those fees, as compensation reasonably ought to be performance driven. Therefore, annual fees range between 1.5% and 2.5% of the committed capital. Levying fees on a committed capital basis prevents the fund managers from going “for volume instead of quality,” i.e., drawing down and investing more capital than he can find remunerative investment opportunities, see [19]. Carried Interest. Carried interest is a bonus entitlement accruing to an investment funds management company or individuals of the management team which — as it is their main source of income — has to provide for a commensurable incentive to strive for a considerable fund performance. It is usually as high as 20% of the profits of a fund and becomes payable once the investors have achieved repayment of their investment in the fund plus a defined hurdle rate, which ranges between 6% and 12%. Carried interest is calculated either on a “fund-as-a-whole” basis, where the overall performance of a fund has to be in excess of the hurdle rate or a “deal by deal” approach with the proceeds of a single transaction necessitating to surpass the hurdle rate. Gross/Net Cash Flows. As illustrated in Fig. 2.3 one can differ between two types of cash flows: (1) Deal cash flows occur between the private equity fund and its portfolio companies; as carried interest and management fees are not yet deducted, they are also referred to as “gross cash flows.” (2) Fund cash flows occur between the private equity fund and the investor (limited partner). Carried interest and management fees had already been deducted at that point; thus, they are also referred to as “net cash flow” or net distributions.

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Figure 2.3 Cash flow structure of a private equity fund.

2.3.4. Fund-of-Funds 2.3.4.1. Structure of a private equity fund-of-funds However, the process of selecting a successful fund is not trivial either, as choosing an apt general partner is crucial to the fund performance which differs immensely. Therefore, the investor may want to outsource this activity to a professional, together with all the other chores entailed by being a limited partner. Hence, a fund-of-funds can be deemed a vehicle that pools a group of investors and uses the capital to assemble a diversified portfolio of funds. The organization of funds-of-funds basically resembles the structure of regular private equity funds (see Fig. 2.4).

2.3.4.2. Advantages Diversification. The possibility of greater diversification is definitely one of the greatest advantages of funds-of-funds, as various studies have shown that they perform similarly to individual funds but with less pronounced extremes. With funds-of-funds participating in up to 100 individual private equity funds averaging about 20 portfolio companies, the investor can engage in about 2000 target enterprises. Furthermore, very specific expertise is required when investing in new industries, emerging regions, or financing stages. As an individual fund cannot specialize in every field, it is certainly pertinent to get exposure to different private equity markets via a fund-of-funds. Resources. Usually both institutional investors and individuals lack the expertise of research, due diligence, ongoing monitoring, reporting, and administration of a private equity fund investment.

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Figure 2.4 Structure of a private equity fund-of-funds.

Liquidity Management. The vast majority of private equity funds do not know either the timing or the size of the distribution cash flows. However, they do follow the pattern with cash out-flows during the first half and in-flows during the second half of a fund’s lifespan. With a fund-of-funds having many individual funds in its portfolio, the aggregate cash flows become more predictable and thus they can agree upon predefined payout schedules, which especially favor the needs of retail customers.

2.3.4.3. Disadvantages The greatest drawback associated with a fund-of-funds is the additional layer of fees the investor is charged. Funds-of-funds charge a management fee customarily ranging between 1.0% and 1.5% of the committed assets and additionally receive carried interest once the hurdle rate is attained. However, funds-of-funds predominantly deduct only 5% of the gains. Furthermore, investors typically have fewer possibilities of wielding influence on the development of their money compared to a single-fund investment. Beyond, funds-of-funds do not offer a tailor-made investment focus which some investors may need in order to offset their underexposure to some countries or industries.

2.3.5. Publicly Traded Private Equity Investing in publicly traded private equity (PTPE), also called “listed private equity” (LPE) if the entity is quoted at a stock exchange, is another approach to get exposure

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to this asset class while maintaining a decent level of liquidity. As private equity companies primarily hold the corporate structure of a limited liability partnership, they are not publicly quoted. However, there are some previously or currently listed companies whose core business is private equity. Furthermore, investment funds which invest a predetermined share in private equity, and specially structured investment vehicles which invest in private equity directly are to be found. Of course, one may also purchase bonds issued by private equity companies or resort to a warrant with a private equity company’s share as underlying value. Moreover, there are certificates with a private equity index or a specific basket comprising private equity portfolios with different investment focuses as underlying values. LPX GmbH is a company based in Switzerland that specializes in constructing indices based on listed private equity (LPE). It provides the global LPE index LPX50, LPX Major Market, and LPX Composite; LPX GmbH also calculates and publishes [17] regional indices such as the LPX Europe as well as style indices such as the LPX Buyout, LPX Indirect, and the LPXVenture. For the portfolio calculations subsequently conducted the LPX 50 is considered. The LPX 50 is a global index that consists of the 50 largest liquid LPE companies covered by LPX. The total return index (as opposed to a price index) is applied here, which means that dividend payments are also included by reinvesting them in stocks.

2.3.6. Secondary Transactions Secondary transactions deal with securities, which are bought and sold after their initial sale, while investors in the primary market purchase their investment instruments directly from the issuer, see [11].

2.3.6.1. Types of secondary transactions It is possible to differentiate between the following types of secondary transactions, see [7]: Limited Partner (LP) Secondary Transactions. This most commonly conducted category of secondary transactions occurs, when the limited partner wants to withdraw his investment from a private equity fund prior to maturity. Portfolio Transfer. “Portfolio transfer” refers to a transaction of a fund’s aggregate portfolio to another company. This type of sale is quite complex and entailed by an in-depth due diligence of all the assets to be relocated. Secondary Buyout. Purchase of selected portfolio companies by a general partner is called “secondary buyout.” Compared to a portfolio transfer, a secondary buyout appears to be less risky, as the purchaser can pick supposedly valuable targets.

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2.3.6.2. Buyer’s motivation There are various reasons why investors choose to sell their private equity stakes. They may have overcommitted capital or overestimated timing and amount of cash distributions and thus need liquidity. There might have been a change in strategy or new regulatory provisions, which induce a fund to decrease exposure to some sectors, countries, or asset classes. Depending on the intensity of pressure to divest the seller faces, there is a good possibility for the buyer to receive a discount. Buying secondary investments gives an opportunity of diversifying of vintage years, which is — due to the great variability of performance depending on the vintage year — a possible diversification approach. Furthermore, secondary investments have proven to display anti-cyclic behavior: Due to higher discounts achieved during underperforming periods of the primary market, secondary transactions do specifically well in such times, enabling a possible reduction of risk when incorporated into a portfolio. Additionally, compared to the fund’s original investors, a secondary buyer gets his investment back faster, as there is only a limited amount of original investment opportunities in the market, it usually takes several years for the complete committed capital to be invested, whereas the secondary investor invests his commitment all at once. Therefore, a secondary transaction is a reasonable means of quickly attaining private equity exposure.

2.4. PRIVATE EQUITY AS ALTERNATIVE ASSET CLASS IN AN INVESTMENT PORTFOLIO After a detailed description of the different types of private equity investments, the benefits of private equity for asset allocation purposes are analyzed in the following. Like every other alternative asset class, private equity is supposed to provide diversification benefits, i.e., reducing the risk of a portfolio while keeping the expected return at a high level, or, respectively, increasing the expected return while not bearing a higher risk. Unfortunately, it is not easy to measure the performance of private equity investments. This is due to the fact that — except for listed private equity vehicles — there is usually only a very limited secondary market. Hence, although it is possible to calculate the return of a private equity investment after a fund is liquidated, it is difficult to find a fair value during the fund’s lifetime. However, these fair values are necessary to identify the risk profile of private equity investments. Therefore, the subsequent study is only based on listed private equity as it provides the only private equity investments, which are liquid enough to have realistic price time series. The following methodology and the results of this section are based on the study which is presented in detail in [2].

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2.4.1. Characteristics of LPE Return Series This study is based on monthly log-returns of bonds (represented by the JP Morgan Global Government Bond Index), stocks (represented by the MSCI World Index), and listed private equity (represented, as described above, by the LPX 50 Index) between January 1998 and December 2007. (Data assigned to an asset class in this study always refers to these according indices.) Table 2.1 provides the empirical statistics of monthly log-returns. The average mean for bonds is exceptionally high, especially compared to the performance of stocks. Combined with the low standard deviation, this would cause a risk–averse investor to allocate an extremely high portion of his portfolio to bonds. In general, the question arises whether past returns can be used as an indicator for the future. Therefore, often experts’ opinions on expected returns are used for asset allocation purposes. To account for both, the impact of past returns and (market and experts’) expectations, the Black–Litterman approach [3] is applied to adjust the empirical returns. It allows to combine absolute and relative forecasts at a confidence level τ with the empirical expected values µemp (which are assigned a weight of 1−τ). The forecasts are expressed through q = P · µ, where each row of P and q stands for one particular forecast. P gives the linear transformations of the asset classes for each forecast, and q the actual values of these forecasts. The Black–Litterman expectations µBL can then (given that P is invertible) be calculated via: µBL = τ · P −1 · q + (1 − τ) · µemp .

(2.1)

The assumptions for the Black–Litterman approach employed here at a confidence level of τ = 0.75 are the following: (1) Bonds are expected to have an annual return of 3.96%. This number is a combination of expected returns of several bond markets

Table 2.1

Empirical Statistics of Monthly Log-Returns.

Empirical statistics

Bonds

Stocks

LPE

Mean (emp.) Mean (BL) Standard deviation Skewness Excess kurtosis Autocorrelation: lag 1 Autocorrelation: lag 2 Autocorrelation: lag 3 Autocorrelation of squared returns (lag 1) 5% critical value for autocorrelation

0.0051 0.0037 0.0190 0.2012 −0.0989 0.1449 −0.0618 0.0759 −0.0622 0.1941

0.0056 0.0061 0.0410 −0.7789 1.1124 0.0862 −0.0959 0.0739 0.0223 0.1941

0.0081 0.0086 0.0651 −0.1106 2.2936 0.3288 0.0501 0.0392 0.1391 0.1941

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(provided by Bloomberg, 30 August, 2008), weighted according to the constitution of the index. (2) The equity risk premium is 3.5% (in accordance with several other studies [8, 9]). (3) The difference of the means of listed private equity and stocks remains the same. The Black–Litterman adjusted means are given in the second line of Table 2.1 beneath the empirical means. It can be observed that the listed private equity index exhibits a lag-1 autocorrelation, which is significant at the 5%-level. Autocorrelation is a possible sign of stale-pricing effects. Stale pricing occurs, if the market adjusts only slowly to new information, which usually is the case when liquidity is low. Of course, the (secondary) market of unquoted private equity funds is highly illiquid, but also the constituents of the LPX might suffer from liquidity problems. Regarding skewness and excess kurtosis, returns of the MSCI and the LPX 50 tend to be non-normally distributed (which can be confirmed by a Jarque–Bera test). Table 2.2 shows the correlations between these three asset classes.

2.4.2. Modeling Return Series with Markov-Switching Processes A common way of handling those cases where non-normality and/or autocorrelation occurs is to simulate the returns by a Monte Carlo method: A model describing the return time series as well as possible is built. As future return time series are not predictable, this model certainly is based on random processes. Therefore, in a next step, numerous (in this case 10,000) possible return paths are simulated with this model. Finally, the required return distributions can be derived from these paths and the portfolios can be optimized.

2.4.2.1. Markov–Switching models Markov–Switching models are capable of fitting the previously mentioned empirical statistics. There are many academic studies emphasizing the benefits of Markov– Switching models in mapping the economic time series. For more details consult Timmermann [25]; there are also various applications of Markov–Switching models

Table 2.2 Empirical Correlation Structure of Monthly Log-Returns. Correlation Bonds Stocks LPE

Bonds

Stocks

1 −0.007867 −0.007867 1 0.012427 0.785161

LPE 0.012427 0.785161 1

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(e.g., in Brunner/Hafner [6]). Here, the Markov–Switching model allowing for stateindependent (first lag) autoregressive dynamics is used. This Markov–Switching process is defined as follows: rt,a = µst ,a + φ · (rt−1,a − µst−1 ,a ) + εt,a

(2.2)

with st denoting the state of markets at time t, µst ,a denoting the mean return of asset a ∈ {1, 2, 3} (where 1 = ˆ bonds, 2 = ˆ stocks, 3 = ˆ LPE) in state st , εt = (εt,1 , εt,2 , εt,3 )T the innovation at time t, εt ∼ N(0, st ), and the autocorrelation parameter φ satisfying |φ| < 1. There are two possible states: If the markets are in state 1 then st = 1, if they are in state 2 then st = 2. The changes of the state st over time are modeled by a Markov chain. The transition probabilities of changing from state i at time t − 1 into state j at time t are given by the matrix:
P=

p11 p21

 p12 , p22

where P(st = j|st−1 = i) = pij .

The whole process is then described by the parameter vector  = (µ1 , µ2 , p12 , p21 , φ, 1 , 2 ) with µ1 , µ2 ∈ R3 , φ ∈ (−1, 1)3 , p12 , p21 ∈ [0, 1] and 1 , 2 ∈ R3×3 the covariance matrices for the two states 1 and 2. This model is described in detail in [2]. Another possible way is to apply univariate processes for each asset and then include the correlation structure between the asset returns by generating multivariate normally distributed residuals. This would allow for all possible combinations of different individual market states, where each asset class can be in one of two states, and therefore has one drawback: As the correlation would only be accounted for by fitting the residuals, the second big part which is able to influence the correlation is neglected, which is the correlation occurring from the state combination. Therefore, instead of modeling independent Markov chains, the ones resorted to in this study are related to each other in the way that only two metastates are allowed for: st = 1 and st = 2, i.e., all assets are in the same state at each point of time t. Of course, the more market states are taken into account, the better the empirical moments and autocorrelations may be matched. However, the (quadratic) increase of the parameter vector itself causes the fitting method to be less efficient. Furthermore, overfitting problems are more likely to occur. Therefore, as few states as possible should be used. In this case, two market states already enable the model to match the empirical parameters in a very good way.

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2.4.2.2. Fitting the parameters The parameters are chosen such that the theoretical statistics of the Markov–Switching process match the empirical statistics best possible. The background for the method of moments, which has been applied for the fitting, has been depicted by Timmermann [25]. The fitting focuses on the first four moments as well as the autocorrelation of lag 1 (which has proven to be significant for LPE). As the autocorrelations for lags 2 and 3 as well as the autocorrelation of the squared returns were not significant for any of the indices, these statistics are not taken into account. The results for the parameter vector  are given in Table 2.3. These parameters show some very interesting characteristics: State 1 describes a market scenario, where bonds perform well, whereas stocks and LPE are in their worse of the two states. State 2 describes a scenario where stocks and LPE perform very well, whereas bonds show a poor performance. It is even more interesting to consider the standard deviations of stocks and LPE in the two states: In state 1, the weaker of the two states, the volatility is much higher than in state 2, where both asset classes have a strong performance. Regarding the autocorrelation parameter φ, LPE shows the highest value of all asset classes. Table 2.4 shows the correlation matrices for the two states st = 1 and st = 2. (These matrices only give the correlation of the error terms εt and not of the returns of the asset classes themselves, such as in Table 2.2.) Even these correlation matrices Table 2.3 Parameter Bonds Stocks LPE

Model Parameters for the Assets.

µ1

µ2

σ1

σ2

φ

0.01254 −0.01236 0.00567

−0.00422 0.02266 0.01114

0.01818 0.04955 0.08727

0.01508 0.02000 0.01527

0.19134 0.11436 0.32983

 P=

Table 2.4



0.4512

0.5488

0.4944

0.5056

State-Dependent Correlations of the Residuals. st = 1

a2 a1

Bonds Stocks LPE

st = 2

Bonds

Stocks

LPE

Bonds

Stocks

LPE

1 0.06316 0.04029

0.06316 1 0.92412

0.04029 0.92412 1

1 0.69471 0.04500

0.69471 1 0.41966

0.04500 0.41966 1

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show some realistic characteristics, e.g., the correlation between the error terms of LPE and stocks is much higher (0.92412) for the bad market state 1 than for the “better” state 2 (0.41966). The parameters are fitted in such a way that the deviation between the empirical statistics of the time series and the theoretical statistics of the model is minimized. The (relative) deviation of the theoretical statistics (determined by the parameters) and the empirical statistics is always smaller than 1% (only the deviation for the kurtosis of bonds is higher (3.5%)).

2.4.2.3. Simulation of return paths Having constructed the model, 10,000 possible scenarios of how the return paths of the assets could develop over 60 months are simulated by applying the previously expounded process and using the fitted parameters. As st is an ergodic Markov chain, there exists a unique stationary distribution. The unconditional probability of such a process being in state 1 is given by π1 = p21 /(p12 + p21 ) and the probability of it being in state 2 is respectively given by π2 = p12 /(p12 + p21 ). In this case, π1 = 0.4739 and π2 = 0.5261. The simulated return paths show the same properties as the empirical time series. The statistics of the simulated paths match the empirical statistics very well. Additionally, most of the assets show a highly significant non-normality. The non-normality will be one of the main issues in the following, because some very popular concepts in portfolio theory (such as Sharpe ratio, mean–variance framework) do only account for the first two moments, i.e., mean and standard deviation. But risk–averse investors are also sensitive to higher moments because they prefer a higher skewness as well as a lower excess kurtosis. Therefore, it will be necessary to apply more complex concepts of portfolio theory.

2.4.3. Listed Private Equity in Asset Allocation It is the aim to determine the optimal fraction an investor should invest in listed private equity, or in this particular case, in the LPX 50 Index. The results described here are an excerpt from [2] and treat the one-year time horizon. However, the described method can be applied to any time horizon which may be of interest, which is also done for three and five years in the cited study.

2.4.3.1. Performance measurement The following section introduces performance measures, which are necessary to evaluate and compare the benefits of different asset allocations.

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Sharpe Ratio. The Sharpe ratio, see [23, 24], was first introduced by William F. Sharpe and is one of the most common and intuitive measures. It is defined as follows: SR(R) =

µR − r σR

(2.3)

with portfolio return R, its mean µR , standard deviation σR , and risk-free rate r. Despite its intuitive formula (expected excess return per unit of risk in terms of standard deviation), it has one major shortcoming: Since risk is only measured in terms of standard deviation, the Sharpe ratio solely uses the first two moments of the return distribution and ignores characteristics of non-normally distributed returns. Therefore, it is not appropriate (or at least not sufficient) for return distributions, which show non-normal characteristics, such as it is the present case. Omega Measure. The Omega measure, see [22], is based on the idea that an investor partition returns into gain and loss, i.e., in returns above and below a return threshold τ. Thus, the Omega measure represents the probability weighted ratio of returns above and below the threshold or, in other words, upside per downside: τ (R) =

E[R − τ]+ . E[τ − R]+

(2.4)

This definition can be transformed into τ (R) = (µR − τ)/(E[τ − R]+ ) + 1. This notation reminds of the Sharpe ratio. But unlike the standard deviation for the Sharpe ratio, the downside potential (as a measure of risk) exhibits the crucial advantage of considering the whole return distribution of the portfolio return R where R is smaller than τ (i.e., “left” of the threshold τ). However, the Omega measure still does not allow to account for risk aversion (as it is not the same as choosing the threshold τ). Score Value. The score value, see [10, 14], corrects the upside by the weighted downside and is defined as follows: ScoreλSc (R) = E[R − τSc ]+ − λSc · E[τSc − R]+ ,

(2.5)

where λSc defines the risk aversion of the investor. A high λSc stands for a high risk aversion and vice versa. This performance measure encompasses the complete return distribution of a portfolio as well as an investor-specific risk aversion parameter λSc . It is obvious that the higher the Score value the better the portfolio under this measure. In this study, the parameter τSc for the Score value is set to be equal to the risk-free rate, i.e., τSc = r. However, the Score value also allows to adjust τSc according to an investor’s individual preferences.

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2.4.3.2. Portfolio optimization frameworks After describing how to gauge the performance of portfolios, some criteria are delineated for optimizing them. Like the Sharpe ratio among performance measures, the mean–variance framework is the most common and intuitive concept in portfolio optimization. Nevertheless, there are some shortcomings of this framework since it only considers (just like the Sharpe ratio) the first two moments of return distributions. Mean–Variance Framework. This very popular method of constructing optimal portfolios by Markowitz [18] assumes that mean and standard deviation embody sufficient information about the return distribution of a portfolio (which is the case when dealing with normality). A portfolio is said to be mean–variance efficient when there exists no other portfolio which has the same (or less) risk and a higher expected return, or the same (or a higher) expected return accompanied by lower risk. Thus, one possibility of ensuring mean–variance efficiency of a portfolio is to minimize the risk under the condition that the expected return of the portfolio is higher or equal to a target return µtarget . (Another approach would be to maximize the expected return under the condition that the risk does not exceed a given upper limit.) To ensure that the entire capital is invested (full-investment constraint), and that short-selling is not allowed (each portfolio weight has to be non-negative), a set Z of all possible portfolios is defined:     3  Z := w ∈ R  wi = 1, wi ≥ 0 , (2.6)  i

where the vector w = (w1 , w2 , w3 ) denotes the portfolio weights. Considering various levels of target returns µtarget , a large number of efficient portfolios can be obtained forming a concave efficient frontier in the risk–return space. It depends on the risk aversion of an investor to decide on which of these portfolios is the optimal one. As only risk–averse investors are considered, this choice is a tradeoff between risk (measured in terms of standard deviation) and return. The actual optimization problem can be formulated as λ (2.7) max wT µ − · wT w w∈Z 2 with µ = (µ1 , µ2 , µ3 )T (where µi denotes the expected return of asset i), the covariance–matrix , and the risk aversion parameter λ. T

Power-Utility Framework. The conception of this model is quite commonly applied (e.g., in Grauer and Hakansson [12]). Within this model, the utility of an investor is defined as follows: 1 (2.8) U(R) = (1 + R)γ , γ < 1 (if γ = 0, then U(R) = ln(1 + R)) γ

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with R = R(w) = i wi Ri denoting the portfolio return (subject to w), Ri the return of asset i, and γ capturing the investor’s attitude toward the risk. The aim is to find those weights w, which maximize the expected utility: max E[U(R(w))].

(2.9)

w∈Z

Omega Measure and Score Value Frameworks. As the motivation for the Omega measure and the Score value is to capture more than the first two moments of the return distributions of a portfolio, the portfolios can be optimized by maximizing each of these performance measures, and thereby taking advantage of their respective characteristics. The restrictions are the same as in the other models, i.e., w ∈ Z.

2.4.3.3. Definition of investor types Optimizing a portfolio requires the risk aversion parameters λ, γ, λSc as well as the return threshold τ. The values of these parameters represent different investor types and are crucial when optimizing the weights of a portfolio. In order to show the effects of private equity in asset allocation, three benchmark portfolios (for three typical investors) are constructed, which only consist of bonds and stocks. According to the presented optimization problems, each typical investor is represented by a vector of parameters (λ, γ, λSc , τ). The parameters are chosen such that each optimization problem leads to the same benchmark portfolio for the corresponding investor. The following benchmark portfolios are chosen to characterize these investors when solely investing in bonds and stocks: A conservative investor A who invests 70% in bonds and 30% in stocks, an investor B whose strategy is an equally weighted portfolio of 50% bonds and 50% stocks and an investor C who is willing to take higher risks and invests only 30% in bonds and 70% in stocks. The parameters are listed in Table 2.5 for the according benchmark portfolios. (Note that the parameters depend on the time horizon.) For those parameters where higher parameter values stand for a higher risk aversion (λ and λSc ), investor A has greater values than investor C, and vice versa (for γ). As τ represents a threshold, which the investor sets to distinguish between upside and downside, investor C demands higher returns to consider them as “upside returns” than investor A. Table 2.5 Risk Aversion Parameters of the Different Investor Types. Investor: λ γ τ λSc

A

B

C

8.1756 3.1042 1.9158 −8.0317 −3.0455 −1.8764 0.0090 0.0332 0.0389 4.5397 2.0463 1.7551

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2.4.3.4. Optimization of portfolios The goal is to examine, if and to what extent investments in listed private equity increase the performance of a portfolio. For this purpose, returns of the assets were simulated, performance measures as well as models to optimize portfolios were introduced and three different investor types defined. This section will apply all these aspects. As stated in the optimization problems, an investor optimizing his portfolio at the beginning of the period of one year is considered. The investor has neither the possibility of reallocating nor materializing the returns until the end of his investment horizon. Hence, no transaction costs occur except for the initial purchase and the liquidation after one year. Due to the high liquidity of the underlying assets, these costs are assumed to be negligible. Four different portfolios for each investor type are considered: Each of them is optimized by one of the four previously described optimization models. Table 2.6 shows the optimal portfolios, which are obtained by applying all optimization models to all possible combinations of investor types A, B, and C for Table 2.6 Portfolio Weights and Performance Measures of Optimal Portfolios (One Year). Framework:

MV

PU

Omega

Score

Portfolio weights

A

Bonds Stocks LPX 50

0.7000 0.3000 0.0000

0.7000 0.3000 0.0000

0.7000 0.3000 0.0000

0.7000 0.3000 0.0000

B

Bonds Stocks LPX 50

0.5394 0.3852 0.0754

0.5441 0.3718 0.0841

0.5363 0.3708 0.0929

0.5334 0.3489 0.1177

C

Bonds Stocks LPX 50

0.3880 0.4434 0.1686

0.3944 0.4252 0.1803

0.3582 0.4282 0.2135

0.3101 0.4114 0.2785

Performance measures

A

Sharpe Omega Score

0.1612 4.4783 0.0165

0.1612 4.4783 0.0165

0.1612 4.4783 0.0165

0.1612 4.4783 0.0165

B

Sharpe Omega Score

0.2034 2.0181 0.0437

0.2038 2.0186 0.0438

0.2052 2.0190 0.0438

0.2071 2.0175 0.0438

Sharpe Omega Score

0.2148 1.7631 0.0493

0.2148 1.7637 0.0494

0.2156 1.7648 0.0495

0.2157 1.7621 0.0496

C

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a time horizon of one year. There are several conclusions to be drawn from this tables: There is almost always the same order of how much weight the performance measures assign to the LPX 50 Index: The fractions increase as follows: mean–variance framework < power–utility framework < Omega measure framework. The highest fraction for the LPX 50 is always recommended by the Score value framework, whereas the mean–variance framework allocates the smallest portions to listed private equity among all optimization frameworks. As theory suggests, the higher the risk aversion of the investor the smaller the investment in listed private equity, because this asset is riskier than bonds and stocks. In fact, the most risk–averse investor A would not consider a private equity investment at all. Investor B puts lower fractions into the LPX 50 (7.5–11.8%) than investor C (16.9–27.9%). Another interesting issue to examine is whether stocks or bonds lose weight when the LPX 50 index is included in the portfolio. Overall, it can be observed that bonds hardly change their weight, and thus almost the whole weight of the LPX 50 is drawn from the stocks. It is rather the case that the weight of bonds slightly increases when taking listed private equity into account. The least risky asset class, bonds, is regarded as a counterpart for the highly risky listed private equity for two reasons: its low risk itself (measured in any risk measure) and the low correlation with both stocks and private equity. Figures 2.5–2.8 show the portfolio allocation and the risk–aversion parameters in different optimization frameworks subject to the standard deviation.

Figure 2.5 Optimal portfolio weights in the mean–variance framework.

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Figure 2.6 Optimal portfolio weights in the power–utility framework.

Figure 2.7 Optimal portfolio weights in the Omega measure framework.

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Figure 2.8 Optimal portfolio weights in the Score value framework.

2.5. CONCLUSION Private equity is a versatile asset class. It comprises investments in companies in different stages (early/expansion/late stage), by choosing an appropriate financing (equity/debt/mezzanine financing) and divestment strategy (e.g., initial public offering or trade sale). Private equity investments are usually long-term investments. Among other reasons, this is due to the fact that a prolonged (and expensive) due diligence period is necessary to obtain the information needed to make an investment decision. Additionally, private equity funds tend to buy rather big stakes of unlisted companies so that they can also control the decisions of the target company’s management. Hence, the general partner’s level of expertise in the business area of the target company during the whole investment process is absolutely crucial for the success of a private equity fund. Furthermore, it is also important for the general partner to be well connected in the relevant markets in order to obtain all the relevant information about a non-listed company. Most investors invest in private equity via funds or even funds-of-funds since they themselves do not have such a high level of expertise. However, since there is no mark-to-market of unquoted companies, it is difficult to find a fair price for a stake in a fund during its lifetime. This yields a very illiquid secondary market and, thus, it is not possible to get reliable time series, e.g., for the risk measurement of a private

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equity fund (or fund-of-funds). For this reason, the conducted study about the benefits of private equity vehicles is limited to listed private equity only. Monthly log returns of the LPX50 total return index showed a significant nonnormality and autocorrelation. Capturing these characteristics is not possible with standard approaches. Therefore, the concept of Markov–Switching models is introduced to model more realistic return time series in order to find optimal asset allocations. Depending on the degree of risk aversion of an investor and the optimization method used, listed private equity should have a weight between 0% and 27.9% in an optimal portfolio, where only the most risk–averse investor type is supposed to ignore listed private equity for a one-year investment horizon.

References [1] Aigner, P, S Albrecht, G Beyschlag, T Friederich, M Kalepky and R Zagst (2008). What drives PE? Analyses of success factors for private equity funds. The Journal of Private Equity, (Fall), pp. 63–85. [2] Aigner, P, G Beyschlag, T Friederich, M Kalepky and R Zagst (2009). Optimal risk-return profiles for portfolios including stocks, bonds, and listed private equity (submitted). [3] Black, F and R Litterman (1990). Asset allocation: Combining investor views with market equilibrium. Goldman Sachs Fixed Income Research. [4] Brealey, R and S Myers (1991). Principles of Corporate Finance. NewYork: McGraw-Hill. [5] Brigl, M, A Herrera, H Liechtenstein, H Meerkatt, J Prats and J Rose (2008). The advantage of persistence, how the best private-equity firms ‘beat the fade.’ The Boston Consulting Group, Inc., and the IESE Business School of the University of Navarra. [6] Brunner, B and R Hafner (2006). Modelling Hedge Fund Returns: An Asset Allocation Perspective, pp. 429–452. Risk Books. [7] Damaschke, D and P Züchner. Private Equity Secondary Market, Grundlagen und Praxis für Investoren, Consultants und Wissenschaftler. Saarbrücken: VDM Verlag Dr. Müller. [8] Dimson, E, P Marsh and M Staunton (2006). The worldwide equity premium: A smaller puzzle. London Business School, Working Paper. [9] Donaldson, G, M Kamstra and L Kramer (2007). Estimating the ex ante equity premium. Rotman School of Management Working Paper No. 07/02. [10] Gong, X, M Huber, S Lanzinner and R Zagst (2006). Zertifikate. Mehrwert für Privatanleger? Zeitschrift für das gesamte Kreditwesen, 22, 1235–1239. [11] Grabenwarter, U and T Weidig (2005). Exposed to the J-Curve, Understanding and Managing Private Equity Fund Investments. London: Euromoney Institutional Investor Plc., [12] Grauer, R and N Hakansson (1986). A half century of returns on levered and unlevered portfolios of stocks, bonds and bills. Journal of Business, 59, 287–318. [13] Grunert, C (2005). Private Equity im Asset Management Institutioneller Investoren. Frankfurt: Bankakademie-Verlag GmbH. [14] Höcht, S, K.-H. Ng, J Wolf and R Zagst (2008). Optimal portfolio allocation with Asian hedge funds and Asian reits. Journal of Services Sciences, 1(1), 36–68. [15] Jesch, Th (2004). Private-Equity-Beteiligungen, Wirtschaftliche, rechtliche und steuerliche Rahmenbedingungen aus Investorensicht. Wiesbaden: Verlag Dr. Th. Gabler.

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[16] Liechtenstein, H and H Meerkatt (2008). Get ready for the private-equity shakeout. The Boston Consulting Group, Inc., and the IESE Business School of the University of Navarra. [17] LPX GmbH. Guide to the LPX Equity Indices. [18] Markowitz, H (1952). Portfolio selection. Journal of Finance, 7(1), 77–91. [19] Meyer, Th and P-Y Mathonet (2005). Beyond the J-Curve, Managing a Portfolio of Venture Capital and Private Equity Funds. West Sussex: John Wiley & Sons. [20] Modigliani, F and M Miller (1958). The cost of capital, corporation finance and the theory of investment. American Economic Review, pp. 261–297. [21] Rouvinez, C (2006). Top quartile persistence in private equity. Private Equity International, pp. 76–79. [22] Shadwick, W and C Keating (2002). A universal performance measure. Journal of Performance Measurement, 6(3), 59–84. [23] Sharpe, WF (1964). Capital asset prices —A theory of market equilibrium under conditions of risk. Journal of Finance, 19, 77–91. [24] Sharpe, WF (1994). The Sharpe ratio. Journal of Portfolio Management, pp. 49–58. [25] Timmermann, A (2000). Moments of markov switching models. Journal of Econometrics, 96, 75–111.

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3

ALTERNATIVE REAL ASSETS IN A PORTFOLIO CONTEXT

WOLFGANG MADER∗ , SVEN TREU† and SEBASTIAN WILLUTZKY‡ risklab GmbH, Seidlstraße 24–24a, 80335 München, Germany ∗ [email protected] † [email protected] ‡ [email protected]

In this chapter, we are introducing the asset class “Alternative Real Assets”. This asset class provides access to “real” investment opportunities, real meaning both partially inflation protected and solid, via limited partnership fund structures. This (sub-)asset class comprises fund vehicles that invest, for example, in infrastructure, shipping, or renewable energy projects. For some of these alternative real assets, financial incentives are provided, such as preferential feed-in tariffs for solar-generated electricity. We discuss alternative real assets, including a comparison to other asset classes and a description of distinctive features, with a focus on photovoltaic investments. The bottom-up modeling approach of individual photovoltaic projects and the approach to incorporate these investments in portfolio analytics are presented. Based on a comprehensive framework, we are finally able to evaluate the benefits of photovoltaic investments in a multi-asset portfolio. Investing in photovoltaic facilities is environmentally supportive. We are able, as well, to show that under reasonable assumptions, financing solar plants is an attractive asset class, both stand-alone and in a portfolio context.

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3.1. INTRODUCTION This chapter introduces the asset class “Alternative Real Assets.” Alternative real assets provide access to investment opportunities in special forms of project finance, usually via limited partnership structures. This asset class comprises fund vehicles that invest, for example, in infrastructure, shipping, or renewable energy projects. For some of these alternative real assets, financial incentives are in place, such as preferential feedin tariffs for solar-generated electricity. The analysis of photovoltaic investments is the focus of this article. A bottom-up modeling of investments in photovoltaic facilities, as well as the approach to incorporate these investments in portfolio analytics, is presented. Based on a comprehensive framework, we are able to evaluate the benefits of photovoltaic investments in a multiasset portfolio. Empirical data and realistic modeling assumptions are the result of a joint research project with a leading provider of closed-end funds in the area of photovoltaic investments. This chapter is organized as follows. Section 3.2 provides an introduction to alternative real assets. In Sec. 3.3, we present our bottom-up approach to model alternative real asset investments. Section 3.4 describes how to incorporate these investments to a portfolio. It shows the results of integrating photovoltaic investments into a traditional portfolio, including changes in portfolio statistics and the corresponding substitution effects. A conclusion is given in the last section.

3.2. OVERVIEW ON ALTERNATIVE REAL ASSETS Broadly speaking, asset classes contain investments with similar features including their risk and return characteristics and their sensitivity toward major market factors. The investment universe is usually divided into traditional asset classes and non-traditional or alternative asset classes that emerged over the last few decades. Traditional asset classes include equities, nominal government, or corporate bonds. Examples for non-traditional or alternative asset classes are private equity or hedge fund investments. The rationale for investing in the different asset classes varies broadly, which is in turn the reason why investors should diversify their asset allocation. For equity investments, a main investment objective is to participate in the growth potential and payout of listed companies. The rationale for bond investments includes income generation, deflation protection, and reduction of portfolio volatility. Traditional real assets include commodities, inflation-linked bonds, and real estate investments (direct or indirect, but not listed real estate investments). The categorization to this asset class subset is usually based on either the inflation protection feature

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such as real return, or the tangibility of the assets (see [3], p. 20, for a definition of real assets). Alternative real assets are represented by investments in shipping, aviation, infrastructure, or renewable energy facilities like photovoltaic or wind power plants. The real assets that provide the necessary cash flows may be roads, ships, or solar plants. Therefore, alternative real assets provide direct exposure to specific projects and the corresponding “real facilities” but no exposure to a listed company’s capital structure. Due to the project finance character, alternative real assets are not very liquid investments, even though secondary markets are already emerging. As revenues of the financed projects are often linked to inflation, most alternative real assets provide some inflation protection. In the following, we will focus on investments in solar power plants. This kind of alternative real assets is characterized by cash flows that are mainly driven by solar radiation and preferential feed-in tariffs. These cash flows are often linked to inflation and show a sensitivity to interest rates due to the partial debt financing of the photovoltaic projects. More details on the modeling of photovoltaic investments are given in the next section.

3.3. MODELING PHOTOVOLTAIC INVESTMENTS 3.3.1. General Approach When developing models for alternative real assets, one might encounter problems due to the special design of this asset class. A top-down approach is usually oversimplifying. It is not appropriate to link alternative real asset returns to the returns of the equity market using the capital asset pricing model (CAPM), as we do not find listed companies that are comparable to these highly individual projects. However, this is the only approach taken in literature concerning the analysis of alternative real assets thus far. For example [12], apply the CAPM approach to shipping investments. For further comprehension, a few words on the photovoltaic technique: photovoltaic is the direct generation of electricity out of global solar irradiance. This method of power generation is generally implemented through semiconductor panels, which are making use of a special case of the photo-electric effect. These solar panels are arranged in large arrays with additional infrastructure, so-called photovoltaic solar power plants. Using sunlight as the major resource, these facilities convert global irradiance into electricity in an environment-friendly way and are therefore categorized as renewable energy power plants. As a matter of fact, the return structure of limited partnership funds financing photovoltaic projects is completely different from stock returns of solar sector companies (a company producing solar panels, for example). On the one hand, we have the direct

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exposure to a specific project with a well-defined revenue and cost structure, on the other hand we have exposure to a broad mix of risk factors that drive the stock returns of a listed company: • • • • • •

Capital structure of the company Balance sheet effects General economic environment Change in exchange rates Differing expectations of market participants Analyst appraisals of profit expectations

Therefore, we believe that a different methodology than the top-down CAPM approach is required, when dealing with this kind of project finance. However, in relevant academic and practitioner publications, frequently top-down approaches are used for evaluating the alternative real assets. Based on expert information from a well-known German initiator of photovoltaic projects (we would like to thank KGAL [KG Allgemeine Leasing GmbH & Co.] for valuable insights to the financial modeling of solar plants), we analyzed the revenue and cost structure reflecting the return profile of individual solar power plants. Thus, we were able to identify crucial and cash flow relevant aspects of photovoltaic projects that should be covered by an appropriate model: • • • • •

Revenues of the sale of electricity Cost of acquisition Operating expenses Interest on debt capital Tax payments and interest subsidy

Modeling these building blocks properly is at the heart of our bottom-up approach. Reference [7] follows a similar bottom-up approach to derive the cash flows resulting from shipping investments. However, they combine this procedure with a CAPMapproach to finally be able to price a shipping investment. Recently, at the stock market exchanges of Hamburg and Hannover, a ship fund index is calculated using a bottom-up methodology. The index calculation is described in [14].

3.3.2. Definition of the Investment Project As we do not focus on evaluating a highly specific photovoltaic project for a certain investor, we set up a typical photovoltaic project that disregards individual, locationdependent features of an existing photovoltaic power station, and instead accentuates general properties of a typical project in a certain region. Solar power plants are usually set up in countries supporting renewable energy projects by an attractive compensation

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for feeding “clean” electricity into the grid. Such feed-in tariffs are guaranteed by local law and paid by the government, securing cash inflow. In some countries, the compensation is even linked to local consumer price indices and accordingly provides partial inflation protection to potential investors. Due to generous (inflation-linked) government compensations and a high level of global irradiance, Spain is a favored country for building up photovoltaic facilities and feeding the produced electricity into the local grid. For further comprehension, global irradiance is the total incoming solar radiation on a horizontal plane at the Earth’s surface, i.e., the incoming solar radiation after scattering effects in the Earth’s atmosphere. Therefore, our representative photovoltaic project is located in Spain and we have chosen a corresponding parametrization. This includes a project duration of 25 years, which corresponds to the maximum duration of the attractive feed-in tariffs. When following a bottom-up approach, it is absolutely necessary to model the project’s properties and the corresponding dependencies (especially the dependencies on stochastic risk factors) with a high level of detail. To be able to model the cash flow structure in an appropriate way, we set up a comprehensive bottom-up framework covering all revenue and cost-related features of the sample solar plant. An overview of the framework components and properties can be found in Table 3.1. A special focus of our analysis is on the simulation of stochastic risk factors that drive the revenue and the cost side of the photovoltaic project. The key risk factors and their dependencies to various components of the bottom-up framework are listed in Table 3.2. These risk factors have to be modeled and simulated in an integrated approach. This allows us to conduct a multi-scenario analysis of the inherent risk factors, the

Table 3.1

Comprehensive Framework for Bottom-Up Modeling.

Revenue

Modeling of the generation of electricity out of solar radiation, and the corresponding feed-in tariffs, including inflation-indexed tariff adjustments

Costs

Considering acquisition and operating costs, as well as inflation-driven increase over time

Financing

Modeling of equity, debt capital, and shareholder debt capital, as well as required liquidity reserves

Tax

Considering tax payments on company level, tax deductibility of interest payments on debt and shareholder debt, and tax loss carry-forwards to future periods

Facility

Considering annual power loss of the solar panels due to degradation effects, liquidation proceeds are on par with deconstruction costs at project termination

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−→

Revenue

Inflation

−→

Revenue, Costs

Nominal interest

−→

Financing

sample photovoltaic project and a portfolio consisting of traditional assets and the photovoltaic project.

3.3.3. Modeling of Risk Factors 3.3.3.1. Economic factors The theoretical framework used for the analysis of alternative real assets in a portfolio context is the risklab Economic Scenario Generator (ESG), see [16]. This model incorporates fundamental macroeconomic factors to describe the evolution of interest rates and equities. Using a cascade structure, it captures the long-term economic relationships while allowing for short-term deviations. This setting allows for an integrated modeling of financial markets, delivering economically meaningful, and consistent scenarios. We assume an arbitrage-free, frictionless financial market in continuous time t ∈ [0, T ∗ ], where the uncertainty in the market is described by the complete filtered probability space (
, F, F, IP ). Details on this framework can be found in [17]. We
t use a non-defaultable money market account defined by P(0, t) = exp( 0 r N (s)ds) as numeraire, where the process {rN (t)}t∈[0,T ∗ ] is the nominal short rate. We assume the existence of a measure Q equivalent to IP, under which all discounted price processes of the financial market under consideration are martingales. The before mentioned cascade structure is now imposed on this foundation to incorporate long-term economic dependencies. The model’s first cascade comprises inflation {i(t)}t∈[0,T ∗ ] and economic growth {w(t)}t∈[0,T ∗ ] , which are modeled by Vasicek processes, introduced in [15]. Under the equivalent martingale measure Q, they are specified as follows: di(t) = [θi − aˆ i · i(t)]dt + σi dWiQ (t) dw(t) = [θw − aˆ w · w(t)]dt + σw dWwQ (t) with the positive real numbers θi , θw , aˆ i , aˆ w , σi , σw and the independent standard Brownian motions WiQ and WwQ . The second cascade contains the interest rate processes. The real short rate {r(t)}t∈[0,T ∗ ] is modeled by a two factor Hull–White model. Its dynamic is specified as dr(t) = [θr (t) + brw · w(t) − aˆ r · r(t)]dt + σr dWrQ (t)

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with the positive real numbers brw , aˆ r , and σr , the time-dependent deterministic function θr and the standard Brownian motion WrQ , independent of the Brownian motions mentioned before. The nominal short rate {rN (t)}t∈[0,T ∗ ] is defined as the sum of real short rate and inflation, i.e., r N (t) = r(t) + i(t). The term structure of interest rates can then be derived by the zero-coupon bond prices obtained from this setting. As shown in [16], p. 4254f, the price of a zero-coupon bond with maturity t < T ≤ T ∗ is given by P(t, T ) = exp(A(t, T ) − B(t, T )r(t) − C(t, T )i(t) − D(t, T )w(t)), where B(t, T ) =

1 (1 − exp(−ˆar (T − t)), aˆ r

1 (1 − exp(−ˆar (T − t)), aˆ i   brw 1 − exp(−ˆar (T − t)) exp(−ˆaw (T − t)) − exp(−ˆar (T − t)) , D(t, T ) = · + aˆ r aˆ w aˆ w − aˆ r  T 1 2 A(t, T ) = (σ B(l, T )2 + σi2 C(l, T )2 + σw2 D(l, T )2 ) − θr (l)B(l, T ) 2 r t  − θi C(l, T ) − θw D(l, T ) dl. C(t, T ) =

Using Girsanov’s Theorem, the model equations can be derived under the real measure IP instead of the equivalent martingale measure Q by replacing WiQ , WwQ , WrQ with the independent standard Brownian motions Wi , Ww , Wr and using the parameters ai = aˆ i − λi σi2 , aw = aˆ w − λw σw2 , and ar = aˆ r − λr σr2 . λi , λw , and λr are obtained by the change of measure, as shown in [10]. As can be seen, the term structure of interest rates is driven by the real short rate process, as well as the underlying macroeconomic factors of economic growth and inflation. The third cascade contains equity assets. The equity prices {StE }t∈[0,T ∗ ] are driven by the following dynamics: dS E (t) = [αE + bEr r(t) − bEi i(t) + bEw w(t)]S E (t)dt + σE S E (t)dWE (t), where bEr , bEi , bEw , and σE are positive real numbers, αE ∈ R, and WE (t) is a standard Brownian motion, independent of those mentioned above.

3.3.3.2. Non-economic factors The final model component needed, which will add to the basic framework described above, is yearly global irradiance on a horizontal surface {Y(t)}t∈{0,1,...,T ∗ } , which is the

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starting point for the generation of electricity in a solar plant. An appropriate starting point for the modeling approach is average monthly global irradiance. Having a higher frequency will, apart from data availability issues, increase the noise component of the model, but not have a different impact on at most monthly simulated cash flows generated by the photovoltaic plant. As the irradiance model component independently adds to the basic framework, and the frequency of our data is also monthly, we develop a discrete-time model. The following sections now present the building blocks for the derivation of a discrete-time model for yearly global irradiance.

3.3.3.3. Historical analysis of monthly global irradiance To model monthly global irradiance on a horizontal surface, {Y M (m)}m∈{1,...,12·T ∗ } , where m indicates the month, we use a simple Angström-type equation, where bright sunshine duration is related to global irradiance. An overview of Angström-type equations to model global irradiance on horizontal surfaces is given in [1]. We follow a parsimonious approach, where we use a sinusoidal function to describe monthly global irradiance. This offers the advantage of not requiring information on bright sunshine hours, day length, and extraterrestrial radiation. We specifically assume monthly global irradiance to follow the process   2π ∗ Y M (m) = A + B sin (3.1) m − F + (m), 12 where A is the average monthly global irradiance, B is the amplitude of seasonal radiation variation, F is the phase angle in radian measure, i.e., the necessary shift of the function on the time axis, and m∗ ∈ {1, 2, . . . , 12} is the number of a specific month in the year, i.e., the number of January is 1, etc., see [2]. The zero-mean process {(m)}m∈{1,...,12·T ∗ } represents random fluctuations around the seasonal trend of monthly global irradiance. We use a least squares method to estimate the parameters A, B, and F in (3.1), and conduct the estimation on annualized monthly global irradiance data ranging over 12 years from 1/1982 to 12/1993 for several locations in Spain. (For nonlinear least squares estimation, see e.g., [9].) Our data source is the online archive of the U.S. Department of Energy’s National Renewable Energy Laboratory and the World Radiation Data Center (http://wrdc-mgo.nrel.gov/). The estimated model produces a high degree of variance explanation, with R2 > 0.9 for all locations. R2 is calculated as 1 − Var(ˆm )/Var(Yˆ (m)), where Yˆ (m) denotes the observed radiation series. Exemplarily for the location Toledo, the fitted series and the original series can be seen in Fig. 3.1. The fitted series is close to the original series with the model explaining 98% of the variation in the original series. The analysis of the model residual time series {ˆm }m∈{1,...,12·T ∗ } shows skewed distributions. The estimated density of residuals for

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Annualized monthly solar irradiance

2500

2000

1500

1000

500

0

4

0

x 10

10

20

30

40 Time in months

50

60

70

80

−3

3.5

Estimated Density

3

2.5

2

1.5

1

0.5

0 −400

−300

−200

−100 0 100 200 Monthly global solar irradiance

300

400

500

Figure 3.1 Left: Original (continuous line) and fitted (dashed line) annualized monthly global radiation series. Right: Estimated density of residuals.

the location Toledo is shown in Fig. 3.1. The density was estimated using Gaussian kernel smoothing, i.e., the density estimator fˆ(Y ) =

 T  1 − (Y −Yˆ 2(t))2 1  2h √ e Th t=1 2π

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 1 was applied to the observed data series, where the bandwidth used was h = 3T4 5 σ, and σ was obtained using the median absolute deviation estimator. (More information on kernel smoothing can be found in [4]). To capture the skewness in the fluctuation around the average seasonal global irradiance, we specify the noise process {m }m∈{1,...,12·T ∗ } to follow a Markov switching model, i.e., M M M (m) = µM 1 + X (m)(µ2 − µ1 )

+ (σ1M + XM (m)(σ2M − σ1M )) · uM (m),

(3.2)

M M M + M M where µM 1 , µ2 ∈ R, σ1 , σ2 ∈ R , {X (m) : X (m) ∈ {0, 1}}m∈{1,...,12·T ∗ } is a timeM homogenous Markov chain and u (m) is the increment of a standard Brownian motion, independent of XM (m). Details on Markov Switching Models can be found in [13]. To obtain parameter estimates, we are applying a maximum likelihood approach to the residual time series. (For the maximum likelihood estimation of Markov switching models, see [8].) The residual distribution from the estimated model, obtained by a Monte Carlo simulation, is shown for Toledo in Fig. 3.2. We conduct a two-sample Kolmogorov–Smirnov test on the simulated and the original monthly global irradiance series, which does not show evidence against both series coming from the same distribution. (For the two-sample Kolmogorov–Smirnov test, see [6].) The out-of-sample performance of the estimated model can be seen in Fig. 3.3, exemplarily for Toledo. The original time series are shown along with the 1%, 50%, and 99% quantiles of a simulation of 1000 paths following (3.1) and (3.2).

9000 8000 7000

Frequency

6000 5000 4000 3000 2000 1000 0 −800

−600

−400 −200 0 200 400 600 Annualized monthly global irradiance results

800

1000

Figure 3.2 Histogram of simulated monthly global irradiance residuals.

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Annulized monthly global irradiance

3500 3000 2500 2000 1500 1000 500 0

0

5

10 Time

15

20

Figure 3.3 Original series (dots) and annualized monthly global irradiance quantiles (barred lines).

3.3.3.4. Monte Carlo analysis of yearly global irradiance We use the results for monthly global irradiance to analyze yearly global irradiance, which is needed for the analysis of the photovoltaic investment. The global irradiance of a specific year is just the sum of the monthly global irradiance over that year. m : m ∈ {1, . . . , 12 · T ∗ } ∧ m Formally, the global irradiance of year t, Y(t), with t ∈ { 12 mod 12 = 0} is given by Y(t) =

12 

Y M (12t − j + 1).

(3.3)

j=1

We now use the model for Y M (m) to obtain the distribution of Y(t) by simulating 1000 trajectories of monthly global irradiance, each over a time span of 25 years. By calculating yearly radiation on the simulated processes according to (3.3), we generate 1000 paths of yearly global irradiance with 25 observations per path. The resulting distribution can be seen in Fig. 3.4 for Toledo. Compared to monthly global irradiance the skewness is reduced, but retained. The results we obtain for yearly global irradiance are similar to the published information on average yearly global irradiance, as available for example through radiation maps shown in [11] or offered on the website www.meteonorm.com, which provides a comprehensive meteorological reference. Table 3.3 below shows the means and relative standard deviations, i.e., the

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Figure 3.4 Histogram of simulated yearly global irradiance. Table 3.3 Yearly Global Irradiance Statistics for Locations in Spain. Location

Mean in KWh/m2

Rel. standard deviation (%)

R2 (%)

Toledo Caceres Logrono Madrid Murcia Mallorca Santander Sevilla

1,655.96 1,661.07 1,407.48 1,602.93 1,698.89 1,556.73 1,205.08 1,658.85

2.67 2.86 3.06 3.05 2.00 1.92 3.12 1.95

95.9 95.3 95.5 96.1 96.9 97.7 94.9 97.2

standard deviation divided by the mean, resulting from the simulation of yearly global irradiance for several sites, as well as the R2 -values of the estimated model. Comparing means and standard deviations with the information in [11] shows them to be in line with their data. The skewness of the distributions for the different locations is very similar; the distributions vary mainly in location and scale. For the following analysis of photovoltaic investments, we choose Toledo as a reference for yearly global irradiance on a horizontal surface, because it shows average characteristics of global irradiance in Spain, and is located close to a large number of existing solar plants.

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3.4. PHOTOVOLTAIC INVESTMENTS IN A PORTFOLIO CONTEXT 3.4.1. Setting the Portfolio Context We analyze the impact of photovoltaic investments in a portfolio context by evaluating the changes in risk/return profiles. Apart from photovoltaic investments, the asset class universe underlying this analysis comprises equity and bonds, represented by a bond index, which is constructed by periodically buying and selling bonds such that a constant modified duration of approximately six years is achieved. (The necessary bond prices are determined by the stochastic evolution of interest rate term structures generated using the integrated economic model.) The specific characteristics of these asset classes in terms of risk (measured by the standard deviation of annual returns) and expected return (measured by the mean of annual returns) over the considered investment horizon of 25 years are given in the following Table 3.4. Three types of investors are characterized by three different initial asset allocations. These allocations, presented in Table 3.5, are the starting point for our analysis. The three allocations represent different risk (and return) preferences with 10%, 20%, and 30% equity allocations. Comparing the risk and return figures for the “10% Equity” portfolio in Table 3.5 to the pure bond profile in Table 3.4, we can already see the benefits of diversification, i.e., an increase in expected return at the same level of expected risk when a 10% allocation in bonds is substituted by equities. Based on the three initial portfolios in Table 3.5, we evaluate the changes in risk and expected return figures when adding a maximum of 5%, 15%, or 25%, and up to 100% of photovoltaic investments. Table 3.4 Traditional Asset Classes. Return p.a. (%)

Risk p.a. (%)

7.7 4.3

14.8 3.5

Equity Bonds

Table 3.5 Initial allocation “10% Equity” “20% Equity” “30% Equity”

Initial Asset Allocations.

Equity (%)

Bonds (%)

Return p.a. (%)

Risk p.a. (%)

10 20 30

90 80 70

4.9 5.3 5.6

3.5 4.1 5.1

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3.4.2. Including Photovoltaic Investments in a Portfolio As a mark-to-market modeling of a photovoltaic investment is impossible without very strong assumptions (e.g., regarding the appropriate discount rate), we make use of an approach that does not require a pricing of the portfolio at each point in time. Furthermore, we attached importance to a realistic modeling of the investment itself. Starting with a predefined initial portfolio consisting of the traditional assets, i.e., equities and bonds, a portion of an investment in a photovoltaic project is added. Incoming cash flows from the photovoltaic project are reinvested into the traditional part of the portfolio. In each time step, the reinvestment is allocated according to the weights of the traditional assets in the portfolio. As a result of this approach, the absolute allocations of the photovoltaic investment decrease over time. A reinvestment of the dividends from the photovoltaic investment in other (new) projects, in order to maintain the relative photovoltaic investment, would imply arbitrary availability and divisibility. An investor would not enter into new photovoltaic projects on an (semi-)annual basis as dividends flow back. We perform the portfolio analysis on log-returns, which offers, for example, the advantage that multi-period log-returns can be calculated as the sum of the single-period log-returns. Due to the before-mentioned mark-to-market problems of photovoltaic investments, we cannot calculate portfolio returns before the end of the investment horizon. Using log-returns allows us to easily transform observable total return characteristics over a period of several years to more common per annum values. Consider therefore portfolio p, which comprises photovoltaic investments. Thus, the portfolio value, and therewith the single-period returns cannot be calculated during the investment horizon, as the value of the photovoltaic investment is unknown during the investment horizon. Assume, nevertheless, that the (unobservable) single-period log-returns {rp,i } for the periods i = 1, . . . , n satisfy Var(rp,i ) = σp2 ∀ i = 1, . . . , n and Cov(rp,i , rp,i−j ) = kp (j) · σp2 for j = ±1, ±2, . . . . total The multi-period or total log-return rp,1 →n is obtained by summing up the single-period log-returns, i.e., total rp,1 →n =

n 

rp,i .

i=1

Given these assumptions, it is easy to show that (for this result compare also [5], p. 49.)    n−1   j total  · kp (j) · σp2 , Var(rp,1 1− (3.4) →n ) = n · 1 + 2 n j=1

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with n   total   Var(rp,i ) + Cov(rp,i , rp,j ) = Var rp,1 →n i =j

i=1

=

n  i=1

=

n 

i−1 

Var(rp,i ) + 2 ·

Cov(rp,i , rp,j )

j=1

 σp2 + 2 ·

i−1 

i=1



(σp2 · kp (i − j))

j=1

 = σp2 n + 2 ·

n  i−1 

 kp (i − j) .

i=1 j=1

Now, defining k := i − j, we have n  i−1 

kp (i − j) =

i=1 j=1

n  i−1 

kp (k)

i=1 k=1

=

n−1  n 

kp (k)

k=1 i=k+1

=

n−1 

(n − k) · kp (k),

k=1

where we used the fact that 1 ≤ i ≤ n, 1 ≤ k ≤ i − 1 ⇔ 1 ≤ k ≤ n − 1, k + 1 ≤ i ≤ n in the second equality. Factoring out n, Eq. (3.4) follows. Using this result, we define a volatility adjustment factor    n−1   j · kp (j) . α := 1 + 2 1− (3.5) n j=1 Thus, knowing the degree of autocorrelation and having for instance yearly periods allow us to calculate the annualized volatility from total volatility by dividing the total volatility by the square root of the number of years adjusted for the degree of autocorrelation by the factor 1/α. Assuming the absence of autocorrelation, only the observable initial portfolio value and the observable portfolio value at the end of the investment horizon are needed to calculate single-period portfolio log-return volatility. The dependence of cash flows generated by a solar power plant over time should mainly be driven by their dependence on inflation and interest rates, which both have a negative impact on the cash flows, and both show positive autocorrelation. Thus, one would expect positive autocorrelation, if any, in the single-period returns

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of a photovoltaic investment. Positive autocorrelation is increasing α defined in (3.5), and therefore would increase the measured total variance of a portfolio including photovoltaic investments. Scaling the total variance by the number of periods will then overstate the single-period variance, which makes our assumption of absence of autocorrelation and therewith the results of our analysis conservative.

3.4.3. Results Tables 3.6–3.8 present the results of our portfolio optimizations for the three initial allocations “10% Equity,” “20% Equity,” and “30% Equity.” Every table shows the resulting optimized portfolios for similar expected risk or similar expected return Table 3.6

Results for the Initial Allocation “10% Equity.”

Allocations Portfolio

Substitution w.r.t. “10% Equity”

Statistics

PV (%) E (%) B (%) Return (%) Risk (%) PV (%) E (%) B (%)

“10% Equity”

0

10

90

4.9

3.5

—

—

—

“HR 5% PV” “HR 15% PV” “HR 25% PV” “LR 5% PV” “LR 15% PV” “LR 25% PV”

5 15 25 5 15 25

13 15 15 7 5 4

82 70 60 88 80 71

5.1 5.4 5.6 4.9 5.0 5.2

3.5 3.5 3.5 3.3 3.1 2.9

+5 +15 +25 +5 +15 +25

+3 +5 +5 −3 −5 −6

−8 −20 −30 −2 −10 −19

Table 3.7

Results for the Initial Allocation “20% Equity.”

Allocations Portfolio

Substitution w.r.t. “20% Equity”

Statistics

PV (%) E (%) B (%) Return (%) Risk (%) PV (%) E (%) B (%)

“20% Equity”

0

20

80

5.3

4.1

—

—

—

“HR 5% PV” “HR 15% PV” “HR 25% PV” “LR 5% PV” “LR 15% PV” “LR 25% PV”

5 15 25 5 15 25

21 21 21 17 12 7

74 64 54 78 73 68

5.4 5.7 5.9 5.3 5.3 5.3

4.1 4.1 4.1 3.8 3.3 3.0

+5 +15 +25 +5 +15 +25

+1 +1 +1 −3 −8 −13

−6 −16 −26 −2 −7 −12

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Alternative Real Assets in a Portfolio Context Table 3.8

Results for the Initial Allocation “30% Equity.”

Allocations Portfolio

Substitution w.r.t. “30% Equity”

Statistics

PV (%) E (%) B (%) Return (%) Risk (%) PV (%) E (%) B (%)

“30% Equity”

0

30

70

5.6

5.1

—

—

—

“HR 5% PV” “HR 15% PV” “HR 25% PV” “LR 5% PV” “LR 15% PV” “LR 25% PV”

5 15 25 5 15 25

30 29 28 27 21 15

65 56 47 68 64 60

5.8 6.0 6.2 5.6 5.6 5.6

5.1 5.1 5.1 4.7 4.1 3.5

+5 +15 +25 +5 +15 +25

0 −1 −2 −3 −9 −15

−5 −14 −23 −2 −6 −10

statistics when we allow the allocation of a maximum weight of 5%, 15%, or 25% to the photovoltaic investment opportunity. The “higher return” or “lower risk” portfolios, denoted by “HR X% PV”, and “LR X% PV,” respectively, are given in the three tables together with the corresponding expected risk and expected return figures. (X denotes the photovoltaic investment proportion in the portfolio.) These efficient portfolios (with weights for photovoltaic (PV), equity (E) and bond (B) investments), compared to the initial allocation, yield the same expected return with less risk, or the same risk with more expected return. The last three columns of Tables 3.6–3.8 present the weights differences between the initial allocations and the new efficient weights, i.e., the substitutional effects due to the introduction of photovoltaic investments. The dominance of portfolios including the photovoltaic investment opportunity is striking. In Table 3.6, we see the results for an initial equity allocation of 10%. The higher return portfolios show an increase in expected return from 20 bp up to 70 bp depending on the maximum quota of photovoltaic investments in the portfolio. For example, the higher return portfolio with a limit of 5% in photovoltaics (“HR 5% PV”) allocates the maximum of 5% to photovoltaic investments, 13% to equities and 82% to bonds, increasing the expected return from 4.9% to 5.1% at the same level of expected risk (3.5%). The lower risk portfolios (“LR X% PV”) decrease the expected risk by up to 60 bp (or approximately 17% in relative terms) at the same level of expected return. With higher photovoltaic limits, the risk and return characteristics cannot even be brought down to the initial levels when selecting portfolios from the efficient set. The minimum expected return for “LR 15% PV” is 5.1%, for “LR 25% PV” it is 25%, respectively.

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In Table 3.7 we see higher return and lower risk portfolios for an initial allocation of 20% in equites. The improvement on the return side ranges from 10 bp to 60 bp and up to 110 bp (or approximately 27% in relative terms) on the risk side. The comparison of the optimized portfolios to a 30% equity portfolio in Table 3.8 reveals up to 60 bp more return and up to 160 bp (or approximately 31% in relative terms) less risk. Interestingly, the optimized portfolio allocations in the case of the “10% Equity” and “20% Equity” show a slight increase in the equity weight for the higher return portfolios, as the photovoltaic investment seems to reduce the portfolio risk substantially. Therefore, the bond allocation is reduced by up to 30% (“HR 25% PV” with “10% Equity” as initial allocation). For the “30% Equity” portfolio, the optimized higher return portfolios show a stable allocation in equities — bonds are substituted for by the allocation to photovoltaic investments. When we optimize the lower risk portfolios, the allocation to photovoltaic investments consistently substitutes equity and bonds. For the “20% Equity” portfolio both substituted at similar amounts, for the “30% Equity” portfolio compared to bonds more equity is substituted, and for “10% Equity” portfolio more bonds are substituted compared to equity.

3.5. CONCLUSION Investing in photovoltaic facilities is environmentally supportive. In addition, photovoltaic investments are an attractive asset class from an investment perspective. In our analysis, the maximum weight allowed is allocated in all of our optimized portfolios. We see a straightforward tendency that, including photovoltaic investments in a portfolio, decreases risk and/or increases expected return. To derive these results, we make use of a comprehensive framework to analyze the attractiveness of photovoltaic investments in a multi-asset portfolio. Based on our bottom-up approach, we are able to identify the quantitative characteristics of photovoltaic investments in a portfolio context. We optimized portfolios including photovoltaics with respect to initial allocations of 10%, 20%, and 30% in equities, respectively. These base allocations reflect different risk and return preferences of representative investors. Depending on the preferences and the maximum allowed photovoltaic weight, optimized higher return portfolios provide increases in expected return of up to 70 bp while the lower risk portfolios decrease the risk figures by up to 1.6% in absolute or by 31% in relative terms. As a result, we find that in optimized higher return portfolios photovoltaic investments substitute bonds — allowing slight increases in the equity quota at the same time. For the lower risk portfolios, the allocation to photovoltaic investments is financed by decreasing bond and equity weights at the same time. The reduction of bond and equity allocations is approximately the same.

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The chosen bottom-up framework allows for the inclusion of alternative real assets when evaluating and optimizing the portfolio allocations. We can examine the portfolio contribution of alternative real assets in general and we are able to customize the modeling for specific investment assets in great detail. Beside the analysis in a portfolio context, this bottom-up modeling also allows ongoing monitoring of specific alternative real assets.

References [1] Akinoglu, BG (2008). Recent advances in the relations between bright sunshine hours and solar irradiation. In V Badescu (ed.), pp. 115–144. Modeling Solar Radiation at the Earth’s Surface Springer Verlag. [2] Alsaad, MA (1991). Correlations for predicting average global solar radiation incident on a horizontal surface in amman. Journal of King Saud University. Engineering Sciences, 3(1) 121–134. [3] Ankrim, EM and CR Hensel (1993). Commodities in asset allocation: A real-asset alternative to real estate?” Financial Analysts Journal, 49(3) 20–29. [4] Bowman, AW and A Azzalini (1997). Applied smoothing techniques for data analysis. Oxford Science Publications. [5] Campbell, J, A Lo, AC MacKinlay (1997). The Econometrics of Financial Markets. Princeton University Press. [6] Conover, WJ (1998). Practical Nonparametric Statistics. Wiley. [7] Drobetz, W, L Tegtmeier and M Topalov (2006). Bewertung von Kommanditanteilen geschlossener Schiffsfonds mit dem Ertragswertverfahren. Finanz Betrieb, 6, 399–412. [8] Hamilton, JD (1994). Time Series Analysis. Princeton University Press. [9] Intriligator, M, R Bodkin and C Hsiao (1996). Econometric Models, Techniques, and Applications. Prentice Hall. [10] Schmidt, B and R Zagst (2000). A three-factor defaultable term structure model. Journal of Fixed Income, 10(2), 63–79. [11] Suri, M et al. (2007). Uncertainty in photovoltaic electricity yield prediction from fluctuation of solar radiation. In Proceedings of the 22nd European Photovoltaic Solar Energy Conference, Milano (preprint). [12] Tegtmeier, L and M Topalov (2006). Die Bedeutung von Schiffsbeteiligungen im Rahmen der Asset Allocation von Privatanlegern. Finanz Betrieb, 7–8, 506–509. [13] Timmerman, A (2000). Moments of Markov switching models. Journal of Econometrics, 96, 75–111. [14] TKL Fonds (2009). Leitfaden zum SFI TKL.Ship Fund Index. [15] Vasicek, O (1977). An equillibrium characterization of the term structure. Journal of Financial Economics, 5, 177–188. [16] Zagst, R, Th Meyer and H Hagedorn (2007). Integrated modelling of stock and bond markets. Journal of International Finance, 19(1), 4252–4277. [17] Zagst, R (2002). Interest Rate Management. Springer Verlag.

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4

THE FREIGHT MARKET AND ITS DERIVATIVES

RÜDIGER KIESEL∗ and PATRICK SCHERER† ∗

Institute of Mathematical Finance, Ulm University, Helmholzstr. 18, 89081 Ulm, Germany, and Centre of Mathematics for Applications, University of Oslo, [email protected] † Institute of Mathematical Finance, Ulm University, Helmholzstr. 18, 89081 Ulm, Germany, [email protected]

Despite the fact that the freight market is one of the oldest markets, it found only limited attention in financial (mathematics) studies. One possible reason may be that this market is very intransparent and only active market participants such as shipowners, dockyards, and commodity traders were fully informed on market activities. With the launch of freight market derivatives, however, many other financial institutions started to actively trade on the freight market, for instance by taking positions on future global trade activities via Forward FreightAgreements and Freight Futures. This article provides an introduction to the freight market and its most commonly used financial assets. We investigate the relationship between the freight market, freight derivatives, and macro-economic variables. Further, we show how these relationships can be used to better explain and predict the future price movements of freight derivatives. To achieve this, we propose to use classical statistical methods, such as Vector Auto

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Kiesel and Scherer Regression and Vector Error Correction Models, to incorporate the information from spot markets and related explanatory variables into the prediction process.

4.1. INTRODUCTION: THE FREIGHT MARKET The oversea shipping market has been established in the 17th century and thus is one of the oldest physical markets. In 2007, the volume of international seaborne trade reached 8.02 billion tons [1]. In 2008, over 80 per cent of the worlds merchandise trade volume was carried by sea, making the maritime transport the backbone of international trade and globalization. Despite its size, it found only limited attention in financial (mathematics) studies and for a long time it failed to attract the focus of investors from outside the traditional shipping market environment. Since the shipping market is exposed to various risks, it is highly volatile. Historically, these risks were mainly due to unsuitable technical equipment, lack of reliable weather forecasts, piracy, etc., which in many cases led to the loss of freight, the ship, and the lives of the crew. While natural risks are still present nowadays, financial risk for shipowners, commodity traders, charterers, and other investors and market participants have become more important.

4.1.1. Vessels The modern fleet of vessels used for overseas shipping is well diversified. We see a large range of modern cargo vessels, ranging from small coasters to very large tankers, but generally we can classify them in terms of the groups given in Table 4.1 below. Not reported in the table are liquefied natural gas (LNG) tankers, general cargo (Gen Cargo), and Roll On-Roll Off (Ro-Ro) carriers, as well as vessels that are able to carry a combination of chemicals and oil products. Furthermore, passenger cruise ships are excluded in the analysis of this paper. In the 2008 report of the United Nations Conference on Trade and Development (UNCTAD [1]), it is stated that the world vessel fleet expanded by 7.2 per cent during 2007 to 1.12 billion deadweight tonnage (DWT) at the beginning of 2008. Deadweight tonnage is a measure of how much weight of cargo a ship can carry. Just to give an impression on prices, a 170,000 DWT dry bulk carrier was priced at $97 million in December 2007 about 39 per cent more than a year before, and 2.4 times the price paid in 2000. Greece, Japan, and Germany are the world’s top three shipowning nations.

4.1.2. Cargo The main cargos of tankers are crude oil products (“dirty oil”) and refined oil products (“clean oil”), chemicals as well as liquefied natural and propane gas (LNG resp. LPG),

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The Freight Market and Its Derivatives Table 4.1 Number of Vessels in Service (IS), on Order (OO), Under Construction (UC) and Their Combined Capacity Measured in Dead Weight Tonnage (DWT). Obtained from Bloomberg on 20/2/2009. Tankers

# IS

# OO

# UC

DWT (#000)

Crude oil Oil product LPG Chemical

2028 1416 486 371

521 341 101 164

73 79 12 30

307,497 47,101 12,684 6860

Carriers

# IS

# OO

# UC

DWT (#000)

6091 80 679

2957 87 195

235 16 44

374,239 17,240 11,187

Containers

# IS

# OO

# UC

DWT (#000)

Container ships

4461

1001

139

161,504

Bulk Ore Vehicle

whereas dry cargo such as ore, coal, bauxite, phosphates, cement, scrap, steel, sugar, grain and rice, cars, containers are shipped via bulk carriers or container ships. Half of the cargos are energy related while container traffic is just over 10% by weight, but much higher in terms of value. Ship sizes are measured in DWT and include Capesize >100,000 DWT, Panamax 60–100,000 DWT (capable to cross the Panama Canal), Supramax 40–60,000 DWT as well as Handysize 10–40,000 DWT.

4.1.3. Routes The biggest trading routes by vessel type and size are given in Table 4.2. In our analysis, we restrict our focus to the three routes given in bold. These are the most frequently served routes, and we were able to obtain a data set of high quality for them. TD3 The “Tanker Dirty Route 3” route from Ras Tanura, Saudi Arabia to Chiba, Japan is the most common trade route into Asia for very large crude carriers. The standard 260,000 ton of non-heat crude carried from Saudi Arabia to Japan on TD3 is approximately two million barrels of oil. As spot price (USD/Ton), we make use of the Bloomberg price series “D27TAGJP Index.” TC2 Being the freight element of the cross Atlantic gasoline arbitrage trade, the “Tanker Clean Route 2” route is the most frequently traded clean product tanker contract. The TC2 route is typically served by modern Panamax or Supramax tankers carrying clean product (most often gasoline and other light distillates) from Rotterdam to New York Harbor. For spot prices, we use the Baltic Clean

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Most Common Routes.

Dirty tanker TD3 TD5 TD7 TD8 TD9 TD11 TD17

Main cargo Ras Tanura, Saudi Arabia to Chiba, Japan West Africa to US coast North sea to Continent Kuwait to Singapore Caribbean to US Gulf CrossMed Baltic to UK/Continent

Crude oil

TC2

Rotterdam to New York

Clean oil

TC4 TC5 TC6

Singapore–Japan AG–Japan CrossMed

Clean tanker

Capesize C3 C4 C5 C7

Tubaro to Beilun Richards Bay, South Africa to Rotterdam Western OZ to Beilun Puerto Bolivar (Ecuador) to Rotterdam

Coil

Panamax P2A P3A

Mediterranean to Far East Japan to South Korea

Tanker Index “BITY Index,” which is an average over all clean tanker routes for the year 2005, and the “CMRTUKUA Index” for the remaining time, because its price history starts in 2006 and all other prices are available from 2005. C4 The most common dry cargo routes are the big coal and iron ore transportation routes from Africa to Europe and from South America and Australia to Asian destinations. We consider the “Capesize Route 4” route from Richards Bay, South Africa to Rotterdam with spot prices represented by the SSY price history “SSYWRBRT Index.”

4.2. FREIGHT RATES: WHAT DRIVES THE MARKET? As it can be seen from the spot prices of the three routes given in Fig. 4.1, volatility is high in the freight market. Typically, the price of shipping one ton of cargo on one of

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The Freight Market and Its Derivatives TD3 freight rates 60 40 20 0 2005

2006

2007

2008

2009

2008

2009

2008

2009

TC2 freight rates 60 40 20 0 2005

2006

2007 C4 freight rates

60 40 20 0 2005

2006

2007

Figure 4.1 Spot price histories in USD/Ton for the period 01/01/2005 to 20/2/2009.

the three routes trades between 15 and 35 USD, but strong deviations from this band have occurred even before 2008. We further observe that the freight market experienced many pronounced cycles of high as well as low rates. Also the financial crisis of 2008/2009 had its effects on the freight market; in line with the stock markets, rates dropped in late 2008 and remained at a low level until spring 2009 when they slightly recovered. The effects of the credit crunch on the freight rates as well as on the entire shipping industry is further discussed in [2]. The authors note that the economic slowdown affected the shipping market more than most other markets, for example causing an 94% drop of the Baltic Exchange Dry Index from May to December 2008. The consequences of this variability are twofold: the typical market participants are in need of hedging instruments to lay off some of the risk of adverse price movements, while new investors and speculators are drawn to the market because large price movements can imply large profits. In order to understand this high variability of the freight rates, we first have to identify the price drivers of the market, i.e., what variables govern the demand and the supply of freight capacity, and therefore the freight rates.

4.2.1. Demand for Shipping Capacity We expect the demand for shipping capacity to be positively correlated with the industrial performance and activities in the country of origin as well as destination. Looking

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at the typical cargos (oil products, chemicals, gas, ore, coal, bauxite, phosphates, cement, scrap, steel, sugar, grain, rice), we expect this relationship to be rather strong. What is the level of industrial production? Has the grain harvest been successful? Are power stations importing more coal? How is the steel industry performing? Commodity demand directly implies demand of shipping capacity. Stock indices and futures on indices can be seen as a proxy for the physical industrial activities. Futures have the additional feature of a time horizon for which expectations of the underlying prices are captured. Further, we include various commodity prices as well as prices of fuel and energy products. Due to the globalization, many goods and commodities are imported from overseas. This is beneficial as a basic principal from economic theory, but additionally on a short-to-medium horizon, exchange rates further influence the benefit. Thus, in the analysis below we include the main FX rates as explanatory variables. Note that the demand of shipping capacity can change rapidly. Extreme economical or environmental events, even exceptional weather conditions, can imply significant and fast changes in the need of energy, building materials, goods, and thus of shipping capacity.

4.2.2. Supply of Shipping Capacity In contrast to the potentially fast-changing demand, adjusting the supply side can only be done gradually. In the short term, supply can be shortened by changing the speed of the vessels, or by not operating them at all. Long-term adjustments can be made by ordering/building more or fewer ships, or by moving the scrapping day of old vessels forward. With historical high demand for shipping capacity lasting until late 2008, the shipping industry responded by ordering new tonnage which led to a constant increase of shipping capacity over the last decades. For example, the order book of crude oil tankers reached its all-times high in late 2008 with a size of 576 vessels. The drop in the order book down to 521 in February 2009 seems relatively small. With the Chinese expansion plans (e.g., found in [3]) in our minds, it seems clear that supply will continue to increase over the next years. In 2010, however, due to new environmental regulations, many single hull tankers will have to be scrapped, or transformed to double hull tankers or bulk carriers which will shorten the likely surplus of tanker capacity in that period. In the analysis below, we take fleet supply into account by including tanker as well as bulk carrier DWT histories into the analysis.

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4.2.3. Costs The main components of a vessel’s costs are typically divided into voyage costs and daily running costs and may include, among other factors, crew wages, overtime, pension contributions, insurance, traveling costs, victualing, insurance, deck and engine room stores and spares, lubricating oil, periodic costs such as dry-docking, special surveys, running repairs, office and/or management costs, port disbursement, canal dues, commissions, taxes, etc. The main component, however, are bunker prices. With bunker fuel accounting for between one-quarter and one-third of the cost of running a vessel, oil price movements directly affect shipowners. We further include interest rate products in the analysis below because the costs of financing ships, financing freight for the duration of the journey, etc. are likely to affect freight rates.

4.3. FREIGHT DERIVATIVES: HEDGING OR SPECULATING? A freight derivative is a financial contract which sets an agreed future price for carrying commodities at sea. Freight derivatives were introduced to give freight market participants an instrument to reduce their various risks. Given the current volatility of the freight markets, managing freight market risk is a significant issue for the shipping industry. For example, by locking in deals well in advance, shipping companies as well as commodity traders achieve to be less exposed to spontaneous adverse price movements on the spot markets. The first contracts in this respect were time charter agreements. Here, the vessel is chartered to a counter party over a certain time span, typically a few years. We now see efforts to open the market to more participants, making it possible to trade not only cash settled (the supertanker “AbQaiq” will not show up at your front door), but also in standard contracts traded on an exchange and without counter party risk. The two main institutions in the field are the Baltic Exchange in London [4] as well as Imarex in Oslo [5], both providing further information on trading freight derivatvies as well as background reports.

4.3.1. Forward Freight Agreement One type of modern freight derivatives are the so-called forward freight agreements (FFAs). These contracts are traded “over the counter” on a principal-to-principal basis. As such, they can be designed very flexible to fit the needs of the two contractual parties. But as a consequence, the FFAs will not be comparable and thus are not traded on any exchange.

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To obtain more standardized products, the contract details can be based on the terms and conditions of the Forward Freight Agreement Brokers Association [6]. The main terms of an agreement cover the following specifications: • • • • •

Contract route: the agreed physical trading route. Contract period: the period for which the prices are fixed. Contract quantity (Q): the amount of the cargo for which prices are fixed. Contract rate (CR): the rate at which differences will be settled. Settlement rate (SR): calculated as the average spot price during the contract period.

On settlement, typically within five days after the contract period, the two parties will settle the contract according to the following simple calculation: the seller of the FFA contract is required to pay the buyer an amount equal to (CR − SR) · Q. Again, due to the inhomogeneity of the freight market, there is no spot price readily available. Instead, daily freight rate assessments of the physical routes are provided by independent organizations such as the Baltic Exchange. Their daily route assessments are based on input from participating ship brokers according to fixtures concluded on the day and market sentiment. Baltic spot rates aim to be “Fair Value” assessments of the market on any day.

4.3.2. Freight Futures In order to allow for clearing and to make it possible for an even broader clientele to trade in freight derivatives, freight futures were introduced. In contrast to the FFAs above, these contracts are always marked-to-market on a daily basis. Again, these are cash-settled contracts with no physical delivery and the settlement is against the average spot price in the delivery period. The biggest exchange for these contracts is the International Maritime Exchange [5]. The disadvantage of the futures contracts is that a perfect hedge might not be possible. Since only main routes are traded with these standardized contracts, there might not be futures for an actually serviced route, and instead a position may, e.g., only be hedged with an available future on a route basket. In our analysis, we include the monthly future prices for the three routes chosen in Sec. 4.1. These are provided by Imarex and we obtained them via Bloomberg. The names and Bloomberg tickers are reported in Table 4.3. The actual underlying spot prices of these futures are the Baltic Exchange Route Assessments, but access to these is only granted to subscribers of their service. Thus, we use the spot prices that we could obtain through the Bloomberg service. After comparing these spot prices with

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The Freight Market and Its Derivatives Table 4.3

Route Spots and Futures.

TD3

Bloomberg ticker

Spot Front month Front month+1 Front Month+2 TC2 Spot 2005 Spot 2006-current Front month Front month+1 Front month+2 C4 Spot Front month Front month+1 Front month+2

FA

D27TAGJP index IFTD3TBM index IFTD3T1M index IFTD3T2M index Bloomberg ticker BITY index CMRTUKUA index IFTC2TBM index IFTC2T1M index IFTC2T2M index Bloomberg ticker SSYWRBRT index IFFDC4BM index IFFDC41M index IFFDC42M index

the front month contract and carefully reading the product specifications of our “spot” prices, we believe to have a good proxy for freight spot prices at hand.

4.4. EXPLANATORY VARIABLES We now try to identify explanatory variables for the observed price movements in freight spot and freight futures rates. Instead of manually setting up an economic model for the relations between variables, we extract the information by means of standard statistical Vector Auto Regression (VAR) and Vector Error Correction Models (VECM). As potential explanatory variables for the spot and future prices of our three routes, we take variables from the groups identified in Sec. 4.2. In more detail, these are reported in Tables 4.4 and 4.5. We consider the indices and futures as forward looking indicators of economic activity. The exchange rates are motivated by the geographic location of the main routes (linking Europe, Africa, Japan, and the United States) and

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Kiesel and Scherer Table 4.4

Potential Explanatory Variables and Bloomberg Tickers I.

Stock index, futures Dow Jones industrial FTSE 100 (UK) DAX (Germany) EURO STOXX 50 Nikkei 225 (Japan) S&P ASX (Australia) Shipping market Baltic Panamax Baltic Capesize Baltic Handymax Baltic Clean Tanker Baltic Dirty Tanker FX EUR vs. USD GP vs. USD Commodities Gold Silver Copper Lead Aluminum Zinc

(Index) INDU UKX DAX SX5E NKY AS51

(Index) GCC 200 (Africa) Future Dow Jones Future FTSE 100 Future DAX Future EuroSTOXX Hedge Fund Index

(Index) BPIY BCIY BHSI BITY BIDY

(Index) Tanker DWT Bulk Carrier DWT Conainership DWT Baltic Dry Index Baltic Supramax

(Curncy) EUR GBP

VESLCODW VESLBKUC VESLCTDW BDIY BSI (Curncy)

JPN vs. USD South Africa vs. USD

(Index) GCM9 SIK9 LPK9 LLK9 LAK9 LXK9

BGCC200 DJH9 Z H9 GXH9 VGH9 HFRXGLE

JPY ZAR (Comdty)

CRB Commodity CRB Metal Index GSCI GSCI Energy GSCI Indust. Mtl Nickel

CRB RIND CRB METL SPGCCI SPGCEN SPGCIN LNK9

the commodities are part of the freight. We expect that the fuel- and energy-related commodities are particularly helpful since their prices are closely monitored economic variables and their dependence on economic activity is high.

4.4.1. Explanatory Power Vector Auto Regression (VAR) models extend the traditional univariate autoregression model to a vector of time series which jointly evolve through time. Besides autocorrelation, the model allows for cross-correlations, thus allowing interaction between the two time series. To assess the actual explanatory power of the potential explanatory variables listed above, we estimate a VAR for the bivariate log returns series of the freight rates and each variable.

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The Freight Market and Its Derivatives Table 4.5

Potential Explanatory Variables and Bloomberg Tickers II.

Energy Brent crude Gas oil RBOB gasoline Crude oil Natural gas Heating oil Oil USA 1M Oil USA 2M Oil USA 3M Oil USA 4M Oil USA 5M Oil USA 6M

(Comdty) COM9 QSK9 XBK9 CLK9 NGK9 HOK9 CL1 CL2 CL3 CL4 CL5 CL6

(Index) Oil Spot UK Oil Spot USA Oil Spot Arab. Gulf Oil Spot Asia Coil Rotterdam 1M Coil Rotterdam 2M Coil Rotterdam 3M Coil Rotterdam 4M Coil Richard’s Bay 1M Coil Richard’s Bay 2M Coil Richard’s Bay 3M Coil Richard’s Bay 4M

Int. rate products

(Index)

LIBOR USD 1M LIBOR USD 3M LIBOR USD 6M LIBOR JPY 1M LIBOR JPY 3M

US0001M US0003M US0006M JY0001M JY0003M

EUCRBRDT USCRWTIC PGCRDUBA APCRTAPI API21MON API22MON API23MON API24MON API41MON API42MON API43MON API44MON (Index)

LIBOR JPY 6M LIBOR GBP 1M LIBOR GBP 3M LIBOR GBP 6M

JY0006M BP0001M BP0003M BP0006M

We introduce the following notation: since it will be clear from the context if spots or futures prices are considered, both the freight spot and freight futures time series are named F = (Ft ), and by ft = log(Ft+1 /Ft ) we refer to their log returns. Similarly, by et = log(Et+1 /Et ), we refer to the log return series of the explanatory variable E = (Et ) under consideration. We work with the following specification of the VAR model.   20 20   µ + aj ft−j + bj et−j  
   f  f ft
t j=1 j=1   (4.1) = +  20 20   et
et   µ +  ce + df e

j t−j

j=1

j t−j

j=1

with an N0, distributed white noise error term 
, and all coefficients being real numbers. Thus, in the model framework used here, we assume that log returns of freight rates and explanatory variables are given by a constant (µf resp. µe ) plus a weighted sum of lagged “old” log returns, plus an error term which for simplicity we assume to be normally distributed.

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We use 20 lags because working with daily data, this number corresponds to one month worth of data. It is desirable to include even more lags, to capture relations which hold over several months, but we found this yields unreliable, unstable results in most cases and thus we do not increase the maximal lag sizes at the moment. One application of the VAR model is that it allows for a Granger Causality test.

4.4.2. Granger Causality A time series e is said to Granger Cause another time series f if the present values of f can be predicted more accurately by using past values of e than by not doing so, see [7]. In terms of the Vector Autoregression model above, the explanatory variables under consideration Granger Cause the freight rates under consideration (spot or futures) if some of the bj coefficients are significantly nonzero. Standard regression theory can be applied for a test of the joint null hypothesis that all of the bj values appearing in Eq. (4.1) jointly equal zero vs. the alternative that at least one does not equal zero. For the test statistics, one calibrates the full and the reduced model (i.e., setting all bj = 0) and compares sums of squared residuals to assess the gain of explanatory power from including the bj variables. The test statistic is GCRT =

(T − 2) · (l − 1) RSSRed − RSSFull , · l RSSFull

and is compared to a quantile of the F(l, (T − 2) · (l − 1)) distribution. Here T stands for the number of observation days, and l stands for the number of lags, e.g., l = 20 for one month of daily data in our specification. The sums of squared residuals SSRFull and SSRRed are defined by SSRFull = T f t=1
t with residuals from the full model in Eq. (4.1), and similarly SSRRed is the sum of squared residuals in the reduced model with bj = 0 for all j. For each of the 71 explanatory variables, we calculate the P-Value for the Granger Causality test between the explanatory variable’s log return series and the log return series of the route’s spot and futures prices. In Table 4.6, we report the top 15 resulting values and five further values that are of interest because of the economic relation between the explanatory variable and the route. Before moving on, let us do a quick plausibility check of the results obtained thus far. TD3 The tankers of this route carry crude oil from Saudi Arabia to Japan. Indeed, as expected the spot prices of oil, priced in Saudi Arabia and Japan, can also be found on the list, as well as the Nikkei 225, the major stock index for Japan.

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Table 4.6 Granger Causality Test Results. For Each of the Three Routes We Report the 15 Most Significant Causality Test P-Values, Averaged over FFA Spot and Futures. TD3

TC2

C4

0.00 0.02 0.04 0.05 0.07 0.09 0.09 0.10 0.11 0.11 0.11 0.12 0.12 0.13 0.14

Dirty Tanker Ind. LIBOR USD 1M Oil Spot Arab. Gulf Natural Gas Oil Spot UK Clean Tanker Ind. GSCI Energy Crude Oil Oil Future USA 1M Oil Future USA 4M Oil Spot Asia Oil Future USA 2M FX GP vs. USD Oil Future USA 3M Brent Crude

0.00 0.01 0.01 0.05 0.05 0.07 0.07 0.07 0.09 0.09 0.09 0.10 0.10 0.11 0.11

Clean Tanker Ind. LIBOR USD 1M FX EUR vs. USD GSCI Energy Dirty Tanker Ind. GSCI Oil Spot USA Oil Future USA 2M CRB Comm. Ind. Natural Gas FX GP vs. USD Oil Future USA 3M Oil Future USA 1M Oil Future USA 5M Oil Spot Asia

0.00 0.00 0.02 0.09 0.10 0.10 0.12 0.15 0.15 0.19 0.21 0.21 0.21 0.22 0.23

Baltic Dry Ind. Capesize Index Dow Jones Indust. CRB Metal Index CRB Comm. Index DAX FTSE 100 Oil Spot Asia EURO STOXX 50 S&P ASX Index Oil Future USA 6M Oil Future USA 3M Coil Rotterdam 2M Crude Oil Aluminum

0.19 0.19 0.43 0.45 0.58

Nikkei 225 Oil Spot USA Dow Jones Indust. FX JPN vs USD Tanker DWT

0.14 0.46 0.49 0.50 0.55

Oil Spot UK Tanker DWT Dow Jones Indust. EURO STOXX 50 DAX

0.25 0.26 0.29 0.30 0.65

Oil Future USA 2M Coil Rotterdam 1M Coil Rich. Bay 2M Coil Rotterdam 3M Bulk Carrier DWT

Further, both the clean and dirty tanker indices from the Baltic Exchange rank among the top variables. TC2 This route’s vessels take oil products from Rotterdam to the United States. Again, oil-related variables are on the list of the top 15 related variables, together with exchange rates and oil futures prices. C4 The capesize bulk carries of the C4 route mainly carry coil from Richards Bay, South Africa to Rotterdam. Besides the energy related-variables, the metal and commodity indices from CRB as well as the Coil Prices are among the top 15 out of the total 71 variables.

4.4.3. Selection Algorithm “Top Five” In what follows, we attempt to improve the prediction of future freight prices, using information available today. To keep things simple, we restrict ourselves to just using five of the above variables. Instead of simply using the top five of Table 4.6, we

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extend the bivariate model from Eq. (4.1) to a multivariate model and take those five variables that minimize the sum of squared residuals of the freight rate log return series. The reason for this additional selection algorithm is that five carefully selected variables typically have a higher combined explanatory power than the five top individual explanatory variables. For example the Baltic Dry Index is very similar to the Baltic Capesize Index, and therefore will not deliver substantial additional information. Instead of trying out all 71 · 70 · 69 · 68 · 67 combinations of explanatory variables, we apply a Greedy scheme which subsequently adds the variable that minimizes the sum SSRFull in the two, three, four and five dimensional model. The resulting top variables for the spot and first month contracts are given in Table 4.7, where we applied the procedure on the entire data set. It is interesting to see that besides the variables with the obvious relations other variables have a high explanatory power as well, for example the LIBOR rates.

4.4.4. Cointegration Another concept of dependence between two or more time series is that of cointegration. In the bivariate case, we say that two time series f and e are cointegrated if each of the series itself is integrated of order one, while some nontrivial linear combination (f, e) · a of the two series is stationary. a is then called cointegration vector. The interest in cointegration theory among the economists is due to the fact that it allows to model stable long-term relations in between non-stationary variables. We can use cointegration analysis in three ways: • Find an equilibrium relation in a system of price series. Table 4.7 Top Five Selected Variables Through the Greedy Scheme for the Entire Data Set. The First Set of the Five Variables Corresponds to FFA Spot Prices and the Second Set to Front Month FFA Futures Prices. TD3

TC2

C4

Top 1 Top 2 Top 3 Top 4 Top 5

Dirty Tanker Ind. Natural Gas FX EUR vs. USD LIBOR USD 1M LIBOR JPY 1M

Clean Tanker Ind. LIBOR USD 1M LIBOR JPY 3M FX EUR vs. USD Copper

Baltic Dry Index LIBOR JPY 1M Gas Oil Dow Jones Indust. GCC 200 (Africa)

Top 1 Top 2 Top 3 Top 4 Top 5

Clean Tanker Ind. LIBOR JPY 1M Dirty Tanker Ind. LIBOR USD 1M Nikkei 225

Clean Tanker Ind. Oil Spot Asia Oil Spot Arab. Gulf LIBOR USD 1M Oil Spot UK

Capesize Index LIBOR GBP 3M Dow Jones Indust. GCC 200 (Africa) LIBOR USD 1M

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• Estimation of future prices can be improved. If the cointegration vector is known or estimated and the equilibrium relation is violated, then a move of the series back to the equilibrium is more likely. • There must exist causality in at least one direction when two series are cointegrated [7], i.e., at least one variable is helpful in predicting another. The models used in this context are of the following form, where F and Ei , i = 1, . . . , 5 denote the freight and explanatory variables respectively,  denotes first order difference. Let yt be given by yt = (log(Ft ), log(Et1 ), log(Et2 ), log(Et3 ), log(Et4 ), log(Et5 )) , then the Vector Error Correction Model (VECM) is given by the following representation: yt = µ +

20 

j · yt−j +  · yt−1 +
t .

j=1

Thus, in this framework, we model the FFA log returns (which equal the first differences of log prices) to be given by the following composition: a constant µf , given by the first entry of the real vector µ, plus a weighted sum of lagged log returns of the FFA series and the lagged log returns of the explanatory variables. The weights of the sum corresponding to the j  -lagged log returns are given by the first row of the real– matrix j . The third summand is given by the first entry of the matrix–vector product of the cointegration matrix  with the lagged log price series. The rows of the cointegration matrix describe the cointegration relations and the rank of the matrix corresponds to the number of cointegration relationships. The last summand is given by the first entry of a six-dimensional white noise error term
, which we again assume to be normally distributed. The explanatory variables are modeled using the same principle. For the estimation of the models we utilize the R package “URCA” by Bernhard Pfaff, where we find a fast estimation procedure based on the seminal papers by Johansen in [8]– [10]. Further tests in the VECM framework are implemented as well. These are typically employed to estimate the cointegration rank and to test if the cointegration vector equals a certain exogeneously given vector, all of which we do not need for our application of the model. The tests are rather involved and the test statistics depend either on the rank or eigenvalues of the cointegration matrix  or on the residual vector
. However, we checked on the rank and always found at least one cointegration relation with the “top five” variables obtained through the selection algorithm above which suffices for our task. As an illustrative example, we find the cointegration relationship between the logarithms of spot prices of the tanker route TD3 and five explanatory variables. In order to be able to check the results, we manually selected five explanatory variables,

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based on obvious economic relations in between the variables. In this example, we consider F E1 E2 E3 E4 E5

TD3 Spot Dow Jones Industrial EURO STOXX 50 Exchange Rate EUR vs. USD Oil Spot Price UK Oil Spot Price USA

Applying the cointegration procedures, we estimate the cointegration vector to be a = (1, −0.38, 0.33, 0.07, 8.33, −9.49) , i.e., the linear combination (F, E1 , E2 , E3 , E4 , E5 ) · a is trend stationary. A possible economic interpretation of this relation can be given as follows: let us look at E1 and E2 only. If the European economy is slowing down compared to the American economy, it follows that −0.38 · log(E1 ) + 0.33 · log(E2 ) will decrease. As a consequence, to keep the above relation trend stationary, the FFA spot price has to increase, which is likely to happen in this example, since more oil is asked for in the United States than in Europe, and thus fright prices rise. A similar interpretation can be given for E4 and E5 . If, for whatever reason, spot prices in Europe fall compared to spot prices in the United States, then 8.33 · log(E4 ) − 9.49 · log(E5 ) will decrease, forcing the FFA rate to increase, to keep the total sum trend stationary. Again, this is likely to happen since it is more profitable to ship oil to New York, where higher prices are paid.

4.5. PREDICTING FREIGHT SPOT AND FUTURES RATES We have made two models available and we will see in this section that both allow for out of sample predictions of future prices. The extension of the VECM over the VAR is the additional balance term which models the cointegration relationship. Since we work with log returns in the VAR model and apply the VECM with its first differences to log prices, the two models become well comparable. For a given calibration day T , we apply the following procedure to draw out of sample forecasts of freight spot and freight futures prices F over the period [T, T + τ]. In the following, we set τ = 100, and therefore make forecasts for the next 100 working days. 1. Preparation: prepare the historical data used for the estimation: for the freight rate F under consideration and the potential explanatory variables: 1.1 Specify the estimation window [t, T ]. 1.2 Exclude the explanatory variables which are too sparse in the estimation window. 1.3 Fill the minor data gaps using Brownian Bridges.

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1.4 Transform the prices to log prices for the VECM model. 1.5 Calculate the log returns for the VAR model. 2. Top Five: out of the remaining pool of cleaned explanatory variables, select the five variables with the biggest combined explanatory power in the estimation window via the algorithm explained in Sec. 4.4.3. 3. Estimate models: estimate the parameters of the VAR and VECM model from Sec. 4.4 in the six-dimensional setup using the data in the estimation window. 4. Draw out of sample forecasts for the period [T, T + τ] and transform these to actual FFA rates, i.e., transform predicted log returns to prices in the VAR model, and the forecasted differenced log prices to actual prices in the VECM model. The predictions occurring in the fourth step above are obtained in the natural way: In two dimensions, the VAR model is set to the specification:   20 20   aj ft−j + bj et−j   µ +
    f f ft
t j=1 j=1   + , =  20 20   et
et    µ + ce + df e

j t−j

j=1

j t−j

j=1

with an N0, distributed error term 
. In order to draw an out-of-sample forecast fT +1 , we set the random error term to zero and increase t in the formula above to get fˆ T +1 = µf +

20 

aj fT +1−j +

j=1

20 

bj eT +1−j .

j=1

Further estimated log returns fˆT +2 , . . . , fˆT +τ are then received recursively, treating the estimated forecasted values needed for the next day forecast as if they were true log returns. The step in the VECM model is carried out similarly: The model is given as yt = µ +

20 

j · yt−j +  · yt−1 +
t ,

j=1

and increasing the time by one day as well as setting the random term to its mean yields the estimate ˆyT +1 = µ +

20  j=1

j · yT +1−j +  · yT .

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90 80 70 60 50 40 30 20 10 0 Parameter Estimation Window 3/1/2005

100

200

300

Forecast Period 400

2/1/2007

6/25/2007

Figure 4.2 Forecast carried out on 2/1/2007 for 100 working days and compared to actual prices.

That is, in two dimensions only we get for the FFA term  log Fˆ T +1 = µf +

20  j=1

aj  log FT +1−j +

20 

bj  log ET +1−j

j=1

+ (1, 1) · log FT + (1, 2) · log ET . Working with all the uncertainties of the freight market, like its sudden price movements in spot as well as futures rates, we cannot expect that our forecasts will yield the correct prices without error. We can however examine if this error is in the range of a very conservative forecast model.

4.6. THE BACKTESTING ALGORITHM We examine the forecast error of our model with the following back test. As an error quantity we define for a τ days ahead prediction carried out on day T the prediction error PE as following:

F

T +τ − Fˆ T +τ PET,τ =

,

FT

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i.e., we compare the absolute value of the relative error of actual rates to model implied predicted rates. Note that PET,0 = 0. For the actual test, we apply an increasing window with the initial window covering the period from 1/1/2005 to 1/1/2007. Starting with this window, we forecast 100 working days and then increase the window by one working day and repeat the forecast. We advocate the use of an increasing window over the use of a moving window because the length of the data set allows robust estimations of the model. For the results described below, we used 20 lags. Increasing the window on a daily basis allows for 550 calibrations until reaching the end of the time series on 20/2/2009. As an additional reference, we include a third very conservative forecast model, where we simply assume that the prices remain constant. In order to demonstrate the developed methods, we perform the following backtest. After preparing the data as described above, we run the following algorithm for the spot and futures rates of the three routes under consideration. 1. Initializing: 1.1. Prepare the data. 1.2. Set the initial window [1, T 0 ] to the period from 1/1/2005 to 1/1/2007. 2. Loop windows: 2.1. In each of the 550 possible steps, increase the window by one working day to get the new window [1, T i+1 ]. 2.2. Estimate the VAR and VECM model parameters using data only available in the window, and draw forecasts as described in Sec. 4.5 for the next 100 working days beyond T i+1 . 2.3. Calculate and save the prediction error PET i+1 ,τ , for τ = 1, . . . , 100 for the VAR and VECM model as well as for constant forecasts. To visualize the results, we average the prediction error PET i ,τ for τ ∈ {1, 2, . . . , 100} over the 550 out-of-sample forecasts drawn from the VAR and VECM model as well as the constant forecasts. The resulting Fig. 4.3 shows the relative error for the TD3 spot prices. Starting with PET,0 = 0 for the “0-days”-ahead prediction, it follows the error of the conservative constant out-of-sample forecast for the first “10-days”-ahead predictions but then the error increases slightly. This shows that the models are able to yield stable forecasts over periods as long as 100 days ahead. Additional benefits of the models over the constant predictions are for example readily available confidence regions and the possibility to check on trends rather than just having constant predictions.

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0.6 0.5 0.4 0.3 0.2 0.1 0

0

10

20

30

Figure 4.3

40

50

60

70

80

90

100

Relative forecast error.

4.7. CONCLUSION After an introduction to the freight market and its two main derivatives, we examined the relation of the highly volatile freight rates with explanatory variables. Incorporating such physical and economical relations in a model is desirable, as it leads to a better understanding of the freight market and may improve out-of-sample forecasts. We proposed the use of two statistical methods to jointly model freight rates and related explanatory variables. After two very favorable years, the freight market currently suffers from the world financial crisis. Large investments over the last years have made shipping capacities available, but freight rates fell to a level which drove many participants out of the physical market as well as its derivatives. However, with a revival of the world’s economy, it is likely to see an upturn of the freight market again.

References [1] [2] [3] [4] [5] [6] [7]

UNCTAD (2008). Review of Maritim Transport. (UNCTAD, 2008). Whittall, C (2009). Shipping out, Risk, March, pp. 52–53. Stopford, M (2007). Oil tanker market. Technical report, Baltic Freight Derivatives Forum. http://www.balticexchange.com/. The Baltic Exchange. http://www.exchange.imarex.com. International Maritime Exchange. http://www.balticexchange.com/default.asp?action=article&ID=18. FFABA. Granger (1969). Investigating causal relations by econometric models and cross-spectral methods, Econometrica, 37, 424–38.

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[8] Johansen, S (1988). Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control, pp. 231–254. [9] Johansen, S (1991). Estimation and hypothesis testing of cointegration vectors in gaussian vector autoregressive models, Econometrica, pp. 1551–1580. [10] Johansen, S. Juselius, K (1990). Maximum likelihood estimation and inference on cointegration with applications to the demand for money, Oxford Bulletin of Economics and Statistics, pp. 169–210.

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ON FORWARD PRICE MODELING IN POWER MARKETS

FRED ESPEN BENTH Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway and University of Agder, School of Management, Serviceboks 422, 4604 Kristiansand, Norway [email protected]

Power forward contracts deliver electricity over a specified period, and can be viewed as a portfolio of forwards with maturity at each time instant in the delivery period. We investigate the implied power forward dynamics from a geometric Brownian motion specification of the forward price, which turns out to have a very complicated structure. Lognormal approximations are argued for, and we demonstrate that they work excellently in many situations. In particular, we focus on the approximation suggested by Bjerksund et al. [8], where the volatility of the power forward is simply the average of the fixed-maturity forward volatility. Although giving a superior model to the moment matched dynamics, it fails to estimate the tails of the power forward distribution in some cases with extreme volatility and mean-reversion. We provide analytical bounds in terms of geometric Brownian motions for the power forward dynamics, and also compare the covariance structure with those implied by a geometric Brownian motion.

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5.1. INTRODUCTION The power markets have in the recent two decades been liberalized in Europe, America, Asia, and Australia. This freeing of trade in power is a process still going on, and gradually attracting investors not having any physical interest in power production or consumption. Large investors like investment banks, hedge funds, and pension funds see opportunities for portfolio diversification and speculation with this new asset class. The power markets are naturally divided into spot and forward products, where the former are usually linked to physical production and consumption, while the latter are financial. Naturally, investors without physical capacity are interested in the forward markets for power. We shall concentrate on the markets for electricity, however, coal, gas, and oil are also an integrated part of the power market being important fuels for power production. Taking the Nordic market for electricity, NordPool, as an example, the power forward contracts are financially settled against the system price in the market. The system price is the “spot” of this market, an auction-based hourly price for nextday delivery of electricity. The forward contracts are both of futures and forward type, and deliver the money-equivalent to receiving the spot over a pre-determined delivery period. Typically, the delivery period can be a week, month, quarter or year ahead, and the forward price is given in terms of Euro/MWh. In this paper, we will be concerned with the analysis of models for the forward price dynamics. The basis for our studies is taken from Benth and Koekebakker [4], Bjerksund et al. [8] and Kiesel et al. [17] (see also Benth et al. [6]), where the Heath– Jarrow–Morton (HJM) technique taken from fixed-income theory is applied to the power forward market. The main obstacle in using the HJM approach to power markets is the fact that the forward contracts do not deliver the underlying spot at a fixed maturity time in the future, but over a delivery period. This creates difficulties in stating reasonable models, which are analytically and empirically tractable. The standard approach is to suppose a dynamics for a (non-existing) fixed maturity contract, and derive the power forward price by averaging over the delivery period. We investigate the properties of such models and approximations, to understand the stochastic dynamics of power forward prices. The motivation for analyzing the price dynamics of power forwards is twofold. First, by investing in such power products one is naturally interested in the risk associated with the strategy chosen. To get hold of this, a proper understanding of the stochastics of the dynamical behavior of prices is required. This entails in an analysis of the volatility of the price. Secondly, there are markets for options on power forwards, both organized and OTC. The arbitrage-free prices of such options are naturally connected to the volatility of the underlying forward as well.

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Our analysis and results are presented as follows: In the next section, we describe the HJM approach to power forwards, recalling some existing results from Benth et al. [6]. In Sec. 5.3, we discuss the stochastic dynamics of power forwards, and link it to approximative geometric Brownian motion processes. In particular, we study the Bjerksund, Rasmussen, and Stensland approximation to the power forward dynamics, discuss the covariance structure of power forwards with different delivery periods and their distributional properties. Several empirical examples are studied. Option pricing is the topic in Sec. 5.4, where we compare the approximation by Bjerksund, Rasmussen, and Stensland with the true prices, both for plain vanilla call options and calendar spreads. We conclude in the final section.

5.2. HJM APPROACH TO POWER FORWARD PRICING Suppose St is the spot price of power at time t, being a stochastic process defined on some filtered probability space (
, F, F, Q) with time t running in the finite time interval [0, Tmax ]. We assume that the filtration F satisfies the usual conditions (see page 10 in Karatzas and Shreve [16]), and supports a Wiener process W = {Wt }t≥0 . Throughout the paper, we work exclusively under a risk–neutral probability Q. A long position entered at time t in a power forward contract settled continuously against the spot price over a period [T1 , T2 ] will yield a profit/loss
T2 e−r(u−t) (Su − Ft (T1 , T2 ))du. (5.1) T1

Here, t ≤ T1 , and Ft (T1 , T2 ) is the forward price at time t which is denoted in Euro/MWh. Furthermore, r is the risk-free interest rate, and the expression (5.1) measures the time t value of the profit/loss. The integration over the delivery period is for convenience. In the real market, there is a summation over hourly prices instead. Solving expression (5.1) for Ft (T1 , T2 ), we define the forward price as  
T2 Ft (T1 , T2 ) = E (5.2) w(u, T1 , T2 )Su du | Ft T1

with w(u, T1 , T2 ) =

re−ru . − e−rT2

e−rT1

(5.3)

If the forward is settled at the end of the delivery period T2 , we use w(u, T1 , T2 ) = 1/(T2 − T1 ) instead (see Sec. 4.1 in Benth et al. [6] for details). Obviously, we implicitly assume that w(u, T1 , T2 )Su is integrable with respect to Q × du on
× [T1 , T2 ].

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By commuting the integration and expectation in (5.2), we obtain
T2 w(u, T1 , T2 )ft (u)du, Ft (T1 , T2 ) =

(5.4)

T1

since by definition the forward price of a contract maturing at time u is ft (u) = E[Su | Ft ]. We note that there are no power forward contracts with fixed maturity time, thus, ft (u) exists only in a mathematical sense. However, the relation (5.4) shows us that a power forward contract may be considered as the weighted average of “standard” forward contracts, where the averaging goes over the delivery times. Alternatively, holding a power forward is equivalent to holding a portfolio of forward contracts delivering at each time in the delivery period. Since we are going to focus on the HJM approach, the spot price process Su will not play any role further in this paper. The idea of the HJM modeling perspective in fixed-income theory is to state an arbitrage-free dynamics of the forward rate process for all maturities. Taking this approach to commodities, one models the forward price dynamics for all maturities u (see Chapter 8 in Clewlow and Strickland [9], say). In the context of power markets, this would entail in modeling the price dynamics of the non-existing contracts ft (u) for all maturities u. On the other hand, one may ask for dynamical models of Ft (T1 , T2 ) for all delivery periods [T1 , T2 ]. However, as it is shown in Benth and Koekebakker [4], we are ensured an arbitrage-free model as long as
T2 Ft (T1 , T2 ) = w(u, T1 , T2 )Ft (u, u)du, (5.5) T1

for all 0 ≤ T1 < T2 . This leads us back to a model for Ft (u, u), which is equivalent to a forward price for a contract with fixed maturity time u. Hence, from this view, it is natural to use the HJM approach to state models for ft (u), and next derive the power forward price Ft (T1 , T2 ) from the relation (5.4). In this paper, we shall investigate the dynamics of the resulting power forward price, and investigate approximations of it in view of option pricing. Since ft (u) has to be a martingale, the dynamics will have the form dft (u) = σ˜ t (u)dWt , where t  → σ˜ t (u) is a stochastic process on [0, u] which is supposed to be integrable with respect to W . The model is defined for general choices of maturity times u, with the only restriction that u ≤ Tmax . To simplify matters, we shall concentrate on a geometric Brownian motion structure, and assume that dft (u) = σt (u)ft (u)dWt .

(5.6)

Here, we let σt (u) be a deterministic volatility function with 0 ≤ t ≤ u ≤ Tmax , and t  → σt (u) is square-integrable on [0, u] for every u ≤ Tmax . We have a model

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specifying the full behavior of the forward curve dynamics driven by a single Brownian motion Wt . We remark in passing that there is no difficulty in assuming a multi-factor model, but to stay focused we refrain from doing so. Note that the volatility σt (u) depends on the maturity time u. This allows us to incorporate the Samuelson effect into the model. The Samuelson effect tells us that the volatility of the forward price converges to the spot volatility as time to maturity tends to zero. This effect is heavily based on a mean-reverting property of the underlying spot price dynamics. In fact, assume that the (risk–neutral) spot price follows the exponential of an Ornstein–Uhlenbeck process Xt , dXt = α(µ − Xt )dt + σdWt , with α, σ, and µ being constants, and α, σ > 0. Then, we easily derive that dft (u) = σe−α(u−t) ft (u)dWt . Thus, σt (u) = σ exp(−α(u − t)), and letting u − t → 0, we find that the forward volatility tends to σ at an exponential speed given in terms of the mean–reversion coefficient α. There is both economical and empirical evidence for an explicit dependence on the maturity time in the volatility function (see Geman [13] and Eydeland and Wolynieck [10]). Other specifications of a term structure of volatility σt (u) for energy and commodity markets have been suggested (see Clewlow and Strickland [9]). For instance, since forwards in the long end of the curve indeed show price volatility contrary to the mean–reversion model discussed above, it has been suggested that the volatility looks like σt (u) = σ0 + σ1 e−α(u−t) ,

(5.7)

for two positive constants σ0 and σ1 . The first term σ0 is coming from a non-stationary factor in the spot prices, like for instance a non-stationary long-term stochastic mean. One may also introduce seasonality in the volatility term structure. See Benth and Koekebakker [4] for a discussion of the various models with references to related work. The volatility model in (5.7) will be our choice in many examples to follow, and serve as a canonical choice of volatility in power markets. The degenerate case when σt (u) = σt gives an explicit geometric Brownian motion dynamics for Ft (T1 , T2 ). Since in that case  
t
1 t 2 ft (u) = f0 (u) exp − σs ds + σs dWs 2 0 0 we find



t 1 t 2 Ft (T1 , T2 ) = w(u, T1 , T2 )f0 (u)du exp − σ ds + σs dWs 2 0 s 0 T1  
t
1 t 2 σ ds + σs dWs . = F0 (T1 , T2 ) exp − 2 0 s 0 


T2





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Hence, we have dFt (T1 , T2 ) = σt Ft (T1 , T2 )dWt .

(5.8)

When σt (u) explicitly depends on u, the simple geometric Brownian motion structure is no longer true. Before we proceed with analyzing the dynamics of Ft (T1 , T2 ), let us discuss how one could infer estimates on the parameters of ft (u) when data for such contracts are not available. One frequently used method is to smoothen the observed power forward curve, that is, to derive a continuous curve of forward prices fˆt (u) from the discretely observed power forwards Fˆ t (T1 , T2 ). A way to construct such a curve is proposed in Benth et al. [5]. From a given a priori seasonal function, a fourth order polynomial spline is fitted using a maximum smoothness criterion so that the implied power forward prices are within the bid–ask spread of the market prices. Another similar approach is described in Fleten and Lemming [11]. For each day, one can construct a time series of such curves fˆt (u), t = 0, 1, . . . , and these can be used to infer estimates on the volatility term structure σt (u) of the process ft (u). Obviously, one also gets estimates on the drift of ft (u) from these data, where this drift is known as the risk premium in the market. A different way to approach the inference problem is to use implied volatilities from quoted option prices in the market. This way to estimate is problematic since we need to derive theoretical option pricing formulas for call and put options, which is not straightforward at all due to the delivery period of the power forwards. Thus, we end up with first having to perform a Monte Carlo simulation of prices in combination with a numerical integration, a procedure which is slow when the goal is to derive implied estimates on the volatility term structure σt (u). To speed up this process, one may use some approximations of the dynamics of Ft (T1 , T2 ) where analytical (Black-76) option pricing formulas become available. The validity of such approximations is the target of the study in this paper. One may leave completely the specification of ft (u), and directly specify the dynamics of the power forwards Ft (T1 , T2 ), only for those being actually traded in the market. This kind of LIBOR model for the power forward market is analyzed in Benth and Koekebakker [4] and Kiesel et al. [17]. In the latter paper, they use it to find implied estimates for the volatility term structure of power forwards in the German EEX market.

5.3. POWER FORWARDS AND APPROXIMATION BY GEOMETRIC BROWNIAN MOTION In this Section, we analyze the time dynamics of power forwards resulting from the dynamics of ft (u). Our standing assumption is the geometric Brownian motion structure of ft (u) defined in (5.6). We further analyze approximations of the power forward

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price dynamics in terms of geometric Brownian motion, in particular the approximation suggested by Bjerksund et al. [8]. Following the calculation on page 170 in Benth et al. [6], we have
T2
T2
t w(u, T1 , T2 )f0 (u)du + w(u, T1 , T2 ) σs (u)fs (u)dWs du Ft (T1 , T2 ) = T1

= F0 (T1 , T2 ) +


t


0

T1 T2

w(u, T1 , T2 )σs (u)fs (u)dudWs , 0

T1

where we have used the stochastic Fubini Theorem (see pages 159, 160 in Protter [20]). Next, applying a straightforward integration by parts yields
T2 w(u, T1 , T2 )σs (u)fs (u)du T1




T2

= σs (T2 )Fs (T1 , T2 ) −

T1

∂σs (u) ∂u




u

w(v, T1 , T2 )fs (v)dvdu. T1

Dividing and multiplying with w(v, T1 , u) in the dv-integral on the right-hand side, and observing that w(v, T1 , T2 )/w(v, T1 , u) is independent of v, gives us the following expression:
T2 w(u, T1 , T2 )σs (u)fs (u)du T1


= σs (T2 )Fs (T1 , T2 ) −

T2 T1

∂σs (u) w(v, T1 , T2 ) Fs (T1 , u)du. ∂u w(v, T1 , u)

Collecting terms together, we have proven: Proposition 5.1. Given the dynamics (5.6) for ft (u), the dynamics of the power forward Ft (T1 , T2 ) is  
T2 ∂σt (u) w(v, T1 , T2 ) Ft (T1 , u) dFt (T1 , T2 ) (5.9) = σt (T2 ) − du dWt , Ft (T1 , T2 ) ∂u w(v, T1 , u) Ft (T1 , T2 ) T1 where w(v, T1 , T2 )/w(v, T1 , u) is independent of v. Although ft (u) has a simple geometric Brownian motion dynamics with maturitydependent volatility, the power forward has a complex dynamics which involves not only the power forward Ft (T1 , T2 ) itself, but the states of all power forwards Ft (T1 , u) for u ∈ (T1 , T2 ]. The geometric Brownian motion structure is thus not inherited from ft (u). We also note that a volatility σt (u), which is not depending on u will have ∂σt (u)/∂u = 0, and we obtain a geometric Brownian motion dynamics as we already saw above in the derivation of (5.8). In the general case, this is a rather complex model, especially from the point of view of simulation and pricing of options since in principle it involves an infinite number of power forwards.

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We may also view the dynamics of Ft (T1 , T2 ) as a geometric Brownian motion with stochastic volatility modeled through a factor process given by ft (u). Appealing to the dynamics of ft (u), and dividing by Ft (T1 , T2 ), we obtain  T2 T w(u, T1 , T2 )σt (u)ft (u)du (5.10) Ft (T1 , T2 )dWt . dFt (T1 , T2 ) = 1 T2 w(u, T , T )f (u)du 1 2 t T1 We see that the dynamics of Ft (T1 , T2 ) has a stochastic volatility described as a ratio of two weighted averages of ft (u). The stochasticity comes in through fluctuations of ft (u) in time t. If σt (u) is rather flat as a function of maturity u, we expect the stochastic volatility to be close to σt . This happens, say, when we have a volatility function for which limu→∞ σt (u) = σˆ t . Then, for delivery periods in the long end of the curve, that is, T1 − t large, we will approximately have
T2
T2 w(u, T1 , T2 )σt (u)ft (u)du ≈ σˆ t w(u, T1 , T2 )ft (u)du, T1

T1

and hence, dFt (T1 , T2 ) ≈ σˆ t Ft (T1 , T2 )dWt . I.e., in the long end of the power forward curve, the prices move approximately according to a geometric Brownian motion with maturity-independent volatility given as the limit of σt (u) with respect to u. In the discussion leading to this result, we have implicitly assumed that ft (u) is well-behaved as a function of u ∈ [T1 , T2 ], for instance, that it is continuous. As an example, consider σt (u) given as in (5.7). Then lim σt (u) = σ0

u→∞

for all t ≥ 0. Thus, in this case, the power forward will for large values of T1 − t behave approximately as a geometric Brownian motion with volatility given by the long-term level σ0 . We note that closer to maturity the power forward will naturally have increasing volatility due to the Samuelson effect of ft (u), and thus its dynamics will depart from the constant volatility geometric Brownian motion. However, one may ask the question whether there exist reasonable geometric Brownian motion models, which approximate the power forward dynamics. Looking at the mean-value theorem in calculus, we have that
T2 ∗ w(u, T1 , T2 )ft (u)du, ft (u (t)) = T1 ∗

for some u (t) ∈ (T1 , T2 ). Hence, Ft (T1 , T2 ) = ft (u∗ (t)),

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which gives an explicit relation between a power forward price dynamics and a forward price ft (u) being a geometric Brownian motion. However, the latter has a parameter u, which in this case will depend on time t. The connection motivates to look for approximative models, which indeed are lognormal processes. Following the argument in Benth and Henriksen [3], we use a Taylor expansion of the logarithmic function to find 1 2 ln(ft (u) + ft (v)) = ln ft (u) + ln ft (v) + (cosh(ln(ft (u)/ft (v))) − 1) 2 + 2 ln(2) + O((ln(ft (u)/ft (v)))4 ). Now, for two nearby delivery times u and v, ft (u) ≈ ft (v) as long as σ(t, u) is sufficiently regular in u. Hence, we see that the logarithm of ft (u) + ft (v) is close to the sum of logarithms of ft (u) and ft (v) and the constant 2 ln(2), which is following a drifted Brownian motion. The crucial observation here is that there is the same Brownian motion Wt driving both ft (u) and ft (v) making them perfectly correlated, and it is only the regularity of σt (u) as a function of maturity that decides how good the approximation is. Indeed, in Henriksen [14], it is demonstrated empirically that a sum of positively correlated lognormal variables are very well approximated by a lognormal variable. Of course, in our case, we have an integral of lognormal variables, but still this discussion points toward the reasonability of geometric Brownian motion approximations for power forwards. We remark that a geometric Brownian motion dynamics for Ft (T1 , T2 ) in general will violate the no-arbitrage condition (5.5) (see Lemma 6.1, page 168 in Benth et al. [6]), and thus cannot be a valid dynamics for the whole power forward curve. However, as dynamics for a contract with a specified delivery period [T1 , T2 ], it is arbitrage-free. Since analytical pricing formulas for plain vanilla options are available for geometric Bronwian motion dynamics, such approximations are particularly interesting.

5.3.1. A Geometric Brownian Motion Dynamics by Volatility Averaging In (5.10), divide and multiply with the weighted average of σ(t, u) in the numerator to obtain
T2 dFt (T1 , T2 ) = w(u, T1 , T2 )σt (u)du T1

 T2

T wσ (t, u, T1 , T2 )ft (u)du Ft (T1 , T2 )dWt , × 1 T2 T1 w(u, T1 , T2 )ft (u)du

(5.11)

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where w(u, T1 , T2 )σt (u) . wσ (t, u, T1 , T2 ) =  T2 T1 w(v, T1 , T2 )σt (v)dv T Since wσ (t, u, T1 , T2 ) is non-negative and T12 wσ (t, u, T1 , T2 )du = 1, it is a weighting function. Hence, the fraction on the right-hand side of the dynamics (5.11) is the ratio between two weighted averages of ft (u) over u ∈ [T1 , T2 ]. As long as wσ (t, u, T1 , T2 ) ≈ w(u, T1 , T2 ), this is close to one. This is equivalent to σt (u) being approximately equal to its weighted average, i.e., being close to flat over the delivery period. We consider an example, where this is true. Consider σt (u) = σ exp(−α(u − t)) and w(u, T1 , T2 ) = 1/(T2 − T1 ). We find αe−αu . − e−αT2 Note that ∂wσ (t, u, T1 , T2 )/∂u = −αwσ (t, u, T1 , T2 ) < 0, and the weighting function is decreasing as a function of u. Its maximal value becomes wσ (t, u, T1 , T2 ) =

e−αT1

wσ (t, T1 , T1 , T2 ) = α(1 − e−α(T2 −T1 ) )−1 , and minimal value wσ (t, T2 , T1 , T2 ) = α(eα(T2 −T1 ) − 1)−1 . Moreover, wσ (t, T1 , T1 , T2 ) − wσ (t, T2 , T1 , T2 ) = α. If α is small, then the variation in wσ is correspondingly small. Furthermore, as long as α(T2 − T1 ) is small, we have eα(T2 −T1 ) ≈ 1 + α(T2 − T1 ). Hence, wσ (t, u, T1 , T2 ) ≈ Thus, we have in this case that
dFt (T1 , T2 ) ≈

1 = w(u, T1 , T2 ). T2 − T 1

T2

w(u, T1 , T2 )σt (u)duFt (T1 , T2 )dWt .

T1

This is the approximation of the power forward price dynamics suggested by Bjerksund et al. [8], and we introduce the BRS-approximation as dFtBRS (T1 , T2 ) = BRS (T1 , T2 )FtBRS (T1 , T2 )dWt t with the BRS-volatility function given by
T2 BRS (T1 , T2 ) = w(u, T1 , T2 )σt (u)du. t T1

(5.12)

(5.13)

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The BRS-approximation will play a central role in the analysis to follow, since this is the benchmark model for many applications and seems to be frequently used in practice. The approximation in (5.12) can also be argued for by looking at the returns of ft (u). Thinking of Ft (T1 , T2 ) as a portfolio of ft (u)’s, it may seem natural that the return from Ft (T1 , T2 ) is the sum of returns from the “assets” in the portfolio, namely ft (u) for u ∈ [T1 , T2 ]. Hence,  
T2
T2 dFt (T1 , T2 ) dft (u) w(u, T1 , T2 ) w(u, T1 , T2 )σt (u)du dWt . ≈ du = Ft (T1 , T2 ) ft (u) T1 T1 In the last equality, we have used that 
T2 
T2 dft (u) du = w(u, T1 , T2 ) w(u, T1 , T2 )σt (u)du dWt . ft (u) T1 T1 In effect, in the above approximation, we are boldly assuming that the self-financing property holds. Thus, we have again reached the approximative model in (5.12). It is of course a question whether the return of a power forward approximately is equal to the weighted average of the return of ft (u) in general.

5.3.2. A Geometric Brownian Motion Dynamics by Moment Matching In this Subsection, we propose and analyze an alternative approximation procedure to the one discussed above. Since the power forward dynamics is the weighted integral over maturity u of ft (u), it is the weighted average of lognormal random variables. In general, such an integral will not be lognormal. However, we may still ask for a geometric Brownian motion dynamics that approximates Ft (T1 , T2 ), and in order to be reasonably close to the distribution of F we construct in this Subsection a model that matches the first two moments. Moment matching is a much-used technique when pricing basket options, since it allows for explicit pricing formulas. Extensive studies of its performance, both theoretical and empirical, are provided in Zacks and Sokos [21], Asmussen and RojasNandayapa [1], Levy [18], Henriksen [14] and Hoedemakers [15]. Define the process Ftm (T1 , T2 ) by m dFtm (T1 , T2 ) = m t (T1 , T2 ) Ft (T1 , T2 )dWt ,

(5.14)

where the subscript “m” indicates that we are looking for a dynamic matching the first two moments of F . The deterministic volatility function t → m t (T1 , T2 ) is assumed to be square integrable for t ∈ [0, T1 ] to ensure that Ftm (T1 , T2 ) is a martingale for

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any fixed T1 , T2 . Our aim is to find F0m (T1 , T2 ) and m t (T1 , T2 ) such that the first two m moments of Ft (T1 , T2 ) match those of Ft (T1 , T2 ) for all 0 ≤ t ≤ T1 < T2 . First, by the martingale properties of Ftm (T1 , T2 ) and ft (u) we find that
E[Ftm (T1 , T2 )] = F0m (T1 , T2 ) =

T2

w(u, T1 , T2 )f0 (u)du = F0 (T1 , T2 ).

T1

Thus, the first moment condition gives F0m (T1 , T2 ) = F0 (T1 , T2 ). We next calculate the second moment of Ft (T1 , T2 ). From integration-by-parts, we have 
T2 2
T2
v w(u, T1 , T2 )ft (u)du = 2 w(v, T1 , T2 )w(u, T1 , T2 )ft (v)ft (u)dudv. T1

T1

T1

Taking the expectation yields
E[Ft2 (T1 , T2 )]

=2

T2




v

w(v, T1 , T2 )w(u; T1 , T2 )E[ft (u)ft (v)]dudv. T1

T1

The expectation inside the double integral on the right-hand side can be calculated by appealing to well-known properties of geometric Brownian motion (note that we have u ≤ v here),  
1 t 2 E[ft (u)ft (v)] = f0 (u)f0 (v) exp − σs (u) + σs2 (v)ds 2 0  
t  × E exp σs (u) + σs (v)dWs 0




1 t 2 2 σ (u) + σs (v)ds = f0 (u)f0 (v) exp − 2 0 s 
t  1 exp (σs (u) + σs (v))2 ds 2 0 
t  σs (u)σs (v)ds . = f0 (u)f0 (v) exp 0

On the other hand, we find 
E[(Ftm )2 (T1 , T2 )]

=

F02 (T1 , T2 ) exp



t 0

2 (m s ) (T1 , T2 )ds

.

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Hence, matching the second moment gives the equation 
F02 (T1 , T2 ) exp
=2

2 (m s ) (T1 , T2 )ds

0

T2




t

v

w(u, T1 , T2 )w(v, T1 , T2 )f0 (u)f0 (v) T1

T1






t

× exp

σs (u)σs (v)ds dudv. 0

To this end, introduce the mapping η → ξ(t, T1 , T2 ; η) of functions η: [T1 , T2 ] → R,
ξ(t, T1 , T2 ; η) =

T2




v

w(u, T1 , T2 )w(v, T1 , T2 )f0 (u)f0 (v) T1

T1

×e

t 0

σs (u)σs (v)ds

η(u)η(v)dudv.

(5.15)

We implicitly suppose that η is sufficiently regular (continuous on [T1 , T2 ], say) in order to make ξ well-defined. Taking logarithms and differentiating both sides with respect to t, gives the formula for m : 2 (m t ) (T1 , T2 ) =

ξ(t, T1 , T2 ; σt (·)) . ξ(t, T1 , T2 ; 1)

(5.16)

For a given volatility function σt (u), we can calculate m t (T1 , T2 ) from (5.16). (T , T ) depends on the initial forward From the definition of (5.15), we see that m 1 2 t curve f0 (u) as well as the volatility σt (u) and the weighting function w(u, T1 , T2 ). The integrand in the double integral forming ξ(t, T1 , T2 ; η) is rather complex due t to the exponentiating of 0 σs (u)σs (v)ds, and will not in general admit any simple (T1 , T2 ). For closed-form expression as is mostly the case for the BRS-volatility BRS t example, choosing the standard volatility σt (u) as in (5.7) leads to a double integral of the exponential of an exponential. Hence, in order to calculate m t (T1 , T2 ), we must perform a numerical integration. Even worse, we need to do such a numerical integration at each time t we want to have the value of Ftm (T1 , T2 ) since the expression inside the integrals is t-dependent. This may become very time consuming in practice. However, when pricing a plain vanilla call option, we get away with one such numerical integration, as we see from the Black-76 Formula (see Sec. 5.4). In any case, the expression for the volatility resulting from moment matching does not seem to be as operable in practice as for instance the BRS-volatility.

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Let us compare m t (T1 , T2 ) with the BRS-volatility in (5.13). Note that we may write 
T2 2 w(u, T1 , T2 )σt (u)du ξ(t, T1 , T2 ; σ t (·, T1 , T2 )), ξ(t, T1 , T2 ; σt (·)) = T1

with σ¯ t (u, T1 , T2 ) =  T2 T1

σt (u)

.

w(v, T1 , T2 )σt (v)dv

If w(u, T1 , T2 )σt (u) ≈ w(u, T1 , T2 ), wσ (t, T1 , T2 ) =  T2 T1 w(v, T1 , T2 )σt (v)dv we find 
ξ(t, T1 , T2 ; σt (·)) ≈

2

T2

w(u, T1 , T2 )σt (u)du

ξ(t, T1 , T2 ; 1).

T1

Recall that if σt (u) is given by (5.7) and w(u, T1 , T2 ) = 1/(T2 − T1 ), this property holds as long as α, the speed of mean–reversion, is small. Hence, BRS (T1 , T2 ). m t (T1 , T2 ) ≈ t

The two ways of finding a “reasonable” geometric Brownian motion specification will therefore in some relevant cases be approximately equal. However, the expression for BRS (T1 , T2 ) is considerably simpler to find, since it allows for analytical expressions t in most interesting cases, which is not the case for m t (T1 , T2 ). We shall therefore concentrate our analysis on the BRS-approximation.

5.3.3. The Covariance Structure Between Power Forwards We include a discussion of the covariance between two power forwards. We compare with the covariance structure implied by a geometric Brownian motion approximation d F˜ t (T1 , T2 ) = t (T1 , T2 )F˜ t (T1 , T2 )dWt , BRS where t (T1 , T2 ) may be one of the two choices m (T1 , T2 ) introt (T1 , T2 ) or t duced. Naturally, we have F˜ 0 (T1 , T2 ) = F0 (T1 , T2 ). Let now s ≤ t, and consider two delivery periods [T1 , T2 ] and [T3 , T4 ] which may be overlapping, but where we suppose T1 ≤ T3 or T2 ≤ T4 . From similar calculations as in the previous Subsection we derive the following Proposition.

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Proposition 5.2. It holds that
Cov(Fs (T1 , T2 ), Ft (T3 , T4 )) =

T2




T4

w(u, T1 , T2 )w(v, T3 , T4 )f0 (u)f0 (v) T1

T3

× (e

s 0

σr (u)σr (v)dr

− 1)dudv,

(5.17)

and Cov(F˜ s (T1 , T2 ), F˜ t (T3 , T4 )) = F0 (T1 , T2 )F0 (T3 , T4 )(e

s 0

r (T1 ,T2 )r (T3 ,T4 )dr

− 1).

(5.18)

Let us compare the two expressions for the covariance. Introduce the weight function w(u, T1 , T2 )f0 (u) w(u, T1 , T2 )f0 (u) , wf (u, T1 , T2 ) =  T2 = F0 (T1 , T2 ) T1 w(v, T1 , T2 )f0 (v)dv to find Cov(Fs (T1 , T2 ), Ft (T3 , T4 ))
= F0 (T1 , T2 )F0 (T3 , T4 )

T2

s 0

σr (u)σr (v)dr

T4

wf (u, T1 , T2 )wf (v, T3 , T4 ) T1

× (e




T3

− 1)dudv,

Jensen’s Inequality for the convex function exp(x) now gives Cov(Fs (T1 , T2 ), Ft (T3 , T4 )) ≥ F0 (T1 , T2 )F0 (T3 , T4 ) × (e

 T4  s  T2 0 ( T wf (u,T1 ,T2 )σr (u)du T wf (u,T3 ,T4 )σr (u)du)dr 1

3

− 1).

Moreover, as long as wf (u, T1 , T2 ) ≈ w(u, T1 , T2 ), we obtain Cov(Fs (T1 , T2 ), Ft (T3 , T4 )) ≥ Cov(FsBRS (T1 , T2 ), FtBRS (T3 , T4 )). Hence, the BRS-approximation underestimates the true covariance between power forwards with different delivery periods and time points. This may have consequences when pricing path-dependent or spread options on power forwards. Note that from the above considerations the moment approximation F m has the same variance as F , however, not necessarily the same covariance structure.

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5.3.4. The Distribution of a Power Forward We end this section by investigating the distribution of a power forward. More specifically, we show that the power forward is bounded from below and above by two lognormal stochastic processes. Proposition 5.3. For all 0 ≤ t ≤ T1 < T2 , it holds that f

c(t, T1 , T2 )FtBRS (T1 , T2 ) ≤ Ft (T1 , T2 ) ≤ Ft (T1 , T2 ) f

where Ft (T1 , T2 ) is the geometric Brownian motion
T2 f f dFt (T1 , T2 ) = wf (u, T1 , T2 )σt (u)duFt (T1 , T2 )dWt . T1

Here, wf (u, T1 , T2 ) = w(u, T1 , T2 )f0 (u)/

 T2 T1

w(v, T1 , T2 )f0 (v)dv and


t  
T2 1 BRS 2 2 (s ) (T1 , T2 ) − c(t, T1 , T2 ) = exp w(u, T1 , T2 )σs (u)du ds 2 0 T1 T exp( T12 w(u, T1 , T2 ) ln f0 (u)du) . × F0 (T1 , T2 ) Proof. To prove the lower bound, use Jensen’s Inequality for the concave function ln x to find
T2
T2 ln w(u, T1 , T2 )ft (u)du ≥ w(u, T1 , T2 ) ln ft (u)du. T1

T1

Now, from the geometric Brownian motion dynamics of ft (u), we have
T2
T2 w(u, T1 , T2 )ft (u)du ≥ w(u, T1 , T2 ) ln f0 (u)du ln T1

T1



1 t T2 w(u, T1 , T2 )σs2 (u)duds 2 0 T1
t
T2 w(u, T1 , T2 )σs (u)dudWs . +

−

0

T1

Thus, the power forward is dominating a lognormally distributed stochastic process Xt (T1 , T2 ), where ln Xt (T1 , T2 ) has mean
T2

1 t T2 w(u, T1 , T2 ) ln f0 (u)du − w(u, T1 , T2 )σs2 (u)duds 2 0 T1 T1

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and variance
t 


2

T2

w(u, T1 , T2 )σs (u)du 0

ds.

T1

The latter is the variance of the logreturn from the BRS-approximation (5.12), where we recall from (5.13) that
BRS (T1 , T2 ) = t

T2

w(u, T1 , T2 )σt (u)du. T1

By the definition of FtBRS (T1 , T2 ) and the deterministic function c(t, T1 , T2 ) in the Proposition, we find after taking exponentials Ft (T1 , T2 ) ≥ c(t, T1 , T2 )FtBRS (T1 , T2 ). This proves the lower bound. From the definition of wf (u, T1 , T2 ), a straightforward calculation gives


T2

w(u, T1 , T2 )ft (u)du T1


t
1 t 2 w(u, T1 , T2 )f0 (u) exp − σ (u)ds + σs (u)dWs du = 2 0 s 0 T1  
T2
t
1 t 2 = F0 (T1 , T2 ) wf (u, T1 , T2 ) · exp − σs (u)ds + σs (u)dWs du. 2 0 0 T1




T2

Again appealing to Jensen’s Inequality, but now using the convex function exp(x), we find


T2 T1

 
t
1 t 2 wf (u, T1 , T2 ) exp − σs (u)ds + σs (u)dWs du 2 0 0 

1 t T2 ≤ exp − wf (u, T1 , T2 )σs2 (u)duds 2 0 T1 
t
T2 wf (u, T1 , T2 )σs (u)dudWs . + 0

T1

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Thus, by the definition of Ft (T1 , T2 ),

2
T2 T2 1 t f w(u, T1 , T2 )ft (u)du ≤ Ft (T1 , T2 ) exp wf (u, T1 , T2 )σs (u)du 2 0 T1 T1  
T2 2 − wf (u, T1 , T2 )σs (u)du ds T1 f

≤ Ft (T1 , T2 ). In the last inequality, we applied Jensen’s Inequality to the convex function x2 . This shows the upper bound, and the Proposition is proved. In the lower bound for Ft (T1 , T2 ), the deterministic function c(t, T1 , T2 ) is less that one. This we see by using Jensen’s Inequality on the two convex functions exp(x) and x2 . This means that the lower bound of Ft (T1 , T2 ) is less than FtBRS (T1 , T2 ), and in theory the power forward price can be below the BRS-approximation. However, there is a geometric Brownian motion bounding the power forward from above. This geometric Brownian motion may even bound the BRS-approximation.

5.3.5. Numerical Analysis of the Power Forward Distribution We investigate the distribution of power forwards from a numerical point of view, and choose for this purpose a volatility function σt (u) given by (5.7). We treat two different scenarios. First, we have a “moderate” market with σ0 = 20%, σ1 = 50%, and α = 0.05, corresponding to approximately a half-life of 14 days. The market is “moderate” because we have a very slow mean–reversion along with a volatility, which is relatively low for markets like electricity. The “spiky” market has σ0 = 20%, σ1 = 300%, and α = 0.2, meaning a half-life of 3.5 days. Hence, we have very fast mean–reversion along with an extreme volatility (although the stationary volatility is the same), on a level not far from what one may observe in electricity markets. Further, it is assumed that f0 (u) = 10 for all delivery times u, and the weight function is w(u, T1 , T2 ) = 1/(T2 − T1 ). This implies that F0 (T1 , T2 ) = 10. We look at two power forward contracts, both starting delivery in 50 days, T1 = 50. The first contract delivers over a week, T2 = 55, and the latter over a quarter, T2 = 110, where we count five trading days in a week. The power forward prices are simulated using 5000 Monte Carlo outcomes, and a Riemann approximation of the integration over the delivery period. In Figs. 5.1–5.3, we have plotted the resulting empirical distributions of the weekly and quarterly power forwards at times t = 1, t = 25, and t = 50. The empirical densities have been fitted using a kernel density estimator. The left column in each figure shows the weekly contracts, while the quarterly are

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Figure 5.1 The distributions from simulations of a weekly (left column) and quarterly (right column) power forward contract at time t = 1 with row 1 showing the “moderate” market scenario, whereas rows 2 shows the “spiky” market scenarios. The BRS approximation is shown in dashed lines, all distributions are plotted with a logarithmic frequency axis.

depicted in the right column. The BRS approximation is included with a dashed line, using a logarithmic scale on the frequency axis. The first row in each figure shows the “moderate” scenario, while the second row depicts the “spiky” case. We observe that the distributions become wider when approaching T1 , in line with the fact that σt (u) is increasing when approaching time to maturity. We notice that the BRS approximation has close to an excellent fit for all sampling times t, the two contract types, and the different market scenarios. This indicates that the BRS approximation works excellently for short- and long-delivery periods, no matter how far or close to start of delivery of the contract we are. However, one may spot a slight underestimation of the true distribution for the weekly contract in the “spiky” market scenario with t = 50. It seems that the BRS approximation for this situation is underestimating in both tails. To investigate this closer, we looked at the distribution of a weekly and monthly power forward, both starting at T1 = 50, and maturing at T2 = 55 and T2 = 70 respectively. Further, since the effect seems to come from the exponential increase

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Figure 5.2 The distributions from simulations of a weekly (left column) and quarterly (right column) power forward contract at time t = 25 with row 1 showing the “moderate” market scenario, whereas row 2 shows the “spiky” market scenarios. The BRS approximation is shown in dashed lines, all distributions are plotted with a logarithmic frequency axis.

in volatlity, we let σ0 = 0 in the “spiky” market scenario. In Fig. 5.4, we plot the distributions for the two contracts in the two market scenarios at time t = 50. We clearly see that the “spiky” market scenario produces a distribution for the power forward, which has heavier tails than the BRS approximation. The weekly contract is most extreme, whereas the monthly only has a heavier right tail. The effect is very small in the “moderate” market scenario, perhaps most pronounced for the weekly contract. It thus seems like that the BRS approximation becomes better when the delivery period increases, and the farther away we are from T1 , the start of delivery. To further analyze the distributional differences between the BRS approximation and the power forward, we include in Fig. 5.5 plots of the 1%, 5%, 95%, and 99% quantiles of the two for all times t = 40 up to t = 50, the start of delivery. We present the weekly and monthly contracts, separating the moderate and spiky market scenarios. The “moderate” market does not show any difference between the quantiles of the power forward and its BRS approximation. Here, the tails are captured excellently by the BRS approximation. The BRS approximation fails seriously, however, on the 95% and 99% quantiles of both, weekly and monthly forward contracts in the “spiky”

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Figure 5.3 The distributions from simulations of a weekly (left column) and quarterly (right column) power forward contract at times t = 50 with row 1 showing the “moderate” market scenario, whereas row 2 shows the “spiky” market scenarios. The BRS approximation is shown in dashed lines, all distributions are plotted with a logarithmic frequency axis.

scenario. In fact, the quantiles are close to 5% underestimated by the BRS aproximation. There is also an underestimation of the lower quantiles, in particular for the weekly contract. From the figure, we also see how the interplay between volatility and mean–reversion makes the “moderate” market scenario being more risky far from delivery, while the “spiky” regime becomes gradually more riskier toward the start of delivery, and eventually, for times t close to 50, being significantly more volatile than the “moderate” scenario. If we would include a contribution from σ0 in the “spiky” scenario, the effects would be reduced. Furthermore, we clearly see that the longer the delivery period, the less volatile are the prices. There is a significant smoothing effect in the length of delivery period. From these examples, we conclude that the BRS approximation works excellently for cases, where the market is not mean–reverting strongly and where the volatility is not high (in relative terms). When we use the approximation for markets with very fast mean–reversion and high volatility, the tails are not well explained by the BRS approximation. It seems to underestimate significantly the extreme quantiles, an effect which will lead to an underestimation of option prices if considering for instance far out-of-the-money calls.

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Figure 5.4 The distributions of a weekly (left column) and monthly (right column) power forward contracts at time t = 50 with row 1 showing the “moderate” market scenario, whereas row 2 shows the “spiky” market scenarios. The BRS approximation is shown in dashed lines, all distributions are plotted with a logarithmic frequency axis.

5.4. PRICING OF OPTIONS ON POWER FORWARDS In this section, we study options on power forwards. We analyze in particular the option prices resulting from the BRS approximation to power forwards. Since this is a geometric Brownian motion model, the Black-76 formula is available for pricing of plain vanilla call and put options. We begin by recalling this formula. In the famous paper by Black [7], the Black-76 Formula was derived for a call option on a forward contract with price dynamics given by a geometric Brownian motion having constant volatility. It is simple to extend for a time-dependent volatility, as is the case for the BRS approximation. Considering the dynamics of FtBRS (T1 , T2 ) in (5.12), we have the following result: Proposition 5.4. Let r be the risk-free interest rate, K the strike, and 0 ≤ T ≤ T1 the exercise time for a call option written on a forward contract with price process FtBRS (T1 , T2 ) defined as in (5.12). Then, the arbitrage-free option price at time t ≤ T is Pt (T, K, r) = e−r(T −t) {FtBRS (T1 , T2 )(d1 ) − K(d2 )}

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18

16

16

14

14

12

12

10

10

8

8

6

6

4 40

45

50

4 40

13

13

12.5

12.5

12

12

11.5

11.5

11

11

10.5

10.5

10

10

9.5

9.5

9

9

8.5

8.5

8 40

45

50

8 40

45

50

45

50

Figure 5.5 The 1%, 5%, 95%, and 99% quantiles of a weekly (top row) and monthly (bottom row) power forward contracts at all times t up to ten days prior to start of delivery. Left column shows the case of “moderate” market, while the right row depicts the results for the “spiky” market. The BRS approximation is shown with slashed lines.

where






d1 = d2 +

T

(BRS )2 (T1 , T2 )ds s t

T )2 (T1 , T2 )ds ln(FtBRS (T1 , T2 )/K) − 0.5 t (BRS s d2 =  T BRS )2 (T , T )ds 1 2 t (s and  is the cumulative standard normal distribution function.

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Proof. See page 239 in Benth et al. [6] for a proof. For many relevant cases like the various choices of σt (u) considered in this paper, )2 (T1 , T2 ) in the expression for the Black-76 price can be anathe integral of (BRS s lytically calculated, and hence the formula provides a very rapid way to price call options. Note that the moment matching volatility m t (T1 , T2 ) defined in (5.16) can be substituted with the BRS volatility in the option pricing formula. However, the definition of m t (T1 , T2 ) is complicated, and for the interesting situations it does not have any analytical expression, and numerical integration must be used to price options. In fact, one must already use a numerical approach to integrate σt (u) and the initial forward curve f0 (u) to obtain the moment-matching volatility m t (T1 , T2 ). This may be time consuming and not practical, in particular if we want to price dynamically or to derive a hedge. In such a situation, we must update the volatility m at each time step, which would be significantly more efficient to do with the BRS approximation. As a numerical example, consider a call option on a power forward with weekly delivery. The fixed-maturity volatility σt (u) is again given by (5.7), and as usual we separate between a “moderate” and a “spiky” market scenario. The latter has parameters specified as σ0 = 0, σ1 = 300%, and α = 0.2, whereas the “moderate” market scenario has σ0 = 20%, σ1 = 50%, and α = 0.05. We know from the previous section that in the “spiky” scenario the BRS approximation failed to capture the tails of the power forward distribution. In this example, we investigate the consequences on call option pricing, and in particular we focus on options which are far-out-of the money. The initial forward curve is f0 (u) = F0 (T1 , T2 ) = 10, where we have used the weight function w(u, T1 , T2 ) = 1/(T2 −T1 ). The options are based on a weekly power forward contract, starting delivery at time T1 = 50. Further, since the deviation of the BRS approximation from the power forward is greatest at start of delivery, we consider options with exercise time T = 50. The risk-free interest rate is set equal to r = 5%. In Table 5.1, we report the results from the numerical example. The power forward option prices are based on 10,000 Monte Carlo simulations, while the BRS Table 5.1 Call Option Prices on Power Forward and Its BRS Approximation. The Power Forward Has a Weekly Delivery from Time T1 = 50, and Exercise Time Is T = 50. Strike

“Moderate” power

BRS

“Spiky” power

BRS

9 10 11 12 13

1.25 0.68 0.33 0.15 0.06

1.25 0.67 0.32 0.13 0.05

1.43 0.90 0.53 0.30 0.17

1.31 0.75 0.39 0.18 0.08

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approximation is resulting from the Black-76 formula. We observe that there is hardly any difference in the “moderate” scenario, in line with the fact that the BRS approximation is fitting the power forward distribution extremely well in this situation. However, when we let the mean–reversion increase and the spot volatility σ1 be very high, there is a significant effect on option prices. We know that the BRS approximation underestimates the tails of the power forward distribution. This results in an underestimation of the option prices when using the BRS approximation and the Black-76 formula. From Table 5.1, we observe that options prices are indeed lower for the BRS approximation. Already for an in-the-money option with strike K = 9, the price is about 9% understimated by the BRS approximation. Looking at the extreme K = 13, we have close to 50% underestimation of the true price. Hence, the BRS approximation may be seriously flawed in situations with a very steep volatility curve. We also know from the analysis of the power forward and its BRS approximation that the covariance structure between different contracts are not matching. In fact, the BRS approximation is also underestimating this. We consider an example of a calendar spread option, where we price options based on the difference between two power forwards with different delivery periods. For the BRS approximation, we have a Margrabe Formula for the price (see Margrabe [19]): Proposition 5.5. The price at time t of a call option on the difference FtBRS (T1 , T2 ) − FtBRS (T3 , T4 ) with exercise time T, where min(T1 , T2 ) ≥ T ≥ t, strike 0, and interest rate r is given by Pt (T, r) = e−r(T −t) {FtBRS (T1 , T2 )(d + t (T )) − FtBRS (T3 , T4 )(d)} where ln(FtBRS (T1 , T2 )/FtBRS (T3 , T4 )) − 0.52t (T ) , t (T )
T

2 BRS 2 t (T ) = (T3 , T4 ) ds, s (T1 , T2 ) − BRS s d=

t

and  is the standard normal cumulative probability distribution function. Proof. We include the proof for convenience. Let for simplicity t = 0, and factorize out by FTBRS (T3 , T4 ) from the price to get E[max(FTBRS (T1 , T2 ) − FTBRS (T3 , T4 ), 0)]   BRS  ˜ FT (T1 , T2 ) dQ BRS − 1, 0 = F0 (T3 , T4 )E max BRS dQ FT (T3 , T4 )

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with ˜ dQ = exp dQ




T 0

BRS (T3 , T4 )dWs − s

1 2




T 0

 2 (BRS ) (T , T )ds . 3 4 s

Appealing to the Girsanov Theorem, we have that ˜ t = dWt − BRS (T3 , T4 )dt dW t ˜ is a Q-Brownian motion on [0, T ]. Now, since 


2 F0BRS (T1 , T2 ) 1 T BRS FTBRS (T1 , T2 ) (T3 , T4 ) ds = · exp − s (T1 , T2 ) − BRS s BRS BRS 2 0 FT (T3 , T4 ) F0 (T3 , T4 ) 
T BRS ˜ , BRS (T , T ) −  (T , T )d W + 1 2 3 4 s s s 0

the result follows by a straightforward calculation using well-known properties of the normal distribution. In our numerical example, we consider two call option contracts. The first is a call on the spread of two weekly contracts, one delivering from T1 = 50 to T2 = 55, and the other over the time interval T1 = 70 to T2 = 75. The second is a call on the spread of a weekly contract and a monthly, where the weekly is as the first power forward above, and the monthly start delivering at T3 = 50, and ends at T4 = 70. The two options are supposed to mimic a spread on power forward delivering far apart, and a spread of two overlapping power forwards. We consider an exercise time T = 50, the beginning of delivery of the first weekly contract. Option prices are calculated in both the “moderate” and “spiky” scenarios described above. The BRS approximation yields analytical prices by the Margrabe formula in Proposition 5.5, while the “true” prices based on power forwards must be simulated by Monte Carlo. We used 100,000 outcomes, a figure that gave sufficient accuracy for our numerical example. The results are reported in Table 5.2. It is hard to detect any significant difference between the real prices and the approximative ones in any of the two markets scenarios. Checking various other exercise times T < 50 did not change the conclusions. Hence, although there is a difference in the covariance structure between the BRS approximation and the power forward, this does not seem to have any significant effect on calendar spreads. Due to the analyticity of the calendar spread option price based on the BRS approximation, this is much more attractive to use in practice since Monte Carlo is very slow. There may be other specifications of the volatility structure that indeed gives significant mispricing of calendar spread options when using the BRS approximation. Inspecting the numbers in Table 5.2 closer, we observe that in all cases except the “spiky” market scenario with two weekly contracts, the BRS approximation gives

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On Forward Price Modeling in Power Markets Table 5.2 Calendar Spread Option Prices on Power Forward and Its BRS Approximation. The Exercise Time is T = 50.

“Moderate” “Spiky”

Week-week power

BRS

Week-month power

BRS

0.22 0.72

0.22 0.73

0.10 0.46

0.10 0.46

an underestimation compared to the “true” price. This understimation is very small, and one must be careful since the power forward spread prices are based on Monte Carlo. However, this is a sign that there may exist specifications, where the BRS approximation gives wrong prices.

5.5. CONCLUSION In this paper, we have analyzed the distributional properties of power forward contracts. Power forward contracts deliver electricity over a specified period, and can be viewed as a portfolio of forwards with maturity at each time instant in the delivery period. We have investigated the implied power forward dynamics from a geometric Brownian motion specification of the forward price. The implied dynamics is not in general a geometric Brownian motion, however, as it turns out, can be reasonably well approximated by one. We show that there exist upper and lower bounds in terms of geometric Brownian motions for the power forwards. We have spent most of our efforts in this paper to consider the BRS approximation, where the power forward dynamics is approximated by a geometric Brownian motion, with a volatility specified as the average of the fixed-delivery forward volatility. Both, analytically and emprically, the approximation works very well. However, if the market has very strong mean–reversion and excessively high volatility (as is the case for many electricity markets), the BRS approximation may underestimate significantly the tails of the power forward distribution. We show that this may have consequences in risk management and option pricing, where the BRS approximation will seriously understimate the true risk and price. On the other hand, it works excellently in more moderate situations. There are also signs of mis-representation of the true covariance structure between power forward contracts by the BRS approximation. By looking at various calendar spread options, it seems that this is not very significant in practice. However, the BRS approximation allows for an analytical Margrabe formula to price calendar spread options, which is far superior in efficiency compared to the Monte Carlo method which must be used to price spread options on power forwards.

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For basket option pricing, one is frequently using moment matching techniques to fit a lognormal variable to a sum of lognormals. This approach can also be used for power forwards, and provides an explicit representation of the approximating volatility. However, this moment-matched volatility is in general very complicated and requires numerical integration to be evaluated, whereas the BRS volatility is analytically available. Also, in many situations, the BRS volatility is a reasonable approximation of the moment-matched volatility. Hence, the BRS approximation is to be preferred among the two. In this paper, we have only treated the case of one driving Brownian motion for the dynamics. In practice, the dependence structure among different power forwards is far more complicated (see Benth et al. [6]), and more factors are called for. Many of the calculations above can be easily extended to cover such models. However, another feature that should be investigated is the non-normality of returns of power forwards. Empirical analysis of power forward prices has shown that the log returns are significantly heavy tailed (see Benth and Koekebakker [4]). Indeed, as shown in Frestad et al. [12], the normal inverse Gaussian (NIG) distribution fits the log returns on power forwards with delivery periods ranging from weekly to yearly extremely well. The four-parameter NIG distribution was introduced in finance by BarndorffNielsen [2], and has later proved to be a suitable distribution class to model the log returns of financial data. It has the normal distribution as a limiting case, and Frestad et al. [12] demonstrate statistically that the estimated NIG distributions of the power forward contracts are far from normal. To model dynamics, matching these statistical observations requires the use of Lévy processes. To indicate how such an extension could be achieved under the HJM paradigm in our context, one could state an exponential Lévy dynamics for the forward process ft (u) as follows: ft (u) = f0 (u) exp(γt (u)Lt ).

(5.19)

Here, Lt is a Lévy process with NIG marginals, say. The maturity dependence of ft (u) is modeled using the deterministic scaling function γt (u), which plays much the same role as the volatility function σt (u). To have a risk–neutral dynamics, we must require that E[exp(γt (u)Lt )] = 1, for all t ≥ 0 and u ≥ t. The naive BRS approximation in this case would be to consider FtBRS (T1 , T2 ) = F0BRS (T1 , T2 ) exp( BRS (T1 , T2 )Lt ), t with




BRS (T1 , T2 ) = t

T2

w(u, T1 , T2 )γt (u)du. T1

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We immediately spot a first problem in having a martingale dynamics of FtBRS (T1 , T2 ). This and other questions on the validity of such an approximation are left for future studies.

References [1] Asmussen, S and L Rojas-Nandayapa (2006). Sums of dependent lognormal random variables: Asymptotics and simulation. Research Report, Thiele Center, University of Aarhus. [2] Barndorff-Nielsen, OE (1998). Processes of normal inverse Gaussian type. Finance and Stochastics, 2(1), 41–68. [3] Benth, FE and PN Henriksen (2008). Pricing basket options using univariate normal inverse Gaussian approximation. Statistical Research Report, No. 3, Department of Mathematics, University of Oslo. To appear in Journal of Forecasting. [4] Benth, FE and S Koekebakker (2008). Stochastic modeling of financial electricity contracts. Energy Economics, 30(3), 1116–1157. [5] Benth, FE, S Koekebakker and F Ollmar (2007). Extracting and applying smooth forward curves from average-based commodity contracts with seasonal variation. Journal of Derivatives, 15(1), 52–66. [6] Benth, FE, J Šaltyt˙e Benth and S Koekebakker (2008). Stochastic Modeling of Electricity and Related Markets, World Scientific. [7] Black, F (1976). The pricing of commodity contracts. Journal of Financial Economics, 3, 167–179. [8] Bjerksund, P, H Rasmussen and G Stensland (2000). Valuation and risk management in the Nordic electricity market. Working Paper, Institute for finance and management sciences, Norwegian School of Economics and Business Administration (NHH). [9] Clewlow, L and C Strickland (2000). Energy Derivatives: Pricing and Risk Management. Lacima Publications. [10] Eydeland, A and K Wolyniec (2003). Energy and Power Risk Management. John Wiley & Sons. [11] Fleten, SE and J Lemming (2003). Constructing forward price curves in electricity markets. Energy Economics, 25, 409–424. [12] Frestad, D, FE Benth and S Koekebakker (2007). Modelling the term structure dynamics in the Nordic electricity swap market. To appear in Journal of Energy. [13] Geman, H (2005). Commodities and Commodity Derivatives. Wiley-Finance, John Wiley and Sons. [14] Henriksen, PN (2008). Lognormal moment matching and pricing of basket options. Statistical Research Report, No. 2, Department of Mathematics, University of Oslo. [15] Hoedemakers, T (2005). Modern Reserving Techniques for Insurance Business. PhD Thesis, Katholieke Universiteit Leuven. [16] Karatzas, I and S Shreve (1991). Brownian Motion and Stochastic Calculus. Springer Verlag. [17] Kiesel, R., G Schindlmayer and R Börger (2009). A two-factor model for the electricity forward market. Quantitative Finance, 9(3), 279–287. [18] Levy, E (1992). Pricing European average rate currency options. Journal of International Money Finance, 11(5), 474–491.

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[19] Margrabe, W (1978). The value of an option to exchange one asset for another. Journal of Finance, 33, 177–187. [20] Protter, Ph (1990). Stochastic Integration and Differential Equations. Springer Verlag. [21] Zacks, S and CP Tsokos (1978). The distribution of sums of dependent log-normal variables. Technical Report No. 31, Defense Technical Information Center.

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6

PRICING CERTIFICATES UNDER ISSUER RISK

BARBARA GÖTZ∗,‡ , RUDI ZAGST∗,§ and MARCOS ESCOBAR†,¶ ∗

HVB-Stiftungsinstitut für Finanzmathematik, Technische Universität München, Boltzmannstrasse 3, 85748 Garching, Germany † Department of Mathematics, Ryerson University, 350 Victoria St. Toronto, M5B 2K3, Ontario, Canada ‡ [email protected]; [email protected] § [email protected]; [email protected] ¶ [email protected]

Certificates have become very popular in Germany, Austria, and Switzerland in the last few years. From a technical and legal point of view they are bonds. Thus, their value actually also depends on the rating and creditworthiness of the issuing company. This aspect is in general neglected in the pricing of these products. In the following, we present a model which overcomes this lack and incorporates the default risk of the issuing company in the pricing. We derive closed-form expressions for index, basket, and bonus certificates under issuer risk in a Black–Scholes model framework. The results are analyzed for different scenarios and compared with valuations in a model which neglects issuer risk.

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6.1. INTRODUCTION Certificates have become very popular with retail customers in Germany, Austria, and Switzerland in the last few years. They are referred to as structured products. The simplest structures are index certificates, which allow a retail customer to invest in an index like the German stock index Deutscher Aktien Index (DAX) or the Dow Jones Euro Stoxx 50. From a legal point of view, those products are bonds and the investors are, thus, creditors of the respective issuer. This rather technical aspect has been disregarded in the investment decision by many retail investors. Even the value at risk figures which are sometimes indicated by banks to show the investors the risk involved in single certificates are based on the assumption of a non-defaultable issuer. However, in the case of an insolvency of the issuer, the investor may lose his total investment regardless of the performance of the underlyings of the certificate. The case of Lehman Brothers shows that this risk can materialize. Certificates differ in this feature from an investment in funds. The investment in a fund is a so-called special property and is not affected at all by the rating and creditworthiness of the issuing company. Issuer risk is the risk of loss on securities and other tradeable obligations because the issuer does not fulfil his contractual obligations due to his insolvency. Up to today this kind of risk has been hardly addressed in the pricing of exotic securities and especially not from a retailer’s perspective but only in connection with regulatory capital requirements like in Basel II. More details about modeling and evaluating counterparty/issuer risk under an economic or regulatory perspective can be found in [1–3]. Pricing securities under counterparty risk can be traced back to Merton [4]. Johnson and Stulz [5] analyzed the counterparty risk in option pricing. They used a firm value model and assumed that the vulnerable option presents the single debt of the company. A huge increase in the derivative’s value, thus, rises the risk of default of the company. This approach is only appropriate when the derivative is the only or the predominant source of funding of the counterparty. Hull and White [6] as well as Jarrow and Turnbull [7] value so-called vulnerable options, options on a bond written by a defaultable party, in a reduced-form model assuming independence between the credit risk of the counterparty and the asset underlying the derivative. Cherubini and Luciano, see [8,9], suggest to use a copula approach to value the counterparty risk in an investment and to allow for a dependence structure between term-structure movements and a default of one of the parties. Klein [10] as well as Klein and Inglis [11] choose a firm-value model to account for the issuer risk and to model the dependencies between the issuing firm and the underlying. We follow their approach in that regard, and condition the payoff of the certificate on the survival of the issuer: the certificate only pays back the total investment

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and gain as long as the issuer has not defaulted, i.e., its asset value has not fallen under a certain barrier. Like Klein [10] and Klein and Inglis [11], we model the correlation between the assets of the issuer and the asset underlying the derivative explicitly. The barrier is exponentially increasing in time, and the issuer can default, anytime before maturity. In the case of default, the investor recovers a constant fraction of the market value of his investment. As we deal with retail products, we furthermore assume that the exotic structures are fully hedged, i.e., all debt owed to the investor has to be seen alongside assets which the company owns. In the case of default, however, these assets do not cover the investors but are part of the insolvency estate. This allows us to assume the boundary as deterministically increasing rather than stochastic, see [11].

6.2. THE MODEL The model formulation is influenced by the CreditGrades framework, for details see [12, 13]. This approach allows us to derive closed-form expressions for index, participation guarantee, bonus guarantee, discount, and bonus certificates under issuer risk in a Black–Scholes framework. The system of processes is defined on a filtered probability space (
, F, F, P). We assume the existence of an equivalent martingale measure. As an immediate consequence, the market is arbitrage-free. The processes are directly formulated under the martingale measure Q. We consider an issuing company i = 1 with an asset value per share at time t of V1,t , an equity price S1,t , and a total debt per share of D1,t . The firm’s value dynamics follow a geometric Brownian motion dV1,t = (rt − d1,t )dt + σ1 dW1 , V1,t

V1,0 = S1,0 + D1,0 .

(6.1)

The company’s debt is deterministic and yields a continuous interest of rs − d1,s .


t D1,t = D1,0 e 0 (rs −d1,s )ds ,

(6.2)

where rs is the risk-free interest rate and d1,s is the dividend yield on the company’s assets. Both processes, rs and d1,s , are assumed to be deterministic. The issuing company defaults if its value falls below the barrier D1,t . Thus, the time of default is defined by the stopping time τ τ = inf{t ∈ (t0 , T ] : V1,t < D1,t }. We denote the equity per share by  S1,t = V1,t − D1,t , S1,t = 0

otherwise.

if τ > t

and,

(6.3)

(6.4)

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This implies that default occurs whenever the stock price S1,t falls to zero. As soon as S1,t reaches zero, it remains there. In this framework, an European option is seen as the corresponding down-and-out barrier option with an absorbing barrier for S1,t set at 0. Hence, an equivalent time to default to τ is ς = inf{t ∈ (t0 , T ] : S1,t ≤ 0}.

(6.5)

The dynamics for S1,t are, prior to default, found by applying Itô to (6.4) and are given by dS1,t = S1,t (rt − d1,t )dt + (D1,t + S1,t )σ1 dW1 .

(6.6)

The stock price of the issuer follows — prior to default — a shifted log-normal distribution. This distribution implies negative stock prices with positive probability, see [13] for more details. The higher the leverage (debt-to-equity ratio), the higher is the probability of default and vice versa. The dynamics of the underlying assets of a certificate are modeled by geometric Brownian motions. The following system is considered to describe the assets of the issuer and the underlying assets of the derivatives: dS1,t = S1,t (rt − d1,t )dt + (D1,t + S1,t )σ1 dW1

(6.7)

dSi,t = Si,t (rt − di,t )dt + Si,t σi dWi

(6.8)

dWi , dWj  = ρij dt,

for i = 2, . . . , n

for i, j ∈ 1, . . . , n, i = j, −1 ≤ ρij ≤ 1,

where σi and ρij are constants, see [14].

6.3. PRICING OF CERTIFICATES UNDER ISSUER RISK 6.3.1. Building Blocks The most popular certificates are composed from simple building blocks such as zerocoupon bonds Zt , investments in an underlying S2,t , call options Ct (S2 , K), and digital options Cd,t (S2 , K) as well as knock-out put options Pk,t (S2 , K, B). The formulas for these components are derived for the case that the issuer can default and has a recovery rate of zero, for a similar model see [15], page 635ff. In the case of no default, the valuation of the building blocks is well known and will be provided for the purpose of completeness. By means of these building blocks defaultable index, guarantee, and bonus certificates can be valued for any assumption of the recovery rate. The proofs for the results in this section will be provided in the Appendix. A main component of index certificates and discount certificates is the investment in the underlying S2 , which is actually worth S2,t at time t, when we assume that the issuer cannot default.

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Proposition 6.1. In the case when the investment in the equity is guaranteed by an D of this investment at issuer who is defaultable with recovery rate zero, the price S2,t time t is given by D = S2D (S1 , S2 , t) S2,t

= e−


T t

rs ds+ατ+β1 y1 +β2 y2 − 12 χ(δ21 +2ρδ1 δ2 +δ22 ) √ χ

× (e(δ1 ψ1 +δ2 ψ2 ) −e

√ (δ1 ξ1 +δ2 ξ2 ) χ

N2 (ψ1 , z2 , ρ)

N2 (ξ1 , z2 , ρ)),

(6.9)

where N2 (x, y, ρ) is the standard bivariate normal distribution function with correlation ρ,  
T S1,t + D1,t , y2 = ln(S2,t e t (rs −d2,s )ds ), y1 = ln D1,t τ = T − t,  σ2 y1 , z1 = σ1 σ1 − ρσ2 β1 = , 2σ1 (1 − ρ2 )  σ1 β1 , δ1 = − σ2

χ = σ1 σ2 τ,  σ1 z2 = y2 , σ2 σ2 − ρσ1 β2 = , 2σ2 (1 − ρ2 )  σ2 δ2 = (1 − β2 ), σ1

1 σ12 − 2σ1 σ2 ρ + σ22 , 4 2(1 − ρ2 ) z1 √ ψ1 = √ + χ(δ1 + ρδ2 ), χ

z2 √ ψ2 = √ + χ(δ2 + ρδ1 ), χ

α=−

z1 2ρz1 − z2 √ √ + χ(δ2 + ρδ1 ). ξ1 = − √ + χ(δ1 + ρδ2 ), ξ2 = − √ χ χ For a proof see A.1. Products, which ensure a repayment of an investment with notional 1, are internally stripped into a zero-coupon bond and several derivatives.
T As known, a non-defaultable zero-coupon bond is priced by Zt = exp(− t rs ds). Whereas when we assume that the issuer can default and — if he defaults — with no recovery, the value of the zero-coupon bond can be computed by, see [13] ZtD = e−


T t

rs ds

(N (d3 ) − ex1 N (d4 )),

with N (.) being the standard cumulative normal distribution function,   S1,t + D1,t x1 x1 1√ 1√ x1 = ln , d3 = √ − χ, d4 = − √ − χ. D1,t χ 2 χ 2

(6.10)

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When the issuer additionally promises a fixed interest rI on the investment this is val
T D = ued, assuming no default, by ZI,t = (1 + rI ) exp(− t rs ds) and accordingly by ZI,t
T x1 exp(− t rs ds)(1+rI )(N (d3 )−e N (d4 )) in a world with defaults and zero recovery. Call options are introduced in the structured products to leverage the investment and to grant the investor a high participation rate in the value of the underlying. If no default of the issuer is assumed the call option with strike K is priced by the well
T known Black–Scholes formula Ct (S2 , K) = S2,t N (d1 ) − K exp(− t rs ds)N (d2 ), √ √ where d1 = (ln(S2,t /K) + (r + 1/2σ22 )(T − t))/(σ T − t) and d2 = d1 − σ T − t. Proposition 6.2. The price CtD at time t of a defaultable call option, when the payment of the call option is guaranteed by an issuer who is defaultable with recovery rate zero, is given by CtD (S1 , S2 , K) = Ke−


T t

rs ds+ατ+β·X

√ (δ1 ν1 +δ2 ν2 ) χ

−e

1

−e

− − − √ (δ− 1 ν1 +δ2 ν2 ) χ

where

 x1 = ln 

2

2

N2 (η1 , η2 , ρ)

N2 (ν1 , ν2 , ρ))

2 − − −2 − 12 χ(δ− 1 +2ρδ1 δ2 +δ2 )

−e

√ χ

(e− 2 χ(δ1 +2ρδ1 δ2 +δ2 ) (e(δ1 η1 +δ2 η2 )

− −

− −

√ χ

(e(δ1 η1 +δ2 η2 )

− N2 (η− 1 , η2 , ρ)

N2 (ν1− , ν2− , ρ))),

 S1,t + D1,t , D1,t

σ2 x1 , σ1  σ1 − δ1 = − β1 , σ2 z1 √ η1 = √ + χ(δ1 + ρδ2 ), χ

z1 =

z1 √ ν1 = − √ + χ(δ1 + ρδ2 ), χ z1 √ − χ(δ1 + ρδ− η− 1 = √ + 2 ), χ

 x2 = ln 

(6.11)

 S2,t
T (rs −d2,s )ds , et K

σ1 x2 , σ2  σ2 − δ2 = − β2 , σ1 z2 √ η2 = √ + χ(δ2 + ρδ1 ), χ z2 =

2ρz1 − z2 √ + χ(δ2 + ρδ1 ), √ χ z2 √ − η− χ(δ2 + ρδ− 2 = √ + 1 ), χ

ν2 = −

z1 2ρz1 − z2 √ − √ − − + χ(δ2 + ρδ− ν1− = − √ + χ(δ− √ 1 + ρδ2 ), ν2 = − 1 ), χ χ for β1 , β2 , δ1 , and δ2 refer to Proposition 6.1. For a proof see A.2. Similarly, the non-defaultable digital option with strike K is priced
T by Cd,t (S2 , K) = exp(− t rs ds)N (d2 ).

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D Proposition 6.3. The price Cd,t at time t of a defaultable digital call option where the payment of one is guaranteed by an issuer who is defaultable with recovery rate zero is given by D Cd,t (S1 , S2 , K)

= e−


T t

− − − rs ds+ατ+β1 x1 +β2 x2 − 12 χ(δ− 1 +2ρδ1 δ2 +δ2 ) 2

− −

√ χ

− −

− e(δ1 ν1 +δ2 ν2 )

2

− −

− −

√ χ

(e(δ1 η1 +δ2 η2 )

− N2 (η− 1 , η2 , ρ)

N2 (ν1− , ν2− , ρ)),

(6.12)

with parameters explained in Propositions 6.1 and 6.2. For a proof see A.3. The bonus certificate includes, beside other derivatives, a knockout put option. The price of a non-defaultable knock-out put option with barrier B and strike K is given by, see also [16], Pk,t (S1 , S2 , K, B) −


T



B K



2β

2 ln

B

− ln

 S2 







− β σ2 (T − t) √ σ2 T − t

B    2(β−1) 
T 2 ln K − ln SK2 B − t d2 ds  N + Se + (1 − β )σ2 (T − t) , √ K σ2 T − t

= Pt (S2 , K) − Ke

t

rs ds

N

K

K

(6.13) where


T

Pt (S2 , K) = Ke− t rs ds N (−d2 ) − S2,t N (−d1 ),     S2 B ∗ x2 = ln , b = ln , K K β =

1 r − d2 , − 2 σ22

1 2 α = − σ22 β , 2

χ = σ22 τ and d1 , d2 as in Proposition 6.2. D at time t of a defaultable knock-out put option, when Proposition 6.4. The price Pk,t the payoff is guaranteed by an issuer who is defaultable with recovery rate zero, is given by
T

∗





∗

D (S1 , S2 , K, B) = Ke(− t rs ds+α τ+β1 x1 +β2 x2 ) Pk,t  √ 2 ∗  ∗  ∗ × h(τ, z∗1 , z∗2 , y1∗ , y2∗ ) max(e−β1 (σ1 1−ρ y1 +σ1 ρy2 )−β2 (y2 σ2 +b)  −β1 (σ1

−e

√

1−ρ2 y1∗ +σ1 ρy2∗ )+(1−β2 )(y2∗ σ2 +b)

, 0)dy2∗ dy1∗ ,

(6.14)

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where

 x1 x∗ − b x∗ − b , z∗2 = 2 −ρ 2 , σ2 σ2 1 − ρ2 σ1   1 σ σ − ρ 12 σ22 − (r − d2 ) 1 σ2 − ρσ1 r − d2 2 1 2  , β2 = − , β1 = 2 2 (1 − ρ )σ1 σ2 2 (1 − ρ )σ2 (1 − ρ2 )σ22   1 1 2 1 1 2 2 α∗ = − σ12 β1 − σ2 − (r − d2 ) β2 + σ12 β1 + σ22 β2 + ρσ1 σ2 β1 β2 . 2 2 2 2 z∗1 =

1



h(τ, z∗1 , z∗2 , y1∗ , y2∗ ) is given in Appendix A.4.

6.3.2. Index Certificates In this section, we price an index certificate with price It at time t under issuer risk. As stated before, the index certificate allows a retail investor to invest in a single stock or stock index with price S2,t at time t. He fully participates in any movement of the underlying. The index certificate does not provide any protection against a decline of the underlying. Internally this certificate is hedged by buying the respective index or single stock. Taking into consideration the issuer risk and assuming a constant recovery rate R in the case of default of the issuer, the index certificate can be valued by
T

(6.15) I(t, S1 , S2 ) = S2,t − (1 − R)e− t rs ds EQ S2,T 1{ς≤T } | Ft , where ς is given in (6.5). The index certificate can be valued by using the building blocks of Proposition 6.1. Thus, the certificate is priced by the following formula D I (t, S1 , S2 ) = RS2,t + (1 − R) S2,t (S1 , S2 ).

(6.16)

The number of parameters rises considerably if the notion of issuer risk is incorporated in the pricing of derivatives. In the following, we show the results of the valuation of the index certificate in different scenarios. The results are compared to prices which neglect issuer risk. For all the scenario computations in this chapter, we have chosen the following instrument parameterizations if not stated differently in the respective examples: • • • • • • •

Stock prices of the issuer and the underlying S1 = S2 = 100 in t = 0 Volatilities of issuer and underlying σ1 = σ2 = 0.4 Correlation between underlying and issuer ρ = 0.3 Risk-free rate of return r = 0.04 Recovery rate R = 0.4 Debt endowment D0 = 50 in t = 0 Maturity of the respective options T = 4

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Pricing Certificates Under Issuer Risk Index certificate 110 Defaultable Non-defaultable 100

Value

90

80

70

60

50

40

0

200

400

Figure 6.1

600 Debt level

800

1000

1200

Impact of debt.

Figure 6.1 shows that for increasing debt of the issuing company the value of the index certificate declines and finally approaches RS2,t . The price of the non-defaultable certificate stays at par regardless of the debt level. This shows that for a retail investor a simple comparison of the prices of index certificates of different issuers is not appropriate in order to find the security with the best price-performance ratio. For his investment decision, the investor has to take the rating and equity to debt ratio into consideration. In Fig. 6.2, the impact of the volatility of the issuer’s asset (the volatility of the underlying is kept to 0.4) and the correlation between the issuer’s asset and the underlying is analyzed. For that reason, we choose a relatively low debt scenario. The negative relationship between the volatility of the issuer’s assets and the price is clearly visible. The probability of a default considerably increases in high volatility scenarios and thus the price decreases. It strikes that the impact of the correlation on the price grows with the volatility: for a very high volatility, a correlation near 1 leads to a considerably higher value than a correlation near −1. In a low volatility and low debt scenario with D0 = 50, the price is not much affected by the level of the correlation as the probability of default is relatively low. However, in a high volatility and low debt scenario with, thus, higher probability of default of the issuer, both assets tend to move in the same direction if the correlation rises. Thus, in contrast to a negative correlation, the probability that the issuer survives and the underlying asset features a positive return goes up. A negative correlation results in a different performance of

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100 90

Value

80 70 60 50 40 0.7

1 0.5 0.6

0

0.5

0.4

0.3

−0.5 0.2

Volatility S1

−1

Correlation

Figure 6.2 Impact of volatility and correlation.

the issuer’s assets and the underlying. Thus, the probability goes up that the certificate matures at a low value or worthless.

6.3.3. Participation Guarantee Certificates In contrast to the index certificate, the participation guarantee certificate offers the investor some risk protection against a decrease of the underlying. However, this protection is financed by a limited profit in cases when the underlying increases. The investor participates in any positive performance of the underlying on the basis of the so-called participation rate. This structure is built up by long positions in a zero-coupon bond with a standardized notional of 1 and stock options whereas the number of stock options is determined by the level of the participation rate. Taking into consideration the issuer risk, the price is indicated by PG(t, S1 , S2 ) = e−


T t

rs ds

(EQ [(1 + p max[S2,T − K, 0]) | Ft ]

− (1 − R)EQ [(1 + p max[S2,T − K, 0])1{ς≤T } | Ft ]),

(6.17)

where ς is as described in (6.5), p is the participation rate of the contract, and K describes the price level where the participation starts. This means that below this

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point the investor does not profit from any increase in the stock price. A defaultable participation guarantee certificate of an issuer with recovery rate R is, thus, priced by PG(t, S1 , S2 ) = R(Zt + pCt (S2 , K)) + (1 − R)(ZtD (S2 ) + pCtD (S1 , S2 , K)).

(6.18)

For the participation guarantee certificate, we compute some scenario values. For these computations, we choose a notional of 88, a start of the participation of K = 100, and a participation rate of p = 50%. In Fig. 6.3, we find the impact of the debt level on the participation guarantee certificate similar to that of the index certificate: as before the price falls especially steeply for smaller D0 levels. In Fig. 6.4, the joint impact of the issuer’s volatility and the correlation is shown. The form of the graphs resembles the graph of Fig. 6.2 and the same explanations apply to explain the form. The effect of the correlation on the price of the participation guarantee certificate is, however, clearly less distinct because the total price of the certificate considerably depends on the value of the zero-coupon bond for which the correlation is irrelevant.

Participation guarantee certificate 100 Defaultable Non-defaultable 90

Value

80

70

60

50

40

0

200

400

Figure 6.3

600 Debt level

800

Impact of debt.

1000

1200

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95 90

Value

85 80 75 70 65 60 0.7

1 0.6

0.5 0

0.5

0.4

0.3

0.2

−1

Volatility S1

−0.5 Correlation

Figure 6.4 Impact of volatility and correlation.

6.3.4. Bonus Guarantee Certificates When investing in a bonus guarantee certificate, one does not directly participate in fluctuations of the value of the underlying. Rather, one receives a fixed interest rate and additionally a bonus payment if the underlying is above the bonus barrier at maturity. The investor could compose this payoff by buying a zero-coupon bond and a digital option: BG(t, S1 , S2 ) = e−


T t

rs ds

(EQ [(1 + rI + 1{S2,T >K} ) | Ft ]

− (1 − R)EQ [(1 + rI +  1{S2,T >K} )1{ς≤T } | Ft ]),

(6.19)

where ς is as in (6.5), rI the basic interest,  the bonus payment rate described in the contract, and K the bonus barrier. In our framework, the risk–neutral price of the bonus guarantee certificate is given by BG(t, S1 , S2 ) = R(ZI,t + Cd,t (S2 , K)) D D + (1 − R)(ZI,t (S2 ) + Cd,t (S1 , S2 , K)).

(6.20)

To show the impact of debt, volatility, and correlation, we structure a bonus guarantee certificate with notional 88, bonus barrier K = 120, basic interest rI = 3.0%, and bonus payment rate = 3.4%.

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Pricing Certificates Under Issuer Risk Bonus guarantee certificate 80 Defaultable Non-defaultable

75 70 65

Value

60 55 50 45 40 35 30

0

200

400

Figure 6.5

600 Debt level

800

1000

1200

Impact of debt.

For the bonus guarantee certificate, we observe the same typical feature of the graph, which shows prices for different debt levels, see Fig. 6.5, as for the other examples: the graph decreases sharply for lower debt levels and approaches R times the price of the analogous certificate of a non-defaultable issuer. In Fig. 6.6 the relationship between price and volatility is of nearly linear kind: the higher the volatility, the lower the price the investor has to pay. The impact of the correlation is less distinct as in the case of index certificates due to the fact that a major part of the value of the bonus guarantee depends on the zero-coupon bond.

6.3.5. Discount Certificates The risk protection of a discount certificate consists in a risk buffer: the investor buys the certificate at a discount on the actual value of the underlying. This risk limitation is again financed by a gain limit. The structure can be hedged by investing in the underlying and writing a call option, i.e., the value is specified by
T

DC(t, S1 , S2 ) = S2,t − Ct (S2 , K) − (1 − R)e− t rs ds  

× EQ S2,T − max[S2,T − K, 0] 1{ς≤T } | Ft ,

(6.21)

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75

Value

70

65

60

1 0.5

55 0.7

0 0.6

0.5

0.4

−0.5 0.3

Volatility S1

0.2

−1

Correlation

Figure 6.6 Impact of volatility and correlation.

where ς is as in (6.5). The discount certificate can be valued by the following formula DC(t, S1 , S2 ) = R(S2,t − Ct (S2 , K)) D + (1 − R)(S2,t (S1 , S2 ) − CtD (S1 , S2 , K)).

(6.22)

We value a discount certificate with K = 120 in different scenarios. With respect to the debt level, the value of the discount certificate does not differ in its characteristics from the certificates analyzed before, see Fig. 6.7. In Fig. 6.8, we look into the dependence of the price on the issuer’s volatility and the correlation. Regarding the volatility, we see the known structure, i.e., the price falls when the volatility increases. The slopes are similar for all correlation scenarios. As the certificate consists of building blocks clearly dependent on the correlation, the impact of the correlation on the value of the certificate is similar to the index certificate example.

6.3.6. Bonus Certificates The investor in this certificate is protected from a decline of the underlying up to a certain point, the protection barrier B. Below this point, the investor fully participates in any fluctuations of the underlying. The same is true for the performance of the underlying beyond the bonus barrier K. The certificate can be fully hedged by an

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Pricing Certificates Under Issuer Risk Discount certificate 75 Defaultable Non-defaultable

70 65

Value

60 55 50 45 40 35 30

0

200

400

600 Debt level

Figure 6.7

800

1000

1200

Impact of debt.

Discount certificate

75 70 65

Value

60 55 50 45 40 35 0.7

1 0.5 0.6

0.5

0 0.4

Volatility S1

0.3

0.2

−1

−0.5 Correlation

Figure 6.8 Impact of volatility and correlation.

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investment in the underlying and by buying a knock-out put option. The respective barrier level is the protection barrier. When the issuer risk is incorporated in the pricing model, the price is specified by BC(t, S1 , S2 ) = S2,t + e−


T t

rs ds

− (1 − R)e−

EQ [max[K − S2,T , 0]1{τ2 >T } | Ft ]


T t

rs ds

EQ [(S2,T + max[K − S2,T , 0]1{τ2 >T } )1{ς≤T } | Ft ], (6.23)

where τ2 = inf(t ∈ (t0 , T ] : S2,t ≤ B),

(6.24)

where ς is as in (6.5). The following formula evaluates the payoff: BC(t, S1 , S2 ) = R(S2,t + Pk,t (S2 , K, B))  D  D + (1 − R) S2,t (S1 , S2 ) + Pk,t (S1 , S2 , K, B) .

(6.25)

Finally, we show some exemplary computations of the bonus certificate. We assumed a protection barrier B = 80 and a bonus barrier K = 120 for our computations. Not surprisingly, the graph, i.e., Fig. 6.9, plotting the debt level of the issuer against the value of the bonus certificates shows the same features as in the examples above. Bonus certificate 110 Defaultable Non-defaultable 100

Value

90

80

70

60

50

40

0

200

400

Figure 6.9

600 Debt level

800

Impact of debt.

1000

1200

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Pricing Certificates Under Issuer Risk Bonus certificate

100 90

Value

80 70 60 50 1

40 0.5 30 2.5

0 2

1.5

1

−0.5 0.5

0

−1

Volatility S1

Correlation

Figure 6.10 Impact of volatility and correlation.

In Fig. 6.10, we see a strong impact of correlation in high volatility scenarios and small influence in lower volatility scenarios. These characteristics can be explained by the fact that correlation between issuer and underlying has a considerable impact on the price of the building blocks of the bonus certificate.

6.4. CONCLUSION We have derived closed-form expressions for index, discount, participation guarantee, and bonus certificates under issuer risk in a Black–Scholes model framework. Our scenario computations clearly depict that, depending on the issuer’s capital soundness, a pricing formula which neglects issuer risk considerably overprices the value of the singular certificate. Thus, for a retail investor, a simple comparison of the prices of certificates of different issuers is not appropriate in order to find the security with the best price-performance ratio. For his investment decision, the investor has to take the rating and equity to debt ratio into consideration.

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A.1. PROOF OF PROPOSITION 6.1

D Proof. S2,t fulfils the following partial differential equation and boundary conditions

 D D D ∂2 S2,t ∂2 S2,t ∂2 S2,t 1 1 2   + σ22 S22 + ρσ1 σ2 (S1 + D1,t )S2 σ1 (S1 + D1,t )2   2 2  2 2 ∂S1 ∂S2 ∂S1 ∂S2     D D D  ∂S2,t ∂S2,t ∂S2,t D + (rt − d1,t )S1 + (rt − d2,t )S2 + = 0, − rt S2,t ∂S ∂S ∂t  1 2    D   S2 (t, 0, S2 ) = 0,    D S2 (T, S1 , S2 ) = S2,T . By introducing, the following transformations S2D (t, S1 , S2 ) = e−


T t

rs ds ατ+β1 y1 +β2 y2 D∗ e S2 (t, y1 , y2 ),

τ = T − t,   S1,t + D1,t y1 = ln , D1,t

χ = σ1 σ2 τ,


T

y2 = ln (S2,t e t (rs −d2,s )ds ),  σ2 z1 = y1 , σ1 1 σ1 − ρσ2 , β1 = 2 (1 − ρ2 )σ1 α=−



σ1 y2 , σ2 1 σ2 − ρσ1 β2 = , 2 (1 − ρ2 )σ2 z2 =

1 σ12 − 2σ1 σ2 ρ + σ22 , 4 2(1 − ρ2 )

the PDE can be reduced to  D∗ 1 ∂2 S2D∗ ∂2 S2D∗ 1 ∂2 S2D∗ ∂S    2 − − − ρ = 0,   2 ∂z21 2 ∂z22 ∂z1 ∂z2  ∂χ S2 (χ, 0, z2 ) = 0,       σ σ  S D∗ (0, z , z ) = (e−β1 σ12 z1 +(1−β2 ) σ21 z2 ). 1 2 2 D∗

(A.1)

This problem can be rewritten in terms of its transition probability density p(χ, z1 , z2 , z1 , z2 ). p(χ, z1 , z2 , z1 , z2 ) is a fundamental solution of the partial differential equation above and satisfies the backward Kolmogorov equation, see [17], page 368f, with initial condition p(0, z1 , z2 , z1 , z2 ) = δ(z1 − z1 )δ(z2 − z2 ),

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see [15], page 493. We apply the method of images, see [18], page 476ff, to restrict the fundamental solution to the area the problem is defined for. This method is appropriate when the region to which the solution should be bounded is highly symmetric: a solution for the free space is first derived (pF (χ, z1 , z2 , z1 , z2 )) and then restricted to the defined region via symmetry, i.e., the principle of reflection. In this case, the image point (z1 , z2 ) of the source point (z1 , z2 ) is given by (z1 , z2 ) = (−z1 , −2ρz1 + z2 ). Thus, the solution is found by p(χ, z1 , z2 , z1 , z2 ) = pF (χ, z1 − z1 , z2 − z2 ) − pF (χ, z1 + z1 , z2 + 2ρz1 − z2 ). In this case the solution for the free space is 2

pF (χ, z1 , z2 , z1 , z2 ) =

2

(z +z −2ρz1 z2 ) 1 − 1 2 e 2(1−ρ2 )χ . 2π 1 − ρ2 χ

The solution of the original Dirichlet problem can be found by computing, see [17], page 364ff, S2D∗ (χ, z1 , z2 )  =

∞



∞

−∞

0



∞

−

p (χ, z1 − F



−∞

∞

0

With

z1 , z2

−

−β1 z2 )e





σ1  z +(1−β2 ) σ2 1

pF (χ, z1 + z1 , z2 + 2ρz1 − z2 )e

 ϕ= =e

∞ −∞

 0

∞



−β1



σ2  z σ1 2

dz1 dz2 

σ1  z +(1−β2 ) σ2 1

σ2  z σ1 2

dz1 dz2 .



pF (χ, z1 , z2 )ek1 z1 +k2 z2 dz1 dz2

χ(k12 +2ρk1 k2 +k22 ) 2

√ lim N2 ( χ(k1 + ρk2 ), z2 , ρ),

z2 →∞

we get Proposition 6.1.

A.2. PROOF OF PROPOSITION 6.2

Proof. The proof is similar to the one for the investment in the underlying. By the means of the Feyman Kac Theorem, see [19], page 143ff, the following PDE can be

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derived:  2 D 1 2 2 ∂2 CD 1 2 ∂2 C D  2∂ C  (S + D ) + S + ρσ σ (S + D )S σ σ 1 1,t 1 2 1 1,t 2  1 2 2  2 2 ∂S1 ∂S2 ∂S12 ∂S22       ∂CD ∂CD ∂CD + (rt − D1,t )S1 + (rt − d2,t )S2 + − rt C D = 0, ∂S ∂S ∂t 1 2      D  C (t, 0, S2 ) = 0,     D C (T, S1 , S2 ) = max(S2 − K, 0). By applying the transformations, which have been introduced before the PDE can be reduced to  D∗ ∗ ∗ ∗ 1 ∂2 CD ∂2 C D 1 ∂2 C D ∂C   − − ρ = 0, −    ∂χ 2 ∂z21 2 ∂z22 ∂z1 ∂z2   ∗ CD (χ, 0, z2 ) = 0,          σ σ σ σ   D∗ −β1 σ1 z1 +(1−β2 ) σ2 z2 −β1 σ1 z1 −β2 σ2 z2 2 1 2 1 −e , 0). C (0, z1 , z2 ) = max(e Using the method of images on the probability density function as before the solution is given by ∗

CD (χ, z1 , z2 ) 

∞

=



0

∞

0

 −

∞

0

 −  0



∞

p (χ, z1 −

z1 , z2

p (χ, z1 +

z1 , z2

F

0 ∞

0

−

pF (χ, z1 − z1 , z2 − z2 )e

∞

F

0 ∞



∞

0

with

 0

=e





σ1  z +(1−β2 ) σ2 1

−

−β1 z2 )e

+

2ρz1

−



σ1  z −β2 σ2 1

−β1 z2 )e

pF (χ, z1 + z1 , z2 + 2ρz1 − z2 )e

∞

ϕ=

−β1

 0

∞



−β1







σ2  z σ1 2

σ2  z σ1 2

dz1 dz2

dz1 dz2 

σ1  z +(1−β2 ) σ2 1

σ1  σ2 z1 −β2





pF (χ, z1 , z2 )ek1 z1 +k2 z2 dz1 dz2

χ(k12 +2ρk1 k2 +k22 ) 2

√ √ N2 ( χ(k1 + ρk2 ), χ(k2 + ρk1 )).

σ2  σ1 z2

σ2  z σ1 2

dz1 dz2

dz1 dz2

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A.3. PROOF OF PROPOSITION 6.3

Proof. The PDE and the boundary conditions can be indicated by  2 D ∂2 CdD 1 2 2 ∂2 CdD 1 2  2 ∂ Cd  (S + D ) + S + ρσ σ (S + D )S σ σ 1 1,t 1 2 1 1,t 2  1 2 2 2 2  2 2 ∂S1 ∂S2 ∂S1 ∂S2       ∂CD ∂CD ∂CD + (rt − d1,t )S1 d + (rt − d2,t )S2 d + d − rt CdD = 0, ∂S1 ∂S2 ∂t       CdD (t, 0, S2 ) = 0,     CdD (T, S1 , S2 > K) = 1. Applying the introduced transformations and using the method of images on the transformed probability density function, one gets ∗

CdD (χ, z1 , z2 ) =

1 K



∞ ∞ 0

 − 0

0

∞ 0

∞

−β1

pF (χ, z1 − z1 , z2 − z2 )e

p (χ, z1 + F

z1 , z2

+

2ρz1

−



σ1  σ2 z1 −β2

−β1 z2 )e





σ2  σ1 z2

dz1 dz2

σ1  z −β2 σ2 1



σ2  z σ1 2

dz1 dz2



A.4. PROOF OF PROPOSITION 6.4

Proof. The barrier option can be denoted by
T

PkD (t, S1 , S2 ) = EQ e− t rs ds max[K − S2,t , 0]1{τ2 >T } 1{τ1 >T } | Ft . The PDE and boundary conditions are given by  ∂2 BPkD ∂2 PkD 1 2 2 ∂2 PkD 1   σ12 (S1 + D1,t )2 + S + ρσ σ (S + D )S σ 1 2 1 1,t 2   2 2 2 2 ∂S22 ∂S1 ∂S2 ∂S12      D D D  ∂P ∂P ∂P   + (rt − d1,t )S1 k + (rt − d2,t )S2 k + k − rt PkD = 0,  ∂S1 ∂S2 ∂t    PkD (t, 0, S2 ) = 0,       PkD (t, S1 , B) = 0,     D Pk (T, S1 , S2 ) = max(K − S2 , 0).

.

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Introducing the following transformations PkD (t, S1 , S2 ) = Ke−


T t

rs ds α∗ τ+β1 x1 +β2 x2∗

e

∗

PkD (t, x1 , x2∗ ),

  1 1 1 1 2 2 α∗ = − σ12 β1 − σ22 − (r − d2 ) β2 + σ12 β1 + σ22 β2 + ρσ1 σ2 β1 β2 , 2 2 2 2 x1 = ln

S1 + D1,t D1,t

τ = T − t,  x∗ − b x1 −ρ 2 , σ2 1 − ρ2 σ1   1 σ σ − ρ 12 σ22 − (r − d2 ) 2 1 2  β1 = , (1 − ρ2 )σ1 σ2 z∗1 =

1

x2∗ = ln

S2 , K

b = ln

B , K



z∗2 =

x2∗ − b , σ2

β2 =

1 σ2 − ρσ1 r − d2 − 2 , 2 2 (1 − ρ )σ2 σ2 (1 − ρ2 )

the PDE can be reduced to  ∗ ∗ ∗ ∂PkD 1 ∂2 PkD 1 ∂2 PkD   − + − = 0,   2 ∂z2  2 ∂z22 ∂τ  1    D∗ ∗   Pk (τ, z1 , 0) = 0,    

 1 − ρ2 ∗ D∗ ∗ Pk τ, z1 , − z1 = 0,   ρ    √   D∗ ∗ ∗ −β1 (σ1 1−ρ2 z∗1 +σ1 ρz∗2 )−β2 (z∗2 σ2 +b)   (0, z , z ) = max(e P  1 2 k     √ 2∗    ∗  ∗ − e−β1 (σ1 1−ρ z1 +σ1 ρz2 )+(1−β2 )(z2 σ2 +b) , 0). This PDE can be solved by integrating the payoff over the probability density function, see [16]  √  ∗  ∗ 2 ∗ ∗ PkD (τ, z∗1 , z∗2 ) = h(τ, z∗1 , z∗2 , y1∗ , y2∗ ) max(e−β1 (σ1 1−ρ y1 +σ1 ρy2 )−β2 (y2 σ2 +b) 



− e−β1 (σ1

√

1−ρ2 y1∗ +σ1 ρy2∗ )+(1−β2 )(y2∗ σ2 +b)

, 0)dy2∗ dy1∗ ,

where ∞ q q 2  − (q2 +qt2 ) nπθt nπθ t e 2τ sin sin I nπϕ , ϕτ n=1 ϕ ϕ τ 1 − ρ2 , ϕ ∈ [0, π], tan ϕ = − ρ

h(τ, z∗1 , z∗2 , y1∗ , y2∗ ) =

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tan θ =

y2∗ , y1∗

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145

θ ∈ [0, ϕ],

z∗2 , θt ∈ [0, ϕ], z∗1  q = y∗1 2 + y∗2 2 ,

tan θt =

qt =

 z∗1 2 + z∗2 2 .

While the region  is given as follow: 

  1 − ρ2 ∗    (−∞, 0) × −∞, − y1 ∪ (0, +∞)   ρ   

     1 − ρ2 ∗    y1 , +∞ ∪ (−∞, 0) (−∞, 0) × −   ρ 

 =    1 − ρ2 ∗   y1 (0, +∞) × 0, −   ρ    

    2  1 − ρ  (−∞, 0) × −  y1∗ , 0  ρ

if {ρ > 0} ∩ {y1∗ > 0} if {ρ > 0} ∩ {y1∗ < 0} if {ρ < 0} ∩ {y1∗ > 0}   if {ρ < 0} ∩ y1∗ < 0 ,

where 0 denotes the values in t. The region  is obtained from the relationships arctan(y2∗ /y1∗ ) ∈ [0, arctan(− 1 − ρ2 /ρ)] and arctan(− 1 − ρ2 /ρ) ∈ [0, π].

References [1] Canabarro, E and D Duffie (2003). Measuring and marking counterparty risk. In: Asset/Liability Management for Financial Institutions, L Tilman (ed.). Institutional Investor Books. [2] Pykhtin, M and S Zhu (2006). Measuring counterparty credit risk for trading products under Basel II, Working Paper. [3] Pykhtin, M and S Zhu (2007). A guide to modeling counterparty credit risk, GARP Risk Review. [4] Merton, RC (1974). On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29, 449–470. [5] Johnson, H and R Stulz (1987). The pricing of options with default risk. Journal of Finance, 42, 267–280. [6] Hull, JC and AD White (1995). The impact on default risk on the prices of options and other derivative securities. Journal of Banking and Finance, 19, 299–322. [7] Jarrow, R and S Turnbull (1995). Pricing derivatives on financial securities subject to credit risk. Journal of Finance, 50, 53–85.

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[8] Cherubini, U and E Luciano (2002). Copula vulnerability. Risk, 15, 83–86. [9] Cherubini, U and E Luciano (2004). Pricing swap credit risk with copulas, Working Paper. [10] Klein, P (1996). Pricing Black–Scholes options with correlated credit risk. Journal of Banking and Finance, 20, 1211–1229. [11] Klein, P and M Inglis (2001). Pricing vulnerable European options when the option payoff can increase the risk of financial distress. Journal of Banking and Finance, 25, 993–1012. [12] Stamicar, R and C Finger (2005). Incorporating equity derivatives into the CreditGrades model, RiskMetrics Group. [13] Sepp,A (2006). Extended CreditGrades model with stochastic volatility and jumps. Wilmott Magazine, 54, 50–62. [14] Hull, JC (2002). Options, Futures, and Other Derivatives, 5th edn. Prentice Hall. [15] Lipton, A (2001). Mathematical Methods for Foreign Exchange: A Financial Engineer’s Approach. Singapore: World Scientific Publishing Co. Pte. Ltd. [16] He, H et al. (1998). Double lookbacks. Mathematical Finance, 8, 201–228. [17] Karatzas, I and SE Shreve (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Berlin: Springer-Verlag. [18] Zauderer, E (2006). Partial Differential Equations of Applied Mathematics. John Wiley & Sons, Inc. [19] Øksendal, B (2003). Stochastic Differential Equations, 6th edn. Berlin: Springer-Verlag.

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7

ASSET ALLOCATION WITH CREDIT INSTRUMENTS

BARBARA MENZINGER∗,‡ , ANNA SCHLÖSSER∗,§ and RUDI ZAGST†,¶ ∗

risklab GmbH, Seidlstr. 24-24a, 80335 Munich, Germany † HVB-Stiftungsinstitut für Finanzmathematik, Technische Universität München, Parkring II, 85748 Garching, Germany ‡ [email protected] § [email protected] ¶ [email protected]

This chapter presents a consistent, scenario-based asset allocation framework for analyzing traditional financial instruments and credit instruments in a portfolio context. Our framework accounts for the distinct return characteristics of credit instruments by incorporating potential defaults into the total return calculation. We generate correlated default times with a Normal Inverse Gaussian one-factor copula. To determine optimal portfolios, we use a mean-variance and a conditional value at risk optimization. Performing a case study for the U.S. market, we find that the mean-variance optimization overestimates the benefits of low-rated credit instruments. Though, optimal portfolios always contain a considerable proportion of credit instruments.

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7.1. INTRODUCTION The credit market experienced an enormous growth over the last years. As part of the credit market, the credit derivatives market was the worldwide fastest-growing derivatives market (see [1]). The market size expanded from USD 180 billion in 1996 via USD 5,021 billion in 2004 to USD 20,207 billion in 2006 and to estimated USD 33,120 billion in 2008 (see [2]). Due to the financial crisis, the growth was rapidly stopped in 2008 and turned into a decline of the market size. Still, the actual notional amount outstanding of the credit default swaps (CDS), which are the most important instruments in this market, exceeded the estimation of the British Bankers’ Association by far with a notional amount outstanding of USD 41,868 billion (see [3]). After a standardization of credit derivative contracts and the introduction of CDS indices, a revolution in terms of liquidity has taken place, which is only one reason why credit instruments are very attractive to investors. In addition, credit instruments such as corporate bonds, credit derivatives, and securitizations often have an appealing risk–return profile allowing to enhance the portfolio return. Furthermore, due to the correlation structure of their returns to those of traditional asset classes such as stocks and government bonds, they offer high potential for diversification. Finally, they allow to manage the credit risk exposure. Even knowing the potential benefits of different credit instruments, investors still have to know how to combine them optimally with traditional asset classes, i.e., they have to decide on the optimal proportion of credit-related products, especially for their individual level of risk–aversion. A reasonable asset allocation including credit instruments needs to account for the distinct return characteristics of these instruments. We exemplarily analyze the return properties of daily log-returns of U.S. Lehman aggregates over the period from 7 November, 2002 to 29 September, 2006. The descriptive statistics of the returns are summarized in Table 7.1, where it becomes obvious that the risk–return profile of credit instruments cannot be sufficiently described by mean and variance alone.

Table 7.1

U.S. aggregate Aaa Aa A Baa Corporates

Descriptive Statistics of U.S. Lehman Aggregates. No. of issuers

Mean (Ann.)

Volatility (Ann.)

Skewness

Excess kurtosis

3560 759 1404 1215 2721

3.50% 4.23% 5.30% 6.99% 5.65%

3.27% 4.18% 4.55% 4.72% 4.55%

−0.1310 −0.2944 −0.1787 −0.0508 −0.1979

1.7812 1.5802 1.4967 1.3991 1.2648

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We observe negatively skewed return distributions and positive excess kurtosis indicating non-normal distribution of return data. This is confirmed by Jarque–Bera tests for normality. The non-normality might be attributed to defaults within the aggregates. The logical consequence is that the shape of the return distributions should be considered in a realistic asset allocation framework. In this chapter, we propose a consistent, scenario-based asset allocation framework that is composed of a simulation, a total return calculation, and an optimization. With the simulation framework, we produce consistent capital-market scenarios for interest rates, credit spreads, and returns of equity indices, on the one hand, and correlated default times, on the other hand. For the scenario generation, we use the four-factor model according to [4] and the model suggested by [5]. To generate non-normal return distributions of credit instruments, we simulate correlated default times for issuers with different ratings. In this context, the one-factor copula approach for modeling correlated default times between reference entities has become very popular. We use this approach for a Normal-InverseGaussian (NIG) one-factor copula as suggested by [6] since it is able to produce more realistic properties for default times than the wide-spread Gaussian version, e.g., a higher probability of joint defaults of different companies. This property can be predominantly observed during a crisis. Then the probability of joint defaults increases and the value of a portfolio of credit instruments can decrease tremendously, as recently seen during the sub-prime crisis. Therefore, it is particularly important that events having a large impact on the portfolio value are modeled realistically. After having determined the return distribution of the government bonds, the equity index, and the credit instruments, we are able to calculate optimal asset allocations according to different optimization criteria. We do not only apply traditional mean-variance portfolio optimization according to [7] but also conditional value at risk (CVaR) optimization. The latter is more appropriate to capture the distinct distributional properties of credit instruments as it takes into account the left tail of a return distribution. We perform a case study, applying the asset allocation framework to the U.S. market. So we can show that investors benefit from adding credit-related products to their portfolio, i.e., with the same level of risk they can generate a higher expected return. This chapter is organized as follows. The second section describes the simulation framework for the risk factors and the correlated default times. In the third section, we give the general conditions for the total return calculations. In particular, we explain the pricing of funded CDS and funded CDS indices. The applied optimization framework is presented in the fourth section. In the fifth section, we provide the model parameters and present the simulation and optimization results. We close in the last section with a summary of the main results.

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7.2. SIMULATION FRAMEWORK The simulation framework consists of a model to simulate risk factors on the one hand and correlated default times on the other hand. First, we introduce a model to simulate risk factors. We simulate the short rate, the growth rate of the Gross Domestic Product (GDP), and the short-rate spread with the extended model of Schmid and Zagst (see [4]). The short inflation and the returns of an equity index are simulated with the integrated market model suggested by [5]. The latter can be embedded into the extended model of Schmid and Zagst. In the following, we assume that markets are frictionless and perfectly competitive, that trading takes place continuously, that there are neither taxes nor transaction costs or informational asymmetries, and that investors act as price takers. We fix a terminal time horizon T ∗ . Uncertainty in the financial market is modeled by a complete probability space (
, G, P) and all random variables and stochastic processes introduced below are defined on this probability space. We assume that (
, G, P) is equipped with three filtrations H, F, and G, i.e., three increasing families of subσ-fields of G. The default time τ of an obligor is an arbitrary non-negative random variable on (
, G, P). For the sake of convenience, we assume that P[τ = 0] = 0 and P[τ > t] > 0 for every t ∈ (0, T ∗ ]. For a given default time τ, we introduce the associated default indicator or hazard function H(t) = 1{τ≤t} , t ∈ (0, T ∗ ]. Let H = (Ht )0≤t≤T ∗ be the filtration generated by the process H. In addition, we define the filtration F = (Ft )0≤t≤T ∗ as the filtration generated by the multi-dimensional standard Brownian motion Wt = (Wr,t , Wω,t , Wu,t , Ws,t ) and G = (Gt )0≤t≤T ∗ as the enlarged filtration G = H ∨ F, i.e., for every t we set Gt = Ht ∨Ft . All filtrations are assumed to satisfy the usual conditions of completeness and rightcontinuity. For the sake of simplicity, we furthermore assume that F0 is trivial. It should be emphasized that τ is not necessarily a stopping time with respect to the filtration F but of course with respect to the filtration G. If we assumed that τ was a stopping time with respect to F, then it would be necessarily a predictable stopping time. We assume that there exists a measure Q ∼ P such that all discounted price processes of the financial instruments are martingales relative to (Q, G).As numéraires,
t we choose the money-market account Bt = e 0 rl dl with rt denoting the non-defaultable short rate. We will assume throughout the chapter that for any t ∈ (0, T ∗ ] the σ-fields FT ∗ and Ht are conditionally independent (under Q) given Ft . Following [8], p. 167 and p. 242, this is equivalent to the assumption that for any t ∈ (0, T ∗ ] and any Qintegrable FT ∗ -measurable random variable X, we have EQ [X | Gt ] = EQ [X | Ft ]. We start by defining all processes under the real-world measure P and will then work under the equivalent martingale measure Q.

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The dynamics of the non-defaultable nominal short rate r is described by a two-factor Hull–White model and is given by drt = (θr,t + br ωt − ar rt )dt + σr dWr,t ,

(7.1)

dωt = (θω − aω ωt )dt + σω dWω,t ,

(7.2)

where ar , br , σr , aω , and σω are positive constants, θω is a non-negative constant, θr,t is a continuous, deterministic Nelson–Siegel function that fits the market term structure at simulation start date. (For details see, for example, [9, 10].) dWr,t and dWω,t are independent standard Brownian motions. ω represents the GDP growth rate so that the interest-rate levels directly depend on general economic conditions. The dynamics of the short-rate spread are described by dst = [θs + bsu ut − bsω ωt − as st ]dt + σs dWs,t ,

(7.3)

dut = [θu − au ut ]dt + σu dWu,t ,

(7.4)

where bsu , bsω , as , σs , au , and σu are positive constants, θs , θu are non-negative constants and dWu,t , dWs,t are independent standard Brownian motions. u is the so-called uncertainty index and can be interpreted as an aggregation of all available information about the quality of the firm, i.e., it represents the firm-specific risk. The higher its value, the lower the firm’s quality. The GDP behaves reversely. If it grows at a higher rate, spreads usually tighten as the probability of default of a firm becomes smaller. We need the dynamics of the short inflation i for the modeling of the equity-return. It is given by dit = (θi − ai it )dt + σi dWi,t ,

(7.5)

where ai and σi are positive constants, θi is a non-negative constant and dWi,t is a standard Brownian motion. Then the continuous return of an equity index RE,t is described by dRE,t = [αE + bEω ωt − bEi it + bER rt ]dt + σE dWE,t ,

(7.6)

where αE ∈ R, bEω , bEi , bER , and σE are positive constants and dWE,t is a standard Brownian motion. Note that this process reflects the so-called “stock return-inflation puzzle” stating that inflation negatively influences stock returns (see for example [11]– [15]). So far, we considered the dynamics of the SDEs in Eqs. (7.1)–(7.4) under the real-world measure P. Though, as we are interested in zero-coupon bond prices as well as interest rates and credit spreads for different terms, we need to know the parameters of the SDEs under the risk–neutral equivalent martingale measure Q. Replacing ar , aω , as , au by aˆ r , aˆ ω , aˆ s , aˆ u and using independent standard Brownian ˆ ω,t , d W ˆ s,t , d W ˆ u,t instead of dWr,t , dWω,t , dWs,t , dWu,t , leads to the ˆ r,t , d W motions d W

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processes under this measure. The relationship between the parameters is given by aˆ k = ak + λk σk2 , where k = r, ω, s, u. We obtain λr , λω , λs , λu when changing measure from P to Q by applying Girsanov’s Theorem (see, for example, [4]). The processes introduced earlier imply analytical formulas for the zero rates at time t to maturity T of non-defaultable and defaultable bonds, R(t, T ) and Rd (t, T ), as well as for the credit spreads, S(t, T ). So, we can determine the complete term structure. The explicit formulas can be found in the Appendix. To simulate correlated default times, we use Li’s approach, who presents an efficient algorithm in [16]. We use it with a one-factor copula as applied for example by [17]. The one-factor copula is a simple, but powerful way to quickly define a correlation structure between several variables. Therefore, it has become very popular and the standard approach in practice to simulate correlated default times. The underlying idea is the following: In reality, more firms default during a recession than during a booming period. This implies that each firm is subject to the same set of macroeconomic environment and that there exists a dependence among the firms. The Gaussian one-factor copula is often applied due to an easy implementation and the appealing properties of a standard normal distribution, such as the stability under convolution. We rather use the NIG copula which is able to overcome some modeling deficiencies of the Gaussian copula, e.g., the lack of tail dependence. The NIG distribution is a mixture of normal and inverse Gaussian distributions. It is a four parameter distribution with very interesting properties. It can produce fat tails and skewness, it is stable under convolution (under certain conditions), and the density function, the distribution function and the inverse distribution function can be computed sufficiently fast (see [6] and [18]). A definition of the NIG distribution and its most important properties can be found in [6]. Before presenting the copula model, we explain the idea of the Large Homogeneous Portfolio (LHP) approach, introduced by [19]. It assumes a constant default correlation structure over the reference credit portfolio, with the same default probabilities and the same recovery rate in case of default, and it models default using a one-factor Gaussian copula. Reference [6] modified the LHP model by replacing the Gaussian distribution with the NIG distribution. Their one-factor copula model is briefly introduced in the following. Consider a homogeneous portfolio of m credit instruments. The standardized asset return of the i-th issuer in the portfolio, Ai , is assumed to be of the form  Ai = aM + 1 − a2 Xi (7.7) with independent random variables 

βγ 2 γ 3 M ∼ N IG α, β, − 2 , 2 α α

 ,

(7.8)

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and

 √ √ √ √ 1 − a2 1 − a2 1 − a2 βγ 2 1 − a2 γ 3 , (7.9) Xi ∼ N IG , α, β, − a a a α2 a α2  where γ = α2 − β2 . Then, the asset returns Ai also follow an NIG distribution with the parameters   βγ 2 γ 3 α β , , − 2 , 2 =: N IG A Ai ∼ N IG (7.10) a a aα aα

and Ai has zero mean and unit variance. The factor M represents the systematic common market factor and Xi represents firm-specific factors. Equation (7.7) defines a correlation structure between the random variables Ai . Then, the correlation between the asset returns of two issuers is given by a2 , in the case of a homogeneous portfolio. Conditional on M the asset returns of different issuers are independent. Let us assume that default at time t occurs when the asset return of obligor i falls below the threshold C(t), i.e., Ai ≤ C(t). Using this copula model, the variable Ai is then mapped to default time τi of the i-th issuer with a percentile-to-percentile transformation as described for example by [17] or [20]: p(t) = P[τi ≤ t] = P[Ai ≤ C(t)] = N IG A (C(t)).

(7.11)

p(t) = P[τi ≤ t] is the real-world distribution of default times estimated with a migration matrix following [20]. According to Eq. (7.11), we conclude that C(t) = N IG −1 A (p(t)) and thus −1 P[τi ≤ t] = P[Ai ≤ N IG −1 A (p(t))] = P[p (N IG A (Ai )) ≤ t].

Now, we can simulate Ai according to Eq. (7.7) and determine the default times via τi = p−1 (N IG(Ai )).

7.3. FRAMEWORK FOR TOTAL RETURN CALCULATION Based on the simulations according to the previous section, we price the following financial instruments: Government and corporate coupon bonds, funded CDS and funded CDS indices. The pricing for all instruments is done under the following conditions. We price the instruments at every simulated time step tk , k ∈ {0, . . . , p}, with t0 < t1 < · · · < tp = T sim , where T sim denotes the end date of the simulation. We assume that the cash flows occurring during the lifetime of the instrument are reinvested in the respective instrument. For simplicity reasons, we assume a constant recovery rate REC of 40% for all credit instruments in the case of a default rather than using a stochastic recovery model. The constant recovery rate of 40% is in accordance with the recovery rates for the iTraxx Europe (see www.indexco.com). The cash flows

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at default time τ are invested in a risk-free cash account and they are compounded at every simulated time step with the default risk-free government rate. We always assume an initial investment of one unit. The pricing and total return calculation of the bonds and the equity index is straightforward. The pricing of the funded CDS and the funded CDS index, however, is not trivial and is therefore derived in the following. A CDS can be issued in a funded version. Then, the investor (protection seller) buys a floating rate note (FRN), which pays a coupon of the three-month LIBOR plus the fixed CDS spread on a quarterly basis. If no credit event occurs, the coupon is paid until maturity. In this case, the investor receives the notional amount of the FRN. If, however, a credit event of the reference entity takes place, the investor receives the recovery value and the contract terminates. The functionality of a funded CDS index is similar to funded CDS. It can be considered as a portfolio of funded CDS. If a credit event of a reference entity takes place, the investor receives the recovery value, but the notional amount of the FRN is reduced by the weight of the reference entity. If we assume, for example, the first default in the reference portfolio containing 125 equally weighted names, like the broadest and most actively traded investment-grade indices CDX.NA.IG and the iTraxx Europe for North America and Europe, respectively, the notional amout is reduced by 1/125 from 100% to 99.2%. The future coupon payments are based on the new notional amount while the coupon rate remains unchanged. We consider CDS and the CDS index from an investor’s perspective. Hence, we view these instruments as an investment rather than as a means of hedging. Furthermore, we use the funded version of CDS and the CDS index for two reasons. First, we can exclude counterparty risk from our considerations. Second, there are investors who are for regulatory restrictions or due to internal investment policies not allowed to enter into unfunded credit-derivative contracts. Before explaining the total return components of these instruments, we briefly describe the relationship between the relevant time steps. At every simulated time step, tk ∈ {0, . . . , T sim }, we price the funded CDS/CDS index. Furthermore, we assume that t1c < · · · < tnc = T denote the spread payment dates, where t1c denotes the next spread payment date following the current pricing day, tk , and T denotes the maturity of the CDS/CDS index. We denote the previous spread payment date or the settlement date, if the first spread payment has not yet been made, by t0c . Spread payments are made in c to tic . arrear — at time tic for the payment period from ti−1 A funded CDS/CDS index has the following total return components: Present value of the pure, default risk-free FRN at tk paying LIBOR (PVFRN ), coupon payments on the notional amount at t0c , comprising LIBOR and the fixed spread S CDS , compounded to the following simulated date tk (PVS ), recovery payments at default time τ, compounded to the following simulated date tk (PVREC ), and the present value of the CDS/CDS index (PVCDS ). PVCDS can be calculated for every pricing day tk . Its value can be either

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positive, or negative, or zero depending on possible changes in the default intensity λk at tk and on changes in the interest-rate curve. At inception of a funded CDS/CDS index, the fixed spread is determined so that PVCDS is equal to zero, i.e., the fixed spread is a fair price for the default intensity of the reference entity. During the term of the contract, however, the default intensity of the reference entity can change, while the fixed spread does not, resulting in a change in the value of PVCDS . To determine the present value of the funded CDS/CDS index at tk , we need to know the expected loss at tk up to every spread payment day, tic , i ∈ {1, . . . , n}, the premium leg and the protection leg. With the constant default intensity model, we calculate the expected loss at tk up to tic according to EL(tk , tic , λk ) = 1−exp(−λk (tic − tk )). This model assumes a constant default intensity, λk , for a given time tk . (This implies that the partial recovery of market value in the extended model of Schmid and Zagst is stochastic over time.) The premium leg is the present value of all expected spread payments. It is calculated according to Prem Leg(tk , T, S CDS , λk ) =

n 

tic S CDS (1 − EL(tk , tic , λk ))P(tk , tic ),

i=1 c where tic = tic −ti−1 , S CDS is the fixed annual spread of the CDS/CDS index, P(tk , tic ) is the discount factor and 1 − EL(tk , tic , λk ) denotes the probability of no default up to time tic . The protection leg is the present value of all expected protection payments made by the protection seller. As we assume a constant recovery rate REC, we can calculate the protection leg by  T Prot Leg(tk , T, λk ) = (1 − REC) P(tk , l) dEL(tk , l, λk ) tk

≈ (1 − REC)

n 

c (EL(tk , tic , λk ) − EL(tk , ti−1 , λk ))P(tk , tic ).

i=1

At issuance of the funded CDS, the fixed annual spread S CDS is determined so that the value of the premium leg equals the value of the protection leg: c (1 − REC) ni=1 (EL(tk , tic , λk ) − EL(tk , ti−1 , λk ))P(tk , tic ) n S CDS = . (7.12) c c c i=1 ti (1 − EL(tk , ti , λk ))P(tk , ti ) The equality of premium leg and protection leg as shown in Eq. (7.12) must also hold for every day, when S CDS is substituted by the quoted par yield spread. For the sake of simplicity, we use the simulated zero spread as the quoted par yield spread. The absolute difference between the spreads, however, is very low. We analyzed historical credit spreads from 31 December, 1991 to 29 September, 2006. The average difference

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in this period was at most 1.5 bps for rating classes AA, A2 and BBB and maturities of up to five years. The worst case on a single day was 4.9 bps. After this substitution, we can solve the equation for λk and determine the constant (implied) default intensity at tk . Let N(tk , τ) denote the notional amount of a funded CDS/CDS index at time step tk . Then, the present value of the CDS/CDS index is given by PVCDS (tk , T, S CDS , λk ) = (Prem Leg(tk , T, S CDS , λk ) − Prot Leg(tk , T, λk ))N(tk , τ).

(7.13)

The calculation of the present values of the other total return components is straightforward. Then, we determine the total return of the investment, RCDS , for every tk > 0 by RCDS (tk−1 , tk ) =

PVCDS (tk , T, S CDS , λk ) + PVFRN (tk ) + PVREC (tk ) + PVS (tk ) − 1. PVCDS (tk−1 , T, S CDS , λk−1 ) + PVFRN (tk−1 ) + PVREC (tk−1 )

7.4. OPTIMIZATION FRAMEWORK For all optimization criteria, we make the following assumptions. There are neither transaction costs nor taxes; all securities can be divided arbitrarily; the portfolios remain unchanged over time; there are n given assets to invest in with returns Ri , i = 1, . . . , n; the expected return of asset i is given by µi := EP [Ri ] and µ := (µ1 , . . . , µn )T ; xi is the portfolio weight of asset i with ni=1 xi = 1, and the portfolio is denoted by x := (x1 , . . . , xn )T .

7.4.1. Mean-Variance Optimization The mean-variance optimization can be viewed as traditional portfolio optimization and is based on the model of [7]. The basic assumption is that investors select their portfolios taking into account only the first two moments of the asset returns — mean and variance — and the correlation between the assets. Let the covariance matrix be denoted by C = (cij )i,j=1,...,n , where cij = Cov[Ri , Rj ], and let the variance be denoted by σi2 := cii > 0. Then, the meanvariance optimization is given by the following optimization problem min x

xT Cx

(7.14)

s.t. µT x ≥ µ, ¯ 1T x = 1, x ≥ 0, where 1 = (1, . . . , 1)T . If we solve the optimization problem in Eq. (7.14) for every possible µ, ¯ we obtain the set of all efficient portfolios. Obviously, the optimization problem in Eq. (7.14) only takes into account mean and variance of a portfolio. This is appropriate, for example, if returns are normally distributed. However, the returns of credit instruments are not normally distributed,

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as seen earlier. Therefore, this criterion is only of limited use for the optimization of portfolios including credit instruments.

7.4.2. CVaR Optimization The conditional value at risk (CVaR) represents the expected value of all losses that exceed a certain value at risk (VaR). Formally, we can define the CVaR as follows. Let 1 − α be the confidence level for the VaR with α ∈ (0, 1). Then, the CVaR of a portfolio’s return R(x) is given by CVaR(x, α) = −EP [R(x) | R(x) ≤ CR (α)] = −EP [R(x) | R(x) ≤ −VaR(x, α)],

(7.15)

where CR (α) denotes the α-quantile of the portfolio’s return distribution R(x). From the formulas above, it becomes evident that the CVaR provides information on the negative tail of a return distribution since it is not only focussed on the α-quantile but also takes into account the shape of its tail. This is of great importance if instruments may default and so produce fat tails. The corresponding CVaR optimization is given by min x

CVaR(x, α)

(7.16)

s.t. µT x ≥ µ, ¯ 1T x = 1, x ≥ 0.

7.5. MODEL CALIBRATION AND SIMULATION RESULTS We fit our model to market data as of 30 September, 2006 (simulation start date). To calibrate the simulation model, we use parameters estimated by [21] and [22]. The parameters for the short rate and the GDP growth rates are given by ar = 0.37867, aˆ r = 0.24782, br = 0.13315, σr = 0.01496, aω = 1.18532, aˆ ω = 0.26847, θω = 0.01583, σω = 0.00601. The credit spread parameters are displayed in Table 7.2. We adjust the estimated parameters aˆ s , aˆ u , and bsu to better meet historical data in terms of average spreads and spread ranges. For the inflation process and the process for the equity-index returns, we use the following parameters (see [5] and [22]): ai = 0.64073, aˆ i = 0.50319, θi = 1.04790, σi = 0.01447. αE = −3.07791, bER = 3.94000, bEi = 9.32436, bEω = 5.10643, σE = 0.16000. For the equity index, we adjust the level of the returns and the standard deviation to better reflect the updated historical data. We want the U.S. CDS index to represent the current composition of the CDX.NA.IG in terms of proportion of different rating classes. However, we assume the U.S. CDS index (short CDX) only to be composed of the rating classes AA, A, and BBB with the weights 12.28%, 39.47%, and 48.25%. We added the AAA rating class

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as aˆ s θs σs bsω au aˆ u θu σu bsu

AA

A2

BBB1

2.96099 2.96099 0.00275 0.00389 0.07373 0.11269 0.02180 0.12173 0.00459 0.98404

2.80727 2.80727 0.00251 0.00309 0.07634 0.11057 0.03210 0.18566 0.00476 0.92214

2.41739 2.41739 0.00238 0.00287 0.09369 0.11048 0.04075 0.18980 0.00489 1.26089

to rating class AA and BB to BBB, as there are only parameters for the rating classes AA, A, and BBB available. Furthermore, we normalized the weights to sum up to one. For the simulation of correlated default times, we use the average one-year migration matrix for the United States, from 1981 to 2005, provided by [23]. The parameters for the one-factor copula as of simulation start date are estimated in accordance with the model introduced by [6]. We use the following parameters a = 0.37029, α = 0.70138, β = 0. So, the correlation a of a single reference entity to the common market factor is equivalent to a correlation between two reference entities of a2 = 0.13711. Having simulated 5,000 scenarios on a quarterly basis with government interest rates, credit spreads and correlated default times, we can now price various financial instruments. All bonds, CDS and the CDX, are assumed to have an initial term of approximately five years. Table 7.3 reports the return characteristics for investment horizons one year and three years. Here, we make some general observations. We see higher expected returns and higher standard deviations or levels of CVaR with decreasing credit quality of the bonds. Besides, we observe that — as soon as defaults have occurred — the distributions of credit instruments become non-normal. This can be shown by some statistics: The skewness of the credit instruments is negative, the excess kurtosis is significantly higher than 0 and minimum values always indicate defaults. The nonnormality can also be verified easily by Jarque–Bera tests. For the one year investment horizon, the AA and A-rated CDS and the CDX have considerably less volatility and a lower CVaR than government and corporate bonds. At the same time, CDS and the CDX have a similar expected return to government bonds. Though, comparing the oneyear and the three-year investment horizon, we make an interesting observation. For the three-year horizon, the CDS have a similar or higher risk than the corporate bonds and a

Bond AA

Bond A

Bond BBB

CDS index

CDS AA

CDS A

CDS BBB

5.55% 5.50% 3.16% 2.55% −4.35% 17.01% 0.0900 0.0276

10.43% 8.25% 19.30% 32.14% −43.08% 88.10% 0.5568 0.4571

6.03% 6.01% 3.35% 3.24% −59.17% 17.18% −1.3820 28.2614

6.19% 6.18% 3.57% 5.34% −58.22% 18.12% −3.3200 59.8199

6.37% 6.51% 4.73% 17.71% −58.17% 19.63% −6.7629 90.1579

5.49% 5.52% 0.74% −2.60% −13.27% 7.59% −5.1385 109.6775

5.75% 5.77% 1.13% −2.64% −57.44% 8.46% −34.6157 1927.6047

5.92% 5.95% 1.70% −0.39% −57.72% 8.54% −30.6476 1135.5933

6.10% 6.27% 3.46% 13.24% −57.28% 10.00% −16.6226 298.5225

16.57% 16.58% 2.67% −9.41% 7.32% 25.13% −0.0188 −0.0459

33.44% 24.29% 52.74% 54.45% −67.75% 439.46% 1.3790 3.8584

18.04% 18.16% 3.86% −1.53% −55.18% 27.26% −9.2575 172.4228

18.55% 18.86% 5.34% 14.83% −55.41% 28.12% −9.8637 131.3360

19.08% 19.99% 8.77% 53.12% −55.20% 30.60% −7.1960 56.2670

16.43% 16.46% 3.33% −5.77% −30.08% 27.25% −0.9070 10.8976

17.20% 17.25% 4.18% 0.65% −54.21% 29.06% −7.3939 123.0516

17.73% 17.93% 5.43% 15.09% −54.58% 30.19% −8.4513 107.1262

18.41% 19.10% 8.06% 51.90% −54.77% 32.05% −7.1697 59.9592

1 Year Mean Median σ CVaR(1%) Min Max Skewness Excess kurtosis

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Table 7.3

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Mean Median σ CVaR(1%) Min Max Skewness Excess kurtosis

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lower return. This phenomenon can be easily explained by the nature of the instruments. Fixed-income instruments are exposed to market risk which is the higher, the longer the time to maturity. CDS and the CDX are very similar to an FRN and only have a very small market risk. They pay the short-term interest rate, which is lower than the medium-term interest rates if the curve is normal. In addition, the outcome of interest rates is more volatile over a longer investment horizon. In Table 7.4, we show the linear correlations between the returns of the financial instruments for the investment horizons one year and three years. We make the following main observations: Government-bond and corporate-bond returns have a high positive correlation. This is reasonable as the main driver of the bond return is the government rate, which is the same for all bonds. Differences in bond returns come from different credit spreads and different default times. Bond and equity-index returns are negatively correlated, which is also known from historical time series. There is a small positive or negative correlation between returns of corporate bonds and funded CDS/CDX. This may seem counterintuitive at first sight. A closer look at the nature of these instruments proves otherwise. We need to differentiate between two opposing effects, which can be attributed to single return components of the funded CDS/CDX. On the one hand, the present value of the CDS PVCDS (see Eq. (7.13)) strongly behaves like the price of the corporate bond. Zero rates being equal, PVCDS and the price of the corporate bond decrease if spreads increase resulting in a high positive correlation. On the other hand, bonds are fixed income instruments with prices that strongly depend on the level and the structure of zero rates. If zero rates are falling, the bond returns increase as bond prices increase. For the FRN linked to the CDS/CDX, it is the other way round. Since its return strongly depends on the floating LIBOR, the return of the FRN tends to decrease with falling interest rates resulting in a high negative correlation between bond returns and returns of funded CDS/CDX. All in all, the two opposing effects cancel each other out to a certain degree resulting in a small positive or negative correlation between corporate bonds and funded CDS/CDX. The returns of the equity index and CDS or the CDX are positively correlated. This is reasonable since both equity-index returns and the three-month government rate are to a large proportion driven by the short rate, which was empirically confirmed by [5]. To conclude, we identify appealing risk–return profiles and low correlations between bonds, the equity index and CDS/CDX. Therefore, investors should benefit from holding a portfolio consisting of traditional financial instruments and credit instruments. After having analyzed the return characteristics, we can turn to portfolio optimization. For this purpose, we consider a one-year and a three-year investment horizon and the following three investment universes: Initial investment universe consisting of government bonds and an equity index; extension by corporate bonds; extension by CDS and the CDX.

Bond AA

Bond A

Bond BBB

CDS index

CDS AA

CDS A

CDS BBB

1.0000 −0.1184 0.9508 0.8824 0.6828 −0.2338 −0.1569 −0.1013 −0.0510

−0.1184 1.0000 −0.1182 −0.1110 −0.0835 0.1124 0.0751 0.0639 0.0104

0.9508 −0.1182 1.0000 0.8391 0.6511 −0.2227 −0.0781 −0.0986 −0.0465

0.8824 −0.1110 0.8391 1.0000 0.6002 −0.1934 −0.1385 −0.0417 −0.0482

0.6828 −0.0835 0.6511 0.6002 1.0000 −0.0104 −0.1061 0.0265 0.0560

−0.2338 0.1124 −0.2227 −0.1934 −0.0104 1.0000 0.2652 0.2802 0.1830

−0.1569 0.0751 −0.0781 −0.1385 −0.1061 0.2652 1.0000 0.1107 0.0632

−0.1013 0.0639 −0.0986 −0.0417 0.0265 0.2802 0.1107 1.0000 0.0395

−0.0510 0.0104 −0.0465 −0.0482 0.0560 0.1830 0.0632 0.0395 1.0000

1.0000 −0.1629 0.7001 0.5086 0.3271 −0.4689 −0.3812 −0.2919 −0.1814

−0.1629 1.0000 −0.1068 −0.1014 −0.0593 0.2651 0.2201 0.1764 0.1045

0.7001 −0.1068 1.0000 0.3543 0.2511 −0.2249 −0.2387 −0.1934 −0.0579

0.5086 −0.1014 0.3543 1.0000 0.2027 −0.1772 −0.1930 −0.0712 −0.0726

0.3271 −0.0593 0.2511 0.2027 1.0000 −0.0816 −0.1103 −0.0901 −0.0209

−0.4689 0.2651 −0.2249 −0.1772 −0.0816 1.0000 0.6809 0.5481 0.4063

−0.3812 0.2201 −0.2387 −0.1930 −0.1103 0.6809 1.0000 0.4113 0.2732

−0.2919 0.1764 −0.1934 −0.0712 −0.0901 0.5481 0.4113 1.0000 0.2070

−0.1814 0.1045 −0.0579 −0.0726 −0.0209 0.4063 0.2732 0.2070 1.0000

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Correlation of Total Returns of Financial Instruments.

Asset Allocation with Credit Instruments

Table 7.4

3 Years b913-ch07 FA

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7.5.1. Mean-Variance Approach With a mean-variance optimization according to the optimization problem in Eq. (7.14) for the relevant investment universes as described above, we obtain efficient frontiers and the corresponding asset allocations. The results are displayed in Fig. 7.1 for the one-year investment horizon and in Fig. 7.2 for an investment horizon of three years. The upper part of the figures show the efficient frontiers, the lower parts of the figures contain the optimal asset allocations along the efficient frontiers. For the sake of readability, the results of the asset’s weights in the portfolio are partly aggregated, i.e., the element “Bonds AA/A/BBB” represents the corporate bonds with a rating of AA, A, or BBB and is the sum of all corporate bond weights in a certain optimal asset allocation, “CDS AA/A/BBB” represents all single-name CDS rated AA, A, or BBB in a certain allocation. The figures exhibit some similar structures. At first, we examine the efficient frontiers. Allowing for corporate bonds in the portfolio optimization leads to an upward shift of the efficient frontiers compared to the initial investment universe. For the same level of risk, a higher expected return can be generated. Allowing additionally for CDS and the CDX leads to a shift of the efficient frontier to the left, i.e., an investor can reduce the portfolio risk. The proportion of the potential risk reduction or return enhancement depends on the investment horizon. In Table 7.5, we show the potential improvement of the investor’s portfolio risk and/or return position exemplarily for a one-year horizon using the minimum-variance portfolio of the initial investment universe as a reference point. Table 7.5 shows that the minimum variance portfolio based on the initial investment universe has an expected return of 5.76% and a standard deviation of 3.04%. This portfolio is compared to equivalent portfolios on the most extended investment universe, i.e., allowing not only for government bonds and an equity index, but also for all relevant credit instruments. The appropriate portfolios can be identified by either holding the expected return constant and looking for the standard deviation of the equivalent portfolio on the efficient frontier of the most extended investment universe, or holding the standard deviation constant and looking for the expected return on the efficient frontier. In the first case, the portfolio’s standard deviation is only 0.70%, in the second case the expected return is 6.72%. The results from the Table 7.5 allow the conclusion that a mean-variance investor can reduce his portfolio’s risk and/or enhance the return by adding credit instruments to the portfolio of government bonds and an equity index. Similar improvements of the risk and/or return position can be observed for the three-year investment horizon. Now, we focus on the mean-variance optimal asset allocations, i.e., the lower parts of Figs. 7.1 and 7.2. In the initial investment universe, optimal allocations only consist of government bonds and an equity index. In the largest investment universe,

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Expected return (%)

10 9 8 7

6 5 4 0

5

10

15

20

25

Standard deviation (%) Equity Index and Government Bonds Equity Index, Government Bonds, Corporate Bonds

18.17

16.99

15.83

14.67

13.52

11.26

12.38

9.09

10.16

8.05

7.03

6.04

5.08

4.17

3.30

2.48

1.75

1.14

0.76

100 90 80 70 60 50 40 30 20 10 0

0.61

Portfolio weights (%)

Equity Index, Government Bonds, Corporate Bondsand CDS/CDX

Standard deviation (%) Equity Index

Government Bonds

CDS AA/A/BBB

Bonds AA/A/BBB

CDX

Figure 7.1 Results of mean-variance optimization, one-year investment horizon.

government bonds are substituted to a large extent with corporate bonds, CDS, and the CDX. As already seen from Table 7.3, AA and A-rated CDS and the CDX have a similar expected return compared to government bonds but a considerably lower standard deviation for a one-year horizon. Therefore, particularly for lower levels of standard deviation, optimal portfolios contain a considerable proportion of CDS and the CDX. For levels of standard deviation higher than 7.64% (one-year horizon) and 10.38% (three-year horizon), an optimal portfolio only consists of corporate bonds and an equity index. For lower levels of standard deviation, corporate bonds are partially replaced by CDS and the CDX. There is only a rather small proportion of government bonds for a low level of risk, due to the risk–return profile of government bonds and CDS.

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31 29 27 25 23 21 19 17 15 0

10

20

30

40

50

60

Standard deviation (%) Equity Index and Government Bonds Equity Index, Government Bonds, Corporate Bonds Equity Index, Government Bonds, Corporate Bonds and CDS/CDX

49.74

46.62

43.50

40.40

37.31

34.23

31.17

28.13

25.13

22.17

19.26

16.38

13.55

8.21

10.81

5.86

3.86

2.49

1.84

1.50

Portfolio weights (%)

100 90 80 70 60 50 40 30 20 10 0

Standard deviation (%) Equity Index

Government Bonds

CDS AA/A/BBB

Bonds AA/A/BBB

CDX

Figure 7.2 Results of mean-variance optimization, three-year investment horizon.

7.5.2. Conditional Value at Risk In the CVaR optimization, we assume α to be 1%, i.e., we consider the mean of the worst 1% of the portfolio return as risk measure. The resulting efficient frontiers and the optimal asset allocations are displayed in Fig. 7.3 for the one-year investment horizon and in Fig. 7.4 for the three-year investment horizon. When comparing the results of the CVaR optimization with those of the mean-variance optimization, the first impression is very similar. In the following, we describe the main results of

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Asset Allocation with Credit Instruments Table 7.5 Comparison of Selected Mean-Variance Optimal Portfolios Based on Two Different Investment Universes, One-Year Investment Horizon. Investment universe Initial Most extended Most extended

Portfolio selection criterion

µ (%)

σ (%)

Minimum variance portfolio Equal portfolio return Equal portfolio standard deviation

5.76 5.76 6.72

3.04 0.70 3.04

the CVaR optimization at first. Then, we describe and explain some differences of mean-variance and CVaR optimization. We begin with the efficient frontiers. Adding credit instruments to a portfolio only consisting of the initial investment universe leads to an upward shift and a shift to the left of the efficient frontier. This indicates that there is an enormous potential for return enhancement or risk reduction. The potential depends on the investment horizon. It is exemplarily illustrated for the minimum-CVaR portfolio as a reference point, in Table 7.6. Table 7.6 reveals that for a one-year investment horizon there is a high potential to either reduce the risk for a given level of return (from a CVaR of 2.10% to a CVaR of only −3.12%), or to enhance the portfolio return for a given level of risk (from 5.79% to 6.78%). Also for a three-year investment horizon there is a high potential to reduce risk and/or enhance return. Next, we analyze the optimal asset allocations for an investor using the CVaRcriterion. In an optimal portfolio, government bonds are partially substituted with corporate bonds, CDS and the CDX compared to the initial investment universe. As explained earlier, AA- and A-rated CDS and the CDX have a similar expected return to government bonds, but a lower CVaR for a one-year horizon. Particularly for low levels of CVaR, corporate bonds are partially substituted with CDS and the CDX. For levels of CVaR higher than 18.29% (one-year horizon) and 14.47% (three-year horizon), an optimal portfolio only consists of corporate bonds and an equity index. If, however, lower levels of CVaR are of interest, optimal allocations are composed of a considerable proportion of credit derivatives. Having a closer look at the resulting optimal allocations of mean-variance and CVaR optimizations, we can identify some differences. For a one-year investment horizon, optimal allocations contain CDS even for high levels of expected returns. While mean-variance optimal portfolios are composed of CDS up to an expected portfolio return of 7.91%, CVaR optimal portfolios hold CDS up to a return level of 8.94%. In contrast, CVaR optimal portfolios contain significantly less BBB-rated, but significantly more AA-rated corporate bonds. For a three-year investment horizon, CVaR optimal portfolios hold CDS up to an expected portfolio return of 25.30%, while mean-variance optimal portfolios only consist of CDS up to returns of 21.53%.

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Expected return (%)

10 9 8 7

6 5 4 −10

−5

0

5

10

15

20

25

30

35

CVaR (%) Equity Index and Government Bonds Equity Index, Government Bonds, Corporate Bonds

31.36

28.97

26.64

24.32

22.03

19.77

17.62

15.58

13.55

9.62

11.57

7.74

5.92

4.27

2.77

1.27

−0.20

−1.60

−2.78

100 90 80 70 60 50 40 30 20 10 0

−3.30

Portfolio weights (%)

Equity Index, Government Bonds, Corporate Bonds and CDS/CDX

CVaR (%) Equity Index

Government Bonds

CDS AA/A/BBB

Bonds AA/A/BBB

CDX

Figure 7.3 Results of CVaR optimization, one-year investment horizon.

Furthermore, the former are composed of a significant larger proportion of government bonds than the latter. Again, we observe a considerably lower proportion of BBBrated corporate bonds for the CVaR optimal portfolios. BBB-rated bonds provide a rather high expected return. Though, the standard deviation is not able to adequately capture the tail events. From these observations, we can conclude that the meanvariance optimization underestimates the benefits of credit derivatives, particularly for longer investment horizons, by only considering the first two moments of the return distribution, and not its tail. Moreover, it overestimates the benefits of BBBrated corporate bonds by only looking at the expected return and volatility but mainly ignoring the tail events. Our analysis show that an investor, optimizing his portfolio

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Asset Allocation with Credit Instruments 35 33 31 Expected return (%)

29 27 25 23 21 19 17 15 −20

−10

0

10

20

30

40

50

60

CVaR (%) Equity Index and Government Bonds Equity Index, Government Bonds, Corporate Bonds

51.57

47.12

42.79

38.55

30.07

34.31

25.93

21.92

18.02

14.17

10.51

6.98

3.78

0.84

−1.94

−4.67

−7.18

−9.34

−11.02

100 90 80 70 60 50 40 30 20 10 0

−12.00

Portfolio weights (%)

Equity Index, Government Bonds, Corporate Bonds and CDS/CDX

CVaR (%) Equity Index

Government Bonds

CDSAA/A/BBB

Bonds AA/A/BBB

CDX

Figure 7.4 Results of CVaR optimization, three-year investment horizon.

with either the mean-variance or the CVaR criterion, can add performance to his portfolio for a given level of risk, or he can reduce risk for a given target return level. This can be realized due to a low correlation between the different instruments and their attractive risk–return profile.

7.5.3. Comparison of Selected Optimal Portfolios We close our analysis of the simulation results with a comparison of selected optimal portfolios. We examine the optimal asset allocation for two representative investors which we denote by risk–averse and risk–affine. Note that the term risk–affine does

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Table 7.7 Investor.

Portfolio selection criterion

µ (%)

CVaR (%)

Minimum-CVaR portfolio Equal portfolio return Equal portfolio standard deviation

5.79 5.79 6.78

2.10 −3.12 2.10

Benchmark Portfolios for a Risk–Averse and a Risk–Affine

Govt. bond Equity index

Risk–averse investor (%)

Risk–affine investor (%)

70.00 30.00

30.00 70.00

not mean that this investor is seeking risk. He is rather willing to bear more risk than the risk–averse investor in compensation for a higher risk premium. Their respective benchmark portfolio is defined on the initial investment universe and is exhibited in Table 7.7. The procedure for selecting the optimal portfolios to compare with the benchmark portfolios is the following. At first, we determine the risk and return characteristics of the benchmark portfolios, denoted as “riskB ” and “returnB ”, where “riskB ” refers to the relevant risk measure: If optimal portfolios are selected using mean-variance approach (MV), the relevant risk measure is standard deviation, in the case of CVaR optimization it is the CVaR. We use these risk values as reference points to find the optimal portfolios with identical risk values on the efficient frontiers of the most extended investment universe based on the results of mean-variance and the CVaRoptimization presented in the previous sections (see Figs. 7.1–7.4). The results are shown in Tables 7.8 and 7.9 for the one-year and the three-year investment horizon, respectively. In the upper parts of the Tables 7.8 and 7.9 the risk–return profiles of the benchmark portfolios and the optimal portfolios (denoted as “risk∗ ”, “return∗ ”) can be found. In addition, the corresponding optimal asset allocations are displayed in the lower part of these tables. Tables 7.8 and 7.9 reveal some interesting characteristics of mean-variance optimal and CVaR optimal portfolios for the risk–averse and risk–affine investor defined in Table 7.7. At first, we focus on some similarities. Mean-variance and CVaR optimization lead to similar expected portfolio returns. Extending the investment universe by credit instruments an investor always can enhance the portfolio return. A risk– averse investor with a one-year investment horizon can increase his portfolio return

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Asset Allocation with Credit Instruments Table 7.8 Comparison of Benchmark Portfolios and Optimal Portfolios for Risk–Averse and Risk–Affine Investor, One-Year Investment Horizon.

RiskB ReturnB Risk∗ Return∗ Govt. bond Equity index Bond AA Bond A Bond BBB CDS index CDS

Risk–averse investor

Risk–affine investor

MV (%)

CVaR (%)

MV (%)

CVaR (%)

5.95 7.01 5.95 7.50

6.52 7.01 6.52 7.47

13.43 8.97 13.43 9.21

20.52 8.97 20.52 9.20

0.00 29.50 0.00 20.38 37.94 0.00 12.18

0.00 31.01 24.05 27.10 6.46 0.00 11.39

0.00 69.82 0.00 0.00 30.18 0.00 0.00

0.00 69.93 0.00 7.16 22.90 0.00 0.00

Table 7.9 Comparison of Benchmark Portfolios and Optimal Portfolios for Risk–Averse and Risk–Affine Investor, Three-Year Investment Horizon. Risk-averse investor

Risk-affine investor

MV (%)

CVaR (%)

MV (%)

CVaR (%)

RiskB ReturnB Risk∗ Return∗

15.63 21.63 15.63 23.13

4.52 21.63 4.52 22.88

36.80 28.38 36.80 29.12

32.76 28.38 32.76 29.18

Govt. bond Equity index Bond AA Bond A Bond BBB CDS Index CDS

0.00 29.04 0.00 23.32 47.64 0.00 0.00

0.00 30.14 24.80 21.86 11.20 0.00 12.00

0.00 69.89 0.00 0.00 30.11 0.00 0.00

0.00 70.89 0.00 15.86 13.25 0.00 0.00

from 7.01% to 7.50% applying the mean-variance optimization and to 7.47% applying the CVaR optimization. A risk–affine investor with a three-year investment horizon increases his portfolio return from 28.38% to 29.12% (MV) or to 29.18% (CVaR). The proportion of the equity index in an optimal portfolio is always rather close to the

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proportion of the benchmark portfolio, i.e., it is approximately 30% for the risk–averse investor and 70% for the risk–affine investor. However, the equity proportion is always slightly higher for investors using the CVaR criterion. The two investors should never allocate funds in government bonds to build an optimal portfolio. The proportion of the government bonds in the benchmark portfolios is rather reallocated to the credit instruments in the optimized portfolios, particularly to corporate bonds and CDS. For the considered levels of risk, the CDS Index does not play a role. For the one-year horizon both optimization approaches lead to a proportion of approximately 11–12% of CDS for the risk–averse investor. Independent of optimization criterion or investment horizon, the risk–affine investor should not invest in CDS. Having a closer look at the proportions of the credit instruments, we can figure out some interesting differences. A risk–averse investor applying the CVaR criterion and having three-year investment horizon should still allocate approximately 12% to CDS, while this investor using the mean-variance criterion should not hold CDS in his portfolio. This indicates that mean-variance optimization underestimates the benefit of credit derivatives for longer investment horizons. For both investment horizons, mean-variance optimal portfolios contain a considerably higher proportion of BBB-rated corporate bonds than the CVaR-optimal portfolios. This means that mean-variance optimization overestimates the benefits of BBB-rated corporate bonds by only taking the expected return and volatility into account but mainly ignoring the tail events. These effects were already explained in the previous section. To sum up, comparing the resulting optimal asset allocations for a representative risk–averse and risk–affine investor, we see that, independent of investment horizon and optimization criterion, an investor always benefits from substituting government bonds by corporate bonds, CDS, and the CDX. The resulting optimal allocations, however, strongly depend on the investor type, the optimization criterion and the investment horizon.

7.6. SUMMARY AND CONCLUSION Constructing portfolios with credit instruments requires an appropriate asset allocation framework in order to account for the distinct return characteristics of these instruments, such as non-normality of the return distribution due to potential defaults. We have presented a consistent, scenario-based asset allocation framework, which is able to determine optimal portfolios consisting of traditional instruments and credit instruments. The entire framework is composed of a simulation, a total return calculation, and an optimization framework. We have suggested a model to simulate consistent capital-market scenarios offering analytical formulas for the whole term structure of interest rates and credit spreads. Furthermore, we have introduced a

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model to simulate correlated default times with an NIG one-factor copula, which is an extension of the Gaussian one-factor copula. The NIG version is appealing since it allows to generate more realistic properties of default times such as a higher probability of joint defaults of different issuers. After the introduction of general conditions for the total return calculations, we have explained how to price CDS. For portfolio optimization we have applied two criteria — the mean-variance and the CVaR optimization. The former can be viewed as traditional portfolio optimization. Its main disadvantage is that it only takes the first two moments of a return distribution into account, which is obviously not appropriate when also allowing for credit instruments. The latter is able to overcome this drawback since it considers the left tail of a distribution. So, it particularly takes defaults into account. Finally, we have presented the model parameters and we have applied our asset allocation framework to the U.S. market for a one-year and a three-year investment horizon. We have found that credit instruments have an appealing risk–return profile and a correlation structure providing a considerable potential for diversification. Moreover, we have found that mean-variance optimization overestimates the benefits of lowrated bonds by only focussing on mean and variance, and mainly ignoring the tail events. Realistic modeling of return characteristics is very important when instruments with specific return characteristics are included in an asset allocation. Otherwise, optimization results are not more than an approximation. Our model provides realistic return distributions. Furthermore, it can be fitted to market data and it can be easily extended by other credit instruments. Therefore, it is appropriate for an asset allocation with credit instruments.

APPENDIX The zero rates R(t, T ) and Rd (t, T ) at time t to maturity T of non-defaultable and defaultable bonds as well as the credit spreads S(t, T ) are given by

R(t, T ) = − Rd (t, T ) = − S(t, T ) = −

T T T

1 [A(t, T ) − B(t, T )rt − D(t, T )ωt ], −t 1 [Ad (t, T ) − B(t, T )rt − Dd (t, T )ωt − Ed (t, T )ut − F d (t, T )st ], −t 1 [Ad (t, T ) − A(t, T ) − (Dd (t, T ) − D(t, T ))ωt −t

− Ed (t, T )ut − F d (t, T )st ],

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with



A(t, T ) =

T





T

t

Ad (t, T ) =

1 2

 1 2 2 2 2 (σ B(l, T ) + σω D(l, T ) ) − θr (l)B(l, T ) − θω D(l, T ) dl, 2 r [σr2 B(l, T )2 + σω2 Dd (l, T )2 + σu2 Ed (l, T )2 + σs2 F d (l, T )2 ]dl

t



T

−

[θr (l)B(l, T ) + θω Dd (l, T ) + θu Ed (l, T ) + θs F d (l, T )]dl,

t

B(t, T ) =

1 (1 − e−ˆar (T −t) ), aˆ r

br D(t, T ) = aˆ r



1 − e−ˆaω (T −t) e−ˆaω (T −t) − e−ˆar (T −t) + aˆ ω aˆ ω − aˆ r

bsω D (t, T ) = D(t, T ) − aˆ s d

bsu E (t, T ) = aˆ s d

F d (t, T ) =





 ,

1 − e−ˆaω (T −t) e−ˆaω (T −t) − e−ˆas (T −t) + aˆ ω aˆ ω − aˆ s

1 − e−ˆau (T −t) e−ˆau (T −t) − e−ˆas (T −t) + aˆ u aˆ u − aˆ s

 ,

 ,

1 (1 − e−ˆas (T −t) ). aˆ s

For an explicit derivation of the zero rates, we refer to [4].

References [1] FitchRatings (2005). Global credit derivatives survey: Risk dispersion accelerates. [2] British Bankers’ Association (2006). BBA credit derivatives report 2006 — executive summary. [3] Bank for International Settlements (2009). BIS quarterly review June 2009: International banking and financial market developments. [4] Antes, S., M Ilg, B Schmid and R Zagst (2008). Empirical evaluation of hybrid defaultable bond pricing models. Journal Applied Mathematical Finance, 15(3), 219–249. [5] Zagst, R., T Meyer and H Hagedorn (2007). Integrated modelling of stock and bond markets. International Journal of Finance, 19(1), 4252–4277. [6] Kalemanova,A., B Schmid and R Werner (2007). The Normal Inverse Gaussian distribution for sythetic CDO pricing. Journal of Derivatives, Spring. [7] Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7, 77–91. [8] Bielecki, T and M Rutkowski (2004). Credit Risk: Modeling, Valuation and Hedging.

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[9] Hull, J and A White (1994). Numerical procedures for implementing term structure models ii: Two-factor models. Journal of Derivatives. [10] Nelson, CR and A Siegel (1987). Parsimonious modelling of yield curves. The Journal of Business, 60(4), 473–489. [11] Arin, KP and A Mamun (2005). Is it inflation or inflation variability? A note on the stock return-inflation puzzle. Finance Letters, 3(6), 19–23. [12] Sellin, P (2001). Monetary policy and the stock market. Theory and empirical evidence. Journal of Economic Surveys, 15(4). [13] Feldstein, M (1980). Inflation and the stock market. American Economic Review, 70, 839–847. [14] Fama, EF (1981). Stock returns, real activity, inflation and money. American Economic Review, 71, 545–565. [15] Friedman, M (1977). Nobel lecture: Inflation and unemployment. Journal of Political Economy, 85, 451–472. [16] Li, DX (2000). On default correlation: A copula approach. Journal of Fixed Income, 9, 43–54. [17] Hull, J and A White (2004). Valuation of a CDO and an n-th to default CDS without a Monte Carlo simulation. Journal of Derivatives. [18] Kalemanova, A and R Werner (2006). A short note on the efficient implementation of the Normal Inverse Gaussian distribution. Working Paper. [19] Vasicek, O (1987). Probability of loss on loan portflio. Memo, KMV Corporation. [20] Bluhm, C (2003). CDO modeling: Techniques, examples and applications. Working Paper. [21] Schmid, B., R Zagst and S Antes (2006). Pricing of credit derivatives. Working Paper. [22] Zagst, R (2006). Integrated risk management — distinguished lecure series 2006. Lecture notes. [23] Standard & Poor’s (2006) Annual 2005 global corporate default study and rating transitions.

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8

CROSS ASSET PORTFOLIO DERIVATIVES

STEPHAN HÖCHT∗,‡ , MATTHIAS SCHERER∗,§ and PHILIP SEEGERER†,¶ ∗

HVB-Stiftungsinstitut für Finanzmathematik, Technische Universität München, Boltzmannstrasse 3, 85748 Garching bei München, Germany † Assenagon GmbH, Theresienhöhe 13 a, 80339 München, Germany ‡ [email protected] § [email protected] ¶ [email protected]

The dependence of extreme financial events among different asset classes is taken under consideration on a portfolio level. For this, a new product group, called cross asset portfolio derivatives, is introduced and explained in the light of related existing products and pricing methods. A classification is presented and features of these products are described. Finally, two modeling and pricing frameworks using multivariate stochastic processes and (hierarchical) copulas, respectively, are suggested.

8.1. INTRODUCTION TO CROSS ASSET PORTFOLIO DERIVATIVES This chapter focuses on the dependence of tail events, i.e., the risk of rare market events occurring interdependently. A current trend in product development is to apply credit 175

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derivative techniques on stylized insurance claims based on various assets. Hence, we aim at extending the well established concepts for the pricing of credit derivatives, such as credit default swaps (CDS), nth-to-default baskets, and collateralized debt obligations (CDO), to other asset classes in the form of new investment products. Asset classes that might serve as underlyings are equity, commodity, interest rates, and foreign exchange. Thereby, we focus on the dependence of tail events among different asset classes in these products. The following definitions were adopted from the world of credit derivatives, see [1], page 8 ff, and generalized to trigger derivatives.

8.1.1. Definitions and Examples A trigger event causes a payment stream in some derivative. The default of a company (credit event) in the case of a CDS or the drop of a stock below a specified trigger level (equity event) in the case of an equity default swap (EDS) are examples. Very unlikely trigger events, such as a steep sudden drop of an equity index (equity tail event) or the sudden default of an AAA-rated company (credit tail event), are referred to as tail events. A trigger derivative is a bilateral derivative security, where the payoff profile depends on the occurrence of a trigger event. The trigger event is defined with respect to one (or several) reference underlying(s). After the trigger event has occurred, a contingent payment is due by one of the contractual parties. In case of an asset trigger derivative, the trigger event is defined with respect to a tradable reference underlying. Single name asset trigger derivatives (SNATD) stand for only one reference underlying in the contract and are represented by trigger swaps (TS ). These are agreements between two contractual parties to exchange the risk of a reference asset hitting a predefined trigger level. More precisely, one party receives periodic premium payments as long as the trigger event has not taken place. In return, the other party receives a contingent payment at the time of the trigger event. Trigger swaps might be interpreted as EDS (or CDS) extended to other asset classes; the difference of the first to the latter being that the underlying does not default on some equity event (or credit event), instead, the trigger event is defined on a tradable asset. In the context of options, TS might also be seen as digital one-touch options or barrier options, see [2], page 561. In contrast to CDS contracts, where the reimbursement rate (RIR), i.e., one minus the recovery rate, is usually not known prior to the credit event, in a trigger swap the reimbursement rate is fixed as part of the contractual specifications. A range from 50% to 100% seems to be realistic. Whereas SNATD relates to a single reference underlying, portfolio asset trigger derivatives (PATD) have payment streams that are linked to a portfolio of underlyings. At the next level, we distinguish between mono asset portfolio derivatives and cross asset portfolio derivatives. Mono asset portfolio derivatives (MAPD) are derivatives

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with a payoff, which is contingent on trigger events referring to an underlying pool of trigger swaps from the same asset class. Generalizing mono asset portfolio derivatives to multiple asset classes leads to cross asset portfolio derivatives (CAPD). Considering the specific payment schemes, we can distinguish between nth-totrigger baskets and collateralized trigger swap obligations (CTSO). The former are portfolio derivatives with a contingent payment depending on the nth trigger event in the underlying portfolio of trigger swaps, the latter are tranched securities which are backed by a pool of TS. In case of a CTSO, the underlying portfolio is composed of I trigger swaps, where I is typically in the range of 50–200. In contrast to synthetic CDO, where the underlying CDS are linked to different credits, two trigger swaps in a CTSO portfolio might refer to the same underlying asset, however, with different trigger events/trigger levels. The portfolio’s notional, i.e., the sum of the TS’s individual notionals, can be divided into different tranches. Similar to a TS providing exposure to the trigger event risk of the reference asset, a tranche of a CTSO provides exposure to the risk of a particular amount of loss on a portfolio of assets. An investor can decide how much trigger event exposure she is willing to take from the portfolio. This is achieved by choosing a specific tranche, i.e., by choosing a particular level of subordination. In the case where only one underlying asset class is considered, these products were already introduced to the market. Examples are collateralized commodity obligations (CCO) and collateralized foreign exchange obligations (CFXO). In the case of a CFXO, with foreign exchange rate TS serving as the underlying portfolio, several TS for each foreign exchange rate (with a cascade of trigger levels) are included in the underlying portfolio. The following example illustrates the general structure of a CTSO and the involved tranche structure. Example 1 (General structure of a CTSO). Consider a CTSO consisting of 100 TS with a notional of one million US$ each. Instead of taking a linear exposure in all TS simultaneously, which corresponds to selling all TS, an investor (insurance seller) decides to invest in the (10%, 20%) tranche of the CTSO with notional size 10 million. The tranche’s boundaries are called attachment and detachment points. The resulting payoff schedule is as follows: the investor receives a regular periodic spread on the remaining outstanding notional of the invested tranche. If one TS in the portfolio is triggered, the portfolio’s notional is reduced to 99 million (assuming a reimbursement rate of 100%) and the subordination of the investor’s tranche is reduced to (9%, 19%). If the 11th trigger in the portfolio occurs, the investor’s tranche is hit for the first time. The tranche’s remaining notional reduces to nine million and a contingent payment of one million has to be paid to the protection buyer. Since spread is paid pro rata to the remaining tranche, spread payments are reduced accordingly in the following. With the 20th trigger occurring in the portfolio, the whole tranche notional is eliminated and further spread payments are stopped. Since the first loss occurs with the 11th trigger

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and the total loss is realized with the 20th, the result is an option-like payoff. This option-type behavior can be used to introduce the leverage into the investment. Due to the so-called waterfall structure of the tranched CTSO, the investor can choose different subordination levels. This choice determines the real trigger event exposure to the portfolio. A less senior tranche bears a higher trigger risk, but bearing this risk is compensated with higher spread on the outstanding notional. In case of a CDO, one often classifies the subordination in equity, mezzanine, and senior tranches. These names intend to reflect the different risk exposures. Example 2 (General structure of an nth-to-trigger basket). Consider a portfolio of 10 TS, with a notional of one million US$ each, and an investor (insurance buyer) who wants to protect herself against the first loss in the portfolio by means of a first-totrigger swap. For this, she pays periodic premium payments on a pre-specified schedule. As soon as one of the TS is triggered, premium payments stop and she receives a contingent payment in the amount of reimbursement rate times the notional. For n > 1, nth-to-trigger derivatives are defined similarly, with the first trigger replaced by the nth trigger. Figure 8.1 shows the classification of cross asset portfolio derivatives within the product group of trigger derivatives. The specific products mentioned denote examples of the respective group.

Figure 8.1 The classification of trigger derivatives.

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8.2. COLLATERALIZED OBLIGATIONS This section provides an economic classification of cross asset portfolio derivatives within the group of collateralized obligations, opposed to the structural classification in Sec. 8.1. We also stress the risks involved and compare those with existing related products in the context of collateralized obligations. We seek to introduce a simple classification, focusing on the major parameters rather than on pure theoretical constructs. We define collateralized obligations (CO) as structured and tranched investments on an underlying pool of assets. Similar to trigger derivatives we distinguish CO with respect to their underlying collateral. We call a structured and tranched investment on an underlying pool of credit assets (e.g., bonds, loans, CDS, and credit-linked notes) a collateralized debt obligation (CDO). Imposing CDO conventions on other asset classes than debt, e.g., equity, foreign exchange, commodity, and fixed income, results in a collateralized asset obligation (CAO). At the next level, we consider the source of funds for the interest paid to the investor of a CO. On the one hand, we have market value CO, where the underlying collateral is a mark-to-market instrument. On the other hand, we have cash flow CO, where the underlying collateral yields a periodic cash flow. Funding is another criterion to classify CO and we distinguish between cash and synthetic funding. The former involves a portfolio of cash assets as underlying and the ownership of the assets is transferred to the legal entity (referred to as special purpose vehicle) issuing the CO’s tranches. The reference assets of a synthetic CO are not owned but swap-like. Using the above classification, a CTSO can alternatively be defined as a synthetic cash flow CAO. Besides CTSO, there exists another class of CAO referencing to other asset classes than credit, namely collateralized fund obligations (CFO). However, as opposed to CTSO, a CFO is a market value cash CAO. Figure 8.2 shows the classification of CO regarding the distinctions made in Sec. 8.2.

8.3. A COMPARISON OF CFO WITH CTSO At first glance, CFO and CTSO share many similarities, but the structural features as well as the involved risks differ quite significantly. The following structural features are presented in [3].

8.3.1. Structural Features of CFO CFO have been known since the beginning of the 90s, referring to underlying indices on a price level, e.g., commodity indices, hedge fund indices, or equity indices. We assume CFO to fit especially the needs of real-money accounts that are interested in

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Figure 8.2 The classification of CO.

new investment technologies based on asset classes, they are familiar with. Traditional CFO, using equity and commodity indices as underlying, did not become as popular as their counterparts: cash flow CO. In 2000–2001 CFO experienced a strong growth, with hedge fund investments being included. The idea was to use a technology, which generates benefits from a combination of highly volatile and uncorrelated trading strategies (long-short equity, convertible arbitrage, event-driven, and fixed income arbitrage). On the negative side, these CFO investments included multiple fees due to the investment in tradable funds. According to [4], securitized hedge fund investments can provide the following benefits: • Attractive financing terms • Transferability of existing portfolios of fund investments • Diversification of the investor base by providing fixed income investors with market exposure linked to hedge fund returns • New opportunities for investors seeking leveraged returns in this asset class In 2003–2004, according to [3], hedge fund-linked CFO activity slowed down due to accounting issues. Still, market interest remained. The only innovative feature in these transactions, compared to usual CO, was the use of an uncommon underlying.

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8.3.2. Structural Features of CTSO From the perspective of portfolio diversification, CTSO offer the opportunity to tap the market for (correlated) trigger event risks. Tail events are the major risks in such structures, while structural features as well as the unfunded nature of such transactions are similar to the synthetic CDO market. Hence, this might be a favorable playing field for hedge funds, proprietary desks, and insurance companies, given the in general unfunded nature of the investments. The major characteristics of CTSO are the following: • • • •

Unfunded nature, trigger swaps as underlying Tail event risk as major risk factor Tail event correlation as major price input No waterfall principle necessary, due to the synthetic structure

8.3.3. The Different Risks In general, while CFO are funded investments, CTSO are unfunded, i.e., swap-like. Whereas the risk in a CFO transaction is basically market risk (volatility of the underlying), it is trigger event risk in case of a CTSO, due to the swap character of the underlying. Hence, a CTSO and a CFO, referring to the same underlyings, bear a different risk/return profile. One can construct a CFO which refers to specific markto-market instruments such as commodity or equity indices. Using the corresponding swap contracts (with some defined trigger event) on these indices as an underlying in a CTSO, the risk profile of the transaction changes. While the CFO reacts highly sensitive to daily mark-to-market changes in the underlying indices, a CTSO investment better withstands small short-term price fluctuations, but is skewed to trigger events. To sum up, the major difference between a CFO and a CTSO is that volatility is the appropriate risk parameter in the former, and tail event risk (depending on the trigger event of the underlying swap) is suitable to describe the risk profile of the latter.

8.3.4. Correlation of Tail Events in CTSO Since the dependence of tail events is a major pricing input, one needs to stress specific correlation assumptions. For example, in CCO, one might argue that specific macroeconomic shocks lead to a trigger event. Tail event correlation in different hedge funds, for instance, is mostly linked to systematic, regulatory, and legal risks. Consequently, a cross-asset CTSO offers the opportunity to benefit from (thus far) non-tradable asset classes like inter-market tail event correlation. The traditional correlation pattern between asset classes is changing in a cross asset CTSO regime, with idiosyncratic risk factors losing in importance compared to

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systematic risk factors. The risk/return profile of CTSO tranches differs significantly from the one of the underlying assets, shifting risk away from volatility toward tail events. The impact of changes in correlation depends on the specific tranche as the following two examples illustrate: (1) A long position in an equity piece of a CCO, where the trigger levels are set below and above the current spot prices, is long in joint probabilities of large price swings in the commodity market, i.e., the buyer of that piece expects rising joint probabilities of large price movements. (2) A short position in a senior piece of a cross asset CTSO is long in joint probabilities of distortions in the underlying markets, i.e., the short seller expects the likelihood of a market crash to rise. Considering inter-market correlation is not new, but the securitization in terms of the aforementioned products is. However, in order to derive the correlation of trigger events in different financial markets, analyzing historical data is not sufficient. Mathematical models, such as structural and/or copula-based models known from credit markets, have to be implemented in order to get an intuition for the traded tail event correlations in CTSO.

8.4. PRICING CROSS ASSET PORTFOLIO DERIVATIVES 8.4.1. Pricing Trigger Swaps In the following, we introduce the basic notations of TS and state a pricing formula for the annualized fair spread sf , which is also known as par spread. This spread is computed such that both contractual parties can enter the TS at zero cost, i.e., without upfront payment, at initiation. • • • • • •

τ: the trigger time of the underlying N: the nominal of the TS RIR: the reimbursement rate of the TS s: the annualized spread of the TS T : the maturity of the contract rt : the deterministic term-structure of risk free interest rates

At initiation of the contract, a payment schedule T = {0 < t1 < · · · < tn = T } is specified, where t0 = 0 is the settlement date. Assuming, for notational simplicity, the usual (for credit derivatives) quarter-yearly premium payments, this implies tk = k/4 for k = 0, . . . , 4T and tk = tk − tk−1 = 1/4. For a given pricing measure Q and the respective risk free interest rates rtk , continuously compounded, the value of the

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expected discounted premium leg (EDPL) of the TS is obtained via:
EDPL = EQ

n 

 N · s · tk · e−rtk tk 11{τ>tk }

k=1

N · s  −rt tk e k Q(τ > tk ). 4 k=1 n

=

(8.1)

The expected discounted trigger leg (EDTL) of the TS is given by
EDTL = EQ RIR · N

n 

 e−rtk tk 11{tk−1 <τ≤tk }

k=1

= RIR · N

n 

e−rtk tk Q(tk−1 < τ ≤ tk ).

(8.2)

k=1

Hence, from the perspective of the protection buyer, the expected value of a TS at initiation of the contract is given by TS(0, T) = RIR · N

n 

N · s  −rk tk e Q(τ > tk ). 4 n

e−rk tk Q(tk−1 < τ ≤ tk ) −

k=1

k=1

The par spread sf , computed to allow both parties to enter the TS at zero cost, is found by solving TS(0, T) = 0 for s, i.e., sf =

RIR

n −rtk tk Q(tk−1 < τ ≤ tk ) k=1 e .  n 1 −rtk tk Q(τ > tk ) k=1 e 4

(8.3)

Equations (8.1) and (8.3) do not account for accrued interest, which might additionally be included as part of the terms of contract. If accrued interest is stipulated, an additional premium payment is added to the premium leg, which is proportional to the time between the trigger event τ and the last payment date tk−1 and is conditional on the event that the trigger falls into the respective period (tk−1 , tk ].

8.4.2. Pricing nth-to-Trigger Baskets We consider a basket of I underlying TS and assume a pricing measure Q to be given. If not stated differently, we assume the same reimbursement rate RIR for all underlying TS. The order statistic of the stochastically dependent trigger times τ1 , . . . , τI is denoted by τ (1) ≤ · · · ≤ τ (n) ≤ · · · ≤ τ (I) . The expected discounted trigger and

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premium legs of an nth-to trigger contract are given by
n   N · s(n) · tk · e−rtk tk 11{τ (n) >tk } EDPL(n) = EQ k=1

=

(n)

EDTL

N ·s 4


(n)

n 

e−rtk tk Q(τ (n) > tk ),

k=1

= EQ RIR · N

n 

(8.4) 

−rtk tk

e

11{tk−1 <τ (n) ≤tk }

k=1

= RIR · N

n 

e−rtk tk Q(tk−1 < τ (n) ≤ tk ).

(8.5)

k=1

The fair spread sf(n) is again obtained from equating both legs and solving for sf(n) . This yields  RIR nk=1 e−rtk tk Q(tk−1 < τ (n) ≤ tk ) (n) sf = . (8.6) n 1 −rtk tk Q(τ (n) > tk ) k=1 e 4

8.4.3. Pricing CTSO In what follows, we introduce a mathematical notation of the terms verbally explained earlier in Example 1. Assume the CTSO portfolio consists of I TS contracts, indexed by i. This portfolio is segmented in J tranches, indexed by j. The conventions regarding the payment schedule are similar to the ones in the previous sections. Moreover, we need • • • • • • • • • •

TSi : a TS relating to underlying i τi : the trigger time of underlying i Ni : the nominal value of TSi RIRi : the reimbursement rate of underlying i lj , uj : the attachment and detachment point of tranche j M: the total nominal value of the CTSO j Mt : the remaining nominal of tranche j at time t Lt : the cumulated loss of the CTSO portfolio up to time t j Lt : the cumulated loss in tranche j up to time t j s : the annualized spread of tranche j

For notational simplicity, the structure is simplified by assuming a homogeneous portfolio with respect to the reimbursement rate and nominal of each TS, i.e., RIR = RIRi and N = Ni for all i ∈ {1, . . . , I}. The crucial quantity for the description of all

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payment streams is the portfolio-loss process Lt , which is defined as Lt = RIR · N

I 

t ∈ [0, T ].

11{τi ≤t} ,

(8.7)

i=1

The total nominal value of the CTSO is given by M = I · N. This nominal value is segmented using the attachment and detachment point of each tranche. More precisely, we define the partition 0 = l1 < u1 = l2 < · · · < uJ−1 = lJ < uJ = M. Based on the overall portfolio loss, the loss affecting tranche j results in j

Lt = min{max{0, Lt − lj }, uj − lj },

t ∈ [0, T ],

j ∈ {1, . . . , J}.

(8.8)

The remaining nominal of tranche j is determined by j

j

Mt = (uj − lj − Lt ),

t ∈ [0, T ],

j ∈ {1, . . . , J}.

(8.9)

Given the payment schedule T = {0 = t0 < t1 < · · · < tn = T } and the pricing measure Q, the expected discounted premium and trigger legs of tranche j are given by
n   j (j) (j) −rtk tk s · tk · e Mtk , j ∈ {1, . . . , J}, EDPL = EQ (8.10)
(j)

EDTL

= EQ

k=1 n 

 −rtk tk

e

j (Ltk

−

j Ltk−1 )

,

j ∈ {1, . . . , J}.

(8.11)

k=1

The fair spread of tranche j of the CTSO results in   j j n −rtk tk e (L − L ) EQ t t k=1 k k−1 (j)   , sf = j n −rtk tk EQ Mtk k=1 tk · e

j ∈ {1, . . . , J}.

(8.12)

8.4.4. Modeling Approaches For the pricing of derivatives on a portfolio (or basket) of TS, the joint distribution of all trigger times (τ1 , . . . , τI ) is required. On a conceptual level, there are two methods to incorporate dependence among the trigger times that seem promising for an adaption from the world of credit derivatives to the current situation. In what follows, both methodologies are introduced and discussed.

8.4.4.1. The structural approach Dependence is introduced to the model by specifying a suitable I-dimensional stochastic process. The univariate marginals either directly represent the respective underlying in the case of a traded asset, e.g., a stock index, or might be interpreted as

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a proxy variable. An example for the latter is the firm-value interpretation used in structural-default models. Typically, the trigger time τi is then defined as some first-passage time of the respective univariate marginal below or above some trigger level. Depending on the choice of univariate model, one has different options to couple the marginals. For instance, if the univariate marginals are driven by Brownian motions, a natural approach is to assume these Brownian components as being correlated. If more general Lévy processes are used, common jumps might also be used to introduce dependence. References for the modeling of stocks and credit products are [2] and [1]. References for the modeling of interest rates and energy, as examples for other underlyings than stocks and credit, are the books [5] and [6], respectively. The major advantage of the structural approach is that it is fully dynamic, since all relevant objects are modeled as marginals of one high-dimensional stochastic process. Besides the static pricing of all univariate and portfolio derivatives at a time, this additionally induces a dynamic which can be useful for the pricing of options on these derivatives. However, the structural approach has some drawbacks. Firstly, it is not obvious how dependence shall be introduced to univariate processes from different model classes, e.g., classical diffusions and pure jump processes. Secondly, analytically solving a non-trivial multi-dimensional first-passage time problem is virtually impossible. This becomes evident when the example of [7] is considered, who succeeds in presenting a (highly non-trivial) formula for the joint probability of two correlated diffusions to remain above some threshold level for a given time. Hence, pricing portfolio trigger swaps requires a Monte Carlo engine, which is slow and biased if correlated processes are sampled on a discrete grid and monitored for some threshold level. Thirdly, for dependent stochastic processes, it is typically not possible to separate the parameters of the marginals from those specifying the dependence structure; an exception are processes purely driven by correlated Brownian motions. This complicates the calibration of the model. Finally, the dependence structure of the resulting vector (τ1 , . . . , τI ) is unknown for all non-trivial examples.

8.4.4.2. The copula approach Combine the marginal distributions of the univariate trigger times by means of a copula, i.e., by means of an I-dimensional distribution function on the unit I-cube. Since an introduction to copulas is beyond the scope of this chapter, the interested reader is referred to the textbooks [8]– [10]. The following approach was first used in the context of credit risk-modelling by [11] and [12]. For each trigger time, it is assumed that the (risk–neutral) marginal distribution is known and abbreviated as Q(τi ≤ t) = p¯ i (t) = 1 − pi (t). On a univariate level, each trigger time admits the

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canonical construction τi = inf{t ≥ 0 : pi (t) ≤ Ui },

Ui ∼ Uniform(0, 1).

(8.13)

This representation is especially convenient for simulations, since sampling τi boils down to sampling the univariate trigger variable Ui and solving, assuming a continuous and strictly decreasing pi , pi (t) = Ui for t. Considering all random times τ1 , . . . , τt at once it is possible to introduce dependence to the model by assuming the vector of trigger times (U1 , . . . , UI ) to be distributed according to some I-dimensional copula C. Note that, by the defining properties of a copula, it is guaranteed that the univariate marginal distributions (of all τi ) are preserved, which is highly convenient in a calibration. If sampling strategies for C are known, it is straightforward to sample from the vector of random times (τ1 , . . . , τI ): sample (U1 , . . . , UI ) ∼ C and locate each univariate trigger time τi . The major advantage of the copula approach is the separation of marginals from the dependence structure. Firstly, this allows the use of different model classes for the univariate marginals. For instance, one might use a classical geometric Brownian motion for one asset class, a pure-jump Lévy process for a second asset class, and an intensity model for the asset class credit. For each of these classes, one can derive univariate distributions for the respective τi ’s, which are then coupled by means of the copula C. Using copulas with tail-dependence and singularities allows to include Armageddon scenarios (with multiple trigger events at a time) and trigger clusters. Secondly, the approach is well-suited for simulations, as long as sampling routines for the copula are known. Thirdly, the calibration of the multivariate model is simplified when the univariate marginals can be calibrated individually, followed by a calibration of the dependence structure in a second step. However, there are some shortfalls of the copula approach that need to be addressed. Firstly, the model is static in the sense that a multivariate distribution of (τ1 , . . . , τI ) is specified, but no dynamic model leading to it. While this is unproblematic for pure pricing problems, it rules out applications such as the simulation of the time evolution of resulting model spreads. Secondly, it is not clear which class of copulas is best suited for modeling the dependence structure of the respective market. As long as there is no liquid market for tail dependence, it is not clear which copula should be preferred and how this copula should be parametrized. Let us briefly give some references for the required sampling of copulas. In high dimensions, sampling strategies for some elliptical copulas (including the Gaussand t-copula) are known, see e.g., [10]. For exchangeable (meaning that all pairwise correlations are identical) Archimedean copulas the standard reference is [13], an important contribution is [14]. Recently, some authors considered hierarchical (nested) Archimedean copulas and sampling strategies for these, see e.g., [15]– [17]. Sampling Marshall–Olkin type copulas in high dimensions is considered in [18].

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8.4.5. An Example for an nth-to Trigger Basket In the following, we illustrate both pricing methodologies using an example for a trigger derivative of medium complexity. Note that this example is more general than cross asset trigger derivatives as introduced in Sec. 8.1, since it combines asset trigger events with credit events. Example 3 (An nth-to trigger basket over two asset classes). In this example, an nth-to trigger basket consisting of six underlying TS is considered. The payment frequency for premium payments is quarterly, the contract settles on 10 February 2009, with a maturity of T = 5 years. The reimbursement rate is set to RIR = 50% for all TS and for the CDS calibration. All TS contribute the same unit nominal to the portfolio. (1) Stock indices: • DAX 30, initial value S1,0 = 4636 • EuroStoxx 50, initial value S2,0 = 2338 • SMI, initial value S3,0 = 5219 An equity event is triggered when some index falls below a trigger level, which is defined as 20% of the initial value, i.e., li = 0.2 · Si,0 . (2) CDS: • 5 year CDS on Allianz SE, initial spread 77.50 bps • 5 year CDS on Linde AG, initial spread 90.00 bps • 5 year CDS on E.ON N, initial spread 63.33 bps Credit events are triggered according to the iTraxx CDS convention.

8.4.5.1. A pricing exercise of Example 3 (structural approach) We model the evolution of each stock index i = 1, 2, 3 using a standard geometric Brownian motion Si . Considering CDS spreads, we follow [19] and assume a simple structural default model with geometric Brownian motion as firm-value process Vi and monitor continuously for default. Since all processes have the same structure and are driven by Brownian motions, it is natural to introduce dependence by assuming correlated Brownian components. Summarizing, we consider the vector-valued process (S1,t , S2,t , S3,t , V1,t , V2,t , V3,t ), where under Q S ), dSi,t = Si,t (rdt + σS,i dWi,t

Si,0 > 0, i = 1, 2, 3,

(8.14)

dVi,t = Vi,t (rdt +

Vi,0 > 0, i = 1, 2, 3.

(8.15)

V σV,i dWi,t ),

All equity events are defined as τi = inf{t ≥ 0 : Si,t ≤ li }, where li = 0.2 · Si,0 for the respective contract. Credit events are taken as τi = inf{t ≥ 0 : Vi,t ≤ di } for

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some suitable default threshold di , i = 1, 2, 3. Solving the model via a Monte Carlo simulation requires the following steps. (1) Initialize the simulation, i.e., choose the number of simulation runs SR and the grid-size . (2) For each simulation run n = 1, . . . , SR do: (a) Simulate the vector-valued process (S1,t , S2,t , S3,t , V1,t , V2,t , V3,t ) on the grid 0 <  < 2 < · · · < T . (b) At each point of the grid, monitor for eventual trigger times below (or above, for other contract specifications) the equity or default thresholds, respectively. (3) Compute and store the discounted premium and trigger leg of the current simulation run n. (4) Estimate the expected discounted premium and trigger leg as the arithmetic mean of the legs in each run. (5) Set the (estimated) fair spread as the quotient of the estimated trigger and premium leg. Note that such a continuous model might underestimate the risk of extreme (joint) movements. This fact is well known in the univariate case, see e.g., Chapter 1 of [20], and is further amplified for joint movements, due to a lack of tail dependence of the multivariate normal distribution. To overcome this, one might try to use jump processes instead of geometric Brownian motions.

8.4.5.2. A pricing exercise of Example 3 (copula approach) We again model the evolution of each stock index using a standard geometric Brownian motion. Considering CDS spreads, we use a simple reduced-form model with constant default intensity. Since dependence is introduced in a second step by means of a copula, we use independent processes of the form S dSi,t = Si,t (rdt + σS,i dWi,t ),

Si,0 > 0, i = 1, 2, 3,

(8.16)

to model the stocks under Q. All equity events are again defined as τi = inf{t ≥ 0 : Si,t ≤ li }. Credit events are defined as τi = inf{t ≥ 0 : exp (−λi t) ≤ Ui }, where U1 , U2 , U3 are dependent (univariate uniform) triggers and λ1 , λ2 , λ3 denote the (constant) default intensities. Solving the model via Monte Carlo requires the following steps. (1) Initialize the simulation, i.e., choose the number of simulation runs SR. (2) For each stock index, compute survival probabilities Q(τi ≥ t) from the parameters of the respective process. For this, the required probability of a Brownian motion

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with drift not falling below some threshold level for a given amount of time is given in [21], page 61. (3) For each simulation run n = 1, . . . , SR do: (a) Simulate the dependent trigger variables (U1S , . . . , U1C , . . . ) ∼ C. (b) Compute the resulting trigger times. (4) Compute and store the discounted premium and trigger leg of the current simulation run n. (5) Estimate the expected discounted premium and trigger leg as the arithmetic mean of the legs in each run. (6) Set the (estimated) fair spread as the quotient of the estimated trigger and premium leg. More details on the copula approach in the context of credit–risk modeling, including asymptotic confidence intervals and the implementation of several nestedArchimedean copulas, are given in [22].

8.4.5.3. Resulting model spreads The input for our calibration are quotes obtained from Reuters on 10 February 2009 (2 p.m.). The continuously compounded interest rate is chosen as r = 0.024. The marginals of the stock indices and individual stocks are calibrated as follows: the parameter σS,i of stock index i is calibrated as implied volatility from out-ofthe-money put options. For this, a five-year maturity and strike close to li would be ideal. Since these were not liquidly traded, we settled with a one year maturity and a strike in the order of two-thirds of the initial value. The results are (σS,1 , σS,2 , σS,3 ) = (38%, 38%, 31%) for the stock indices and (65%, 42%, 47%) for the individual stocks. Note that these out-of-the money options typically trade at a higher implied volatility compared to at-the-money options. The resulting triggersurvival probabilities for the stock indices are (90%, 90%, 97%). Considering CDS spreads, the stock price process is taken as a reference for the firm-value process and is calibrated in the very same way as the stock index. In a second step, the default threshold di is chosen such that the five-year CDS spread, being monotonically increasing in di , is matched. Note that CDS spreads are computed under the simplifying assumptions of no accrued interest and a deterministic recovery rate R = (1 − RIR) = 50%. Translated in terms of probabilities, this corresponds to a five-year implied survival probability of about (92%, 91%, 94%), respectively, for the three companies. For the copula approach, the marginal intensity λi for the ith CDS is similarly chosen such that the intensity model matches the observed five-year CDS spread. Since market prices of correlated tail risk are not (yet) observable, a calibration to existing derivatives is not possible. Hence, we computed model prices for a large

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spectrum of dependence structures. For this, we assumed a homogeneous correlation ρ1 among each pair of stock processes and firm-value processes, i.e., dWi,t dWj,t = ˆ j,t = ρ1 dt, i  = j, and a homogeneous correlation ρ2 < ρ1 among stocks and ˆ i,t d W dW ˆ j,t = ρ2 dt, i ∈ {1, 2, 3}. For the copula framework, firm-value processes, i.e., dWi,t d W we chose the Gauss copula with a block-matrix as correlation structure such that again the groups stocks and credit are correlated with ρ1 , and stocks to credit are correlated with ρ2 < ρ1 . Moreover, we included a partially nested Archimedean copula of the Gumbel family, see [15] and [16]. We use two levels and three members in each of the two (homogeneous) groups, i.e., C(u1 , . . . , u6 ) = C(C(u1 , u2 , u3 ; ψ1 ), C(u4 , u5 , u6 ; ψ1 ); ψ2 ) = ψ2 (ψ2−1 (ψ1 (ψ1−1 (u1 ) + ψ1−1 (u2 ) + ψ1−1 (u3 ))) + ψ2−1 (ψ1 (ψ1−1 (u4 ) + ψ1−1 (u5 ) + ψ1−1 (u6 )))),

ui ∈ [0, 1], (8.17)

where ψi (t) = exp(−t 1/ϑi ) and ϑi ∈ [1, ∞), i = 1, 2. A sufficient condition for (8.17) being a copula, is that the nodes ψ2−1 ◦ ψ1 have completely monotone derivatives. For the nested Gumbel copula, this condition is equivalent to ϑ2 ≤ ϑ1 . NestedArchimedean copulas might be interpreted as follows: any two members of the groups stocks and credit are coupled via an inner Archimedean copula, and any two members of different groups are coupled via an outer Archimedean copula. Nested Archimedean copulas are appealing for the modeling of cross asset portfolio derivatives, since members of the same asset class can be coupled via inner copulas, which are then connected via one outer copula. For simplicity, we choose a single parameter ϑ1 for both groups (stocks and credit) and a second parameter ϑ2 ≤ ϑ1 for the outer copula. This implies that the dependence within a group is at least as large as the dependence among members of different groups. The resulting model prices as functions of the dependence parameters are presented in Figs. 8.3 through 8.5. An interpretation of the resulting model spreads is given in what follows: • An interesting observation is the monotonicity of spreads with respect to the dependence ordering within a group (stocks and credit) and in between the groups. On a qualitative level, these results are parallel for all pricing methods and therefore interpreted at once — ignoring minor differences. First-to-trigger spreads are decreasing in dependence, which is consistent over all models and the two parameters of dependence. The reason behind this observation is that multiple trigger events, but also no trigger event at all, are more likely for a large correlation when trigger probabilities are kept fix. The second-to-default spread is increasing in group dependence but slowly decreasing in dependence among stocks and credit. The third-to-trigger spread is increasing in group dependence and almost flat in extra-group dependence.

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Figure 8.3 Model spreads from the structural approach (25,000 runs).

The {4,5,6}th-to trigger spreads are all increasing in both dependence parameters, since more dependence (of either sort) makes multiple defaults more likely, with according influence on spreads. • Comparing the structural approach and the Gauss-copula approach to the Gumbel copula shows that kth-to trigger spreads for higher k are much larger for the latter. This is explained by the positive upper-tail dependence of the Gumbel family. Heuristically speaking, this property means that the trigger variables Ui have a positive probability for being simultaneously close to one. Translated into the pricing framework, this implies multiple trigger events within one simulation run, hence, positive spreads for higher kth-to trigger spreads. Extending the simple structural

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Figure 8.4 Model spreads from the Gauss-copula model (250,000 runs).

model using jump processes might resemble this property. Another observation is that spreads for the limiting case of independence agree within all models (up to a small Monte Carlo noise). • A final remark addresses the required computation time for both approaches. The structural approach is based on the simulation of a path of a multivariate process (on a fine grid) in each simulation run. Compared to the copula approach, which only requires a sample from the respective copula in each run, it is therefore not surprising that the computation time for the evaluation of one option price with 250,000 simulation runs differs massively. Our implementation in Matlab on a standard PC requires about 3813 s for the structural and 11 and 18 s for the two copula approaches, respectively.

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Figure 8.5 Model spreads from the Gumbel-copula model (250,000 runs).

8.5. OUTLOOK Innovative instruments in the credit derivatives universe allow investors to construct positions which refer to specific aspects of credit risk, e.g., spread volatility or (correlated) default risk. While the standard valuation approach for these instruments is still skewed to a pure transaction-driven style, i.e., ignoring interdependencies with other instruments and markets, we expect the portfolio perspective to soon become more important. For such a portfolio treatment, cross asset portfolio derivatives become

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interesting. Allowing for collateralized products referring to other asset classes than debt offers new opportunities for investments and risk management. Cross asset portfolio derivatives typically smooth the risk/return profile of volatile asset classes. Whereas the classical Capital Asset Pricing Model (CAPM) is still the appropriate model to optimize the underlyings regarding return, risk, and correlation, this portfolio management style changes in the case of cross asset portfolio derivatives. In the CAPM world, the risk/return optimization is already implemented in the market portfolio, which is defined as being efficient, while correlation between the portfolio constituents is defined as the co-movement of asset prices. The individual risk perception is simply reflected by the investment share of the market portfolio, and the risk-free asset. In a cross asset portfolio derivatives world, the underlying portfolio is selectable without the usual µ/σ-constraints. Correlation is rather joint trigger probability and, e.g., the specific tranche investment in case of a CTSO is done in line with the individual risk perception. Correlation, defined as co-movement of assets, is replaced by the correlation of tail events, which induces a different view on the risk/return optimization. Systematic shocks, like the sub-prime crisis starting in 2007, are a crucial risk factor for a cross asset portfolio derivative, whereas market fluctuations and cyclical moves are of minor importance for the performance of such derivatives. To sum up, cross asset portfolio derivatives might be an interesting investment product or portfolio management tool. There should not be any concerns regarding demand/supply patterns. Both, demand and supply for tail event risk is immense, given the huge variety of players in this market and, e.g., the recent financial crisis or the rise/decline of the oil price. Traditional tail-event risk-driven companies (insurance sector) are probably skewed to selling this kind of risk to players like hedge-funds, banks, and fund managers, who can implement the attractive trading positions, but also optimize the multi-asset portfolios, using cross asset trigger derivatives.

8.6. CONCLUSION An extension of popular portfolio credit derivatives to more general trigger derivatives is presented. The introduced class of cross asset portfolio trigger derivatives allows to securize the risk of correlated tail events in the form of swap-like insurance claims. Two pricing methodologies, transferred from the world of credit derivatives, are discussed and illustrated by an example of medium complexity. It is shown how sensitive model prices are with respect to the assumed concept of dependence. On the one hand, this should be seen as a word of warning from the perspective of model risk. On the other hand, one should keep in mind that typical cross asset portfolios contain exactly these correlated tail risks. Hence, a securitization of correlated tail risks might be interesting

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for both insurance seller and buyer. A standardized market of these derivatives would then allow to calibrate the models to the market’s perception.

Acknowledgments We thank Dr. Jochen Felsenheimer of Assenagon and Dr. Philip Gisdakis of UniCredit MIB for their initial thoughts on this product group. Moreover, we thank Marius Hofert (Universität Ulm) for valuable remarks on earlier versions of the manuscript. Views expressed in this paper are those of the authors and do not necessarily reflect the positions of the respective employers.

References [1] Schönbucher, PJ (2003). Credit Derivatives Pricing Models: Models, Pricing and Implementation, 1st edn. Wiley Finance Series, John Wiley and Sons, Inc. [2] Hull, J (2004). Options, Futures, and Other Derivatives. London: Prentice-Hall International, Inc. [3] Felsenheimer, J (2006). The wonderful world of CAOs (strategy update). Technical report, UniCredit MIB Global Credit Research. [4] Standard and Poor’s (2006). CDO spotlight: Global criteria for securitizations of funds of hedge funds. Standard and Poor’s, Structured Finance. [5] Brigo, D and F Mercurio (2001). Interest Rate Models — Theory and Practice. Springer, Finance, Berlin: Springer. [6] Benth, FE, JS Benth and S Koekebakker (2008). Stochastic Modeling of Electricity and Related Markets. World Scientific. [7] Zhou, C (2001). An analysis of default correlations and multiple defaults. Review of Financial Studies, 14, 555–576. [8] Joe, H (1997). Multivariate Models and Dependence Concepts. New York: Chapman & Hall. [9] Nelsen, RB (1998). An Introduction to Copulas, 1st edn. Berlin: Springer. [10] McNeil, AJ, R Frey and P Embrechts (2005). Quantitative Risk Management. Princeton Series in Finance, Princeton University Press. [11] Li, DX (2000). On default correlation: A copula function approach. The Journal of Fixed Income, 9(4), 43–54. [12] Schönbucher, PJ and D Schubert (2001). Copula-dependent defaults in intensity models. URL http://papers.ssrn.com/sol3/papers.cfm?abstract_id= 301968. Working Paper. [13] Marshall, AW and I Olkin (1988). Families of multivariate distributions. Journal of the American Statistical Association, 83(403), 834–841. URL http://www. jstor.org/stable/pdfplus/2289314.pdf. [14] McNeil, AJ and J Neslehova (2009). Multivariate Archimedean copulas, d-monotone functions and l1-norm symmetric distributions. The Annals of Statistics, 37, 3059–3097. [15] McNeil, AJ (2008). Sampling nested Archimedean copulas. Journal of Statistical Computation and Simulation, 78(6), 567–581. [16] Hofert, M (2008). Sampling Archimedean copulas. Computational Statistics and Data Analysis, 52(12), 5163–5174. doi:10.1016/j.csda.2008.05.019.

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[17] Hofert, M (2008). Efficiently sampling Archimedean copulas. Working Paper. [18] Mai, JF and M Scherer (2009). Lévy-frailty copulas. Journal of Multivariate Analysis, 100(7), 1567–1585. [19] Black, F and J Cox (1976). Valuing corporate securities: Some effects of bond indenture provisions. Journal of Finance, 31, 351–367. [20] Cont, R and P Tankov (2003). Financial Modelling with Jump Processes, 1st edn. Chapman & Hall/CRC Press. [21] Musiela, M and M Rutkowski (2004). Martingale Methods in Financial Modelling (Stochastic Modelling and Applied Probability). Springer. [22] Hofert, M and M Scherer (2010). CDO pricing with nested Archimedean copulas. To appear in Quantitative Finance.

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Part II

Alternative Strategies

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9

DYNAMIC PORTFOLIO INSURANCE WITHOUT OPTIONS

DOMINIK DERSCH Rumfordstr. 6, 80469 München, Germany [email protected]

Dynamic portfolio strategies are an interesting alternative to classical option-based investment and protection strategies. One of the most prominent techniques is Constant Proportion Portfolio Insurance (CPPI). In this chapter, we provide a review of various techniques and formulate a general framework for investment and protection strategies. The common feature of this strategy is that it empowers the investor to replicate various option like pay-off profiles without the usage of options. These strategies may replicate a simple floor type or advanced path-dependent look-back options that implement all-time-high strategies with a given participation rate. We illustrate the different strategies that employ features like various types of lock-in, trailing, leverage, and risky portfolio strategies with historical simulations. We include features that allow the simulation under realistic market conditions taking into account transaction costs and the avoidance of excessive rebalancing through transaction filters. We discuss the use of exchange traded funds (ETF) to invest in broadly diversified multi-asset portfolios. The goal of this chapter was to illustrate different protection strategies and to show how a practical implementation of these strategies could look like. This chapter can serve as a guideline for simple spread-sheet models.

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9.1. INTRODUCTION Institutional investors who require portfolio protection or a minimum absolute performance target are facing the problem to trade off risk and return. On the one hand, exposure to high return risky asset classes is desirable; on the other hand, the downside potential that accompanies higher risk should be kept at the investor’s level of comfort. Holistically, the tasks of asset allocation and risk management are closely linked and should not be performed in two separate steps, as this leaves the investor most likely with sub-optimal solutions. However, a fairly large arsenal of risk management techniques has been developed that work completely independent of the investment strategy and respective asset classes. The sole requirements are that the asset classes are investible and allow — at least in theory — a liquid market and continuous trading. The selection of the underlying depends on the specific requirements of the investor with respect to holding period, taxation, investment guidelines, etc., and the asset class itself. Examples of different underlyings are direct investments in stocks or bonds, futures, and funds. It is important to note that the investment universe and the respective protection strategy have to be closely aligned. Investments with large bid–offer spreads would require a strategy with a low reallocation frequency. For the sake of simplicity, this chapter focuses on this two-step approach described above: Step 1 asset allocation, Step 2 risk management framework. Throughout this chapter, the first step of asset allocation is simulated by investing in a set of indices. The performance of each investment is assumed to follow the performance of the index time-series. The considered investment universe includes stock, bond, commodity, and hedge fund indices. Our choice of the investment universe poses no restriction on the second step of the investment process — the risk management framework. Within the fairly general requirements of “investability” and liquidity, any asset classes and (propriety) trading system may be embedded in this risk management framework. The recent history of financial markets posed a huge challenge to portfolio managers as reflected in massive declines in asset values, historical highs in volatility, and a break down in correlations. Dynamic portfolio strategies are an interesting alternative to classical option-based investment and protection strategies that allow one to cope with the market turmoil. One of the most prominent techniques is Constant Proportion Portfolio Insurance (CPPI) [2, 3]. CPPI went out of fashion because of a number of drawbacks like the fixed time horizon, the inability to take profits and recover from a major draw-down. One of the major shortcomings — the pro-cyclical behavior — has been blamed for huge market movements and the stock market crash of 1987. However, a number of advanced features allow one to overcome the shortcomings of the first generation model and allow a practical application in asset management.

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A comparison of option-based strategies — mentioned above — and CPPI is shown in [1]. The mature market for exchange traded funds (ETF) gives access to a huge universe of different asset classes that may be traded with high liquidity and low bid–offer spreads. This enables the investor to apply advanced portfolio insurance strategies as described above and investment in a broadly diversified universe. The remainder of this chapter is structured as follows. In the next two sections, we review simple portfolio strategies and plain vanilla CPPI and illustrate them with historical simulations. In Sec. 9.4, we formulate a more general framework of CPPI with various features like different types of lock-in, trailing, leverage, and risky portfolio rebalancing strategies. We include features that allow simulation under realistic market conditions taking into account transaction costs and the avoidance of excessive rebalancing through transaction filters. We further show that our framework also includes an extension of CPPI named TIPP [4]. In Sec. 9.5, the strategies are illustrated using historical simulations. In particular, we investigate how the different strategies cope with the historical market evolution. In Sec. 9.6, we discuss the use of exchangetraded funds (ETF) to implement protection strategies. This chapter concludes with final remarks on different risk transfer mechanisms of option strategies versus dynamic portfolio strategies.

9.2. SIMPLE STRATEGIES 9.2.1. Buy-and-Hold Probably the most simple and most common risk management framework is the buyand-hold strategy. Not just among retail investors, either deliberately or not deliberately buy-and-hold is a widespread approach. Many ambitious strategies will eventually drift into passive sit-and-wait strategies as the investor sits out long periods of negative performance or hesitates to take profits on time. Cheekily, buy-and-hold or strategic trades very often stem from short-term tactical trades turned bad. The only reason why this strategy remotely qualifies as some kind of protection strategy is that in the absence of leverage the total loss is limited to the initial investment. However, buy-and-hold has performed well over long periods and across many asset classes.

9.2.2. Stop-Loss A stop-loss strategy is the first non-trivial step toward a risk management framework. Here, we distinguish between an investment target IT on a present value base and a target on a given time horizon T . In the first case, the position has to be switched into a

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risk-free investment matching the investment horizon if the portfolio value falls below the target value at any time PVportfolio (t) > IT .

(9.1)

In the second case, the portfolio value must not fall below PVportfolio (t) > e−r(T −t) · IT

(9.2)

PVportfolio (t) − e−r(T −t) · IT > 0.

(9.3)

or equivalently

Here, T − t is the time until the investment horizon is reached and r is the risk-free rate of this time span. The second case is slightly more complex. Here, we must ensure that the investor reaches his/her investment target at maturity. Therefore, the portfolio value must not fall below a certain floor value given by the right hand side of Eq. (9.2). In case, the stop-loss level is reached, the portfolio must be liquidated and invested in the risk-free asset. Investing the floor value with the risk-free rate will ensure the given target value at maturity. The implementation of a stop-loss strategy requires monitoring both the portfolio performance and the total return of the riskfree investment. The risk-free investment is usually implemented with treasury bills or bonds with a maturity matching the investment horizon. The re-investment of coupons paid until maturity must also be considered. In case, the stop-loss level is reached, a single portfolio re-allocation occurs. Strictly speaking, stop-loss is therefore not a dynamic strategy. In the above analysis, we assume that the investments and the target level are in the same consolidation currency. Otherwise, the respective FX spot or forward rates have to be additionally monitored. Please note that a stop-loss strategy with a target value of zero is the same as the above buy-and-hold strategy.

9.2.3. The Bond Floor Strategy The stop-loss strategy bears the risk that the portfolio value may be exposed to large volatility. In addition, it carries a short fall or gap risk. This is the risk that the stop level is missed in large market movements and the investor is left with a final portfolio value below the target. The bond floor strategy takes a more cautious approach. Here, we only invest in the risky portfolio the amount we are willing to lose in the first place. Let us take Eq. (9.3) at the beginning of the investment horizon PV portfolio (t = t0 ) − e−rT · IT = C.

(9.4)

C is the amount given by the difference between the initial investment and the present value of the target amount. Here, we assume an investment in a risk-free bond with a

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maturity matching the investment horizon. The bond floor strategy is usually implemented with a target value close to 100%. According to the above equation, a given investment of EUR 100, a target value of EUR 101 in one year, and a risk-free rate of 2.5% would leave us with a risk budget of EUR 1.5 and a bond investment of EUR 98.5. That means EUR 1.5 is invested with a buy-and-hold strategy in a risky investment. A total loss of the risky investment would still guarantee the target amount of 101 EUR in one year. For a target value close to 100%, the bond floor strategy — as compared to the stop-loss strategy — swings the pendulum in the other direction of extreme risk aversion with the consequences of little upside potential beyond the target value. There are two things worth mentioning: The bond floor must not be larger than the amount that may be earned with the risk-free investment and a bond floor of zero is the same as the above buy-and-hold strategy.

9.2.4. Plain Vanilla CPPI CPPI tries to bridge the gap between high risk and high risk aversion of the above strategies. The Constant Proportion Portfolio Insurance technique [2] was first introduced by Black and Jones in 1987. It may be seen as a further generalization of stop-loss and bond floor and it is literally a dynamic strategy. Similar to both strategies, a target level and an investment horizon are given. In contrast to the bond floor strategy, we assume that a total loss of the risky investment is highly unlikely. We rather accept that the risky investment may fall by a factor of 1/M within a given time horizon. This means that if we invest twice (e.g., M = 2) the amount given by the right hand side of Eq. (9.3) in the risky portfolio, we still meet the investment target if the investment will lose less than 50% of the initial value. This is the key idea of CPPI. The strategy may be summarized by the following set of steps: (1) Calculate the current risk budget C(t) similar to Eq. (9.4) according to C(t) = PV portfolio (t) − e−r(H−t) · T,

(9.5)

with PV portfolio (t = t0 ) = N0 . (2) Calculate the exposure E(t) invested in the risky portfolio according to E(t) = M · C(t).

(9.6)

(3) Rebalance the portfolio by investing the amount E(t) in the risky portfolio and the remaining amount in the risk-free investment. (4) Wait until
t has passed and go back to Step (2). (5) Repeat the above steps until the end of the investment horizon T is reached. In case C(t) according to Eq. (9.5) is zero, the portfolio will be allocated in the riskfree investment until the end of the horizon. In that case, only the target value is achieved.

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Dersch Comparison of Simple Protection Strategies and Their Relation to CPPI. Risk appetite

Upside potential

Short fall risk

Relation to CPPI

Buy-and-hold

High

High

No

CPPI with zero target

Stop-loss

Medium, depends on stop level

High, depends on stop level

Yes

CPPI with very large multiplier and target equal to stop-loss level

Bond floor

Low, depends on bond floor level

Low, depends on bond floor level

No

CPPI with target level equals bond floor and multiplier M = 1

CPPI

Low, depends on target level

Medium, depends on target level

Yes, but small Yes

Depending on the parameterization, CPPI allows one to implement a given risk appetite: The larger the multiplier and the lower the target level, the higher the risk appetite. It is easy to show that CPPI contains the above strategies buy-and-hold, stop-loss, and bond floor for different settings of target level and multiplier. The discounted target level is also called the floor F(t) = E−r(T −t) · IT.

(9.7)

The floor is the present value of the target. Table 9.1 summarizes the risk and reward characteristics for the above strategies and shows the link to CPPI.

9.3. HISTORICAL SIMULATION I In this section, we show sample simulations for the above protection strategies. Our risky portfolio is the Dow Jones Euro Stoxx 50. The simulation covers a period of close to 10 years. For comparative reasons, we use similar protection levels where applicable. • • • • •

Table 9.2 summarizes the simulation parameter. Figure 9.1 shows the buy-and-hold strategy. Figure 9.2 shows the stop-loss strategy. Figure 9.3 shows the bond floor strategy on the DJ Euro Stoxx 50 index. Figure 9.4 shows the CPPI strategy with a target level of 100% and a multiplier of 6. The floor is calculated using a discount rate of 3.25%. • Table 9.3 summarizes the simulation results.

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Simulation Parameters of Simple Strategies.

Simulation period Investment Risk-free and risky investment Target Discount rate Multiplier

4 January, 1999 — 12 December, 2008 EUR 100 mn EUR Overnight liquidity, DJ Euro Stoxx 50 100% (except buy-and-hold) 3.25% (except buy-and-hold) 6 (CPPI only)

180 160 140 120 100 80 60 40 20 0 1999

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Figure 9.1 Buy-and-hold strategy on the DJ Euro Stoxx 50 performance index for an initial portfolio value of EUR 100 mn. The performance of the portfolio (in EUR mn) is the performance of the index.

In this market environment, the bond floor strategy seems to perform best over the complete period both in the absolute return and the size of the worst draw-down. The three other strategies all reach the same maximal portfolio amount of EUR 156.3 mn (up 56%), but fail to benefit at maturity. The stop-loss and CPPI strategy show a very similar picture. They are both stopped out during the sharp market decline in mid-2002 and realize a slight loss as compared to the initial portfolio value. This translates into a slightly negative annual return of −0.17% (CPPI) and — 0.30% (stop-loss). Buyand-hold ranks last with respect to final portfolio amount annual return, and suffers the worst portfolio draw-down of 65% of the previous all-time high value. Stop-loss and CPPI both fail the target by a small amount. This may be due to two reasons. The first is fundamental and is caused by rapid market movements when the position may only be liquidated below the theoretical stop level. We can reduce this fundamental risk by monitoring the position intraday. The second reason is caused by a simplification of our simulation framework. Monitoring the stop-loss or floor level requires one to monitor the zero coupon bond with a maturity equal to the remaining

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Dersch 180 DJ EUR STOXX 50

risk-free

160 140 120 100 80 60 40 20 0 1999

2000

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Figure 9.2 Stop-loss strategy on the DJ Euro Stoxx 50 performance index with a target level of 100% at maturity. The risk-free investment is EUR overnight liquidity. The y-axis shows the portfolio value in EUR mn.

160 DJ EUR STOXX 50

risk-free

140 120 100 80 60 40 20 0 1999

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Figure 9.3 Bond floor strategy on the DJ Euro Stoxx 50 index with a target level of 100% at maturity. The risk-free investment is EUR overnight liquidity. The y-axis shows the portfolio value in EUR mn.

investment horizon. As a simplification, a fixed discount rate is used in our simulation framework to calculate the stop-loss level and floor. Our simplified risk-free investment is EUR overnight liquidity rather than the corresponding zero bond with matching investment horizon. We may fail to reach the target if the overnight investment fails to earn — over the remaining investment horizon — the return implied by the floor level.

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Dynamic Portfolio Insurance Without Options 180 DJ EUR STOXX 50

risk-free

floor

160 140 120 100 80 60 40 20 0 1999

2000

2001

2002

2003

2004

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2008

Figure 9.4 CPPI strategy on the DJ Euro Stoxx 50 performance index with a target level of 100% at maturity. The risk-free investment is EUR overnight liquidity. The floor (black line) is assumed to follow an annual rate of 3.25%. The y-axis shows the portfolio value in EUR mn. Table 9.3 Historical Simulation of Four Different Protection Strategies Using the Dow Jones Euro Stoxx 50. We Calculate the Annual Return Over the Simulation Period and the Worst Draw-Down. A Draw-Down of, e.g., 65% Means That the Portfolio Lost 65% of Its Previous All-Time-High. Strategy

Buy-and-hold

Stop loss

Bond floor

Ann. return Reached protection Worst draw-down

−1.69% NA 64.64%

−0.30% Yes 48.47%

2.03% No 27.26%

CPPI −0.17% Yes 47.34%

9.4. ADVANCED FEATURES The above simulations illustrate different protection strategies. However, they are not suitable for practical use for a number of reasons: • Transaction costs may have an impact on the real world performance. • Frequent rebalancing may cause excessive transaction costs and should therefore be constrained by transaction filters. • Investors require more sophisticated protection strategies like lock-in of gains. • Investors may wish a leveraged exposure to the risky portfolio. • For non-trivial risky portfolios, e.g., more than one risky asset, different rebalancing strategies for the risky portfolio may be applied.

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In our simulation environment, we implemented a fairly general framework that takes into account the above features.

9.4.1. Transaction Costs Independent of asset classes and markets, transaction costs is a relevant factor that may significantly impact the performance of a trading system. Trading systems that look good on paper, e.g., paper trading, may fail in reality because transaction costs were not considered. The impact of transaction costs is more dominant for frequent trading and for long investment periods. The latter is important because of the compounding effect. Transaction costs that reduce the portfolio amount in an early stage are no longer available for future investment. The impact is hard to estimate and therefore has to be simulated. In our simulation environment, we model transaction costs as a percentage of the transaction volume. Different transaction costs can be set for the risk-free and the risky assets and for buying and selling the asset. The bid–offer spreads of ETF, for example, vary widely from 2 bp to up to 100 bp depending on the asset class and time. This corresponds to a transaction cost of 0.0001–0.005 times the transaction volume. The transaction cost is only half because the bid–offer spread is paid on the full round trip to get in and out of the asset.

9.4.2. Transaction Filter With the ability to model transaction costs, the impact can be analyzed and optimized. Here, we have to trade off flexibility versus rigidity — rapid adaptation to changes in the market environment on the one hand with the downside of a large number of transactions and high transaction costs. On the other hand, less frequent trading reduces the transaction costs but poses the risk that the system is not flexible to respond to large market moves. The introduction of transaction filters , λ, and
t allows the investor to trade off these two effects. Strictly speaking a rebalancing in the plain vanilla CPPI is required whenever Eq. (9.6) is violated. We calculate the deviation after time t = t
+
t has passed according the following equation: E(t) (9.8) 1 − buy ≤ target ≤ 1 + sell . E (t) A rebalancing is performed only if the target exposure Etarget (t) deviates from the current exposure E(t) beyond the given boundaries. Typically, the parameter  for buy and sell are in the range of [0,0.2], with sell slightly smaller than buy , in order to react quicker to a market downturn. Larger  defines a higher threshold for rebalancing,  = 0 requires instant rebalancing.

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The volatility filter λ scales the size of the rebalance — either buy or sell — if a rebalancing is triggered according to Eq. (9.8).
E = λ · |Eold (t) − Enew (t)|,

(9.9)

with λ in the range of [0.7,1.0]. For λ smaller than one, we follow only a fraction of the rebalancing amount. The idea behind the volatility filter is that the market trades in a range rather than follows a trend. The re-observation period
t is also a transaction filter: The larger
t, the more time elapses between a test of Eq. (9.8). The re-observation period has to be in line with . Small  and a large
t or  very close to one and small
t do not fit well together. Typically,
t is in the range of one day. This implies that we test Eq. (9.8) at close, but ignore intra-day movements outside the range defined by . This reflects the observation that the intra-day volatility is typically larger than the day-to-day volatility.

9.4.3. Lock-in Levels The plain vanilla CPPI does not protect any gains. To remedy this weakness, different strategies to lock-in gains by raising the target level have been proposed. The observation period for lock-in may be given by
t, its multiples (k ·
t), or by any other discrete lock-in dates like every week or month. To implement lock-in, we have to distinguish between the lock-in trigger and the lock-in action. The trigger can be a simple trigger in time as described above, or a trigger caused by a certain portfolio level or given by both. The different lock-in actions are performed if a lock-in trigger is reached. They are summarized as follows: • Discrete lock-in steps: X% gain in the portfolio amount is locked in by raising the floor defined in Eq. (9.7). The discrete lock-in steps may refer to the fraction of the initial notional at the beginning of the investment period, like every EUR 100,000 or to the notional at the previous lock-in level (compounding lock-in), like 10% of the portfolio amount at the last lock-in. • Newly reached all-time-high levels are locked in by raising the floor with respect to the previous high level. • Trailing: Y % gain in the portfolio amount is locked in by raising the target level by the given gain. The discrete lock-in steps may refer to the initial target level at the beginning of the investment period or refer to the previous lock-in level (compounding lock-in). • All-time-high trailing: Newly reached all-time-high levels are locked in by raising the target level with respect to the previous high level. The difference between the first two and the last two lock-in actions is that the lock-in is applied to the floor level rather than the target level.

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The advantage of a lock-in is that gains that have been reached in the past are protected. On the other hand, a lock-in may also cause a reduction in the exposure due to a rising floor level and therefore limits participation in a future rise in the risky portfolio.

9.4.4. Leverage and Constrain of Exposure The lock-in levels discussed above support the requirement to protect gains in the risky portfolio. On the other hand, investors may request further upside potential through higher exposure to the risky portfolio. We may introduce leverage by rewriting Eq. (9.6) E(t) = min{Emax (t), M · C(t)}.

(9.10)

In the absence of leverage, we define Emax (t) = PV portfolio (t).

(9.11)

This means that the exposure to the risky portfolio may not be larger than the current portfolio value. We cannot invest more than our current notional amount. More generally, we may write Emax (t) = K · PV portfolio (t).

(9.12)

For K > 1, we allow leverage up to a certain level. For 0 < K < 1, we constrain the maximum exposure to the risky portfolio. The latter has a similar effect as a lock-in. In addition, we may scale Emax with respect to the initial investment PV (t0 ) rather than the current portfolio value. The leverage and constraint in the above description refer to the risky portfolio. Similarly, we may define leverage with respect to the risk-free investment defined by Riskfree(t) = max{Riskfreemin (t), PV portfolio (t) − E(t)}.

(9.13)

In the absence of leverage, we find Riskfreemin (t) = 0.

(9.14)

This means that it is not allowed to borrow funds in order to invest in the risky portfolio. More generally, we may write: Riskfree(t) = k · PVportfolio (t).

(9.15)

For k < 0, we allow borrowing and thus leverage up to a certain level. For 0 < k < 1, we request a minimum amount to be invested in the risk-free asset, which constrains the exposure to the risky portfolio. This has a similar effect as a lock-in. As proposed above, we may also scale the risk-free exposure with respect to the initial investment PV (t0 ) rather than the current portfolio value.

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In our framework, we implemented leverage and constraint of the risky portfolio and the risk-free asset. Both features may be used at the same time. The weaker criterion determines the overall portfolio leverage or constraint.

9.4.5. Rebalancing Strategies for the Risky Portfolio Up to now, we have not discussed how a rebalancing of the risky portfolio is propagated to the portfolio constituents. In principle, we could perform a complete portfolio optimization in each step and adjust the portfolio weights accordingly. This may include Markowitz, VaR, or CVaR optimization with different types of linear, non-linear constraints, or boundary conditions. For the sake of simplicity, we show three types of simple portfolio rebalancing strategies: • Balanced: Here, the notional of all N risky portfolio constituencies is rebalanced in accordance with the initial weights defined at the start of the strategy. This approach reduces the amount of the above-average-performing risky assets and increases the amount of the below-average-performing risky assets. The balanced approach is an anti-cyclical profit-taking strategy that assumes a mean-reverting market within the universe of the risky assets. • Proportional: Here, the risky portfolio constituencies are rebalanced proportional to current weights. This strategy implements a simple trend-following approach with a soft competition among the risky assets. • Squared: Here, the risky portfolio constituencies are rebalanced proportional to the square of the current weight. Similar to the proportional strategy, this approach implements a trend follower but with fierce competition among the risky assets due to the squared-weighting factor.

9.4.6. CPPI and Beyond The described framework allows implementation of a wide range of different strategies customized to the risk appetite and investment guidelines of the investor. These strategies are independent of the invested asset classes. The advanced features may also make the simple strategies buy-and-hold, stop-loss, and bond floor more flexible. When these strategies are extended from a static to a dynamic portfolio approach, the transaction filters are also very useful. The advanced features also offer the opportunity to remedy the weaknesses of the original CPPI, like the lack of protecting gains, the fixed time horizon, and the limited potential to recover from a large draw-down. The guaranteed target level at a fixed time horizon is a feature that may not be necessarily required by an investor as this requirement has also a significant downside: There is little upside potential to take

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advantage of a market recovery if the risk budget is strongly depleted after a downturn in the risky portfolio. The situation is worse if this happens in the early stage of the strategy. One possibility to remedy this weakness is the minimum exposure CPPI. Here, the allocation in the risky portfolio is held at a minimum guaranteed level. This ensures participation over the complete lifetime of the strategy. But the participation comes at a cost. An additional option has to be purchased to guarantee the target level at maturity. In this chapter, we focus on portfolio strategies without options and therefore follow a different approach. If we forgo the requirement of a target level at a fixed time horizon, we can instead attempt to secure — at any time — a fraction, e.g., 80% of a past portfolio value. In our advanced CPPI framework, this approach can be implemented by setting the discount rate in Eq. (9.7) to zero. Now target and floor are the same. In order to provide a risk budget, the initial target value has to be below 100% of the notional amount invested. Adding the lock-in type all-time-high trailing results in a strategy that has been described as Time-Invariant Portfolio Protection (TIPP) [4]. The main characteristics are • No fixed investment horizon is required. • Guaranteed instantaneous target level with an initial value below the investment amount. • Guarantee to recover from a market rebound because the risk-free portfolio continuously generates a new risk budget. • Implements a sequence of all-time-high look-back options with a participation rate, which equals the target level but without the requirement of a fixed expiry date. • The present value of the portfolio may fall with a growing investment horizon in falling or sideward-moving markets. The last point is a drawback of the approach. However, the investor can redeem the investment at any time.

9.5. HISTORICAL SIMULATION II In this section, we illustrate the above-advanced features with individual simulations and analyze the impact on the performance in the current volatile market environment.

9.5.1. Transaction Costs and Transaction Filter We mentioned above that it is important to include transaction costs in a simulation in order to get an idea of the performance under realistic conditions.

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Dynamic Portfolio Insurance Without Options Table 9.4 Simulation Parameters to Study the Impact of Transaction Filters on Transaction Costs. Four Different Simulations are Carried Out. Simulation period Investment Target Multiplier Risk-free investment Risky investment Discount rate Trans. filter (Sim. 1 and 3) Trans. cost (Sim. 1 and 2)

4 January, 1999–26 December, 2006 EUR 100 mn 100% 3 EUR overnight liquidity DJ Euro Stoxx Select Dividend 30 3.25% (buy) = 0.12, (sell) = 0.08, λ = 0.85,
t = 1 day Risky investment 60 bp, risk-free investment 1 bp

In the following, we carry out four simulations with and without transaction filters combined with and without transaction costs. The risky asset is the DJ Euro Stoxx Selected Dividend 30 index. The simulation parameters are summarized in Table 9.4. To illustrate the impact of transaction costs, we selected a period of a rising market. The transaction costs are set to 60 bp of the transaction volume for the risky investment and to 1 bp for the risk-free investment. The transaction costs are deducted from the portfolio at the time of the transaction. In our simulation, we ignore the effect of slippage and partial execution. Figure 9.6 shows one out of the four simulations using transaction filters and considering transaction costs. Table 9.5 summarizes a comparison of different simulations with and without transaction filters and transaction cost. Without the transaction filter, there is a huge impact of transaction costs. The difference is EUR 54.45 mn. This means that the naïve approach — Simulation 4 — would suffer a drop of more than 20% of the final portfolio value in case transaction costs have to be taken into account (Simulation 2). Using the transaction filter results in much smaller dependency on transaction costs. The difference here is only Table 9.5 Comparison of Different Historical CPPI Simulations With (Without) Transaction Filters, Simulations 1 and 3 (Simulations 2 and 4) and With (Without) Transaction Costs, Simulations 1 and 2 (Simulations 3 and 4). For the Different Combinations, We Show the Total Number of Rebalancing Steps, the Portfolio Value at Maturity and the Total Transaction Costs. Simulation

Trans. filter

Trans. costs

# of transactions

PV final in EUR (mn)

Transaction cost in EUR (mn)

1 2 3 4

Yes No Yes No

Yes Yes No No

66 1923 52 1688

230.64 215.40 254.53 269.85

−2.04 −8.84 — —

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EUR 23.90 mn (9.4%). The above simulation clearly reveals the damaging effect of transaction cost in the absence of the transaction filter. There is a primary effect caused by pure transaction costs that are deducted from the portfolio. This accounts for EUR −2.04 mn (Simulation 1, with transaction filter) and EUR −8.84 mn (Simulation 2, without transaction filter). There is a secondary effect that shows the total effect of transaction cost. A comparison of the final portfolio values shows that in case of transaction filters the difference between portfolio values of EUR −23.90 mn contains transaction costs in the amount of EUR −2.04 mn. In the absence of transaction filters, the performance gap is EUR −54.45 mn, where EUR 8.84 mn is the pure transaction cost. In this context, the transaction cost ratio tcr =

PV (tc) − PV (ntc) TC

(9.16)

is a useful ratio to measure the secondary effect of transaction cost. Here, PV (tc) is the performance with transaction cost and PV (ntc) without transaction cost and TC the pure transaction cost incurred over the observation period. A tcr close to one means that the performance difference with and without transaction cost is mainly caused by pure transaction cost. A tcr of two means that the transaction cost causes a performance reduction of twice the amount of the pure transaction cost. In our simulation, the tcr is equal to 11.71 (with transaction filter) and 6.16 (without transaction filter). The tcr depends on the compounding effect and the leverage of exposure. In a CPPI simulation with a multiplier of 3, every EUR of risk capital (cushion) changes the exposure to the risky portfolio by EUR 3. Transaction costs that reduce the exposure early in the investment period may have a tremendous impact on the performance later on. Ironically, in a falling market, transaction costs may even have a positive impact in the presence of a high multiplier as they may force the early reduction of the exposure and save the portfolio from otherwise higher losses. We also found that if the floor level is reached early in a simulation, transaction costs become less significant as it makes no difference whether a further falling market or transaction costs are the cause for a decline in portfolio value. In general, we conclude that the effect of transaction costs is not very easy to estimate. Here, simulations shed light on the effect.

9.5.2. Lock-in Levels The sample simulation of the simple strategies illustrated a fundamental drawback of the protection strategies: the inability to take profits. An historical simulation on the DJ Euro Stoxx 50 index illustrated how the gains of more than 50% that were accumulated in the year 2000 are wiped out again (please compare Fig. 9.4). In order to remedy this weakness, we introduced a number of different lock-in strategies that are designed to protect gains. In this section, we perform historical simulations to evaluate the performance of different lock-in strategies. In order to get a direct comparison, we

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Table 9.6 Parameter Settings for Three Different Advanced CPPI Simulations. The Simulation Parameters Differ only in the Discount Rate for the Floor, the Lock-In Trigger, and Lock-In Action (the Last Three Rows). Simulation period Investment Target Multiplier Risk-free Risky investment Transaction filter Transaction cost Discount rate Lock-in trigger Lock-in action

4 January, 1999–12 December, 2008 EUR 100 mn 80% 6 EUR overnight liquidity DJ Euro Stoxx 50 (buy) = 0.12, (sell) = 0.08, λ = 0.85,
t = 1 day Risky investment 8 bp, risk-free investment 1 bp 3.25% (Simulations 1 and 2), 0% (Simulation 3) NA (Simulation 1, no lock-in), monthly, all-time-high (Simulations 2 and 3) NA (Simulation 1, CPPI without lock-in), trail all-time-high by moving up the floor (Simulations 2 and 3)

perform our simulation on the Dow Jones Euro Stoxx 50 index. Table 9.6 shows the simulation parameters. For comparative reasons, we first show the CPPI simulation without lock-in (Fig. 9.5). As compared to the previous simulation shown in Fig. 9.4, we use a transaction filter, consider transaction costs, and a target of only 80% as compared to 100%. 250 DJ EUR STOXX Select Dividend 30

risk-free

floor

200

150

100

50

0 1999

2000

2001

2002

2003

2004

2005

2006

Figure 9.5 Historical CPPI simulation under realistic condition on the DJ Euro Stoxx Selected Dividend 30 index using transaction filters and transaction costs. For a detailed specification, please refer to the text above. The y-axis shows the portfolio value in EUR mn.

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Dersch 180 DJ EUR STOXX 50

risk-free

floor

160 140 120 100 80 60 40 20 0 1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

Figure 9.6 Historical CPPI simulation on the DJ Euro Stoxx 50. The simulation parameters are shown in Table 9.6. The areas show the allocation of the two assets. The black line is the floor level. The y-axis shows the portfolio value in EUR mn.

Qualitatively, we find a similar result in Fig. 9.6 as before. The decline in the stock market leads to a complete switch into the risk-free asset at the end of the investment horizon. All previous gains are wiped out again. Due to the lower target of 80%, the portfolio is still invested in the risky asset after the sharp decline in 2002 as compared to a target level of 100%. Figure 9.7 illustrates the impact of the all-time-high trailing. The floor is moved upwards with a rising market. The initial target level of 80% trails the complete upward market movement of 56%, resulting in a final portfolio amount of 133.8% of the initial value. The simulation clearly demonstrates that trailing allows one to protect gains that have been previously accumulated. The floor value is slightly missed for the reasons already mentioned above. As a consequence, the increased target level causes a complete exit from the risky investment until the end of the investment horizon. The strategy protects gains, but is unable to recover from the draw-down. This weakness is removed in the next simulation shown in Fig. 9.8. Compared to the previous simulation, we now discount the floor level with zero. As a result, the floor is a horizontal line shifted upwards when new all-time-highs are reached. The floor level marks the guarantee level. Discounting the floor with a rate of zero reduces the risk budget because the floor is guaranteed instantaneously and not at maturity. The first half of the simulation period shown in Fig. 9.8 is similar to the simulation in Fig. 9.7. The floor trails the rising index value and the following decline results in a

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Dynamic Portfolio Insurance Without Options 180 DJ EUR STOXX 50

risk-free

floor

160 140 120 100 80 60 40 20 0 1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

Figure 9.7 Historical CPPI simulation on the DJ Euro Stoxx 50 using the all-time-high trailing with monthly lock-in triggers. The further simulation parameters are shown in Table 9.6. The areas show the allocation of the two assets. The y-axis shows the portfolio value in EUR mn. The black line is the floor level. The floor value is slightly missed for reasons mentioned above. 200 DJ EUR STOXX 50

risk-free

floor

180 160 140 120 100 80 60 40 20 0 1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

Figure 9.8 Historical CPPI simulation on the DJ Euro Stoxx 50. The simulation parameters are shown in Table 9.6. The areas show the allocation of the two assets. The black line is the floor level. The y-axis shows the portfolio value in EUR mn.

reallocation to the risk-free investment. Here, this reallocation occurs earlier because of the reduced risk budget and we therefore do not reach the same all-time-high in the portfolio value. The second half of the simulation shows a significantly different picture as compared to Fig. 9.7. The strategy participates in the rising market starting

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in 2003. In 2005, the strategy is again allocated in the risky investment to a very high proportion for a very short period of time. The guarantee level is moved upwards again. As a result, we reach a final portfolio value of EUR 167.50 nm. For the given example, the simulation shown in Fig. 9.8 is superior to the other strategies because of two reasons: • The ability to recover from the floor. • The target value is guaranteed instantaneously and not at a pre-set maturity. The reason for the rising from the dead like behavior of this strategy is the fact that even on the floor the risk-free asset continuously regenerates a cushion. Supported by a large multiplier, the exposure is quickly scaled up again. By the end of 2005, a new all-time-high levels are reached. They are trailed with a participation rate which equals the target level (80%). Table 9.7 summarizes the results for the three different simulations. The TIPP strategy performs best as compared to the two other strategies.

9.5.3. The Use of Leverage Lock-in is a conservative feature. It allows the protection of past gains. On the other hand, each lock-in reduces the risk budget and therefore may constrain future exposure to the risky asset. Leverage has an opposite effect. It increases the exposure by borrowing risk-free and investing it in the risky portfolio. It therefore allows a leveraged investment in the risky portfolio. In the following, we illustrate leverage with one example on the Credit Suisse/Tremont Investable Hedge Fund Index. The index consists of 60 different hedge funds that represent 10 different strategies. For more information on the index, please see [6]. There exist institutional and retail products on this index. Table 9.8 shows the parameter settings for this simulation. Please note that the rebalancing frequency is monthly and we assume borrowing at the risk-free overnight rate. Table 9.7 Comparison of Different Historical CPPI Simulations With and Without Lock-In (Columns 2 and 3) and with Lock-In and Flat Floor (Column 4). For the Three Combinations, We Show the Portfolio Return, the Final Value and the Final Floor Value. The Floor Value for Simulation 2 is Slightly Missed for the Reasons Mentioned Above. Strategy

(1) CPPI

(2) CPPI with lock-in

Ann. return End value EUR mn Floor Final in %

−2.25% 79.76 80.00%

2.97% 133.78 136.26%

(3) TIPP: lock-in and flat floor 5.33% 167.53 166.80%

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Dynamic Portfolio Insurance Without Options Table 9.8

Parameters for an Historical Advanced CPPI Simulation With Leverage.

Simulation period Investment Target Multiplier Risk-free Risky investment Transaction filter Transaction cost Discount rate Lock-in trigger Lock-in action Leverage

4 January, 2000–4 December, 2008 USD 1 mn 80% 6 USD overnight liquidity Credit Suisse Tremont Investable index (buy) = 0.12, (sell) = 0.08, λ = 0.85,
t = 1 day Risk-free investment 1 bp, risky investment 50 bp 0% Portfolio PV increased by 10% of previous value tested on a monthly base Increase target by 10% of initial portfolio PV Up to 60% of the initial investment of USD 1 mn may be borrowed at any time to increase the exposure to the risky asset

2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 2000

CS Trem. Inv. 2001

2002

risk-free 2003

2004

floor 2005

portfolio 2006

2007

index 2008

Figure 9.9 Historical advanced CPPI simulation on the Credit Suisse Tremont Investable Index. The y-axis shows the portfolio value in EUR mn. The leverage is limited to USD 0.6 mn. The areas show the allocation of the two assets. Leverage shows up in a negative allocation of the risk-free asset (light gray area). The gray-and-black-dashed lines are the performance of the index and portfolio, respectively. The black line is the floor.

Figure 9.9 shows an historical simulation of the above strategy. The light gray area represents the risk-free investment. Until 2008, we find a leveraged investment. The allowed leverage of USD 600,000 is utilized to the maximum level at certain times in 2000, 2005, 2006, 2007, and January 2008. Each increase of the target level

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(black line) is followed by a reduction of leverage. In this example, the leveraged investment results in an outperformance of the hedge fund index (compare dashed black line versus dashed gray line). That means that the proportion based on leverage yields an additional positive return after borrowing and transaction costs. The final portfolio value is USD 1.57 mn. This corresponds to an annual return of 5.19%. The current floor level is USD 1.52 mn. There has been no short position in the risk-free investment since October 2008. Currently, the portfolio still holds a 30% investment in hedge funds. In this example, the lock-in feature mitigates the effect of leverage. Without lock-in, the leverage budget of USD 600,000 mn would be fully utilized from 2001 until the end of the simulation period. A higher leverage together with lock-in would not make a significant difference as the lock-in and multiplier affect the maximum amount to be borrowed (data for both simulations are not shown). This emphasizes the requirement that strategy parameters are interdependent and have to be carefully adjusted.

9.5.4. CPPI on a Multi-Asset Risky Portfolio In this example, we illustrate a risky portfolio of different asset classes represented by different performance indices, namely, equity (DAX), fixed income (Rex), and commodities (Dow Jones AIG Commodity Index) classes. For the three assets, there exist exchange-traded funds. The three asset classes have been selected based on the low historical correlation of daily log-returns. In this example, we make use of the portfolio rebalancing feature proportional. The simulation parameters are shown in Table 9.9. Table 9.9 Parameter Settings for an Historical Advanced CPPI Simulation on a Multi Asset Portfolio Including the DAX, Rex, and Dow Jones AIG. Simulation period Investment Target Multiplier Risk-free Risky investment

Transaction filter Transaction cost Discount rate Lock-in trigger Lock-in action Rebalancing strategy

3 January, 2000–12 December, 2008 EUR 100 mn 80% 6 EUR overnight liquidity, DAX (initial weight 30%), Rex (initial weight 50%), Dow Jones AIG Commodity (initial weight 20%) (buy) = 0.12, (sell) = 0.08, λ = 0.85,
t = 1 day Risky investment 20 bp, risk-free investment 1 bp 0% Monthly, all-time-high Trail all-time-high by moving up the floor Proportional

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DAX

risk-free

DJ AIG Commodity Index

floor

140 120 100 80 60 40 20 0 2000

2001

2002

2003

2004

2005

2006

2007

2008

Figure 9.10 Advanced historical CPPI simulation on a risky portfolio of three different lowcorrelated asset classes. The four different shaded areas indicate the allocation over time. The black line is the floor level.

Figure 9.10 illustrates how the allocations in different asset classes evolve over time. From the end of 2000 until the beginning of 2005, the target level is constant. There are two periods with a full investment in the risky asset (2000 and the end of 2004 until the end of 2005). The final portfolio value is EUR 131.32 mn. This corresponds to a return of 3.0% p.a. mainly attributed to the years 2000 and 2005–2007. The present value time weighted asset allocation is 45% Rex, 21% AIG Commodity Index, 17% DAX, and 17% risk-free. This contrasts the current allocation of 22% Rex, 7% AIG Commodity Index, 6% DAX, and 66% risk-free. This is intuitive as we are currently very close to the floor level of EUR 123.95 mn.

9.6. IMPLEMENT A DYNAMIC PROTECTION STRATEGY WITH ETF The strategies shown in this chapter may be implemented by a direct investment in shares or baskets of shares, bonds, or by an indirect investment via Futures. Exchange Traded Funds (ETF) are an attractive alternative investment vehicle. ETF offers an investment in a wide range of different asset classes. All sample investments used in this chapter — except the Hedge Fund and Commodities index — may be implemented by investing in a corresponding ETF. The ETF on the Dow Jones Stoxx Selected Dividend 30 closely replicates the corresponding index. The index composition implements a portfolio strategy in its own sense as the portfolio constituencies are dynamically

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Comparison of Various Characteristics of ETF’s Please See Also [5].

ETF

ISIN

Bid-offer spread in bp

Management fee in bp pa

eb.rexx Goverm. Germany 5.5–10.5 DJ Euro Stoxx 50 DJ Stoxx 50 DJ Euro Stoxx Select. Dividend 30 DJ Stoxx Selected Dividend 30 Dow Jones-AIG Commodity

DE 000 628 949 9 DE 000 593 395 6 DE 000 593 394 9 DE 000 263 528 1 DE 000 263 529 9 DE 000 A0H 0728

7 8 43 56 72 99

15 15 50 30 30 45

adjusted from the Dow Jones Stoxx 600 universe based on their dividend yield and dividend consistency. For an exact definition of the index, please refer to the corresponding description of Dow Jones. Due to the nature of the index creation, the portfolio follows a more conservative anti-cyclical profit-taking strategy. Generally, the underlying rational of the index, its dynamic and characteristics have to be taken into account when setting the parameters of the dynamic protection strategy. Table 9.10 compares typical bid–offer spreads and management fees of various ETF. The shown bid–offer spreads are snapshots and may vary from day to day. For further information, please compare [5]. Bid–offer spreads reflect the characteristics of the underlying and are influenced among other factors by liquidity and taxation issues. Larger bid–offer spreads in the investment would favor the transaction filter with larger ,
t, and small λ.

9.7. CLOSING REMARKS We have demonstrated how the main drawback of CPPI, namely, its pro-cyclical behavior, the lack of recovering potential once the floor has been hit, and the fixed investment horizon can be overcome by introducing a number of advanced features. The above strategies may be implemented with simple spread sheet models avoiding the usage of options. However, the investor must be aware of the different risk aspects related to this approach. We illustrated how dynamic portfolio strategies empower an investor to replicate fairly complex option profiles including path dependent look-back options. Under certain conditions, the investor may save hedging costs, e.g., the option premium. But the savings come at a price. The investor is left with the risk that the strategy will miss the investment target. Buying an option allows one to lock in the implied volatility at the time of purchase. If the realized volatility over the lifetime is higher than the implied volatility of the option, the option is superior. Buying insurance — in the form

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of options — is therefore fairly expensive in the current market environment; however, if such a strategy was implemented one and a half years ago, an option-based strategy back then looks very cheap now in terms of strikes and volatility.

Acknowledgment The author is grateful to Thorsten Weinelt and UniCredit Research to support this work, Peter Hieber for layout and formating, and to David Dakshaw for proof reading the manuscript. Views expressed in this chapter are those of the author and do not necessarily reflect positions of UniCredit Research.

References [1] Bertrand, P and JL Prigent (2001). Portfolio insurance strategies: Obpi versus cppi. CERGY Working Paper, 30. [2] Black, F and R Jones (1987). Simplifying portfolio insurance. The Journal of Portfolio Management, 14(1), 48–51. [3] Black, F and AF Perold (1992). Theory of constant proportion portfolio insurance. Journal of Economic Dynamics and Control, 16, 403–426. [4] Estep, T and M Kritzman (1988). Tipp: Insurance without complexity. The Journal of Portfolio Management, 14(4), 38–42. [5] iShares (2008). [6] Credit Suisse Tremont (2008).

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10

HOW GOOD ARE PORTFOLIO INSURANCE STRATEGIES?

SVEN BALDER∗ and ANTJE MAHAYNI† Mercator School of Management, University of Duisburg-Essen, Lotharstr. 65, 47057 Duisburg, Germany ∗ [email protected] † [email protected]

Portfolio insurance strategies are designed to achieve a minimum level of wealth while at the same time participating in upward moving markets. The most prominent examples of dynamic versions are option-based strategies with synthetic put and constant proportion portfolio insurance strategies. It is well known that, in a Black/Scholes type model setup, these strategies can be achieved as optimal solution by forcing an exogenously given guarantee into the expected utility maximization problem of an investor with CRRA utility function. The CPPI approach is attained by the introduction of a subsistence level, the OBPI approach stems from an additional constraint on the terminal portfolio value. We bring these results together in order to explain when and why OBPI strategies are better than CPPI strategies and vice versa. We determine the utility losses, which are caused by introducing a terminal guarantee into the unconstrained maximization approach. In addition, we focus on utility losses, which are due to market frictions such as discrete-time trading, transaction costs, and borrowing constraints.

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10.1. INTRODUCTION Portfolio strategies which are designed to limit downside risk and at the same time to profit from rising markets are summarized in the class of portfolio insurance strategies. Among others, [6] and [26] define a portfolio insurance trading strategy as a strategy which guarantees a minimum level of wealth at a specified time horizon, but also participates in the potential gains of a reference portfolio. Principally, one can distinguish between two types of portfolio insurance strategies. A risky portfolio (or benchmark index) is combined either with a risk-free asset or with a financial derivative. In particular, the first class includes dynamic versions of option-based portfolio insurance (OBPI), stop-loss strategies, buy-and-hold strategies and constant proportion portfolio insurance. The second class is mainly characterized by protective put strategies, either in a static or rolling sense. Notice that, with the exception of the buy and hold and the protective put, the above strategies are all dynamic in the sense that they afford portfolio adjustments during the investment horizon. The concept of (synthetic) option-based portfolio insurance is already introduced in [15] and [33]. The constant proportion portfolio insurance (CPPI) is introduced in [12]. For the evolution of portfolio insurance, we refer to [33]. The popularity of portfolio insurance strategies can be explained by various reasons. On the side of institutional investors, there are regulatory requirements including return guarantees as well as requisitions on the risk profile. For example, [1] considers the problem of an institution optimally managing the market risk of a given exposure by minimizing its Value-at-Risk using options. Amongst early papers on the optimality of portfolio insurance are also [7] and [31]. More recently, [22] justifies the existence of guarantees from the point of an investor through behavioral models. In particular, they use cumulative prospect theory as an example, where guarantees can be explained by a different treatment of gains and losses, i.e., losses are weighted more heavily than gains, cf. [30] and [42]. Unfortunately, the justification of guarantees is less clear assuming that the investor’s preferences can be described using the [43] framework of expected utility. Dating back to [37], it is well known that in a Black/Scholes model setup and a constant relative risk aversion (CRRA) utility function, the expected utility maximizing trading rule is a constant mix strategy, i.e., a strategy where a constant fraction of wealth is invested into the risky asset. In this case, an investment weight below one implies that assets are bought when the asset price decreases. This is in sharp contrast to portfolio insurance. In order to honor a terminal guarantee, the asset exposure is to be reduced if the price of the risky asset decreases. Technically, it is straightforward to achieve CPPI and OBPI strategies as the optimal solution of a modified utility maximization problem, which is based on an exogenously given guarantee. CPPI strategies are optimal for an investor, who derives utility from the difference between the terminal strategy value

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and a given subsistence level. In contrast, the OBPI is optimal for a CRRA investor if one exogenously adds the restriction that the terminal portfolio value is above the floor. These results are well known in the literature. Without postulating completeness, we refer to the works of [5, 6, 13, 16, 17, 21, 24, 26–28, 40, 41]. [20] considers the inverse problem. They analyze if a specific dynamic strategy can be explained by solving the maximization problem of an expected utility maximizing investor, i.e., they analyze if a given investment strategy is consistent with expected utility maximization. In particular, they show that a strategy which implies a path-dependent payoff is not consistent with utility maximization in a Black/Scholes-type model. Another strand of the literature analyzes the robustness properties of stylized strategies. Concerning the robustness of option hedges, we refer the reader to [2, 8, 23, 25, 29, 34, 35]. The properties of continuous-time CPPI strategies are also studied extensively in the literature, cf. [13] or [14]. A comparison of OBPI and CPPI (in continuous time) is given in [9]. [44] also compares OBPI and CPPI strategies. In particular, they derive parameter conditions implying second- and third-order stochastic dominance of the CPPI strategy. The literature also deals with the effects of jump processes, stochastic volatility models and extreme value approaches on the CPPI method, cf. [10, 11]. An analysis of gap risk, i.e., the risk that the guarantee is violated, is provided in [3] and [19]. Gap risk is implied by introducing jumps into the model or by market frictions such as discrete time trading. In practice, the gap risk was already observable during the 1987 crash. In addition, the crash is sometimes even explained or seen to be supported by the portfolio protection mechanisms. However, there are also contradicting opinions, cf. [32]. Finally, there is also a a wide strand of empirical papers, which measure the performance of portfolio insurance strategies. For example, we refer to [18] who give an extensive simulation comparison of popular dynamic strategies of asset allocation. The following paper mitigates between expected utility maximization and the comparison of stylized strategies. We start with an exposition of the three optimization problems which imply constant mix, CPPI, and OBPI strategies as optimal. Instead of giving a further justification for the existence of guarantees, we use the (well-known) results of the optimization problems to explain the main differences between portfolio insurance mechanisms. Comparing the terminal payoffs shows that both portfolio insurance strategies, CPPI and OBPI, result in payoffs, which consist of a fraction of the payoff of a constant mix strategy (which is optimal for the unconstrained CRRA investor) and an additional term due to the guarantee. The additional term provides an intuitive way to explain the main advantage of the OBPI approach as compared to the CPPI approach. Intuitively, it is clear that the fraction of wealth which is put into the optimal unconstrained strategy is linked to the price of the guarantee, i.e., the fraction is less than one. In the case of the CPPI approach, the additional term is simply the guarantee itself, i.e., the payoff of an adequate number of zero bonds. In contrast, the

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additional term implied by the OBPI is a put option where the (synthetic) underlying is given by a fraction of the constant mix strategy and the strike is equal to the guarantee. Obviously, the put is cheaper than the zero bonds. Therefore, an investor who follows the OBPI approach puts a larger fraction of his wealth into the unconstrained optimal portfolio than an investor who follows the CPPI approach. To asses the utility costs of forcing a guarantee into the unconstrained problem, i.e., the utility costs from having to use a suboptimal strategy, we compare the certainty equivalents of the different strategies and calculate the loss rates. In addition, we explain one major drawback of the OBPI method, which is due to the kink in the payoff-profile caused by the option component. The terminal value of the OBPI is equal to the guarantee if the put expires in the money. In contrast to the CPPI method, this implies a positive point mass for the event that the terminal value is equal to the guarantee. The probability is given by the real world probability that the terminal asset price is below the strike of the put. Intuitively, it is clear that this can cause a high exposure to gap risk, i.e., the risk that the guarantee is violated, if market frictions are introduced. We illustrate this effect by taking trading restrictions and transaction costs into account. It turns out that the guarantee implied by the CPPI method is relatively robust. In contrast, the probability that the guarantee is not reached under the corresponding synthetic discrete-time OBPI strategy is rather high. The outline of the paper is as follows. In Sec. 10.2, we review the well-known optimization problems yielding constant mix, CPPI, and OBPI strategies as optimal solutions. We compare the optimal strategies and resulting payoffs, and we discuss some advantages (disadvantages) of the different portfolio insurance methods. In Sec. 10.3, we consider the utility losses caused by the introduction of strictly positive terminal guarantees for a CRRA investor. In particular, we compare CPPI and OBPI strategies according to their implied loss rate. We consider the effects of market frictions in Sec. 10.4 where we focus on the loss rates, which are implied by discrete-time trading and transaction costs. In addition, we compare the effects of these market frictions on the protection mechanisms of CPPI and OBPI. In Sec. 10.5, we address the topic of borrowing constraints and consider the capped version of CPPI strategies. Section 10.6 concludes the chapter.

10.2. OPTIMAL PORTFOLIO SELECTION WITH FINITE HORIZONS All stochastic processes are defined on a stochastic basis (
, F, F, P), which satisfies the usual hypotheses. We consider two assets. The riskless zero bond B with maturity T grows at a constant interest rate r, i.e., dBt = Bt rdt where BT = 1. The evolution of the risky asset S, a stock or benchmark index, is given by a geometric Brownian

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motion dSt = St (µdt + σdWt ),

S0 = s,

(10.1)

where W = (Wt )0≤t≤T denotes a standard Brownian motion with respect to the real world measure P. µ and σ are constants and we assume that µ > r ≥ 0 and σ > 0. In particular, dSt = St (rdt + σdWt∗ ),

S0 = s,

(10.2)

where W ∗ = (Wt∗ )0≤t≤T denotes a standard Brownian motion with respect to the equivalent martingale measure Q, i.e.,
 µ−r dQ 1 µ−r 2 = e− 2 ( σ ) t− σ Wt . (10.3) dP t A continuous-time investment strategy or saving plan for the interval [0, T ] can be represented by a predictable process (πt )0≤t≤T . πt denotes the proportion of the portfolio value at time t, which is invested in the risky asset S. In the following, we also refer to πt as the portfolio weight at time t. W.l.o.g., we consider strategies which are self-financing, i.e., money is neither injected nor withdrawn during the investment horizon [0, T ]. Thus, the fraction of wealth which is invested at time t in the zero bond B is given by 1 − πt . Let V = (Vt )0≤t≤T denote the portfolio value process associated with the strategy π, then the dynamics of V are given by
 dSt dBt where V0 = x. + (1 − πt ) (10.4) dVt (π) = Vt πt St Bt For the above model assumptions, it follows dVt (π) = Vt [(πt (µ − r) + r)dt + πt σdWt ]

where V0 = x.

(10.5)

Alternatively, the strategies can be represented by the number of shares. Let ϕt = (ϕt,S , ϕt,B )0≤t≤T where ϕt,S denotes the number of risky assets and ϕt,B the number of zero bonds with maturity T , which are held at time t. In particular, we have ϕt,S =

πt Vt St

and

ϕt,B =

(1 − πt )Vt , Bt

(10.6)

where Bt denotes the t-price of the zero bond maturing at T . Traditionally, a strategy specification via the portfolio weights is used in the context of portfolio optimization while the convention of stating the number of shares is normally preferred in the context of hedging. In the case of a finite investment horizon T and no intermediate consumption possibilities, the relevant optimization problem is given by sup EP [u(VT (π))] subject to Eq. (10.5),

π∈

(10.7)

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where  denotes the set of all self-financing trading strategies. The utility function u (u ∈ C2 ) is assumed to be strictly increasing and concave, i.e., u > 0 and u < 0. In the following, we recall the well-known optimization problems, which justify three basic strategy classes: constant mix (CM) strategies, constant proportion portfolio insurance (CPPI) strategies, and option-based portfolio insurance (OBPI) strategies. In contrast to a CM strategy, which is exclusively specified by a constant portfolio weight m, i.e., πtCM = m, portfolio insurance strategies incorporate a guarantee component which, in the simplest case, is given by an amount GT , which is to be honored at the end of the investment horizon T (T > 0). While a CPPI strategy is value-based in the sense that the portfolio weights are exclusively specified by the current portfolio value (and the present value of the guarantee), the (dynamic) OBPI approach is payoff and model dependent. Here, the investment decisions are, in a complete model, given in terms of the delta hedge of an option payoff. Formally, the three strategy classes are CM = {π ∈ |πt = m, m ≥ 0}     Vt − e−r(T −t) GT CPPIG   = π ∈  πt = m ,m ≥ 0 Vt    t St , OBPIG = π ∈  πt = Vt  ∂ t = EQ [e−r(T −t) (h(ST ) − GT )+ |Ft ] . ∂St

(10.8) (10.9)

(10.10)

Notice that for GT = 0, we have CM = CPPI G . However, we refer to CPPI and OBPI versions, where the guarantee is not a strategy parameter but GT > 0 is exogenously given. In particular, we assume that GT < V0 erT . For the OBPI approach, h is a functional which has to reflect the initial budget constraint. In practice, h is typically a linear function of ST and the resulting payoff equals a protective put strategy, where an initial investment in risky assets is protected by corresponding puts. In general, h can be an arbitrary function, e.g., a power function. In [39], a general discussion of how to choose h optimally with respect to different utility functions is given. Table 10.1 summarizes the optimization problems, which are suited to justify the three strategy classes. Problem (A) is derived from the classic Merton problem, Table 10.1 Problem (A) (B) (C)

Benchmark Optimization Problems.

Utility function (γ > 0, γ = 1) 1−γ

uA (VT ) = VT /(1 − γ) uB (VT ) = (VT − GT )1−γ /(1 − γ) 1−γ uA (VT ) = VT /(1 − γ)

Additional constraint

Optimal strategy

None None VT ≥ GT

CM CPPI OBPI

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cf. [37]. While Merton additionally considers optimal consumption, the optimization problem (A) is limited to the terminal wealth. Nevertheless, due to the separability of the investment and the consumption decision, the optimal investment strategy is equal. Problem (B) introduces a subsistence level GT such that uB belongs to the class of HARA utility functions. Problem (C) consists of the CRRA utility function uA which is also used in problem (A) but poses an additional constraint on the terminal value of the strategy, i.e., the constraint that the terminal strategy value must be above or equal to the terminal guarantee GT . The solutions of the optimization problems and their corresponding proofs are well known in the literature such that we omit some technical parts of the proofs and refer to the literature given in the introduction. Intuitively, it is clear that the solutions of problems (B) and (C) are modifications of the classic Merton problem (A), where a guarantee is exogenously forced into the optimization problem, respectively the solution. Basically, the subsistence level in (B) results in the optimization problem of (A) if the value process is reduced by the present value of the terminal guarantee, i.e., the optimization problem (A) is given in terms of the cushion process. Technically, the solution of problem (C) is more involved. However, the solution of (C) is intuitive in the sense that the constraint on the terminal value features an European option on the optimal payoff of (A), where the initial investment must take into account the price of the option.

10.2.1. Problem (A) Two observations simplify the optimization problem (A) to a large extend. (i), the constant relative risk aversion implies that the attitude toward financial risk is independent of the initial wealth level. (ii), the problem is independent of the current asset price St . This follows from the stationarity of the increments of St . From (i), one can conclude that the optimal terminal value VT (π∗ ) is linear with respect to the initial wealth. Let Vˆ 1 (t, T ) denote the optimal terminal value at T for current wealth Vt = 1. Consider now two dates t1 < t2 . Then, EP [uA (Vt2 (π∗ ))] = EP [EP [uA (Vt2 (π∗ ))|Vt1 (π∗ )]] = EP [EP [uA (Vt1 (π∗ ) · Vˆ 1 (t1 , t2 ))]|Vt1 (π∗ )] = EP [uA (Vt1 (π∗ ))]EP [(Vˆ 1 (t1 , t2 ))1−γ ], i.e., the optimal portfolio weight in t < t1 must be equal for the two investment horizons. This implies that the solution of problem (A) can be obtained by restricting the strategy set to constant mix (CM) strategies such that it is enough to consider the

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maximization problem sup EP [uA (VT (π))] subject to Eq. (10.5).

π∈CM

Notice that for π ∈ CM , Eq. (10.5) simplifies to dVtCM = VtCM [(r + m(µ − r))dt + mσdWt ] r+m(µ−r)− 12 m2 σ 2

VTCM = V0CM e(

i.e., Inserting σWT = ln

ST S0

VTCM

)T +mσWT .

(10.11) (10.12)

− (µ − 21 σ 2 )T gives

=

1 2 2 1 2 V0CM e(m(µ−r)+r− 2 m σ )T −m(µ− 2 σ )T




ST S0

m

= φ(V0CM , m)STm where




1 φ(x, m) := x S0

(10.13) m

1 2 e(1−m)(r+ 2 mσ )T .

(10.14)

Notice that, as a function of the terminal asset price ST , the payoff VTCM is concave for m < 1, linear for m = 1 and convex for m > 1. The expected utility is equal to EP [uA (VTCM )] = =

φ(V0CM , m)1−γ (1−γ)m ] EP [ST 1−γ (V0CM )1−γ (1−γ)(r+m(µ−r)− 1 γm2 σ 2 )T 2 e . 1−γ

(10.15)

Finally, it is straightforward to show that argmaxm EP [uA (VTCM )] =

µ−r =: m∗ . γσ 2

(10.16)

10.2.2. Problem (B) Consider now the (modified) portfolio planning problem of an investor, who derives the utility from the difference between the portfolio value and a given subsistence level. Let C = (Ct )0≤t≤T denote the cushion process, where Ct := Vt − e−r(T −t) GT . If a constant proportion m of the cushion is invested in the risky asset, one obtains analogously to the Eqs. (10.11) and (10.13) dCt = Ct [(r + m(µ − r))dt + mσdWt ], C0 = V0 − e−rT GT

(10.17)

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and CT = φ(C0 , m)STm , as well as argmaxm EP [uA (CT )] = argmaxm EP [uA (φ(C0 , m)STm )] =

µ−r = m∗ . γσ 2

Notice that EP [uA (CT )] = EP [uB (VT )]. Using the fact that the optimal payoff does not depend on the asset price path (cf. for example [20]) implies that the optimal ∗ investment proportion πt,B of problem (B) is ∗ = πt,B

m∗ Ct m∗ (Vt − e−r(T −t) GT ) = . Vt Vt

(10.18)

Obviously, problem (A) and (B) coincide in the case that GT = 0. However, GT > 0 implies a reduction in the investment proportion, i.e., ∗ ∗ ≤ m∗ = πt,A . πt,B

(10.19)

The cushion dynamics given by Eq. (10.17) immediately implies Vt = e−r(T −t) GT + φ(C0 , m; t)Stm , i.e., Vt ≥ e−r(T −t) GT .

10.2.3. Problem (C) The additional constraint VT ≥ GT together with the path-independency of the optimal ∗ can be represented as follows solution implies that the optimal payoff VT,C ∗ = max{h(ST ), GT } VT,C

= h(ST ) + [GT − h(ST )]+ = GT + [h(ST ) − GT ]+ .

(10.20)

Thus, the terminal value of the strategy can be interpreted in terms of an option on the payoff h such that the optimal strategy π∗ is given in terms of the delta hedge, i.e., π ∗ ∈ OBPI . In the special case that GT = 0, the optimization problem reduces to the ∗ classic Merton case, i.e., it holds h(ST ; GT = 0) = φ(V0 , m∗ )STm . For GT > 0, the optimal solution affords a reduction of the initial investment from V0 to V˜ 0 (V˜ 0 < V0 ) ∗ such that h(ST ) = φ(V˜ 0 , m∗ )STm . The remaining money V0 − V˜ 0 is then used to buy the put with payoff [GT − h(ST )]+ . A proof for the optimality of the payoff

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∗ ∗ VT,C = GT + [φ(V˜ 0 , m∗ )STm − GT ]+ is given in [24]. The concavity of uA (x) = (x1−γ )/(1 − γ) implies that for any payoff Vˆ T with Vˆ T ≥ GT

∗ ∗ ∗ uA (Vˆ T ) − uA (VT,C ) ≤ uA (VT,C )(Vˆ T − VT,C )  ∗  ∗ m∗ ∗ −γ −γm ˆ  ˜  φ(V0 , m ) ST (VT − VT,C ) for ST >

=

 −γ   GT (Vˆ T − GT )

∗

for STm

GT ˜ φ(V0 , m∗ ) , GT ≤ φ(V˜ 0 , m∗ )

i.e., −γm∗

∗ uA (Vˆ T ) − uA (VT,C ) ≤ φ(V˜ 0 , m∗ )−γ ST

∗ (Vˆ T − VT,C ) ∗

−γm −γ − GT )+ (Vˆ T − GT ). − (φ(V˜ 0 , m∗ )−γ ST

(10.21)

∗ Now, consider a portfolio VT (ε) = εVT,A + (1 − ε)V˜ T , where V˜ T is the terminal wealth of any other strategy with the same initial wealth. The first-order condition of optimization problem (A) implies  ∂EP [u(VT (ε))]  ∗ = EP [u (VT (ε))(VT,A − V˜T )]|ε=1  ∂ε ε=1 −γm∗

= EP [φ(V0 , m∗ )−γ ST

∗ (VT,A − V˜ T )] = 0.

∗ Therefore, adding and subtracting the optimal unconstrained solution VT,A to the first term on the right hand side of inequality (10.21) and taking expectations gives −γm∗

φ(V˜ 0 , m∗ )−γ (EP [ST

−γm∗

∗ (Vˆ T − VT,A )] + EP [ST

∗ ∗ (VT,A − VT,C )]) = 0.

∗ ∗ Together with Vˆ T ≥ GT a.s. it follows EP [uA (Vˆ T ) − uA (VT,C )] ≤ 0 such that VT,C is indeed the optimal solution w.r.t. problem (C). ∗ ∗ = GT + [φ(V˜ 0 , m∗ )STm − GT ]+ can be replicated by a selfThe payoff VT,C financing strategy where the initial investment can be represented by the expected discounted payoff under the uniquely defined equivalent martingale measure Q, i.e., ∗ V0 = e−rT EQ [GT + (φ(V˜ 0 , m∗ )STm − GT )+ ]


∗ −rT ∗ = e GT + φ(V˜ 0 , m )EQ e−rT STm −

GT φ(V˜ 0 , m∗ )

+  .

V˜ 0 has to be determined such that the initial cushion C0 = V0 −e−rT GT exactly finances φ(V˜ 0 , m∗ ) power call options with power p = m∗ and strike K = (GT )/(φ(V˜ 0 , m∗ )),

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i.e.,
V0 − e−rT GT = φ(V˜ 0 , m∗ )PO 0, S0 ; m∗ ,

G

 (10.22)

φ(V˜ 0 , m∗ )

where PO(t, St ; p, K) denotes the t-price of a power call with power p, strike K, and maturity T . The pricing formula for power calls, cf. [45] or [36], is given by PO(t, St ; p, K) := e−r(T −t) EQ [(ST − K)+ |Ft ]  p 2 − (1−p)σ (T −t) 2 −r(T −t)  St e = e e−r(T −t) p



  √ St × N h1 t, √ T − t − (1 − p)σ p K

 St − KN h2 t, √ . p K

(10.23)

N denotes the one-dimensional standard normal distribution function. The functions h1 and h2 are given by h1 (t, z) =

ln z + (r + 12 σ 2 )(T − t) ; √ σ T −t

√ h2 (t, z) = h1 (t, z) − σ T − t.

(10.24)

Recall that V˜ 0 ≤ V0 . A lower bound on V˜ 0 follows with [STm − GT /φ(V˜ 0 , m∗ )]+ ≤ ∗ STm , i.e., ∗


φ(V˜ 0 , m∗ )PO 0, S0 ; m∗ ,

G φ(V˜ 0

, m∗ )



∗

≤ φ(V˜ 0 , m∗ )EQ [e−rT STm ] = V˜ 0

and Eq. (10.22). This implies V0 − e−rT GT ≤ V˜ 0 ≤ V0 .

(10.25)

∗ Consider now the optimal portfolio weight πt,C of the dynamic strategy, i.e.,

∗ πt,C

=

  T St φ(V˜ 0 , m∗ )PO t, St ; m∗ , φ(V˜G,m ∗) 0

Vt

  T φ(V˜ 0 , m∗ )PO t, St ; m∗ , φ(V˜G,m St ∗) 0 =  , e−r(T −t) GT + φ(V˜ 0 , m∗ )PO t, St ; m∗ , φ(V˜ G,m∗ ) 0

(10.26)

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where PO :=

∂PO (t, St ; p, K). ∂St





FA

PO

Differentiating (10.23) immediately gives

σ2

St e 2 p(T −t) (t, St ; p, K) = p e−r(T −t)

p−1



  √ St N h1 t, √ T − t . − (1 − p)σ p K (10.27)

Finally, we compare the optimal investment weight implied by problem (C) with the one of the unconstrained solution given by m∗ . Notice that 

∗

STm −

GT φ(V˜ 0 , m∗ )

+

∗

≥ STm −

GT φ(V˜ 0 , m∗ )

implies that the denominator of the right hand side of Eq. (10.26) is larger than 1 ∗ 2 ∗ ∗ ∗ φ(V˜ 0 , m∗ )EQ [e−r(T −t) STm |Ft ] = φ(V˜ 0 , m∗ )Stm e(m −1)(r+ 2 m σ )(T −t)

such that
∗ ≤ PO t, St ; m∗ , πt,C

 ∗ GT 1 ∗ 2 ∗ St(1−m ) e(1−m )(r+ 2 m σ )(T −t) . φ(V˜ 0 , m∗ )

In addition, Eq. (10.27) immediately gives
PO



GT t, St ; m , ˜ φ(V0 , m∗ ) ∗

 ≤m

∗




St e−r(T −t)

m∗ −1

e− 2 m 1

∗

(1−m∗ )σ 2 (T −t)

.

Together, we have ∗ πt,C ≤ m∗ .

(10.28)

Analogously to the problem (B), the terminal constraint in problem (C) also gives rise to a reduction of the optimal unconstrained portfolio weight m∗ .

10.2.4. Comparison of Optimal Solutions Recall that the optimal terminal payoffs VT∗ do not depend on the asset price path, but can be specified as a function of the terminal asset price ST . The optimal payoffs for

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the optimization problems (A), (B), and (C) are summarized as follows ∗

∗ = φ(V0 , m∗ )STm VT,A

(10.29) ∗

∗ VT,B = GT + φ(V0 − e−rT GT , m∗ )STm

V0 − e−rT GT ∗ VT,A V0  + ∗ = GT + φ(V˜ 0 , m∗ )STm − GT = GT +

∗ VT,C

∗ ∗ = φ(V˜ 0 , m∗ )STm + [GT − φ(V˜ 0 , m∗ )STm ]+ +

V˜ 0 ∗ V˜ 0 ∗ = V + GT − VT,A . V0 T,A V0

(10.30) (10.31)

(10.32)

∗ corresponds to φ(V0 , m∗ ) power claims with power m∗ , where the The payoff VT,A ∗ number φ(V0 , m ) depends on the initial investment and the optimal investment weight m∗ . The optimization problem (B) introduces a subsistence level which implies that the number of power claims with power m∗ must be reduced to afford the risk-free investment, which is necessary to honor the guarantee. In consequence, the portfolio weight is lower than in the case of problem (A), cf. Inequality (10.19). The link between the solutions of (A) and (B) is even more explicit if one considers the number of shares in the asset which are held. Notice that the cushion dynamics, cf. Eq. (10.17) implies that CtCPPI /C0CPPI = VtCM /V0 if the multiplier m of the CPPI is equal to the portfolio weight of the CM strategy. In particular, this implies that the value of the cushion is proportional to the value of the CM strategy, i.e., CtCPPI = (C0 /V0 )VtCM . This is also true for the number of assets ϕt,S , i.e., CPPI ϕt,S =

C0 CM ϕ V0 t,S

CPPI = GT + ϕt,B

C0 CM ϕ , V0 t,B

(10.33) (10.34)

where ϕt,B denotes the number of zero bonds with maturity T . In particular, it holds that C0 CM V . (10.35) V0 t The CPPI strategy can thus be interpreted as a buy-and-hold strategy of a constant mix strategy with an additional investment into GT zero bonds, cf. also Eq. (10.31). A similar reasoning applies to the solution of (C), which can be interpreted as a buy-andhold strategy of a constant mix strategy with an additional investment into a put with strike GT , cf. Eq. (10.32). Obviously, the put is worth less than GT zero bonds such that one can buy and hold more CM strategies in the case of the option based approach, VtCPPI = e−r(T −t) GT +

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i.e., V˜ 0 /V0 ≥ C0 /V0 . Intuitively, it is thus clear that the OBPI approach gives a better result than the CPPI approach with respect to a utility function, which favors the CM strategy with portfolio weight m∗ . ∗ which honors the guarantee GT and In general, a modification of the payoff VT,A with t0 -price equal to V0 can be represented by

 + +    G G T T ∗ ∗ = V0 . −β subject to EQ e−rT GT + α VT,A −β GT + α VT,A α α (10.36) The CPPI approach corresponds to β = 0 and gives a smooth payoff-profile. β = 1 results in the OBPI approach with a kinked payoff-profile. As a consequence of the ∗ ∗ ∗ guarantee GT , both payoffs VT,B and VT,C are higher (lower) than VT,A for low (high) ∗ terminal asset prices. However, the smooth solution VT,B implies that the intersec∗ ∗ occurs at a higher asset price ST if compared to the intersection of VT,C tion with VT,A ∗ and VT,A . Let si,j (i  = j, i, j ∈ {A, B, C}) denote the terminal asset price ST such that ∗ ∗ . Equation (10.30) immediately gives VT,i = VT,j ∗ ∗ = VT,B VT,A

⇔

∗ VT,A = V0 erT

such that 1 2 sA,B = S0 e(r+ 2 (m−1)σ )T .

(10.37)

With Eq. (10.32), V˜ 0 ≤ V0 and V0 ≥ e−rT GT it follows 1

1

sA,C = S0 e m (g−r)T e(r+ 2 (m−1)σ

2

)T

1

= e m (g−r)T sA,B

(10.38)

where g := T −1 (ln GT − ln V0 ) ≤ r. Finally, one obtains 1 1 1 2 sB,C = S0 e m (ν−r)T e(r+ 2 (m−1)σ )T = e m (ν−r)T sA,B

(10.39)

where ν := T −1 (ln GT − ln(V˜ 0 − C0 )) ≥ r. Therefore sA,C ≤ sA,B ≤ sB,C . An illustration of the payoffs and the intersection points is given in Fig. 10.1. Table 10.2 Model paramter S0 = 1 σ = 0.15 r = 0.03 µ = 0.085

Basic Parameter Constellation. Strategy parameter V0 = 1 T = 10 γ = 1.2 m = m∗ = 2.037

Terminal guarantee GT = 1

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5

Payoff

4

3

2

1

0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Terminal asset price ∗ (solid line), V ∗ (dotted line), and V ∗ (dashed line), Figure 10.1 Optimal payoffs VT,A T,B T,C where the parameters are given as in Table 10.2.

We end this section by emphasizing one important consequence for the two protection mechanisms implied by the smooth and the kinked solutions, i.e., implied by the assumptions that marginal utility jumps gradually or discontinuously to infinity. ∗ Notice that the smooth payoff VT,B implies that there is no probability mass on the ∗ = GT ) = 0. In event that the terminal value is equal to the guarantee GT , i.e., P(VT,B ∗ contrast, for the kinked payoff VT,C , it holds 
GT ∗ m∗ P(VT,C = GT ) = P ST ≤ φ(V˜ 0 , m∗ )     ST = P ln ≤ ln S0

m∗

GT φ(V˜ 0 ,m∗ )

S0

 .

Using the definition of φ, cf. Eq. (10.14), and m∗ = (µ − r)/(γσ 2 ) yields    2 ˜ ln e−rTV0GT − 12 γλ T √   ∗ + λ T, P(VT,C = GT ) = 1 − N  √ λ T γ

(10.40)

(10.41)

where λ := (µ − r)/σ. Intuitively, it is clear that a positive point mass on the event {VT = GT } might indicate that the corresponding strategy is more sensitive to the introduction of gap risk, which is caused by asset price jumps. This problem is considered in Sec. 10.4,

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where trading restrictions in the sense of discrete time trading and transaction costs are introduced.

10.3. UTILITY LOSS CAUSED BY GUARANTEES 10.3.1. Justification of Guarantees and Empirical Observations There are a few comments necessary concerning the justification of guarantees. The optimization problems (B) and (C) are already based on an exogenously postulated guarantee such that one might doubt their capacity to give a meaningful justification of guarantees. However, there are some arguments, which are in favor of a subsistence level. Similar reasonings are true with respect to optimization problem (C). In consequence, to some extent, guarantees can be explained with respect to the assumption that the investor’s preferences can be described using the [43] framework of expected utility. More recently, [22] justifies the existence of guarantees through behavioral models. In particular, they use cumulative prospect theory as an example. In the following, we do not give further justifications for the existence of guarantees or the popularity of portfolio insurance strategies but take them as given. However, we think in terms of utility losses caused by guarantees and compare the smooth and kinked payoff solutions. One possibility which is consistent with empirical observations is given by measuring the utility losses of guarantees with respect to a CRRA utility function, where the parameter of risk aversion is assumed to be above 1 (γ > 1). For the validity of CRRA utility functions and the parameter of risk aversion, we refer to [38] and the literature given herein.

10.3.2. Utility Loss The performance of the strategy π can be measured by its associated expected utility, which can in turn be described by the certainty equivalent. It is defined as the certain amount, which makes the investor indifferent between achieving this certain amount (at T ) or using the strategy π, i.e., the time T certainty equivalent CET of the strategy π is defined by u(CET (π)) = EP [u(VT (π))].

(10.42)

Consider for example the utility function uA (x) = x1−γ /(1 − γ) and a CM strategy with optimal investment proportion m∗ , i.e., (1−γ)m∗

CE T,A (πCM ) = φ(V0 , m∗ )(EP [ST

1

]) 1−γ .

(10.43)

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p

With EP [ST ] = S0 exp{(pµ − 12 p(1 − p)σ 2 )T } it follows ∗

CET,A (πCM ) = φ(V0 , m∗ )S0m e(m ∗

= V0 e(r+m

∗

µ− 12 γσ 2 m∗ (1−m∗ (1−γ)))T

(µ−r)− 12 γ(m∗ )2 σ 2 )T

1

= V0 erT e 2γ (

µ−r 2 ) T σ

(10.44)

.

Analogously, one obtains CE T,B (πCPPI ) = GT + (V0 − e−rT GT )e(r+m 1

∗

= CE T,A (πCM ) + GT (1 − e 2γ (

(µ−r)− 12 γ(m∗ )2 σ 2 )T µ−r 2 ) T σ

(10.45)

).

Thus, the certainty equivalent is lower for an investor with subsistence level than an investor without. We can also calculate the utility loss of an investor, who follows a suboptimal strategy. It is described by the loss rate lT,i (π) of the strategy π and the utility function i (i ∈ {A, B, C}) where lT,i (π) :=

ln



∗  CET,i CET,i (π)

T

(10.46)

.

∗ CE ∗T,i denotes the certainty equivalent of the optimal strategy πi∗ = (πt,i )0≤t≤T while CE T,i (π) the one of the suboptimal strategy π = (πt )0≤t≤T . The loss rates with respect to the utility function uA (x) = x1−γ /(1 − γ) are summarized in Table 10.3 for CM, CPPI, and OBPI strategies with strategy parameter m. Notice that a loss rate, which is higher than the one of a risk-free investment, i.e., πtCM = 0, implies that the associated strategy is prohibitively bad. This critical loss rate is equal to 12 γ(σm∗ )2 , cf. Table 10.3. For m = m∗ , it is obvious that lT,A (πOBPI ) < lT,A (πCPPI ) because πCPPI is suboptimal w.r.t. the optimization problem (C) where

Table 10.3

Loss Rates with Respect to uA (x) = x 1−γ /(1 − γ ).

Strategy π

Loss rate lT,A (π) 2 1 2 ∗ γσ m − m 2

πtCM = m Vt − e−r(T −t) GT Vt  T PO t, St ; m, G ˜

∗

EP [(φ(V0 , m∗ )STm )1−γ ] 1 ln (1 − γ)T EP [(GT + φ(V0 − e−rT GT , m)STm )1−γ ]

πtCPPI = m

πtOBPI =

φ(V0 ,m)

Vt

 St

∗

EP [(φ(V0 , m∗ )STm )1−γ ] 1 ln (1 − γ)T EP [(GT + [φ(V˜ 0 , m)STm − GT ]+ )1−γ ]

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0.06

0.06

0.05

0.05

0.04

0.04

Lossrate

Lossrate

244

FA

0.03

0.03

0.02

0.02

0.01

0.01

0.00

2.04

4

m

6

0.00

2.04

4

6

m

Figure 10.2 Loss rates w.r.t. u = uA for CPPI (solid lines), OBPI (dashed) and CM (dotted) strategies with varying parameter m. The parameters are given in Table 10.2. The investment horizon is T = 10 years (left graph) and T = 20 years (right graph).

uC = uA and πOBPI is optimal. It holds lT,A (πCP ) = 0 < lT,A (πOBPI ) < lT,A (πCPPI ). The utility loss implied by the guarantee is higher for the CPPI than for of the OBPI. This is illustrated in Fig. 10.2. In addition, notice that the loss rates of the CM strategies are symmetric in the sense that for  > 0, a strategy parameter m = m∗ +  implies the same loss rate as the parameter m = m∗ − . In contrast, OBPI strategies yield a lower loss rates in the case of m = m∗ +  than for m = m∗ − . Intuitively, this is clear since the protection feature implies that the portfolio weights of the portfolio insurance strategies are too low compared to the optimal investment proportion m∗ , cf. Inequalities (10.19) and (10.28). For the comparison of CPPI and OBPI, it is important to keep in mind that m∗ is the optimal OBPI parameter for u = uA . In contrast, m∗ is not the optimal CPPI parameter for u = uA , but for u = uB which includes a subsistence level. In order to compare the loss rates implied by CPPI and OBPI w.r.t. u = uA , it is thus necessary to consider the maximization problem   (GT + φ(V0 − e−rT GT , m)STm )1−γ . max EP [uA (VT (π))] = max EP m π∈CPPI 1−γ (10.47) Figure 10.3 illustrates the loss rates for CPPI and OBPI strategies with varying parameter m. In addition, the minimal loss rates are summarized in Table 10.4, i.e., the loss rates for OBPI strategies with m = m∗ and CPPI for m = m∗∗ , where m∗∗ is the optimal solution for the maximization problem (10.47). Although the OBPI strategy with parameter m = m∗ is, per construction, the uA -utility maximizing portfolio insurance

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0.06

0.06

Lossrate

0.08

Lossrate

0.08

0.04

0.04

0.02

0.02

0.00

0.00 2.04

4

6

8

10

12

14

16

18

1.63

4

6

8

m

10

12

14

16

18

m

Figure 10.3 Loss rates implied by CPPI (solid lines) and OBPI (dashed lines) for varying m, cf. Table 10.3. The risk aversion is γ = 1.2 (left figure) and γ = 1.5 (right figure). The other parameters are given in Table 10.2.

Table 10.4 Minimal Loss Rates (uA -Optimal Strategy Parameter m) for Varying T and γ . The Other Parameters are Given in Table 10.2. Strategy

γ\T

1

2

5

10

20

CPPI

1.2

0.040 (11.32)

0.035 (7.83)

0.026 (4.91)

0.018 (3.57)

0.010 (2.73)

OBPI

1.2

0.037 (2.04)

0.031 (2.04)

0.022 (2.04)

0.014 (2.04)

0.007 (2.04)

CPPI

1.5

0.031 (10.60)

0.026 (7.25)

0.019 (4.45)

0.013 (3.16)

0.007 (2.36)

OBPI

1.5

0.028 (1.63)

0.023 (1.63)

0.015 (1.63)

0.009 (1.63)

0.005 (1.63)

CPPI

1.8

0.024 (10.03)

0.020 (6.80)

0.014 (4.10)

0.009 (2.86)

0.005 (2.08)

OBPI

1.8

0.021 (1.34)

0.017 (1.34)

0.011 (1.34)

0.007 (1.34)

0.003 (1.34)

strategy, the additional loss of the CPPI strategy with parameter m = m∗∗ which is measured by the difference of its loss rate and the one of the OBPI is rather low. Notice that the differences remain approximatively equal for varying time horizons and risk aversion parameters.

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10.4. UTILITY LOSS CAUSED BY TRADING RESTRICTIONS AND TRANSACTION COSTS It is important to notice that in practice the concept of portfolio insurance is impeded by market frictions. The protection mechanism of portfolio insurance implies that the asset exposure has to be reduced when the asset price decreases. A sudden drop in the asset price, where the investor is not able to adjust his portfolio adequately, causes a gap risk, i.e., the risk that the terminal guarantee is not achieved. One illustrative and meaningful approach to capture the gap risk is given by the introduction of trading restrictions in the sense of discrete-time trading and transaction costs. Notice that discrete-time trading is one possibility to introduce gap risk. In contrast to adding jumps in the dynamics of the risky asset, it also allows to take into account for transaction costs, which are of practical importance if one decides whether to use the CPPI or OBPI protection mechanism. In particular, we consider transaction costs which are proportional to a change in the position of the risky asset. The proportionality factor is denoted by θ. As before, we assume that the present value of the terminal guarantee GT prevailing at T is lower than the initial investment V0 , i.e., V0 > e−rT GT . Let τ n denote a sequence of equidistant refinements of the interval [0, T ], i.e., τ n = n n n < tnn = T }, where tk+1 − tkn = T/n for k = 0, . . . , n − 1. {t0 = 0 < t1n < · · · < tn−1 To simplify the notation, we drop the superscript n and denote the set of trading dates with τ instead of τ n . The restriction that trading is only possible immediately after tk ∈ τ implies that the number of shares held in the risky asset is constant over the intervals [ti , ti+1 ] for i = 0, . . . , n − 1. However, the fractions of wealth which are invested in the risky asset and the zero bond change as underlying prices fluctuate. Thus, it is necessary to consider the number ϕS of shares held in the risky asset and the number ϕB of zero bonds with maturity T , i.e., the tupel ϕ = (ϕS , ϕB ).

10.4.1. Discrete-Time CPPI Along the lines of [3], we consider a discrete-time CPPI version ϕτ = (ϕSτ , ϕBτ ) which is for t ∈ ]tk , tk+1 ] and k = 0, . . . , n − 1 defined by   mCtτk 1 τ τ τ ϕt,S := max , 0 , ϕt,B := (V τ − ϕt,S Stk ). (10.48) Stk Btk tk Notice that we do not allow for short positions in the risky asset, i.e., the asset exposure is bounded below by zero. However, similar as for the simple (continuous-time) CPPI, the discrete-time CPPI does not include short sale restrictions on the riskless asset. The terminal value of the discrete-time CPPI is given by 

min{s,n}

VTτ

−rT

= GT + (Vt0 − e

GT )

i=1




 Sti m − K(m) er(T −min{ts ,T }) , Sti−1

(10.49)

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where K(m) := (m − 1)er n /m and ts := min{tk ∈ τ|Vtk < e−r(T −tk ) GT }. We set ts = ∞ if the minimum is not attained. In particular, it holds that ts := min{tk ∈ τ|Stk /Stk−1 < K(m)}. For m > 1, the value of the simple CPPI can drop below the floor. Therefore, the discrete-time CPPI version introduces a gap risk, i.e., the risk that the guarantee is violated. For a detailed analysis of the gap risk, we refer to [3]. Besides discrete-time trading, we take also transaction costs into account. Along the lines of [13], we assume that the transaction costs are financed by a reduction of the asset exposure arising in the case without transaction costs. This can be justified by the argument that the protection feature of the CPPI is based on a prespecified risk-free investment such that the introduction of transaction costs must not change the number of risk-free bonds, which are prescribed by the CPPI method (without transaction costs). Thus, the discrete-time CPPI version with transaction costs ϕτ,TA = (ϕSτ,TA , ϕBτ,TA ) is, for t ∈ ]tk , tk+1 ] and k = 0, . . . , n − 1, defined by ! mCtτ,TA 1 τ,TA τ,TA k+ τ ϕt,S := max , 0 , ϕt,B := (V τ,TA − ϕt,S Stk ), Stk Btk tk+ T

(Ctτ,TA := Vtτ,TA − e−r(T −tk ) GT ) denotes the portfolio (cushion) value where Vtτ,TA k+ k+ k+ immediately after tk , i.e., the value net of transaction costs, which are proportional to the asset price Stk . First, consider the portfolio value Vtτ,TA before transaction costs, k := V and for k = 1, . . . , n i.e., Vtτ,TA t0 0 := ϕtτ,TA Stk + ϕtτ,TA Btk Vtτ,TA k k ,S k ,B = max

mCtτ,TA k−1+ Stk−1

! , 0 Stk + (Vtτ,TA − ϕtτ,TA Stk−1 )er(tk −tk−1 ) k ,S k−1+


= m max{Ctτ,TA , 0} k−1+

 Stk − er(tk −tk−1 ) + Vtτ,TA er(tk −tk−1 ) . k−1+ Stk−1

(10.50)

Consider now the adjustment to the proportional transaction costs, which are due immediately after the trading dates. Assuming that the transaction costs are also due at t0 and that the asset positions are transferred into a cash position immediately after tn is consistent to the following definitions τ,TA Vtτ,TA := Vt0 − ϕS,t θSt0 = Vt0 − mθCtτ,TA 0+ 0+ 0+

(10.51)

:= Vtτ,TA − |ϕtτ,TA − ϕtτ,TA |θStk Vtτ,TA k k ,S k+ k+ ,S =

Vtτ,TA k

  St − mθ max{Ctτ,TA , 0} − max{Ctτ,TA , 0} k k+ k−1+ S

tk−1

   

(10.52)

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Vtτ,TA := Vtτ,TA − ϕtτ,TA θStn n n ,S n+ = Vtτ,TA − mθ max{Ctτ,TA , 0} n n−1+

Stn . Stn−1

(10.53)

With Eq. (10.51), it immediately follows Ct0 + = Ct0 /(1 + θm). Using Eqs. (10.52) and (10.50) implies that Ctk + > 0 (k = 0, . . . , n − 1) and θ < 1/m, it holds Ctk+1 + = Ctk+1

   Stk+1   − mθ max{Ctk+1 + , 0} − Ctk + S 

(10.54)

tk


  St 1 + θ Stk+1 m − 1 rT T  n  for er n ≤ k+1 − m e Ctk +   1 + θm S 1 + θm Stk  tk     
  1 − θ Stk+1 m − 1 rT otherwise − m e n = Ctk +  1 − θm Stk 1 − θm     
   S St m − 1 rT  T  Ctk + (1 − θ)m tk+1 − (m − 1)er n e n > k+1 . for Stk m(1 − θ) Stk (10.55) T

For Ctk + ≤ 0, it follows Ctk+1 + = Ctk+1 = er n Ctk + . It is worth mentioning that the < GT } corresponds to the event that the adjusted cushion drops below event {Vtτ,TA n+ zero during the investment horizon. Notice that for Ctk + > 0  {Ctk+1 + < 0} ⇔

Stk+1 m − 1 rT < e n Stk m(1 − θ)

 =: Ak+1 .

Since the complementary of the event {∪n−1 k=0 Ak+1 } is given by the event that all asset T price increments are above er n (m − 1)/(m(1 − θ)), it follows with the assumption that the asset price increments are independent and identically distributed that P(Vtτ,TA n+

n

St1 m − 1 rT n e < GT ) = 1 − P > St0 m(1 − θ) = 1 − (N (d2TA (θ)))n

(10.56)

where d2TA (θ)

:=

ln

(1−θ)m m−1

+ (µ − r) Tn − 21 σ 2 Tn .  σ Tn

(10.57)

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10.4.2. Discrete-Time Option-Based Strategy According to Eq. (10.26), the (continuous-time) self-financing and duplicating strategy for the T -payoff GT + φ(V˜ 0 , m)[STm − GT /φ(V˜ 0 , m)]+ is given by 
GT PO ˜ ϕt,S := φ(V0 , m) (10.58) t, St ; m, φ(V˜ 0 , m) ϕt,B := GT +

  φ(V˜ 0 , m)PO t, St ; m, φ(VG˜ T,m) − ϕt,S St 0

B(t, T)

,

(10.59)

where PO is defined as in Eq. (10.27). We consider as a discrete-time version of an arbitrary continuous-time trading strategy ϕτ = (ϕSτ , ϕBτ ) with respect to the trading dates τ ϕtτ := ϕtk

for t ∈ ]tk , tk+1 ]

and

for all t ∈ [0, T ].

Setting V0 (ϕ; τ) := V0 (ϕ), the value process V(ϕ; τ) which is associated with ϕτ is Vt (ϕ; τ) = ϕtk ,S St + ϕtk ,B e−r(T −t)

for t ∈ ]tk , tk+1 ]

and

0 ≤ k ≤ n − 1.

In general, the discrete-time version of a continuous-time strategy is not self-financing. In particular, there are in- or out-flows from the portfolio which occur immediately after a trading date tk+1 (k = 0, . . . , n − 1). Formally, the costs of discretization (ϕ; τ) which occur immediately after the trading date tk+1 are defined by ξtdis k+1 ξtdis (ϕ; τ) := Vtk+1 (ϕ) − Vtk+1 (ϕ; τ) k+1

(10.60)

= (ϕtk+1 ,S − ϕtk ,S )Stk+1 + (ϕtk+1 ,B − ϕtk ,B )e−r(T −tk+1 ) . Notice that negative costs refer to inflows while positive costs imply that further money is needed to continue the strategy. Taking proportional transaction costs into account also implies that there are trans(ϕ; τ), which occur immediately after a trading date tk+1 , i.e., action costs ξtTA k+1 ξtTA (ϕ; τ) := |ϕt0 ,S |θSt0 0 ξtTA (ϕ; τ) = |ϕtk+1 ,S − ϕtk ,S |θStk+1 k+1

for k = 0, . . . , n − 2

and ξtTA (ϕ; τ) := |ϕtn−1 ,S |θStn . n Defining the payoff V¯ T (ϕ; τ) according to the assumption that the inflows into the strategy are lent according to the interest rate r and outflows are saved according to r

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gives

er(T −t0 ) + V¯ T (ϕ; τ) := Vtn (ϕ) − ξtTA 0

n−1 "

 (ξtdis (ϕ; τ) k+1

+

ξtTA (ϕ; τ))er(tn −tk+1 ) k+1

.

k=0

(10.61)

10.4.3. Comments on Utility Loss and Shortfall Probability The loss rates w.r.t. u = uA as well as the shortfall probabilities of the above discretetime versions are illustrated in Figs. 10.4 and 10.5. Notice that the introduction of a

Loss rate

Shortfall probability

0.010

0.008

0.006

0.004

0.002

0.000 3

4

m

5

6

7

m

Figure 10.4 Loss rates w.r.t. u = uA (shortfall probabilities) implied by continuous-time CPPI (solid line), monthly CPPI without transaction costs (dashed lines) and monthly CPPI with θ = 0.01 (dotted line) for varying m. The parameter setup is given in Table 10.2.

0.8

Shortfall probability

0.10

Loss rate

0.08

0.06

0.04

0.6

0.4

0.2

0.02

0.0

0.00 2.03704

4

m

6

2.04

4

6

m

Figure 10.5 Loss rates w.r.t. u = uA (shortfall probabilities) in the Merton setup implied by continuous-time OBPI (solid line), monthly OBPI without transaction costs (dashed lines) and monthly OBPI with θ = 0.01 (dotted line) for varying m. The parameter setup is given in Table 10.2.

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gap risk, i.e., a strictly positive shortfall probability, gives a loss of minus infinity in the case of u = uB and u = uC . Observe, that for both strategies, OBPI and CPPI, the loss which is in the first instance caused by time-discretizing the strategies is rather low. However, there is a huge impact caused by transaction costs. The effect is even more pronounced for the OBPI than for the CPPI method. Intuitively, it is clear that in the limit to continuous-time trading, the transaction costs eat up the cushion, which allows a risky investment. The convergence of the value (cushion) process of the CPPI is for example analyzed in [3]. Recall that loss rates above the one of a risk-free investment are prohibitive. In the case of the basic parameter constellation, this critical value w.r.t. u = uA is equal to 0.5γ(σm)2 = 0.056. In particular, if the strategy parameter m is not chosen in a cautious way, portfolio insurance can get prohibitively bad because of transaction costs. Consider now the shortfall probability. Recall that, in contrast to the continuoustime CPPI strategy, the continuous-time OBPI results in a strictly positive probability that the payoff is not above the terminal guarantee. Thus, it is to be expected that the shortfall probability of the OBPI is more sensitive to discrete-time trading and transaction costs than the one of the CPPI, cf. the right graphs presented in Figs. 10.4 and 10.5. This effect is also illustrated in Fig. 10.6, where the distribution function of the discrete-time versions of OBPI with m = m∗ and CPPI with m = m∗∗ is plotted. However, to some extend the sensitivity of the OBPI to the gap risk measured by the shortfall probability is also caused by the dicretization scheme. While the transaction costs are financed via the cushion for the CPPI, this is not true in the case of the discrete-time OBPI.

1.0

Distribution function

Distribution function

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.0

0.2

1

2

4

Terminal payoff

6

8

0.0

1

2

4

6

8

Terminal payoff

Figure 10.6 Distribution functions of terminal values for OBPI (left) and CPPI (right) with transaction costs (dotted line) and without transaction costs (solid line).

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10.5. UTILITY LOSS CAUSED BY GUARANTEES AND BORROWING CONSTRAINTS To be practically relevant, it is necessary to take borrowing constraints into account. Notice that the basic CPPI approach and an OBPI strategy, which is based on synthesizing a power option, can result in arbitrarily high investment weights. Regarding the OBPI, it is not clear how borrowing constraints should be incorporated. One possibility is given by setting m = 1, i.e., referring to a standard option instead of a power option. Although this is often done in practice, it will not be considered in the following. Instead we focus on incorporating borrowing constraints into the classic CPPI strategy. This straightforwardly results in the capped CPPI, which we abbreviate with CCP. The capped CPPI strategy ϕCCP = (ϕSCCP , ϕBCCP ) is defined by CCP = ϕt,S

min(ωVtCCP , mCtCCP ) , St

CCP ϕt,B =

CCP VtCCP − ϕt,S St

Bt

,

(10.62)

where CtCCP := VtCCP − e−r(T −t) GT and w (w ≥ 1) denotes the restriction on the investment proportion. In the following, we refer to borrowing constraints in the strict sense, i.e., we assume that ω is set equal to one. It is worth mentioning that the borrowing constraints introduce a path-dependence and the payoff implied by the capped CPPI version cannot be stated as a function of the terminal asset price as it is the case without borrowing constraints. Intuitively, it is to be expected that the path dependence yields an additional utility loss in a Black/Scholes type model setup. In order to calculate the loss rate, we consider the distribution of the terminal value of the CCP strategy. Let f denote the density function of the terminal value of the capped CPPI. Then it holds # E[uA (πCCP )] =

∞

uA (v)f(v)dv GT


# CE T,A (π

CCP

)=

∞

v

1−γ

1  1−γ f(v)dv

GT

such that 

 r+m+ (µ−r)− 21 γ(m∗ σ)2 )T ( 1  V0 e  lT,A (πCCP ) = ln   . 1  1−γ $∞ T 1−γ f(v)dv v GT

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For details on the distribution of the capped CPPI, we refer to [4]. Basically, the distribution can be obtained by considering the process (X)0≤t≤T , which is given by the dynamics dXt = (Xt )dt + dWt , where  1 µ−r    σ − 2 mσ (x) = µ − r 1   − σ σ 2

x≤0 and x>0

 1 (m − 1)V 0    σ ln mG 0 X0 =  1 (m − 1)C0   ln mσ G0

mC0 ≥ V0 . mC0 < V0

Along the lines of [4], one can show that the value process (VtCCP )0≤t≤T and cushion   process CtCCP 0≤t≤T are, for ω = 1, given by

VtCCP

 m σXt   e = Gt m − 1  1 + 1 eσmXt  m−1

Xt ≥ 0 (10.63) Xt < 0

and

CtCCP = Gt

 
m σXt   − 1 Xt ≥ 0 e  m−1   

1 emσXt m−1

.

Xt < 0

In particular, it holds

P[VtCCP

 (m−1)v   ln mGt 1    p dv   σv σ      (m − 1)(v − Gt ) ∈ dv] = ln     1 Gt    dv  p     σm   σm(v − Gt )

v≥

m Gt m−1

v<

m Gt m−1

where p(x) := PX0 (Xt ∈ dx). The loss rate implied by the capped CPPI is illustrated in Fig. 10.7. Notice that for m → ∞, the capped CPPI converges to the stop-loss strategy, cf. [13]. Obviously, the loss rate converges, too. It is interesting to observe that the loss-rate of the capped CPPI is smaller than the loss-rate of the stop-loss-strategy for m ≥ m∗∗ .

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0.06

0.06

Loss rate

0.08

Loss rate

0.08

0.04

0.04

0.02

0.00

FA

0.02

2.04

4

6

8

10

12

0.00

2.04

m

4

6

8

10

12

m

Figure 10.7 Loss rates w.r.t. u = uA implied by CPPI (solid lines) and capped CPPI (dashed lines) for varying m. The risk aversion is γ = 1.2 (left figure) and γ = 1.5 (right figure), the other parameters are as in Table 10.2. The constant line gives the loss rate of a stop-loss strategy.

10.6. CONCLUSION The popularity of portfolio insurance strategies including a strictly positive guarantee component with respect to a fixed investment horizon can be explained by various reasons like regulatory requirements or behavioral finance models. To some extend, the justification of positive guarantees is also possible assuming that the investor’s preferences can be described using the von Neumann-Morgenstern framework of expected utility. Dating back to Merton, it is well known that in a Black/Scholes model setup and for a CRRA utility function, the optimal strategy is to invest a constant fraction of wealth into the risky asset. Such a constant mix strategy implies that, for an investment proportion m > 1 (m < 1), additional asset are bought (sold) if the asset price increases. In particular, for m > 1 the resulting payoff is convex in the asset price so that a constant mix strategy can, at least technically, be classified as a portfolio insurance strategy. However, the payoff is floored by zero. In theory, it is straightforward to achieve optimal strategies yielding payoffs with a positive floor. Here, the expected utility is maximized under the additional constraint that the terminal portfolio value must be above a strictly positive terminal guarantee. Alternatively, the floor can be achieved if the utility is measured in terms of the difference of the portfolio value and the guarantee instead of the portfolio wealth itself, i.e., if a utility function with a subsistence level is used. The modified optimization problems help to understand the most prominent approaches of portfolio insurance strategies, i.e., CPPI and OBPI strategies. The modifications which are imposed on the unconstrained optimization problem give interesting modifications for the payoffs. Using the unrestricted optimization problem as a benchmark, the constant mix strategy and its associated payoff with floor zero defines

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also a benchmark for OBPI and CPPI strategies. The CPPI results from a subsistence level, i.e., the utility is measured in terms of the difference of portfolio value and guarantee instead of the portfolio value itself. In contrast, the OBPI results from the additional constraint that the terminal payoff is above the guarantee or floor. Considering the associated payoffs, both approaches result in payoffs which consist of a fraction of the payoff of the constant mix strategy and an additional term stemming from the guarantee. Intuitively, it is clear that the fraction is linked to the price of the guarantee, i.e., the fraction is less than one. The main difference between OBPI and CPPI can easily be explained by the additional term. In the case of the CPPI approach, the additional term is simply the guarantee itself, i.e., the payoff of the adequate number of zero bonds. In contrast, the additional term implied by the OBPI is based on a put on the fraction of the constant mix payoff with strike equal to the guarantee. Obviously, the put is cheaper than the zero bond itself. Thus, the OBPI fraction which is held of the optimal unrestricted payoff is higher than in the case of the CPPI method. This is a major advantage in terms of the associated utility costs, i.e., the loss in expected utility which is caused by the introduction of a strictly positive guarantee. The utility costs are measured and illustrated in terms of a loss rate linking the certainty equivalents of the strict portfolio insurance strategies to the certainty equivalent of the optimal solution. One major drawback of the OBPI method is due to its kinked payoff-profile. The terminal value of the OBPI is equal to the guarantee if the put expires in the money. In contrast to the CPPI method, this implies a positive point mass that the terminal value is equal to the guarantee. This relevant probability is given by the real world probability that the terminal asset prices are below the strike of the put. Intuitively, it is clear that this can cause a high exposure to gap risk, i.e., the risk that the guarantee is violated, if market frictions are introduced. We illustrate this effect by taking trading restrictions and transaction costs into account. It turns out that the guarantee implied by the CPPI method is relatively robust. However, the probability that the guarantee is not reached under the corresponding synthetic discrete-time OBPI strategy is rather high. Finally, we tackle the question of borrowing constraints. In a strict sense, borrowing constraints imply that the proportion of asset must not be above one. In the case of the CPPI method, the capped CPPI which simply states that the asset proportion is adequately capped according to the borrowing constraint is of high practical relevance. We give the distribution of the capped CPPI and illustrate the corresponding loss rate, i.e., the loss which is due to borrowing constraints.

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11

PORTFOLIO INSURANCES, CPPI AND CPDO, TRUTH OR ILLUSION?

ELISABETH JOOSSENS∗,‡ and WIM SCHOUTENS†,§ ∗

†

Joint Research Centre of the European Commission Katholieke Universiteit Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Leuven, Belgium ‡ [email protected] § [email protected]

Constant proportion portfolio insurance (CPPI) and constant proportion debt obligations (CPDO) strategies have recently created derivative instruments, which try to protect a portfolio against failure events and have only been adopted in the credit market for the last couple of years. Since their introduction, CPPI strategies have been popular because they provide protection while at the same time they offer high yields. CPDOs were only introduced into the market in 2006 and can be considered as a variation of the CPPI with as main difference the fact that CPDOs do not provide principal protection. Both CPPI and CPDO strategies take investment positions in a risk-free bond and a risky portfolio (often one or more credit default swaps). At each step, the portfolio is rebalanced and the level of risk taken will depend on the distance between the current value of the portfolio and the necessary amount needed to fulfill all the future obligations. We first analyze in detail the dynamics of both investment strategies and afterwards test the safetyness of both products under a multivariate Lévy setting. More precise we first propose a quick way to calibrate a multivariate Variance Gamma (VG) process on correlated spreads, which can then be used to quantify the gap risk for CPPIs and CPDOs.

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11.1. INTRODUCTION Financial institutions try to protect their portfolios against failure events and derivative instruments are a possible solution. Derivative instruments are a fast-growing market in which alternative investment strategies such as Constant Proportion Portfolio Insurance (CPPI) and Constant Proportion Debt Obligation (CPDO) are created. Although these two recently developed instruments function in different ways when deciding on their investment strategy, both investment funds attempt to provide a portfolio insurance. More precisely, their strategy is to invest only a part of the capital in a risky asset and to invest the remainder in a safe way. The total value of the portfolio at each time step will influence the position taken in the risky asset. The decisions taken on the risk position at each time step aim to allow the investor of a CPPI or CPDO to recover, at maturity, a given percentage of their initial capital, which allows them to benefit from a capital guarantee while participating in the upside of an underlying asset. It could happen that the promised return is not achieved. In this case, for the CPPI structure, the bank will have to cover the losses at maturity while for the CPDO structure, the CPDO will unwind and the investor will not receive the promised amount at maturity but only the remainder amount at the time of unwinding. CPPI structures clearly safeguard a given percentage of the invested capital for the investor, while for CPDO the investor appears to be taking a risk. In the past, this risk has always been seen as very small and CPDOs have been sold as very safe. Here, we will study those two products in depth in order to answer the question whether CPDO and CPPI are really as safe and attractive as they seem. First, the concept of credit risk will be introduced. This is the risk that, after agreeing on a certain contract, one of the involved parties will not fulfil its financial obligations (such as paying a premium). Often the quality and price of financial products will heavily depend on this risk. Different ways to model this risk are presented and can be used to price financial products. Next, a credit default swap (CDS) is introduced. This financial instrument tries to provide a protection for credit risk by transferring it. In exchange for a predefined cost, also called “spread”, a second party will cover the losses one might suffer due to credit risk. The costs for such an insurance will clearly depend on the size of the risk and the impact of the possible losses. Hence, the price can be determined using the above-mentioned models for credit risk. There are no obligations to trade credit default swaps so their changes in credit spread can be used for speculations, for example: CDS spreads can be used as risky assets in investment structures such as CPPI and CPDO. The third section will be dedicated to a particular type of insurance portfolio, the CPPIs. They are a derivative security with capital guarantee using a dynamic trading strategy in order to incorporate the performance of a certain underlying product such

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as a simple stock or a CDS. They were introduced more than 10 years ago and are frequently used. The investment structure of CPPIs has been and continues to be a popular topic for research. First, a multivariate jump-driven model, which can be applied for pricing credit derivatives such as CPDOs, is discussed in detail in Sec. 11.4. Next, in Sec. 11.5, a more general overview of the other recent developments regarding CPPIs is given. All papers could be classified under three different fields of interest. A first group of papers concentrates on proposing different models for pricing. The second group concentrates on the estimation of the leverage factor in order to fix an upper bound for the gap-risk, and the third group tries to extend the structure even more in order to include an extra safety factor. One of the recent new products based on the idea of a CPPI is the CPDO or constant proportion debt obligation, which is discussed in Sec. 11.6. CPDOs are used for credit portfolios comprising exposures to credit indices such as iTraxx and CDX. The CPDO structure borrows many features from the CPPI structure, such as the constant proportion. The main goal of a CPDO is to produce a high-yielding product and this is achieved through a high degree of leverage. Contrary to a CPPI investment strategy, leverage will be increased when the net asset value of the portfolio decreases and descends below the target amount, but leverage will be decreased when the net asset value of the portfolio increases and approaches the target. Once a CPDO reaches this target amount, it will completely de-leverage. In this section, we will not only focus on the dynamics of the structure but we will also give a short overview of the research concentrating on this topic and discuss the question of the “safeness” of this structure as has been highlighted recently in the news. We will conclude by discussing in more depth the differences and similarities of CPPI and CPDO. This should give an even better insight into the structure of both financial instruments.

11.2. CREDIT RISK AND CREDIT DEFAULT SWAPS This section is intended to give a short introduction to the main financial concepts, which will play a role in this work. First, the concept of credit risk is introduced and different ways to model this risk are discussed. Next, a short introduction on credit derivatives, i.e., more precise credit default swaps, is provided.

11.2.1. Credit Risk Credit risk refers to the risk that a specified reference identity does not meet its credit obligations within a specified time horizon T (called maturity). In other words, whenever two or more parties sign an agreement, there is a risk that one of them will not

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meet its obligations. A simple example is the case, where a single person signs a loan with a bank. Here, it can happen that the person does not repay his debt according to the agreement. In such a case, we say that default will have occurred. The risk that such an event will happen is called credit risk and will always be spread over a certain time length. Taking a more global perspective, in finance we do not only deal with the situation of a person and a loan but it will always be possible to characterize the credit risk in terms of the following components: the obligor, the set of criteria defining the default, and a time interval over which the risk is spread. When, for instance, we talk about bonds, their default can be defined in several ways. It could, for example, be bankruptcy but it might also be a rating downgrade of the company or failure to pay an obligation (such as a coupon). But it could also concern the value of a firm — here a firm’s value is linked to the value of its financial assets. In general one will look at the firm’s asset value V = Vt , 0 ≤ t ≤ T and default will be defined as a boundary condition on the asset value. For example a default event will occur if the value falls below a certain fixed level L within the time horizon. Figure 11.1 presents two possible paths 250

200

V

t

150

100

50

0

0

1

2

3

4

τ

5 T

6

7

8

9

10

Figure 11.1 Two possible paths for a firm’s asset value over time with T = 10. When the black line occurs the obligor will default (when Vt < L) in the gray case the obligor will survive.

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for the firm value over time [0, 10] modeled through Black–Scholes, where µ = 0.05, σ = 0.4, and S0 = 100 as explained below. In the case of the black line, the value of the firm will fall below the lower bound, which is fixed at 20 just before t = 5 and hence will default. In the case that the firm value follows the gray path, no default will occur before T = 10. As one will try to protect oneself from defaults, the size of credit risk will be one of the crucial factors when determining the prices and hence the techniques for estimating the probability of default of a reference entity within time T will be very important. Developing new models for the estimation of credit risk is an important topic in the field of finance. The strong interest in this topic is also linked to the fact that in the last couple of years the volume of instruments linked to credit risk traded on the market has increased exponentially. Besides the increase in investments, there has also been interest due to the Basel II Accord, which encourages financial institutions to develop methods for assessing their risk exposure. Credit risk models are usually classified into two categories: reduced-form models, including intensity-based models, and structural models. Intensity-based models, also known as hazard-rate models, focus directly on modeling the default probability. The main idea of these models lies in the fact that at any moment in time (as long as the contract is running) there is a probability that an obligor might default. Default is defined at the first jump of a counting process with a certain intensity. In practice, the models assume that the intensities of the default times follow a certain process (stochastic or deterministic) and under those conditions the underlying default model can be constructed. This intensity of the process depends heavily on the firm’s overall health and on the situation of the market. The structural models, also known as firm-value models, link default events to the value of the financial assets of the firm, such as in the example presented above. Credit risk will hence depend on the model used for the value of the financial assets of the firm and the criteria used for a default. This approach will almost always be used in the remainder of this chapter when modeling the credit default. A common way to model the time evolution of assets uses the following diffusion process: dSt = St (µdt + σdWt ),

S0 > 0,

(11.1)

where Wt is a standard Brownian motion. Here, µ and σ > 0 are the so-called drift and volatility factors. The path of a Brownian motion is continuous and can easily be simulated when one discretizes the time using very small steps t. The value of a Brownian motion at time {nt, n = 1, 2, . . .} is obtained by sampling a series of independent standard

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normal random numbers {νn , n = 1, 2, . . .} and setting: W0 = 0

and Wnt = W(n−1)t +

√ tνt .

The solution of the SDE given in Eq. (11.1) is called geometric Brownian motion and is given by St = S0 exp((µ − σ 2 /2)t + σWt ).

(11.2)

The above way to model price changes of financial products is also referred to as the Black–Scholes model. The main advantages of using the Black–Scholes model is that it is easy to understand. Calculation of prices of derivatives under the model is moreover typically not that time-consuming. A drawback is that it assumes normality of the log returns of the financial assets, which is often not true in reality. Moreover, the Black–Scholes model does not capture the possibility of sudden jumps, which do occur in real live and often cause extra credit risk. Hence, a more flexible stochastic process is often required to model reality in a better way. It would be good to keep some properties of the Brownian motion such as independence and stationarity of the increments but to drop the constraints of normality and continuity of the paths. To create such a process, we must restrict ourselves to the group of infinitely divisible distributions. For each infinitely divisible distribution (with characteristic function φ(u)), a stochastic process can be defined which starts at zero and has independent and stationary increments such that the distribution of the increments over [s, s + t], s, t ≥ 0 has (φ(u))t as characteristic function. Such processes are called Lévy processes, in honour of Paul Lévy, a pioneer of the theory. Definition 11.1. Lévy process: A cadlag stochastic process X = {Xt , t ≥ 0} defined on a probability space (, F, P) is a Lévy process if the following conditions hold: (1) Xt is a continuous process P-almost surely: ∀ε > 0 :

lim P(|Xt+h − Xt | ≥ ε) = 0.

h→0

(2) X0 = 0. (3) The process has stationary increments. (4) The process has independent increments. Lévy process became very popular and are still more and more used in practice. Examples of Lévy process are the (compound) Poisson process, the Gamma process, the inverse Gaussian process, and the Variance Gamma process (VG). An in-depth study of Lévy process in finance and how they can be applied can be found in [1]. Next, we will define an important group of credit derivatives, which are often used in practice.

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11.2.2. Credit Default Swaps (CDS) Credit Default Swaps (CDSs) are very simple credit derivatives and have a big share in the market of credit derivatives. Credit derivatives can be defined as the group of all derivatives, whose payoffs are affected by the default of a specified reference entity (or a basket of entities). They are often used to hedge, transfer, or manage the risk and can hence be considered as an insurance against default. The main idea of credit derivatives is that credit risk is transferred without reallocating the ownership of the underlying asset(s). This way they provide a certain protection against decreasing solvency or default of the underlying asset(s). CDSs in particular are designed to isolate the risk of default on a credit obligation. A CDS is a bilateral agreement, where the protection buyer transfers the credit risk of a reference entity to the protection seller for a determined amount of time T . In exchange for this shift of risk, the protection buyer will make predetermined payments to the protection seller. These payments will occur until the end of the contract (the time of maturity) T unless a default event occurs before the time to maturity. If default of the reference entity occurs, the protection seller will cover the losses (or part of the losses) of the protection buyer due to the default of the underlying entity and the contract will be terminated. The yearly rate paid by the protection buyer to enter a CDS contract against failure is called the CDS spread. Spreads are almost always quantified in bp where bp stands for “basis point” and is equal to 0.01%. The amount of the spread will reflect the riskiness of the underlying credit, if the probability of default increases also the cost of the CDS (and hence the spread) will increase. We note that CDSs are often also used to speculate on changes in credit spread. Figure 11.2 presents the cash flows for two possible scenarios (default at time t = 7 or no default) for an example. We consider the case where a person owns a zero-coupon defaultable bond of a company with a face value F = 10,000 Euro and maturity T = 10 years. Suppose that this person would like to cover himself against the possible default of the bond. He can buy this protection by entering into a CDS contract. A possible situation would be that the contract requests an annual payment of an amount of 400 bp from the protection buyer to be protected against the default. In return, the protection seller will cover the loss, which might result from defaulting. The amount of the loss will be equal to the difference between F and the recovery value after default. We hence take into account that when a default event occurs, the total amount will not automatically be lost completely, but part of the value might be recovered. The concept of recovery can be understood through the following example. In the case that a company goes bankrupt, there are creditors claiming against the assets of the company, and the owner of the bond is one of those creditors. The assets are sold by a liquidator and the profits are used to meet the claims as far as possible. Historically values of the recovery rate fall between 20%

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and 50%. For the current example, we assume that the recovery rate will be equal to R = 40%. The annual amount paid by the protection buyer in this example is hence equal to 400 bp · 10,000 = 400 Euro and the payment of the protection seller in case of default will be F(1 − R) = 6000 Euro. In Fig. 11.2, it is assumed that in the second scenario the bond defaulted at the beginning of the seventh year. Pricing models for CDSs based on Lévy processes can be found in [2] and [3]. In practice, CDS are not only used to reallocate the risk of a single asset (the so-called single name CDS), but also a basket of assets might be considered. A credit default swap index is a credit derivative used to hedge credit risk or to take a position on a basket of credit entities. There are currently two main families of CDS indices: CDX and iTraxx. CDX indices contain North American and Emerging Market companies, whereas iTraxx contains companies from the rest of the world. One of the most widely traded index is the iTraxx Europe index composed of the most liquid 125 CDS referencing European investment grade credits. There are also significant volumes, in nominal values, of trading in the HiVol index. HiVol is a subset of the main index consisting of what are seen as the most risky 30 constituents at the

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time the index is constructed. Typically every six months new series of CDS indices are issued. Indices are like CDSs not only used as insurance against default risk but are also traded on the market in a speculative way.

11.3. PORTFOLIO INSURANCES Portfolio Insurances are capital guarantee derivative securities that embed a dynamic trading strategy in order to make a contribution to the performance of a certain underlying product (e.g., an asset, a CDS or a CDS index, …). One particular type is considered here, the constant proportion portfolio insurance e.g., [4], investing partially in a risk-free way and combining this with a risky asset. The family of Constant Proportion Portfolio Insurance consists of investments for which the amount necessary for guaranteeing a repayment of a fixed amount N at maturity T is invested in a risk-free way, typically a bond, B, and only the exceeding amount will be invested in one or more risky assets, S (i) . This way an investor can limit its downside risks and maintain some upside potential. This type of portfolio insurance has first been introduced by [5] and [6]. The product manager will take larger risks when his strategy performed well. But if the market went against him, he will reduce the risk rapidly. The following factors play a key role in the risk strategies an investor will take: • Price: The current value of the CPPI. The value at time t ∈ [0, T ] will be denoted as Vt . • Floor: The reference level to which the CPPI is compared. This level will guarantee the possibility of repaying the fixed amount N at maturity T , hence it could be seen as the present value of N at maturity. Typically this is a zero-coupon bond and its price at time t will be denoted as Bt . • Cushion: The cushion is defined as the difference between the price and the floor: Cushion = Price − Floor. • Cushion%: It is defined as the ratio of the cushion over the price. • Multiplier: The multiplier is a fixed value, which represents the amount of leverage an investor is willing to take. • Investment level: It is the percentage invested in the risky asset portfolio; this also known as the exposure and is for each step fixed at: e = Multiplier × Cushion%. • “gap” risk: This is the probability that the CPPI value will fall under the floor.

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The level of risk an investor will take at each time t is equal to the investment level as long as the value of the CPPI exceeds the floor. For any time t, the future investment decision will be made according to the following rule: • If Vt ≤ Floor = Bt , we will invest the complete portfolio into the zero-coupon bond. • If Vt > Floor, we will invest a proportion equal to e in the risky asset portfolio. It can easily be shown that under the assumption that the underlying asset price follows a Black–Scholes model with continuous trading, there is no risk of going below the floor and that the expected return at maturity of the CPPI is equal to [7] E[VT ] = N + (V0 − Ne−rT ) exp(rT + m(µ − r)T ). In practice, however, it is known that the probability of going below the floor is nonzero. It might, for instance, happen that during a sudden downside move or due to overnight changes, the fund manager might not be able to adjust the portfolio in time. In the case of an event where the actual portfolio value falls under the floor, at maturity the issuer will have to cover the difference between the actual portfolio value and the guaranteed amount N. It is therefore of importance for the issuer of a CPPI note to be able to quantify this risk, also called gap risk. We will present an example of a possible cash flow for a CPPI with maturity T equal to 10 years. For the sake of simplicity, we will consider only one underlying asset with prices St and a risk-free asset, a zero-coupon bond Bt with a constant interest rate r = 5%. We also assume that the initial price of the asset is equal to S0 = 100 and the prices over time are simulated from a Variance Gamma model, which will be presented later (see Eq. (11.4)) with parameters σ = 0.5, ν = 0.25, and θ = 0.026. For the CPPI process, the leverage or multiplier is fixed at 2.5 and the starting capital is 100. We also consider that the CPPI at maturity repays the investor with at least the initial capital. Figure 11.3 presents two examples of possible scenarios for the simple CPPI. In the example on the left, the value of the CPPI will always stay above the floor, while in the second example (on the right) at time τ a sudden drop of the risky asset will result in a CPPI value below the floor, which is the gap risk. Note that different scales are applied for the presentation of the results. In this example, as the repayment at maturity of the initial value should be insured, the floor will be 100 exp(−r(10 − t)) at each time t. For each step, the value of the cushion is calculated and the portfolio is re-balanced according to the risk exposure. The re-balancing is such that the bigger the difference between the CPPI value and the floor, the higher the cushion value and the more risk one will take. The process will stop at maturity or once a drop of the asset value occurs of such a level that the CPPI value falls below or hits the floor. If such a drop happens the product manager will put the risk exposure to zero and only invest in a risk-free way until maturity.

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Figure 11.3 Value of the risky asset (top), CPPI performance (central), and cushion (bottom) in case the CPPI value stays above the floor (left) and in case a sudden downwards jump occurs and the CPPI value drops bellow below the floor (right).

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Recent papers discuss different aspects of this group of investment strategies. In the next section, an overview of the main findings is given.

11.4. MODELING OF CPPI DYNAMICS USING MULTIVARIATE JUMP-DRIVEN PROCESSES This section is based on [8] and presents a possible way of modeling credit CPPI structures. More precise, a dynamic Lévy model (VG) is set up for a series of correlated spreads. A recent innovation is the possibility to trade swaptions, an option granting its owner the right but not the obligation to enter into an underlying swap (e.g., a CDS), on indices. We will make use of these instruments to calibrate the underlying dynamic spread models. The parameters of the model actually come from a two-step calibration procedure. First, a joint calibration of the correlated spreads on swaptions and second by a correlation matching procedure. For the joint calibration, we make use of equity-like pricing formulas for payers and receivers swaptions based on characteristic functions and Fast Fourier Transform methods. To obtain the required correlation, we set by a closed-form matching procedure the models correlation exactly equal to a prescribed (e.g., historical) correlation. We then have a model in place that can generate very fast correlated spread dynamics under jump dynamics. The calibrated model can be used to price a whole range of exotic structures. We illustrate this by pricing credit CPPI structures. The CPPI structures considered in this section are such that the invested capital is put in a risk-free bond and a position is taken on credit derivatives indices (usually protection is sold). As discussed before, important in the handling of CPPIs is assessing the risk that spreads of the underlying credit indices jump and lead to so-called gap risk. Because of the built in jump dynamics, a better assessment of this gap risk is made possible.

11.4.1. Multivariate Variance Gamma Modeling Next, a dynamic Lévy model, more precise a Multivariate Variance Gamma (MVG) model, is set up for a series of correlated spreads. As this model can generate correlated spreads in fast way, it can be applied in order to price different exotic structures such as the CPPI. For the construction of the Multivariate Variance Gamma model, the authors start from the Univariate Variance Gamma process. The Variance Gamma process was introduced in the financial literature by [9]. The characteristic function of the Variance Gamma (VG(σ, ν, θ)) distribution is defined as follows: 
1 2 2 −1/ν . φVG (u; σ, ν, θ) = 1 − iuθν + σ νu 2

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The distribution is known to be infinitely divisible and by standard Lévy theory one thus can build out of such a distribution a process. The VG process is defined as X(VG) = {Xt(VG) , t ≥ 0} with parameters σ, ν > 0, and θ. It is a process, which starts at zero has stationary and independent increments, and for which the increments √ (VG) − Xs(VG) follow a VG(σ t, ν/t, tθ) distribution over the time [s, t + s] (for more Xs+t details see [1]). Theoretical background on Lévy process can be found in [10] and [11]. For some historical background on the VG model see [12]. Another way of constructing the VG process is by the technique of time changing. Here, we will start from a Gamma process. Recall that the density function of a Gamma(a, b) distribution is given by ba a−1 x exp(−xb), x > 0, and a, b > 0, (a) where (.) is the Gamma function. It is known that the distribution is infinitely divisible and hence using the Gamma distribution one can build a process with independent and stationary Gamma increments. Defining G = {Gt , t ≥ 0} as a Gamma process with parameters a = b = 1/ν, the resulting Gt will follow a Gamma(at, b) distribution and E[Gt ] = t. A VG process can be constructed by time-changing a Brownian Motion with drift. More precisely, one can show (for more details see [1]) that the process fGamma (x; a, b) =

Xt(VG) = θGt + σWGt ,

t ≥ 0,

(11.3)

where W = {Wt , t ≥ 0} is a standard Brownian motion independent from the Gamma process, is indeed a VG process with parameters (σ, ν, θ). The construction can be interpreted as if we now look at a Black–Scholes world but now measured according to a new business clock (Gamma time). It has proven to be very successful in the univariate setting, as the underlying VG distribution can take into account, in contrast to the Normal distribution, skewness, and excess-kurtosis. In [13], a VG model was used to accurately fit CDS curves. With this alternative definition, we have an easy way to simulate a sample path of the VG processes, which can be obtained by sampling a standard Brownian motion and a Gamma process. The Gamma process can easily, like the Brownian motion, be simulated at time points {nt, n = 1, 2, . . .} with t small. First generate independent Gamma(at, b) random numbers {gn , n = 1, 2, . . .}. Then, the Gamma process can be constructed by G0 = 0

and Gnt = G(n−1)t + gn ,

n ≥ 1.

Similarly to Eq. (11.2) for the Brownian motion, the value over time of an asset using the dynamics of a Variance Gamma process can now be modeled by St = S0 exp(ωt + θGt + σWGt ), with ω = ν−1 log(1 − σ 2 ν/2 − θν).

(11.4)

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We work with multivariate extensions of the model along the technique described in [14] and [15]. Additionally, we want to note that in [9], a symmetric version of a multivariate VG is initiated. To build the multivariate VG process a Gamma process G = {Gt , t ≥ 0} is needed, where the parameters a and b are both fixed at 1/ν. Also an N-dimensional Brownian motion is introduced W = {W t , t ≥ 0}, where W t = (Wt(1) , Wt(2) , . . . , Wt(N) ). It is assumed that this process is independent of the Gamma process and that the Brownian motions have a correlation matrix given by ρW = (ρijW , i, j, = 1, . . . , N) (j)

(i) ρijW = E[W1 , W1 ].

A multivariate VG process X = {X t = (Xt(1) , . . . , Xt(N) ), t ≥ 0} is defined as Xt(i) = θi Gt + σi WG(i)t . Note that there is dependence between the Xt(i) ’s due to two causes: they are all constructed by a time change with a common Gamma time. This will mean that the processes will all jump together, but jumps’ sizes can be different. Moreover, there is dependency also built in via the Brownian motions. A straightforward calculation shows that the correlation between two components is given by ρij =

(j) (j) θi θj ν + σi σj ρijW E[X1(i) X1 ] − E[X1(i) ]E[X1 ] =  .   (j) σi2 + θi2 ν σj2 + θj2 ν Var[X1(i) ] Var[X1 ]

(11.5)

This clearly shows that, even if we assume the Brownian components to be independent of each other, one still obtains a correlation between the different components because of the common gamma time. Suppose that we need to model, as later on will be the case, the evolution of Ncorrelated spreads. Take for example the evolution of the iTraxx Europe main index and its overseas opponent the Dow Jones CDX.NA.IG main index and/or the spreads of their HiVol subsets. We assume the following correlated dynamics for the evolution of N-dependent spreads: St(i) = S0(i) exp(ωi t + θi Gt + σi WG(i)t ) = S0(i) exp(ωi t + Xt(i) ), where

i = 1, . . . , N,


 1 ωi = ν−1 log 1 − σi2 ν − θi ν . 2

These mean-correction terms, ωi , are in place because one can then easily show that the spread processes are mean-reverting in the sense that we have for every t ≥ 0 E[St(i) ] = S0(i) .

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11.4.2. Swaptions on Credit Indices The two most liquidly traded index swaptions types are payers and receivers. These are typically European style. A payer/receiver option holder has on expiry the right but not the obligation to buy/sell protection on the underlying index at the strike level. If a default happens among the index constituents prior to option expiration, both the buyer of a payer or the seller of a receiver option can trigger on expiry a credit event (so-called non-knockout feature). The payoff of an index swaption at expiry has two components: payoff due to difference between expiry spread level and the strike and payoff due to any default losses. For short-maturity options, the later is very unlikely to happen and is often ignored. Let us introduce some notation. Denote by T the (payer or receiver) swaption maturity (typical 3, 6, or 9 months) and by T ∗ the index maturity (typically 5, 7, or 10 years). Let us denote with At the risky annuity for maturity t (i.e., the present value of 1 bp of the fee leg). The forward annuity is denoted by A(T, T ∗ ) and is the forward annuity from swaption maturity to index maturity as of the trade day (t = 0). We have of course that At = A(0, t) and A(T, T ∗ ) = AT ∗ − AT . The forward spread as of the trade day (i.e., at time t = 0) with 0 ≤ T < T ∗ , F0 = F0 (T, T ∗ ) is the forward spread from swaption maturity T to index maturity T ∗ and is given by St At − Ss As . F0 (s, t) = At − A s

11.4.2.1. Black’s model The market standard for modeling credit spreads options is a modification of the Black’s formula for interest rate swaptions (see ( [16])). It models spread dynamics in the same way as in Eq. (11.2), where µ = 0. Black’s formula for the value of a payer/receiver swaption with maturity T and strike value K is given by Payer(T, K) = A(T, T ∗ )(F0 N(d1 ) − KN(d2 )), Receiver(T, K) = A(T, T ∗ )(KN(−d2 ) − F0 N(−d1 )), where √ log(F0 /K) + σ 2 T/2 and d2 = d1 − σ T . √ σ T If the payer swaption is non-knockout, as is typically the case for the index swaptions we are dealing with, we adjust the forward spread to account for the non-knockout feature of index options. We account for this additional protection by increasing the d1 =

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forward spread by the cost of this protection. More precisely, the adjusted forward spread is given by (adj)

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(T, T ∗ ) = F0 (T, T ∗ ) +

S T ∗ AT , AT ∗ − A T

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(adj)

N(d1 ) − KN(d2 )),

Receiver(T, K) = A(T, T ∗ )(KN(−d2 ) − F0

(adj)

N(−d1 )),

where (adj)

d1 =

log(F0

/K) + σ 2 T/2 √ σ T

and

√ d2 = d1 − σ T .

Summarizing, we want to remark on the striking connection with vanilla option prices in equity. Basically, pricing comes down to pricing under a Black–Scholes regime with no interest rates and no dividends. However, the model has all the deficiencies of the Black–Scholes framework: no-jumps, light-tails, symmetric underlying distribution, etcetera.

11.4.2.2. The variance gamma model Completely similar to the equity setting. The Black–Scholes dynamics are replaced with the better performing jump dynamics of a Variance Gamma process. We now model the spread dynamics as St = S0 exp(ωt + θGt + σWt ) = S0 exp(ωt + Xt ). Pricing of vanillas has already been worked out in full detail in equity settings by [17] and by a slight adaption to the credit setting we can very fast calculate payer and receiver swaptions using the Carr–Madan formula in combination with fast Fourier transform methods. More precisely, Payer(T, K) = A(T, T ∗ )  ×

exp(−α log(K)) π

+∞

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φ(v − (α + 1)i; T ) dv, α2 + α − v2 + i(2α + 1)v

where the characteristic function of the log of the adjusted forward spread process at maturity T is given by (adj)

φ(u; T ) = E[exp(iu(log F0

+ ωT + XT ))],

which is known analytically in many Lévy settings, including VG.

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11.4.3. Spread Modeling by Correlated VG Processes We now have a multivariate spread model available, and a fast pricers for the standard payer and receiver swaptions. Using these fast pricers on these index swaption, one can calibrate the model in a two steps procedure. (1) We will make sure that our model reproduces the swaption market data as best as possible by doing a joint calibration using our fast FFT pricer. This step will give parameter estimators for the σi , ν, and θi . (2) We put in place the exact correlation structure we want, by calculating (using a closed-form formula) the correlation matrix of the underlying driving standard Brownian motions. Note, that the matrix calculated in the this fashion is not necessary positive-definite. In such a situation, one can then as a kind of ad hoc method look for the closed positive-definite matrix and work with that one. The model is illustrated by a worked out example, where we price a CPPI structure.

11.4.3.1. The pricing of CPPIs We work out the details of credit CPPI by an example, where positions will be taken in four highly correlated indexes and a predefined trading strategy is in place. In our example, we take positions in the following index products: • • • •

iTraxx Europe Main on the run (5 years) unfunded iTraxx Europe HiVol on the run (5 years) unfunded DJ CDX.NA.IG Main on the run (5 years) unfunded DJ CDX.NA.IG HiVol on the run (5 years) unfunded

The swap rates for the above products are highly correlated as can be seen from Fig. 11.4. The corresponding correlation matrix of spreads itself and the correlation matrix of the corresponding daily log returns based on observations from the 21st June 2004 until 13th March 2007 are given in Tables 11.1 and 11.2, respectively. We start with a portfolio of say 100M EUR and an investment horizon of six years. We want to have the principal of the initial investment protected. Therefore, we calculate the bond floor as the value of a risk-free bond that matures at the end of the investment horizon. Suppose we take r = 0.04 and use compound interest rates, then the bond-floor is initially at 78.6628M EUR. Suppose we set the leverage at m = 30. Initially, the cushion is thus at 21.3372M EUR. Multiplying the cushion with the constant leverage factor of 30 gives the risky exposure that we are going to take, namely, 640.12M EUR. We are taking the following positions: • Sell protection on iTraxx Europe Main on the run (5 years) for half of the risky exposure.

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30 • Buy protection on iTraxx Europe HiVol on the run (5 years) for 12 125 of the risky exposure. • Sell protection on DJ CDX.NA.IG Main On the run (5 years) for half of the risky exposure. 30 • Buy protection on DJ CDX.NA.IG HiVol On the run (5 years) for 12 125 of the risky exposure.

The risky exposure is actually taken in position in iTraxx Europe Main and DJ CDX.NA.IG Main. By buying some protection on their HiVol components, the actual risky exposure is reduced and one is only exposed to the non-HiVol names in both indices. This is often done to eliminate as much as possible default risk. If a name defaults, it will most likely be a HiVol name. The initial 100M EUR are put on a risk-free bank account at a compound rate of four percent. Suppose the current quotes for the four components of our portfolio are these as given in Table 11.3. We rebalance after regular times, say daily and continue doing this until maturity or until we have a negative cushion at a rebalancing date. In that case, all positions are closed. We can however not pay back the principal amount since the portfolio value is below the bond-floor. This is called gap risk (see Fig. 11.5). One of the aims of the model is to calculate the gap risk or in other words the present value of these gaps. We assume the following correlated VG dynamics for the spreads: St(1) = S0(iTraxxMain) exp(ω1 t + θ1 Gt + σ1 WG(1)t ) St(2) = S0(iTraxxHiVol) exp(ω2 t + θ2 Gt + σ2 WG(2)t ) St(3) = S0(CDXMain) exp(ω3 t + θ3 Gt + σ3 WG(3)t ) St(4) = S0(CDXHiVol) exp(ω4 t + θ4 Gt + σ4 WG(4)t ), where Gt is a common Gamma Process such that Gt ∼ Gamma(t/ν, 1/ν) and Wt(i) are correlated standard Brownian motions with a given correlation matrix ρW = (ρijW ). Table 11.3 Current Quotes for the Four Components of Our Portfolio in bp. t=0 iTtraxx Main iTraxx HiVol CDX Main CDX HiVol

24.625 48.75 37.5 88.5

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The log-returns in this model have a correlation structure as in Eq. (11.5). We will first perform a joint calibration on swaptions of the individual indices. This determines the parameters ν, θi , and σi , i = 1, . . . , 4. Next, we match the historical correlations with ρij by setting   ρij σi2 + θi2 ν σj2 + θj2 ν − θi θj ν ρijW = . σi σj Hence, we are able to match quite accurately all the individual spread dynamics by correlated jump processes and moreover are able to impose a correlation structure completely matching the observed historical correlation. Indeed for our example, the result of the calibration can be found in Fig. 11.6. In order to match with the required correlation, which we have taken from the log-return historical correlation from 21 June 2004–13 March 2007 as shown in Table 11.2, we need to set the Brownian correlation matrix equal to   1.0000 0.9265 0.4935 0.3352 0.9265 1.0000 0.4470 0.3247  ρW =  0.4935 0.4470 1.0000 0.8688 . 0.3352 0.3247 0.8688 1.0000 In Fig. 11.7, one sees a typical picture of the correlated moves under the multivariate VG jump dynamics.

11.4.3.2. Gap risk As already mentioned in the previous section, the gap risk under Black’s model under continuous rebalancing is zero. Of course, because the continuous paths of the Brownian Motion, the bond floor is never crossed but at maximum hit. One could artificially rebalance only periodically, say quarterly, in order to generate some gap risk. Much more natural and conform reality is to rebalance continuously (or daily), but to include jump dynamics in the model. Indeed, if jumps are present in the spread dynamics, then the portfolio value can suddenly jump below the bond-floor. Hence, the price of this gap risk is not zero anymore. The price to cover against the gap risk, estimated on 100,000 Monte Carlo simulations of the multivariate VG model in our example for instance is around 5 bp per year. Finally, we want to note that the model is not restricted to a credit setting, but one can pimp the model to a hybrid setting. One can set up multivariateVG dynamics, where for example equity indices and stock dynamics are combined with credit dynamics. This is possible in case fast vanilla pricers are available under a univariate VG model, like is the case for equity vanillas and credit swaptions.

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Calibration on swaptions. Figure 11.6

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11.5. RECENT DEVELOPMENTS FOR CPPI CPPIs received and still receive a lot of attention, not only from banks but also from academia. Here, we try to give an overview of the results discussed in a list of papers. Three main topics of research can be identified: the behavior of CPPI, limiting the risk, and building insurances for this risk. The first group of papers study in detail the behavior of CPPI strategies under specific conditions for the underlying portfolio. One way of pricing was already presented in detail in the previous section. The next group of papers try to measure the risk factors involved. In the first paper discussed here, an upper bound for the multiplier m is searched for in such a way that the investment in the risky portfolio is maximized under the condition that the gap risk must stay under a certain limit. The second discussion presents an extended way to calculate the VaR and GVaR of the CPPI portfolio. The last group of papers concentrate on possible ways of extending the CPPI in such a way that an insurance against the small but existing gap risk is build in. The price and size of such an extra insurance will depend on the probability of hitting the floor and hence ways to quantify this risk are also discussed.

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For each of the groups, we discuss one example in more detail in the following three subsections.

11.5.1. Portfolio Insurance: The Extreme Value Approach to the CPPI Method In this paper, published by Bertrand and Prigent [18] in 2002, extreme value theory (EVT) is applied to CPPIs. They aim to find a multiplier as high as possible while ensuring that the portfolio value will always be above the floor at a given probability level (typically 99%). If the obtained multiplier is applied, in practice one would maximize its risk under the constraint that the gap risk stayed below a certain level (here 1%). The methods applied in the paper focus on the opposite of the relative jump of the risky asset at time t + 1: Xt+1 =

St − St+1 . St

To determine an upper bound for the multiplier, we only need to concentrate on the positive values of X. First a general and very strong upper bound for m is constructed which is then relaxed considering different assumptions. The proposed methods are then applied to calculate empirical estimates for the upper bound for m using S&P 500 index during the period January 1969–September 1997 as risky assets under the different assumptions. As an initial step, it is shown that in order to keep the cushion positive at all times the upper bound on the multiple needs to be the following: m≤

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where n indicates the size of the dataset. This strong condition is then relaxed by applying quantile hedging. In order to get a result for the multiplier, the possible values of Xk are first truncated to the interval [a, b] and Xk[a,b] denotes the truncated jumps (Xk 1a≤Xk ≤b ) with their corresponding arrival times Tk[a,b] (i.e., the sequence of times at which Xt takes values in the interval [a, b]). The values of a and b will be chosen in such a way that the values of Xk will all fall in the interval. It is assumed that inter-arrival times of the truncated values are exponentially distributed with parameter λ[a,b] . Under this condition, it is concluded that m≤

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If the distribution F [a,b] is not known, two different cases can be considered. Either one can assume that no prior information is available or, alternatively, one might be able to fix an upper bound due to anticipation. If no historical information is available, extreme value statistics will be applied on the maximum value of any time period T  (0 < T  < T ). EVT states that the distribution of the maximum of n observations will converge to a limiting distribution with two parameters. After applying maximum likelihood estimation the parameters of this limiting extreme value distribution, an approximation of the upper bound of m is obtained. In the other case, where some prior information is available, a portfolio manager might fix a maximal drop level b based on his believes. Under this assumption of a maximal drop level, the calculation of the upper bound for m becomes more complicated but the portfolio manager will be able to use a higher multiplier m. Finally, changes of the upper bound under the different assumptions are compared when using S&P 500 index as a risky asset.

11.5.2. VaR Approach for Credit CPPI In [19], the authors discuss the need to build an overnight profit and loss distribution of the portfolio in order to guarantee the capital of a CPPI. Based on this P&L distribution, one could measure the underlying risks and adjust the multiplier in order to limit gap risk. Credit CPPIs strategies invest in risky assets based on an index or single name CDSs, an index or bespoke tranches or spread options. The latest generation of credit CPPIs involve generic portfolios of credit derivatives and the credit index alone might not be a good proxy of the risk. As a solution, it is proposed to measure the impact of each risk factor separately. The different factors identified are spreads, jump-to-default and credit correlation, and the possibility of second-order risk factors is also included. Those should then be combined to get an idea of the global risk. Next, it is argued that as historical data are often of low quality when looking at portfolios of credit derivatives (such as CDSs but also CDO tranches). It is not recommended to use them in order to compute the risks. It might be possible to find a distribution for each of the risks, using VaR, separately, but not for the sum, hence GVaR will be used for estimating the risks. First, the problem of estimating spreadVaR and defaultVaR for a portfolio of CDSs will be tackled. Here, a model for the spread dynamic is proposed. The correlation caused by the fact that spreads of CDSs are driven by global, regional, and sectoral factors is included in order to estimate the profit and loss coming from spread moves. For the default VaR of a CDS portfolio, the aim is to find the profit and loss coming from overnight defaults. It is assumed that defaults are correlated through a Gaussian copula but no detailed discussion is presented in [19].

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To finish the discussion, portfolios with CDOs are included. For the spread VaR and default VaR, the previous results can be re-used. Besides those, also the VaR of the profit and loss coming from daily moves of the correlation smile is of importance. Here a model-free approach is proposed. The above risk measures for each factor are finally combined using GVaR to give a global estimation of risk. This is supposed to be stable and to offer great netting.

11.5.3. CPPI with Cushion Insurance In [20], under the assumption that the dynamics of the value of the risky asset follow the classic diffusion process given in Eq. (11.1), the authors discuss the situation where an investor does not invest the total amount available but only a part of the initial amount, qV0 , into a CPPI. He will use the remainder amount, (1 − q)V0 , to create an extra insurance on the CPPI. This extra insurance will be build in such way that it pays out a value K in case the value of the CPPI hits an adjusted floor Fta = (F0 + a)ert with a > 0 and r the continuous interest rate. As the adjusted floor lies above the initial floor, this protection will avoid creating gap risk up to a certain level. This approach seems to be acceptable as the adjusted floor will, first of all, prevent the CPPI portfolio value coming too close to the floor and hence gap risk will only remain possible in the event of a large downward jump. Second, for the calculation of the risk exposure a manager will only consider an adjusted cushion (= price-adjusted floor), which is smaller than the original cushion. These two factors will make a jump creating gap risk less likely. The first time the CPPI hits the adjusted floor is denoted as Ta1 . If Ta1 > T , only qV0 is invested into the CPPI, otherwise at Ta1 the amount K is paid out and will from then on be invested in the CPPI portfolio together with the current value of the CPPI portfolio (VTa1 ). In order to calculate the costs linked to this system, the authors first calculate the possibility that the CPPI portfolio hits the modified floor under Black’s model, and hence that the investor will receive the amount K. Using this probability (for given values of a, m, and q) together with the principle that expected cost and outcome must be equal, the correct value of K can be computed. Now all the parameters have been obtained and properties of this investment strategy are studied using the “greeks”. Greeks are the quantities representing the market sensitivities, the name is used because the parameters are often denoted by Greek letters. The Delta, for example, measures the sensitivity to changes in the price of the underlying asset. The paper concludes that the Delta of the modified CPPI portfolio is similar to one of the simple CPPI portfolio. Next, the same exercise is performed assuming that the underlying assets dynamics are not given by Eq. (11.1) but by a Lévy process defined as follows dSt = St− [µ(t, St )dt + σ(t, St )dWt + δ(t, St )dlt ],

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where (Wt )t is a standard Brownian motion, independent from the Poisson process with measure of jumps (lt )t . It is also assumed that the expected value of jumps in every finite interval [0, t] is finite and equal to λt (λ ≥ 0). The paper concludes by extending the insurance in the sense that not only the first time but every time the CPPI value reaches the adjusted floor, the amount K will be invested in the CPPI portfolio.

11.6. A NEW FINANCIAL INSTRUMENT: CONSTANT PROPORTION DEBT OBLIGATIONS Constant Proportion Debt Obligations (CPDOs) first appeared in August 2006 and are a variation on the CPPI structure. They are used for credit portfolios comprising the exposures to credit indices such as CDX and iTraxx. The CPDO’s risk exposure, just as with the CPPI, is determined using a constant proportion approach and rebalances its portfolio between the credit portfolio and a safe asset. The CPDO structure does this with the aim of producing a high-yielding “AAA”-rated product. A CPDO funds itself through the issuance of long-term debt paying timely coupon and principal on the notes. The promised coupon is a spread above for example LIBOR. The combination of a high coupon payment and the high rating have made CPDOs very popular products. First, we will try to explain the structure in detail. CPDOs are currently also a popular topic in the news and a small overview is provided the next subsection. This section is concluded presenting an example on rating of CPDOs.

11.6.1. The Structure Constant Proportion Debt Obligations are structures which use, as suggested by their name, a constant proportion approach for their risk exposure and re-balance their portfolio at every time step between the credit portfolio and a safe asset. The CPDO structure takes leveraged exposure to a risky asset by selling protection on individual names or indices (CDS or indices on CDS). The risky exposure ensures that there is enough spread to meet the promised liabilities and also covers the costs and potential losses that the transaction will absorb. Risk positions will be taking in function of the value of the CPDO. If the structure is not performing well, it will increase its risk exposure (up to a pre-defined maximum leverage level) in order to allow for recovery from the negative performance by increasing the income from the risky asset to rebuild the portfolio’s value. The following factors play a key role in the risk strategies an investor will take: • Net Asset Value (NAV ): This is the current value of the CPDO. It will be the sum of the safe investment and the market value of the risky portfolio.

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• Riskfree: The amount of the total CPDO value invested in a risk-free way. • PV(liabilities): The present value of the current liabilities will be the sum of all discounted coupons still to be paid and the discounted value of the final principal amount. • Shortfall: The shortfall is the capital, which is missing to fulfill all the future obligations. It is defined as the difference between the present value of the liabilities and the net asset value, Shortfall = PV(liabilities) − NAV. • CDS premium: The premium is the amount to be paid to the protection seller (such as a CDS or CDS index) in exchange for the insurance. An increasing value of the spread refers to an increased default probability of the underlying asset and hence it will be negative as a default will lead to a decreasing NAV . Conversely, an increasing spread results in a higher income. • PV(CDS premium): The present value of all future premium payments up to maturity. • Leverage: The leverage refers to the degree of risk which will be taken at each time step. A maximum level is often fixed at 15. The leverage is hence defined as 

Shortfall , max(Leverage) , (11.6) Leverage = min β PV(CDS premium) where β is a multiplier. Similar as in [21], page 18, in this document β will be fixed at 1/Riskfree. • Cash-in: In case the NAV is equal to or exceeds the target value (PV(liabilities)), the necessary amount to cover all future liabilities is reached and hence the risky exposure and leverage will be put to zero. From this point onwards the NAV will be completely invested at the risk-free rate, with coupon and fees being paid until maturity. • Cash-out: if there are substantial losses and the NAV falls below a certain threshold (often fixed at 10% of the initial investment), it will be said that cash-out has occurred. In such a situation, the CPDO will unwind and the investor will receive the remaining value. The investment strategy follows the following steps. At every time-step, one should check the discounted value of the future obligations (coupon and principal payment), which is the amount which one tries to reach. Next, as the CPDO exposes itself to risk by selling protection (CDSs or CDS indices), the mark-to-market of this risky investment is checked and compared to the price paid at the previous time step for this investment. An increase in the spread of the risky CDS means that the underlying insurance has become more expensive and linked to this it has become more likely that the seller of the protection will have to cover a

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future loss. Compared to the previous time-step, this can be considered as a loss for the protection seller, as he appears to be underpaid for the protection it provides, and a gain for the protection buyer. Besides this gain or loss linked to the protection spread, the costs (or incomes) related to the protection also need to be taken into account. Those costs are referred to as the fee which is equal to the sum of all the CDS premium payments made by the protection buyer for the insurance within the time step, taking into consideration the possibility of a default event. The total value of the CPDO (NAV ) at time t + 1 will hence be equal at the accumulated value of the cash, invested risk free at time t, plus the value of the risky asset at time t + 1 augmented with the gains (or losses) made by investing and the fees collected (or paid) for the insurance in the time period [t, t + 1]. From this value, we should subtract the coupon payments which need to be made to the CPDO investor. Based on the above value, and before going to the next step, the new leverage is determined and according to this leverage new investment positions are taken in the risky asset(s). If the NAV increases, the shortfall decreases and hence the leverage will go down; whereas if the NAV decreases, the shortfall will increase and the leverage will go up in order to try to fix the previous negative performance. The above steps are repeated at each time step until cash-in or cash-out occurs, or until maturity. As described in the definition, cash-in occurs when the total value of the CPDO reaches or exceeds the current value of the future obligations. In this case, the seller of a CPDO is sure to be able to fulfil all its future obligations and will from then on only invest in a risk-free way. The probability for a CPDO that such an event takes place will have a big impact on its rating. Cash-out occurs when the value of the CPDO hits or shoots below a lower bound which is fixed at time t = 0. If this happens, the CPDO will unwind and the investor will receive the remaining proceed. Such an event could be called the gap risk and is comparable to the gap risk of a CPPI. The risk of a cash-out event cannot be excluded, since in case of under-performance more risk will be taken, which increases even more the possibility of arriving below the lower bound when a downward jump occurs. A similar situation as in Sec. 11.3 is used in order to present the CPDO functioning in an example. For the risky asset, a CDS-index is considered with starting spread S0 = 100, and the spreads follow the same Variance Gamma model as before. The risk-free interest rate is fixed at r = 5%. For the CPDO structure, the coupon payment is fixed at r + 2% and the maximum leverage is equal to 15. In this example, a cash-out level of 10% of the initial investment is considered. Figure 11.8 presents two examples of possible scenarios of a CPDO. In the left column, the target level is reached before maturity at time t around 8 from that point onwards all cash will be invested in a risk-free way. For the second example, in the right column, the CPDO value will drop

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below the cash-out level just before time t = 5. This is similar to what, in the CPPI case, we refer to as a gap. If such an event occurs, the CPDO will unwind and the investor will receive all remaining cash. In practice, risk positions will almost always be taken in CDS indices (iTraxx and CDX). The benefits of this choice are summarized on Slide 10 of [22]. However, questions about safety in the sense of the correctness of the high triple-A rating do remain and this sensitive topic will be discussed in the next two subsections. Some literature on CPDOs can be found in [21, 23–28].

11.6.2. CPDOs in the Spotlight Recently, it has become clear that CPDOs are not as safe as is often thought. In real life, cash-out events have occurred and these events have also received attention in the media. As a result, the safety of CPDOs has been put up for discussion. On 16th November 2007, for example Moody’s Investors Service downgraded its ratings on six CPDOs, one even to a “junk” rating of Ba2. The downgrading was done because of the continuing spread widening on the financial names underlying these CPDOs. On 28th November 2007, the first CPDO unwinding was announced. This unwinding shows the controversial credit product’s potential for volatility, and moreover it has raised the question of whether the probability they will pay off is as high as implied in a triple-A rating. And also more recently, on 25th January 2008, Moody’s Investors Service confirms that two more series of notes from structured deals backed by financial companies were liquidated after losing investors approximately 90% of their investment. Besides the two unwindings, it also discusses a list of downgrading which has occurred. Many more cases of downgrading and even unwinding have taken place in the last year.

11.6.3. Rating CPDOs Under VG Dynamics A short example has already been presented in this section to explain the dynamics of the CPDO structure. In this section, another example is presented in which positions are taken in two high correlated indices: the iTraxx (iTraxx Europe Main on the run (5 years) unfunded) and CDX (DJ CDX.NA.IG Main on the run (5 years) unfunded). As trading position, we will always invest half of the risky exposure in each index. As presented in [8] and discussed in detail in Sec. 11.4, a Multivariate Variance Gamma model is used to model a series of correlated spreads. First, the calibration is performed using swaption prices based on those indices in order to construct a model

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Figure 11.9 First graph: value of the risky asset (CDS index); second: the corresponding CPDO performance; third: shortfall at each time step; final: leverage taken at each time.

for their dynamics. The resulting model is then applied to simulate the evolution over time of a CPDO. Assume that the risk-free interest rate is fixed r = 0.04. For the CPDO structure the coupon payment is fixed at r + 2% and the maximum leverage is equal to 15. The initial quotes for our portfolio are iTraxx is quoted 24.625 bp and CDX is quoted 37.5 bp. The top graph of Fig. 11.9 presents a typical picture of the correlated spread evolution under the multivariate VG jump dynamics, which is obtained after calibration. The corresponding path of the CPDO is presented in the second graph and the third and fourth graphs present the corresponding shortfall and leverage positions at each time. In most of the current research documents, it is considered that the spreads follow a Brownian motion which do not allow sudden jumps and hence it is less likely to hit the barrier. If jumps are present in the spread dynamics, the value of the portfolio can suddenly jump under the floor. Based on 10,000 Monte Carlo simulations using the model as obtained from the calibration, we find that the likelihood of occurrence of a cash-out event is around 1.94%, which is not as negligible as might be suggested in

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the literature. It looks that this percentage is closer to what is also observed in practice, as discussed before.

11.7. COMPARISON BETWEEN CPPI AND CPDO When CPDOs were created, they were considered as a variation of the CPPI. They borrow certain features such as a “constant proportion” approach to determine the leverage and the re-balancing of the portfolio between the credit portfolio and the safe asset. On the other hand, they are also very different. Initially, the CPDO value will be below the target value, while the CPPI manager tends to invest only the amount exceeding the floor, which is needed to make the principal payment at maturity in a risky way. Once decisions on the risky exposure need to be taken the idea is that a CPDO investor will increase its risk as it is performing negatively while the CPPI will decrease its risk as it is not performing well and approaches the floor level. In other words, at each time step, the CPPI investor takes risk exposure positions based on the amount of surplus the portfolio value has with respect to the floor value. The CPDO investor, on the other hand, will at each time take risk exposure proportional to the amount the CPDO portfolio is lacking in order to reach the target value. Once the CPDO value reaches the target value, the manager will stop investing in a risky way as there is sufficient capital to pay out all future liabilities and there is no more need to create capital in a risky way. A CPPI manager will try to optimize its profit but will stop taking risk at the moment that the CPPI value touches the floor, as he is afraid to fall below by taking more risk. Also the behavior at the final stage of the contract is different. A CPPI will, at maturity, irrespective of the performance of the risky asset, receive the principal, together with any positive return generated from the risky asset. In case of loss, when the CPPI portfolio falls below the floor, the losses are covered by the seller of a CPPI so that the investor will still receive the principal. An investor can hence always be sure of receiving the principal and, in the case of good performance, even more. For the CPDO, on the other hand, a target value is aimed for and in the case of a sufficiently well performing risky asset a cash-in event will occur and the investor will receive all promised coupon and principal payments. But when the risky asset does not perform well and a cash-out event occurs, the CPDO will unwind before maturity. In such a situation only the remaining amount will be paid out to the investor. Investors will want to know the size of this risk at the time of investing in a CPDO and they will use its rating as an indication. This way, the rating of a CPDO becomes important and will have an effect on the price of a CPDO. In practice, the risk of a cash-out event has apparently been under-estimated.

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11.8. CONCLUSIONS As the market of structured credit products grew over the last years, also the request of protection mechanisms in structured credit transactions stays high and hence, also here, a continuous evolution can be observed. CPPI and CPDO are recently developed alternative investment strategies which aim to provide a protection. CPPIs first came into use around 10 years ago and promise a pre-defined principal payment at maturity. A constant proportion rule is applied to decide the investment strategy. At every time step, the investment in the underlying risky asset and the safe asset is re-balanced in order to optimize the profit. CPDOs were introduced in 2006 and are intended to be safe, high-yielding instruments. A similar constant proportion rule is used for their investment strategies and they will invest in a risky asset by selling protection (such as CDX and iTraxx). Similarly to the CPPI, re-balancing will be done at every time step until the targeted value has been reached. The aim of this chapter was to create an in depth view of the dynamics and risks linked to both alternative investment strategies. Hence, first we have tried to explain step by step how they both function and how investment decisions are made. Understanding the dynamics well helps to identify remaining “safety gaps” and allows an investor to get an idea about the size and possibility of experiencing such a gap. For the CPPI, the possibility that the value of the total portfolio will fall below the floor exists and will create a loss, while for the CPDO a loss occurs when a cash-out event occurs. In both cases, there is a strong interest in quantifying this risk. In a first group of papers, the researchers concentrate on quantifying the risk using specific conditions. Next, some propose ways to limit the multiplier factor in order to limit the risk, while other papers suggest the possibility of investing on an insurance to cover this risk. But also for those new developments a good and robust way of quantifying the risk is necessary. It could be concluded that the gap risk for CPPIs should not be neglected but safety nets can be used to avoid suffering from it. As CPDOs are still very new, the field of research is still limited, and as they have only been used on the market for a couple of years, their performance in the real world has only been observed over a short time. As recent experience has shown, in real life CPDOs do not seem to be as safe as they were expected to be. Clearly, there is still a strong need to quantify the risk of cash-out events in a more realistic manner, and a great deal of research remains to be done in this field.

References [1] Schoutens, W (2003). Lévy Processes in Finance: Pricing Financial Derivatives. John Wiley & Sons.

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[2] Cariboni, J (2007). Credit Derivatives Pricing Under Lévy Processes. PhD thesis, Katholieke Universiteit Leuven. [3] Madan, D and W Schoutens (2007). Break on through to the single side. Technical Report, Section of Statistics Technical Report 07-07, (2007). URL http://perswww.kuleuven.be/ ∼u0009713/LevyCDS.pdf. [4] Overhaus, M, A Bermudez, H Buehler, A Ferraris, C Jordinson and A Lamnouar (2007). Equity Hybrid Derivatives. John Wiley & Sons. [5] Black, F and R Jones (1987). Simplifying portfolio insurance. Journal of Portfolio Management, 48–51. [6] Perold,A (1986). Constant portfolio insurance. Unpublished manuscript, Harvard Business School. [7] Cont, R and P Tankov (2007). Constant proportion portfolio insurance in presence of jumps in asset prices, SSRN eLibrary. URL http://ssrn.com/paper=1021084. [8] Garcia, J, S Goossens and W Schoutens (2008). Let’s Jump Together: Pricing of Credit Derivatives. Risk Magazine, September. [9] Madan, D and E Seneta (1990). The variance Gamma model for share market returns. Journal of Business, 63, 511–524. [10] Bertoin, J (1996). Lévy Processes. Vol. 121, Cambridge Tracts in Mathematics, Cambridge: Cambridge University Press. [11] Sato, K (2000). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press. [12] Seneta, E (2007). The early years of the variance-gamma process. Advances in Mathematical Finance. Boston: Birkhäuser. [13] Cariboni, J and W Schoutens (2004). Pricing credit default swaps under Lévy models. UCS Report 2005-02, K. U. Leuven. [14] Luciano, E and W Schoutens (2006). A multivariate jump-driven financial asset model. Quantitative Finance, 6(5), 385–402. [15] Leoni, P and W Schoutens (2008). Multivariate smiling, Wilmott Magazine, March Issue. [16] Pederson, C (2004). Introduction to Credit Default Swaptions, Lehman Brothers. [17] Carr, P and D Madan (1999). Option pricing and the fast fourier transform. Journal of Computational Finance, 2(4), 61–73. [18] Bertrand, P and J Prigent (2002). Portfolio insurance: The extreme value approach to the CPPI method. Finance, 23(01A13), 68–86. [19] Brun, J and L Prigneaux (2007). VaR approach for credit CPPI and counterparty risk. Quant Congress, Quant Congress. [20] Prigent, JL and F Tahar (2005) CPPI with cushion insurance. SSRN eLibrary. URL http://ssrn.com/paper=675824. [21] Standard&Poor’s (2007). Quantitative modeling apporach to rating index CPDO structures. Technical Report, Standard&Poor’s. [22] ABN-AMRO (2007). Surf CPDO: A breakthrough in synthetic credit investments. Structured Products Forum-Tokyo. [23] Varloot, E, G Charpin and E Charalampidou (2006). CPSO an asset class on its own or a glorified bearish rated equity, UBS Investment Research — European Structured Credit. [24] Lucas, D and R Manning (2007). A CPDO primer. UBS Investment Research — CDO Insight. [25] Varloot, E, G Charpin and E Charalampidou (2007). CPDO insights on rating actions and manager impacts using a new formula. UBS Investment Research — European Structured Credit.

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[26] Cont, R and C Jessen (2008). Constant propoertion debt obligations. Financial Risks Inernational Forum. [27] Saltuk, Y and J Goulden (2007). CPDOs and the upcoming roll. Technical Report, JP Morgan. [28] Linden, A, M Neugebauer, S Bund, J Schiavetta, J Zelter and R Hardee (2007). First generation CPDO: Case Study on Performance and Rating. Structured credit global special report, Derivative Fitch.

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ON THE BENEFITS OF ROBUST ASSET ALLOCATION FOR CPPI STRATEGIES

KATRIN SCHÖTTLE∗,‡ and RALF WERNER†,§ ∗

MEAG MUNICH ERGO AssetManagement GmbH, Oskar-von-Miller-Ring 18, 80333 München, Germany † Hypo Real Estate Holding AG, Unsöldstrasse 2, 80538 München, Germany ‡ [email protected] § [email protected]

In recent years, new ideas for the robustification of the traditional Markowitz frontier have appeared in the literature. Based on one of these ideas — the so-called robust counterparts — we introduce the concept of the robustified efficient frontier. As mean– variance efficient portfolios are frequently used as risky assets for CPPI strategies, we investigate the behavior of such strategies under estimation risk. Based on a toy example, we explain the main idea how the concept of a robustified frontier can be used to improve the performance of CPPI strategies. For this purpose, we compare the theoretical performance of CPPI strategies based on the original and the robust efficient frontier.

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12.1. MOTIVATION During the last years, dynamic investment strategies have gained more and more importance in quantitative asset management. Until now, traditional asset management was mainly based on the choice of a strategic asset allocation (SAA), which has been fixed for one year in most cases. Nowadays, there is a move toward dynamic asset allocation (DAA), mainly driven by portfolio insurance concepts and the influence of option pricing theory on investment strategies, like the best-of-two concept of [9]. Further, it has been observed that the utilization of a dynamically rebalanced allocation allows for a better-tailored risk/return profile for the investor, as described in [1]. In the following study, we focus on one of the most popular portfolio insurance concepts — a constant proportion portfolio insurance (CPPI) strategy. This concept was probably first introduced by Perold in 1986, see [24] (see also [25] as well as [5]), followed by a large variety of theoretical and empirical studies, comparison to alternative strategies, etc. For a recent and extensive overview of related work, we refer to the book [27], Chapter 9 and the references therein. Meanwhile, the properties as well as the pros and cons of CPPI like strategies are well known and discussed in detail, see also [27], Chapter 9. However, in our opinion, one rather relevant practical issue has not yet found much attention by researchers — the influence of estimation risk on the performance of CPPI strategies. In contrast to this observation, there is a good amount of research on the influence of estimation risk on static one-period mean– variance optimization, see [8] and the references contained therein. Hence, we focus on the investigation of the influence of this estimation risk on the performance of CPPI strategies as well as a mitigation of this risk by the robustification of mean–variance efficient portfolios. The remainder of this paper is organized as follows: first, we introduce the traditional CPPI strategy. A few numerical examples will show that the choice of the risky asset has a significant impact on the investment performance of the CPPI strategy and that especially the allocation according to mean–variance optimal portfolios adds value to the strategy. Thereafter, it is shown that this behavior may deteriorate in the presence of estimation risk. Therefore, robustified mean–variance efficient portfolios are introduced and their impact on the investment performance is investigated.

12.2. THE FINANCIAL MARKET In contrast to most academic research, our exposition is based on a discrete time framework instead of a continuous time setting. This setup was chosen for several reasons. First, in continuous time, CPPI strategies only exhibit a shortfall risk in the presence of discontinuous price paths (i.e., in jump models). Second, estimation risk is much

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more important in discrete time. Third and most important, practical implementations are always based on discrete time setups.

12.2.1. The Basic Financial Market In order to properly formulate the standard CPPI strategy and to highlight the aforementioned issues with this strategy, we introduce the following basic financial market. Definition 12.1. Let us assume that we are given a discrete time arbitrage-free financial market (
, F, F, P) with time steps t and final horizon T . Further, assume that on this market, there are I tradeable assets, e.g., stocks, bonds, or equity indices, with I prices {St }t=0,...,T , St ∈ R+ following some adapted (discrete time) stochastic process. This process will be specified in more detail in Assumption 12.3. Remark 12.2. Please note that the probability measure P represents the true real world measure and should not be confused with some martingale measure. Furthermore, we do not assume that the market is complete. Assumption 12.3. For the remainder of this paper, we will assume that the discrete one-period asset returns St are i.i.d. in time and follow a multivariate elliptically contoured distribution, i.e., Si,t − Si,t−t (12.1) St ∼ E(µ, , ψ) with Si,t := Si,t−t I independent of t, with parameters µ ∈ R+ ,  ∈ SI+ , and characteristic generator ψ  with 2ψ (0) = −1. Further, we assume that St is bounded, i.e.,

St ∈ [µ − 1 , µ + 1 ],

especially St > −11.

Since St is bounded, it possesses finite moments, which means that E[St ] = µ,

Cov[St ] = −2ψ (0)  = 

and the characteristic function of St −µ is given by u → ψ(u u). For more details on elliptical distributions, we refer to [11] or [10]. Remark 12.4. In Assumption 12.3, the one-period mean µ and covariances  are used to describe the distribution of St . However, if not explicitly stated otherwise, all tables and figures are scaled to annualized numbers through division by t. Remark 12.5. We need to point out that the usual assumption of multivariate norI mal (or student t) returns contradicts the assumption of St ∈ R+ . We have cured this by restricting the distribution of St to [µ − 1 , µ + 1 ]. For practical purposes, P[Si,t ≤ −1] is negligibly small anyway for sufficiently small t, even if we were in a multivariate normal or student t framework.

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As we want to focus on estimation risk, we assume in the following that the characteristic generator ψ is known (i.e., no model risk is involved) but that the parameters µ and  are not given a priori. Hence, one has to rely on estimators for these quantities. Therefore, we will only consider parameter uncertainty and no model uncertainty for the ease of this exploration. For the formulation of the standard CPPI strategy, we need to make some additional assumptions about the financial market. First, we introduce two riskless assets in the financial market; a cash account and a zero bond. Second, we define what in practice is known as a risky asset. Whereas the cash account and the zero bond enhance the financial market by two additional assets, all risky assets will be completely reproducible by the existing tradeable assets (see Formula (12.2) below) and do therefore not enlarge the market.

12.2.2. The Riskless Asset Assumption 12.6. As mentioned above, on top of the described assets, we need the following additional (locally) riskless assets: • The market includes a cash account, also called (locally) riskless asset. For brevity and notational convenience, we denote the process of the cash account with Ct instead of including it in the already defined process St . The main feature of the cash account is that Ct+t = exp(rt t)Ct ,

t = 0, . . . , T − t,

where the one-period rate rt is assumed to be known at time t, hence Ct+t is Ft -measurable, i.e., predictable. In this sense, we may speak of the cash account as locally riskless asset. • The market is equipped with a zero bond Zt with maturity T , i.e., ZT = 1. Although the price evolution of this zero bond is stochastic in its very nature, zero bonds are still subsumed within the class of riskless assets as the payment at maturity is known in advance. We will see in the following that for the standard CPPI strategy, only the zero bond Z is actually necessary, the cash account C will only become important when dealing with the problem of inter-temporal risk budgeting or in case of general liabilities, see Remark 12.20.

12.2.3. The Risky Asset As risky asset, we understand a basket consisting exclusively of the original I assets building the basic financial market. In general, the risky asset is assumed

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to have a constant asset mix, i.e., it is described by a constant vector of weights I , w1 = 1 (i.e., no short sales are allowed). The risky asset is usually deterw ∈ R+ mined by the client of the asset manager or the asset manager herself and remains fixed throughout the whole investment period. In order to obtain fixed weights, a permanent rebalancing in this risky asset has to be assumed. In this sense — as already mentioned above — the risky asset can be seen as a mutual fund, which is constructed within the financial market. The risky asset is usually established at the inception of the CPPI strategy and remains fixed throughout the life time. Under the assumption of rebalancing at each time t, the price process {Xt }t=0,...,T of the risky asset is given by Xt+t = Xt · (1 + wSt+t ) t = 0, . . . , T − t.

(12.2)

In analogy to St in Eq. (12.1), we set Xt =

Xt − Xt−t , Xt−t

and

µX := E[Xt ] = w µ, σX2 := Cov[Xt ] = ww. I Please note that due to w ∈ R+ it holds that Xt is bounded as well, i.e., Xt ∈ [µX − 1, µX + 1], and especially Xt > −1.

Remark 12.7. In practice, there are a few different ways how the asset allocation for the risky asset can be determined: • The most simple way uses the naïvely diversified portfolio, i.e.,
w=

1 1 ,..., I I



1 = 1, I

i.e., wi =

1 . I

This portfolio is often used as a benchmark against which other methodologies are tested and compared. The advantage of this choice is that it does not rely on any estimate of future returns or volatilities (i.e., covariance matrix of asset returns). • More sophisticated allocations are based on the traditional Markowitz optimization. At the inception of the CPPI strategy (i.e., at t = 0), an asset allocation is chosen from the efficient frontier based on mean–variance optimization. For this approach, estimates for the expected returns µ and the corresponding covariances  need to be available. Given these estimates, superior investment results should be obtained due to the usage of efficient portfolios. The most popular choices are — The global minimum variance portfolio — The maximum Sharpe ratio portfolio

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More details on the setup of the mean–variance optimization will be given in the next subsection. Besides these initially fixed asset allocations, dynamically updated allocations are also possible within the framework of CPPI strategies. As these are also subject to the same estimation issues as the above myopic portfolio choice, we later focus on this simple myopic setup.

12.2.4. Classical Mean–Variance Analysis The traditional mean–variance optimization was first introduced by Markowitz in 1952, see [21]. The basic idea is that a portfolio is solely characterized by the two figures risk (mostly measured in terms of the variance or volatility) and expected return. Since an investor is seeking for an allocation with little risk and high expected return, a trade-off between these two conflicting aims has to be made. In many practical applications, the set of assets describing the financial market is supposed to stem from a multivariate normal distribution, where the expectation and covariance matrix thereof are then used to express the risk and return of the portfolios in the optimization problem. The Markowitz approach is naturally applicable as well in the more general case of multivariate elliptical distributions, see [15]. In the following, we use the Markowitz optimization framework to determine particular portfolios from the efficient frontier, which are then used as underlying risky asset in CPPI strategies. The asset universe under consideration consists of the initial I tradeable assets only. As we will see, the move to efficient portfolios of tradeable assets has a beneficial impact on the performance of the CPPI strategy. To determine all efficient portfolios, i.e., portfolios lying on the efficient frontier, we consider the following family of portfolio optimization problems: Definition 12.8. The classical mean–variance portfolio optimization problem is given by √ min (1 − λ) ww − λw µ, (Pλ ) w∈W

where W ⊂ {w ∈ RI | w1 = 1} is assumed to be non-empty, convex, and comI | w1 = 1}, which will be pact. This is, for example, fulfilled by W = {w ∈ R+ used in the following. The parameter λ ∈ [0, 1] expresses the trade-off between risk (i.e., volatility) and return of the portfolio. The optimal solutions w∗cl (λ) of (Pλ ) for given trade-off parameter λ span the efficient frontier   w∗cl (λ)w∗cl (λ), w∗cl (λ) µ 0≤λ≤1

in the (σ, µ)-diagram, see Fig. 12.1.

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Excess return (p.a.)

0.105

0.1

0.095

0.09

Efficient frontier Min. variance portfolio Naive portfolio Max. sharpe ratio portfolio

0.085

0.08 0.13

0.14

0.15

0.16

0.17 0.18 Volatility (p.a.)

0.19

0.2

0.21

0.22

Figure 12.1 Markowitz efficient frontier and selected portfolios.

Remark 12.9. Based on the assumption that W ⊂ {w ∈ RI | w1 = 1} and since  is positive definite, it is easy to show that there exists at least one solution for 0 ≤ λ < 1 and that the solution w∗cl (λ) is indeed unique, see [29]. Remark 12.10. Two portfolios have been of particular interest thus far: (1) The minimum variance portfolio w∗cl (0), i.e., the portfolio at the left end of the frontier where λ = 0 (2) The maximum Sharpe ratio portfolio — in this context — given by w∗cl (λS ) where λS = arg max  λ∈[0,1]

w∗cl (λ) µ w∗cl (λ)w∗cl (λ)

.

We have illustrated both portfolios together with the naïve portfolio in Fig. 12.1. Remark 12.11. Under the above assumptions, problem (Pλ ) is equivalent to the classical Markowitz formulation max w∈W

w µ

(MVσ )

s.t. ww ≤ σ 2 , where σ and λ are in a monotonous one-to-one relationship. For an exact definition of the equivalence, we refer to [32].

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12.2.5. The Trading Strategy For the mathematical definition of a self-financing trading strategy in the above market setup, let us refer to [12], Sec. 5.1. Our setup differs only in notation from their framework, as we prefer that at time t the portfolio composition (holdings or lot sizes) ϕt ∈ RI+3 shall denote the holdings after trading at time t. This means that the portfolio allocation remains constant throughout the period [t, t +t[ and equals ϕt . A particular trading strategy is thus determined by the way the portfolio composition ϕt ∈ RI+3 at time t is determined based on known quantities at time t. Seen as a stochastic process, this means that ϕt is an Ft -measurable process, i.e., adapted. For notational convenience, we split the trading strategy ϕt ∈ RI+3 into several components, based on the underlying assets:   ϕt = ϕtS , ϕtC , ϕtZ , ϕtX ∈ RI × R × R × R for the first I tradeable assets S, the cash acount C, the zero bond Z, and the risky asset X, resp. Remark 12.12. In the following, the CPPI strategy will only invest in the zero bond and in the risky asset, i.e., we have ϕtS = 0 as well as ϕtC = 0. Nevertheless, the tradeable assets S are required to construct the risky asset X. Definition 12.13. Let us further introduce a few notations: • The corresponding portfolio value at time t is given as   Vt := ϕtS St + ϕtC · Ct + ϕtZ · Zt + ϕtX · Xt . • Let us denote the difference in the holdings before and after trading, the traded lots, as ϕt := ϕt − ϕt−t . • The corresponding trading volume at time t can be computed as vt :=

I

S |ϕi,t | · Si,t + |ϕtC | · Ct + |ϕtZ | · Zt + |ϕtX | · Xt .

i=1

12.3. THE STANDARD CPPI STRATEGY Based on the riskless and the risky asset, we can define the main features of a CPPI strategy. Its core idea lies in the dynamic allocation of capital between the riskless and the risky asset. The proportion of the riskless asset in the portfolio is chosen in such a way that a prespecified minimum level of wealth is guaranteed at time T . This final

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minimum level of wealth Vmin is called floor. In general, it is expressed in absolute terms or relative to the initial wealth V0 : Vmin := f · V0 ,

with 0 < f <

1 , Z0

(12.4)

for example f = 0.95, 1.00, 1.02, . . . . The condition f · Z0 < 1 stems from the absence of arbitrage in the financial market. It is easy to see that investing Lt := Vmin ·Zt into the zero bond Z at time t yields a final wealth of at least Vmin . This observation is the basis for the simple version of the CPPI strategy.

12.3.1. The Simple Case Algorithm 12.14. Given all information at time t, the most simple CPPI strategy can then be stated as follows: (1) Compute the present value of the floor Lt = Zt · Vmin = f · Zt · V0 . (2) Compute the exposure Et := (Vt − Lt )+ . (3) Invest Vt − Et in the zero bond and Et in the risky asset, i.e., ϕtX =

Et , Xt

ϕtZ =

V t − Et . Zt

Let us start with a feasible initial value of the floor L0 , cf. (12.4). Then it holds V0 > E0 = V0 − L0 > 0 and therefore ϕ0X =

E0 , X0

ϕ0Z =

V0 − E0 L0 = = Vmin . Z0 Z0

A closer investigation of the simple CPPI strategy yields that — independent of the returns of the risky asset or the zero bond — the lot sizes ϕtZ remain constant over time: ϕtZ = ϕ0Z = Vmin . Therefore, since the strategy is self-financing, it must also hold that ϕtX = ϕ0X . This means that in its simple version the CPPI strategy does not need to rebalance the portfolio — here we mean rebalancing between riskless and risky asset, the risky asset itself may of course need some rebalancing. Remark 12.15. In general, the above simple CPPI strategy is not very well suited to achieve the returns exceeding the risk free rate. Table 12.1 illustrates that although the underlying risky asset has a rather broad annual return distribution ranging at least from −0.20 to 0.50, the annual return of the CPPI remains within tight bounds from 0.01 to 0.06. This is immediately clear from the property of the simple CPPI strategy that it is just a buy-and-hold strategy between the zero bond and the risky asset, i.e., at each time t the fraction fZ0 is invested in the zero bond, whereas only 1 − fZ0 is

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Performance of Selected Rebalancing and CPPI Strategies.a

Strategy

E

Std

Skewness

Excess kurtosis

1% pctl.

99% pctl.

Reb. naïve Reb. minvar Reb. maxsharpe CPPI naïve CPPI minvar CPPI maxsharpe Zero bond

0.1192 0.1196 0.1290 0.0374 0.0374 0.0382 0.0305

0.1710 0.1505 0.1586 0.0134 0.0117 0.0124 —

0.4524 0.4009 0.4198 0.4524 0.4009 0.4198 —

0.3603 0.3275 0.3217 0.3603 0.3275 0.3217 —

−0.2255 −0.1903 −0.1953 0.0105 0.0132 0.0128 —

0.5771 0.5163 0.5502 0.0731 0.0684 0.0710 —

a This

table gives the average performance of the three risky assets and the zero bond in comparison to those of the corresponding three simple CPPI strategies (with f = 0.95) based on a Monte Carlo simulation with 25,000 random paths. Besides the expected performance (E), we have added the standard deviation (std), the skewness, and the excess kurtosis as well as the 1% and 99% percentile of VT /V0 − 1 for a better comparison.

invested in the risky asset: L0 V0 − L0 V0 − Z0 · Vmin Vt = · Zt + · Xt = Vmin · Zt + · Xt Z0 X0 X0 
Zt Xt . = V0 · fZ0 + (1 − fZ0 ) Z0 X0 Nevertheless, using more information on the distribution of the return of the risky asset, the CPPI strategy can be refined to obtain better investment returns. Remark 12.16. The simulation and optimization setup for the results in Table 12.1 and all subsequent calculations is as follows: • The continuous risk free interest rate is assumed to be 3% p.a., i.e., the zero bond has a discrete return of r = 3.05% p.a. • The excess returns of six individual tradeable assets over the risk-free return of the zero bond are given by   µann,exc = 4.68%, 10.56%, 6.36%, 10.56%, 9.48%, 8.52% , expressed in annualized terms. • The (annualized) covariance matrix of the financial assets is   0.0359 0.0188 0.0122 0.0115 0.0228 0.0199 0.0188 0.0588 0.0322 0.0247 0.0272 0.0158     0.0122 0.0322 0.0460 0.0182 0.0214 0.0108 ann =  . 0.0115 0.0247 0.0182 0.0590 0.0202 0.0079   0.0228 0.0272 0.0214 0.0202 0.0431 0.0172 0.0199 0.0158 0.0108 0.0079 0.0172 0.0220

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• The simulations of the discrete one-period asset returns are based on a normal distribution N (µt, t) with µ = µexc + r. The normal distribution has been cut off to guarantee Assumption 12.3 in the sense of Remark 12.5 simply by neglecting any violating scenario.

12.3.2. The General Case The main idea of the standard CPPI strategy is to leverage the investment in the risky asset and to actively balance between the risky asset and the zero bond. The leverage factor m, by which the investment in the risky asset is increased in comparison to the simple version, is usually called multiplier. The introduction of this multiplier m leads to the CPPI strategy in its standard form: Algorithm 12.17. Given all the necessary information at time t, the standard CPPI strategy can be formulated as (1) (2) (3) (4)

Compute the present value of the floor Lt = Zt · Vmin = f · Zt · V0 . Compute the cushion or risk budget Bt := (Vt − Lt )+ . Compute the leveraged exposure as Et := min(mBt , Vt ). Invest Vt − Et in the zero bond and Et in the risky asset, i.e., Et V t − Et ϕtX = , ϕtZ = . Xt Zt

Remark 12.18. In the above version, the exposure is bounded by the current portfolio wealth, however, it is possible to introduce an upper boundary level 0 < b < 1 and to restrict Et by Et = min(mBt , bVt ). Remark 12.19. The cap of Et by Vt in the standard CPPI strategy restricts an investment in the risky asset beyond the wealth of the strategy, i.e., short positions in the zero bond are prohibited. Note that this cap was not necessary in the simple version of the CPPI strategy. Remark 12.20. In more general settings, Lt is not linked to a deterministic zero bond investment any more, but given by a more general stochastic liability (hence the abbreviation Lt ). As long as this liability Lt is investable, e.g., by so-called replicating portfolios, CPPI strategies can still be formulated. For example, if a money market fund is to be beaten, f · Ct may be chosen as liability. As we see from Fig. 12.2 and Table 12.2, the multiplier m has a significant influence on the performance of the investment strategy. We can observe that the average return E[VT /V0 − 1] increases with increasing m up to a certain level. From there onwards, no significant increase in average performance is possible. We can also observe that the standard deviation behaves accordingly.

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Expected Return Volatility 5

10

15 Multiplier m

20

25

30

Figure 12.2 Average return and standard deviation of the standard CPPI strategy on the naïve risky asset with f = 0.95 for different multipliers m.

Table 12.2 Descriptive Statistics of the Standard CPPI Strategy on the Naïve Risky Asset with f = 0.95 for Different Multipliers m. Multiplier m=1 m=3 m=5 m=7 m=9 m = 11 m = 13 m = 15 m = 17 m = 19 m = 21 m = 23 m = 25 m = 30

E

Std

Skewness

Excess kurtosis

0.0374 0.0531 0.0707 0.0837 0.0909 0.0944 0.0961 0.0969 0.0973 0.0976 0.0978 0.0979 0.0980 0.0981

0.0134 0.0493 0.0972 0.1288 0.1451 0.1537 0.1585 0.1613 0.1632 0.1645 0.1655 0.1662 0.1667 0.1677

0.4524 1.4819 1.9288 1.5671 1.3436 1.2221 1.1504 1.1012 1.0673 1.0417 1.0227 1.0076 0.9965 0.9761

0.3603 3.8084 4.8104 2.5131 1.5518 1.1190 0.8890 0.7365 0.6338 0.5610 0.5093 0.4701 0.4412 0.3899

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Remark 12.21. This behavior can easily be explained by a deeper investigation of Algorithm 12.17, Step (3). If the multiplier m is increased step by step, the leveraged exposure Et will equal Vt as long as there is at least a cushion of Vt /m — which diminishes to 0 if m → ∞. This means that the CPPI strategy is fully invested in the risky asset until the risk budget is lost. As soon as Bt ≤ 0, the investment is completely shifted from the risky asset to the riskless asset. In other words, for m → ∞, the CPPI strategy converges to a simple stop-loss strategy, which invests into the risky asset until the wealth is below the floor Lt . As soon as the cushion is lost, the CPPI strategy cannot recover as it is fully invested in the zero bond for the rest of the time and thus VT ≤ LT = Vmin . In contrast to the simple CPPI strategy, for m > 1, it might happen that VT ≤ LT , i.e., the floor is no longer guaranteed. This probability of a shortfall is investigated in more detail in the next subsection. Please note that this cannot happen in continuous time as long as the price paths are continuous, see [27]. In addition to the multiplier m, also the floor f has a significant impact on the performance and the risk of the strategy. In Tables 12.3 and 12.4, an overview of the dependence is given. It can be observed that with increasing multiplier the expected return of the CPPI strategy increases, since more capital can be invested in the risky asset which accounts for higher returns. The same line of argumentation holds for the floor: the lower the floor, the more budget for riskier investments and thus the higher the expected return. Table 12.3 Expected Return of the Standard CPPI Strategy on the Naïve Risky Asset for Different Multipliers m and Different Floors f .

m=1 m=3 m=5 m=7 m=9 m = 11 m = 13 m = 15 m = 17 m = 19 m = 21 m = 23 m = 25 m = 30

f = 0.90

f = 0.925

f = 0.95

f = 0.975

f = 0.99

f = 1.00

0.0417 0.0672 0.0913 0.1029 0.1072 0.1091 0.1099 0.1103 0.1105 0.1107 0.1107 0.1108 0.1109 0.1110

0.0395 0.0602 0.0816 0.0946 0.1005 0.1031 0.1043 0.1050 0.1053 0.1055 0.1057 0.1057 0.1057 0.1058

0.0374 0.0531 0.0707 0.0837 0.0909 0.0944 0.0961 0.0969 0.0973 0.0976 0.0978 0.0979 0.0980 0.0981

0.0352 0.0461 0.0587 0.0701 0.0776 0.0819 0.0844 0.0854 0.0858 0.0858 0.0858 0.0859 0.0859 0.0861

0.0339 0.0419 0.0512 0.0606 0.0677 0.0721 0.0747 0.0761 0.0768 0.0769 0.0770 0.0767 0.0765 0.0764

0.0331 0.0390 0.0461 0.0537 0.0600 0.0644 0.0670 0.0684 0.0692 0.0696 0.0696 0.0694 0.0693 0.0694

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Table 12.4 Standard Deviation of the Standard CPPI Strategy on the Naïve Risky Asset for Different Multipliers m and Different Floors f .

m=1 m=3 m=5 m=7 m=9 m = 11 m = 13 m = 15 m = 17 m = 19 m = 21 m = 23 m = 25 m = 30

f = 0.90

f = 0.925

f = 0.95

f = 0.975

f = 0.99

f = 1.00

0.0216 0.0799 0.1366 0.1588 0.1660 0.1687 0.1701 0.1710 0.1715 0.1720 0.1723 0.1726 0.1728 0.1731

0.0175 0.0646 0.1191 0.1468 0.1587 0.1638 0.1664 0.1680 0.1691 0.1698 0.1704 0.1708 0.1712 0.1718

0.0134 0.0493 0.0972 0.1288 0.1451 0.1537 0.1585 0.1613 0.1632 0.1645 0.1655 0.1662 0.1667 0.1677

0.0092 0.0340 0.0706 0.1032 0.1229 0.1347 0.1422 0.1470 0.1503 0.1526 0.1541 0.1553 0.1563 0.1579

0.0067 0.0248 0.0527 0.0827 0.1038 0.1172 0.1260 0.1321 0.1364 0.1394 0.1417 0.1434 0.1448 0.1469

0.0051 0.0187 0.0400 0.0665 0.0875 0.1017 0.1114 0.1180 0.1229 0.1265 0.1292 0.1313 0.1330 0.1361

Analogously, with larger investments in the risky asset (i.e., higher multiplier and/or lower floor), the volatility also increases.

12.3.3. Shortfall Probability of CPPI Strategies Definition 12.22. For a given CPPI strategy according to Algorithm 12.17 with parameters f and m, the shortfall probability SP(f, m) and the expected shortfall ES(f, m) are defined as SP(f, m) := P[VT ≤ LT ] = P[VT ≤ Vmin ], ES(f, m) := E[VT − LT | VT ≤ LT ]. The corresponding stopping time τ := inf{t | Vt ≤ Lt } is called shortfall time. In case no shortfall is experienced by the CPPI strategy, we have τ = ∞. We say that the CPPI strategy has reached the absorbing state if Vt ≤ Lt , as from then onwards the strategy is fully invested in the zero bond. Tables 12.5 and 12.6 illustrate the interrelation of the shortfall probability and the expected shortfall with the multiplier m and the floor f for a fixed risky asset (the naïve one). As expected, it can be observed that the shortfall probability increases with increasing multiplier m. The floor influences the shortfall probability in a similar way: the

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Table 12.5 Shortfall Probability of the Standard CPPI Strategy on the Naïve Risky Asset for Different Multipliers m and Different Floors f .

m=1 m=9 m = 11 m = 13 m = 15 m = 17 m = 19 m = 21 m = 23 m = 25 m = 30

f = 0.90

f = 0.925

f = 0.95

f = 0.975

f = 0.99

f = 1.00

0 0 0 0.08% 0.48% 1.82% 3.80% 6.68% 9.16% 11.64% 15.86%

0 0 0.02% 0.16% 0.70% 2.62% 5.78% 10.10% 13.74% 17.56% 24.04%

0 0 0.02% 0.26% 1.14% 3.88% 8.30% 14.22% 19.62% 24.94% 33.42%

0 0 0.02% 0.28% 1.78% 5.62% 12.00% 20.12% 28.22% 35.44% 46.94%

0 0 0.02% 0.32% 2.08% 6.60% 14.22% 24.08% 34.06% 43.26% 56.54%

0 0 0.02% 0.38% 2.32% 7.32% 16.00% 27.20% 38.52% 48.54% 63.02%

Table 12.6 Expected Shortfall of the Standard CPPI Strategy on the Naïve Risky Asset for Different Multipliers m and Different Floors f .

m=1 m=9 m = 11 m = 13 m = 15 m = 17 m = 19 m = 21 m = 23 m = 25 m = 30

f = 0.90

f = 0.925

f = 0.95

f = 0.975

f = 0.99

f = 1.00

— — — −0.0008 −0.0030 −0.0025 −0.0026 −0.0026 −0.0027 −0.0028 −0.0034

— — −0.0015 −0.0030 −0.0026 −0.0022 −0.0023 −0.0023 −0.0025 −0.0027 −0.0033

— — −0.0011 −0.0022 −0.0029 −0.0024 −0.0023 −0.0023 −0.0025 −0.0027 −0.0034

— — −0.0008 −0.0016 −0.0022 −0.0022 −0.0021 −0.0021 −0.0022 −0.0024 −0.0031

— — −0.0006 −0.0012 −0.0018 −0.0018 −0.0019 −0.0020 −0.0022 −0.0025 −0.0031

— — −0.0004 −0.0022 −0.0018 −0.0015 −0.0017 −0.0019 −0.0021 −0.0023 −0.0031

closer the floor to the initial wealth, the higher the shortfall probability. Further, the expected shortfall remains approximately constant. Remark 12.23. In practice, the multiplier m is deduced from the consideration of the shortfall risk by a Monte Carlo simulation. The multiplier is chosen as large as possible to obtain a sufficient average investment return, while keeping a pre-determined level of shortfall risk, i.e., low shortfall probability and low expected shortfall. For this purpose, practitioners usually run simulation studies like the above to obtain a suitable multiplier. It should be noted that there are analytical approximations available

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(see [27]), which link the multiplier to the volatility or percentiles of the one-period return distribution. However, these approximations are only valid if Et = mBt , i.e., for sufficiently small multipliers, and do not longer hold if Et is capped at Vt . In the latter case, the approximation largely overstates the potential shortfall risk. Therefore, one has to rely on Monte Carlo simulations.

12.3.4. Improving CPPI Strategies Thus far, we have only used the naïve portfolio as risky asset. We already pointed out that it should be possible to improve the performance of the CPPI strategy in the same manner as the one-period portfolio performance can be improved by mean–variance optimization. If we replace the naïve portfolio (characterized by the weekly parameters µX = 0.16%, σX = 2.1%) by a mean–variance efficient portfolio with the same level of risk (volatility) and higher average return (µY = 0.19%, σY = 2.1%), we expect an improved average performance of the CPPI strategy and along with this a decreased shortfall risk. This choice of the mean–variance efficient portfolio is illustrated in Fig. 12.3. As we see in Fig. 12.4, the average return of the CPPI strategy is indeed significantly increasing, while the risk stays nearly at the same level. Therefore, we can

0.11

Excess return (p.a.)

0.105

0.1

0.095

0.09 Efficient frontier Min. variance portfolio Naive portfolio Max. sharpe ratio portfolio Naive portfolio efficient

0.085

0.08 0.13

0.14

0.15

0.16

0.17 0.18 Volatility (p.a.)

0.19

0.2

0.21

0.22

Figure 12.3 Markowitz efficient frontier including mean–variance efficient portfolio.

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0.3

0.25

Exp. return naive Exp. return efficient Vol naive Vol efficient Shortfall prob. naive Shortfall prob. efficient

0.2

0.15

0.1

0.05

0

5

10

15 Multiplier (m)

20

25

30

Figure 12.4 Improvement of the standard CPPI (f = 0.95) by replacing the naïve portfolio with an efficient one.

conclude that mean–variance optimization can actually improve the performance of CPPI strategies. Since mean–variance optimization offers a complete efficient frontier, it is not clear which portfolio thereon should be chosen as risky asset. It is further not obvious how the performance of CPPI strategies on these different risky assets should be compared. We have therefore picked a rather practical approach toward this question. For each CPPI strategy, we choose the largest multiplier — cf. Table 12.7 — such that the shortfall risk is still smaller than 1% and compare their average performance and standard deviation, see Tables 12.8 and 12.9. As risky assets, we haven chosen the Table 12.7 Multipliers for Selected CPPI Strategies for Different Floors f , Such that the Shortfall Probability Is Below 1%.

CPPI naïve CPPI naïve efficient CPPI minvar CPPI maxsharpe

f = 0.95

f = 0.975

f = 0.99

f = 1.00

14.50 14.50 16.75 16.00

14.00 14.00 16.00 15.50

13.75 14.00 16.00 15.25

13.75 13.75 15.75 15.25

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CPPI naïve CPPI naïve efficient CPPI minvar CPPI maxsharpe

f = 0.95

f = 0.975

f = 0.99

f = 1.00

0.0968 0.1092 0.1016 0.1085

0.0850 0.0958 0.0900 0.0960

0.0754 0.0848 0.0805 0.0854

0.0676 0.0754 0.0724 0.0768

Table 12.9 Standard Deviation of Selected CPPI Strategies for Different Floors f and Multiplier m, Such that the Shortfall Probability is Below 1%.

CPPI naïve CPPI naïve efficient CPPI minvar CPPI maxsharpe

f = 0.95

f = 0.975

f = 0.99

f = 1.00

0.1607 0.1661 0.1473 0.1555

0.1448 0.1511 0.1359 0.1437

0.1285 0.1360 0.1234 0.1301

0.1141 0.1209 0.1107 0.1172

already introduced canonical candidates from the previous section — the minimum variance portfolio, the maximum Sharpe ratio portfolio, the naïve portfolio, and its mean–variance efficient replacement. It can be observed that the three mean–variance optimal portfolios clearly dominate the inefficient naïve portfolio, both in terms of expected return and Sharpe ratio. Comparing the efficient portfolios with each other is not as easy. Taking the point of view of the asset manager, the risk of the asset manager is clearly the cost of shortfall. These costs are roughly given by the product of shortfall probability and expected shortfall. With shortfall probability below 1% and expected shortfall roughly equal to 30 basis points, these costs are below one basis point. On the other hand, as the asset manager wants to offer a strategy with a high expected return, the maximum Sharpe ratio portfolio is clearly the best choice. This portfolio offers a similar return as the efficient version of the naïve portfolio, but possesses smaller standard deviation, and should hence be preferred. Remark 12.24. Although the above analysis looks very promising, a few details should not be overlooked. • The CPPI strategy needs to be implemented in the simulation framework as close as possible to the real-world setup, as distinct features of such strategies may influence the results of simulation studies. Among these are for example trading filters (mainly directly linked to the traded lots ϕt ), fees (usually depending on Vt ), slippage effects (depending on the traded volume vt ), ratchets, allowance for short positions, additional risks from overlay management, planned cash injections, etc.

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• The joint distribution of the zero bond and the risky asset has to be modeled rather exactly. Especially the tail behavior plays a major role in the quantification of the shortfall risk, whereas the estimation of the expected return is important for the communication of the investment target return. • The choice of the composition of the risky asset and thus the multiplier m are subject to the same estimation risk. If the parameters of the return distribution are uncertain, it is well known that the mean–variance efficient portfolios have a rather poor performance (see next section). In this case, the originally valid idea of using optimized portfolios may revert into the contrary. Unfortunately, while the first point can be covered with sufficient accuracy, the estimation of both the distribution and the distribution parameters are always subject to uncertainty. As shown in [4], [7] or [16] mean–variance efficient portfolios strongly depend on the input data. Due to the inherent uncertainty in the estimates, the resulting portfolios as well as their estimated returns and volatilities are thus not reliable, see for example [23]. Thus, a natural question in this context is: how much can the results differ in the presence of estimation risk? As already mentioned in the introduction, we focus on the estimation risk for the parameters and assume for simplicity that the distribution family itself is known. Remark 12.25. Bertrand and Prigent have considered a possible estimation of the multiplier by extreme value theory in [3]. Although this research is probably the closest one to this study, its line of thinking is solely concerned with shortfall risk. They concentrate on the choice of the multiplier by looking at the corresponding percentiles of the one-period return distributions by extreme value theory. Our study goes into a different direction, as we simultaneously cover shortfall risk and expected return, but do so via parameter uncertainty.

12.3.5. CPPI Strategies Under Estimation Risk For the illustration of estimation risk, we consider a very simple example. We still assume that the asset returns follow the originally specified distribution, however, the asset manager has to estimate these parameters, for example from an historical ˆ instead sample. Therefore, the asset manager has to work with estimates µ ˆ and  of the original parameters µ and . We have chosen a typical example of µ ˆ and ˆ which represents all stylized facts known about estimation risk. In Fig. 12.5, the , solid line represents the efficient frontier based on the original parameters µ and , which are unknown to the investor. Results (see Table 12.10) corresponding to portfolios based on this efficient frontier will be tagged optimal or original. The dashed line in Fig. 12.5 represents the expectation of the asset manager, i.e., the efficient

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0.18

Excess return (p.a.)

0.16

efficient frontier based on original parameters efficient frontier based on random parameters random parameters only for optimization minimum variance portfolio maximum Sharpe ratio portfolio

0.14

0.12

0.1

0.08

0.06 0.1

0.12

0.14

0.16 Volatility (p.a.)

0.18

0.2

0.22

Figure 12.5 Original and estimated efficient frontiers.

ˆ This situation will lead to the frontier based on the random parameters µ ˆ and . numbers tagged perceived or random. Finally, the dotted line is the efficient frontier the investor will actually obtain, i.e., optimizing the portfolios according to the ˆ but realizing the risk–return profile with the true market expected parameters µ ˆ and , parameters µ and . Table 12.10 summarizes the return and volatility characteristics of the minimum variance and the maximum Sharpe ratio portfolios on the three frontiers. Therefore, in Fig. 12.5, two kinds of estimation risk can be noticed. First, the ˆ and thus return µ asset allocation is calculated based on µ ˆ and  ˆ X and volatility σˆ X are obtained based on the optimized allocation. As shown in [17], these are Table 12.10 Illustration of the Estimation Risk of the Minimum Variance Portfolio and the Maximum Sharpe Ratio Portfolio.

Return minimum variance Return maximum sharpe ratio Volatility minimum variance Volatility sharpe ratio

Actual

Perceived

Optimal

0.0800 0.0733 0.1355 0.1539

0.0823 0.1147 0.1175 0.1323

0.0836 0.0924 0.1336 0.1393

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biased estimators compared to the optimal values µX and σX and on average it holds that µ ˆ X > µX and σˆ X < σX . This also gets obvious from Fig. 12.5. Further, as the wrong asset allocation is implemented, an inferior portfolio performance can be achieved in the market, see also [17] for more details. This can again be observed in Fig. 12.5, where the true original frontier dominates the portfolios calculated by the asset manager. Although we have picked only one representative toy example, in [17], it is shown that this represents the average situation. Taking the estimates µ ˆ ˆ together with the corresponding risky asset compositions, the simulation of and  the standard CPPI can be performed as usual Table 12.11. From this, optimal multipliers are obtained for both portfolios. The multipliers have been chosen in such a way that the corresponding shortfall probabilities are roughly 1%. This choice of multipliers yields the following perceived performance and standard deviation, see Tables 12.12 and 12.13. Based on these multipliers and the wrong allocation, we run the simulation again, this time with the true market parameters. Then, instead of the above figures, we obtain the corresponding actual figures, see Tables 12.14–12.16. Table 12.11

CPPI minvar CPPI maxsharpe

Table 12.12

f = 0.95

f = 0.975

f = 0.99

f = 1.00

19.50 17.50

18.75 16.50

18.25 16.25

18.00 16.00

Perceived Expected Return of Selected CPPI Strategies.

CPPI minvar CPPI maxsharpe

Table 12.13

Multipliers m of Selected CPPI Strategies.

f = 0.95

f = 0.975

f = 0.99

f = 1.00

0.1048 0.1358

0.0941 0.1205

0.0847 0.1075

0.0766 0.0960

Perceived Standard Deviation of Selected CPPI Strategies.

CPPI minvar CPPI maxsharpe

f = 0.95

f = 0.975

f = 0.99

f = 1.00

0.1331 0.1574

0.1265 0.1496

0.1163 0.1384

0.1057 0.1264

Table 12.14 Actual Shortfall Probabilities of Selected CPPI Strategies.

CPPI minvar CPPI maxsharpe

f = 0.95

f = 0.975

f = 0.99

f = 1.00

0.0409 0.0631

0.0446 0.0584

0.0447 0.0595

0.0449 0.0568

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CPPI minvar CPPI maxsharpe

f = 0.95

f = 0.975

f = 0.99

f = 1.00

0.0978 0.0880

0.0868 0.0774

0.0778 0.0691

0.0704 0.0625

Table 12.16 Actual Standard Deviation of Selected CPPI Strategies.

CPPI minvar CPPI maxsharpe

f = 0.95

f = 0.975

f = 0.99

f = 1.00

0.1493 0.1608

0.1391 0.1456

0.1267 0.1304

0.1145 0.1163

The most noteworthy differences are both in the expected performance and in the shortfall probability. Although the shortfall probability was believed to be below 1%, it significantly increased to about 5%. This is not only worrying from the point of view of the asset manager, who actually faces higher costs to cover shortfall, but as well from the perspective of the client, who has a much higher probability of ending up in the absorbing state than intended. Even more troublesome is the expected return of the formerly superior maximum Sharpe ratio portfolio. Due to the error maximization, the expected return of this portfolio turned out to be unreliable and the gap to the perceived performance is significant. In contrast to this, the minimum variance portfolio shows a rather stable behavior in terms of expected return. Based on this toy example, it is already obvious that a more prudent choice of multiplier and a consideration of the estimation risk in the composition of the risky asset is absolutely necessary. Therefore, in the next section, we will introduce a framework which not only provides more robust asset allocations, but at the same time gives prudent estimates of risk and return which can be used in the composition of the CPPI strategy.

12.4. ROBUST MEAN–VARIANCE OPTIMIZATION AND IMPROVED CPPI STRATEGIES In the last years, several attempts have been made to improve the estimation of mean–variance efficient portfolios. A meanwhile pretty well-known way to robustify portfolios is the resampling technique introduced and made popular by Michaud, see [23], based on concepts as, e.g., given in [16]. Another rather popular approach is the usage of robust estimators, as, e.g., investigated in [18], [19] or [26]. The third line of thinking is based on the robust counterpart idea introduced by Ben-Tal and Nemirovski,

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see [2]. Several authors have considered instances varying from theoretical to practical settings, see e.g. [6], [13], [20], [31] and [33].

12.4.1. Robust Mean–Variance Analysis Depending on the availability of historical data (length of sample period, frequency of sampling, etc.) and the utilized estimation routine for µ and , the resulting estimates, ˆ can vary, i.e., they are uncertain. Now, instead of only considering denoted by µ ˆ and , these uncertain point estimates, an entire range of possible parameter realizations — a (convex and compact) uncertainty set U — is taken into account and the portfolio is optimized with respect to the worst outcome within U. Definition 12.26. The robust counterparts to problems (Pλ ) are given by √ min max (1 − λ) w Cw − λw r w∈W (r,C)∈U

(RPλ )

with U being the (joint) uncertainty set for the unknown parameters (µ, ). In analogy to the classical setting, the optimal solution of (RPλ ) will be denoted by w∗rob (λ). Remark 12.27. It can be shown that the resulting robust portfolios are unique for 0 ≤ λ < 1, see [29], Proposition 5.2, for general convex and compact uncertainty sets U. Most applications of the robust counterpart approach are based on statistical confidence sets around the corresponding point estimates, see [2], [6], [13], [20], [22], or [32]. In general, the most intuitive confidence sets coming from statistics are confidence ellipsoids, as for example nicely motivated in [22], Sec. 2.4.3. Typically, the center of the ellipsoid is determined as the estimated value of the point estimate and its shape is described using an according estimate for the estimator’s covariance matrix. In addition, the size of the uncertainty set is calculated from an appropriate percentile to achieve the desired level of confidence. Alternatively, several different expert opinions (or likewise different estimators) can be used to form an uncertainty set, see [20]. In most practical instances, the robustification is focused on the uncertainty in the mean, while the covariance matrix is assumed to be known with certainty, which is well supported by a number of theoretical results on the impact of both uncertainties.

12.4.2. Uncertainty Sets Via Expert Opinions or Related Estimators In the following, we assume that we have several different estimators available for the uncertain parameter, for example from experts’ forecasts. Alternatively, different estimators may be derived by different estimation routines. The standard maximum

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likelihood estimator, although very popular, is not the only unbiased consistent estimator, which may be used for elliptical distributions. In fact, there exist several other common estimators for the mean of an elliptical distribution used by practitioners. For example one could consider the following four consistent estimators for the return besides the maximum likelihood estimator: • The median, i.e., the 50% quantile of the data sample • The average of the 25% and the 75% quantile as well as two robust estimators, • The Huber estimator, see [14] • The trimmed mean, which is defined as the maximum likelihood estimator of the sample reduced by the α-percent smallest and largest values (i.e., the outliers) As shown in [30], ellipsoidal uncertainty sets have nicer theoretical properties — in terms of smoothing and robustness properties — than polyhedral ones. Thus, instead of using the (polyhedral) convex hull ˆ U = conv(r1 , . . . , r5 ) × {}    5 5   ˆ = r r = αi ri , with αi ≥ 0, αi = 1 × {}  i=1

i=1

of these five estimators, denoted by r1 , . . . , r5 , to define the uncertainty set U, we prefer to use the ellipsoid ˆ U = {r | (r − r¯ ) C¯ −1 (r − r¯ ) ≤ δ2 } × {}, where r¯ is the average of the five different estimates and C¯ = D2 with D being the diagonal matrix with the standard deviations of the individual estimates from their common average. In this case, the size δ of the ellipsoid is determined such that all estimates are contained within. In other words, the uncertainty set for the return is given by the smallest ellipsoid centered at the common average of the various estimators and shaped by their standard deviations. Combining everything and solving the inner maximization problem in (RPλ ) analytically, the robust portfolios w∗rob (λ) can be obtained by solving   ˆ + λ[δ w Cw ¯ − w r¯ ]. min (1 − λ) w w w∈W

This means that on top of the investment risk, i.e., risk from the stochastic asset returns, an additional penalty for estimation risk is considered in the optimization. This penalty grows linearly with the uncertainty in the estimators, i.e., with δ, as well as linearly in the importance λ of the return term. Although this is a very appealing approach

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in practical settings, for the aim of the study, it is easier to work with an alternative specification of uncertainty sets.

12.4.3. Uncertainty Sets Via Confidence Sets As already mentioned, the most prominent choice for the uncertainty set lies in the traditional confidence ellipsoid around the maximum likelihood estimator µ, ˆ i.e., ˆ ˆ −1 (r − µ) ˆ  ˆ ≤ δ2 } × {}, U1 := {r | (r − µ)

(12.6)

where δ is chosen appropriately to the target confidence level of the uncertainty set. Again, no uncertainty of the covariance matrix is considered. A more advanced version of the confidence set also considers the uncertainty of the covariance matrix. In the latter case, applying similar ideas as in [22], it is possible to show (see [32]) that the natural confidence ellipsoid around the joint maximum ˆ approximately has the following form likelihood estimator for both (µ, ˆ )  −1    ˆ 2 (C − ) ˆ  ˆ − 12 2 ≤ δ2 . ˆ −1 (r − µ) ˆ  ˆ + 2 ·  U2 = (r, C) | (r − µ) (12.7) tr Using any one of the above uncertainty sets a surprising result can be proved by a close investigation of the necessary and sufficient first-order optimality conditions (i.e., the KKT conditions), see [28] or [32]: the robust efficient frontier is exactly the classical efficient frontier up to a certain risk (i.e., volatility) level depending on the choice of the parameter δ. In other words, the original classical efficient frontier is robust in itself, as long as we do not move too far to the right. This can be interpreted as if robustification is able to identify the unreliable upper part of the efficient frontier which is cut off. The following theorems, taken from [32] formally state the described result. Let δ H(θ) = 1 − θ + θ √ , S   K(θ) = max (1 − θ) 1 + δ



κ∈[0,1]

2 (1 − κ) + θδ S−1



 κ , S

where in this context S denotes the number of observations used for the point estimation. Although K is given in terms of an optimization problem, it is computed rather easily by any one-dimensional optimization routine. Theorem 12.28. Let U = U1 as in (12.6) and let θ ∈ [0, 1]. Then the optimal solution w∗rob (θ) of (RPθ ) equals the optimal solution w∗cl (λ) of the original problem (Pλ ) for λ = H (θ) = θ/(θ + H(θ)).

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Theorem 12.29. Let U = U2 as in (12.7) and let θ ∈ [0, 1]. Then the optimal solution w∗rob (θ) of (RPθ ) equals the optimal solution w∗cl (λ) of the original problem (Pλ ) for λ = K (θ) = θ/(θ + K(θ)). Remark 12.30. The most important consequences of these theorems are • The cut off risk level for the robust frontier can be determined by solving the classical formulation with 1 √ < 1. λmax = H (1) = K (1) = 1 + δ/ S The optimal portfolio w∗cl (λmax ) then determines the cut off risk level as  ˆ ∗cl (λmax ). σmax = w∗cl (λmax )w • Considering the special point θ = 0, i.e., the (robust) minimum variance portfolio, the corresponding λ is also zero, i.e., the minimum variance portfolio is already robust in itself. For U1 the robust counterpart reduces to  ˆ min ww, w∈W

which is obvious as the minimum variance portfolio does not depend on the uncertain parameter µ. If uncertainty of the covariance matrix is considered, then the robust counterpart reduces to    2    ˆ ˆ w w. min K(0) w w = min 1 + δ w∈W w∈W S−1 for the uncertainty set U2 . This means that the consideration of the uncertainty of the covariance matrix is reflected in the factor K(0). Since this factor is larger than 1, a higher volatility (i.e., a larger objective value) is expected for the minimum variance portfolio in the robust setting. This increase in the risk can be interpreted as a measure for estimation risk and can be incorporated to obtain an improved estimation of the multiplier m. • The results of both theorems hold true due to the fact that the matrices for measuring the portfolio’s risk and for describing the shape of the uncertainty set U have the same structure and hence the expressions for the risk and the penalty for estimation risk can be combined. Otherwise we would end up as in the previous subsection, where different penalty terms for estimation risk appear in the objective function. The effect of such a robustification for an investor is that her position on the efficient frontier moves to the left, toward the minimum variance portfolio. Hence, robustification leads to more conservative portfolio choices that are nevertheless efficient — simply with respect to a different trade-off parameter.

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12.4.4. Usage and Implications for CPPI Strategies The above robust optimization framework can now be used to improve the performance of CPPI strategies under estimation risk. For this purpose, we suggest the following methodology: Algorithm 12.31. For a given robustification parameter δ and estimated parameters ˆ based on S historical observations, we proceed in the following steps to µ ˆ and  determine an appropriate allocation of the risky asset w together with a corresponding multiplier m: (1) Calculate the robust efficient frontier according to the previous section. Choose an arbitrary portfolio w from the robust frontier, preferably the robust minimum variance or the robust maximum Sharpe ratio portfolio. (2) Determine the robustified parameters 1 ˆ w, µ ˜ =µ ˆ − δ√  ˆ S ww

ˆ ˜ = K(0)2 · . 

(3) Run the CPPI strategy with the robustified parameters and fix the multiplier m as usual, i.e., at an acceptable level of shortfall risk. The choice K(0) is deduced from the fact that this factor appears in the robustified version of the minimum variance problem and represents the penalty for the uncertainty in . For all other portfolios, K(λ) is smaller and could be calculated, but for simplicity, we use K(0) instead. The correction of the expected returns is motivated by the fact that the worst-case return in the confidence set is given by µ, ˜ although this is a rather pessimistic approach which may need some refinement. If we apply Algorithm 12.31 to the previous setting for realistic choices of robustification parameters, δ = 1 and S = 52, we obtain µ ˜ and K(0) as in Table 12.17. It can be observed that the robustified return µ ˜ is overly pessimistic and cannot be used to estimate the expected performance of the portfolios or the CPPI strategy. Nevertheless, it is still well suited for the determination of the multiplier m, as for this purpose, an adverse market setting leads to a conservative choice. Further, the volatility plays the major role in the setting of the multiplier anyway. Table 12.17 µ µ ˆ µ ˜ K(0)

0.15% 0.16% −0.13% 1.0945

Period Returns for the Tradeable Assets.

0.26% 0.17% −0.17%

0.18% 0.33% 0.05%

0.26% 0.16% −0.13%

0.24% 0.24% −0.09%

0.22% 0.21% −0.07%

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Schöttle and Werner Table 12.18 Multipliers m of Robust CPPI Strategies Based on Robustified Parameters.

CPPI robust minvar CPPI robust maxsharpe

f = 0.95

f = 0.975

f = 0.99

f = 1.00

16.25 15.75

15.75 15.25

15.75 15.25

15.50 15.00

Using the robustified allocations for the risky assets and running the CPPI simulation with the robustified parameters, we get the following multipliers, illustrated in Table 12.18. The multiplier was again chosen in such a way that the corresponding shortfall probability is below 1%. Comparing these multipliers to those in Table 12.11, we see that the above multipliers are significantly smaller than those derived from the estimated, but non-robustified, parameters. Remark 12.32. From the illustration of the robust frontier, see Fig. 12.6, it can be noticed that the robust frontier is usually rather short. This means that large deviations from the minimum variance portfolios are usually not possible within the robust framework. Therefore, we do not expect large differences between the (robust) minimum variance and the robust maximum Sharpe ratio portfolio.

0.11

Excess return (p.a.)

0.105

0.1

0.095

0.09

0.085 Efficient frontier Robust efficient frontier 0.08 0.13

0.14

0.15

0.16

0.17 0.18 Volatility (p.a.)

0.19

0.2

0.21

Figure 12.6 Illustration of the classical and the robust efficient frontier.

0.22

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12.4.5. CPPIs with Robust Asset Allocations After we have calculated both the (robustified) risky allocations and the corresponding multipliers, the CPPI strategy can be simulated as usual with the true market parameters. From this simulation, we obtain the actual shortfall probabilities and the actual average performance of the CPPI strategy. According to our expectation, the robustification, which resulted in a decrease of the corresponding multiplier as well as a more prudent asset allocation, yields very promising results on shortfall probabilities. In contrast to the significantly increased probabilities for the non-robustified allocations we now have comparable shortfall probabilities as intended, see Table 12.19. Before comparing actual expected returns of selected (robust) CPPI strategies, we recall in the following Tables 12.20 and 12.21 the optimal and perceived expected returns of non-robustified CPPI strategies as already given in Tables 12.8 and 12.12. Now finally, the actual expected returns of the considered non-robust and robust CPPI strategies are summarized in Table 12.22. It can be observed that the two CPPI strategies based on minimum variance portfolios lead to almost identical expected returns. Eventually, the almost identical returns are not surprising since the allocations for the risky asset used in the respective CPPI strategies are identical, only the multipliers differ. Nevertheless, this difference in the multiplier is responsible for a rather improved Table 12.19 Actual Shortfall Probabilities of Selected CPPI Strategies.

CPPI minvar CPPI maxsharpe CPPI robust minvar CPPI robust maxsharpe

Table 12.20

f = 0.975

f = 0.99

f = 1.00

0.0409 0.0631 0.0077 0.0093

0.0446 0.0584 0.0081 0.0099

0.0447 0.0595 0.0098 0.0110

0.0449 0.0568 0.0094 0.0096

Optimal Expected Return of Selected CPPI Strategies.

CPPI minvar CPPI maxsharpe

Table 12.21

f = 0.95

f = 0.95

f = 0.975

f = 0.99

f = 1.00

0.1016 0.1085

0.0900 0.0960

0.0805 0.0854

0.0724 0.0768

Perceived Expected Return of Selected CPPI Strategies.

CPPI minvar CPPI maxsharpe

f = 0.95

f = 0.975

f = 0.99

f = 1.00

0.1048 0.1358

0.0941 0.1205

0.0847 0.1075

0.0766 0.0960

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CPPI minvar CPPI maxsharpe CPPI robust minvar CPPI robust maxsharpe

f = 0.95

f = 0.975

f = 0.99

f = 1.00

0.0978 0.0880 0.0973 0.0945

0.0868 0.0774 0.0861 0.0833

0.0778 0.0691 0.0771 0.0744

0.0704 0.0625 0.0695 0.0670

Table 12.23 Actual Standard Deviation of Selected CPPI Strategies.

CPPI minvar CPPI maxsharpe CPPI robust minvar CPPI robust maxsharpe

f = 0.95

f = 0.975

f = 0.99

f = 1.00

0.1493 0.1608 0.1473 0.1503

0.1391 0.1456 0.1351 0.1368

0.1267 0.1304 0.1222 0.1229

0.1145 0.1163 0.1091 0.1093

shortfall probability for the robust minimum variance portfolio compared to the nonrobust one. In case of the CPPI strategies based on maximum Sharpe ratio portfolios, the expected returns of the robust versions are significantly higher compared to the non-robustified risky asset allocations and much closer to the optimal ones. However, even with robustification, CPPI on maximum Sharpe ratio portfolios actually cannot beat those on the minimum variance portfolio, which in turn still beats the naïve strategy. This means that portfolio optimization still adds value to the investment process, but that estimation risks have to be taken very seriously. For completeness, Table 12.23 contains the actual standard deviations of selected non-robust and robust CPPI strategies. As expected, all robust versions have a lower standard deviation than their corresponding non-robust strategy.

12.5. CONCLUSION In this study, we have shown that under full information CPPI strategies should rely on mean–variance efficient portfolios as risky assets. Based on Monte Carlo simulations, suitable multipliers can be found. We have demonstrated that estimation risk may have a significant impact on the performance of the CPPI strategy. A potential remedy by robust mean–variance optimal portfolios seems to be beneficial for the overall performance of the strategy, especially for the shortfall risk of the asset manager and the average return. Nevertheless, our results also show that optimization beyond the minimum variance portfolio is heavily influenced by estimation risk. Based on these

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preliminary results, it can be concluded that a more detailed and thorough analysis needs to be carried out to investigate estimation risk and its cures in a broader context.

Acknowledgments The authors want to express their gratitude to the editors for the opportunity to make this contribution. Further, the second author appreciates the hospitality at the HVB Institute for Mathematical Finance, TU München, especially the inspiring annual research seminars which fostered part of the research on robust allocations.

Disclaimer The views expressed in this article are the authors’ personal opinions and should not be construed as being endorsed by MEAG MUNICH ERGO AssetManagement GmbH or Hypo Real Estate Holding AG.

References [1] Allianz Global Investors (2005). Privatizing pensions. [2] Ben-Tal, A and A Nemirovski (1998). Robust convex optimization. Mathematics of Operations Research, 23(4), 769–805. [3] Bertrand, P and JI Prigent (2002). Portfolio insurance: The extreme value theory of the CPPI method. Finance, 23(2), 69–86. [4] Best, MJ and RR Grauer (1991). On the sensitivity of mean–variance-efficient portfolios to changes in asset means: Some analytical and computational results. The Review of Financial Studies, 4(2), 315–342. [5] Black, F and R Jones (1987). Simplifying portfolio insurance. Journal of Portfolio Management, 14(1), 48–51. [6] Ceria, S and RA Stubbs (2006). Incorporating estimation errors into portfolio selection: Robust portfolio construction. Journal of Asset Management, 7(2), 109–127. [7] Chopra, VK and WT Ziemba (1993). The effect of errors in means, variances and covariances on optimal portfolio choice. Journal of Portfolio Management, 19(2), 6–11. [8] DeMiguel, V and F Nogales (2006). Portfolio selection with robust estimates of risk. SSRN Working Paper. [9] Dichtl, H and C Schlenger (2004). Aktien oder Renten? — Das Langfristpotenzial der Best of Two-Strategie. Die Bank, 12, 809–813. [10] Fang, K-T, S Kotz and K-W Ng (1990). Symmetric Multivariate and Related Distributions. London: Chapman and Hall. [11] Fang, K-T and Y-T Zhang (1990). Generalized Multivariate Analysis. Beijing: Science Press. [12] Föllmer, H and A Schied (2004). Stochastic Finance — An Introduction in Discrete Time. de Gruyter Studies in Mathematics.

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[13] Goldfarb, D and G Iyengar (2003). Robust portfolio selection problems. Mathematics of Operations Research, 28(1), 1–38. [14] Huber, P-J (1981). Robust Statistics. New York: John Wiley & Sons. [15] Ingersoll, J-E (1987). Theory of Financial Decision Making. Maryland: Rowman & Littlefield. [16] Jorion, P (1992). Portfolio optimization in practice. Financial Analysts Journal, 48(1), 68–74. [17] Kan, R and DR Smith (2008). The distribution of the sample minimum–variance frontier. Mangement Science, 54(7), 1364–1380. [18] Kempf, A and C Memmel (2003). On the estimation of the global minimum variance portfolio. SSRN Working Paper. [19] Lauprête, GJ,AM Samarov and RE Welsch (2002). Robust portfolio optimization. Metrika, 55(2), 139–149. [20] Lutgens, F (2004). Robust Portfolio Optimization. PhD Thesis, Maastricht University. [21] Markowitz, H (1952). Portfolio selection. Journal of Finance, 7(1), 77–91. [22] Meucci, A (2005). Risk and Asset Allocation. Berlin: Springer. [23] Michaud, RO (1998). Efficient Asset Management. Boston: Harvard Business School Press. [24] Perold, A (1986). Constant portfolio insurance. Unpublished manuscript. [25] Perold, A and W Sharpe (1988). Dynamic strategies for asset allocation. Financial Analysts Journal, 44, 16–27. [26] Perret-Gentil, C and M-P Victoria-Feser (2004). Robust mean–variance portfolio selection. FAME Research Paper no. 140. [27] Prigent, J-L (2007). Portfolio Optimization and Performance Analysis. Chapman & Hall. [28] Scherer, B (2007). Can robust portfolio optimisation help to build better portfolios? Journal of Asset Management, 7(6), 374–387. [29] Schöttle, K (2007). Robust Optimization with Application in Asset Management. Dissertation Technische Universität München. [30] Schöttle, K and R Werner (2005). Costs and benefits of robust optimization. Working Paper. [31] Schöttle, K and R Werner (2006). Towards reliable efficient frontiers. Journal of Asset Management, 7(2), 128–141. [32] Schöttle, K and R Werner (2009). Robustness properties of mean–variance portfolios (to appear in: Optimization). [33] Tütüncü, RH and M Koenig (2004). Robust asset allocation. Annals of Operations Research, 132(1–4), 157–187.

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ROBUST ASSET ALLOCATION UNDER MODEL RISK

PAULINE BARRIEU∗ and SANDRINE TOBELEM† Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE England, UK ∗ [email protected] † [email protected]

In this chapter, we propose a robust asset allocation methodology, when there is some ambiguity concerning the distribution of asset returns. The investor considers several prior models for the assets distribution and displays an ambiguity aversion against them. We have developed a two-step ambiguity robust methodology that offers the advantage to be more tractable and easier to implement than the various approaches proposed in the literature. This methodology decomposes the ambiguity aversion into a model-specific ambiguity aversion as well as relative ambiguity aversion for each model across the set of different priors. The optimal solutions inferred by each prior are transformed through a generic absolute ambiguity function ψ. Then, the transformed solutions are mixed together through a measure π that reflects the relative ambiguity aversion of the investor for the different priors considered. This methodology is then illustrated through the study of an empirical example on European data.

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13.1. BACKGROUND In this chapter, we aim at characterizing and constructing a methodology for robust portfolio allocation under model risk, i.e., when the investors consider different models for asset returns distribution to take their allocation decision. More precisely, let us consider a financial market with N risky assets and a riskfree asset. An investor wants to allocate her wealth among these assets, by choosing φ ≡ (φ0 , . . . , φN ), the vector of weights for the risk-free asset and the N risky assets. The standard framework for investment decision making has been developed by Markowitz in [1], where there is no model uncertainty. The optimal portfolio allocation is obtained as the solution of the following optimization program: φ∗ ≡ argmax EP [u(Xφ , λ)],

(13.1)

φ

where u is a Von-Neumann Morgenstern utility function characterizing the investor’s preferences and parametrized by the risk aversion parameter λ, and Xφ stands for the terminal value of the portfolio at a given time horizon. In this setting, P stands for the only prior (or model for the distribution of the assets returns) the investor has, and is known without ambiguity. Hence, the risk of the investor is perfectly quantifiable through the know-ledge of the distribution P. The agent may also consider different models Q in a finite set of possible models Q. In this case, the investment problem is modified according to the subjective view π(Q) of the investor on each model Q. More precisely, π(Q) represents the subjective likelihood of the model Q for the investor. The investor operates a linear blending of the different models, weighted by their subjective probability π(Q) to be the “real” model. Under each model, the investor considers the objective-expected utility of her future wealth. Across all priors, the investor considers the subjective-expected value of the expected utilities under the different models. Such a framework is referred to as Subjective Expected Utility (SEU) and was first introduced by Savage in [2]. The optimal portfolio allocation is then obtained as
EQ [u(Xφ , λ)]π(Q). (13.2) φ∗ ≡ argmax φ

Q∈Q

Note that in this framework, even if the agent has several priors as reference, he/she does not show any aversion toward the co-existence of different models to take her decision. He or she is neutral toward model uncertainty as there exists no ambiguity about the set of models considered and their likelihood to occur. However, as shown by Ellsberg in [3], the decision makers show a more averse behavior when betting on events for which outcomes are ambiguous (i.e., when there is also some uncertainty regarding the underlying model) than when betting on events for which the outcomes are only risky (i.e., the underlying model is well known).

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A financial illustration of the Ellsberg paradox is the risk premium paradox: investors tend to invest more in their local market even though the expected return is lower than for foreign markets, the reason being that the investors add an ambiguity premium to risky assets (investors prefer investing in assets located in their geographical zone, because they can better apprehend their return distribution). For this reason, the SEU framework fails to take into account this additional source of aversion of investors. Various approaches have been developed in the literature to take into account this aversion toward model uncertainty in the investment decision process. Among them, Gilboa and Schmeidler in [4] proposed a min–max approach leading to a very conservative decision rule based upon the worst case model. More recently, Klibanoff et al. in [5] introduced a generalized model. They consider an increasing, concave transformation function  characterizing the investor ambiguity aversion through a parameter γ. The optimal weights vector is then determined as
{EQ [u(Xφ , λ)], γ}π(Q). (13.3) φ∗ ≡ argmax φ

Q∈Q

The main feature of this model is that it unifies all the previous approaches accounting for model ambiguity. However, this theoretical approach can be very challenging to implement in practice for different reasons, including the calibration of the various parameters. Indeed, no distinction is made between specific ambiguity aversion for a given model (“How good is this specific model to represent the reality?”) and general ambiguity aversion for the whole class of models (“How much can all the models explain the reality?”). Moreover, solving explicitly Program (13.3) can be extremely difficult, even numerically, especially in the multi-dimensional case or when adding some constraints on the portfolio allocation. Note that Klibanoff et al. only give a simple numerical example for a portfolio with three assets in their paper [5], whereas practitioners often consider portfolios with hundreds of assets. We have compared their example to our methodology in [6] and we also provide a more complex theoretical example that can be solved in close form if using our methodology. Finally, such an approach lacks some flexibility in the sense that if the investor considers a new model, he or she has to re-compute the program entirely. To overcome those limitations, we propose a robust, general framework for decision making under uncertainty. We are not aiming at finding the optimal solution for a given criterion but more at finding a robust solution.

13.2. A ROBUST APPROACH TO MODEL RISK We propose a new approach to model ambiguity that is altogether more flexible, easier to compute, more robust and tractable than the methods proposed in the literature. This

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Ambiguity Robust Adjustment (ARA) approach is independent of the set of models considered, as well as of the choice criterion (i.e., the value function they consider to determine their preferred asset allocation) to define the optimal portfolio under each model Q ∈ Q, for Q a finite set of models. More precisely, we proceed in two steps and introduce a distinction between two types of ambiguity: • Absolute ambiguity: this refers to the ambiguity the investor has for a given model. We first solve the optimization problem assuming that the model considered is the true model. We thus compute a distorted expected value of the deduced optimal weights, transformed by an Absolute Ambiguity Robust Adjustment (AARA) function denoted ψ. Note that the absolute adjustment is made on the solution and not on the choice criterion. This allows us for some additional flexibility in the use of each model. • Relative ambiguity: this expresses the relative ambiguity the investor has among his or her different models: in a second step, we aggregate the adjusted optimal weights computed for each model through a RelativeAmbiguity RobustAdjustment (RARA) function, denoted π. Let us denote φQ ≡ argmaxφ EQ [u(Xφ , λ)]. Then the ARA portfolio allocation φARA ≡ (φiARA )i∈{1,...,N} is obtained as
φiARA ≡ ψ{φQi , γ}π(Q), i ∈ {1, . . . , N} (13.4) N

Q∈Q

and φ0ARA ≡ 1 − i=1 φiARA . Due to its specific nature, the risk-free asset has no model risk associated with it (its future value is known with certainty). Therefore, it plays a specific role in the ambiguity-adjusted optimal asset allocation. It can be assimilated to a refuge value in the following sense: the more the investor is averse to ambiguity, the more he or she will invest in the risk-free asset. In this sense, as the “disinvested” part of the wealth from the risky assets is transferred to the risk-free asset, the adjusted weight of the risk-free asset corresponds to the amount of money the investor is reluctant to invest in risky assets because of her aversion toward model risk. We describe below the characteristics of the functions ψ and π.

13.2.1. The Absolute Ambiguity Robust Adjustment The idea behind the AARA adjustment is that the investor, because she has doubts about the optimal weights generated by a given model, wishes to scale down those weights and especially the biggest absolute weights that could entail the biggest risks in her portfolio.

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The function ψ must satisfy some key properties to be consistent with the rationality of the investor: • Definition of the function ψ: The investor treats the absolute ambiguity aversion with the same type of transformation across all the different models (ψ is the same for all the models). What distinguishes the absolute ambiguity aversion transformation across the models is the specific ambiguity aversion parameter γ Q the investor attributes to each model. As the optimal weights obtained for each model φQ are bounded by 1, ψ(1, γ Q ) represents the maximum weight the investor will assign to any asset after the AARA transformation. We note aQ ≡ ψ(1, γ Q ) and a ≡ maxQ∈Q aQ . Therefore, ψ is defined on the set of optimum model dependent weights [−1; 1] × Q onto a set [−a; a] of transformed weights. Note that the investor can set the value of aQ and deduce the value of γ Q depending on the explicit form he or she chooses for ψ.  ψ : [−1; 1] × Q → [−a; a] (13.5) (φ, Q): → ψ(φQ , γ Q ) The following properties apply for the risky assets only, i.e., for i ∈ {1, . . . , N}. • Monotonicity: One of the key characteristics of ψ is its monotonicity property. ψ preserves the relative order of the optimal weights (φiQ )1≤i≤N deduced by a given model Q, so that the relative preference of the investor toward the different risky assets given a model Q is preserved through the transformation ψ. • Convexity: The function ψ is concave on [0; 1] and convex on [−1; 0], so that the function ψ reduces more the absolute largest weights given by the optimized portfolios under each model considered. The convexity scale is parametrized through the aversion coefficient γ Q : the bigger the aversion coefficient γ Q the more averse the investor is to large weights inferred by Q. Also the convexity of ψ will depend upon the investor ambiguity aversion toward the model Q ∈ Q considered. The function ψ has an S-shape, as it penalizes more the largest positive and negative weights (this is a particular case of the convexity property of ψ). For all assets i and all models Q, we have |ψ(φiQ )| ≤ |φiQ |. The absolute ambiguity adjusted weights are smaller than the optimal weights computed under a given model Q in absolute terms. Some additional properties can be considered depending on the assumption made on the investor preferences and trading constraints. • Symmetry: In a context where short selling is possible, there is no reason to differentiate the long or short weights of the same magnitude in terms of ambiguity aversion.

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The function ψ is then an odd function symmetric around zero. We can assume that a long-short investor has the same aversion to positive or negative weights of the same absolute value: ∀φi ∈ [−1; 1], ψ(−φi ) = −ψ(φi ). The AARA function penalizes the scale of the optimal weights of a given model without discriminating between negative and positive weights. • Invariant point: There is no ambiguity aversion for a zero weight: ψ(0) = 0. If the model Q assigns no weight on a given asset, the transformation ψ should not modify the “neutrality” of the model Q toward this asset. • Limit behavior: When the investor is infinitely averse to ambiguity, it will prevent her from trading as he or she trusts none of his or her models, and therefore all the portfolio weights should be defaulted to zero. On the contrary, if the investor is neutral to ambiguity, the function ψ should leave the model-dependent weights invariant. We used a very similar function than the one applied by Klibanoff et al. to account for ambiguity. The function ψ can be any classical S-Shape function, which has the nice property of being concave (convex for negative values), symmetric, and monotonic (similar attributes as for classical utility functions). What really characterizes the function ψ is its ambiguity parameter γ that accounts for the concavity of the function ψ and therefore the ambiguity aversion of the investor (Fig. 13.1).

Figure 13.1 ψ for different values of the ambiguity aversion parameter γ.

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An example for the function ψ is  1 − exp−γx   ,   γ ψ(x, γ) ≡  expγx −1   ,  γ

0 ≤ x ≤ 1, (13.6) −1 ≤ x ≤ 0.

13.2.2. Relative Ambiguity Robust Adjustment Once the optimal solutions have been computed for each prior Q and have been independently adjusted for ambiguity aversion through the AARA function ψ, we need to aggregate them across all priors in the set Q. The RARA function takes into account the ambiguity aversion of each prior relative to the whole class of priors Q. Such an adjustment is made through a mixture measure π. The RARA function π(Q) represents the likelihood or degree of confidence the decision maker has for the adjusted result given under Q when knowing all the adjusted results for all the other priors. Therefore, π(Q) can be seen as a subjective weight given by the decision maker to the adjusted solution for the model Q. π(Q) will therefore always be non-negative. If the decision maker does not trust at all the prior Q relatively to the other priors, he or she will simply set the weight to zero. If on the contrary he or she fully trusts the prior Q, relatively to the other priors, then the weight for all the other priors will be zero. The weight π(Q) is not necessarily one, since the agent may still believe that he or she does not have the full understanding of the situation. More formally we have Definition 13.1 (RARA). The measure π : Q → [0; 1] is a Relative Ambiguity Robust Adjustment (RARA) measure if:
π(Q) ≤ 1. ∀ Q ∈ Q, 0 ≤ π(Q) ≤ 1 and Q∈Q

After the transformation through the AARA function ψ and the RARA function π, the ARA weight of any risky assets is defined as
ψ(φiQ , γ Q )π(Q) ∀i ∈ [1; N], φiARA = Q∈Q

and for the risk-free asset, the optimal weight is φ0ARA = 1 −

N
i=1

φiARA .

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13.2.3. ARA Parametrization The investor’s aversion to ambiguity is dynamic in the sense that, depending on the period considered, he or she will be more or less confident about his or her models and the overall set of models he or she considers. Therefore, we allow the function π and the ambiguity aversion parameter γ to dynamically adapt and expand or contract the total investment size whether the total ambiguity aversion decreases or increases over time (the ambiguity parameter γ and the measure π can be re-parametrized at every decision time). As pointed out by Epstein and Schneider [7], the ambiguity aversion of an investor is not monotonically decreasing over time. Our RARA function allows the investor to adjust her portfolio weights dynamically, depending on his or her overall belief of how much his or her models can explain the true distribution P. The approach is rather different from a classical Baysian updating approach, where the investor learns more about the underlying model with any new information flowing in the stock price returns. In a Baysian framework, the investor believes that once he or she gets enough information, he or she will ultimately converge toward the true model and therefore is gradually and monotonically more and more confident about her model. Under model ambiguity however, this is not the case. The investor can become more or less confident over time, in a non-monotonic way. He or she does not assume that more information can systematically give his or her more confidence about his or her model. Many methods could be used in order to calibrate the different measures π(Q), as well as the ambiguity aversion parameter γ Q for a given model Q. We propose a simple empirical methodology that takes into account the relative historical performance of the different models: First, we compute a number of performance measures (the Sharpe, Sortino, Gain Loss or Win Lose ratios, as described in Sec. 13.4.2) on the different models considered, evaluated over a given time window. The measure π can then be computed as a weighted average of the performance measures, whereas the ambiguity aversion parameter γ can be parametrized as the inverse of those performance measures. More formally, we will consider the following parametrization for our empirical example in Sec. 13.4: if we denote by PMpQ a given performance measure p for a given model Q, we have γQ ≡

1

(13.7)

PMpQ

and π(Q) ≡ 

PMpQ

P∈Q

PMpP

.

(13.8)

Here we see that the measure π is relative, as it takes into account all the different models performances, whereas γ is absolute, as it only considers the performance of a given

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model. Note that the absolute ambiguity parameter is −γ rather than γ, according to the shape of the function ψ given in Sec. 13.2.1. Both the relative ambiguity parameter π and the absolute ambiguity parameter −γ are proportional to the performance measure considered: the higher the performance measure, the higher the parameters. Note that it is for the sake of simplicity that we chose to parametrize similarly π and γ, what matters is that the absolute and relative ambiguity aversion be positively correlated with the performance measure considered.

13.3. SOME DEFINITIONS RELATIVE TO THE AMBIGUITY-ADJUSTED ASSET ALLOCATION To compare different asset allocations for different models, and also the impact of the ambiguity aversion to the different weights assigned to each asset, we present in the following section some properties and give some definitions. We also propose a measure of distance between two different asset allocations that we will use to compare different asset allocations in our empirical example developed in the last section. The Ambiguity Robust Adjustment refers to the combine adjustment by the AARA to the optimal weights independently computed for each prior and by the RARA performed to combine those adjusted weights. These measures can be used to describe how to measure the ambiguity aversion of an investor toward a model in general, and more specifically toward a single asset. For the following definitions, let φQ be the asset allocation conditional on model Q ∈ Q and φARA be the ARA asset allocation. Definition 13.2 (Portfolio Distance). Let us consider two models Q1 and Q2 in the set of priors Q. We define the distance measure δ between the two models as δ(φQ1 , φQ2 ) =

N


|φiQ1 − φiQ2 |

i=0

δ(φQ1 , φQ2 ) represents the turnover value to rebalance the investor’s portfolio from the asset allocation φQ1 to the asset allocation φQ2 . Definition 13.3 (Weighted Ambiguity Adjustment). We denote by WAAi the ARAweighted ambiguity adjustment of an investor toward the asset i:
|φiARA − φiQ |π(Q) ∀i ∈ [1, N], WAAi ≡ Q∈Q

the ambiguity aversion of an investor toward an ambiguous asset i. WAAi represents the average difference between the ARA weight for the asset i and the optimal weights for the asset i under the different priors. For two ambiguous

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assets Si and Sj ,j, i ∈ [1, N], if WAAi < WAAj , we say that the investor shows more ambiguity aversion toward the asset j than toward the asset i. Definition 13.4 (Absolute Ambiguity Adjustment). We define the value of the Absolute Ambiguity Adjustment (AAA) of an investor toward the model Q as AAA(Q) ≡

N


|φiQ − ψ(φiQ , γ Q )|.

i=1

AAA(Q) represents the theoretical turnover to rebalance the investor’s asset allocation on the risky assets i = 1, . . . , N from the optimal weights obtained if the prior Q is assumed to be equal to the unknown real model P to the Absolute Ambiguity Adjusted portfolio, taking into account the investor absolute aversion against his or her prior Q. We deduce that the total value of the investor’s Weighted Ambiguity Adjustment toward all her priors is AAA(Q) ≡

N



|φiQ − ψ(φiQ , γ Q )|π(Q).

Q∈Q i=1

Definition 13.5 (Relative Ambiguity Adjustment). We define the value of the Relative Ambiguity Adjustment (RAA) of an investor as    N 
  ARA
Q  − φi π(Q) , RAA(Q) ≡ φi  i=1  Q∈Q which represents effectively the turnover between the Robust Ambiguity Portfolio and the Subjective Expected Utility Portfolio (the SEU portfolio allocation is defined as:  φiSEU ≡ Q∈Q φiQ π(Q)).

13.4. EMPIRICAL TESTS In this section, we detail the results of some tests we run to evaluate the performance of ambiguity robust portfolios, compared to other classical optimized portfolios (we consider optimized portfolios as well as other type of factor model portfolios, and therefore we mainly focus on the ambiguity transformation and not on the initial optimization under each prior). The tests were run on European assets. We collected daily close prices from January 2000 to April 2008 (equivalent to 2167 business days). Our set of securities is the set of Eurostoxx 600 constituents that were trading across the whole period. For this empirical study, we consider a null risk-free rate (the cash is not remunerated), as we consider a portfolio total return, as would be done for strategies without benchmarks such as hedge funds or proprietary groups strategies. We assume that our transactions costs, fees, and slippage correspond to three basis point of the daily

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turnover of the strategy (i.e., the total costs and fees are assumed to be in percentage three basis point of the daily turnover, for instance if the turnover of a strategy at a given date is 50% and its return is 10 basis points, the costs and fees amount to 1.5 basis points return and therefore the startegy return for this day after cost is 8.5 basis points). We run a back test on historical data when the investor re-balances daily her portfolio by re-estimating the different models and re-setting her optimal investment weights over an estimation window of 120 days (we choose a daily rebalancing, as we consider daily returns for the asset time series). The performances of the different strategies were then evaluated. We find that our Ambiguity Robust approach allows the investor to achieve superior performance compared to that achieved by classically optimized portfolios.

13.4.1. Portfolios Tested We consider several models and their outcomes. More precisely, for each of them, we computed portfolio weights using an estimation window, as if each one was the true one. For the minimum-variance portfolio and the mean–variance portfolio, we compute these weights according to the classical Markowitz framework. For the other portfolios, we give some details below on how we compute the different weights. We focus here on how the investor deals with her model ambiguity after the different model portfolio weights have been computed. We use an estimation window to estimate the following portfolio weights: • The equally weighted portfolio (EW): gives an equal weight to all the risky assets. We define the EW portfolio asset allocation as ∀i ∈ {1, . . . N} : φtEW,i =

1 . N

• The minimum variance portfolio (MN): is the fully invested Markowitz efficient portfolio with minimum variance, obtained when the investor minimizes the expected variance of the portfolio (i.e., minφ φ φ, where  represents the covariance matrix of the stock returns). The MN portfolio allocation is defined as φtMN =

−1 t−w,t−1 1 1 −1 t−w,t−1 1

where t−w,t−1 is the empirical covariance matrix estimated over the window [t − w, t − 1] and 1 is the N-vector of ones. • The mean variance portfolio (MV): is the fully invested, maximum Sharpe, mean– variance Markowitz efficient portfolio, obtained when the investor maximizes the empirical quadratic expected utility (i.e., maxφ µφ −φ φ, where µ is the empirical

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mean and  the empirical covariance matrix of the stock returns, note that we consider a risk aversion equal to 1). The MV allocation is defined as φtMV =

−1 t−w,t−1 µt−w,t−1 1 −1 t−w,t−1 µt−w,t−1

,

where t−w,t−1 is the empirical covariance matrix estimated over the window [t − w, t − 1] and µt−w,t−1 is the empirical vector of mean returns estimated over the same window. • The CAPM portfolio weights (CA): we base our CAPM portfolio on the Jensen alphas. We estimate the CAPM betas over the estimation window. ConsiderM ing rt−w:t−1 as the vector of the Eurostoxx 600 market returns over the period [t − w, t − 1], the beta of the risky asset i is therefore estimated at time t as βti ≡

i M cov(rt−w:t−1 , rt−w:t−1 ) . M var(rt−w:t−1 )

We then compute the Jensen alpha as the difference between the observed return at time t of the asset i and the beta-adjusted market return: αit ≡ rti − βti rtM . We then define the CAPM weights as the weighted average alphas across all the risky assets considered: αi φtCA,i ≡ N t j=1

j

αt

.

• The CAPM uncertain portfolio (UC): we define the UCAPM weights as the CAPM weights adjusted by the variance of the CAPM residuals: φtUC,i

αit var(αi. )

≡ N

j

αt j=1 var(αj. )

.

We also consider three fundamental portfolios based on stock specific financial ratios: • The price earning portfolio (PE): we compute the relative price earning ratio (PER) return of a stock among its sector peers and we give positive weights to the lower PER stocks and negative weights to the higher ones. • The cash flow portfolio (CF): we compute the relative cash flow ratio (CFR) return of a stock among its sector peers and we give positive weights to the lower CFR stocks and negative weights to the higher ones. • The price to book portfolio (PB): we compute the relative price to book ratio (PBR) return of a stock among its sector peers and we give positive weights to the lower PBR stocks and negative weights to the higher ones.

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More formally, we compute the relative financial ratio FR of the asset i among its peers assets j in the sector S: i ≡ µ(FR.,S αFR,i,S t t ) − FRt ,

where µ(FR.,S t ) is the average of the given financial ratio among the assets in the sector S. And therefore we get the following asset allocation, by scaling the alphas per sector, so that the investor invests the same amount of money in each sector: φtFR,i ≡ N S

αFR,i t FR,j,S

j=1

|αt

|

,

where FR stands either for PE, CF, or PB and NS is the number of assets belonging to sector S. • The subjective expected utility portfolio (SEU): we define by π the vector of Relative Ambiguity Aversion Weights given to the models considered. In the present  empirical example, we assume that Q∈Q π(Q) = 1, therefore the SEU portfolio is defined as the different models weights weighted by π.
∀i ∈ {1, . . . , N} : φtSEU,i ≡ φQ,i π(Q). Q∈Q

• The ambiguity robust portfolio (RA): finally, the optimal ambiguous portfolio is defined as the different models weights adjusted by the Absolute Ambiguity Adjustment ψ and weighted by the vector of Relative Ambiguity Aversion Weights π:
∀i ∈ {1, . . . , N} : φtARA,i ≡ ψ(φQ,i , γ Q )π(Q). Q∈Q

13.4.2. Performance Measures In order to parametrize the absolute ambiguity parameter γ of the function ψ and the relative ambiguity function π, to compute the SEU and RA portfolios, we use different portfolio performance measures: φ Given a portfolio allocation φ, we denote by rt the return of this portfolio at time t. • Sharpe ratio (SHR): that represents at time t the ratio of the empirical mean return of a portfolio over its empirical standard deviation over the period [t − w, t − 1]. We denote: φ

φ

SHRt ≡

µ(rt−w,t−1 ) φ

σ(rt−w,t−1 )

.

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• Sortino Price ratio (SOR): that represents at time t the ratio of the empirical mean return of a portfolio over the empirical standard deviation of its negative returns over the period [t − w, t − 1]: φ µ(rt−w,t−1 )

φ

SORt ≡

φ

φ

σ(rt−w,t−1 {rt−w,t−1 < 0})

.

• Gain Loss ratio (GLR): which is the ratio of total positive returns over total negative returns:  φ {rt−w,t−1 > 0} φ GLRt ≡  φ . {rt−w,t−1 < 0} • Winner Loser ratio (WLR): similar to the Gain Loss ratio, it is the ratio of the number of total positive returns over the number of total negative returns:  φ φ rt−w,t−1 {rt−w,t−1 > 0} φ . WLRt ≡  φ φ rt−w,t−1 {rt−w,t−1 < 0} The following two measures are used to compare different portfolio performances. • Certain equivalent (CER): corresponds to the equivalent risk-free return of the portfolio return (the portfolio return adjusted for its risk aversion-adjusted standard deviation), as used by DeMiguel et al. in their comparative study of portfolios performance [8]: φ

φ

φ

CERt ≡ µ(rt−w,t−1 ) − σ 2 (rt−w,t−1 ). • Turnover (T/O): corresponds to the change in portfolio weights from one rebalance period to the next. The investor aims to reduce the turnover as trading implies costs (exchange fees, price impact, etc.): φ

T/Ot ≡

N


i |φti − φt−1 |.

i=0

As mentioned in Sec. 13.2.3, we consider as an estimate for π the relative performance measure of a model within the class of models. Similarly, γ is estimated by the inverse absolute performance measure of a model (the worse the performance, the bigger the ambiguity aversion). Note that if the performance measure of a model is zero, then π and γ are defaulted to zero.

13.4.3. Results First we describe the individual performances of the eight models considered. Then, we show how the Subjective Expected Utility portfolios of Savage (as defined in Eq. (13.2) and where the single strategies are linearly weighted by the weighted average of their

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different performance measures) outperform the individual strategies. We use the SEU portfolios as benchmarks for our ARA methodology, as the Klibanoff et al. model is almost impossible to compute for a large universe of assets. Finally, we display the performances of the ambiguity robust portfolios, parametrized by the four different performance measures considered. We conclude that the ambiguity robust portfolios outperform by far the non-ambiguous SEU portfolios.

13.4.3.1. Performances of the different models We display in Table 13.1 the statistics of the different strategies, computed over the over the period January 2000 to April 2008 and annualized. We use the four performance measures described previously to parametrize π and γ and construct our ambiguity robust portfolios. At each date t, we compute the

Table 13.1 UC µ (%) µ ¯ (Bps) σ (%) max(µ) (Bps) min(µ) (Bps) SHR SOR GLR (%) WLR (%) CER (Bps) T/O (%)

8.71 0.43 3.95 210.82 −139.12 0.27 0.40 104.94 100.60 0.39 135.01 EW

µ (%) µ ¯ (Bps) σ (%) max(µ) (Bps) min(µ) (Bps) SHR SOR GLR (%) WLR (%) CER (Bps) T/O (%)

69.74 3.41 15.26 470.80 −540.82 0.56 0.72 110.48 119.16 2.94 2.89

Strategies Performances. PE 11.04 0.54 4.38 193.83 −217.76 0.31 0.44 105.88 102.52 0.50 141.51 MN 35.03 1.71 6.77 199.81 −317.57 0.63 0.75 112.05 123.96 1.62 21.85

CF 5.42 0.26 5.06 225.39 −205.43 0.13 0.19 102.48 99.11 0.21 140.83 MV 10.68 0.52 4.75 133.63 −219.94 0.27 0.32 104.98 122.98 0.48 28.60

PB 1.89 0.09 4.93 234.74 −234.40 0.05 0.07 100.88 98.72 0.04 141.59 CA 13.95 0.68 4.71 182.75 −162.10 0.36 0.52 106.77 101.00 0.64 135.31

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performance measure of the different strategies over the historical returns between date t − 120 and t − 1. Even though the strategies UC and CA returns underperform the EW strategy return over the whole period considered (the equally weighted portfolio has the biggest total return of almost 70%), there exist some periods where the reverse is true (in 2002 for instance, the CA and UC portfolios perform better). This confirms that the relative weight of each models should be dynamic, as proposed in our ambiguity robust methodology. In order to have numbers of similar magnitude, we scale the inverse Sharpe and Sortino measures by a reference ratio of three annualized: in practice, it means that an investor considers a portfolio for which the return is three times as big as its risk as benchmark. We have also capped the performance measures by three to prevent a strongly performing model dominating the others. Formally, we have: 3 3 ), 3) and SOR = min(max(0, SOR ), 3). SHR = min(max(0, SHR

13.4.3.2. SEU portfolio In Table 13.2 are displayed the performances and statistics for the SEU portfolios that weight the different models linearly with the measure π (respectively computed with the four different performance measures Sharpe, Sortino, Gain Loss and Win Lose Ratios), without considering the absolute ambiguity adjustment from the function ψ. The SEU portfolios outperform almost all the single strategy portfolios. The SEU portfolios outperform the individual strategies, however, we improve greatly these performances with our Ambiguity Robust approach as we show below.

13.4.3.3. Ambiguity robust portfolios We have computed the performance of the four different Ambiguity Robust portfolios (where the ambiguity parameters γ and π are estimated using the four different Table 13.2

SEU Strategies Performances. SHR

Total return (%) Mean daily return (Bps) Volatility (%) Max return (Bps) Min return (Bps) SHR SOR GLR (%) WLR (%) CER (Bps) T/O (%)

77.59 3.79 7.06 326.58 −411.38 1.34 1.52 128.85 135.45 3.69 76.33

SOR 75.26 3.68 6.92 322.47 −412.40 1.33 1.50 128.59 135.09 3.58 78.26

GLR 55.72 2.72 6.89 310.73 −275.82 0.99 1.25 119.60 121.06 2.63 112.96

WLR 59.58 2.91 6.95 321.65 −270.02 1.05 1.33 120.95 123.23 2.81 112.65

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Robust Asset Allocation Under Model Risk Table 13.3 Ambiguity Robust Strategies Performances. SHR µ (%) µ ¯ (Bps) σ (%) max(µ) (Bps) min(µ) (Bps) SHR SOR GLR (%) WLR (%) CER (Bps) T/O (%) RAA AAA

158.37 7.74 5.22 271.44 −206.80 3.71 5.59 196.40 158.16 7.68 137.35 0.25 0.24

SOR 157.04 7.67 5.27 279.05 −209.66 3.64 5.44 193.90 155.50 7.62 138.88 0.23 0.14

GLR 151.82 7.42 5.70 262.80 −189.94 3.25 4.66 181.26 155.36 7.35 136.28 0.34 0.20

WLR 149.43 7.30 6.11 264.90 −186.81 2.98 4.22 172.80 155.49 7.23 134.73 0.37 0.35

performance measures: Sharpe, Sortino, Gain Loss and Win Lose ratios). In Table 13.3, we display the statistics of the Ambiguity Robust portfolios estimated with the γ and π measures. The RA portfolios outperformed the best CA strategy portfolio in terms of all four different performance measures considered. The Absolute Ambiguity Aversion (AAA) and the Relative Ambiguity Aversion (RAA) measures allow us to quantify our ambiguity adjustment. The biggest ambiguity adjustments are made for the less performing strategies (based on Win Lose and Gain Loss ratios). As we can see, the Ambiguity Robust Portfolios outperform all the SEU portfolios, meaning that the ψ adjustments enhance the performance of ambiguity averse investors portfolios.

13.5. CONCLUSION Our aim was to provide the reader with a simple and easy to implement methodology for practitioners who want to allocate in a robust way their portfolio of assets when they have several models for the asset returns distributions but they are ambiguous about each of them (i.e., they do not fully trust them). We propose a simple, robust, and systematic method of combining the different weights computed conditionally per model. The concrete methodology we develop enables to account for ambiguity via practical empirical measures, such as the performance measures of each conditional portfolio. Our method is very different from the approach proposed in the literature and in particular that of Klibanoff et al. as we do not proceed to a unique complex optimization that is extremely challenging to solve in practice. Our approach is more practical and

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very industry-oriented: practitioners can easily define the optimal weights for each of the models they consider, then they can mix those prior conditional weights, taking into account the absolute and relative ambiguity they have against each model, as reflected by the concavity of ψ and the weight π(Q) attributed to each model Q. The parametrization we propose for the Absolute Ambiguity Aversion and the Relative Ambiguity Aversion in our empirical example is by no means optimal. However, we have shown that it can greatly enhance the performance of some classical portfolio strategies. The proposed robust ambiguity portfolio is more stable, with a lower turnover, and less risky, with a higher certainty equivalent ratio, than unadjusted portfolios as well as SEU portfolios. The ARA methodology smoothens and reduces the risk of classical strategies. One of the main features of the ARA methodology is to provide a dynamic adjustment for model uncertainty. As the parametrization depends on the past performance of the strategy, it would be interesting to study how the ARA model would behave in major downturns, and adapt to different market regimes.

References [1] Markowitz, H (1952). Portfolio selection. Journal of Finance, 7, 77–91. [2] Savage, L (1954). The Foundations of Statistics. New York: Wiley. [3] Ellsberg, D (1961). Risk, ambiguity and the savage axiom. Quarterly Journal of Economics, 75, 643–669. [4] Gilboa, I and D Schmeidler (1989). Maxmin expected utility with a non-unique prior. Journal of Mathematical Economics, 18, 141–153. [5] Klibanoff, P, M Marinacci and S Mukerji (2005). A smooth model of decision making under ambiguity. Econometrica, 73, 1849–1892. [6] Barrieu, P and S Tobelem (2009). Robust asset allocation under model risk. Risk Magazine. [7] Epstein, L and M Schneider (2007). Learning under ambiguity. Review of Economic Studies, 74(4), 1275–1303. [8] DeMiguel, V, L Garlappi and R Uppal (2009). Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy? Review of Financial Studies, forthcoming.

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SEMI-STATIC HEDGING STRATEGIES FOR EXOTIC OPTIONS∗

HANSJÖRG ALBRECHER†,§ and PHILIPP MAYER‡,¶ †

Institute of Actuarial Science, University of Lausanne, Quartier UNIL-Dorigny, Bâtiment Extranef, 1015 Lausanne, Switzerland ‡ Department of Mathematics, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria § [email protected] ¶ [email protected]

In this chapter, we give a survey of results for semi-static hedging strategies for exotic options under different model assumptions and also in a model-independent framework. Semi-static hedging strategies consist of rebalancing the underlying portfolio only at certain pre-specified timepoints during the lifetime of the hedged derivative, as opposed to classical dynamic hedging, where adjustments have to be made continuously in time. In many market situations (and in particular in times of limited liquidity), this alternative approach to the hedging problem is quite useful and has become an increasingly popular research topic over the last years. We summarize the results on barrier options as well as strongly path-dependent options such as Asian or

∗ Supported

by the Austrian Science Fund Project P18392. 345

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Albrecher and Mayer lookback options. Finally, it is shown how perfect semi-static hedging strategies for discretely observed options can be developed in quite general Markov-type models.

14.1. INTRODUCTION In the famous work [10] of 1973, it was shown how a standard European option can be replicated by dynamically trading the underlying asset and investing in the riskless bond. The so-called delta hedging strategy is (with some modifications) still among the most widely used methods to manage and reduce the risk inherent in writing an option. However, already in [60] static “hedging” methods were applied to relate different option prices, and also the Put–Call–Parity, which at least goes back to [76], can be interpreted as a consequence of a particular static hedging strategy. Recently, in [39], some pitfalls of the dynamic replication strategies were outlined and it was argued in favor of the more robust static replication methods, which in the meantime attained quite some attention in the practical and academic literature, starting in the mid-1990’s with [14, 37]. The semi-static hedging approach structurally differs from the dynamic counterpart in terms of hedging instruments and trading times. While dynamic hedges base on the assumption of being able to trade continuously in time, semi-static portfolios do not need to be adjusted dynamically but only at pre-specified (stopping) times. Furthermore, in contrast to classic dynamic hedging, they typically use plain vanilla (or other liquidly traded) options as hedging instruments. Studies comparing the performance of semi-static and dynamic hedging strategies (as for example undertaken in [43, 64, 79]) indicate that the semi-static ones are in many situations more robust and able to outperform their dynamic counterparts. Here, we aim to give an overview of the most prominent semi-static hedging strategies for options written on some asset (e.g., a stock or an exchange rate), for which some standard European options are liquid. In this chapter, we focus on fully discrete strategies, i.e., we exclude the possibility of dynamic hedging (for combined dynamic-static hedging strategies see [52, 53]). In principle, the outlined methods directly translate to the case where the options are written on baskets of assets, but one has to be aware that plain vanilla options with the basket as the underlying are needed to apply the strategies in practice. Strategies, for which the composition of the portfolio is not changed after the initial composition except investing gains in the riskless asset are called static, and those with a finite number of transactions are termed semi-static (semi-static strategies can be further subclassified as done in [55]). We also distinguish between model-dependent and model-independent strategies. The former depend on assumptions on the asset price process (like continuity) or need

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a fully specified model, while the latter are correct in any market, which is frictionless in the following sense: Definition 14.1 (Frictionless market). A market is called frictionless, if • • • • • • •

The no-arbitrage assumption holds. All investors are price-takers. All parties have the same access to relevant information. There are no transaction costs, taxes, or commissions. All assets are perfectly divisible and can be traded at any time. There are no restrictions on short-selling. Interest rates for borrowing and lending are identical.

Throughout the text, we assume frictionless markets. Although the methods work for time-dependent interest rates as well, we (mainly for notational convenience) further assume that the riskless interest rate is a constant r. Definition 14.2 (Weak and strong path-dependence). An option is said to be weakly path-dependent if the dynamics of the option price depend only on asset price and time, i.e., the stochastic differential equation (SDE) for the option price differs from the SDE of a vanilla option only in the boundary conditions. Otherwise the option is said to be strongly path-dependent. While for weakly path-dependent options, quite general methods are available, the theory is somewhat more involved for strongly path-dependent options. Nevertheless, we present hedging strategies for lookback options, as well as for Asian options, and discretely sampled options in general. The rest of the chapter is structured as follows. Section 14.2 introduces a method to synthesize arbitrary (sufficiently regular) payoffs that depend solely on the asset price at some later time T . In Sec. 14.3 techniques for weakly path-dependent options are outlined. The focus lies in particular on barrier options, which are the most liquid and best understood exotic options of this kind. Section 14.4 deals with two examples of strongly path-dependent options (the lookback and the Asian option) and shows some robust hedging strategies. In Sec. 14.5, discretely sampled options, which may be weakly path-dependent (e.g., discretely monitored barrier options) or strongly pathdependent (e.g., Asian options) are considered. Finally, Sec. 14.6 concludes and points out some potential future research topics.

14.2. HEDGING PATH-INDEPENDENT OPTIONS Now consider options with a path-independent payoff. To fix ideas let us denote the price process of some asset S up to time T by ST = {Ft }0≤t≤T and the payoff function

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by p, so that p(ST ) = p(ST ). Moreover we assume that there is a riskless bond available which pays off 1 at time T .

14.2.1. Plain Vanilla Options with Arbitrary Strikes are Liquid First we suppose that standard European options with maturity T are liquid for all strike levels K ≥ 0, and the payoff function p is assumed to be twice differentiable. Then the Taylor expansion implies
S p(S ) = p(K∗ ) + p (K∗ )(S − K∗ ) + p (x)(S − x)dx. K∗

Since (S − x) = (S − x)+ − (x − S )+ , we can rewrite this to obtain p(S ) = p(K ∗ ) + p (K∗ )(S − K∗ )
∞
 + + p (x)(S − x) dx + K∗

K∗

p (x)(x − S )+ dx.

(14.1)

0

Note that (14.1) holds for any K ∗ and even extends to cases where p is not twice differentiable (e.g., for a convex payoff function, p should then be understood as left derivative and p as a positive measure). For hedging purposes, we can rewrite (14.1) as p(S ) = p(K∗ ) + p (K∗ )(F − K ∗ ) + p (K∗ )(S − F )
K∗
∞ p (x)(S − x)+ dx + p (x)(x − S )+ dx, + K∗

0

where F denotes the forward price of the asset. In this way, the payoff of the contingent claim p is decomposed into four parts, two of which are hedgeable by static positions in the riskless bond and a forward on the asset. The other two are synthesized by “infinitesimal” positions in standard European options. In total we get the following static hedging strategy (cf. [22]): Payoff

Hedged by positions in

p(K ∗ ) + p (K∗ )(F − K ∗ )

(p(K ∗ ) + (F − K ∗ )p (K ∗ )) bonds

p (K∗ )(ST − F )  ∞  + K ∗ p (x)(ST − x) dx  K∗  + 0 p (x)(x − ST ) dx

p (K∗ ) forwards p (x)dx calls struck at x

(∀x ∈ [K∗ , ∞))

p (x)dx puts struck at x

(∀x ∈ [0, K∗ ))

This means that if European calls and puts with maturity T are liquid for all strikes, any (sufficiently regular) contingent claim on the time-T -price of S can be replicated

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perfectly — regardless of the model assumptions and in particular also in incomplete market models (this fact was already noticed e.g., by [11, 47, 62, 71]). A nice application of Formula (14.1) was given in [22], where it was shown that a variance swap written on some forward F can be replicated by a contingent claim with payoff ln(FT ) and a dynamic trading strategy involving the forward (see also e.g., [18, 19, 38, 42] for further details). It is worth noting that (14.1) also yields a pricing formula for the contingent claim. Denote by C(Ft , t; K, T ) and P(Ft , t; K, T ) the call and put price at current time t, respectively. Since due to the first fundamental theorem of asset pricing (see [35]) the price of the claim in an arbitrage-free setup is given as the risk-neutral expectation (conditioned on Ft ) of the discounted payoff, we have (the forward contract has value 0 by definition) EQ [e−r(T −t) p(ST )|Ft ] = e−r(T −t) p(K∗ ) + e−r(T −t) (Ft − K∗ )p (K∗ )
K∗
∞ p (x)C(Ft , t; x, T )dx + p (x)P(Ft , t; x, T )dx, (14.2) + K∗

0

where interchanging the order of integration is justified whenever the integrands in this last formula are absolutely integrable. A straightforward corollary to the above formula is the familiar Put-Call-Parity C(Ft , t; K, T ) = e−r(T −t) (Ft − K) + P(Ft , t; K, T ),

(14.3)

which is obtained by setting p(S ) = (S − K)+ and choosing K∗ > K (Ft denotes the time t-forward price of the asset). Note that (14.1) furthermore implies the following static hedging portfolio for the call payoff: • A long position in a put with the same strike K and maturity T as the call • A long position in a forward contract • Investing exp(−rT )(Ft − K) in the bond It is easy to see that the payoff of this portfolio at time T is the same as provided by the vanilla put, and the hedging strategy is therefore correct.

14.2.2. Finitely Many Liquid Strikes The derivation of the perfect hedging strategies in the last section relied on the assumption that standard European options are liquid for arbitrary strikes. In reality, of course, there are only a finite number of maturity/strike combinations available on the market. The first question in this more realistic context is whether the option prices themselves are consistent with the no-arbitrage assumption. This question is, for instance. addressed in [21] and in a general setting in [32] (see also [28]).

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Now, given a set of traded option prices, one can derive no-arbitrage bounds for the price of contingent claims by solving a semi-definite optimization problem (see [9,51], where also more general options are considered). The dual of this problem is then to find the most expensive sub- and the cheapest super-replicating strategy using static positions in liquid options. The latter question is obviously highly relevant in practical applications, as it shows an (in some sense) optimal way to cover the payoff in any case. Alternatively the right hand side of (14.1) may be approximated by a sum, where the sampling points correspond to the liquid strikes. However, then one cannot ensure that the payoff of the hedging portfolio is sufficient to cover the payoff of the contingent claim, unless the weights of the liquid options are chosen with great care.

14.3. HEDGING BARRIER AND OTHER WEAKLY PATH DEPENDENT OPTIONS The prototype of a weakly path-dependent option is the barrier option, which, starting from [14], has attracted by far the most attention in the literature of semi-static hedging. The payoff of a barrier option generally depends on whether the asset price has reached some pre-specified region up to maturity or not. For example, the Down-and-Out call (DOC) with parameters strike K, barrier B < S0 , and maturity T has the payoff p(ST ) = (ST − K)+ 11{inf 0≤t≤T Ft >B} ,

(14.4)

where 11A denotes the indicator function of the set A. Analogously the payoff of a Down-and-In call (DIC) is defined by p(ST ) = (ST − K)+ 11{inf 0≤t≤T Ft ≤B} . In a similar manner the payoff of Up-and-Out and Up-and-In calls (UOC and UIC) are defined by p(ST ) = (ST − K)+ 11{sup0≤t≤T Ft
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Since the plain vanilla option payoff obviously dominates the barrier option payoff, a first simple super-replicating portfolio for the considered barrier option is a long position in a standard European option with the same strike and the same maturity. Let us denote the time-t-price of a Down-and-Out and a Down-and-In call with strike K, barrier B, and maturity T by DOC(Ft , t; K, T, B) and DIC(Ft , t; K, T, B), respectively. Since the sum of the payoff of a DOC and a DIC with the same parameters equals the payoff of a plain vanilla call, we have by the law of one price C(Ft , t; K, T ) = DOC(Ft , t; K, T, B) + DIC(Ft , t; K, T, B).

(14.5)

Analogous equations hold for UOC and UIC prices and naturally for the corresponding put options as well. In the remainder of this section, the DOC serves as the illustrating example. Extensions to other option types are often analogous. In practice, early-exercise or cancellation features additionally play a role. The interested reader is referred to [65] for an optimization based treatment of superand sub-replicating strategies for American options and to [33, 34] or [8] for hedging strategies of installment options. Let us assume that the barrier option with maturity T is written on the time T forward price FT of an asset and let us use the forward price as the underlying. For convenience, the prices of all options are w.l.o.g. expressed with respect to this forward price in the rest of the section. Any barrier option can be adapted to this notation, simply by replacing the original constant barrier by a time-dependent one. The first hitting time τ is defined as the first time for which the forward asset price process is at or below the (now possibly timedependent) barrier B(t), τ = inf{t ≤ T : Ft ≤ B(t)}, with the convention inf ∅ = ∞.

14.3.1. Model-Dependent Strategies: Perfect Replication First we assume that the barrier B(t) ≡ B for the forward is constant and we suppose the forward price process Ft to be continuous. The following is a nice and conceptionally simple semi-static hedging strategy for a DOC written on the forward price of an asset (with strike K, maturity T , and barrier B = K at the strike): • Buy a standard European call struck at K and maturing at T and sell a European put with the same parameters. • The first time the forward price touches the barrier close the positions. • If the barrier is never hit, hold the portfolio until expiry.

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This perfectly replicates the payoff of the DOC, as follows by considering the two different sacenarios: • The barrier is not reached until maturity: then the payoff of the portfolio is the same as the one of the DOC, i.e., (ST − K), as necessarily ST = FT ≥ B = K; • The barrier is hit at τ < T : then C(Fτ , τ; K, T ) = P(Fτ , τ; K, T ), because of the Put–Call–Parity (14.3) (recall that Fτ = K). Thus the cost of closing the positions is 0 and the payoff of the DOC is replicated also in this case. The price of the DOC is hence given by DOC(F0 , 0; K, T, K) = C(F0 , 0; K, T ) − P(F0 , 0; K, T ).

(14.6)

This hedging strategy is rather simple (and — due to the involved continuity assumption — model-dependent). But it is not straight-forward to extend this kind of strategy to other types of barrier options (an exception is the barrier exchange option, see [57]). Thus there is the need for a more general approach. One of the most famous semi-static hedging strategies for barrier options was originally developed by Carr and Bowie [14] (see also [15–17]) and is based on a relation between standard European calls and puts with different strikes: Assumption 1. Denote the current time by t. For any T ≥ t the implied volatility smile (for maturity T ), plotted as a function of the log-moneyness, is symmetric. More precisely: let K and H√ be such that the geometric mean of K and H is the current forward price Ft , i.e., KH = Ft , then σimp (K) = σimp (H ). Theorem 14.1 (Put-Call-Symmetry, [17]). Given √a frictionless market, under Assumption 1 the following holds for all K, H with KH = Ft :  K ∀T ≥ t. C(Ft , t; K, T ) = · P(Ft , t; H, T ), H We shortly sketch the proof of the above theorem: recall that the price of a call in the Black-Scholes model (with volatility σ) is given by   C(Ft , t; K, T ) = e−r(T −t) Ft (d+ (Ft /K)) − K(d− (Ft /K)) , where Ft is the forward price of the asset, d± (x) =

log(x) ± 12 σ 2 (T − t) , √ σ T −t

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and √ (x) is the cumulative distribution function of the normal distribution. Due to KH = Ft , the above is easily seen to be equivalent to   K −r(T −t)  H(−d− (Ft /H )) − Ft (−d+ (Ft /H )) e C(Ft , t; K, T ) = H  =

K P(Ft , t; H, T ). H

Thus as long as the implied volatility for vanilla options with strike K and H is identical (which is Assumption 1), the theorem remains valid. Necessary and sufficient conditions on an asset price model to fulfill Assumption 1 are given in [20]. A first example is of course the Black-Scholes model. Another is the following local volatility model for the forward price: dFs = Fs σ(Fs , s)dWsQ ,

s > t,

where Ws is a standard Brownian motion and σ(Fs , s) = σ(Ft2 /Fs , s).

(14.7)

Condition (14.7) implies that the local volatility at a fixed time s plotted as a function of the log-returns is symmetric around 0, which explains also the symmetry of the implied volatility surface. Starting from Theorem 14.1, in [14] it is investigated how the Put–Call–Symmetry can be used to hedge barrier options (a similar relation between different call and put prices, termed Put–Call–Reversal, was used in [6] to hedge long-term call options). Consider a DOC with strike K, maturity T , and barrier B < K. By definition, the payoff of this option is (ST − K)+ , if the barrier was never hit and 0 otherwise. In order to hedge the final payoff, a vanilla call with the same strike and maturity as the DOC should be bought. Clearly this is a super-hedge and another vanilla option can be sold such that one obtains a portfolio with value 0 at the barrier. Assuming the conditions for the Put–Call–Symmetry at the first hitting time τ, we know that a put with the same maturity and strike B2 /K fulfills the geometric mean condition of the theorem. Therefore C(B, τ; K, T ) =

K P(B, τ; B2 /K, T ). B

Hence a portfolio consisting of a long position in the corresponding vanilla call and a short position in K/B puts with strike B2 /K has value 0, if the forward price is exactly at the barrier. Consequently, a replicating portfolio for a DOC is to take a long position in a standard European call with strike K and maturity T and a short position in K/B puts with strike B2 /K and the same maturity. The corresponding trading strategy is to hold the portfolio until expiry, if the barrier is never hit, or to close the positions (at 0

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cost) at τ ≤ T . Note that for the correctness of the replication strategy it is necessary that the forward price is exactly B at the first hitting time (which explains the continuity assumption). Summing up, under the no-arbitrage principle this leads to the equation: K (14.8) P(F, 0; B2 /K, T ). B This hedging procedure can be extended to other types of barrier options but then typically becomes more complicated. For example, for an UOC one would also need to take a position in a binary call (i.e., an option, which pays 1 if the asset price at maturity exceeds the strike, and 0 otherwise) and the latter one can be hedged by the strategies outlined in the previous section. If the assumption of a constant barrier on the forward price is relaxed, super- and sub-replication strategies can be derived. For the details we refer to [17] (see also the nice summary [69]). In [56], the case when only a finite number of options with prespecified strikes are available is considered and conditions under which a strategy based on the Put–Call–Symmetry is a semi-static superhedge are identified. Another hedging strategy for barrier options has its roots in papers of Derman et al. [36,37]. In order to point out the main idea of the Derman–Ergener–Kani (D–E–K) algorithm, a binomial model is considered first, i.e., the asset price process is assumed to follow a binomial tree. In this setting the D–E–K-algorithm for a DOC with maturity T , strike K, and barrier B is as follows: DOC(F0 , 0; K, T, B) = C(F0 , 0; K, T ) −

• Identify the boundary nodes, i.e., the maturity and the barrier nodes and denote the resulting time grid by t1 < · · · < tn ; • Buy a standard European option replicating the payoff of the barrier option if the barrier boundary is not hit during the lifetime of the barrier option. In the case of a DOC with B < K this would be a standard European call with the same strike and maturity as the barrier option; • Choose n options with different maturities Ti and strikes Ki having payoff 0 if the barrier was not hit. In the example of a DOC, one could for instance choose puts with strikes Ki ≤ B and maturities Ti ≤ T . Then solve the linear equation system Vi · w + C(B, ti ; K, T ) = 0,

i = 1, . . . , n

(14.9)

for w, where Vi := (P(B, ti ; K1 , T1 ), P(B, ti ; K2 , T2 ), . . . , P(B, ti ; Kn , Tn )) denotes the vector of the time-ti -prices of the n additional options and · is the scalar product. Note that due to (14.9) the portfolio consisting of the option replicating the final payoff and wi options with strike Ki and maturity Ti can be sold at zero cost at every barrier node and thus the semi-static trading strategy of holding the portfolio until expiry in

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case the barrier has not been hit, or to close the positions at the first hitting time, is a perfect replication of the barrier option. In the binomial framework the D–E–K-algorithm thus gives a perfect hedge for all kinds of barrier options — in particular also for double-barrier options and options with a time-dependent barrier — as long as enough different standard European options are liquid (i.e., at least as many as there are barrier nodes). However, since the prices in (14.9) are model-dependent (as they have to be calculated using a model — in the above case the binomial model), the entire portfolio is model-dependent. For continuous-time models, the approach outlined in [37] is to discretize (in time) the asset price process and hedge the values at the discrete monitoring times. In [78], the effectiveness of such a discretization for stochastic volatility is examined and it is found that it works well for a small volatility of volatility, but not satisfactory in the opposite case combined with discontinuous final payoff of the barrier option (as for an UOC). We come back to better alternatives in the next section. A possible limit strategy for continuous-time price models can be achieved by assuming standard options with all strikes and maturities to be liquid. This strategy was developed by Andersen et al. [5] (see also [61]), who observed that the key assumption on the price model is that the price of European options, besides their parameters, only depends on the current time t and asset price Ft (this assumption for instance rules out stochastic volatility models). The approach is, as the D–E–K-algorithm, very powerful in terms of the barrier structure and allows for arbitrary (sufficiently regular) functions of time. For the ease of exposition we assume that the forward price again follows a local volatility model in the risk-neutral world (the method can be extended to more general Markov-type models), i.e., dFt = σ(Ft , t)dWt , Ft

(14.10)

where σ is now just assumed to be positive and sufficiently regular to admit a unique solution of Eq. (14.10). Consider again the standard example of a DOC on the forward price of an asset with dynamics as specified in (14.10). Then the following PDE specifies the price function G(Ft , t) of a DOC with strike K, barrier B, and maturity T started at time t (and has in particular not reached the barrier up to the time t): 1 Gt (F, t) + σ 2 (F, t)F 2 GFF (F, t) = 0, 2 G(F, t) = 0,

t < T, F > B t < T, F ≤ B

G(F, T ) = (F − K) where we assumed again for simplicity, that B < K.

+

∀F,

(14.11)

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Then G(F, t) = DOC(F, t; K, T, B), in case the asset price has not crossed the barrier before t. If the function σ in (14.11) is sufficiently regular, G(F, t) is twice differentiable with respect to F , except at F = B, where the (formal) second derivative is given by δ(F − B)G+ F (B, t) (δ denoting the + Dirac Delta measure and GF the right derivative). Using (14.11) and a generalized Meyer–Itô-formula (see [68]), one finds:
T 11{Ft >B} GF (Ft , t)Ft σ(Ft , t)dWt (FT − K)+ = G(F0 , 0) + 0

1 + 2




T

0

B G+ F (B, t)dLt ,

where LBt is the local time of F at B and hence fulfills 1 →0 

LBt = lim




T

0

11{B≤Fs ≤B+} σ 2 (Fs , s)Fs2 ds

a.s.

Rearranging the terms gives:
T
1 T + + G(F0 , 0) + 11{Ft >B} GF (Ft , t)dFt = (FT − K) − G (B, t)dLBt . 2 0 F 0

(14.12)

(14.13)

Using (14.12) and (14.13), one might deduce the following semi-static hedge for the DOC: take a long position in a call with strike K and maturity T and short positions in G+ F (B(t), t)dt options with maturity t (∀ 0 ≤ t ≤ T ) and payoff 11{B≤Ft ≤B+} σ 2 (Ft , t)Ft2 / ( very small). Note that the latter payoff can, for any  > 0, be approximated arbitrarily closely by standard options (we assumed options for all maturities and strikes to be liquid). The corresponding trading strategy is to hold the portfolio until expiry, if the barrier is not reached and to sell the portfolio at the first hitting time τ. This is in fact a nearly perfect replication strategy, which can be seen by considering the limiting case for  → 0: if the barrier is not reached, then the payoff of the replicating portfolio equals the payoff of the call option and thus the payoff of the DOC. On the other hand, with (14.13) we have  
T  E G(F0 , 0) + 11{Ft >B(t)} GF (Ft , t)dFt Fτ 0 +




T

= E[(FT − K) |Fτ ] − E 0

  1 + B G (B, t)dLt Fτ , 2 F

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which can be reformulated as
τ
1 τ + 11{Ft >B(t)} GF (Ft , t)dFt + G (B, t)dLBt G(F0 , 0) + 2 0 F 0  
T  1 + B = C(B(τ), τ; K, T ) − E GF (B, t)dLt Fτ , 2 τ where the right-hand-side is exactly the time-τ-price of the portfolio. Note that at this point it is crucial that the prices of European options only depend on the asset price at time τ in order to interpret the expectations as prices. Since
τ
1 τ + 11{Ft >B(t)} GF (Ft , t)dFt + G (B, t)dLBt = G(B(τ), τ) G(F0 , 0) + 2 0 F 0 and G(B(τ), τ) = 0 by definition, the value of the portfolio at time τ is 0. Thus we can close the positions at zero cost and replicate the payoff of the DOC. This replication strategy can be generalized to all kinds of terminal payoffs, barrier regions, rebates, and also to jump-diffusion processes. We refer to [5] for details. An interesting difference to hedging strategies using the Put–Call–Symmetry is that here not only the correctness, but also the strategy itself depends on the model. More precisely: the number G+ F (B(t), t) has to be calculated using a model, while the strategy outlined before is independent of the exact specification of the local volatility, as long as Assumption 1 is fulfilled.

14.3.2. Model-Dependent Strategies: Approximations The hedging strategies in the previous sections were perfect if the model was correct, i.e., the risk of the option could be completely eliminated. In particular, this gave further insight into the underlying models and into relations between barrier options and standard European options. However, the perfect replication could only be achieved at the cost of more or less severe assumptions like liquidity of arbitrary standard European options or restrictions on the model. In this section a different and in some sense more pragmatic approach to semi-static hedging is discussed. In the previous section we discussed an algorithm for a binomial model (although already Derman et al. generalized the approach to continuous time models). A key point of this generalized D–E–K algorithm is that the barrier is hit exactly and that the prices of the European options only depend on the asset price. This assumption is too restrictive, as for example stochastic volatility or price jumps cannot be accommodated in this framework. Therefore, a lot of research in recent years has been devoted to generalize the assumptions, but keeping the main idea of matching the barrier option value at the first hitting time. Examples of such studies are [4,44,46,58,59,63,64,75].

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To outline the basic ideas of these generalizations, we focus again on a DOC. As in the D–E–K algorithm, we take a long position in a call with the same maturity and strike as the DOC to cover the payoff at maturity in the case the barrier was not hit. Thus the aim is to take positions in other options, such that the value of the portfolio is 0 if the barrier is hit. Furthermore the additional options should not have any payoff if the asset price stays above the barrier during the lifetime of the barrier option. In a binomial world this could be done by solving equation system (14.9). For general asset price dynamics, this equation system has to be replaced by something more general and the method of choice of the above cited studies is to consider different scenarios for the portfolio value at the hitting time. To fix ideas, let  denote the set of possible scenarios for the evolution of the barrier option. Then an element θ ∈  consists of • The time when the barrier is hit (the hitting time) • The undershoot of the asset price under the barrier • The form of the implied volatility surface at the hitting time (i.e., The prices of the used European options at the hitting time) Thus we have to solve Vθ · w + Cθ (Fτθ , τθ ; K, T ) = 0,

θ ∈ ,

(14.14)

where Vθ is the price vector of the options and Cθ (Fτθ , τθ ; K, T ) is the price of the call option at the hitting time τθ under scenario θ. Note that the asset price Fτθ ≤ B(τθ ) is allowed to undershoot the barrier. As  is an infinite set in general, (14.14) consists of a continuum of linear equations, which is obviously not feasible. Hence the set  has to be discretized to make (14.14) a finite-dimensional equation system. This can be done by assuming a model and using Monte-Carlo simulation (see e.g., [46, 63, 64, 75]) or by an a-priori choice of “reasonable” scenarios [4, 59]. The implied volatility surface at the hitting time, i.e., Vθ and Cθ , can be calculated using a pre-specified model, or can be allowed to vary in some reasonable manner (e.g., by allowing the parameters of the model to take a set of values [59]) in order to make the hedge robust with respect to model risk. After a discretization of (14.14), the number of equations, albeit finite, may be huge, if a large number of different scenarios is considered. However, since the number of scenarios is directly linked to the error due to discretization, it should in fact be chosen large in order to obtain robust results. As in general only a limited number of different options is liquid, we cannot hope to be able to solve even a discretized version (14.14) exactly. To overcome this problem, several approaches are feasible. [75] (see also [63, 64]) proposes to minimize a risk measure of the hedging error under the budget condition that the price of the hedging portfolio is smaller or equal to the one of the barrier option. Alternatively, in [46,58,59] it is suggested to modify the equation

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to an inequality, such that the trading strategy becomes a super-replication and then minimize the cost of this hedge.

14.3.3. Model-Independent Strategies: Robust Strategies In the preceding section the problem of making the hedging strategies robust with respect to model risk was already mentioned. Now let us sharpen the question: what can be deduced on the price of the barrier option if the market is solely assumed to be frictionless? In particular, robust super- and sub-replicating strategies for barrier options are considered. Those strategies are called robust because they over- (respectively under-) hedge the barrier option in any frictionless (and in particular arbitragefree) market model. For barrier options, these were first developed by Brown et al. [12] (for some recent extensions see [29, 30]). In order to outline the approach, assume again a forward market model and a constant barrier on the forward price (the ideas can be extended to time-dependent barriers, but the price intervals implied by the hedges might increase). Consider again the DOC with strike K, maturity T , and barrier B. If K ≥ B, then for any γ˜ > K a sub-replicating portfolio for the DOC is given by: • A long position in a call with strike K and maturity T • A short position in (γ˜ − K)/(γ˜ − B) puts with strike B and maturity T • A short position in (K − B)/(γ˜ − B) calls with strike γ˜ ∨ K The corresponding trading strategy is to hold the portfolio until expiry if the barrier is not reached and otherwise to unwind the positions at the first hitting time. To show the sub-replication property of this semi-static strategy, let us again distinguish the two cases: • If the barrier is not hit, the payoff of the strategy is (ST − K)+ −

K−B ˜ + ≤ (ST − K)+ , (ST − γ) γ˜ − B

where the right-hand side corresponds to the payoff of the DOC; • If the barrier is hit at time τ, then Ft = B − x, where x ≥ 0 denotes the (possible) undershoot. Hence for the value Vτ of the portfolio we find Vτ = C(B − x, 0; K, T ) −

γ˜ − K K−B P(B − x, 0; B, T ) − C(B − x, 0; γ, ˜ T) γ˜ − B γ˜ − B

≤ C(B − x, 0; K, T ) −

K−B γ˜ − K C(B − x, 0; B, T ) − C(B − x, 0; γ, ˜ T) γ˜ − B γ˜ − B

≤ 0,

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where the first inequality follows from the Put–Call–Parity (14.3) and the second one from the convexity (in K) of the call price (or alternatively by a direct look at the final payoff of the last portfolio). The portfolio is thus a subreplicating one for any γ˜ and hence γ˜ can be chosen such that the initial value V of the portfolio is maximized. Since V = C(F0 , 0; K, T ) −

γ˜ − K K−B P(F0 , 0; B, T ) − C(F0 , 0; γ, ˜ T) γ˜ − B γ˜ − B

= C(F0 , 0; K, T ) − P(F0 , 0; B, T ) +

K−B ˜ T )), (P(F0 , 0; K, T ) − C(F0 , 0; γ, γ˜ − B

the optimal γ˜ is given by γ˜ = argmaxβ>K

P(F0 , 0; B, T ) − C(F0 , 0; β, T ) . β−B

If K < B, then a sub-replicating portfolio for the DOC is given by: • A long position in a forward contract (with forward price F0 ) • A long position in F0 − K bonds • A short position in (B − K)/(γ − B) puts with maturity T and strike γ > B The trading strategy is the same as before and the subreplication property is again shown by distinguishing two cases: • If the barrier is not hit, we have for the payoff of the strategy (ST − F0 ) + (F0 − K) −

B−K (γ − ST )+ ≤ (ST − K), γ −B

where the right-hand side corresponds to the payoff of the DOC, since B > K; • If the barrier is hit at time τ, then Ft = B − x. Thus the time-τ-value of the long position in the forward is exp(−r(T − τ)) (B − x − F0 ) and hence for Vτ we find Vτ = e−r(T −τ) (B − x − F0 ) + e−r(T −τ) (F0 − K) −

B−K P(B − x, τ; γ, T ) γ −B

≤ e−r(T −τ) (B − K) −

B−K P(B, τ; γ, T ) γ −B

≤ e−r(T −τ) (B − K) −

B − K −r(T −τ) e (γ − B) = 0, γ −B

where the first inequality follows from the monotonicity of the put price and the second one from the Put–Call–Parity (14.3).

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Similarly as in the case K > B it can be shown, that the optimal choice for γ, i.e., the one maximizing the initial value of the portfolio, is given by γ = argminβ>B

P(F0 , β, 0, T ) . β−B

Super-replicating portfolios are in both cases found by omitting the barrier feature in the final payoff, i.e., they are given as the trivial super-hedges consisting of • A long position in a call with maturity T and strike K, if K ≥ B • A long position in a call with maturity T and strike B and in B − K binary calls (a path-independent option with payoff 11{ST ≥B} ) with strike B, if K < B Of course the sub- and super-hedging strategies also imply price bounds on the DOC: If K ≥ B we have C(F0 , 0; K, T ) −

γ˜ − K K−B ˜ T) P(F0 , 0; B, T ) − C(F0 , 0; γ, γ˜ − B γ˜ − B

≤ DOC(F0 , 0; K, T, B) ≤ C(F0 , 0; K, T ), while in the case K < B we have B−K e−rT (F0 − K) − P(F0 , 0; γ, T ) ≤ DOC(F0 , 0; K, T, B) γ −B ≤ C(F0 , 0; B, T ) + (B − K)BC(F0 , 0; B, T ), ˜ γ where BC(Ft , t; B, T ) denotes the time-t-price of a binary call with strike B, and γ, are as before. The particular role of these sub- and super-replication strategies is identified in the following result: Theorem 14.3.1 ( [12]). Assume that European options for all strikes with maturity T are liquid and that there are no options available with maturity less than T. Then the bounds on the price of the DOC are sharp. This means that there are forward price processes that are martingales and consistent with the prices of the European options for which the price of the DOC is given by the lower (resp. upper) bound. In those market models, the hedging strategies are perfect. The proof of this theorem is closely related to the Skorokhod problem (see also [67]).

14.4. HEDGING STRONGLY PATH-DEPENDENT OPTIONS In this section we consider strongly path-dependent options. In particular we focus on lookback and Asian options. For lookback options we show how barrier options (and

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the corresponding hedging strategies) can be used to hedge them. For Asian options robust hedging strategies are shown, which are related to hedging strategies for basket options. A perfect (model-dependent) strategy for discretely sampled options (DSO), which include Asian or cliquet options, is outlined in Sec. 14.5.

14.4.1. Lookback Options Let us consider a floating strike lookback option (LO) with terminal payoff p(ST ) = ST − mT , where mT = inf 0≤s≤T Ss . This kind of lookback option can be hedged robustly in terms of barrier options, as was first observed by Carr et al. [17] (see also [26, 27] for static hedging strategies of similar insurance products). The hedging strategies are not exact but rather suband super-replications and hence impose bounds on the price of the lookback option. Along the way to obtain the hedging strategies for the lookback option, we also deal with some other exotic options with gradually increasing complexity. A roll-down call (RDC) with maturity T and strike K0 has two barriers H1 > H2 . In contrast to a double-barrier option, these two barriers are both below the initial asset price and the strike. The payoff of the RDC is specified as follows: if neither of the two barriers is hit, then the payoff is (ST − K0 )+ . If the nearer barrier (H1 ) is hit, the strike of the option is rolled down to this barrier and the second barrier becomes an out-barrier. So hitting H1 the RDC becomes a DOC with strike H1 and out-barrier H2 . For some more details concerning roll-down-calls and also roll-up-puts, see [45]. For the purpose of hedging the lookback option, the definition should be extended as follows. Let H1 > H2 > · · · > Hn be a decreasing sequence of barriers all below the current spot price and below the initial strike K0 . If no barrier is hit, then the payoff is again (ST − K0 )+ . If the first barrier is hit, the strike rolls down to some level K1 ∈ [H1 , K0 ]. If the second barrier is hit, the strike rolls down to a certain strike K2 ∈ [H2 , K1 ]. This rolling down process is repeated until the asset price reaches (or undershoots) Hn — the out-barrier. Thus hitting Hn knocks out the option. The option defined above is called extended roll-down call (ERDC) and admits the following decomposition in terms of DOC’s: ERDC(F0 , 0; K0 , T, {Ki }1≤i≤n−1 , {Hi }1≤i≤n ) = DOC(F0 , 0; K0 , T, H1 ) +

n−1 

 DOC(F0 , 0; Ki , T, Hi+1 ) − DOC(F0 , 0; Ki , T, Hi ) .

(14.15)

i=1

This portfolio indeed matches the payoff of the ERDC exactly, since if Hi+1 < mt ≤ Hi (recall mt = inf 0≤s≤t Ss ), the sum in (14.15) starts with i + 1 and the leading term

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DOC(F0 , 0; K0 , T, H1 ) is replaced by DOC(Ft , t; Ki , T, Hi+1 ). Thus the ERDC can be perfectly replicated by a finite number of barrier options. Note that the representation above is model-independent and the semi-static hedge is perfect in all frictionless models. Therefore, to build a semi-static hedging strategy for ERDC’s using European options, the corresponding hedging strategies for DOC’s can be used. The next option we want to consider is the ratchet call. The ratchet call (RC) is an ERDC with initial strike K0 and the strikes Ki equal the barrier levels Hi for 1 ≤ i ≤ n. Furthermore it cannot be knocked out. Thus the RC can be written as follows: RC(F0 , 0; K0 , T, {Ki }1≤i≤n ) = DIC(F0 , 0; Kn , T, Kn ) + ERDC(F0 , 0; K0 , T, {Ki }1≤i≤n−1 , {Ki }1≤i≤n ), where DIC stands for a down-and-in call. Using the model-independent representation (14.15) of the ERDC, we find: RC(F0 , K0 , 0, T, {Ki }1≤i≤n ) = DOC(F0 , 0; K0 , T, K1 ) + DIC(F0 , 0; Kn , T, Kn ) +

n−1 

 DOC(F0 , 0; Ki , T, Ki+1 ) − DOC(F0 , 0; Ki , T, Ki ) .

(14.16)

i=1

Consider now the floating strike lookback call, defined at the beginning of the section. Recall that the payoff at maturity T is given by ST −mT , where mT is the minimum price of the asset up to T . This type of lookback calls can be seen as RC with a continuum of roll-down barriers and strikes. Due to this continuum of roll-down barriers we are not able to find an exact hedge, but we give super- and sub-hedges for lookback calls. It is clear that an RC with K0 = F0 and Kn = 0 undervalues the LO, because the strike of the lookback option at time t can only be below or equal the strike of the RC. Therefore a model-independent sub-hedge for a LO is given by: LO(F0 , 0; T ) ≥ RC(F0 , 0; F0 , T, {Ki }1≤i≤n ) = DOC(F0 , 0; F0 , T, K1 ) + DIC(F0 , 0; Kn , T, Kn ) +

n−1 

 DOC(F0 , 0; Ki , T, Ki+1 ) − DOC(F0 , 0; Ki , T, Ki ) .

i=1

(14.17) Again the barrier options might be hedged using the strategies of Sec. 14.3. Adding more roll-down barriers obviously increases the quality of the sub-hedge, i.e., a tighter bound is obtained.

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In order to find a super-hedge for the LO, an ERDC with Ki = Hi+1 for 1 ≤ i ≤ n and Hn+1 = 0 can be used. Such an ERDC is a super-hedge, because its strike is below or equal to the strike of the LO throughout the lifetime of the option. Hence we have LO(F0 , 0; T ) ≤ ERDC(F0 , 0; H1 , T, {Hi+1 }1≤i≤n , {Hi }1≤i≤n+1 ) = DOC(F0 , 0; H1 , T, H1 ) +

n+1 

 DOC(F0 , 0; Hi , T, Hi ) − DOC(F0 , 0; Hi , T, Hi−1 ) .

(14.18)

i=2

Another portfolio having the same payoff as the one stated above is given by: LO(F0 , 0; T ) ≤ C(F0 , 0; H1 , T ) − P(F0 , 0; H1 , T ) +

n−1 

(Hi − Hi+1 ) DIB(F0 , 0; Hi , T )



i=1

+ Hn DIB(F0 , 0; Hn , T ),

(14.19)

where DIB is a down-and-in digital option that pays 1 at expiry if the barrier was hit before maturity. As for the sub-hedge, the bounds become tighter the more Hi are added. Note that (14.17) as well as (14.18) and (14.19) are model-independent superand sub-hedging portfolios, respectively. For fixed-strike lookback options one can use similar arguments and digital options to replicate the payoff (see [13]). Then again, the hedging strategies for weakly pathdependent options can be used. It is worth noting that in [48] the best model-independent bounds for lookback options is calculated. However, the ideas are similar to the ones used for the derivation of the robust strategies for hedging barrier options and we omit the details here.

14.4.2. Asian Options Let K be the strike, {ti }1≤n the monitoring times, and T the maturity of an Asian call (AC). Then the payoff p of the AC at T is given by

p(ST ) =

1 St − K n i=1 i n

+

+

n 1 = St − nK . n i=1 i

(14.20)

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Taking the operator (·)+ inside the sum in (14.20) clearly gives an upper bound and hence a static super-replication strategy for the AC in terms of European calls, i.e.,

n

+ Sti − nK

n  + = (St1 − K1 ) + · · · + (Stn − Kn ) ≤ (Sti − Ki )+ , (14.21)

i=1

i=1

n

whenever i=1 Ki = nK. A simple first choice for the strikes of the calls is Ki = K ∀ 1 ≤ i ≤ n, which is, however, not optimal. The cheapest (an hence best) choice was originally found in [74] for complete markets and later generalized in [1, 3] (see also [25, 80]) and it turns out that the concept of comonotonicity is a helpful tool to this end: Definition 14.3 (Stop-loss transform). Let FX (x) be the distribution function of a non-negative random variable X. Then the stop-loss transform FX (m) is defined by
∞ (x − m)dFX (x) = E[(X − m)+ ], m > 0. FX (m) = m

Definition 14.4 (Comonotone random vector). Let (X1 , X2 , . . . , Xn ) be a nonnegative random vector with marginal distribution functions FX1 , FX2 , . . . , FXn . The vector is called comonotone, if the joint distribution function FX1 ,X2 ,...,Xn is given by FX1 ,X2 ,...,Xn (x1 , x2 , . . . , xn ) = min{FX1 (x1 ), FX2 (x2 ), . . . , FXn (xn )}. Note that the right-hand side of the above equation is a copula (often called lower Frechet-copula). For a general introduction to this field and proofs of the properties used below, see [40, 41]. To simplify notation, let us assume that the marginal distribution functions are strictly increasing. Let (X1 , . . . , Xn ) be a non-negative random vector with marginal distribution functions FXi and suppose (Y1 , . . . , Yn ) to be the comonotone vector with the same marginal distributions. Setting S C = ni=1 Yi the following holds: FS−1 C (x) =

n

FX−1i (x),

0 ≤ x ≤ 1.

(14.22)

i=1

A crucial result is the following: FSC (m) =

n

  FXi FX−1i (FS C (m)) ,

m ≥ 0.

(14.23)

i=1

Let (Y1 , . . . , Yn ) be the comonotone vector with the marginal distribution functions FSt1 , . . . , FStn and S C the sum of the Yi ’s. The following inequality holds for all

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n i=1

Ki = nK:

FSC (nK) = E

n

+  ≤

Yi − nK

n n   E (Yi − Ki )+ = Sti (Ki ).

i=1

i=1

i=1

Using Eq. (14.23) and the relation Sti (Ki ) = exp(rti )C(F0 , 0; Ki , T ) we finally find: n

exp(rti )C(F0 , 0; FS−1 (FS C (nK)), T ) ti

i=1

≤

n

exp(rti )C(F0 , 0; Ki , T ).

i=1

As a consequence, the optimal choice for the call strikes in (14.21) is: Ki = FS−1 (FS C (nK)). t i

Note that n

FS−1 (FS C (nK)) = FS−1 C (FS C (nK)) = nK, t i

i=1

for all strictly increasing marginal cumulative distribution functions. Since the inverse function of FS C is given by (14.22) and the right-hand side is strictly increasing in x, it is computationally straight-forward to calculate FS C (nK). Thus this hedging strategy is also simple to evaluate. Numerical studies carried out in [1] for popular Lévy models and in [3] for stochastic volatility models suggest that the obtained bounds become tighter the deeper the options are in the money. Of course the concept of comonotonicity can also be applied to other types of options, for which the payoff depends on a sum of possibly dependent random variables, e.g., basket options. For more details we refer to [25, 49, 50]. In those papers, in particular, the (realistic) case is considered, when there are only finitely many strikes liquid in the market and one has to find the optimal combination of those. This last question and especially the problem of finding a sub-replicating portfolio (which is associated to a lower price bound for theAsian option) is also addressed in [2]. Lower price bounds on the AC price were first considered by Curran [31] and Rogers and Shi [70], who pioneered a quite accurate method to determine lower price bounds in the Black-Scholes model based on the following idea: Jensen’s inequality gives + n +

n  1  1 St − K ≥ E Sti |Z − K , (14.24) n i=1 i n i=1 where Z is an arbitrary random variable. In the Black-Scholes model the choice

n  n1  Z= Si i=1

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or variants thereof are very popular (because the distribution of the geometric mean is explicitly available in the Black-Scholes model) and leads to tight lower price bounds, since the arithmetic and geometric average are strongly correlated (see [31, 66, 70, 77, 80]). However, in contrast to upper price bounds based on the concept of comonotonicity, this method in general does not imply a subreplication strategy. Nevertheless, in [2] it is shown that using Z = St1 together with (14.22) yields a robust sub-replicating portfolio consisting of ni=1 e−r(T −ti ) /n calls with maturity t1 n and strike nK/( j=1 er(tj −t1 ) ). The corresponding trading strategy is to do nothing when S1 ≤ nK/ ni=1 er(ti −t1 ) or to buy ni=1 e−r(T −ti ) /n assets in the case that n r(ti −t1 ) . The cost for this trade is exactly the payoff of the options S1 > nK/ i=1 e in the portfolio plus Ke−r(T −t1 ) , which one should borrow. Then at each monitoring time ti one should sell e−r(T −ti ) /n assets and invest the gain in the riskless bank account. At maturity T of the Asian call, the payoff of the trading strategy is

n  1 St − K 11{ ni=1 er(ti −t1 ) St1 >nK} , n i=1 i which is clearly dominated by the payoff of the AC.

14.5. CASE STUDY: MODEL-DEPENDENT HEDGING OF DISCRETELY SAMPLED OPTIONS In Sec. 14.2 a hedging portfolio for path-independent options was presented. As the payoff of discretely monitored options (DSO) depends only on a finite number of monitoring times, such options can be understood as path-independent between the monitoring times and it is possible to construct a semi-static hedging strategy with adaptations of the portfolio only at the monitoring times. This kind of strategies was developed by Carr and Wu [24] and Joshi [55] (see also [54]). Similar to Sec. 14.2, we assume that standard European options are liquid for all strikes and monitoring times of the DSO. However, to apply the techniques for the path-independent options we need the extra assumption that the asset price process is Markovian. In particular, the prices of standard European options may only depend on the current option price. We consider a recursion algorithm that can be used to price such options and to obtain some Greeks (namely the Delta and the Gamma) of the exotic option through the corresponding ones of plain vanilla options. This algorithm is based on the assumption that the price of the exotic option depends, in addition to the current price of the asset, only on some summary statistic of the historic asset prices at the monitoring times which is measurable with respect to the filtration generated by the asset price and updated only at the monitoring times of the option. If more than one statistic is needed

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to describe the price of the exotic option, the method is still feasible, but for notational convenience we focus on the case with a single one. As an illustrating example, an AC with strike K, monitoring times t1 , . . . , tn , and maturity T = tn is considered, but any other similar DSOs (like e.g., cliquets, or discretely monitored barrier and lookback options) could serve as well. Let us denote the summary statistic of a generic DSO at time ti by Xi . Due to the assumptions on X, Xi is Fti -measureable and Xt = Xi for ti ≤ t ≤ ti+1 , i = 0, 1, . . . , n − 1. Furthermore we assume that Xi+1 = f(Xi , Sti+1 ),

(14.25)

which is fulfilled for all DSOs we are aware of. For the AC, Xi is the running average Ai of the asset prices at time ti , i.e., 1 St , i j=1 j i

Xi := Ai =

∀ 1 ≤ i ≤ n.

Obviously here assumption (14.25) is fulfilled, since Ai+1 =

i 1 Ai + St , i+1 i + 1 i+1

∀ 1 ≤ i ≤ n − 1.

Due to (14.25) we have for the payoff of the generic DSO p(ST ) = g(Xn−1 , Stn ) for some terminal payoff function g. Now, since Xn−1 is known at time tn−1 , we can use (14.1) to find
∞ p(ST ) = g(Xn−1 , K∗ ) + gS (Xn−1 , K∗ )(S − K∗ ) + gSS (Xn−1 , x)(S − x)+ dx
+

K∗

K∗

gSS (Xn−1 , x)(x − S )+ dx,

(14.26)

0

where gS and gSS denote the first and second derivative of g with respect to its second argument. Note that, as in Sec. 14.2, the above describes a static hedge for the DSO at the time tn−1 and therefore also its time-tn−1 -value Vtn−1 is settled. Vtn−1 (Xtn−1 , Stn−1 ) = g(Xn−1 , K ∗ ) + gS (Xn−1 , K ∗ )(Ftn−1 − K∗ )
+

∞

K∗


+

gSS (Xn−1 , x)C(Stn−1 , tn−1 ; x, T )dx

K∗

gSS (Xn−1 , x)P(Stn−1 , tn−1 ; x, T )dx,

0

where Ftn−1 is the time-tn−1 -forward price.

(14.27)

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For the AC we have  p(ST ) =

n−1 1 An−1 + ST − K n n

+

=

1 (ST − (nK − (n − 1)An−1 ))+ n

and Vtn−1 (Xtn−1 , Stn−1 ) =

1 C(Stn−1 , tn−1 ; nK − (n − 1)An−1 , T ), n

where C(Ft , t; K, T ) := e−r(T −t) (Ft − K) for K < 0 with Ft denoting again the forward price. The Markov property together with (14.27) then implies that the value Vtn−1 of the DSO at time tn−1 only depends on Xtn−1 and Stn−1 and using again (14.25) we actually have (with a slight abuse of notation) Vtn−1 (Xtn−1 , Stn−1 ) = Vtn−1 (Xtn−2 , Stn−1 ). Thus we can use the right-hand side of (14.27) as the new payoff function of a (modified) DSO and iterate the procedure until we reach the current time t0 and obtain a replicating portfolio. The associated hedging strategy is to hold this portfolio until expiry t1 of the standard European options involved and invest the payoff of these options to form the new replicating portfolio for the European-type option Vt2 (X1 , St2 ). This strategy is of course self-financing. Note that the replicating portfolio does not change until t1 , and thus one can calculate the price of the DSO, as well as the derivatives with respect to the asset price, with this portfolio. Unfortunately the derivatives with respect to other parameters of the DSO price in general cannot be calculated in the same manner, since changing those parameters would also have an effect on the portfolio. It is worth noting that the weights of the standard options in the hedging portfolio at the current time t0 are determined by the Gamma of the DSO at time t1 . The derivation of this hedging strategy of course relies on the Markov assumption on the model and the liquidity of arbitrary standard European options. However, even if those assumptions may fail in practice, the approach gives an idea of how to obtain approximative hedging strategies. More precisely: if neither the Markov assumption nor the liquidity assumption is fulfilled, one might not be able to hedge the DSO perfectly. However, if alternatively a hedging strategy is chosen in order to minimize a certain risk measure for the hedging error in (14.26), the semi-static strategy might still outperform classical dynamic hedging strategies. Recently, this kind of approach was applied e.g., in [7].

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14.6. CONCLUSION AND FUTURE RESEARCH Semi-static hedging strategies can be a valuable alternative to classic dynamic replication approaches in various situations, often leading to a better performance and reduced model risk. Although over the last years various results have been obtained, there are many open research questions. From a theoretical perspective, it might be interesting to extend the discussed sharp price bounds to other classes of exotic options, like cliquet options. Also, improved price bounds in terms of other liquid options beyond standard European options are needed. It could be rewarding to investigate which price bounds can be obtained if American put options, variance/volatility swaps, or credit default swaps are liquid. A recent paper in this direction is [23]. Another line of extension is to model the vanilla option prices themselves. Recently, in [73] dynamics for option prices were found that are consistent with the no-arbitrage assumption and one might be able to use this kind of modelling to design more advanced trading strategies with options. Among further future research topics is the quantitative and systematic comparison of hedging performance of static and dynamic strategies, including historical backtesting of the strategies.

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15

DISCRETE-TIME VARIANCE-OPTIMAL HEDGING IN AFFINE STOCHASTIC VOLATILITY MODELS

JAN KALLSEN∗,§ , JOHANNES MUHLE-KARBE†,¶ , NATALIA SHENKMAN‡, and RICHARD VIERTHAUER∗,∗∗ ∗

Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Christian-Albrechts-Platz 4, 24098 Kiel, Germany † Fakultät für Mathematik, Universität Wien, Austria Nordbergstr. 15, 1090 Wien, Austria ‡ Lehrstuhl für Energiehandel und Finanzdienstleistungen Universität Duisburg-Essen, Universitätsstraße 12, 45141 Essen § [email protected] ¶ [email protected]  [email protected] ∗∗ [email protected]

We consider variance-optimal hedging when trading is restricted to a finite time set. Using Laplace transform methods, we derive semi-explicit formulas for the varianceoptimal initial capital and hedging strategy in affine stochastic volatility models. For the corresponding minimal expected-squared hedging error, we propose a closedform approximation as well as a simulation approach. The results are illustrated by computing the relevant quantities in a time-changed Lévy model.

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15.1. INTRODUCTION A classical question in Mathematical Finance is how the issuer of an option can hedge her risk by trading in the underlying. To tackle this problem in incomplete markets, we consider variance-optimal hedging, cf. [5, 18, 22] and the references therein for a survey of the extensive literature. Variance-optimal hedging of a contingent claim H means that one minimizes the expected squared hedging error E[(v0 + ϕ • ST − H)2 ] over all initial endowments v0 and trading strategies ϕ, where ϕ • ST represents the cumulated gains resp. losses from trading ϕ up to the expiry date T of the claim. In this chapter, we consider the above problem in affine stochastic volatility models. These generalize Lévy processes by allowing for volatility clustering and are capable of recapturing most of the stylized facts observed in stock price time series. For Lévy processes, variance-optimal hedging has been dealt with using PDE methods by [6] and by employing Laplace transform techniques in [4,10]. The approach of [10] has subsequently been extended to affine models by [12,13,17] if the discounted asset price is a martingale and by [14] in the general case. However, whereas [4, 10] incorporate both continuous and discrete rebalancing, the results for affine processes have focused on continuous trading thus far. The present study complements these results by showing how to deal with discretetime variance-optimal hedging in affine models. Since only finitely many trades are feasible in reality, this analysis is important in order to answer the following questions: (1) How should discrete rebalancing affect the investment decisions of the investor, i.e., to what extent should she adjust her hedging strategy? (2) How can one quantify the additional risk resulting from discrete trading, i.e., by how much does the hedging error increase? The general structure of variance-optimal hedging in discrete time has been thoroughly investigated by [21]. However, examples of (semi-) explicit solutions seem to be limited to the results of [4, 10] for Lévy processes and [2] for some specific diffusion models with stochastic volatility. Here, we show how to extend the Laplace transform approach of [10] to general affine stochastic volatility models. Similarly as in [12, 13], we focus on the case where the discounted asset price process is a martingale. Numerical experiments using the results of [10] and [14] indicate that the effect of a moderate drift rate on hedging problems is rather small. This article is organized as follows. In Sec. 15.2, we summarize for the convenience of the reader the general structural results of [21] on variance-optimal hedging in discrete time, reduced to the case where the underlying asset is a martingale. Subsequently, we explain how the Laplace transform approach can be used in general discrete-time

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models in order to obtain integral representations of the objects of interest. Section 15.4 turns to the computation of the integrands from Sec. 15.3 in affine stochastic volatility models. We show how to compute all integrands in closed form for the optimal initial capital and hedging strategy. This parallels results for continuous-time hedging in [4, 10] and [12, 13] and for discrete-time hedging in [4, 10]. Somewhat surprisingly, the expressions for the corresponding hedging error turn out to be considerably more involved than in the continuous-time case and cannot be computed in closed form. We propose two approaches to circumvent this problem: First, we determine a closed form approximation, whose error becomes negligible as the number of trades tends to infinity. As an alternative, we put forward a simple Monte–Carlo scheme to approximate the hedging error via simulation. Section 15.5 contains some numerical examples for the time-changed Lévy models introduced by [3].

15.2. DISCRETE-TIME VARIANCE-OPTIMAL HEDGING Let T > 0 be a fixed time horizon, N ∈ N, T0 := {t0 , t1 , . . . , tN }, and T := T0 \{0}, where tn = nT/N for n = 0, . . . , N. Denote by (, F, (Ft )t∈T0 , P) a filtered probability space with discrete time set T0 . For simplicity, we assume that the initial σ-field F0 is trivial. As for an introduction to financial mathematics in this discrete setup, the reader is referred to the textbook of Lamberton and Lapeyre [15]. The logarithm X of the discounted stock price process S = S0 exp(Xt ),

S0 ∈ R+ ,

t ∈ T0 ,

is supposed to be the second component of an adapted process (y, X), where X0 is normalized to zero. The first component y models stochastic volatility or, more accurately, stochastic activity in the model. Throughout, we suppose that E[ST2 ] < ∞, as well as E[St2n |Ftn−1 ] > 0,

t∈T,

(15.1)

to rule out degenerate cases. Our goal is to compute the variance-optimal hedge for a given contingent claim H in the following sense. Definition 15.1. We say that (v0 , ϕ) is an admissible endowment/strategy pair, if v0 ∈ R and ϕ = (ϕt )t∈T is a predictable process (i.e., ϕt is Ft−1 -measurable) such that
ϕt St ∈ L2 (P). ϕ • ST := t∈T

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An admissible endowment/strategy pair (v0 , ϕ ) is called variance-optimal for a contingent claim with discounted payoff H ∈ L2 (P) at time T , if it minimizes the expected squared hedging error (v0 , ϕ)  → E[(v0 + ϕ • ST − H)2 ] over all admissible endowment/strategy pairs (v0 , ϕ). In this case, we refer to v0 as the variance-optimal initial capital and call ϕ variance-optimal hedging strategy. As noted in the introduction, we restrict ourselves to the case where the stock price is a martingale. Assumption 15.2. The stock price process S is a square-integrable martingale. In this case, the variance-optimal capital and strategy can be represented as follows. Proposition 15.3. Let H ∈ L2 (P). Then the variance-optimal endowment/strategy pair for H is given by v0 = V0 ,

ϕtn =

E[Vtn Stn |Ftn−1 ] − Vtn−1 Stn−1 , E[St2n |Ftn−1 ] − St2n−1

tn ∈ T ,

where Vtn := E[H|Ftn ],

tn ∈ T0

denotes the option price process of H. The corresponding minimal expected squared hedging error is given by
J0 := E[VT2 − V02 ] − E[ϕtn (E[Vtn Stn |Ftn−1 ] − Vtn−1 Stn−1 )]. tn ∈T

Proof. This follows from [21, Sec. 4.1] by making use of the martingale properties of S and V .
Notice that if the initial capital is fixed at v0 ∈ R rather than being part of the optimization problem, the same strategy ϕ is still optimal if S is a martingale (cf. [21, Sec. 4.1]). However, the corresponding hedging error increases by (v0 − V0 )2 in this case.

15.3. THE LAPLACE TRANSFORM APPROACH In order to derive formulas that can be computed in concrete models, we use the Laplace transform approach, which has been introduced to variance-optimal hedging by [10]. The key assumption on the contingent claim is the existence of an integral representation in the following sense.

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Assumption 15.4. Suppose that the payoff function of the claim is of the form H = f(ST ) for some function f : (0, ∞) → R, such that  f(s) =

R+i∞

sz l(z)dz, R−i∞

for l : C → C and R ∈ R such that x → l(R + ix) is integrable and E[exp (2RXT )] < ∞.

Example 15.5. Most European options admit an integral representation of this kind. For example, for the European call with payoff function f(s) = (s − K)+ , we have, f(s) =

1 2πi



R+i∞

sz R−i∞

K1−z dz, z(z − 1)

for any R > 1 by [10, Lemma 4.1]. More generally, the Bromwich inversion formula as in [10, Theorem A.1] ascertains that l is typically given by the bilateral Laplace transform of x  → f(exp(x)), cf. [10] for more details and examples. Henceforth, we only consider contingent claims satisfying Assumption 15.4. In this case, Proposition 15.3 can also be written in integral form. Theorem 15.6. We have H ∈ L2 (P) and the corresponding option price process is given by  Vtn =

R+i∞

V(z)tn l(z)dz,

tn ∈ T 0 ,

R−i∞

for the square-integrable martingales V(z)tn := E[STz |Ftn ],

tn ∈ T0 .

Moreover, the variance-optimal hedging strategy for H can be represented as  ϕtn

=

R+i∞

R−i∞

E[V(z)tn Stn |Ftn−1 ] − V(z)tn−1 Stn−1 l(z)dz, E[St2n |Ftn−1 ] − St2n−1

tn ∈ T .

Proof. The first assertion follows from Assumption 15.4 and Fubini’s Theorem along the lines of [13, Lemma 3.3 and Proposition 3.4]. The second can be derived analogously using the Cauchy–Schwarz Inequality and E[St2n ] < ∞, tn ∈ T .
For the hedging error, Proposition 15.3 and Theorem 15.6 yield the following similar integral representation.

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Corollary 15.7. For tn ∈ T and z1 , z2 ∈ R + iR, let J1 (z1 , z2 ) := V(z1 )T V(z2 )T − V(z1 )0 V(z2 )0 , J2 (tn , z1 , z2 ) :=

E[V(z1 )tn Stn |Ftn−1 ] − V(z1 )tn−1 Stn−1 E[St2n |Ftn−1 ] − St2n−1 × (E[V(z2 )tn Stn |Ftn−1 ] − V(z2 )tn−1 Stn−1 ).

If, for tn ∈ T ,  ∞ ∞ E[|J2 (tn , R + ix1 , R + ix2 )|]|l(R + ix1 )||l(R + ix2 )|dx1 dx2 < ∞, −∞

(15.2)

−∞

the minimal expected squared hedging error is given by  R+i∞  R+i∞ E[J1 (z1 , z2 )]l(z1 )l(z2 )dz1 dz2 J0 = R−i∞

−

R−i∞


 tn ∈T

R+i∞

R−i∞



R+i∞

E[J2 (tn , z1 , z2 )]l(z1 )l(z2 )dz1 dz2 .

R−i∞

15.4. APPLICATION TO AFFINE STOCHASTIC VOLATILITY MODELS Theorem 15.6 and Corollary 15.7 show that in order to compute semi-explicit formulas of the discrete variance-optimal capital, hedging strategy and hedging error, one must be able to compute conditional exponential moments of the process X. This suggests to consider models whose moment generating function E[exp(uXtn )] is known in closed form. Here we use affine processes in the sense of [7]. Assumption 15.8. Suppose that (yt , Xt )t∈T0 is the restriction to discrete time of a semimartingale which is regularly affine w.r.t. y in the sense of [7, Definitions 2.1 and 2.5]. This means that the characteristic function of (y, X) has exponentially affine dependence on y, i.e., there exist mappings j : T × iR2 → C, j = 0, 1 such that, for t ≥ s and (u1 , u2 ) ∈ iR2 , E[eu1 yt +u2 Xt |Fs ] = exp(0 (t − s, u1 , u2 ) + 1 (t − s, u1 , u2 )ys + u2 Xs ).

(15.3)

Example 15.9. By [7, Theorems 2.7 and 2.12], a continuous-time semimartingale (y, X) is affine if and only if its local dynamics expressed in terms of the infinitesimal generator resp. the differential characteristics depend on y in an affine way. Moreover, the functions 0 , 1 can be determined by solving some generalized Riccati equations.

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In [11], it is shown that a large number of stochastic volatility models from the empirical literature fit into this framework. Examples include the models of Heston [9] and Barndorff-Nielsen and Shephard [1] as well as their extensions to time-changed Lévy models by [3]. A particular specification of this general class of models is given by the following OU-time-change model: Xt = L t ys ds , 0

dyt = −λyt dt + dZt ,

y0 > 0,

for a mean reversion speed λ > 0, a Lévy process L with Lévy exponent ψL , and an increasing Lévy process Z with Lévy exponent ψZ , i.e., E[euLt ] = exp(tψL (u)),

E[euZt ] = exp(tψZ (u)),

∀u ∈ iR.

In this case, (y, X) is affine by [11, Sec. 4.4] and in view of [11, Corollary 3.5], we have 1 (t, u1 , u2 ) = e−λt u1 + 

1 − e−λt L ψ (u2 ), λ

t

0 (t, u1 , u2 ) =

ψ Z (1 (s, u1 , u2 ))ds. 0

If y is chosen to be a Gamma-OU process with stationary Gamma(a, b) distribution (see [20] for more details), we have ψZ (u) = (λau)/(b − u) and 0 can be determined in closed form as well. By [12, Proposition 3.6], we have 0 (t, u1 , u2 )    b − 1 (t, u1 , u2 ) aλ  L  + tψ (u ) if bλ = ψL (u2 ), b log  2  b − u1  bλ − ψ L (u2 ) =    b  λt  −aλ − 1) + t if bλ = ψL (u2 ), (e λu1 − ψ L (u2 ) where log denotes the distinguished logarithm in the sense of [19, Lemma 7.6], i.e., the branch is chosen such that the resulting function is continuous in t. To compute exponential moments of X such as V(z)t = E[STz |Ft ], z ∈ R + iR, we need Eq. (15.3) to remain valid on a suitable extension of iR2 . The following sufficient condition is taken from [13]. Assumption 15.10. Suppose that for all tn ∈ T0 , the mappings (u1 , u2 ) → j (tn , u1 , u2 ), j = 0, 1 admit analytic continuations to the strip S := {z ∈ C2 : Re(z) ∈ (−∞, (M ∨ 0) + ) × ((2R ∧ 0) − , (2R ∨ 2) + )}, for some > 0 and M := sup{21 (T − tn , 0, r) : r ∈ [R ∧ 0, R ∨ 0], tn ∈ T0 }.

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The existence of the analytic extensions in Assumption 15.10 is difficult to verify in general. For affine diffusion processes, [8, Theorem 3.3] shows that it suffices to establish that solutions to the corresponding Riccati equations exist on [0, T ]. In the presence of jumps, the situation is more involved and one has to work on a case-by-case basis. For time-changed Lévy processes, this has been carried out in detail by [12]. Example 15.11. By the proof of [12, Theorems 3.3, 3.4], Assumption 15.10 holds in the OU-time-change models from Example 15.9, if the Lévy exponents ψ L and ψZ admit analytic extensions to {z ∈ C : Re(z) ∈ ((2R ∧ 0) − , (2R ∨ 2) + )} resp. {z ∈ C : Re(z) ∈ (−∞, M + )} for some > 0. For example, if L is chosen to be an NIG process with Lévy exponent



ψL (u) = uµ + δ( α2 − β2 − α2 − (β + u)2 ), for µ ∈ R, δ, α > 0, β ∈ (−α, α) in the Gamma-OU-time-change model from Example 15.9, one easily shows that the Lévy exponents ψ Z and ψ L admit analytic extensions to {z ∈ C : Re(z) ∈ (−∞, b)} resp. {z ∈ C : Re(z) ∈ (−α − β, α − β)}. Consequently, checking the validity of Assumption 15.10 amounts to verifying M < b,

2R > −α − β,

2R ∨ 2 < α − β,

for M = ((1 − e−λT )/λ)2 max{ψL (R ∧ 0), ψ L (R ∨ 0)} ≥ 0 in this case. By [7, Theorem 2.16(ii)], Assumption 15.10 implies that the exponential moment formula (15.3) holds for all z ∈ S. In particular, S is square-integrable. We proceed by providing sufficient and essentially necessary conditions that ensure the validity of the martingale Assumption 15.2 and the non-degeneracy condition (15.1). Assumption 15.12. Assume that the martingale conditions   T T , 0, 1 = 1 , 0, 1 = 0 0 N N are satisfied and suppose that for  T δ0 := 0 , 0, 2 , N

 δ1 := 1

(15.4)

T , 0, 2 , N

we have δ0 , δ1 ≥ 0 and δ0 > 0

or δ1 yt > 0 a.s. for all t ∈ T .

(15.5)

Example 15.13. For OU-time-change models, the martingale conditions (15.4) read as ψL (1) = 0, i.e., exp(L) has to be a martingale. For example, in the NIG-OU models from Example 15.11, this means



µ = δ( α2 − (β + 1)2 − α2 − β2 ).

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As for the non-degeneracy condition (15.5), the term δ0 + δ1 y is actually bounded away from zero in most applications. (1) In OU-time-change models satisfying the conditions of Example 15.11, 1 (s, 0, 2) = ψL (2)(1 − exp(−λs))/λ > 0, unless L is deterministic. More T/N over, we have 0 (T/N, 0, 2) = 0 ψZ (1 (s, 0, 2))ds, which is also positive by [19, Theorem 21.5]. Since (yt )t∈T0 is bounded from below by exp(−λT )y0 > 0, the term δ0 + δ1 y is bounded away from zero in this case. In particular, (15.5) is satisfied. (2) Now suppose that the Ornstein–Uhlenbeck process y is replaced by a square-root process √ dyt = κ(η − yt )dt + σ yt dWt ,

y0 > 0,

where κ, η, σ > 0 and W denotes a standard Brownian motion. Subject to certain regularity conditions [cf. 12, Assumption 4.2], the proof of [12, Theorems 4.3, 4.4] and a comparison argument show that 1 (s, 0, 2) > 0 for s > 0. This in t turn yields 0 (t, 0, 2) = κη 0 1 (s, 0, 2)ds > 0 for t > 0 and hence δ0 , δ1 > 0. Since y is positive, this shows that δ0 + δ1 y is bounded away from zero for these CIR-time-change models as well and (15.5) holds. From now on, Assumptions 15.10 and 15.12 are supposed to be in force. Combined with Theorem 15.6, Assumption 15.10 allows us to compute the variance-optimal initial capital v0 and the variance-optimal hedging strategy ϕt at time t by performing single numerical integrations. Theorem 15.14. For tn ∈ T0 and z ∈ R + iR, we have V(z)tn = Stzn exp(0 (T − tn , 0, z) + 1 (T − tn , 0, z)ytn ). Moreover, for tn ∈ T ,  ϕtn

=

R+i∞ R−i∞

V(z)tn−1 Stn−1



exp(κ0 (tn , z) + κ1 (tn , z)ytn−1 ) − 1 l(z)dz, exp(δ0 + δ1 ytn−1 ) − 1

where, for j = 0, 1,  T δ j = j , 0, 2 , N   T T , 1 (T − t, 0, z), z + 1 − j , 1 (T − t, 0, z), z . κj (t, z) := j N N

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Proof. The formula for V(z) follows immediately from Assumption 15.10 and [7, Theorem 2.16(ii)]. Analogously, we obtain E[St2n |Ftn−1 ] − St2n−1 = St2n−1 (exp(δ0 + δ1 ytn−1 ) − 1)

(15.6)

and E[V(z)tn Stn |Ftn−1 ] − V(z)tn−1 Stn−1 = V(z)tn−1 Stn−1 (exp(κ0 (tn , z) + κ1 (tn , z)ytn−1 ) − 1),

(15.7)

with 

T , 1 (T − t, 0, z), z + 1 N  T −0 T − t + , 0, z + 0 (T − t, 0, z), N   T T , 1 (T − t, 0, z), z + 1 − 1 T − t + , 0, z . κ1 (t, z) = 1 N N

κ0 (t, z) = 0

By the martingale property of V(z), we have V(z)tn−1 = E[V(z)tn |Ftn−1 ]. Together with [7, Theorem 2.16(ii)], this establishes the semiflow property e0 (T −tn−1 ,0,z)+1 (T −tn−1 ,0,z)ytn−1 = e0 (T −tn ,0,z)+0 (T/N,1 (T −tn ,0,z),z)+1 (T/N,1 (T −tn ,0,z),z)ytn−1 , for tn ∈ T and z ∈ R + iR. Insertion into (15.7) yields the assertion.




We now consider the expression for the minimal expected squared hedging error in Corollary 15.7. The first term J1 represents the variance of an unhedged exposure to the option. In view of Assumption 15.10, it can be computed by evaluating a double integral with the following integrand. Lemma 15.15. For z1 , z2 ∈ R + iR, we have E[J1 (z1 , z2 )] = V(z1 + z2 )0 − V(z1 )0 V(z2 )0 . Proof. This is due to the martingale property of V(z1 + z2 ).




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We now turn to the second term in the formula for the hedging error in Corollary 15.7. Suppose for the moment that (15.2) holds. By Eqs. (15.6) and (15.7), we have J2 (tn , z1 , z2 ) = V(z1 )tn−1 V(z2 )tn−1 ×

(eκ0 (tn ,z1 )+κ1 (tn ,z1 )ytn−1 − 1)(eκ0 (tn ,z2 )+κ1 (tn ,z2 )ytn−1 − 1) . eδ0 +δ1 ytn−1 − 1

In view of Corollary 15.7, it therefore remains to compute E[J2 (tn , z1 , z2 )] = (I(κ1 (tn , z1 ) + κ1 (tn , z2 ), tn , z1 , z2 )eκ0 (tn ,z1 )+κ0 (tn ,z2 ) + I(0, tn , z1 , z2 ) − I(κ1 (tn , z2 ), tn , z1 , z2 )eκ0 (tn ,z2 ) − I(κ1 (tn , z1 ), tn , z1 , z2 )eκ0 (tn ,z1 ) ) × S0z1 +z2 e0 (T −tn−1 ,0,z1 )+0 (T −tn−1 ,0,z2 ) , where I(u, tn , z1 , z2 )  := E

e(u+1 (T −tn−1 ,0,z1 )+1 (T −tn−1 ,0,z2 ))ytn−1 +(z1 +z2 )Xtn−1 . exp(δ0 + δ1 ytn−1 ) − 1

Unfortunately, I(u, tn , z1 , z2 ) can only be computed explicitly in some very special cases, unlike for continuous-time variance-optimal hedging. For example, if N = 1, i.e., for static hedging, the sum in Corollary 15.7 only consists of the term J2 (T, z1 , z2 ), which is also deterministic in this case. Hence, we obtain E[J2 (T, z1 , z2 )] = V(z1 )0 V(z2 )0

(eν0 (z1 )+ν1 (z1 )y0 − 1)(eν0 (z2 )+ν1 (z2 )y0 − 1) , e0 (T,0,2)+1 (T,0,2)y0 − 1

for νj (z) = j (T, 0, z + 1) − j (T, 0, z), j = 0, 1, which allows to compute the static hedging error by evaluating a double integral. For Lévy processes, we have δ1 = 0. Hence the denominator in the expression for I reduces to a constant in this case and the expectations can be computed using (15.3). This leads to the formula obtained in [10]. For affine models, one can verify that as the number of trading times N tends to infinity, the argument of the expectation in the expression for I converges to an

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expression of the form a + bytn−1 exp(uytn−1 + (z1 + z2 )Xtn−1 ). c + dytn−1 The expectation of this term can then be calculated, cf. the proof of [13, Theorem 4.2] for more details. If the set of trading times is finite, it does not seem possible to calculate I in closed form. We discuss two ways to circumvent this problem and tackle the computation of the hedging error. The first is to use the approximation exp(δ0 + δ1 yt ) − 1 ≈ δ0 + δ1 yt , which seems reasonable as the number N of trading dates tends to infinity, because both δ0 = 0 (T/N, 0, 2) and δ1 = 1 (T/N, 0, 2) converge to zero in this case. We obtain the following first-order approximation. Theorem 15.16. Suppose that for any t ∈ T0 , the following holds. (1) The mappings (u1 , u2 )  → j (T −t, u1 , u2 ), j = 0, 1 admit analytic extensions to S  := {z ∈ C2 : Re(z) ∈ (−∞, (M  ∨ 0) + ) × ((2R ∧ 0) − , (2R ∨ 2) + )}, for some > 0 and M  := M ∨ 21 (T/N, M/2, R + 1). (2)

 E

exp(M  yt + 2RXt ) < ∞. δ0 + δ1 yt

(15.8)

Then (15.2) is satisfied and for n = 1, . . . , N, z1 , z2 ∈ R + iR, u ∈ 1 (T − tn−1 , 0, z1 ) + 1 (T − tn−1 , 0, z2 ) + {κ1 (tn , z1 ) + κ1 (tn , z2 ), κ1 (tn , z1 ), κ1 (tn , z2 ), 0}, we have  exp(uytn−1 + (z1 + z2 )Xtn−1 ) E δ0 + δ1 ytn−1  −uδ0 /δ1  1   δ1 δ0 e   + us exp us + χ0 (s) + χ1 (s)y0 ds if δ0 , δ1 = 0,    δ1 δ0 δ1 0        1 1 + us = 1 exp(ϑ0 (s) + ϑ1 (s)y0 )ds if δ0 = 0,  δ1 0 s         1   exp(0 (tn−1 , u, z1 + z2 ) + 1 (tn−1 , u, z1 + z2 )y0 ) if δ1 = 0, δ0

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where, for j = 0, 1,  T , 0, 2 , δ j = j N

 δ1 χj (s) := j tn−1 , log(s) + us, z1 + z2 , δ0

ϑj (s) := j (tn−1 , log(s) + us, z1 + z2 ). Proof. In view of (15.5), we have exp(δ0 + δ1 yt ) − 1 > δ0 + δ1 yt , which combined with (15.6) yields

E[V(z1 )tn Stn |Ftn−1 ] − V(z1 )tn−1 Stn−1 1

E[|J2 (tn , z1 , z2 )|] <

δ0 + δ1 ytn−1 Stn−1

E[V(z2 )tn Stn |Ftn−1 ] − V(z2 )tn−1 Stn−1

.

×

S

(15.9)

tn−1

By Assumption 15.10, [7, Theorem 2.16(ii)] and Jensen’s Inequality, we have

V(zj )tn−1 Stn−1

≤ S R e0 (T −tn−1 ,0,R)+1 (T −tn−1 ,0,R)ytn−1 +RXtn−1

0

Stn−1 

≤ S0R e0 (T −tn−1 ,0,R) e 2 M ytn−1 +RXtn−1 , 1

for j = 1, 2. Likewise, it follows from Jensen’s Inequality that, for j = 1, 2,

E[V(zj )tn Stn |Ftn−1 ]



S tn−1

≤ E[|V(zj )tn |Stn |Ftn−1 ]S0−1 e−Xtn−1 ≤ S0R E[e0 (T −tn ,0,R)+1 (T −tn ,0,R)ytn +(R+1)Xtn |Ftn−1 ]e−Xtn−1 ≤ S0R e0 (T −tn ,0,R) E[e 2 Mytn +(R+1)Xtn |Ftn−1 ]e−Xtn−1 1

= S0R e0 (T −tn ,0,R) e0 (T/N,M/2,R+1)+1 (T/N,M/2,R+1)ytn−1 +RXtn−1 

≤ S0R e0 (T −tn ,0,R)+0 (T/N,M/2,R+1) e 2 M ytn−1 +RXtn−1 , 1

where we have used Assumption 15.10 and [7, Theorem 2.16(ii)] for the equality. Together with (15.9), this implies |J2 (tn , z1 , z2 )| ≤ C

exp(M  ytn−1 + 2RXtn−1 ) , δ0 + δ1 ytn−1

for some constant C > 0 which does not depend on ω and z1 , z2 . Consequently, (15.8) and Assumption 15.4 yield (15.2). The second part of the assertion now follows along the lines of the proof of [13, Theorem 4.2] under the stated assumptions.


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Example 15.17. In most applications the denominator in (15.8) is actually bounded away from zero (cf. Example 15.13). In this situation, (15.8) follows immediately from Condition 1 of Theorem 15.16 and [7, Theorem 2.16(ii)]. In view of Theorem 15.16, the hedging error can be approximated by a sum of triple integrals with known integrands. Notice that because 1 δ0 + δ1 yt +

1 (δ 2 0

+ δ 1 yt

)2

=

1 1 − , δ0 + δ1 yt δ 0 + 2 + δ1 y t

a second-order approximation based on 1 exp(δ0 + δ1 yt ) − 1 ≈ δ0 + δ1 yt + (δ0 + δ1 yt )2 2 follows directly from Theorem 15.16. Instead of using the closed-form approximation proposed above, one can eschew semi-explicit computations and instead calculate the hedging error using a Monte–Carlo simulation as in [6]: (1) Simulate K ∈ N independent trajectories (y(ωk ), X(ωk )), k = 1, . . . , K of (y, X) and compute the realizations S(ωk ) = S0 exp(X(ωk )) and H = f(ST (ωk )) of S and H. (2) Calculate the values of v0 and ϕtn (ωk ), tn ∈ T using numerical integration to evaluate the formulas from Theorem 15.14. (3) Compute the realized squared hedging errors  J0 (ωk ) = v0 +




2 ϕtn (ωk )Stn (ωk ) − f(ST (ωk )) .

tn ∈T

(4) Use the empirical mean

1 K

K k=1

J0 (ωk ) as an estimator for J0 .

In addition to its simplicity, this approach has the advantage of approximating the entire distribution of the hedging error, rather than just its mean. On the other hand, computation time is increased.

15.5. NUMERICAL ILLUSTRATION In order to illustrate the applicability of our formulas and examine the effect of discrete trading, we now investigate a numerical example. More specifically, we consider the NIG-Gamma-OU model from Examples 15.9 and 15.11. As for parameters, we use the

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values estimated in [16] using the generalized method of moments, adjusting the drift rate µ of L in order to ensure the martingale property of S: β = −16.0,

α = 90.1,

λ = 2.54,

δ = 85.9,

a = 0.847,

µ = 15.0,

b = 17.5.

By Example 15.11, Assumption 15.10 and the prerequisites of Theorem 15.16 are satisfied for European call options and R = 1.1. Henceforth, we consider a European call with discounted strike K = 100 and maturity T = 0.25 years. The results for the variance-optimal initial hedge ratio ϕ0 for N = 1 (static hedging) and N = 12 (weekly rebalancing) are shown in Figs 15.1 and 15.2. For static hedging, the impact of discretization seems to be quite pronounced, in particular for out-of-the-money options. Also notice that this effect turns out to be substantially bigger for the NIG-Gamma-OU than for the Black–Scholes model. For weekly rebalancing, the effect of discretization on the initial hedge ratio already becomes marginal. More specifically, the difference between the discrete- and continuous-time variance-optimal hedging strategies is barely visible in Fig. 15.2. Figure 15.3 shows a simulated path of the discrete variance-optimal hedges for N = 1, 3, 12, 60.

Figure 15.1 Variance-optimal initial hedge ratios for N = 1.

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Figure 15.2 Variance-optimal initial hedge ratios for N = 12. Hedging strategy (Strike = 100, T = 0.25, N = 1, 3, 12, 60 ) 1

0.9

0.8

Hedge ratio

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

N=1 N=3 N=12 N=60 0.05

0.1

0.15

0.2

0.25

Time

Figure 15.3 A simulated path of optimal hedge ratios for N = 1, 3, 12, 60.

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We now turn to the minimal expected squared hedging error, which is depicted for N = 1 (static hedging) to N = 60 (daily rebalancing) in Fig. 15.4. As the number N of trading dates tends to infinity, the discrete hedging errors approach the respective continuous-time limits both in the Black–Scholes model and in the NIG-Gamma-OU model. Naturally, this limit vanishes in the complete Black–Scholes model. As noticed above, the static hedging error for N = 1 can be computed without using any approximations. For N ≥ 2, the discrete-time hedging error in the given NIG-Gamma-OU model is approximated surprisingly well by the sum of the respective continuoustime hedging error and the corresponding discrete-time hedging error in the Black– Scholes model. In fact, the maximal absolute difference is smaller than 0.045. If such an approximation can be used for the specific model at hand, computation time can often be drastically reduced by evaluating the formulas from [10, 12] instead of Theorem 15.16. Note that the discrete hedging errors in the NIG-Gamma-OU model have been approximated using Theorem 15.16. Since the corresponding results for a simulation study using one million Monte–Carlo runs differ by less than 2.5% for N = 1, . . . , 60, we do not show them here. However, in Fig. 15.5, we use the results of the Monte– Carlo study to depict an approximation of the distribution of the hedging error for

Figure 15.4 Minimal expected squared hedging errors.

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Figure 15.5 Approximated distribution of the hedging error for N = 1, 12, 60.

N = 1, N = 12, and N = 60. Apparently, not only the variance of the hedging error, but also its law depend crucially on the rebalancing frequency.

Acknowledgments The first and fourth authors gratefully acknowledge financial support through Sachbeihilfe KA 1682/2-1 of the Deutsche Forschungsgemeinschaft.

References [1] Barndorff-Nielsen, O and N Shephard (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. Journal of the Royal Statistical Society, Series B, 63, 167–241. [2] Bertsimas, D, L Kogan and A Lo (2001). Hedging derivative securities and incomplete markets: An -arbitrage approach. Operations Research, 49, 372–397. [3] Carr, P, H Geman, D Madan and M Yor (2003). Stochastic volatility for Lévy processes. Mathematical Finance, 13, 345–382. ˇ [4] Cerný, A (2007). Optimal continuous-time hedging with leptokurtic returns. Mathematical Finance, 17, 175–203.

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ˇ [5] Cerný, A and J Kallsen (2007). On the structure of general mean–variance hedging strategies. The Annals of Probability, 35, 1479–1531. [6] Cont, R, P Tankov and E Voltchkova (2007). Hedging with options in presence of jumps. In Stochastic Analysis and Applications: The Abel Symposium 2005 in honor of Kiyosi Ito, F Benth, G Di Nunno, T Lindstrom, B Øksendal and T Zhang, (eds.), pp. 197–218. Berlin: Springer. [7] Duffie, D, D Filipovic and W Schachermayer (2003). Affine processes and applications in finance. The Annals of Applied Probability, 13, 984–1053. [8] Filipovi´c, D and E Mayerhofer (2009).Affine diffusion processes: Theory and applications. Radon Series on Computational and Applied Mathematics, 8, 1–40. [9] Heston, S (1993). A closed-form solution for options with stochastic volatilities with applications to bond and currency options. The Review of Financial Studies, 6, 327–343. [10] Hubalek, F, L Krawczyk and J Kallsen (2006). Variance-optimal hedging for processes with stationary independent increments. The Annals of Applied Probability, 16, 853–885. [11] Kallsen, J (2006). A didactic note on affine stochastic volatility models. In From Stochastic Calculus to Mathematical Finance, Y. Kabanov, R Liptser, and J Stoyanov (eds.), pp. 343– 368. Berlin: Springer. [12] Kallsen, J and A Pauwels (2009). Variance-optimal hedging for time-changed Lévy processes. To appear in Applied Mathematical Finance. [13] Kallsen, J and A Pauwels (2009). Variance-optimal hedging in general affine stochastic volatility models. To appear in Advances in Applied Probability. [14] Kallsen, J and RVierthauer (2009). Quadratic hedging in affine stochastic volatility models. Review of Derivatives Research, 12, 3–27. [15] Lamberton, D and B Lapeyre (1996). Stochastic Calculus Applied to Finance. London: Chapman & Hall. [16] Muhle-Karbe, J (2009). On Utility-Based Investment, Pricing and Hedging in Incomplete Markets. Dissertation, Technische Universität, München. [17] Pauwels, A (2006). Varianz-optimales Hedging in affinen Volatilitätsmodellen. Dissertation, Technische Universität, München. [18] Pham, H (2000). On quadratic hedging in continuous time. Mathematical Methods of Operations Research, 51, 315–339. [19] Sato, K (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press. [20] Schoutens, W (2003). Lévy Processes in Finance. New York: Wiley. [21] Schweizer, M (1995). Variance-optimal hedging in discrete time. Mathematics of Operations Research, 20, 1–32. [22] Schweizer, M (2001). A guided tour through quadratic hedging approaches. In Option Pricing, Interest Rates and Risk Management, E Jouini, J Cvitanic, and M Musiela (eds.), pp. 538–574. Cambridge: Cambridge University Press.

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INDEX

absolute ambiguity robust adjustment (AARA), 330 affine process, 376, 380 alternative asset classes, 52 Alternative real assets, 51–54, 69 ambiguity, 327–337, 339–344 ambiguity robust adjustment (AARA), 330, 333 ambiguity robust adjustment (RARA), 330 Angström-type equation, 58 asset allocation, 147–149, 162, 165, 168, 170, 171 asset class, 3, 4, 8–13, 17–19 Autocorrelation, 65, 66 aviation, 53

calendar spread option, 118, 119 capital-market scenarios, 149, 170 CCO, 177 CDS/credit default swaps, 259–261, 265–267, 270, 271, 283, 285–290 CDX, 261, 266, 272, 275–277, 285, 289, 290, 292 CFXO, 177 CIR-time-change model, 383 closed-end funds, 52 cointegration, 84–86 collateralized asset obligation, 179 collateralized commodity obligation, 177 collateralized debt obligation, 176 collateralized fx obligation, 177 collateralized trigger swap obligations, 177 comonotone random vector, 365 conditional value at risk optimization/CVaR optimization, 147, 157, 164–169, 171 confidence ellipsoid, 317, 319 constant proportion portfolio insurance (CPPI), 227, 228, 232 copula approach, 186, 187, 189, 190, 193 corporate bonds, 148, 158, 160, 162, 163, 165, 166, 170 correlated default times, 147, 150, 152, 158

backtesting, 88 barrier options, 345, 347, 350, 352–355, 359, 361–364 Black–Litterman expectation, 9, 10, 17 Black-76, 98, 105, 114, 116, 117 Bonus (Guarantee) Certificates, 134 borrowing constraints, 227, 230 bottom-up approach, 52, 54, 55, 68 Brownian motion, 263, 264, 271, 272, 275, 277, 285, 290

395

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396 CPDO/constant porportion debt obligation, 259–261, 285–292 CPPI, 201–203, 205–207, 209–211, 213–224 CPPI strategies, 295, 296, 300, 304, 305, 311, 321, 323, 324 CPPI/constant proportion portfolio insurance, 259–261, 267–270, 275, 278, 281, 282, 284, 285, 289, 291, 292 Credit default swap index/CDS index, 154, 155, 157, 161, 169 Credit default swap/CDS, 148, 149, 153–155, 157, 158, 160–163, 165, 169, 170 CreditGrades, 125 cross asset portfolio derivatives, 175–177, 182, 191, 195 cushion, 267–269, 275, 277, 282, 284, 305, 307 default risk, 123 defaultable knock-out put options, 126 discount certificates, 126 discretely sample options, 347, 362, 367 dynamic asset allocation, 296 dynamic portfolio insurance, 201 Economic Scenario Generator, 56 efficient portfolios, 295, 296, 299, 300, 312, 313, 316, 324, 325 estimation risk, 295, 296, 298, 313, 314, 316, 320, 321, 324, 325 ETF, 201, 203, 210, 223, 224 European options, 348, 349, 355, 357, 358, 361, 363, 367, 369, 370 (extended) roll-down call, 362 expected shortfall, 308, 309, 312 expected-squared hedging error, 375 explanatory variables, 72, 79–82, 84–86, 90 exposure, 303, 305, 307 extreme value theory (EVT), 282 floor, 303, 305, 307–309 forward freight agreement, 77, 78 freight derivatives, 71, 77, 78

Index freight futures, 71, 78, 79, 81, 86 freight rates, 74, 75, 77, 80–82, 90 Gamma distribution, 271 Gamma-OU process, 381 gap risk, 229, 230, 241, 246, 247, 251, 259, 268, 270, 277–279, 281–284, 287, 292 Global irradiance, 53, 55–58, 60–62 granger causality, 82, 83 guarantees, 228–230, 242 Heath–Jarrow–Morton, 94 Hull-White model, 56 index certificates, 124, 126, 135 inflation, 51–53, 55–57, 65 inflation protection, 52, 53, 55 infrastructure, 51–53 issuer risk, 123–125, 130, 132, 139 iTraxx, 261, 266, 272, 275–277, 285, 289, 290, 292 Jensen’s Inequality, 107–110 Lévy process, 264 Laplace transform approach, 376, 378 leverage, 261, 267, 268, 275, 285–288, 290, 291 LIBOR model, 98 limited partnership fund, 51, 53 log-normal (shifted), 126 lookback options, 346, 347, 364, 368 Macroeconomic factors, 56, 57 Margrabe Formula, 117 Markov-Switching model, 3, 4, 11, 12, 19 Markov transition matrix, 4, 6, 7 maximum Sharpe ratio portfolio, 299, 301, 312, 314, 316, 321, 322 mean-variance optimization, 299, 300, 310, 311 method of moments, 11 minimum variance portfolio, 299, 301, 312, 316, 320, 322–324 model-independent strategies, 359 Monte Carlo simulation, 13

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Index multi-asset portfolio, 51, 52, 68 multiplier, 305–313, 315, 316, 320–324 multivariate variance gamma model, 289 NIG process, 382 Normal-inverse gaussian one-factor copula/NIG one-factor copula, 147, 149, 152, 171 nth-to-trigger basket, 177, 178, 183 oil, 72–74, 76, 77, 81–84, 86 option-based portfolio insurance (OBPI), 228, 232 OU time-change model, 382, 383 Participation Guarantee Certificates, 132 path-independent payoff, 347 photovoltaic, 51–56, 58, 61–64, 66–69 Portfolio, 51, 52, 56, 63–69 portfolio insurance, 259, 267 portfolio optimization, 13, 149, 156, 160, 162, 171, 231 power forward, 93–96, 98–103, 107, 108, 110–120 power markets, 93, 94, 96, 97 Put-Call-Parity, 346, 349, 352, 360 ratchet call, 363 real return, 53 renewable energy, 51–53 robust counterpart, 295, 316, 317, 320 robust mean-variance optimization, 316 Samuelson effect, 97, 100 semi-static hedging strategies, 345, 352, 370 shipping, 51–54 Shipping market, 72, 75, 80 shortfall probability, 250, 251, 308, 309, 311, 312, 316, 322, 324 socially responsible investing (SRI), 3, 4 solar plants, 51, 53, 54

b913-Index

FA

397 static hedging, 385, 389, 391 static hedging strategies, 362, 370 stop-loss transform, 365 strong path-dependence, 347 structural approach, 185, 186, 188, 192, 193 sub-replication (sub replicating strategies), 354, 359 subjective expected utility (SEU), 328 super-replication (super replicating strategies), 359, 361, 362, 365 sustainability, 3, 4, 6–8, 18 sustainability score, 6, 7, 18 swaptions, 270, 273–275, 279, 280 Term structure of interest rates, 57 TIPP, 203, 214, 220 total variance, 66 trading route, 73, 78 transaction costs, 227, 230, 242, 246, 247, 249–251 trigger derivative, 176, 178, 179, 188, 195 uncertainty set, 317–320 utility, 227–230, 232–234, 240–244, 246, 250 functions, 232, 233, 237, 242 loss, 230, 243–245, 250, 251 variance gamma process, 264, 270, 274 variance-optimal hedging strategy, 378, 379, 383 variance-optimal initial capital, 375, 378, 383 Vasicek process, 56 vector auto regression model, 72, 79, 80 vessel, cargo, 72 vulnerable options, 124 weak (strong) path-dependence, 347, 350, 364 wind power plants, 53