Readings for the Financial Risk Manager Vol II

Readings for the Financial Risk Manager Volume 2

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Readings for the Financial Risk Manager Volume 2

RENÉ M. STULZ, EDITOR RICH APOSTOLIK, EDITOR GLOBAL ASSOCIATION OF RISK PROFESSIONALS, INC.

John Wiley & Sons, Inc.

Copyright © 2005 by Global Association of Risk Professionals, Inc. All rights reserved. Please see Credits for additional copyright and source information. In all instances, permission credits and source information appear directly on any reprinted chapter. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, 201-748-6011, fax 201-748-6008, or online at http://www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at 800-762-2974, outside the United States at 317-572-3993 or fax 317-572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, visit our web site at www.wiley.com. ISBN-13 978-0-471-78297-1 ISBN-10 0-471-78297-1 Printed in the United States of America.

Contents

Editors’ Note Acknowledgments READING 57

Computing Value-at-Risk Philippe Jorion Reproduced with permission from Value at Risk, 2nd ed. (New York: McGraw-Hill, 2001): 107–128.

READING 58

VaR Methods Philippe Jorion Reproduced with permission from Value at Risk, 2nd ed. (New York: McGraw-Hill, 2001): 205–230.

READING 59

Liquidity Risk Philippe Jorion Reproduced with permission from Value at Risk, 2nd ed. (New York: McGraw-Hill, 2001): 339–357.

READING 60

Credit Risks and Credit Derivatives René M. Stulz Reproduced with permission from Risk Management and Derivatives (Mason, Ohio: South-Western, 2003): 571–604.

READING 61

Extending the VaR Approach to Operational Risk Linda Allen, Jacob Boudoukh, and Anthony Saunders Reproduced with permission from Understanding Market, Credit and Operational Risk: The Value at Risk Approach (Oxford: Blackwell Publishing, 2004): 158–199.

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READINGS FOR THE FINANCIAL RISK MANAGER

READING 62

Case Studies Reto Gallati Reproduced with permission from Risk Management and Capital Adequacy (New York: McGraw-Hill, 2003): 441–493.

READING 63

What Is Operational Risk? Douglas G. Hoffman Reproduced with permission from Managing Operational Risk (New York: John Wiley & Sons, 2002): 29–55.

READING 64

Risk Assessment Strategies Douglas G. Hoffman Reproduced with permission from Managing Operational Risk (New York: John Wiley & Sons, 2002): 181–212.

READING 65

Operational Risk Analysis and Measurement: Practical Building Blocks Douglas G. Hoffman Reproduced with permission from Managing Operational Risk (New York: John Wiley & Sons, 2002): 257–304.

READING 66

Economic Risk Capital Modeling Douglas G. Hoffman Reproduced with permission from Managing Operational Risk (New York: John Wiley & Sons, 2002): 375–403.

READING 67

Capital Allocation and Performance Measurement Michel Crouhy, Dan Galai, and Robert Mark Reproduced with permission from Risk Management (New York: McGraw-Hill, 2001): 529–578.

Contents

READING 68

The Capital Asset Pricing Model and Its Application to Performance Measurement Noël Amenc and Véronique Le Sourd Reproduced with permission from Portfolio Theory and Performance Analysis (West Sussex: John Wiley & Sons, 2003): 95–102, 108–116.

READING 69

Multi-Factor Models and Their Application to Performance Measurement Noël Amenc and Véronique Le Sourd Reproduced with permission from Portfolio Theory and Performance Analysis (West Sussex: John Wiley & Sons, 2003): 149–194.

READING 70

Fixed Income Security Investment Noël Amenc and Véronique Le Sourd Reproduced with permission from Portfolio Theory and Performance Analysis (West Sussex: John Wiley & Sons, 2003): 229–252.

READING 71

Funds of Hedge Funds Jaffer Sohail Reproduced with permission. Lars Jaeger, ed., The New Generation of Risk Management for Hedge Funds and Private Equity Investments (London: Euromoney Books, 2003): 88–107.

READING 72

Style Drifts: Monitoring, Detection and Control Pierre-Yves Moix Reproduced with permission. Lars Jaeger, ed., The New Generation of Risk Management for Hedge Funds and Private Equity Investments (London: Euromoney Books, 2003): 387–398.

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READING 73

Risk Control Strategies: The Manager’s Perspective Pierre-Yves Moix and Stefan Scholz Reproduced with permission. Jaffer Sohail, ed., Funds of Hedge Funds (London: Euromoney Books, 2003): 219–233.

APPENDIX

FRM Suggested Readings for Further Study Credits About the CD-ROM

Editors’ Note

he objective of this volume is to provide core readings recommended by the Global Association of Risk Professionals’ Financial Risk Manager (FRM®) Committee for the 2005 exam that are not available on the first Readings for the Financial Risk Manager CD-ROM. The FRM Committee, which oversees the selection of reading materials for the FRM Exam, suggests 100 readings for those registered for the FRM Exam and any other risk professionals interested in the critical knowledge essential to their profession. Fifty-five of these recommended readings appear on the Readings for the Financial Risk Manager CD-ROM* and 17 appear on this CD-ROM. While every attempt has been made by GARP to obtain permissions from authors and their respective publishers to reprint all materials on the FRM Committee’s recommended reading list, not all readings were available for reprinting. A list of those readings that are not reprinted on either the Readings for the Financial Risk Manager CD-ROM or this CD-ROM can be found in the Appendix of this CD-ROM. In every instance, full bibliographic information is provided for those interested in referencing these materials for study, citing them in their own research, or ultimately acquiring the volumes in which the readings first appeared for their own risk management libraries. GARP thanks all authors and publishers mentioned—particularly those who graciously agreed to allow their materials to be reprinted here as a companion text to the Financial Risk Manager Handbook, Third Edition, by Philippe Jorion. We hope these books of readings prove to be of great convenience and use to all risk professionals, including those enrolled for the FRM Exam.

T

*The Editors note that Reading 56, which appears on the first Readings for the Financial Risk Manager CD-ROM, is not on the suggested reading list for the 2005 FRM Exam. To avoid confusion, we have labeled the first reading on Volume 2 as Reading 57, so that each suggested reading, whether current or dormant, has its own unique assigned number.

ix

Acknowledgments

his second volume of Readings for the Financial Risk Manager was made possible through the work of the Global Association of Risk Professionals’ FRM Committee. To choose the readings for the FRM Exam, the Committee reviewed an extremely large number of published works. The readings selected were chosen because they meet high expositional standards and together provide coverage of the issues the Committee expects candidates to master. GARP’s FRM Exam has attained global benchmark status in large part because of the hard work and dedication of this core group of risk management professionals. These highly regarded professionals have volunteered their time to develop, without a historical road map, the minimum standards that risk managers must meet. The challenge to successfully implement this approach on a global basis cannot be overstated. GARP’s FRM Committee meets regularly via e-mail, through conference calls, and in person to identify and discuss financial risk management trends and theories. Its objective is to ensure that what is tested each year in the FRM Exam is timely, comprehensive, and relevant. The results of these discussions are memorialized in the FRM Study Guide. The Study Guide, which is revised annually, clearly delineates in a topical outline the base level of knowledge that a financial risk manager should possess in order to provide competent financial risk management advice to a firm’s senior management and directors. FRM Committee members represent some of the industry’s most knowledgeable financial risk professionals. The following individuals were the Committee members responsible for developing the 2005 FRM Study Guide:

T

Dr. René Stulz (Chairman) Richard Apostolik Juan Carlos Garcia Cespedes Dr. Marcelo Cruz Dr. James Gutman Kai Leifert Steve Lerit, CFA

x

Ohio State University Global Association of Risk Professionals Banco Bilbao Vizcaya Argentaria Risk Maths, Inc. Goldman Sachs International Credit Suisse Asset Management New York Life Investment Management

xi

Acknowledgments

Michelle McCarthy Dr. Susan Mangiero Michael B. Miller Peter Nerby Dr. Victor Ng Dr. Elliot Noma Gadi Pickholz Robert Scanlon Omer Tareen Alan Weindorf

Washington Mutual Bank BVA, LLC Fortress Investment Group Moody’s Investors Service Goldman Sachs & Co. Asset Alliance Corporation Ben Gurion University of the Negev Standard Chartered Bank Microsoft Corporation Starbucks Coffee Company

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CHAPTER

5

Computing Value at Risk

The Daily Earnings at Risk (DEaR) estimate for our combined trading activities averaged approximately $15 million. J.P. Morgan 1994 Annual Report

P

erhaps the greatest advantage of value at risk (VAR) is that it summarizes in a single, easy to understand number the downside risk of an institution due to financial market variables. No doubt this explains why VAR is fast becoming an essential tool for conveying trading risks to senior management, directors, and shareholders. J.P. Morgan, for example, was one of the first users of VAR. It revealed in its 1994 Annual Report that its trading VAR was an average of $15 million at the 95 percent level over 1 day. Shareholders can then assess whether they are comfortable with this level of risk. Before such figures were released, shareholders had only a vague idea of the extent of trading activities assumed by the bank. This chapter turns to a formal definition of value at risk (VAR). VAR assumes that the portfolio is “frozen” over the horizon or, more generally, that the risk profile of the institution remains constant. In addition, VAR assumes that the current portfolio will be marked-to-market on the target horizon. Section 5.1 shows how to derive VAR figures from probability distributions. This can be done in two ways, either from considering the actual empirical distribution or by approximating the distribution by a parametric approximation, such as the normal distribution, in which case VAR is derived from the standard deviation. Section 5.2 then discusses the choice of the quantitative factors, the confidence level and the horizon. Criteria for this choice should be guided by the use of the VAR number. If VAR is simply a benchmark for risk, 107

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the choice is totally arbitrary. In contrast, if VAR is used to set equity capital, the choice is quite delicate. Criteria for parameter selection are also explained in the context of the Basel Accord rules. The next section turns to an important and often ignored issue, which is the precision of the reported VAR number. Due to normal sampling variation, there is some inherent imprecision in VAR numbers. Thus, observing changes in VAR numbers for different estimation windows is perfectly normal. Section 5.3 provides a framework for analyzing normal sampling variation in VAR and discusses methods to improve the accuracy of VAR figures. Finally, Section 5.4 provides some concluding thoughts. 5.1 COMPUTING VAR

With all the requisite tools in place, we can now formally define the value at risk (VAR) of a portfolio. VAR summarizes the expected maximum loss (or worst loss) over a target horizon within a given confidence interval. Initially, we take the quantitative factors, the horizon and confidence level, as given.

5.1.1 Steps in Constructing VAR Assume, for instance, that we need to measure the VAR of a $100 million equity portfolio over 10 days at the 99 percent confidence level. The following steps are required to compute VAR: ■ ■

■

■

■

Mark-to-market of the current portfolio (e.g., $100 million). Measure the variability of the risk factors(s) (e.g., 15 percent per annum). Set the time horizon, or the holding period (e.g., adjust to 10 business days). Set the confidence level (e.g., 99 percent, which yields a 2.33 factor assuming a normal distribution). Report the worst loss by processing all the preceding information (e.g., a $7 million VAR).

These steps are illustrated in Figure 5–1. The precise detail of the computation is described next.

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CHAPTER 5 Computing Value at Risk

FIGURE

109

5–1

Steps in constructing VAR.

Mark position to market

Measure Set variability of time risk factors horizon

Set confidence level

Report potential loss

Value

Value

Frequency

Value

VAR σ

10 days Time

−α

Horizon

Horizon

Sample computation: $100M

x

15%

x

(10/252)

x

2.33

=

$7M

5.1.2 VAR for General Distributions To compute the VAR of a portfolio, define W0 as the initial investment and R as its rate of return. The portfolio value at the end of the target horizon is W  W0 (1  R). As before, the expected return and volatility of R are  and . Define now the lowest portfolio value at the given confidence level c as W*  W0 (1  R*). The relative VAR is defined as the dollar loss relative to the mean: VAR(mean)  E(W)  W*  W0 (R*  )

(5.1)

Sometimes VAR is defined as the absolute VAR, that is, the dollar loss relative to zero or without reference to the expected value: VAR(zero)  W0  W*  W0R*

(5.2)

In both cases, finding VAR is equivalent to identifying the minimum value W* or the cutoff return R*. If the horizon is short, the mean return could be small, in which case both methods will give similar results. Otherwise, relative VAR is conceptually more appropriate because it views risk in terms of a deviation

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from the mean, or “budget,” on the target date, appropriately accounting for the time value of money. This approach is also more conservative if the mean value is positive. Its only drawback is that the mean return is sometimes difficult to estimate. In its most general form, VAR can be derived from the probability distribution of the future portfolio value f(w). At a given confidence level c, we wish to find the worst possible realization W* such that the probability of exceeding this value is c: c



∞

f(w) dw

(5.3)

W*

or such that the probability of a value lower than W*, p  P(w  W*), is 1  c: 1c



W*

∞

f(w) dw  P(w  W*)  p

(5.4)

In other words, the area from ∞ to W* must sum to p  1  c, for instance, 5 percent. The number W* is called the quantile of the distribution, which is the cutoff value with a fixed probability of being exceeded. Note that we did not use the standard deviation to find the VAR. This specification is valid for any distribution, discrete or continuous, fat- or thin-tailed. Figure 5–2, for instance, reports J.P. Morgan’s distribution of daily revenues in 1994. To compute VAR, assume that daily revenues are identically and independently distributed. We can then derive the VAR at the 95 percent confidence level from the 5 percent left-side “losing tail” from the histogram. From this graph, the average revenue is about $5.1 million. There is a total of 254 observations; therefore, we would like to find W* such that the number of observations to its left is 254  5 percent  12.7. We have 11 observations to the left of $10 million and 15 to the left of $9 million. Interpolating, we find W*  $9.6 million. The VAR of daily revenues, measured relative to the mean, is VAR  E(W)  W*  $5.1 million  ($9.6 million)  $14.7 million. If one wishes to measure VAR in terms of absolute dollar loss, VAR is then $9.6 million.

5.1.3 VAR for Parametric Distributions The VAR computation can be simplified considerably if the distribution can be assumed to belong to a parametric family, such as the normal dis-

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CHAPTER 5 Computing Value at Risk

FIGURE

111

5–2

Distribution of daily revenues. 20

Number of days VAR=$15 million

Average=$5 million

5% of Occurrences 15

10

5

0

<-25

-20

-15

-10

-5 0 5 10 15 Daily revenue ($ million)

20

tribution. When this is the case, the VAR figure can be derived directly from the portfolio standard deviation using a multiplicative factor that depends on the confidence level. This approach is sometimes called parametric because it involves estimation of parameters, such as the standard deviation, instead of just reading the quantile off the empirical distribution. This method is simple and convenient and, as we shall see later, produces more accurate measures of VAR. The issue is whether the normal approximation is realistic. If not, another distribution may fit the data better. First, we need to translate the general distribution f(w) into a standard normal distribution ( ), where has mean zero and standard deviation of unity. We associate W* with the cutoff return R* such that W*  W0(1  R*). Generally, R* is negative and also can be written as |R*|.

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Further, we can associate R* with a standard normal deviate  0 by setting |R*|   



(5.5)

It is equivalent to set 1c



W*

∞

f(w) dw 



|R*|

∞

f(r) dr 





∞

( ) d

(5.6)

Thus the problem of finding a VAR is equivalent to finding the deviate  such that the area to the left of it is equal to 1  c. This is made possible by turning to tables of the cumulative standard normal distribution function, which is the area to the left of a standard normal variable with value equal to d: N(d) 



d

∞

( ) d

(5.7)

This function also plays a key role in the Black-Scholes option pricing model. Figure 5–3 graphs the cumulative density function N(d ), which increases monotonically from 0 (for d  ∞) to 1 (for d  ∞), going through 0.5 as d passes through 0. To find the VAR of a standard normal variable, select the desired left-tail confidence level on the vertical axis, say, 5 percent. This corresponds to a value of   1.65 below 0. We then retrace our steps, back from the  we just found to the cutoff return R* and VAR. From Equation (5.5), the cutoff return is R*    

(5.8)

For more generality, assume now that the parameters  and  are expressed on an annual basis. The time interval considered is t, in years. We can use the time aggregation results developed in the preceding chapter, which assume uncorrelated returns. Using Equation (5.1), we find the VAR below the mean as  VAR(mean)  W0(R*  )  W0t

(5.9)

In other words, the VAR figure is simply a multiple of the standard deviation of the distribution times an adjustment factor that is directly related to the confidence level and horizon.

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CHAPTER 5 Computing Value at Risk

FIGURE

113

5–3

Cumulative normal probability distribution. 1

N(d)

1.65σ 0.5

c = 5% confidence level 0.05 0

-3

-2

-1 0 1 d=Standard normal variable

2

3

When VAR is defined as an absolute dollar loss, we have   t) VAR(zero)  W0R*  W0(t

(5.10)

This method generalizes to other cumulative probability functions (cdf) as well as the normal, as long as all the uncertainty is contained in . Other distributions will entail different values of . The normal distribution is just particularly easy to deal with because it adequately represents many empirical distributions. This is especially true for large, welldiversified portfolios but certainly not for portfolios with heavy option components and exposures to a small number of financial risks.

5.1.4 Comparison of Approaches How well does this approximation work? For some distributions, the fit can be quite good. Consider, for instance, the daily revenues in Figure 5–2. The standard deviation of the distribution is $9.2 million. According

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FIGURE

5–4

Comparison of cumulative distributions. 1

Cumulative probability

0.5

Normal distribution Actual distribution 5% 0

<-25

-20

-15

-10

-5 0 5 10 15 Daily revenue ($ million)

20

to Equation (5.9), the normal-distribution VAR is   (W0)  1.65  $9.2 million  $15.2 million. Note that this number is very close to the VAR obtained from the general distribution, which was $14.7 million. Indeed, Figure 5–4 presents the cumulative distribution functions (cdf) obtained from the histogram in Figure 5–2 and from its normal approximation. The actual cdf is obtained from summing, starting from the left, all numbers of occurrences in Figure 5–2 and then scaling by the total number of observations. The normal cdf is the same as that in Figure 5–3, with the horizontal axis scaled back into dollar revenues using Equation (5.8). The two lines are generally very close, suggesting that the normal approximation provides a good fit to the actual data.

5.1.5 VAR as a Risk Measure VAR’s heritage can be traced to Markowitz’s (1952) seminal work on portfolio choice. He noted that “you should be interested in risk as well as

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return” and advocated the use of the standard deviation as an intuitive measure of dispersion. Much of Markowitz’s work was devoted to studying the tradeoff between expected return and risk in the mean-variance framework, which is appropriate when either returns are normally distributed or investors have quadratic utility functions. Perhaps the first mention of confidence-based risk measures can be traced to Roy (1952), who presented a “safety first” criterion for portfolio selection. He advocated choosing portfolios that minimize the probability of a loss greater than a disaster level. Baumol (1963) also proposed a risk measurement criterion based on a lower confidence limit at some probability level: L    

(5.11)

which is an early description of Equation (5.10). Other measures of risk have also been proposed, including semideviation, which counts only deviations below a target value, and lower partial moments, which apply to a wider range of utility functions. More recently, Artzner et al. (1999) list four desirable properties for risk measures for capital adequacy purposes. A risk measure can be viewed as a function of the distribution of portfolio value W, which is summarized into a single number (W): ■

■

■

■

Monotonicity: If W1  W2, (W1)  (W2), or if a portfolio has systematically lower returns than another for all states of the world, its risk must be greater. Translation invariance. (W  k)  (W)  k, or adding cash k to a portfolio should reduce its risk by k. Homogeneity. (bW)  b(W), or increasing the size of a portfolio by b should simply scale its risk by the same factor (this rules out liquidity effects for large portfolios, however). Subadditivity. (W1  W2)  (W1)  (W2), or merging portfolios cannot increase risk.

Artzner et al. (1999) show that the quantile-based VAR measure fails to satisfy the last property. Indeed, one can come up with pathologic examples of short option positions that can create large losses with a low probability and hence have low VAR yet combine to create portfolios with larger VAR. One can also show that the shortfall measure E(X|X  VAR),

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which is the expected loss conditional on exceeding VAR, satisfies these desirable “coherence” properties. When returns are normally distributed, however, the standard deviation–based VAR satisfies the last property, (W1  W2)  (W1)  (W2). Indeed, as Markowitz had shown, the volatility of a portfolio is less than the sum of volatilities. Of course, the preceding discussion does not consider another essential component for portfolio comparisons: expected returns. In practice, one obviously would want to balance increasing risk against increasing expected returns. The great benefit of VAR, however, is that it brings attention and transparency to the measure of risk, a component of the decision process that is not intuitive and as a result too often ignored. 5.2 CHOICE OF QUANTITATIVE FACTORS

We now turn to the choice of two quantitative factors: the length of the holding horizon and the confidence level. In general, VAR will increase with either a longer horizon or a greater confidence level. Under certain conditions, increasing one or the other factor produces equivalent VAR numbers. This section provides guidance on the choice of c and t, which should depend on the use of the VAR number.

5.2.1 VAR as a Benchmark Measure The first, most general use of VAR is simply to provide a companywide yardstick to compare risks across different markets. In this situation, the choice of the factors is arbitrary. Bankers Trust, for instance, has long used a 99 percent VAR over an annual horizon to compare the risks of various units. Assuming a normal distribution, we show later that it is easy to convert disparate bank measures into a common number. The focus here is on cross-sectional or time differences in VAR. For instance, the institution wants to know if a trading unit has greater risk than another. Or whether today’s VAR is in line with yesterday’s. If not, the institution should “drill down” into its risk reports and find whether today’s higher VAR is due to increased volatility or larger bets. For this purpose, the choice of the confidence level and horizon does not matter much as long as consistency is maintained.

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5.2.2 VAR as a Potential Loss Measure Another application of VAR is to give a broad idea of the worst loss an institution can incur. If so, the horizon should be determined by the nature of the portfolio. A first interpretation is that the horizon is defined by the liquidation period. Commercial banks currently report their trading VAR over a daily horizon because of the liquidity and rapid turnover in their portfolios. In contrast, investment portfolios such as pension funds generally invest in less liquid assets and adjust their risk exposures only slowly, which is why a 1-month horizon is generally chosen for investment purposes. Since the holding period should correspond to the longest period needed for an orderly portfolio liquidation, the horizon should be related to the liquidity of the securities, defined in terms of the length of time needed for normal transaction volumes. A related interpretation is that the horizon represents the time required to hedge the market risks. An opposite view is that the horizon corresponds to the period over which the portfolio remains relatively constant. Since VAR assumes that the portfolio is frozen over the horizon, this measure gradually loses significance as the horizon extends. However, perhaps the main reason for banks to choose a daily VAR is that this is consistent with their daily profit and loss (P&L) measures. This allows an easy comparison between the daily VAR and the subsequent P&L number. For this application, the choice of the confidence level is relatively arbitrary. Users should recognize that VAR does not describe the worst-ever loss but is rather a probabilistic measure that should be exceeded with some frequency. Higher confidence levels will generate higher VAR figures.

5.2.3 VAR as Equity Capital On the other hand, the choice of the factors is crucial if the VAR number is used directly to set a capital cushion for the institution. If so, a loss exceeding the VAR would wipe out the equity capital, leading to bankruptcy. For this purpose, however, we must assume that the VAR measure adequately captures all the risks facing an institution, which may be a stretch. Thus the risk measure should encompass market risk, credit risk, operational risk, and other risks.

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The choice of the confidence level should reflect the degree of risk aversion of the company and the cost of a loss exceeding VAR. Higher risk aversion or greater cost implies that a greater amount of capital should cover possible losses, thus leading to a higher confidence level. At the same time, the choice of the horizon should correspond to the time required for corrective action as losses start to develop. Corrective action can take the form of reducing the risk profile of the institution or raising new capital. To illustrate, assume that the institution determines its risk profile by targeting a particular credit rating. The expected default rate then can be converted directly into a confidence level. Higher credit ratings should lead to a higher VAR confidence level. Table 5–1, for instance, shows that to maintain a Baa investment-grade credit rating, the institution should have a default probability of 0.17 percent over the next year. It therefore should carry enough capital to cover its annual VAR at the 99.83 percent confidence level, or 100  0.17 percent. Longer horizons, with a constant risk profile, inevitably lead to higher default frequencies. Institutions with an initial Baa credit rating have a default frequency of 10.50 percent over the next 10 years. The same credit rating can be achieved by extending the horizon or decreasing the confidence level appropriately. These two factors are intimately related.

T A B L E 5–1

Credit Rating and Default Rates

Default Frequency Desired Rating

1 Year

10 Years

Aaa Aa A Baa Ba B

0.02% 0.05% 0.09% 0.17% 0.77% 2.32%

1.49% 3.24% 5.65% 10.50% 21.24% 37.98%

Source: Adapted from Moody’s default rates from 1920–1998.

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5.2.4 Criteria for Backtesting The choice of the quantitative factors is also important for backtesting considerations. Model backtesting involves systematic comparisons of VAR with the subsequently realized P&L in an attempt to detect biases in the reported VAR figures and is described in a later chapter. The goal should be to set up the tests so as to maximize the likelihood of catching biases in VAR forecasts. Longer horizons reduce the number of independent observations and thus the power of the tests. For instance, using a 2-week VAR horizon means that we have only 26 independent observations per year. A 1-day VAR horizon, in contrast, will have about 252 observations over the same year. Hence a shorter horizon is preferable to increase the power of the tests. This explains why the Basel Committee performs backtesting over a 1-day horizon, even though the horizon is 10 business days for capital adequacy purposes. Likewise, the choice of the confidence level should be such that it leads to powerful tests. Too high a confidence level reduces the expected number of observations in the tail and thus the power of the tests. Take, for instance, a 95 percent level. We know that, just by chance, we expect a loss worse than the VAR figure in 1 day out of 20. If we had chosen a 99 percent confidence level, we would have to wait, on average, 100 days to confirm that the model conforms to reality. Hence, for backtesting purposes, the confidence level should not be set too high. In practice, a 95 percent level performs well for backtesting purposes.

5.2.5 Application: The Basel Parameters One illustration of the use of VAR as equity capital is the internal models approach of the Basel Committee, which imposes a 99 percent confidence level over a 10-business-day horizon. The resulting VAR is then multiplied by a safety factor of 3 to provide the minimum capital requirement for regulatory purposes. Presumably, the Basel Committee chose a 10-day period because it reflects the tradeoff between the costs of frequent monitoring and the benefits of early detection of potential problems. Presumably also, the Basel Committee chose a 99 percent confidence level that reflects the tradeoff between the desire of regulators to ensure a safe and sound financial system and the adverse effect of capital requirements on bank returns.

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Even so, a loss worse than the VAR estimate will occur about 1 percent of the time, on average, or once every 4 years. It would be unthinkable for regulators to allow major banks to fail so often. This explains the multiplicative factor k  3, which should provide near absolute insurance against bankruptcy. At this point, the choice of parameters for the capital charge should appear quite arbitrary. There are many combinations of the confidence level, the horizon, and the multiplicative factor that would yield the same capital charge. The origin of the factor k also looks rather mysterious. Presumably, the multiplicative factor also accounts for a host of additional risks not modeled by the usual application of VAR that fall under the category of model risk. For example, the bank may be understating its risk due to a short sample period, to unstable correlation, or simply to the fact that it uses a normal approximation to a distribution that really has more observations in the tail. Stahl (1997) justifies the choice of k based on Chebyshev’s inequality. For any random variable x with finite variance, the probability of falling outside a specified interval is P(|x   | r)  1/r2

(5.12)

assuming that we know the true standard deviation . Suppose now that the distribution is symmetrical. For values of x below the mean, P[(x  )  r]  12 1/r2

(5.13)

We now set the right-hand side of this inequality to the desired level of 1 percent. This yields r(99%)  7.071. The maximum VAR is therefore VARmax  r(99%). Say that the bank reports its 99 percent VAR using a normal distribution. Using the quantile of the standard normal distribution, we have VARN  (99%)  2.326

(5.14)

If the true distribution is misspecified, the correction factor is then VARmax 7.071 k  

 3.03 VARN 2.326

(5.15)

which happens to justify the correction factor applied by the Basel Committee.

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5.2.6 Conversion of VAR Parameters Using a parametric distribution such as the normal distribution is particularly convenient because it allows conversion to different confidence lev) is els (which define ). Conversion across horizons (expressed as t also feasible if we assume a constant risk profile, that is, portfolio positions and volatilities. Formally, the portfolio returns need to be (1) independently distributed, (2) normally distributed, and (3) with constant parameters. As an example, we can convert the RiskMetrics risk measures into the Basel Committee internal models measures. RiskMetrics provides a 95 percent confidence interval (1.65) over 1 day. The Basel Committee rules define a 99 percent confidence interval (2.33) over 10 days. The adjustment takes the following form: 2.33   4.45VARRM VARBC  VARRM

10 1.65 Therefore, the VAR under the Basel Committee rules is more than four times the VAR from the RiskMetrics system. More generally, Table 5–2 shows how the Basel Committee parameters translate into combinations of confidence levels and horizons, taking an annual volatility of 12 percent, which is typical of the DM/$ T A B L E 5–2

Equivalence Between Horizon and Confidence Level, Normal Distribution, Annual Risk  12 Percent (Basel Parameters: 99 Percent Confidence over 2 Weeks)

Confidence Level c

Number of S.D. 

Baseline 99% 57.56% 81.89% 86.78% 95% 99% 99.95% 99.99997%

2.326 0.456 0.911 1.116 1.645 2.326 3.290 7.153

Horizon t

2 1 3 2 4 2 1 1

weeks year months months weeks weeks week day

Actual S.D.  t

Cutoff Value  t

2.35 12.00 6.00 4.90 3.32 2.35 1.66 0.76

5.47 5.47 5.47 5.47 5.47 5.47 5.47 5.47

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exchange rate (now the euro/$ rate). These combinations are such that t. For instance, a 99 percent they all produce the same value for  confidence level over 2 weeks produces the same VAR as a 95 percent confidence level over 4 weeks. Or conversion into a weekly horizon requires a confidence level of 99.95 percent. 5.3 ASSESSING VAR PRECISION

This chapter has shown how to estimate essential parameters for the measurement of VAR, means, standard deviations, and quantiles from actual data. These estimates, however, should not be taken for granted entirely. They are affected by estimation error, which is the natural sampling variability due to limited sample size. Users should beware of the limited precision behind the reported VAR numbers.

5.3.1 The Problem of Measurement Errors From the viewpoint of VAR users, it is important to assess the degree of precision in the reported VAR. In a previous example, the daily VAR was $15 million. The question is: How confident is management in this estimate? Could we say, for example, that management is highly confident in this figure or that it is 95 percent sure that the true estimate is in a $14 million to $16 million range? Or is it the case that the range is $5 million to $25 million. The two confidence bands give quite a different picture of VAR. The first is very precise; the second is rather uninformative (although it tells us that it is not in the hundreds of millions of dollars). This is why it is useful to examine measurement errors in VAR figures. Consider a situation where VAR is obtained from the historical simulation method, which uses a historical window of T days to measure risk. The problem is that the reported VAR measure is only an estimate of the true value and is affected by sampling variability. In other words, different choices of the window T will lead to different VAR figures. One possible interpretation of the estimates (the view of “frequen^ ^ and  are samples from an untist” statisticians) is that these estimates  derlying distribution with unknown parameters  and . With an infinite number of observations T → ∞ and a perfectly stable system, the estimates should converge to the true values. In practice, sample sizes are limited, either because some series, like emerging markets, are relatively recent or because structural changes make it meaningless to go back too

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far in time. Since some estimation error may remain, the natural dispersion of values can be measured by the sampling distribution for the pa^ ^ and  . We now turn to a description of the distribution of starameters  tistics on which VAR measures are based.

5.3.2 Estimation Errors in Means and Variances When the underlying distribution is normal, the exact distribution of the ^ is distributed sample mean and variance is known. The estimated mean  normally around the true mean ^  N(, 2/T ) 

(5.16)

where T is the number of independent observations in the sample. Note that the standard error in the estimated mean converges toward 0 at a rate  as T increases. of 1/T As for the estimated variance ^ 2, the following ratio has a chi-square distribution with (T  1) degrees of freedom: (T  1) ^ 2

 2(T  1) 2

(5.17)

In practice, if the sample size T is large enough (e.g., above 20), the chisquare distribution converges rapidly to a normal distribution, which is easier to handle:





2 ^ 2  N 2, 4

T1

(5.18)

As for the sample standard deviation, its standard error in large samples is se(^ )  

 2T 1

(5.19)

For instance, consider monthly returns on the DM/$ rate from 1973 ^  0.15 percent, ^  3.39 percent, to 1998. Sample parameters are  with T  312 observations. The standard error of the estimate indicates how confident we are about the sample value; the smaller the error, the ^ ^ is se( )  ^ 1/T  more confident we are. One standard error in  ^   0.19 percent. Therefore, the point estimate of   0.15 3.39 1/312 percent is less than one standard error away from 0. Even with 26 years of data,  is measured very imprecisely.

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In contrast, one standard error for ^ is se(^ )  ^ 1/2T   3.39   0.14 percent. Since this number is much smaller than the es1/624 timate of 3.39 percent, we can conclude that the volatility is estimated with much greater accuracy than the expected return—giving some confidence in the use of VAR systems. As the sample size increases, so does the precision of the estimate. To illustrate this point, Figure 5–5 depicts 95 percent confidence bands around the estimate of volatility for various sample sizes, assuming a true daily volatility of 1 percent. With 5 trading days, the band is rather imprecise, with upper and lower values set at [0.41%, 1.60%]. After 1 year, the band is [0.91%, 1.08%]. As the number of days increases, the confidence bands shrink to the point where, after 10 years, the interval narrows to [0.97%, 1.03%]. Thus, as the observation interval lengthens, the estimate should become arbitrarily close to the true value.

FIGURE

5–5

Confidence bands for sample volatility.

Daily volatility (%) 1.5

1

0.5

5

10

20

30 60 90 126 252 Observation interval (days)

504 1260 2520

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Finally, ^ can be used to estimate any quantile (an example is shown in Section 5.1.4). Since the normal distribution is fully characterized by two parameters only, the standard deviation contains all the information necessary to build measures of dispersion. Any -based quantile can be derived as q^   ^

(5.20)

At the 95 percent confidence level, for instance, we simply multiply the estimated value of ^ by 1.65 to find the 5 percent left-tail quantile. Of course, this method will be strictly valid if the underlying distribution is closely approximated by the normal. When the distribution is suspected to be strongly nonnormal, other methods, such as kernel estimation, also provide estimates of the quantile based on the full distribution.1

5.3.3 Estimation Error in Sample Quantiles For arbitrary distributions, the cth quantile can be determined empirically from the historical distribution as q^ (c) (as shown in Section 5.1.2). There is, as before, some sampling error associated with the statistic. Kendall (1994) reports that the asymptotic standard error of q^ is se(q^ ) 

 c(1  c)

T f(q)2

(5.21)

where T is the sample size, and f() is the probability distribution function evaluated at the quantile q. The effect of estimation error is illustrated in Figure 5–6, where the expected quantile and 95 percent confidence bands are plotted for quantiles from the normal distribution. For the normal distribution, the 5 percent left-tailed interval is centered at 1.65. With T  100, the confidence band is [1.24, 2.04], which is quite large. With 250 observations, which correspond to 1 year of trading days, the band is still [1.38, 1.91]. With T  1250, or 5 years of data, the interval shrinks to [1.52, 1.76]. These intervals widen substantially as one moves to more extreme quantiles. The expected value of the 1 percent quantile is 2.33. With 1 year of data, the band is [1.85, 2.80]. The interval of uncertainty is about 1. Kernel estimation smoothes the empirical distribution by a weighted sum of local distributions. For a further description of kernel estimation methods, see Scott (1992). Butler and Schachter (1998) apply this method to the estimation of VAR.

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FIGURE

5–6

Confidence bands for sample quantiles. 3

Quantile-normal distribution Expected

T=100

T=250

T=1250

2

1

1%

5% Left-tail probability

10%

twice that at the 5 percent interval. Thus sample quantiles are increasingly unreliable as one goes farther in the left tail. As expected, there is more imprecision as one moves to lower lefttail probabilities because fewer observations are involved. This is why VAR measures with very high confidence levels should be interpreted with extreme caution.

5.3.4 Comparison of Methods So far we have developed two approaches for measuring a distribution’s VAR: (1) by directly reading the quantile from the distribution q^ and (2) by calculating the standard deviation and then scaling by the appropriate factor ^ . The issue is: Is any method superior to the other? Intuitively, we may expect the -based approach to be more precise. Indeed, ^ uses information about the whole distribution (in terms of all

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T A B L E 5–3

Confidence Bands for VAR Estimates, Normal Distribution, T  250 VAR Confidence Level c

Exact quantile Confidence band Sample qˆ -Based, ˆ

99%

95%

2.33

1.65

[1.85, 2.80] [2.24, 2.42]

[1.38, 1.91] [1.50, 1.78]

squared deviations around the mean), whereas a quantile uses only the ranking of observations and the two observations around the estimated value. And in the case of the normal distribution, we know exactly how to transform ^ into an estimated quantile using . For other distributions, the value of  may be different, but we should still expect a performance improvement because the standard deviation uses all the sample information. Table 5–3 compares 95 percent confidence bands for the two meth2 ods. The -based method leads to substantial efficiency gains relative to the sample quantile. For instance, at the 95 percent VAR confidence level, the interval around 1.65 is [1.38, 1.91] for the sample quantile; this is reduced to [1.50, 1.78] for ^ , which is much narrower than the previous interval. A number of important conclusions can be derived from these numbers. First, there is substantial estimation error in the estimated quantiles, especially for high confidence levels, which are associated with rare events and hence difficult to verify. Second, parametric methods provide a substantial increase in precision, since the sample standard deviation contains far more information than sample quantiles. Returning to the $15.2 million VAR figure at the beginning of this chapter, we can now assess the precision of this number. Using the parametric approach based on a normal distribution, the standard error of this )  number is se(q^ )    se(^ )  1.65  $9.2 million 1/(2254 $0.67. Therefore, a two-standard-error confidence band around the VAR 2. For extensions to other distributions such as the Student, see Jorion (1996).

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estimate is [$13.8 million, $16.6 million]. This narrow interval should provide reassurance that the VAR estimate is indeed meaningful. 5.4 CONCLUSIONS

In this chapter we have seen how to measure VAR using two alternative methodologies. The general approach is based on the empirical distribution and its sample quantile. The parametric approach, in contrast, attempts to fit a parametric distribution such as the normal to the data. VAR is then measured directly from the standard deviation. Systems such as RiskMetrics are based on a parametric approach. The advantage of such methods is that they are much easier to use and create more precise estimates of VAR. The disadvantage is that they may not approximate well the actual distribution of profits and losses. Users who want to measure VAR from empirical quantiles, however, should be aware of the effect of sampling variation or imprecision in their VAR number. This chapter also has discussed criteria for selection of the confidence level and horizon. On the one hand, if VAR is used simply as a benchmark risk measure, the choice is arbitrary and only needs to be consistent. On the other hand, if VAR is used to decide on the amount of equity capital to hold, the choice is extremely important and can be guided, for instance, by default frequencies for the targeted credit rating.

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CHAPTER

9

VAR Methods

In practice, this works, but how about in theory? Attributed to a French mathematician

Value at risk (VAR) has become an essential component in the toolkit of risk managers because it provides a quantitative measure of downside risk. In practice, the objective should be to provide a reasonably accurate estimate of risk at a reasonable cost. This involves choosing among the various industry standards a method that is most appropriate for the portfolio at hand. To help with this selection, this chapter presents and critically evaluates various approaches to VAR. Approaches to VAR basically can be classified into two groups. The first group uses local valuation. Local-valuation methods measure risk by valuing the portfolio once, at the initial position, and using local derivatives to infer possible movements. The delta-normal method uses linear, or delta, derivatives and assumes normal distributions. Because the deltanormal approach is easy to implement, a variant, called the “Greeks,’’ is sometimes used. This method consists of analytical approximations to first- and second-order derivatives and is most appropriate for portfolios with limited sources of risk. The second group uses full valuation. Fullvaluation methods measure risk by fully repricing the portfolio over a range of scenarios. The pros and cons of local versus full valuation are discussed in Section 9.1. Initially, we consider a simple portfolio that is driven by one risk factor only. This chapter then turns to VAR methods for large portfolios. The best example of local valuation is the delta-normal method, which is explained in Section 9.2. Full valuation is implemented in the historical 205

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simulation method and the Monte Carlo simulation method, which are discussed in Sections 9.3 and 9.4. This classification reflects a fundamental tradeoff between speed and accuracy. Speed is important for large portfolios exposed to many risk factors, which involve a large number of correlations. These are handled most easily in the delta-normal approach. Accuracy may be more important, however, when the portfolio has substantial nonlinear components. An in-depth analysis of the delta-normal and simulation VAR methods is presented in following chapters, as well as a related method, stress testing. Section 9.5 presents some empirical comparisons. Finally, Section 9.6 summarizes the pros and cons of each method. 9.1 LOCAL VERSUS FULL VALUATION

9.1.1 Delta-Normal Valuation Local-valuation methods usually rely on the normality assumption for the driving risk factors. This assumption is particularly convenient because of the invariance property of normal variables: Portfolios of normal variables are themselves normally distributed. We initially focus on delta valuation, which considers only the first derivatives. To illustrate the approaches, take an instrument whose value depends on a single underlying risk factor S. The first step consists of valuing the portfolio at the initial point V0  V(S0)

(9.1)

along with analytical or numerical derivatives. Define 0 as the first partial derivative, or the portfolio sensitivity to changes in prices, evaluated at the current position V0. This would be called modified duration for a fixed-income portfolio or delta for a derivative. For instance, with an atthe-money call,   0.5, and a long position in one option is simply replaced by a 50 percent position in one unit of underlying asset. The portfolio  simply can be computed as the sum of individual deltas. The potential loss in value dV is then computed as V dV   0 dS  0  dS S

(9.2)

which involves the potential change in prices dS. Because this is a linear relationship, the worst loss for V is attained for an extreme value of S.

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If the distribution is normal, the portfolio VAR can be derived from the product of the exposure and the VAR of the underlying variable: VAR   0   VARS   0   (S0)

(9.3)

where  is the standard normal deviate corresponding to the specified confidence level, e.g., 1.645 for a 95 percent confidence level. Here, we take (dS/S) as the standard deviation of rates of changes in the price. The assumption is that rates of changes are normally distributed. Because VAR is obtained as a closed-form solution, this method is called analytical. Note that VAR was measured by computing the portfolio value only once, at the current position V0. For a fixed-income portfolio, the risk factor is the yield y, and the price-yield relationship is dV  D*V dy

(9.4)

where D* is the modified duration. In this case, the portfolio VAR is VAR  (D*V )  ()

(9.5)

where (dy) is now the volatility of changes in the level of yield. The assumption is that changes in yields are normally distributed, although this is ultimately an empirical issue. This method is illustrated in Figure 9–1, where the profit payoff is a linear function of the underlying spot price and is displayed at the upper left side; the price itself is normally distributed, as shown in the right panel. As a result, the profit itself is normally distributed, as shown at the bottom of the figure. The VAR for the profit can be found from the exposure and the VAR for the underlying price. There is a one-to-one mapping between the two VAR measures. How good is this approximation? It depends on the “optionality’’ of the portfolio as well as the horizon. Consider, for instance, a simple case of a long position in a call option. In this case, we can easily describe the distribution of option values. This is so because there is a one-to-one relationship between V and S. In other words, given the pricing function, any value for S can be translated into a value for V, and vice versa. This is illustrated in Figure 9–2, which shows how the distribution of the spot price is translated into a distribution for the option value (in the left panel). Note that the option distribution has a long right tail, due to the upside potential, whereas the downside is limited to the option premium. This shift is due to the nonlinear payoff on the option.

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FIGURE

9–1

Distribution with linear exposures.

Payoff

Frequency

Spot price

Spot price

Frequency

VAR Payoff

Here, the cth quantile for V is simply the function evaluated at the cth quantile of S. For the long-call option, the worst loss for V at a given confidence level will be achieved at S*  S0 S0, and VAR  V(S0) V(S0 S0)

(9.6)

The nonlinearity effect is not obvious, though. It also depends on the maturity of the option and on the range of spot prices over the horizon. The option illustrated here is a call option with 3 months to expiration. To obtain a visible shift in the shape of the option distribution, the volatility was set at 20 percent per annum and the VAR horizon at 2 months, which is rather long. The figure also shows thinner distributions that correspond to a VAR horizon of 2 weeks. Here, the option distribution is indistinguishable from the normal. In other words, the mere presence of options does not necessarily invalidate the delta-normal approach. The quality of the approximation depends on the extent of nonlinearities, which is a function of the

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FIGURE

209

9–2

Transformation of distributions.

Option value

Spot price

Distribution of option values

2 weeks

Distribution of spot prices

2 months

type of options, of their maturities, as well as of the volatility of risk factors and VAR horizon. The shorter the VAR horizon, the better is the deltanormal approximation.

9.1.2 Full Valuation In some situations, the delta-normal approach is totally inadequate. This is the case, for instance, when the worst loss may not be obtained for extreme realizations of the underlying spot rate. Also, options that are near expiration and at-the-money have unstable deltas, which translate into asymmetrical payoff distributions. An example of this problem is that of a long straddle, which involves the purchase of a call and a put. The worst payoff, which is the sum of the premiums, will be realized if the spot rate does not move at all. In general, it is not sufficient to evaluate the portfolio at the two extremes. All intermediate values must be checked.

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The full-valuation approach considers the portfolio value for a wide range of price levels: dV  V(S1) V(S0)

(9.7)

The new values S1 can be generated by simulation methods. The Monte Carlo simulation approach relies on prespecified distributions. For instance, the realizations can be drawn from a normal distribution, dS/S  N(0, 2)

(9.8)

Alternatively, the historical simulation approach simply samples from recent historical data. For each of these draws, the portfolio is priced on the target date using a full-valuation method. This method is potentially the most accurate because it accounts for nonlinearities, income payments, and even timedecay effects that are usually ignored in the delta-normal approach. VAR is then calculated from the percentiles of the full distribution of payoffs. Computationally, this approach is quite demanding because it requires marking-to-market the whole portfolio over a large number of realizations of underlying random variables. To illustrate the result of nonlinear exposures, Figure 9–3 displays the payoff function for a short straddle that is highly nonlinear. The resulting distribution is severely skewed to the left. Further, there is no direct way to relate the VAR of the portfolio to that of the underlying asset. The problem is that these simulation methods require substantial computing time when applied to large portfolios. As a result, methods have been developed to speed up the computations. One example is the grid Monte Carlo approach, which starts by an exact valuation of the portfolio over a limited number of grid points.1 For each simulation, the portfolio value is then approximated using a linear interpolation from the exact values at the adjoining grid points. This approach is especially efficient if exact valuation of the instrument is complex. Take, for instance, a portfolio with one risk factor for which we require 1000 values V(S1). With the grid Monte Carlo method, 10 full valuations at the grid points may be sufficient. In contrast, the full Monte Carlo method would require 1000 full valuations. 1. Picoult (1997) describes this method in more detail.

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FIGURE

211

9–3

Distribution with nonlinear exposures.

Payoff

Frequency

Spot price

Spot price

Frequency

VAR

Payoff

9.1.3 Delta-Gamma Approximations (the “Greeks’’) It may be possible to extend the analytical tractability of the delta-normal method with higher-order terms. We can improve the quality of the linear approximation by adding terms in the Taylor expansion of the valuation function: V 1 2V V dV   dS   dS2  dt    S 2 S2 t 1  dS  dS2 dt    2

(9.9)

where is now the second derivative of the portfolio value, and is the time drift, which is deterministic.

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For a fixed-income portfolio, the price-yield relationship is now 1 dV  (D*V) dy  (CV) dy2    2

(9.10)

where the second-order coefficient C is called convexity and is akin to . Figure 9–4 describes the approximation for a simple position, a long position in a European call option. It shows that the linear model is valid only for small movements around the initial value. For larger movements, the delta-gamma approximation creates a better fit. We use the Taylor expansion to compute VAR for the long-call option in Equation (9.6), which yields VAR  V(S0) V(S0 S0)  V(S0) [V(S0) ( S) 1/2 ( S)2] (9.11)     (S) 1/2 (S)2 This formula is actually valid for long and short positions in calls and puts. If is positive, which corresponds to a net long position in options, the second term will decrease the linear VAR. Indeed, Figure 9–4 FIGURE

9–4

Delta-gamma approximation for a long call.

Current value of option

10

Actual price

5 Delta+gamma estimate Delta estimate

0 90

100 Current price of underlying asset

110

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shows that the downside risk for the option is less than that given by the delta approximation. If is negative, which corresponds to a net short position in options, VAR is increased. This transformation does not apply, unfortunately, to more complex functions V(S), so we have to go back to the Taylor expansion [Equation (9.9)]. The question now is how to deal with the random variables dS and dS2. The simplest method is called the delta-gamma-delta method. Taking the variance of both sides of the quadratic approximation [Equation (9.9)], we obtain 2(dV)  22(dS) (1/2 )22(dS2) 2(1/2 ) cov(dS, dS2)

(9.12)

If the variable dS is normally distributed, all its odd moments are zero, and the last term in the equation vanishes. Under the same assumption, one can show that V(dS2)  2V(dS)2, and the variance simplifies to 2(dV)  22(dS) 1/2[ 2(dS)]2

(9.13)

Assume now that the variables dS and dS2 are jointly normally distributed. Then dV is normally distributed, with VAR given by VAR    (S)2

1/2 ( S22 )2

(9.14)

This is, of course, only an approximation. Even if dS was normal, its square dS2 could not possibly also be normally distributed. Rather, it is a chi-squared variable. A further improvement can be obtained by accounting for the skewness coefficient , as defined in Chapter 4.2 The corrected VAR, using the so-called Cornish-Fisher expansion, is then obtained by replacing  in Equation (9.14) by    1⁄6(2 1)

(9.15)

There is no correction under a normal distribution, for which skewness is zero. When there is negative skewness (i.e., a long left tail), VAR is increased.3 The second method is the delta-gamma–Monte Carlo method, which creates random simulations of the risk factors S and then uses the Taylor 2. Skewness can be computed as   [E(dV3) 3E(dV2)E(dV) 2E(dV)3]/3(dV) using the third moment of dV, which is E(dV3)  (9/2)2 S 44 (15/8) 3S 66. 3. See also Zangari (1996).

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approximation to create simulated movements in the option value. This method is also known as a partial-simulation approach. Note that this is still a local-valuation method because the portfolio is fully valued at the initial point V0 only. The VAR can then be found from the empirical distribution of the portfolio value. In theory, the delta-gamma method could be generalized to many sources of risk. In a multivariate framework, the Taylor expansion is dV(S)  dS 1/2(dS) (dS)   

(9.16)

where dS is now a vector of N changes in market prices,  is a vector of N deltas, and is an N by N symmetrical matrix of gammas with respect to the various risk factors. While the diagonal components are conventional gamma measures, the off-diagonal terms are cross-gammas, or i,j  2V/SiSj. For instance, the delta of options also depends on the implied volatility, which creates a cross-effect. Unfortunately, the delta-gamma method is not practical with many sources of risk because the amount of data required increases geometrically. For instance, with N  100, we need 100 estimates of , 5050 estimates for the covariance matrix , and an additional 5050 for the matrix , which includes second derivatives of each position with respect to each source of risk. In practice, only the diagonal components are considered. Even so, a full Monte Carlo method provides a more direct route to VAR measurement for large portfolios.

9.1.4 Comparison of Methods To summarize, Table 9–1 classifies the various VAR methods. Overall, each of these methods is best adapted to a different environment: ■

■

■

For large portfolios where optionality is not a dominant factor, the delta-normal method provides a fast and efficient method for measuring VAR. For portfolios exposed to a few sources of risk and with substantial option components, the “Greeks’’ method provides increased precision at a low computational cost. For portfolios with substantial option components (such as mortgages) or longer horizons, a full-valuation method may be required.

It should be noted that the linear/nonlinear dichotomy also has implications for the choice of the VAR horizon. With linear models, as we

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T A B L E 9–1

Comparison of VAR Methods

Valuation Method Risk Factor Distribution Analytical Simulated

Local Valuation Delta-normal Delta-gamma-delta Delta-gamma-MC

Full Valuation Not used Monte Carlo (MC) Grid MC Historical

have seen in Chapter 4, daily VAR can be adjusted easily to other periods by simple scaling by a square root of time factor. This adjustment assumes that the position is constant and that daily returns are independent and identically distributed. This time adjustment, however, is not valid for options positions. Since options can be replicated by dynamically changing positions in the underlying assets, the risk of options positions can be dramatically different from the scaled measure of daily risk. Therefore, adjustments of daily volatility to longer horizons using the square root of time factor are valid only when positions are constant and when optionality in the portfolio is negligible. For portfolios with substantial options components, the full-valuation method must be implemented over the desired horizon instead of scaling a daily VAR measure.

9.1.5 An Example: Leeson’s Straddle The Barings’ story provides a good illustration of these various methods. In addition to the long futures positions described in Chapter 7, Leeson also sold options, about 35,000 calls and puts each on Nikkei futures. This position is known as a short straddle and is about delta-neutral because the positive delta from the call is offset by a negative delta from the put, assuming most of the options were at-the-money. Leeson did not deal in small amounts. With a multiplier of 500 yen for the options contract and a 100-yen/$ exchange rate, the dollar exposure of the call options to the Nikkei was delta times $0.175 million. Initially, the market value of the position was zero. The position was

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FIGURE

9–5

Leeson’s straddle.

Payoff ($ millions)

0

Delta estimate

-50

Exact value

-100

Delta+gamma estimate

-150 17,000

18,000

19,000

20,000

21,000

Nikkei index

designed to turn in a profit if the Nikkei remained stable. Unfortunately, it also had an unlimited potential for large losses. Figure 9–5 displays the payoffs from the straddle, using a BlackScholes model with a 20 percent annual volatility. We assume that the options have a maturity of 3 months. At the current index value of 19,000, the delta VAR for this position is close to zero. Of course, reporting a zero delta-normal VAR is highly misleading. Any move up or down has the potential to create a large loss. A drop in the index to 17,000, for instance, would lead to an immediate loss of about $150 million. The graph also shows that the delta-gamma approximation provides increased accuracy. How do we compute the potential loss over a horizon of, say, 1 month? The risks involved are described in Figure 9–6, which plots the frequency distribution of payoffs on the straddle using a full Monte Carlo simulation with 10,000 replications. This distribution is obtained from a revaluation of the portfolio after a month over a range of values for the Nikkei. Each replication uses full valuation with a remaining maturity of

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FIGURE

217

9–6

Distribution of 1-month payoff for straddle.

Frequency

VAR

-300

-200

-100

0

Payoff ($ millions)

2 months (the 3-month original maturity minus the 1-month VAR horizon). The distribution looks highly skewed to the left. Its mean is $1 million, and the 95th percentile is $139 million. Hence the 1-month 95 percent VAR is $138 million. How does the “Greeks’’ method fare for this portfolio? First, let us examine the delta-gamma-delta approximation. The total gamma of the position is the exposure times the sum of gamma for a call and put, or $0.175 million  0.000422  $0.0000739 million. Over a 1-month horizon, the standard deviation of the Nikkei is S  19,000  20 percent/12   1089. Ignoring the time drift, the VAR is, from Equation (9.13), [ (S) ]  1.65  ($0.00 00739 million  1089 )    1.65  $62 million  $102 million

VAR  

1  2

2 2

1  2

2 2

This is substantially better than the delta-normal VAR of zero, which could have fooled us into believing the position was riskless. Using the Cornish-Fisher expansion and a skewness coefficient of 2.83, we obtain a correction factor of   1.65 16 (1.652 1)( 2.83)

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 2.45. The refined VAR measure is then 2.45  $62 million  $152 million, much closer to the true value of $138 million. Finally, we can turn to the delta-gamma–Monte Carlo approach, which consists of using the simulations of S but valuing the portfolio on the target date using only the partial derivatives. This yields a VAR of $128 million, not too far from the true value. This variety of methods shows that the straddle had substantial downside risk. And indeed the options position contributed to Barings’ fall. As January 1995 began, the historical volatility on the Japanese market was very low, around 10 percent. At the time, the Nikkei was hovering around 19,000. The options position would have been profitable if the market had been stable. Unfortunately, this was not so. The Kobe earthquake struck Japan on January 17 and led to a drop in the Nikkei to 18,000, shown in Figure 9–7. To make things worse, options became more expensive as market volatility increased. Both the long futures and the straddle positions lost money. As losses ballooned, Leeson increased his exposure in a desperate attempt to recoup the losses, but to no avail. On February 27, the Nikkei dropped further to 17,000. Unable to meet the mounting margin calls, Barings went bust. FIGURE

9–7

The Nikkei’s fall.

Nikkei index 20,000

Kobe earthquake Barings collapse

19,000

18,000

17,000

16,000 Jan

Feb

Mar

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9.2 DELTA-NORMAL METHOD

9.2.1 Implementation If the portfolio consisted of only securities with jointly normal distributions, the measurement of VAR would be relatively simple. The portfolio return is N

Rp,t 1   wi,tRi,t 1

(9.17)

i1

where the weights wi,t are indexed by time to recognize the dynamic nature of trading portfolios. Since the portfolio return is a linear combination of normal variables, it is also normally distributed. Using matrix notations, the portfolio variance is given by 2(Rp,t 1)  wt t 1wt

(9.18)

where t 1 is the forecast of the covariance matrix over the VAR horizon. The problem is that VAR must be measured for large and complex portfolios that evolve over time. The delta-normal method, which is explained in much greater detail in a subsequent chapter, simplifies the process by ■ ■

■ ■ ■

Specifying a list of risk factors Mapping the linear exposure of all instruments in the portfolio onto these risk factors Aggregating these exposures across instruments Estimating the covariance matrix of the risk factors Computing the total portfolio risk

This mapping produces a set of exposures xi,t aggregated across all instruments for each risk factor and measured in dollars. The portfolio VAR is then xt t 1 xt VAR  

(9.19)

Within this class of models, two methods can be used to measure the variance-covariance matrix . It can be solely based on historical data using, for example, a model that allows for time variation in risk. Alternatively, it can include implied risk measures from options. Or it can

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FIGURE

9–8

Delta-normal method. Historical data

Option data

Volatility, correlation model

Option model

Estimated future volatilities, correlations

Securities delta model

Delta valuation

Delta positions

Estimated value changes

use a combination of both. As we saw in the preceding chapter, optionsimplied measures of risk are superior to historical data but are not available for every asset, let alone for pairs of assets. Figure 9–8 details the steps involved in this approach.

9.2.2 Advantages The delta-normal method is particularly easy to implement because it involves a simple matrix multiplication. It is also computationally fast, even with a very large number of assets, because it replaces each position by its linear exposure. As a parametric approach, VAR is easily amenable to analysis, since measures of marginal and incremental risk are a by-product of the VAR computation.

9.2.3 Problems The delta-normal method can be subject to a number of criticisms. A first problem is the existence of fat tails in the distribution of returns on most financial assets. These fat tails are particularly worrisome precisely because VAR attempts to capture the behavior of the portfolio return in the

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left tail. In this situation, a model based on a normal distribution would underestimate the proportion of outliers and hence the true value at risk. As discussed in Chapter 8, some of these fat tails can be explained in terms of time variation in risk. However, even after adjustment, there are still too many observations in the tails. A simple ad hoc adjustment consists of increasing the parameter  to compensate, as is explained in Chapter 5. Another problem is that the method inadequately measures the risk of nonlinear instruments, such as options or mortgages. Under the deltanormal method, options positions are represented by their “deltas’’ relative to the underlying asset. As we have seen in the preceding section, asymmetries in the distribution of options are not captured by the deltanormal VAR. Lest we lead you into thinking that this method is inferior, we will now show that alternative methods are no panacea because they involve a quantum leap in difficulty. The delta-normal method is computationally easy to implement. It only requires the market values and exposures of current positions, combined with risk data. Also, in many situations, the delta-normal method provides adequate measurement of market risks. 9.3 HISTORICAL SIMULATION METHOD

9.3.1 Implementation The historical simulation method provides a straightforward implementation of full valuation (Figure 9–9). It consists of going back in time, such as over the last 250 days, and applying current weights to a time-series of historical asset returns: N

Rp,k   wi,tRi,k

k  1, . . ., t

(9.20)

i1

Note that the weights wt are kept at their current values. This return does not represent an actual portfolio but rather reconstructs the history of a hypothetical portfolio using the current position. The approach is sometimes called bootstrapping because it involves using the actual distribution of recent historical data (without replacement). More generally, full valuation requires a set of complete prices, such as yield curves, instead of just returns. Hypothetical future prices for scenario k are obtained from applying historical changes in prices to the current level of prices: S*i,k  Si,0 Si,k

i  1, . . ., N

(9.21)

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FIGURE

9–9

Historical simulation method. Historical returns

Securities model

Full valuation

Portfolio positions

Distribution of values

A new portfolio value V*p,k is then computed from the full set of hypothetical prices, perhaps incorporating nonlinear relationships V*k  V(S*i,k). Note that to capture vega risk, due to changing volatilities, the set of prices can incorporate implied volatility measures. This creates the hypothetical return corresponding to simulation k: V*k V0  Rp,k   V0

(9.22)

VAR is then obtained from the entire distribution of hypothetical returns, where each historical scenario is assigned the same weight of (1/t). As always, the choice of the sample period reflects a tradeoff between using longer and shorter sample sizes. Longer intervals increase the accuracy of estimates but could use irrelevant data, thereby missing important changes in the underlying process.

9.3.2 Advantages This method is relatively simple to implement if historical data have been collected in-house for daily marking-to-market. The same data can then be stored for later reuse in estimating VAR.

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Historical simulation also short-circuits the need to estimate a covariance matrix. This simplifies the computations in cases of portfolios with a large number of assets and short sample periods. All that is needed is the time series of the aggregate portfolio return. The method also deals directly with the choice of horizon for measuring VAR. Returns are simply measured over intervals that correspond to the length of the horizon. For instance, to obtain a monthly VAR, the user would reconstruct historical monthly portfolio returns over, say, the last 5 years. By relying on actual prices, the method allows nonlinearities and nonnormal distributions. Full valuation is obtained in the simplest fashion: from historical data. The method captures gamma, vega risk, and correlations. It does not rely on specific assumptions about valuation models or the underlying stochastic structure of the market. Perhaps most important, it can account for fat tails and, because it does not rely on valuation models, is not prone to model risk. The method is robust and intuitive and, as such, is perhaps the most widely used method to compute VAR.

9.3.3 Problems On the other hand, the historical simulation method has a number of drawbacks. First, it assumes that we do have a sufficient history of price changes. To obtain 1000 independent simulations of a 1-day move, we require 4 years of continuous data. Some assets may have short histories, or there may not be a record of an asset’s history. Only one sample path is used. The assumption is that the past represents the immediate future fairly. If the window omits important events, the tails will not be well represented. Vice versa, the sample may contain events that will not reappear in the future. And as we have demonstrated in Chapter 8, risk contains significant and predictable time variation. The simple historical simulation method presented here will miss situations with temporarily elevated volatility.4 Worse, historical simulation will be very slow to incorporate structural breaks, which are handled more easily with an analytical methods such as RiskMetrics. 4. A simple method to allow time variation in risk proceeds as follows: First, fit a time-series model to the conditional volatility and construct historical scaled residuals. Second, perform a historical simulation on these residuals. Third, apply the most recent volatility forecast to the scaled portfolio volatility. For applications, see Hull and White (1998).

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This approach is also subject to the same criticisms as the movingwindow estimation of variances. The method puts the same weight on all observations in the window, including old data points. The measure of risk can change significantly after an old observation is dropped from the window.5 Likewise, the sampling variation of the historical simulation VAR will be much greater than for an analytical method. As is pointed out in Chapter 5, VAR is only a statistical estimate and may be subject to much estimation error if the sample size is too short. For instance, a 99 percent daily VAR estimated over a window of 100 days produces only one observation in the tail, which necessarily leads to an imprecise VAR measure. Thus very long sample paths are required to obtain meaningful quantiles. The dilemma is that this may involve observations that are not relevant. A final drawback is that the method quickly becomes cumbersome for large portfolios with complicated structures. In practice, users adopt simplifications such as grouping interest rate payoffs into bands, which considerably increases the speed of computation. Regulators also have adopted such a “bucketing’’ approach. But if too many simplifications are carried out, such as replacing assets by their delta equivalents, the benefits of full valuation can be lost. 9.4 MONTE CARLO SIMULATION METHOD

9.4.1 Implementation Monte Carlo (MC) simulations cover a wide range of possible values in financial variables and fully account for correlations. MC simulation is developed in more detail in a later chapter. In brief, the method proceeds in two steps. First, the risk manager specifies a stochastic process for financial variables as well as process parameters; parameters such as risk and correlations can be derived from historical or options data. Second, fictitious price paths are simulated for all variables of interest. At each horizon considered, the portfolio is marked-to-market using full valuation as in the historical simulation method, V*k  V(S*i,k). Each of these “pseudo’’ realizations is then used to compile a distribution of returns, 5. To alleviate this problem, Boudoukh et al. (1998) propose a scheme whereby each observation Rk is assigned a weight wk that declines as it ages. The distribution is then obtained from ranking the Rk and cumulating the associated weights to find the selected confidence level.

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FIGURE

225

9–10

Monte Carlo method. Historical/implied data Model parameters

Stochastic model

Future rates

Securities model

Full valuation

Portfolio positions

Distribution of values

from which a VAR figure can be measured. The method is summarized in Figure 9–10. The Monte Carlo method is thus similar to the historical simulation method, except that the hypothetical changes in prices Si for asset i in Equation (9.20) are created by random draws from a prespecified stochastic process instead of sampled from historical data.

9.4.2 Advantages Monte Carlo analysis is by far the most powerful method to compute VAR. It can account for a wide range of exposures and risks, including nonlinear price risk, volatility risk, and even model risk. It is flexible enough to incorporate time variation in volatility, fat tails, and extreme scenarios. Simulations generate the entire pdf, not just one quantile, and can be used to examine, for instance, the expected loss beyond a particular VAR. MC simulation also can incorporate the passage of time, which will create structural changes in the portfolio. This includes the time decay of options; the daily settlement of fixed, floating, or contractually specified cash flows; or the effect of prespecified trading or hedging strategies.

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These effects are especially important as the time horizon lengthens, which is the case for the measurement of credit risk.

9.4.3 Problems The biggest drawback of this method is its computational time. If 1000 sample paths are generated with a portfolio of 1000 assets, the total number of valuations amounts to 1 million. In addition, if the valuation of assets on the target date involves itself a simulation, the method requires a “simulation within a simulation.’’ This quickly becomes too onerous to implement on a frequent basis. This method is the most expensive to implement in terms of systems infrastructure and intellectual development. The MC simulation method is relatively onerous to develop from scratch, despite rapidly falling prices for hardware. Perhaps, then, it should be purchased from outside vendors. On the other hand, when the institution already has in place a system to model complex structures using simulations, implementing MC simulation is less costly because the required expertise is in place. Also, these are situations where proper risk management of complex positions is absolutely necessary. Another potential weakness of the method is model risk. MC relies on specific stochastic processes for the underlying risk factors as well as pricing models for securities such as options or mortgages. Therefore, it is subject to the risk that the models are wrong. To check if the results are robust to changes in the model, simulation results should be complemented by some sensitivity analysis. Finally, VAR estimates from MC simulation are subject to sampling variation, which is due to the limited number of replications. Consider, for instance, a case where the risk factors are jointly normal and all payoffs linear. The delta-normal method will then provide the correct measure of VAR, in one easy step. MC simulations based on the same covariance matrix will give only an approximation, albeit increasingly good as the number of replications increases. Overall, this method is probably the most comprehensive approach to measuring market risk if modeling is done correctly. To some extent, the method can even handle credit risks. This is why a full chapter is devoted to the implementation of Monte Carlo simulation methods.

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9.5 EMPIRICAL COMPARISONS

It is instructive to compare the VAR numbers obtained from the three methods discussed. Hendricks (1996), for instance, calculated 1-day VARs for randomly selected foreign currency portfolios using a delta-normal method based on fixed windows of equal weights and exponential weights as well as a historical simulation method. Table 9–2 summarizes the results, which are compared in terms of percentage of outcomes falling within the VAR forecast. The middle column shows that all methods give a coverage that is very close to the ideal number, which is the 95 percent confidence level. At the 99 percent confidence level, however, the delta-normal methods seem to underestimate VAR slightly, since their coverage falls short of the ideal 99 percent. Hendricks also reports that the delta-normal VAR measures should be increased by about 9 to 15 percent to achieve correct coverage. In other words, the fat tails in the data could be modeled by choosing a

T A B L E 9–2

Empirical Comparison of VAR Methods: Fraction of Outcomes Covered

Method Delta-normal Equal weights over 50 days 250 days 1250 days Delta-normal Exponential weights:   0.94   0.97   0.99 Historical simulation Equal weights over 125 days 250 days 1250 days

95% VAR

99% VAR

95.1% 95.3% 95.4%

98.4% 98.4% 98.5%

94.7% 95.0% 95.4%

98.2% 98.4% 98.5%

94.4% 94.9% 95.1%

98.3% 98.8% 99.0%

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distribution with a greater  parameter. A Student t distribution with four to six degrees of freedom, for example, would be appropriate. As important, when the VAR number is exceeded, the tail event is, on average, 30 to 40 percent greater than the risk measure. In some instances, it is several times greater. As Hendricks states, “This makes it clear that VAR measures—even at the 99th percentile—do not bound possible losses.’’ This empirical analysis, however, examined positions with linear risk profiles. The delta-normal methods could prove less accurate with options positions, although it should be much faster. Pritsker (1997) examines the tradeoff between speed and accuracy for a portfolio of options. Table 9–3 reports the accuracy of various methods, measured as the mean absolute percentage error in VAR, as well as their computational times. The table shows that the delta method, as expected, has the highest average absolute error, at 5.34 percent of the true VAR. It is also by far the fastest method, with an execution time of 0.08 seconds. At the other end, the most accurate method is the full Monte Carlo, which comes arbitrarily close to the true VAR, but with an average run time of 66 seconds. In between, the delta-gamma-delta, delta-gamma–Monte Carlo, and grid Monte Carlo methods offer a tradeoff between accuracy and speed. An interesting but still unresolved issue is, How would these approximations work in the context of large, diversified bank portfolios? There is very little evidence on this point. The industry initially seemed to prefer the analytical covariance approach due to its simplicity. With the rapidly decreasing cost of computing power, however, there is now T A B L E 9–3

Accuracy and Speed of VAR Methods: 99 Percent VAR for Option Portfolios

Method Delta Delta-gamma-delta Delta-gamma-MC Grid Monte Carlo Full Monto Carlo

Accuracy: Mean Absolute Error in VAR (%)

Speed: Computation Time, s

5.34 4.72 3.08 3.07 0

0.08 1.17 3.88 32.19 66.27

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229

a marked trend toward the generalized use of historical simulation methods. 9.6 SUMMARY

We can distinguish a number of different methods to measure VAR. At the most fundamental level, they separate into local (or analytical) valuation and full valuation. This separation reflects a tradeoff between speed of computation and accuracy of valuation. Delta models can use parameters based on historical data, such as those implemented by RiskMetrics, or on implied data, where volatilities are derived from options. Both methods generate a covariance matrix, to which the “delta’’ or linear positions are applied to find the portfolio VAR. Among full-valuation models, the historical simulation method is the easiest to implement. It simply relies on historical data for securities valuation and applies the most current weight to historical prices. Finally, the most complete model, but also the most difficult to implement, is the Monte Carlo simulation approach, which imposes a particular stochastic process on the financial variables of interest, from which various sample paths are simulated. Full valuation for each sample path generates a distribution of portfolio values. Table 9–4 describes the pros and cons of each method. The choice of the method largely depends on the composition of the portfolio. For portfolios with no options (nor embedded options) and whose distributions are close to the normal pdf, the delta-normal method may well be the best choice. VAR will be relatively easy to compute, fast, and accurate. In addition, it is not too prone to model risk (due to faulty assumptions or computations). The resulting VAR is easy to explain to management and to the public. Because the method is analytical, it allows easy analysis of the VAR results using marginal and component VAR measures. For portfolios with options positions, however, the method may not be appropriate. Instead, users should turn to a full-valuation method. The second method, historical simulation, is also relatively easy to implement and uses actual, full valuation of all securities. However, its typical implementation does not account for time variation in risk, and the method relies on a narrow window only. In theory, the Monte Carlo approach can alleviate all these technical difficulties. It can incorporate nonlinear positions, nonnormal distributions, implied parameters, and even user-defined scenarios. The price to pay for this flexibility, however, is heavy. Computer and data require-

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T A B L E 9–4

Comparison of Approaches to VAR

Features Positions Valuation Distribution Shape Time-varying Implied data Extreme events Use correlations VAR precision Implementation Ease of computation Accuracy Communicability VAR analysis Major pitfalls

Delta-Normal

Historical Simulation

Monto Carlo Simulation

Linear

Full

Full

Normal Yes Possible Low probability Yes Excellent

Actual Possible No In recent data Yes Poor with short window

General Yes Possible Possible Yes Good with many iterations

Yes Depends on portfolio Easy Easy, analytical Nonlinearities, fat tails

Intermediate Yes

No Yes

Easy More difficult Time-variation in risk, unusual events

Difficult More difficult Model risk

ments are a quantum step above the other two approaches, model risk looms large, and value at risk loses its intuitive appeal. As the price of computing power continues to fall, however, this method is bound to take on increasing importance. In practice, all these methods are used. A recent survey by Britain’s Financial Services Authority has revealed that 42 percent of banks use the covariance matrix approach, 31 percent use historical simulation, and 23 percent use the Monte Carlo approach. The delta-normal method, which is the easiest to implement, appears to be the most widespread. All these methods present some advantages. They are also related. Monte Carlo analysis of linear positions with normal returns, for instance, should yield the same result as the delta-normal method. Perhaps the best lesson from this chapter is to check VAR measures with different methodologies and then to analyze the sources of differences.

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CHAPTER

14

Liquidity Risk

LTCM then faced severe market liquidity problems when its investments began losing value and the fund attempted to unwind some of its positions. President’s Working Group on Financial Markets, 1999

T

raditional value at risk (VAR) models rely on market prices, since marking-to-market is widely viewed as the ultimate discipline. VAR, however, assumes that the portfolio is “frozen” over the horizon and that market prices represent achievable transaction prices. This marking-to-market approach is adequate to quantify and control risk but may be more questionable if VAR is supposed to represent the worst loss over a liquidation period. The question is how VAR can be adapted to deal with liquidity considerations. As we saw in Chapter 1, liquidity risk can be grouped into asset liquidity and funding liquidity risk. The former relates to the risk that the liquidation value of the assets differs significantly from the current mark-to-market value. The latter refers to the risk that an institution could run out of cash and is unable to raise new funds to meet its payment obligations, which could lead to formal default. Thus liquidity considerations should be viewed in the context of liabilities. This chapter discusses recent developments that adapt traditional VAR measures to liquidity considerations. First, Section 14.1 analyzes asset and funding liquidity risk in some detail. Asset liquidity risk can be evaluated by the price impact of the liquidation. Funding liquidity risk, in contrast, deals with cash resources as well as potential cash requirements. Next, Section 14.2 provides insight into asset liquidity risk by comparing liquidation strategies. One strategy is immediate liquidation, which 339

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may incur very large costs due to the price impact but insulates the portfolio from price volatility risk. Another strategy is that of liquidation at a constant rate, which minimizes trading costs but leaves the portfolio exposed to volatility risk. Optimal execution strategies best balance these low-cost and volatility requirements. Section 14.3 then discusses measures of funding liquidity risk proposed by the Counterparty Risk Management Policy Group (CRMPG). These measures show that even though an institution can have zero traditional VAR, different swap credit terms can generate very different cash requirements. Next, Section 14.4 is devoted to an analysis of the Long-Term Capital Management (LTCM) debacle. LTCM failed because of its lack of diversification and funding liquidity risk and asset liquidity risk, which were simply due to its sheer size. Finally, Section 14.5 provides some concluding comments.

14.1 DEFINING LIQUIDITY RISK

14.1.1 Asset Liquidity Risk Asset liquidity risk, sometimes called market/product liquidity risk, arises when a transaction cannot be conducted at prevailing market prices due to the size of the position relative to normal trading lots. Liquidity can be measured by a price-quantity function. This is also known as the market impact effect. Highly liquid assets, such as major currencies or Treasury bonds, are characterized by deep markets, where positions can be offset with very little price impact. Thin markets, such as exotic OTC derivatives contracts or emerging market equities, are those where any transaction can quickly affect prices. This price function is illustrated in Figure 14–1.1 The starting point is the current mid price, which is the average of the bid and ask quotes and is generally used to mark the portfolio to market. The bid-ask spread is $0.25, valid up to some limit, say, 10,000 shares. This is sometimes called the normal market size. For quantities beyond this point, however, the sale price is a decreasing function of the quantity, reflecting the price pressure required to clear the market. Conversely for the purchase price. 1. In what follows, we ignore the fixed component of trading costs, i.e., commissions and taxes.

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CHAPTER 14 Liquidity Risk

FIGURE

341

14–1

Price-quantity function.

Price $100.5

Ask price $100.0

Bid price

$99.5

0 10,000

50,000 Quantity transacted

100,000

The relationship is assumed to be linear, although it could take another shape with parameters that vary across assets and, possibly, across time. In this case, selling 100,000 shares would incur a cost of about 60 basis points per share. In practice, the size of the sale would have to be measured relative to some metric such as the median daily trading volume. For a widely traded stock such as IBM, for instance, selling 4 percent of the daily trading volume incurs a cost of about 60 basis points. Here, liquidity can be measured usefully by the price-quantity function combined with the existing position of the institution. If the position is below 10,000 shares, then market liquidity is not a major issue. In contrast, if the institution holds a number of shares worth several days of normal trading volume, liquidity should be of primary concern. In addition to varying across assets, liquidity is also a function of prevailing market conditions. This is more worrying, because markets seem to go through regular bouts of liquidity crises. Most notably, liquidity in bond markets dried up during the summer of 1998 as uncertainty about defaults led to a “flight to quality,” i.e., increases in the prices of Treasuries relative to those of other bonds. A similar experience occurred

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during the 1994 bond market debacle, at which time it became quite difficult to deal in eurobonds or mortgage-backed securities. Traditionally, asset liquidity risk has been controlled through position limits. The goal of position limits is to limit the exposure to a single instrument, even if it provides diversification of market risk, in order to avoid a large market impact in case of forced liquidation.

14.1.2 Funding Liquidity Risk Cash-flow/funding liquidity risk refers to the inability to meet payment obligations when the institution runs out of cash and is unable to raise additional funds. Often, this forces unwanted liquidation of the portfolio. It should be noted that funding risk arises from the use of leverage, whereby institutions borrow to expand their assets. This type of risk must be analyzed in the context of the asset and liability structure of the institution. Looking at the asset side, potential demands on cash resources depend on ■ ■

■

Variation margin requirements, due to marking-to-market Mismatch in the timing of collateral payments, due to the fact that even if an institution is perfectly matched in terms of market risk, it may be forced to make a payment on a position without having yet received an offsetting payment on a hedge Changes in collateral requirements, due to requests by lenders to increase the amount of collateral they require

Here, some words of explanation are in order. An example of mismatch in cash flows is an institution that has two economically hedged positions structured with different credit terms, such as a one-way markto-market swap and a two-way mark-to-market swap. In the former, the institution is required to make payments if the position loses money; it will not, however, receive intermediate payments if the position gains. In contrast, under two-way swaps, payments can be made or received if the position loses or gains money. Because of the asymmetry in the hedge, the institution will be subject to mismatches in the timing of collateral payments if the first swap loses money. Even with two-way mark-tomarket agreements, there can be some uncertainty in the cash-flow payments due to operational errors or discrepancies in the valuation of the swaps.

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343

Next, changes in collateral requirements can arise due to requests by lenders to raise their haircuts. Normally, brokers require collateral that is worth slightly more than the cash loaned, by an amount known as a haircut, designed to provide a buffer against decreases in the collateral value. Unless contract terms forbid it, brokers can raise their haircuts if the market becomes more volatile, creating demands on cash. These examples provide yet another illustration of the complex interaction between different types of risks. Marking-to-market has been developed primarily to control credit risk. The problem is that it may create another risk, cash-flow risk. Further, if the institution does not carry enough cash to meet its margin calls, it may be forced to liquidate holdings at depressed prices, thereby creating asset liquidity risk. Looking at the liability side is also important, however. The institution may be able to meet margin calls by raising funds from another source, such as a line of credit or new equity issues. The problem is that it may be difficult to raise new funds precisely when the institution is faring badly and needing it most. Conversely, the institution also must evaluate the likelihood of redemptions, or cash requests from equity holders or debt holders. However, this is most likely to occur when the institution appears most vulnerable, thereby transforming what could be a minor problem into a crisis. It is also important to avoid debt covenants or options that contain “triggers” that would force early redemption of the borrowed funds. 14.2 DEALING WITH ASSET LIQUIDITY RISK

Trading returns typically are measured from midmarket prices. This may be adequate for measuring daily profit and loss (P&L) but may not represent the actual fall in value if a large portfolio were to be liquidated. The question is how to measure risk more properly, which can give insights into how to manage this risk. Traditional adjustments are done on an ad hoc basis. Liquidity risk can be loosely factored into VAR measures by ensuring that the horizon is at least greater than an orderly liquidation period. Generally, the same horizon is applied to all asset classes, even though some may be more liquid than others. Sometimes, longer liquidation periods for some assets are taken into account by artificially increasing the volatility. For instance, one could mix a large position in the dollar/yen with another one in the dollar/Polish

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zloty, both of which have an annual volatility of 10 percent, by artificially increasing the second volatility in the VAR computations.

14.2.1 Bid-Ask Spread Cost More formally, one can focus on the various components of liquidation costs. The first and most easily measurable is the quoted bid-ask spread, defined in relative terms, that is, [P(ask)  P(bid)] S   P(mid)

(14.1)

Table 14–1 provides typical spreads. We see that spreads vary from a low of about 0.05 percent for major currencies, large U.S. stocks, and onthe-run Treasuries to much higher values when dealing with less liquid currencies, stocks, and bonds. Treasury bills are in a class of their own, with extremely low spreads. These spreads are only indicative, since they depend on market conditions. Also, marketmakers may be willing to trade within the spread. T A B L E 14–1

Typical Spreads and Volatility

Volatility, % Asset Currencies Major (euro, yen, etc.) Emerging (floating) Bonds On-the-run Treasuries Off-the-run Treasuries Corporates Treasury bills Stocks U.S. Average, NYSE Average, all countries

Spread, % (Bid-Ask)

Daily

Annual

0.05–0.20 0.50–1.00

0.3–1.0 0.3–1.9

5–15 5–30

0.03 0.06–0.20 0.10–1.00 0.003–0.02

0.0–0.7 0.0–0.7 0.0–0.7 0.0–0.1

0–11 0–11 0–11 0–11

0.05–5.00 0.20 0.40

1.3–3.8 1.0 1.0–1.9

20–60 15 15–30

Note: Author’s calculations. Cost of trades excludes broker commissions and fees. See also Institutional Investor (November 1999).

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At this point, it is useful to review briefly the drivers of these spreads. According to market microstructure theory, spreads reflect three different types of costs: order-processing costs, asymmetric information costs, and inventory-carrying costs. Order-processing costs cover the cost of providing liquidity services and reflect the cost of trading, the volume of transaction, the state of technology, and competition. With fixed operating costs, these orderprocessing costs should decrease with transaction volumes. Asymmetric information costs reflect the fact that some orders may come from informed traders, at the expense of marketmakers who can somewhat protect themselves by increasing the spread. Finally, inventory-carrying costs are due to the cost of maintaining open positions, which increases with higher price volatility, higher interest rate carrying costs, and lower trading activity or turnover. If the spread were fixed, one could simply construct a liquidityadjusted VAR from the traditional VAR by adding a term: LVAR  VAR  L1  (W)  1/2(WS)

(14.2)

where W is the initial wealth, or portfolio value. For instance, if we have $1 million invested in a typical stock, with a daily volatility of   1 percent and spread of S  0.25 percent, the 1-day LVAR at the 95 percent confidence level would be LVAR  ($1 million  1.645  0.01)  1/2($1 million  0.0025)  $16,450  $1250  $17,700 Here, the correction factor is relatively small, accounting for 7 percent of the total. This adjustment can be repeated for all assets in the portfolio, leading to a series of add-ons, 1⁄2 i | Wi | Si. This sequence of positive terms increases linearly with the number of assets, while the usual VAR benefits from diversification effects. Thus the relative importance of the correction factor will be greater for large portfolios. A slightly more general approach is proposed by Bangia et al. (1999), who consider the uncertainty in the spread. They characterize the distribution by its mean S and standard deviation S. The adjustment considers the worst increase in the spread at some confidence level: 1   S)] LVAR  VAR  L2  (W)  [W(S 2

(14.3)

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At the portfolio level, one could theoretically take into account correlations between spreads. In practice, summing the individual worst spreads provides a conservative measure of the portfolio worst spread. Although this approach has the merit of considering some transaction costs, it is not totally satisfactory. It only looks at the bid-ask spread component of these costs, which may be appropriate for a small portfolio but certainly not when liquidation can affect market prices. Also, if the decision is taken to liquidate the portfolio immediately, the VAR horizon should be very short. What is needed is a model that takes into account the cost and benefit of trading strategies.

14.2.2 Trading Strategies The extension based on bid-ask spreads, while an improvement over traditional VAR calculation, very much ignores the market impact factor, which can be significant. To some extent, this can be mitigated by suitable execution strategies. These should be taken into account when computing a liquidity-adjusted VAR. To simplify, let us assume a linear price-quantity function. For a sale, P(q)  P0(1  kq)

(14.4)

7

Assume that P0  $100 and k  0.5  10 . Say that we start with a position of 1 million shares of the stock. If we liquidate all at once, the price drop will be P0 kq  $100  (0.5  107)  1,000,000  $5, leading to a total price impact of $5 million. In contrast, we could decide to work the order through at a constant rate of 200,000 shares over 5 days. Define n as the number of days to liquidation. In the absence of other price movements, the price drop will be $1.0, leading to a total price impact of $1 million, much less than before. Immediate liquidation creates quadratic costs: C1(W )  q  [P0  P(q)]  kq2P0

(14.5)

whereas uniform liquidation creates lower costs: C2(W )  q  [P0  P(q/n)]  k(q2/n)P0

(14.6)

The drawback of liquidating more slowly is that the portfolio remains exposed to price risks over a longer period. The position profiles are illustrated in Figure 14–2. Under the immediate sale, the position is liquidated before the end of the next day, leading to a high cost but min-

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CHAPTER 14 Liquidity Risk

FIGURE

347

14–2

Profile of execution strategies. 1,000,000

Position

Uniform sales: minimum impact 500,000

Minimum L-VAR Immediate sale: minimum risk 0

0

1

2 3 4 Calendar time (days)

5

6

imum risk. Under the uniform sale, the position is sold off in equal-sized lots, leading to low costs but high risk. The key is to choose a strategy that offers the best cost-risk tradeoff. To analyze the risk profile of these strategies, define  as the daily volatility of the share price, in dollars. We assume that sales are executed at the close of the business day, in one block. Hence, for the immediate sale, the price risk or variance of wealth is zero, V1(W )  0. For the uniform sale, the portfolio variance can be computed assuming independent returns over n days as



1 V2(W )  2q2 1   n

  2



2





1 2      1  (n  1)  n 1 1 1  2q2 n  1   1   (14.7) 3 n 2n

1  1  2  n

 





For example, with n  5, the correction factor between braces is 1.20. Thus constant liquidation over 5 days is equivalent to the markingto-market risk of a position held over 1.2 days. It is interesting to note

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PART 3 Value-at-Risk Systems

that the 10-day fixed horizon dictated by the Basel Committee is equivalent to a constant liquidation over 31 days. Execution strategies need not be limited to these two extreme cases. More generally, we can choose a strategy x that leads to an optimal tradeoff between execution costs and price risk: minx[Cx(W )  Vx(W )]

(14.8)

where reflects the aversion to price risk. Lawrence and Robinson (1997), for instance, propose a simple solution, which is to minimize over n, using Equations (14.7) and (14.6). Almgren and Chriss (1999) provide a more complete closed-form solution for efficient execution strategies in the cost-risk plane. An optimal trajectory is described in Figure 14–2. Note that the strategy can be described by its half-life, which is the time required to liquidate half the portfolio. For the optimal strategy here, this takes 1 day. This leads to the concept of liquidity-adjusted VAR, or implementation shortfall, which is )  C(W ) LVAR   V(W

(14.9)

where  corresponds to the confidence level c. In other words, this LVAR measure takes into account not only liquidation costs but also the best execution strategy such that LVAR will not be exceeded more than a fraction c of the time. Figure 14–3 compares various VAR measures for different speeds of execution. The “static” 1-day and 5-day VARs correspond to the usual mark-to-market VAR measures with 30 percent annual volatility. Under these conditions, the daily volatility is 1.9 percent, and the 1-day VAR is $3.1 million for this $100 million portfolio. With a 25-basis-point spread and no market impact, the spread adjustment is small, at $125,000 only. This, however, ignores price impact. Instead, the LVAR measure incorporates the total execution cost and price risk components in a consistent fashion. As we extend the length of liquidation, the execution cost component decreases, but the price risk component increases. Here, the total LVAR is minimized at a half-life of 1 day. In this case, a 5-day static VAR would provide a conservative measure of liquidation VAR. The real benefit of this approach is that it draws attention to market impact effects in portfolio liquidation. It also illustrates that execution strategies should account for the tradeoff between execution costs and price volatility.

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CHAPTER 14 Liquidity Risk

FIGURE

349

14–3

Liquidity-adjusted VAR. 10

VAR ($ millions)

9 8

Static-5 days

7 6 5

LVAR

4 3 2

Price risk

1

Execution cost

Static-1 day

0

0

1

10 Length of liquidation (half-life, in days)

For instance, there are other ways to implement the liquidation. In the case of stock portfolios, for example, the portfolio manager could cut its price risk by immediately putting in place a hedge with stock index futures. In this case, the remaining price risk is “specific” to the security. Orders to sell could then be transmitted so as to minimize their price impact.

14.2.3 Practical Issues In practice, the computational requirements to adjust the conventional VAR numbers are formidable. The method requires a price-quantity function for all securities in the portfolio. Combined with the portfolio position, this yields an estimate of the price impact of a liquidation, as well as the optimal time to liquidation. Table 14–2 provides an example of such an analysis, as provided by Morgan Stanley (MS) for a four-country $50 million equity portfolio. The data for Switzerland are expanded at the individual stock level. To estimate the total impact cost, we need information about the historical bid-ask spreads, the median trading volume, and recent volatility. The

350

Source: Morgan Stanley (1999).

France Germany Switzerland Novartis Swatch Nestle CS Group U.K. Total

Asset 19,300,182 19,492,570 19,300,182 5,351,851 64,678 1,752,009 1,165,797 5,860,371 50,004,974

Value (US$)

Market Impact Cost Report

T A B L E 14–2

184,063 322,550 9,355 1,630 400 935 6,390 424,373 940,341

Shares Held

Portfolio

104.9 60.4 572.1 1,453.6 161.7 1,873.8 182.4 13.8 53.2

Price 19.9 26.1 12.5 11.7 32.9 6.4 22.2 20.2 21.6

Spread (bp)

123,554 42,559 76,004 978,168

Median Volume

Cost Analysis

1.3% 2.5% 1.1% 1.3% 0.9% 1.2% 0.7% 0.6% 1.7%

Shares/ Volume

18.2 29.3 9.5 8.8 15.5 7.3 14.1 17.6 21.5

Impact Cost (bp)

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CHAPTER 14 Liquidity Risk

351

portfolio relative size is then defined as the number of shares held as a percentage of median trading volume. MS then computes the total impact cost as a function of half the bid-ask spread and of this size variable. Here, the total cost of immediate (1-day) liquidation is estimated to be 21.5 basis points. This can be compared to the daily markingto-market volatility of this portfolio, which is 110 basis points. Thus, if the portfolio were to be liquidated at the end of the next day, the worst LVAR loss at the 95 percent confidence level would be about $50 million  (1.65  1.1 percent  0.22 percent)  $50 million  2.0 percent  $1.0 million. The relative importance of liquidity would no doubt be much greater for a larger portfolio. In addition, the framework presented in the preceding section allows the investor to evaluate the cost and risk of various trading strategies. 14.3 GAUGING FUNDING LIQUIDITY RISK

Measuring funding liquidity risk involves examining the asset-liability structure of the institution and comparing potential demands on cash with the available supply of equivalent instruments. Some lessons are available from the Counterparty Risk Management Policy Group (CRMPG), which was established in the wake of the LTCM near failure to strengthen practices related to the management of market, counterparty credit, and liquidity risk.2 The CRMPG proposes to evaluate funding risk by comparing the amount of cash an institution has at hand with to what it could need to meet payment obligations. It defines cash liquidity as the ratio of cash equivalent over the potential decline in the value of positions that may create cash-flow needs. Table 14–3 illustrates an example of an institution that has a position in two offsetting swaps. Since it is perfectly hedged, it has zero traditional VAR, or market risk. Yet different swap credit terms create funding risk. In the case where the two swaps are both marked-to-market, any cash payment in one swap must be offset by a receipt on the other leg. The only risk is that of a delay in the receipt. Assume the worst move on a $100 million swap at the 99 percent level over 1 day is $1.1 million. 2. The CRMPG consists of senior-level practitioners from the financial industry, including many banks that provided funding to LTCM.

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PART 3 Value-at-Risk Systems

T A B L E 14–3

Computing Funding Liquidity Ratio

Case 1 Assets Cash Liabilities Equity Derivatives Long 10-year swap Short 10-year swap Cash equivalent Funding VAR Ratio

Case 2

$5

$5

$5

$5

$100, two-way marked-to-market $100, two-way marked-to-market $5 $1.1 (1-day) 4.5

$100, unsecured $100, two-way marked-to-market $5 $3.5 (10-day) 1.4

Let us consider this is the worst cash need. The funding ratio is then $5/$1.1  4.5, rather high, which indicates sufficient cash coverage. In the case where one of the swaps is not secured by markingto-market, the risk is that of a decrease in the value of the mark-to-market swap, which would not be offset by a cash receipt on the other leg until settlement. We now need to consider a longer horizon VAR, say, 10 days, which gives $3.5, for a funding ratio of 1.4. This seems barely enough to provide protection against funding risk. Thus some of the elements of traditional VAR can be used to compute funding risk, which can be quite different from market risk when the institution is highly leveraged. 14.4 LESSONS FROM LTCM3

The story of Long-Term Capital Management (LTCM) provides a number of lessons in liquidity risk. LTCM was founded by Meriwether in 1994, who left Salomon Brothers after the 1991 bond scandal. Meriwether took with him a group of traders and academics and set up a hedge fund that tried to take advantage of “relative value” or “convergence arbitrage” trades, betting on differences in prices, or spreads, among closely related securities. 3. This draws on Philippe Jorion, “How Long-Term Lost Its Capital,” Risk (September 1999).

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14.4.1 LTCM’s Leverage Since such strategies tend to generate tiny profits, leverage has to be used to create attractive returns. By December 1997, the total equity in the fund was $5 billion. LTCM’s balance sheet was about $125 billion. This represented an astonishing leverage ratio of 25:1. Even more astonishing was the off-balance-sheet position, including swaps, options, and other derivatives, that added up to a notional amount of $1.25 trillion. This represents the total of gross positions, measured as the sum of the absolute value of the trade’s notional principals. To give an idea of the magnitude of these positions, the BIS reported a total swap market of $29 trillion in 1998. Hence LTCM’s swap positions accounted for 2.4 percent of the global swap market. Many of these trades, however, were offsetting each other, so this notional amount is practically meaningless. What mattered was the net risk of the fund. LTCM, however, failed to appreciate that these gross positions were so large that attempts to liquidate them would provoke large market moves.

14.4.2 LTCM’s “Bulletproofing” LTCM was able to leverage its balance sheet through sale-repurchase agreements (repos) with commercial and investment banks. Under repo agreements, the fund sold some of its assets in exchange for cash and a promise to purchase them back at a fixed price on some future date. Normally, brokers require collateral that is worth slightly more than the cash loaned, by an amount known as a haircut, designed to provide a buffer against decreases in the collateral value. LTCM, however, was able to obtain unusually good financing conditions, with next-to-zero haircuts, since it was widely viewed as “safe” by its lenders. In addition, the swaps were subject to two-way marking-to-market. On the supply side, LTCM also had “bulletproofed” itself against a liquidity squeeze. LTCM initially had required investors to keep their money in the fund for a minimum of 3 years. The purpose of this socalled lockup clause, which was very unusual in the hedge fund industry, was to avoid forced sales in case of poor performance. LTCM also secured a $900 million credit line from Chase Manhattan and other banks. Even though LTCM had some protection against funding liquidity risk, it was still exposed to market risk and asset liquidity risk.

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354

PART 3 Value-at-Risk Systems

FIGURE

14–4

LTCM’s returns. 60%

(Percent after fees)

40% 20% 0% -20% -40%

LTCM Fund

-60%

U.S. Stocks LTCM, cumulative, 1998

-80% -100% 1994 1995 1996 1997 Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

14.4.3 LTCM’s Downfall LTCM’s strategy profited handsomely from the narrowing of credit spreads during the early years, leading to after-fees returns above 40 percent, as shown in Figure 14–4. Troubles began in May and June of 1998. A downturn in the mortgage-backed securities market led to a 16 percent loss in LTCM’s capital. Then came August 17. Russia announced that it was “restructuring” its bond payments—de facto defaulting on its debt. This bombshell led to a reassessment of credit and sovereign risks across all financial markets. Credit spreads, risk premia, and liquidity spreads jumped up sharply. Stock markets dived. LTCM lost $550 million on August 21 alone. By August, the fund had lost 52 percent of its December 31 value. With assets still at $126 billion, the leverage ratio had increased from 28:1 to 55:1. LTCM badly needed new capital. It desperately tried to find new investors, but there were no takers. In September, the portfolio’s losses accelerated. Bear Stearns, LTCM’s prime broker, faced a large margin call from a losing LTCM

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CHAPTER 14 Liquidity Risk

355

T-bond futures position. It then required increased collateral, which depleted the fund’s liquid resources. LTCM was now caught in a squeeze between funding risk, as its reserves dwindled, and asset risk, as the size of its positions made it impractical to liquidate assets. From the viewpoint of the brokers, a liquidation of the fund would have forced them to sell off tens of billions of dollars of securities and to cover their numerous derivatives trades with LTCM. Because lenders had required next-to-zero haircuts, there was a potential for losses to accrue while the collateral was being liquidated. The potential disruption in financial markets was such that the New York Federal Reserve felt compelled to act. On September 23, it organized a bailout of LTCM, encouraging 14 banks to invest $3.6 billion in return for a 90 percent stake in the firm. These fresh funds came just in time to avoid meltdown. By September 28, the fund’s value had dropped to $400 million only. Investors had lost a whopping 92 percent of their year-to-date investment.

14.4.4 LTCM’s Liquidity LTCM failed because of its inability to measure, control, and manage its risk. This was due in no small part to the fact that LTCM’s trades were rather undiversified. LTCM was reported to have lost about $1.5 billion from interest rate swaps positions and a similar amount from short positions on equity volatility. As we show later, this was a result of an ill-fated attempt to manage risk through portfolio optimization. Table 14–4 describes the exposure of various reported trades to fundamental risk factors. All the trades were exposed to increased market volatility. Most were exposed to liquidity risk (which is itself positively correlated with volatility). Many were exposed to default risk. To illustrate the driving factor behind LTCM’s risks, Figure 14–5 plots the monthly returns against monthly changes in credit spreads. The fit is remarkably good, indicating that a single risk factor would explain 90 percent of the variation up to the September bailout. Thus there was little diversification across risk factors. In addition, LTCM was a victim of both asset and funding liquidity risk. Although it had taken some precautions against withdrawal of funds, it did not foresee that it would be unable to raise new funds as its performance dived. The very size of the fund made it difficult to organize an orderly portfolio liquidation.

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356

PART 3 Value-at-Risk Systems

T A B L E 14–4

Exposure of LTCM’s Portfolio to Risk Factors

Loss If Risk Factor Increases Trade

Volatility

Default

Illiquidity

Yes Yes

Yes

Yes

Long interest rate swap Short equity options Long off-the-run/short on-the-run Treasuries Long mortgage-backed securities (hedged) Long sovereign debt

FIGURE

Yes Yes Yes

Yes Yes

Yes

14–5

Explaining LTCM returns. 60%

Return after fees

40% 20% Jul

0%

May

Jun

-20% -40%

To Sep-98

-60%

After bail-out

Aug

-80%

Sep

-100% -0.4

-0.2

0

0.2

Change in corporate bond spread

0.4

0.6

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CHAPTER 14 Liquidity Risk

357

The episode also raised questions about the soundness of the brokers’ risk-management systems. The brokers lulled themselves into thinking that they were protected because their loans were “fully collateralized.” Even so, their loans carried no haircuts and were exposed to the risk that LTCM could default at the same time as the collateral lost value. One of the lessons of this near disaster was to accelerate the integration of credit and market risk management. 14.5 CONCLUSIONS

This chapter has shown how to account for liquidity risk. Originally, the implicit assumption behind VAR was that the assets can be liquidated before they have moved by more than the amount estimated. In practice, this concept can be refined to account for liquidity risk. Asset liquidity risk can be traced to the price impact of the liquidation of a large position relative to normal market sizes. We have seen that different execution strategies can be compared on the basis of a liquidityadjusted VAR, which requires an estimate of the price impact function. Funding liquidity risk, in contrast, considers the available supply of cash equivalents as well as potential claims that may arise due to leverage. Here again, the VAR concept can be altered to estimate the risk that a portfolio could run out of cash. Whether liquidity risk is relevant very much depends on the liability structure of the investment. Consider, for instance, an investment in “illiquid” assets such as exotic collateralized mortgage obligations. If the investment is funded by short-term borrowing, the investor runs a liquidity risk and should mark the investment at its liquidation value. In contrast, if the investment is funded by unleveraged equity, marking-tomarket is an appropriate measure of risk. As for other applications of VAR, the main benefit of these analyses is not so much to come up with one summary risk number but rather to provide a systematic framework to think about alternative execution strategies or marking-to-market arrangements. While LVAR may be somewhat difficult to measure, some rules of thumb are useful. We do know that bid-ask spreads are positively correlated with volatility. A position in illiquid assets will incur greater execution costs as volatility increases. Thus liquidity risk can be mitigated by taking offsetting positions in assets, or businesses, that benefit from increased volatility or have positive vega.

At the end of this chapter, you will:

1. Understand the main approaches to price risky debt. 2. Know how to evaluate credit risks quantitatively for individual credits and for portfolios.

3. Have learned what credit derivatives are and how to use them.

4. Have tools to assess the credit risks of derivatives.

From Risk Management and Derivatives, 1st edition, by René M. Stulz. © 2003. Reproduced with permission ofThis South-Western, division of Thomson www.thomsonrights.com. Fax 800 chapter has beenareproduced with permission from Riskby Management and Derivatives, René M. 730-2215. Stulz, by of From Risk Management and Derivatives, 1st Learning: edition, René M. Stulz. © 2003. by Reprinted with published permission Thomson South-Western, Mason, Ohio 2003 (© Thomson South-Western, all rights reserved). South-Western, a division of Thomson Learning: www.thomsonrights.com. Fax 800 730-2215. From Risk Management and Derivatives, 1st edition, by René M. Stulz. © 2003. Reproduced with permission of South-Western, a division of Thomson Learning: www.thomsonrights.com. Fax 800 730-2215.

Part 3

Beyond Plain Vanilla Rirk Munagemmt

Consider Credit Bank Corp. It makes loans to corporations. For each loan, there is some risk that the borrower will default, in which case Credit will not receive all the payments the borrower promised to make. Credit has to understand the risk of the individual loans it makes, but it must also be able to quantify the overall risk of its loan portfolio. Credit has a high franchise value and wants to protect that franchise value by making sure that the risk of default on its loans does not make its probability of financial distress too high. Though Credit knows how to compute VaR for its trading portfolio, it cannot use these techniques directly to compute the risk of its loan portfolio. Loans are like bonds-credit never receives more from a borrower than the amounts the borrower promised to pay. Consequently, the distribution of the payments received by borrowers cannot be lognormal. To manage the risk of its loans, Credit must know how to quantify the risk of default and of the losses it makes in the event of default both for individual loans and for its portfolio of loans. At the end of this chapter, you will know the techniques that Credit can use for this task. We will see that the Black-Scholes formula is useful to understand the risks of individual loans. Recently, a number of firms have developed models to analyze the risks of portfolios of loans and bonds. For example, J.I? Morgan has developed CreditMetricsTM along the lines of its product kskMetricsTM. We discuss this model in some detail.

A credit risk is the risk that someone who owes money might fail to make promised payments. Credit risks play two important roles in risk management. First, credit risks represent part of the risks a firm tries to manage in a risk management program. If a firm wants to avoid lower tail outcomes in its income, it must carefully evaluate the riskiness of the debt claims it holds against third parties and determine whether it can hedge these claims and how. Second, the firm holds positions in derivatives for the express purpose of risk management. The counterparties on these derivatives can default, in which case the firm does not get the payoffs it expects on its derivatives. A firm taking a posirion in a derivative must therefore evaluate the riskiness of the counterparty in the position and be able to assess how the riskiness of the counterparty affects the value of its derivatives positions. Credit derivatives are one of the newest and most dynamic growth areas in the derivatives industry. At the end of 2000, the total notional amount of credit derivatives was estimated to be $8 10 billion; it was only $180 billion two years before. Credit derivatives have payoffs that depend on the realization of credit risks. For example, a credit derivative could promise to pay some amount if Citibank defaults and nothing otherwise; or a credit derivative could pay the holder of Citibank debt the shortfall that occurs if Citibank defaults on its debt. Thus firms can use credit derivatives to hedge credit risks.

18. I. Credit risks as options Following Black and Scholes (1973), option pricing theory has been used to evaluate default-risky debt in many different situations. The basic model to value risky debt using option pricing theory is the Merton (1974) model. To understand this approach, consider a levered firm that has only one debt issue and pays

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Chapter 18

Credit Ri~ksand Credit Deriva~iver

no dividends. Financial markets are assumed to be perfect. There are no taxes and no bankruptcy costs, and contracts can be enforced costlessly. Only debt holders and equity holders have claims against the firm and the value of the firm is equal to the sum of the value of debt and the value of equity. The debt has no coupons and matures at T. At date T, the firm has to pay the principal amount of the debt, F. If the firm cannot pay the principal amount at T, it is bankrupt, equity has no value, and the firm belongs to the debt holders. If the firm can pay the principal at T, any dollar of firm value in excess of the principal belongs to the equity holders. Suppose the firm has issued debt that requires it to make a payment of $100 million to debt holders at maturity and that the firm has no other creditors. If the total value of the firm at maturity is $120 million, the debt holders receive their promised payment, and the equity holders have $20 million. If the total value of the firm at maturity is $80 million, the equity holders receive nothing and the debt holders receive $80 million. Since the equity holders receive something only if firm value exceeds the face value of the debt, they receive VT - F if that amount is positive and zero otherwise. This is equivalent to the payoff of a call option on the value of the firm. Let VT be the value of the firm and ST be the value of equity at date T. We have at date T:

To see that this works for our example, note that when firm value is $120 million, we have ST equal to Max($l20M - $100M, 0), or $20 million, and when firm value is $80 million, we have ST equal to Max($80M -$100M, 0), or $0. Figure 18.1 graphs the payoff of the debt and of the equity as a function of the value of the firm. If the debt were riskless, its payoff would be the same for any value of the firm and would be equal to F. Since the debt is risky, when the value of the firm falls below F, the debt holders receive less than F by an amount equal to F - VT. The amount F - VT paid ifVT is smaller than F, Max(F - VT, O), corresponds to the payoff of a put option on VT with exercise price F. We can therefore think of the debt as paying F for sure minus the payoff of a put option on the firm with exercise price F:

where DT is the value of the debt at date T. Equation (18.2) therefore tells us that the payoff of risky debt is equal to the payoff of a long position in a risk-free zerocoupon bond with face value F and a short position on a put option on firm value with exercise price F. This means that holders of risky debt effectively buy riskfree debt but write a put option on the value of the firm with exercise price equal to the face value of the debt. Alternatively, we can say that debt holders receive the value of the firm, VT, minus the value of equity, ST. Since the payoff of equity is the payoff of a call option, the payoff of debt is the value of the firm minus the payoff of a call option with exercise price equal to the principal amount of the debt.

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Part 3

"7-7-

e .-.-Ts .-

'iF@we 18.1

-r---w--

=-----%-T.w?-?+..% Debt and equity payoffs when debt is risky 7-

- ~ ~ - - - % . m 7 = , . - %

----aF---.--T+y=

--

F is the debt prlnc~palamount andV(T) 1s the value of the flrm at date T.

To price the equity and the debt using the Black-Scholes formula for the pricing of a European call option, we require that the value of the firm follow a lognormal distribution with a constant volatility o, the interest rate r be constant, trading take place continuously, and financial markets be perfect. We do not require that there is a security that trades continuously with value V. All we need is a portfolio strategy such that the portfolio has the same value as the firm at any particular time. We use this portfolio to hedge options on firm value, so that we can price such options by arbitrage. We can write the value of equity as S(V, F, T, I-) and use the formula to price a call option to obtain: Merton's formula for the value of equity

Let S(V, F, T, t ) be the value of equity at date t, V the value of the firm, F the face value of the firm's only zero-coupon debt maturing at T, (T the volatility of the value of the firm, P,(T) the price at t of a zero-coupon bond that pays $1 at T, and lV(d) the cumulative distribution function evaluated at d. With this notation, the value of equity is:

yv, F, T, t ) = V N ( ~ )- P ~ ( T ) F N -( ~O K )

When V is $120 million, F is $100 million, T is equal to t + 5, P,(T) is $0.6065, and o is 20 percent, the value of equity is $60.385 million. From our understanding of the determinants of the value of a call option, we know that equity increases in value when the value of the firm increases, when firm volatility in-

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Chapter 18

Credit Risks and Credit Derivatives

creases, when time to maturity increases, when the interest rate increases, and when the face value amount of the debt falls. Debt can be priced in two different ways. First, we can use the fact that the payoff of risky debt is equal to the payoff of risk-free debt minus the payoff of a put option on the firm with exercise price equal to the face value of the debt:

where p(V, F, T, t ) is the price of a put with exercise price F on firm value V. F is $100 million and P,(T) is $0.6065, so P,(T)F is $60.65 million. A put on the value of the firm with exercise price of $100 million is worth $1.035 million. The value of the debt is therefore $60.65M - $1.035M, or $59.615 million. The second approach to value the debt involves subtracting the value of equity from the value of the firm:

We subtract $60.385 million from $120 million, which gives us $59.615 million. The value of the debt is, for a given value of the firm, a decreasing function of the value of equity, which is the value of a call option on the value of the firm. Everything else equal, therefore, the value of the debt falls if the volatility of the firm increases, if the interest rate rises, if the principal amount of the debt falls, and if the debt's time to maturity lengthens. To understand the effect of the value of the firm on the value of the debt, note that a $1 increase in the value of the firm affects the right-hand side of equation (18.5) as follows: It increases V by $1 and the call option by $6, where 6 is the call option delta. 1 - 6 is positive, so that the impact of a $1 increase in the value of the firm on the value of the debt is positive and equal to $1 - $6. 6 increases as the call option gets more in the money. Here, the call option corresponding to equity is more in the money as the value of the firm increases, so that the impact of an increase in firm value on debt value falls as the value of the firm increases. Investors pay a lot of attention to credit spreads. The credit spread is the difference between the yield on the risky debt and the yield on risk-free debt of same maturity. If corporate bonds with an A rating have a yield of 8 percent while Tbonds of the same maturity have a yield of 7 percent, the credit spread for A-rated debt is 1 percentage point. An investor can look at credit spreads for different ratings to see how the yields differ across ratings classes. An explicit formula for the credit spread is: Credit spread

=

- ( T1t ) l n ( F ) -

where r is the risk-free rate. For our example, the risk-free rate (implied by the zero-coupon bond price we use) is 10 percent. The yield on the debt is 10.35 percent, so the credit spread is 35 basis points. Not surprisingly, the credit spread falls as the value of the debt rises. The logarithm of D/F in equation (18.6) is

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Part 3

Beyond Plain Vanilla Risk Management

multiplied by -[l/(T - t ) ] .An increase in the value of debt increases D/F, but since the logarithm of D/F is multiplied by a negative number, the credit spread falls. With debt, the most we can receive at maturity is par. As time to maturity lengthens, it becomes more likely that we will receive less than par. However, if the value of the debt is low enough to start with, there is more of a chance that the value of the debt will be higher as the debt reaches maturity if time to maturity is longer. Consequently, if the debt is highly rated, the spread widens as time to maturity gets longer. For sufficiently risky debt, the spread can narrow as time to maturity gets longer. This is shown in Figure 18.2. Helwege and Turner (1999) show that credit spreads widen with time to maturity for low-rated public debt, so that even low-rated debt is not risky enough to lead to credit spreads that narrow with time to maturity. It is important to note that credit spreads depend on interest rates. The expected value of the firm at maturity increases with the risk-free rate, so there is less risk that the firm will default. As a result, credit spreads narrow as interest rates increase.

18.1.1. Finding firm value and firm value volatility Suppose a firm, Supplier Inc., has one large debt claim. The firm sells a plant to a very risky third party, In-The-Mail Inc., and instead of receiving cash it receives a promise from In-The-Mail Inc. that it will pay $100 million in five years. We want to value this debt claim. IfV,, 5 is the value of In-The-Mail Inc. at maturity of the debt, we know that the debt pays F - Max(F - V,. 5, 0) or V, 5 Max(V,+ 5 - F, 0). If it were possible to trade claims on the value of In-The-Mad Inc., pricing the debt would be straightforward. We could simply compute the value of a put -onIn-The-Mail Inc. with the appropriate exercise price. In general, we cannot directly trade a portfolio of securities that represents a claim to the whole firm; In-The-Mail Inc.'s debt is not a traded security, +

The fact that a firm has some nontraded securities creates two problems. First, we cannot observe firm value directly and, second, we cannot trade the firm to hedge a claim whose value depends on the value of the firm. We can solve both problems. Remember that with Merton's model the only random variable that affects the value of claims on the firm is the total value of the firm. Since equity is a call option on firm value, it is a portfolio consisting of 6 units of firm value plus a short position in the risk-free asset. The return on equity is perfectly correlated with the return on the value of the firm for small changes in the value of the firm because a small change in firm value of AV changes equity by 6AV. We can therefore use equity and the risk-free asset to construct a portfolio that replicates the firm as a whole, and we can deduce firm value from the value of traded claims on the value of the firm. To do that, however, we need to estimate the 8 of equity. To compute the 6 of equity from Merton's formula, we need to know firm value V, the volatility of firm value, the promised debt payment, the risk-free interest rate, and the maturity of the debt. If we have this information, computing delta is straightforward. Otherwise, we can estimate these variables using information we have. If we have an estimate of 6 and we know the value of the firm's equity, then we can solve for firm value and the volatility of firm value as long as we know the promised debt payment and the maturity of the debt. This is be-

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Credit Rrskj and Credif Derivatzves

Chapter 18

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cause, in this case, we have two unknowns, firm volatility and firm value, and two equations, the Merton equation for the value of equity and the Merton equation for the equity's delta. We can solve these equations to find firm volatility and the value of the firm. Having these values, we can then solve for the value of the debt. (In practical applications, the Merton model is often used for more complicated capital structures. In this case, the promised debt payment is an estimate of the amount of firm value over some period of time such that if firm value falls below that amount, the firm will be in default and have to file for bankruptcy.) Suppose we do not know 6. One way to find 8 is as follows. We assume Inthe-Mail Inc. has traded equity and that its only debt is the debt it owes to

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Part 3

Beyotld Plain Vanilla Risk Management

Supplier Inc. The value of the debt claim in million dollars is D(V, 100, t + 5, t ) . The value of a share is $14.10 and there are 5 million shares. T h e value of the firm's equity is therefore $70.5 million. The interest rate is 10 percent per year. Consequently, in million dollars, we have:

We know that equity is a call option on the value of the firm, so that S(V, 100, t + 5 , t ) = c(V, 100, t +5, t ) .We cannot trade V because it is the sum of the debt we own and the value of equity. Since the return to V is perfectly correlated with the return on equity, we can form a dynamic portfolio strategy that pays V,, 5 at t + 5. We will see the details of the strategy later, but for now we assume it can be done. If we know V, we can use equation (18.7) to obtain the firm's implied volatility, but the equation has two unknowns: V and the volatility of the value of the firm, o.To solve for the two unknowns, we have to find an additional equation so that we have two equations and two unknowns. Suppose that there are options traded on the firnl's equity. Suppose further that a call option on one share with exercise price of $10 and maturity of one year is worth $6.72. We could use the option pricing formula to get the volatility of equity and deduce the volatility of the firm from the volatility of equity. After all, the option and the equity both depend on the same random variable, the value of the firm. The difficulty with this is that the Black-Scholes formula does not apply to the call option in our example because it is a call option on equity, which itself is an option when the firm is levered. We therefore have an option on an option, or what is called a compound option. The Black-Scholes formula applies when equity has constant volatility, but the equity in our example cannot have constant volatility if firm value has constant volatility. For a levered firm where firm value has constant volatility, equity is more volatile when firm value is low than when it is high, so that volatility falls as firm value increases. This is because, in percentage terms, an increas; in firm value has more of an impact on equity ;hen the value of equity is extremely low than when it is extremely high-even though the equity's 6 increases as firm value increases.

A compound call option gives its holder the right to buy an option for a given exercise price. Geske (1979) derives a pricing formula for a compound option that we can use. Geske's formula assumes that firm value follows a lognormal distribution with constant volatility. Therefore, if we know the value of equity, we can use Geske's formula to obtain the value of firm volatility This formula is presented in Technical Box 18.1, Compound option formula.

18.1.2. Pricing the debt of In-The-Mail Inc. We now have two equations that we can use to solve for V and a: the equation for the value of equity (the Black-Scholes formula), and the equation for the value of an option on equity (the compound option formula of Technical Box 18.1). These two equations have only two unknowns. We proceed by iteration. Suppose we pick a firm value per share of $25 and a volatility of 50 percent. With this, we find that equity should be worth $15.50 and that the call price should be $6.0349. Consequently, the value of equity is too high and the value

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Chapter 18

Credit risk^ and Credit Dmivativa

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From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Part 3

Beyond Plain Valzilla Rijk Management

of the call is too low. Reducing the assumed firm value reduces both the value of equity and the value of rhe call, so that it brings us closer to the value of equity we want but farther from the value of the call we want. We therefore need to change both firm value and some other variable, taking advantage of the fact that equity and a call option on equity have different greeks. Reducing our assumed firm value reduces the value of equityas well as the value of the call, and increases volatility, which increases the value of equity and the value of the call. In this case, we find an option value of $7.03 and a value of equity of $13.30. Now, the value of equity is too low and the value of the option is too high. This suggests that we have gone too far with our reduction in firm value and increase in volatility. A value of the firm of $21 per share and a volatility of 68.36 percent yield the right values for equity and for the option on equity. Consequently, the value of the debt per share is $6.90. The value of the firm is therefore $105 million. It is divided between debt of $34.5 million and equity of $70.5 million. Once we have the value of the firm and its volatility, we can use the formula for the value of equity to create a portfolio whose value is equal to the value of the firm and thus can replicate dynamically the value of the firm just using the risk-free asset and equity. Remember that the value of the firm's equity is given by a call option on the value of the firm. Using the Black-Scholes formula, we have:

Inverting this formula, the value of the firm is equal to:

Note that we know all the terms on the right-hand side of this e uation. Hence, an investment of 1/ N ( d )of the firm's equity and of N (d- a T - t)FIN(d) units of the zero-coupon bond is equal to the value of the firm per share. Adjusting this portfolio dynamically over time insures that we have V(T) at maturity of the debt. We can scale this portfolio so that it pays off the value of the firm per share. With our example, the portfolio that pays off the value of the firm per share has an investment of 1.15 shares, and an investment in zero-coupon bonds worth $7.9 1.

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18.1.3. Subordinated debt In principle, subordinated debt receives a payment in the event of bankruptcy only after senior debt has been paid in full. Consequently, when a firm is in poor financial condition, subordinated debt is unlikely to be paid in full and is more like an equity claim than a debt claim. In this case, an increase in firm volatility makes it more likely that subordinated debt will be aid off and hence increases the value of subordinated debt. Senior debt always falls in value when firm volatility increases.

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Chapter 18

Credit Riskj and Credit Derivatives

To understand the determinants of the value of the subordinated debt, consider then the case where the senior debt and the subordinated debt mature at the same date. F is the face value of the senior debt and U is the face value of the subordinated debt. Equity is an option on the value of the firm with exercise price U + F since the shareholders receive nothing unless the value of the firm exceeds U + F. Figure 18.3 shows the payoff of subordinated debt as a function of the value of the firm. In this case, the value of the firm is:

D denotes senior debt, SD subordinated debt, and S equity. The value of equity is given by the call option pricing formula:

By definition, shareholders and subordinated debt holders receive collectively the excess of firm value over the face value of the senior debt, F, if that excess is positive. Consequently, they have a call option on V with exercise price equal to F. This implies that the value of the senior debt is the value of the firm V minus the value of the option held by equity and subordinated debt holders:

Having priced the equity and the senior debt, we can then obtain the subordinated debt by subtracting the value of the equity and of the senior debt from the value of the firm:

Both debt claims are zero-coupon bonds.They mature at the same tirne.The subordinated debt has face value U and the senior debt has face value F. Firm value isV(T).

/ Equity

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Part 3

Beyond Plain Vanilla Risk Management

With this formula, the value of subordinated debt is the difference between the value of an option on the value of the firm with exercise price F + U and an option on the value of the firm with exercise price F. Consider a firm with value of $1 20 million. It has junior debt maturing in five years with face value of $50 million and senior debt maturing in five years with face value of $100 million. The interest rate is 10 percent and the volatility is 20 percent. In this case, we have F = $100 million and U = $50 million. The first call option is worth $60.385 million. It is sin~plythe equity in the absence of subordinated debt, which we computed before. T h e second call option is worth $36.56 million. The value of the subordinated debt is therefore $60.385M - $36.56M = $23.825 million. Using our formula for the credit spread, we find that the spread on subordinated debt is 4.83 percent, which is more than 10 times the spread on senior debt of 35 basis points. The fact that the value of subordinated debt corresponds to the difference between the value of two options means that an increase in firm volatility has an ambiguous effect on subordinated debt value. As shown in Figure 18.4, an increase in firm volatility can increase the value of subordinated debt. Subordinated debt is a portfolio that has a long position in a call option that increases in value with volatility and a short position in a call option that becomes more costly as volatility increases. If the subordinated debt is unlikely to pay off, the short position in the call is economically unimportant. Consequently, subordinated debt is almost similar to equity and its value is an increasing function of the volatility of the firm. Alternatively, if the firm is unlikely to ever be in default, then the subordinated debt is effectively like senior debt and inherits the characteristics of senior debt. The value of subordinated debt can fall as time to maturity decreases. If firm value is low, the value of the debt increases as time to maturity increases because there is a better chance that it will pay something at maturity. If firm value is high and debt has low risk, it behaves more like senior debt. Similarly, a rise in interest rates can increase the value of subordinated debt. As the interest rate rises, the value of senior debt falls so that what is left for the subordinated debt holders and equity increases. For low firm values, equity gets little out of the interest rate increase because equity is unlikely to receive anything at maturity, so the value of subordinated debt increases. For high firm values, the probability that the principal will be paid is high, so the subordinated debt is almost risk-free, and its value necessarily falls as the interest rate increases.

118.1.4. The pricing of debt when interest rates change randomly Unanticipated changes in interest rates can affect debt value through two channels. First, an increase in interest rates reduces the present value of promised coupon payments absent credit risk, and hence reduces the value of the debt. Second, an increase in interest rates can affect firm value. Empirical evidence suggests that stock prices are negatively correlated with interest rates. Hence, an increase in interest rates generally reduces the value of debt both because of the sensitivity of debt to interest rates and because on average it is associated wi.th an adverse shock to firm values. When we want to hedge a debt position, we therefore have to take into account the interaction between interest rate changes and firm value changes.

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Credit Rrsks and Cvedrt Deriuatrves

Chapter 18

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We consider the pricing of risky debt when the spot interest rate follows the Vasicek model discussed in Chapter 14. The change in.the spot interest rate over a period of length At is:

where r , is the current spot interest rate and E, is a random shock. When A is positive, the interest rate reverts to a long-run mean of k. With equation (18.14) describing how the interest rate evolves, the price of a zero-coupon bond at t that pays $1 at T, Pt(T),is given by the Vasicek model.

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Part 3

Beyond Pkain Vanilla Rirk Management

Suppose value and interest rate changes are correlated. Shimko, Tejima, and van Deventer (1993) show that with these interest rate dynamics the value of risky dcbt is:

To see how interest rate changes affect the price of debt, we price debt of face value of $100 million maturing in 5 years on a firm worth $120 million as we did before. When we assumed a fixed interest rate of 10 percent and a firm volatility of 20 percent, we found that the value of the debt was $59.6 15 million in section 18.1. We choose the parameters of the Vasicek model to be such that, with a spot interest rate of 10 percent, the price of a zero-coupon bond that pays $1 in 5 years is the same as with a fixed interest rate of I0 percent. This requires us to assume k to be 14.21 percent, the interest rate volatility 10 percent, and the mean reversion parameter 0.25. The volatility of firm value is 20 percent as before and the correlation between firm value changes and interest rate changes is -0.2. With these assumptions, the debt is then $57.301 1 million. Figure 18.5 shows how the various parameters of interest rate dynamics affect the price of the debt. We choose a firm value of $50 million, so that the firm could not repay the debt if it matured immediately. In Panel A, the value of the debt falls as the correlation between firm value and interest rate shocks increases. In this case, firm value is higher when the interest rate is high, so that the impact of an increase in firm value on the value of the debt is more likely to be dampened by a simultaneous interest rate increase. In Panel B, an increase in interest rate volatility and an increase in the speed of mean reversion reduce debt value. With high mean reversion, the interest rate does not diverge for very long from its long-run mean, so that we are closer to the case of fixed interest rates. However, with our assumptions, the long-run mean is higher than the current interest rate. Figure 18.6 shows that the debt's interest rate sensitivity depends on the volatility of interest rates. At highly volatile interest rates, the value of the debt is less sensitive to changes in interest rate. Consequently, if we were to hedge debt against changes in interest rates, the hedge ratio would depend on the parameters of the dynamics of interest rates.

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Credzt Risks and Credit Derivatzves

Chapter 18

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1 8.1.5. VaR and credit risks Once we have a pricing model for the valuation of default-risky debt held by the firm, we can incorporate credit risk into the computation of firm-wide risk. Suppose we use the Merton model. Default-risky debt depends on firm value, which itself is perfectly correlated with the debtor's stock price. This makes the debtor's equity one additional risk factor. Suppose we want to compute a VaR measure for the firm, and the firm has just one risky asset: its risky debt. One way is to cornpute the deha-VaR by transforming the risky debt into a portfolio of the risk-free

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Part 3

Beyond Plain VanzLla R ~ s kManagemenr

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bond and of the debtor's equity if we are computing a VaR for a short period of time. A second way is to compute the Monte Carlo VaR by simulating equity returns and valuing the debt for these equity returns. If the firm has other assets, we must consider the correlations among the asset returns. Conceptually, the inclusion of credit risks in computations of VaR does not present serious difficulties. All the difficulties are in the implementation-but they are serious. The complexities of firm capital structures create important obstacles to valuing debt, and often debt is issued by firms with no traded equity, There are thus alternative approaches to debt pricing.

18.2. Beyond the Met-ton model Corporations generally have many different types of debt with different maturities, and most debt makes coupon payments when not in default. T h e Merton model approach can be used to price any type of debt. Jones, Mason, and Rosenfeld (1984) test this approach for a panel of firms that include investment-grade firms as well as firms below investment grade. They find that a naive model predicting that debt is riskless works better for investment-grade debt than the Merton model. In contrast, the Merton model works better than the naive model for debt below investment grade. As ~ o i n t e dout by Kim, Ramaswamy, and Sundaresan (1993), however, Merton's model fails to predict credit spreads large enough to match empirical data. They point out that from 1926 to 1986, AAA spreads ranged from 15 to 215 basis points, with an average of 77, while BAA spreads ranged from 5 1 to 787 basis points, with an average of 198. Yet they show that Merton's model cannot generate spreads in excess of 120 basis points. There are important difficulties in implementing the Merton model when debt makes coupon payments or a firm has multiple debt issues that mature at

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Chapter 18

Cvedil Risks and Credit Derivatrves

different dates. Consider the simple case where debt makes one coupon payment u at t' and pays F + u at T. We know how to value the coupon payment u since it is equivalent to a risky zero-coupon debt payment at t'. Valuing the right to F + u to be received at T is harder because it is contingent on the firm paying u at t ' .Taking the viewpoint of the equity holders simplifies the analysis. After the equity holders have paid the coupon at t', their claim on the firm is the same as the claim they have in our analysis in section 18.1, namely a call option on the firm with maturity at T with exercise price equal to the promised payment to the debt holders (which here is F + u). Consequently, by paying the coupon at t' the equity holders acquire a call option on the value of the firm at T with exercise price F + u. The value of equity at t is the present value of the call option equity holders acquire at t' if they pay the coupon. This is the present value of the payoff of a European call option maturing at t', since they get Max(S,q - u, 0), where St#is the value of equity at t'. Hence, at t, the equity holders have a call option on the equity value at t' with exercise price u-they have an option on an option, or a compound option. The value of debt at t is firm value minus a compound option. If we had an additional coupon at t", so that t' < t" < T, we would have to subtract from V at t a n option on an option on an option. This creates a considerable computational burden in computing debt value. This burden can be surmounted, but it is not trivial to do so. In practice, this difficulty is compounded by the difficulty that one does not know V because of nontraded debt. Another difficulty with the Merton model is that default is too predictable. Remember that to obtain prices of debt in that model, we make the Black-Scholes assumptions. We know that with these assumptions firm value cannot jump. As a result, default cannot occur unless firm value is infinitesimally close to the point where default occurs. In the real world, default is often more surprising. For instance, a run on a bank could make its equity worthless even though before the run its equity value was not close to zero. These problems have led to the development of a different class of models that take as their departure point a probabil& of default that evolves over time according to a well-defined process. Under this approach, the probability of default can be posirive even when firm value significantly exceeds the face value of the debt-this is the case if firm value can .jump. . The economics of default are modeled as a black box. Default either happens over an interval of time or it does not. Upon default, the debt holder receives a fraction of the promised claim. 'l'he recovery rate is the fraction of the principal recovered in the event of default. This recovery rate can be random or certain. Let's look at the simplest case and assume that the process for the probability of default is not correlated with the interest rate process and recovery in the event of default is a fixed fraction of the principal amount, 0, which does not depend on time. The bond value next period is D,, A , + u if the bond is not in default, where u is the coupon. If the debt is in default, its value is OF. In the absence of arbitrage opportunities, the bond price today, D,, is simply the expected value of the bond next period computed using risk-neutral probabilities discounted at the risk-free rate. Using qas the risk-neutral probability of default, it must be the case that:

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Part 3

Beyond Plain Vanilla Risk Manugemcnt

In this equation, the value of the nondefaulted debt today depends on the value of the nondefaulted debt tomorrow. To solve this problem, we therefore start From the last period. In the last period, the value of the debt is equal to the principal amount plus the last coupon payment, F + u, or to OF. We then work backward to the next-to-the-last period, where we have:

If we know q and 0, we can price the debt in the next-to-the-last period and continue to keep worlung backward to get the debt value. How can we obtain the probability of default and the amount recovered in the event of default? If the firm has publicly traded debt, we can infer these parameters from the price of the debt. Alternatively, we can infer risk-neutral probabilities of default and recovery rates from spreads on debt with various ratings. Different applications of this approach can allow for random recovery rates as well as for correlations between recovery rates and interest rates or correlations between default rates and interesr rates. The empirical evidence shows that this approach works well to price swap spreads and bank subordinated debt.

18.3. Credit risk models There are several differences between measuring the risk of a portfolio of debt claims and measuring the risk of a portfolio of other financial assets. First, because credit instruments typically do not trade on liquid markets where we can observe prices, we cannot generally rely on historical data on individual credit instruments to measure risk. Second, the distribution of returns differs. We cannot assume that continuously compounded returns on debt follow a normal distribution. The return of a debt claim is bounded by the fact that investors cannot receive more than the principal payment and the coupon payments. In statistical terms, this means that the returns to equity are generally symmetric, while the returns to debt are skewed-unless the debt is deeply discounted. . The third difference is that firms often have debt issued by creditors with no

traded equity. A fourth difference is that typically debt is not marked to market in contrast to traded securities. When debt is not marked to market, a loss is recognized only if default takes place. Consequently, when debt is not marked to market, a firm must be able to assess the probability of default and the loss made in the event of default.

A number of credit risk models resolve some of the difficulties associated with debt portfolios. Some models focus only on default and on the recovery in the event of default. The most popular model of this type is CreditRisk+, from Credit Suisse Financial Products. It is based on techniques borrowed from the insurance industry for the modeling of extreme events. Other models are based on the marked-to-market value of debt claims. CreditMetricsTM is a risk model built on the same principles as those of RiskMetrics-. The purpose of this risk model is to provide the distribution of the value of a portfolio of debt claims, which leads to a VaR measure for the portfolio. The KMV model is in many ways quite similar to the CreditMetricsTM model, except that it makes direct use of the Merton

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Credit Risks and Credit Derivatives

Chapter 18

model in computing the probability of default.' All these models can be used to compute the risk of portfolios that include other payoffs besides those of pure debt contracts. For example, they call illclude swap contracts. As a result, these models talk about estimating the risk of obligors-all those who have legal obligations-to the firm-rather than debtors. To see how we can use the Merton model for default prediction, remember that it assumes that firm value is lognormally distributed Gith constant volatility and that the firm has one zero-coupon debt issue. If firm value exceeds the face value of debt at maturity, the firm is not in default. We want to compute the probability that firm value will be below the face value of debt at maturity of the debt becaise we are interested in forecasting the likelihood of a default. ib cornpute this probability, we have to Isnow the expected rate of return of the firm since the higher that expected rate of return, the less likely it is that the firm will be in default. Let p, be the expected rate of return of the firm value. In this case, the probability of default is simply:

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0llT-t

"1

where N denotes the cumulative normal distribution, F is the face value of the debt, V the value of the firm, T the maturity date of the debt, and o the volatility of the rate of change of V. Consider the case where a firm has value of $120 million, debt with face value of $100 million and maturity of five years, the expected rate of change of firm value is 20 percent, the volatility is 20 percent, and the interest rate is 5 percent. The probability that the firm will default is 0.78 percent. Figure 18.7 shows how the probability of default is related to volatility and firm value. As volatility increases, the probability of default increases. It can be substantial even for large firm values compared to the face value of the debt when volatility is high. We use the same approach to compute the firm's expected loss if default occurs. The loss if default occurs is often called loss given default or LGD. Since default occurs when firm value is Jess than the face value of the debt, we have to compute the expected value of V given that it is smaller than F. The solution is: Expected loss

FN

ln(F)

- ln(V)

- -(T - t ) + 0 . 5 0 2 ( ~-

(18.191

0.JT-t

1 Oldrich A. Vas~cek,Credit Valuat~on,KMV Corporation, 1984

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Part 3

,-.&-P-W..fi p

;

y .

'FGure

-----.

18.7

.

,

Beyond Plain Vanzlku Risk Manngmnent

.

-

7

u

,

- .-

-

-

. z *,-..

= - -

Probability of default

. - ., '

. ..

*

.- -

.

- - - .-,

The firm has debt with face value of $100 million due in five years.The firm's expected rate of return is 20 percent.

Firm value

501-.

The expected loss is $1 00,6 14. Figure 18.8 shows how the expected loss depends on firm value and its volatility.

18.3.1. CreditRisk+ CreditRisk+ allows only two outcomes for each firm over the risk measurement period: default and no default. If default occurs, the creditor experiences a loss of fixed size. The probability of default for an obligor depends on its rating, the realization of K risk factors, and the sensitivity of the obligor to the risk factors. The risk factors are common across all obligors, but sensitivity to the risk factors differs across obligors. Defaults across obligors covary only because of the K risk factors. Conditional on the risk factors, defaults are uncorrelated across obligors. The conditional probability of default for an obligor is the probability of default given the realizations of the risk factors, while the unconditional pobability of default is the probability obtained if we do not know the realizations of the risk factors. For example, if there is only one risk factor, say, macroeconomic activity, we would expect the conditional probability of default to be higher when macroeconomic activity is poorer. The unconditional probability of default in this case is the probability when we do not know whether macroeconomic activity is poor or not.

I f p i ( x ) is the probability of default for the ith obligor conditional on the realizations of the risk factors, and x is the vector of risk factor realizations, the model specifies that:

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Chapter 18

Credzt Rzsk~and Credzt Dertvatiues

,

- -

The expected loss and its dependence on volatility

> - < . , -

..

?=~;trr.rvc18.8

The firm has value of $120 million, the face value o f the debt is $100 million, the expected rate of change of firm value is 20 percent, and the interest rate is I 0 percent.

.4 F i r m volatility

where ~ q ; is the ) unconditional probability of default for obligor i given that it belongs to grade G. A natural choice of the grade of an obligor would be its public debt rating if it has one. Often, obligors may not have a rating, or the rating of the company may not reflect the riskiness of the debt. Bank debt has different covenants than public debt, which makes it easier for banks to intervene when the obligor becomes riskier. As a result, bank debt is less risky than otherwise comparable public debt. The bank internal evaluation system often grades loans on a scale from one to ten. A bank internal grading system could be used to grade obligors.

'The risk factors can take only positive values and are scaled so that they have a mean of one. 'l'he model also assumes that the risk factors follow a specific statistical distribution (the gamma distribution). If the kth risk factor has a realization above one, this increases the probability of default of firm i in proportion to the obligor's exposure to that risk factor measured by w;k. Once we have computed the probability of default for all the obligors, we can get the distribution of the total number of defaults in the portfolio. The relevant distribution is the distribution of losses. The model expresses the loss upon default for each loan in standardized units. A standardized unit could be $1 million. The exposure to the ith obligor, v ( i ) , would be an exposure of v(i) standardized units. A mathematical function gives the unconditional probability of a loss of n standardized units for each value of n. We can also get the volatility of the probabi1i.t~of a loss of n standardized units since an unexpectedly high realization of the vector of risk factors will lead to a higher than expected loss. The Creditask+ model is easy to use, largely because of some carefully chosen assumptions about the formulation of the probability of default and the distribution of the risk factors. The statistical assumptions of the model are such that

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

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Beyomd Plu~nVanilla Risk Manngenzeni

an increase in the volatility of the risk factor has a large impact on the tail of the distribution of the risk factor. Gordy (2000) provides simulation evidence on the CreditRisk+ model. His base case has 5,000 obligors and a volatility of the risk factor of 1. The distribution of grades for obligors is structured to correspond to the typical distribution for a large bank according to Federal Reserve Board statistics. He assumes that the loss upon default is equal to 30 percent of the loan for all loans. Losses are calculated as a percentage of outstanding loans. For a lowquality portfolio (the model rating is BB), the expected loss is 1.872 percent and its volatility is 0.565 percent. The distribution of losses is skewed, so the median of 0.769 percent is much lower than the expected loss. There is a 0.5 percent probability that the loss will exceed 3.320 percent. As the volatility of the risk factor is quadrupled, the mean and standard deviation of losses are essentially unchanged, but there is a 0.5 percent probability that the loss will exceed 4.504 percent.

18.3.2. CreditMetricsTM J.R Morgan's CreditMetrics-

offers an approach to evaluate the risk of large portfolios of debt claims on firms with realistic capital structures. To see how the CreditMetricsTMapproach works, we start from a single debt claim, show how we can measure the risk of the claim with the approach, and then extend the analysis to a portfolio of debt claims.2 Consider a debt claim on Almost Iffy Inc. We would like to measure the risk of the value of the debt claim in one year using VaR. To do that, we need to know the fifth quantile of the distribution of the value of the debt claim if we use a 5 percent VaR.

Our first step in using the CreditMetricsTM approach is to figure out a rating class for the debt claim. Say that we decide the claim should have a rating BBB. Almost I f v s debt could remain at that rating, could improve if the firm does better, or could worsen if default becomes more likely. There is a historical probability distribution that a claim with a BBB rating will move to some other rating. Across claims of all ratings, the rating transition matrix presented in Table 18.1 gives us the probability that a credit will migrate from one rating to another over one year. Such matrices are estimated and made available by rating agencies. For a debt claim rated BBB, there is a 1.17 percent probability that the debt claim will have a B rating next year. To obtain the distribution of the value of the debt claim, we compute the value we expect the claim to have for each rating in one year. Using the term structure of bond yields for each rating category, we can get today's price of a zero-coupon bond for a forward contract to mature in one year. Table 18.2 provides an example of one-year forward zero curves. The rows of the table give us the one-year forward discount rates that apply to zero-coupon bonds maturing in the following four years. We assume coupons are promised to be paid exactly in one year and at the end of each of the four subsequent years. Say that the coupon is $6. We can use the

2 The CreditMetricsTM Technical Manual, available on RiskMetrics website, analyzes the example used here in much greater detail. The data used here is obtained from that manual.

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Credzt Rzsks and Cvedzt Dertvatives

Chapter 18

Initial rating- AAA -

Rating at yeafiend (%)

AA

A

BBB

BB

B

CCC

Default

AAA AA A

90.8 1

8.33

0.68

0.06

0.12

0

0.70

90.65

7.79

0.64

0.06

0.14

0 0.02

0 0

0.09

2.27

9 1.05

5.52

0.70

0.26

0.0 1

0.06

BBB

0.02

0.33

5.95

86.93

5.30

1.17

0.12

0.18

0

0.1 I

0.24

0.43

6.48

83.46

4.07

5.20

0.22

0

0.22

1.30

2.38

1 1.24

64.86

19.79

B

CCC

forward zero curves to compute the value of the bond for each possible rating next year. For example, using Table 18.2, if the bond migrates to a RB rating, the present value of the coupon to be paid two years from now as of next year is $6 discounted at the rate of 5.55 percent. If the bond defaults, we need a recovery rate, which is the amount received in the event of default as a fraction of the principal. Suppose that the bond is a senior unsecured bond. Using historical data, the recovery rate for this type of bond is 5 1.13 percent. We can compute the value of the bond for each rating class next year and assign a probability that the bond will end up in each one of these rating classes. Table 18.3 shows the result of such calculations. A typical VaR measure would use the fifth percentile of the bond price distribution, which is a BB rating and a value of $102.02. The mean value of the bond is $107.09, so that the fifth percentile is $5.07 below the mean. Say that the price today is $108. There is a 5 percent chance we will lose at least $5.98.

---

-

---- ----, One-year forward zero curves f o r various rating classes (%)

7

Rating class AAA

-rr-

? --lr-.-*..---r.c"r----.

Year I

Year 2

Year 3

Year 4

3.60

4.17

4.73

AA

3.65

4.22

4.78

5.12 5.17

A BBB BB B CCC

3.72

4.32

4.93

5.32

4.10

4.67

5.25

5.63

5.55

6.02

6.78

7.27

6.05

7.02

8.03

8.52

15.05

15.02

14.03

13.52

------rrm

-IF-

-.- --

%bLe

---.". 18.2 n-

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Part 3

Tvmw,,

iTglh".m~~,F,lRy-TF--~~.v----mymm**rr-

Table 78.2

Beyond Plain Vanzlla RtsR Managanmt

nt-,

r---v-..

+----

- . - .-- --

-,?e"-." -.,-.+.

-->*"

z:=v

--

-V"q,

-r

7-

,.-.

vz-.-.

The value of the debt claim across rati classes and associated probabilities Y e a ~ e n dratine

AAA AA A BBB BB

B

CCC Default

Probability (?A)

0.02 0.33 5.95 86.93 5.30 1.17 0.12 0.18

Bond value plus coupon($)

109.37 1 09.19 108.66 107.55 102.02 98.10 83.64 51.13

If we have many claims, we have to make an assumption about the correlations among the various claims. If we know the correlations, we can measure the risk of a portfolio of debt claims using the distribution of the portfolio value. For example, suppose that we have an AAA bond and a B bond whose migration probabilities are independent. That is, knowing that the B bond migrates from a B rating to a BB rating provides no information about the likelihood that the AAA bond will migrate to an AA rating. We compute the probabilities of the transitions for each bond independently and multiply them to obtain the joint probability. Using the transition probability matrix in Table 18.1, we know that the probability of a B bond moving to a BB rating is 6.48 percent and that the probability of an AAA bond moving to an AA rating is 8.33 percent. The probability of these two events happening at the same time is the product of the probabilities of the individual events, 0.0648 x 0.0833 = 0.0054, or 0.54 percent. We can compute the value of the portfolio for that outcome. Once we have computed the value of the portfolio for each possible outcome as well as the probability of each outcome, we have a distribution for the value of the portfolio, and we can compute the fifth percentile to obtain a VaR measure. If the probabilities of the two bonds moving to a particular rating are not independent, the probability that the B bond moves to BB given that the AAA bond moves to AA is not the product of the probability of the B bond moving to BB and the probability of the AAA bond moving to AA. We have to know the probability that the two bonds will move that way. In other words, we need to know the joint distribution of bond migrations. Note that the values of the portfolio for each possible outcome are the same whether the bond migrations are independent or not. The probabilities of the various outcomes differ depending on the migration correlations. Once we know the probabilities of each outcome, we can compute a distribution for the bond portfolio and again compute its VaR. The major difficulty using the CreditMetricsTM approach is computing the joint distribution of the migrations of the bonds in the portfolio. One way is to use historical estimates for the joint probabilities of bond migrations. In other words, we could figure how often AAA bonds move to an AA rating and B bonds

Chapter 18

Credid Rijks and Credij Derivatives

move to a BB rating. This historical frequency would give us an estimate of the probability that we seek. Once we have the joint probability distribution of transitions for the bonds in the portfolio, we can compute the probability distribution of the portfolio. In general, though, the historical record of rating migrations will not be enough. The correlation among the rating migrations of two bonds depends on other factors. For example, firms in the same industry are more likely to migrate together. To improve on the historical approach, CreditMetricsTM proposes an approach based on stock returns. Suppose that a firm has a given stock price, and we want to estimate its credit risk. From the rating transition matrices, we know the probability of the firm moving to various ratings. Using the distribution of the stock return, we can compute ranges of returns corresponding to the various ratings-if there is a 5 percent probability of default, a default event corresponds to all stock returns that have a probability of at least 95 percent of being exceeded over the period over which credit risk is computed. Proceeding this way we can produce stock returns corresponding to the various rating outcomes for each firm whose credit is in the portfolio. The correlations between stock returns can then be used to compute probabilities of various rating outcomes for the credits. For instance, if we have rwo stocks, we can compute [he probability that one stock will be in the BB rating range and the other in the AA rating range. With a large number of credits, using stock returns to compute the joint distribution of outcomes is time-consuming. To simplify the computation, CreditMetricsTM recommends using a factor modej in which stock returns depend on country and industry indices as well as on unsystematic risk. The CreditMetricsTM technical manual shows how to implement such a model.

18.3.3. The KMV model KMV derives default probabilities using the "Expected Default Frequency" for each obligor from an extension of equation (18.18). KMV computes similar probabilities of default, but assumes a slightly more complicated capital structure in doing so. With KMV's model, the capital structure includes equity, short-term debt, long-term debt, and convertible debt. KMV then solves for the firm value and volatility. One advantage of the KMV approach is that probabilities of default are obtained using the current equity value, so that any event that affects firm value translates directly into a ching; in the probability bf default. Ratings change only with a lag. Another advantage is that probabilities of default change continually rather than only when ratiigs chang;. An increase in equity value reduces thk probability of default. In the CreditMetricsTM approach, the value of the firm can ;hang substantially, but the probability of d e f d t may remain the same because the firm's rating does not change.

KMV uses an approach inspired by the CAPM to obtain the expected growth of firm values that is required to implement equation (18.18) and uses a factor model to simplify the correlation structure of firm returns. The assumptions used imply an analytical solution for the loss distribution, so that simulation is not needed to compute a credit VaR with the KMV model.

--

Part 3

Beyorid Plain Vanrllcl Rirk Management

18.3.4. Some difficulties with credit portfolio models The credit portfolio models just discussed present an important advance in measuring credit risk. At the same time, however, the models as presented have obvious limitations. Some have addressed some of these limitations in implementing the models and other models have been developed trying to avoid some of these limitations, but these models as described are the most popular. Models in their most common implementations do not take into account changes in interest rates or credit spreads. Yet, we know that the value of a portfolio of debt can change both because of changes in default risk and changes in interest rates or credit spreads. Nor do the models do much to take into account current economic conditions. As the economy moves from expansion to recession, the distribution of defaults changes dramatically. For example, default numbers reached a peak in 199 1, a recession year, then fell before reaching another peak in 2001, another recession year. Further, the transition correlations increase in recessions. Models that use historical transition matrices cannot take into accounc changing economic conditions.

1 8.4. Credit derivatives Credit derivatives are financial instruments whose payoffs are contingent on credit risk realizations. For most credit derivatives, the payoff depends on the occurrence of a "credit event" for a reference entity. Generally, a credit event is (1) failure to make a required payment, (2) restr;cturing that makes any creditor worse off, (3) invocation of cross-default clause, and (4) bankruptcy. Generally, the required payment or the amount defaulted will have to exceed-a minimum value (e.g., $10 million) for the credit event to occur.3 Credit derivatives are designed as hedging instruments for credit risks. Consider a bank that has credit exposure to many obligors. Before the advent of loan sales and credit derivatives, banks managed their credit risk mostly through diversification. The problem with that approach to managing credit risk is that it forces a bank to turn down customers with which it has valuable relationships. With a credit derivative, a bank can make a loan to a customer and then hedge part or all of this loan by buying a credit derivative. A highly visible example of such a way to use credit derivatives is discussed in Box 18.2, Citigroup and Enron. Except for a futures contract discussed later, credit derivatives are not traded on exchanges. They are over-the-counter instruments. However, firms can also issue securities publicly that provide them with credit protection. The simplest credit derivative is a put that pays the loss on debt due to default at maturity. A put on the firm value with the same maturity as the debt and with an exercise price equal to the face value of the debt is a credit derivative, called a credit default put, that compensates its holder for che loss due to default if such a loss occurs. The put gives its holder the option to receive the exercise price in exchange of the debt claim. Since the put pays the loss incurred by the debt holder if default takes place, a portfolio of the risky debt and the put option is

3 For a description of the documentation of a credit derivative trade, see "Inside a credit trade," Derivatives Strategy, 1998 (December), 24-28.

Credit Risks and Credzt Derivatives

Chapter 18

derable e'xposure Citigroup h 1 in 200(1. At tha t time, % was a successful company. It had equity capitafization in excess. of $50 billion ar the start of the year. Its net income for the year was $979 milliofi. Eriron's senior unsecured debt's rating was upgraded, m that it finished the year rated, -. . BAA1 by Moody's and BBB+ by Standard and 'Poor's. Despite all rhis, group chose to issue securities for $1.4 billion from August 2000 to May t h a ~effectively hedged Citigroup's exposure to Enrom.

,

Enron's senior unsecured debt kept its ratings until October 2001. In December 2001, Enron's rating was a D; it was bankrupt. Citigroup had a loan exposure of $1.2 billion. It also had some insurance-related obligations. It had c dlaterd for about Ihalf of the loan -exposure. Most likiely, its p 1 :losses we1re covereld by the securities it had i,ssued.

.

. TL,llr;ac ,, ,*..".*.,,. a, r;uuluco

- .

vvorked' as fol.lows. C i t i b a n ~uadccd a trust. 1m a L ~LbLL 13sued five-year notes with.fixed interest paymenrs. The proceeds were iri\irested in high-quality debt. If Ehron did not go bankrupt, the investors would rt:ceive A- principal after five yeax. Tf Enron did " go bankrupt, Citigroup had the ,.:,-At " swap En1:on's deb t tk Citigyoup for the secu rities in Ithe trust. 'A

7.

1

~ of 7.37 percel~ tAt Citigrou]3 prom is^ed a c o uJon . the: time, B1 were promising 8.07 percent. However, according to a presentation by EI~ron's treasurer to. Standard and Poor's in 2000, Enron debt was trading above iits rating, which-led him to pitch a rating of AA. At the same time, he explaineii that the off-balance sheet debt was not material to E n ~ o n . Source Daniel Altman,"How C~t~group hedged bets on Enron," New York T~mes,February 8,2002

equivalent to holding default-free debt since the risk of the debt is offset by the purchase of the put. We already priced such a put when we valued In-The-Mail's deb.t, since that debt was worth risk-free debt minus a put. The holder of In-TheMail debt was short a put on firm value; by buying the credit default put, the holder of the debt eliminates his credit risk by acquiring an offsetting position in the same put. The most popular credit derivatives involve swap contracts.* One type of contract is called a credit default swap. With this swap, party A makes a fixed annual payment to party B, while party B pays the amount lost if a credit event occurs. For example, if Supplier Inc. in our example wants to get rid of the credit risk of In-The-Mail Inc., it can enter a credit default swap with a bank. It makes fixed payments to the bank. If In-The-Mail Inc. defaults at maturity, the bank pays Supplier Inc. the shortfall due to default (face value minus fair value of the debt at the time of default). The credit default swap for Supplier Inc. is effectively equivalent to buying a credit default put but paying for it by installment. Note

4

Dominrc Baldwin, "Business is boom~ng," Credrt Risk Spec~alReport,

Risk, April 1999, 8.

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Part 3

Beyo~zdPlain Vanilla Risk Manngement

that the debt will in general require interest payments before maturity and covenants to be respected. If the obligor fails in fulfilling its obligations under the debt contract, there is a credit event. With the credit event, the default payment becomes due. The credit default swap can have physical delivery, so that Supplier Inc. would sign over the loan to the bank in the event of a default and would receive a fixed payment. Physical delivery is crucial for loans that do not have a secondary market, but physical delivery of loans involves tricky and time-consuming issues. Borrowers often object to having their loans signed over. The settlement period for a credit default swap with physical delivery tends to be longer than for a bond trade. If the transfer of a loan faces objections from the borrower, the settlement might extend beyond 30 days.5 A credit default exchange swap requires each party to pay the default shortfall on a different reference asset. Two banks might enter a credit default exchange swap for Bank A to pay the shortfall on debt from Widget Inc. and Bank B to pay the shortfall on debt from In-The-Mail Inc. This way, BankA reduces its exposure to In-The-Mail Inc. and Bank B reduces its exposure to Widget Inc. Another popular structure is the total return swap. The party seeking to buy insurance against credit risks receives the return on a risk-free investment and pays the return on an investment with default risk. Suppose a bank, the protection buyer, owns a debt claim worth $80 million today that pays interest of $6 million twice a year in the absence of default for the next five years. In a total return swap, the bank pays what it receives from the debt claim every six months. Assuming that the issuer of the debt claim is not in default, the bank pays $6 million every six months. If the issuer does not pay interest at some due date, then the bank pays nothing. In return, the bank might receive six-month LIBOR on $80 million. At maturity, the obligor repays the principal if he is not in default. Suppose the principal is $100 million. In this case, the bank gets a payment at maturity of $20 million corresponding to the final payment of $100 million minus the initial value of $80 million to the swap counterparty. Or, if the obligor is in default and pays only $50 million, the protection buyer receives from the swap counterparty $80 million minus $50 million, or $30 million. This total return swap guarantees to the bank the cash flows equivalent to the cash flows of a risk-free investment of $80 million. Pricing a total return swap is straightforward, since it is effectively the exchange of a risky bond for a default-free bond. At initiation, the two bonds have to have the same value. Another credit derivative is a futures contract. The Chicago Mercantile Exchange Quarterly Bankruptcy Index (QBI) futures contract has been traded since April 1998. The QBI is the total of bankruptcy filings in U.S. courts over a quarter. Most bankruptcies are filed by individuals, which makes the contract appropriate to hedge portfolios of consumer debts, such as credit card debt. The contract is cash settled and the index level is the number of bankruptcy filings in thousands during the quarter preceding contract expiration. At matu-

5

Dwight Case, "The dev~l'sIn the details," Risk, August 2000, 26-28.

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Chapter 18

rity, the futures price equals the index level. The settlement variation is the change in the futures price times $1,000. The minimum increment in the futures price is $0.025.

18.5. Credit risks of derivatives Since the value of an option is never negative whereas a swap can alternate beween positive and negative values, it is not surprising thar the credit risks of options are easier to evaluate than the credit risks of swaps. An option with default risk is called a d n e r a b l e option. At maturity, the holder of an option receives the promised payment only if the writer can make the payment, Suppose rhe writer is a firm with value V and the option is a European call on a stock with price S. The exercise price of the call is K. Without dehull: 'isk, the option holder receives Max(S - K, 0 ) at maturity. With a vulnerable oprion, the holder receives the promised payment only if it is smaller than V, so that the payoff of the call is: Max[Min(V, S - Kl, 01

(18.21)

The current value of the call with default risk is just the present value of this payment. There is no closed-form solution for su& an optLon, but it is not diffi;cdt to evaluate its value using a Monte Carlo simulation. The cotrelacion between the value of the fum and the value of the option's underlying asset plays an extremely important role in valuation of the vulnerable option. Suppose that V and S are strbngly negatively correlated. In this case, it could be that the option has lictle value because V is low when the option pays off. I f V and S are strongly positively correlated, the option might have almost no credit risk because V is always high when S is high. If an option has credit risk, it becomes straightforward to write an option contract that eliminates that credit risk. The appropriate credit derivative is one that & pays. the difference between a call without default risk and the vulnerable call:

If we can price the vulnerable call, we can also price the credit derivative that insures the call.

An alternative approach is to compute the probability of default and apply a recovery rate if default occurs. In this case, the option is a weighted average of an option without defaulc risk and of rhc present value of the payoff if default occurs. Say that the option can default only at maturity and do& ;o with probability p. If default occurs, the holder receives a fraction z of the vaIue of the option. In this case, the value of the option today is (1 - p) c + pzc, where c is the value of the option without default risk. This approach provides a rather simple way to incorporate credit risk in the value of the option when the probability of default is independent of the value of the underlying asset of the option. Say that the probability of default is 0.05 and the recovery rate is 50 percent. In this case, the vulnerable call is worth

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Beyond Plain Vanilla Risk Managonat

- 0.05) c +

0.05 x 0.5 x c, which is 97.5 percent of the value of the call without default risk.

(I

As we saw in Chapter 16, it is often the case that the counterparties to a swap enter margin arrangements to reduce default risk. Nevertheless, swaps can entail default risk for both counterparties. Netting means that the payments between the two counterparties are netted out, so that only a net payment has to be made. We assume that netting takes place and that the swap is treated like a debt claim. If the counterparty due to receive net payments is in default, that counterparty still receives the net payments. This is called the full two-way payment covenant. In a limited two-way payment covenant, the obligations of the counterparties are abrogated if one party is in default. With these assumptions, the analysis is straightforward when the swap has only one payment. Suppose a market maker enters a swap with a risky credit. The risky credit receives a fixed amount F at maturity of the swap-the fixed leg of the swap-and pays S. S could be the value of equity in an equity swap or could be a floating rate payment determined on some index value at some point after the swap's inception. Let V be the value of the risky credit net of all the debt that is senior to the swap. In this case, the market maker receives S-F in the absence of default risk. This amount can be positive or negative. If the amount is negative, the market maker pays F-S to the risky credit for sure. If the amount is positive, the market maker receives S-F if that amount is less than V, The swap's payoff to the market maker is:

The payment that the risk-free counterparty has to make, F, is chosen so that the swap has no value at inception. Since the risk-free counterparty bears the default risk, in that it may not receive the promised payment, it reduces F to take into account the default risk. To find F, we have to compute the present value of the swap payoff to the market maker. The first term in equation (18.23) is minus the value of a put with exercise price F on the underlying asset whose value is S. The second term is the present value of an option on the minimum of two risky assets. Both options can be priced. The correlation between V a ~ l dS plays a crucial role. As this correlation falls, the value of the put is unaffected, but the value of the option on the minimum of two risky assets falls because for a low correlation it will almost always be the case that one of the assets has a low value. Swaps generally have multiple payments, however, so this approach will work only for the last period of the swap, which is the payment at T, At the payment date before the last payment date, T - At, we can apply our approach. At T - 2A t, however, the market maker receives the promised payment at that dare plus the promise of two more payments: the payment at T - At and the payment at T. The payment at T - A t corresponds to the option portfolio of equation (18.23), but at T - At the market maker also has an option on the payment of date T which is itself a portfolio of options. In this case, rather than having a compound option, we have an option on a portfolio of options. Valuation of an option on a

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Cvedit Rixkx and Creclit Derivatives

Chapter 18

-

portfolio of options is difficult to handle analytically, but as long as we know the dynamics that govern the swap payments in the default-free case as well as when default occurs, we can use Monte Carlo analysis.

18.6. Summary We have developed merhods to evaluate credit risks for individual risky claims, for portfolios of risky claims, and for derivatives. The Merton model allows us to price risky debt by viewing it as risk-free debt minus a put written on the firm issuing the debt. The Merton model is practical mostly for simple capital structures with one debt issue that has no coupons. ,Other approaches to pricing risky debt model the probability of default and then discount the risky cash flows from debt using a risk-neutral distribution of the probability of default. Credit risk models such as the ~ r e d i t ~ i s kmodel, + the CreditMetricsTM model, and the KMV model provide approaches to estimating the VaR for a portfolio of credits. Credit derivatives can be used to hedge credit risks.

credit default swap, 597 credit event, 596 credit risk, 572 credit spread, 575 CreditMetricsTM,588 KMV model, 595

?&died

loss given default (LGD), 5 89 obligors, 589 rating transition matrix, 592 recovery rate, 593 vulnerable option, 599

@estio~s

1. Vlrhy is equity an option? 2. Why are you doing the equivalent of writing a put when you buy risky debt?

3. What is Merton's model for the pricing of debt?

4. What is a compound option? 5. How can you use the risk-neutral distribution of default to price coupon debt?

6. Why is it not reasonable to estimate the VaR of a portfolio of debt using 1.65 times the volatility of the return of the portfolio?

7. What is CreditRisk+? 8. What is CreditMetricsm?

9. What is a credit-default swap? 10. Why is it more difficult to evaluate the credit risk of a swap than to evaluate the credit risk of a risky zero-coupon bond?

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

@----

Part 3

Beyond Plain Vanilla Rirk Managnnent

1 . Debt can be priced using the Merton model. You are told that an increase in the volatility of firm value can increase the value of senior discount debt on the firm because in some cases debt is like equity and equity increases in value as firm volatility increases. Is this argument correct? 2. Consider a firm with a debt payment due in five years. The debt is not traded, so that firin value does not correspond to a traded asset. The assumptions that make the Merton model work hold. The longest maturity option traded on the firm's equity matures in one year. Assume that you know the volatility of the return of the firm and you know the firm value. How could you form a portfolio strategy that pays the same amount as the debt in five years? Extra credit: Provide portfolio weights if firm value volatility is 50 percent, interest rate is constant at 5 percent, firm value is $1 billion, and face value of debt is $1.2 billion.

3. Given your answer in question 2, what are the implications for your portfolio strategy of an unexpected increase in the firm's stock price caused by an increase in firm value? Extra credit: Using the data of the previous question, show how the portfolio changes if equity value increases by 10 percent because of a change in firm value.

4. Assume again that the assumptions that make the Merton model work hold, except that the firm pays a known dividend before maturity of its only debt, which is discount debt. How can you quantify the impact of a known future dividend payment on the value of discount debt? Extra credit: Using the data in the extra credit portion of question 3, assume a dividend payment of $1 00 million in three years. Wliat is the impact of that payment on firm value?

5. Suppose that you compute a one-year VaR of a portfolio using the deltaVaR approach. The assumptions that make the Merton model work hold. You want to include in your VaR computation discount debt that matures in sixteen months. Somebody points out to you that your VaR computation has a significant probability that the discount debt will pay more than par. Is that right?

6 . Attending a conference on credit risk, an attendee tells you that he computes the-probability of default for a firm as the cumulative normal that multiplies P,(T)F in equation (18.3). Has he discovered a better mousetrap or does he have a biased estimate of the probability of default? If he has a biased estimate, what is the bias? Extra credit: Using the data from question 3, compute the probability of default and the expected loss for the debt. How does your answer differ from the one proposed by the conference attendee?

7. At the same conference, you find an old friend. She emphatically declares that, since her firm does not mark loans to market, she uses CreditRisk+

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Chapter 18

Credit R i s b and Credit Deriuutzver

because CreditMetricsTM would be useless for her. What arguments could you make for her to use CreditMetricsm?

8. You are told by your boss that there is no reason for your balk to use a credit risk model because the bank has thousands of obligors, so that default risk is diversifiable. Is this argument correct if Merton's model applies to each of the obligors?

9. In an advertisement for credit derivatives, it is argued that the benefit of a credit default swap for the protection seller is that he "earns investment income with no funding cost."6 How would you analyze the impact on a bank of selling credit protection through a credit default swap?

10. You want to enter a swap with a bank. The bank does not want to make the trade because their credit risk model implies that the swap has too much credit risk. The bank agrees to enter the swap if it is marked to market, but your boss objects to such an arrangement. Why does the bank find such an arrangement more agreeable than a swap that is not marked to market? What arguments could your boss have to reject the arrangement?

Black and Sclioles (1 973) had a brief discussion of the pricing of risky debt. Merton (1974) provides a detailed analysis of the pricing of risky debt using the Black-Scholes approach. Black and Cox (1976) derive additional results, including the pricing of subordinated debt and the pricing of debt with some covenants. Geske (1977) demonstrates how to price coupon debt using the compound option approach. Stulz and Johnson (1985) show the pricing of secured debt. Longstaff and Schwartz (1995) extend the model so that default takes place if firm value falls below some threshold.. Their model takes into account interest rate risk as well as the possibility that strict priority rules will not be respected. Arnin and Jarrow (1992) price risky debt in the presence of interest rates changing randomly using the Black-Scholes approach with a version of the HeathJarrow-Morton model. Duffie and Singleton (1999) provide a detailed overview and extensions of the approaches that model the probability of default. Applications of this approach show that it generally works quite well. Perhaps the easiest application to follow is the work of Das and Tufano (1996). They extract probabilities of default from historical data on changes in credit ratings. Armed with these probabilities and with historical evidence on recovery rates and their correlations with interest rates, they price corporate debt. lnstead of using historical estimates of default probabilities and recovery rates, they could have extracted .ihese parameters from credit spreads and their study discusses how this could be done. Jarrow and Turnbull (1995) build an arbitrage model of risky debt where the probability of default can be obtained from the firm's credit spread curve. Jarrow, Lando, and Turnbull (1997) provide a general approach using credit ratings. Another

6 Barclays Cap~tal,"Applying credit derivatives to emerging markets," Cred~tR ~ s kSpecial Report, Advertising Supplement, Risk, November 1998.

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

Part 3

Beyond Plaifz Vanilla Risk Manage?nent

interesting application is Duffie and Singleton (1997), who use this approach to price credit spreads embedded in swaps. Madan and Unal (1998) price securities of savings and loan associations. They show how firm-specific information can bc incorporated in the default probabilities. These various approaches to pricing risky claims have rather weak corporate finance underpinnings. They ignore the fact that firms act differently when their value falls and that they can bargain with creditors. Several recent papers take strategic actions of the debtor into account. Leland (1994) models the firm in an intertemporal setting taking into account taxes and the ability to change volatility. Anderson and Sundaresan (1996) take into account the ability of firms to renegotiate on the value of the debt. Deviations from the doctrine of absolute priority by the courts are described in Eberhart, Moore, and Roenfeldt (1990). Crouhy, Galai, and Mark (2000) provide an extensive comparative analysis of the CreditRisk+, CreditMetricsm, and KMV models. Gordy (2000) provides evidence on the performance of the first two of these models. Jarrow and Turnbull (2000) critique these models and develop an alternative. Johnson and Stulz (1987) were the first to analyze vulnerable options. A number of papers provide formulas and approaches to analyzing the credit risk of derivatives. Jarrow and Turnbull (1995) provide an approach consistent with the use of the HJM model. Jarrow and Turnbull (1997) show how the approach can be implemented to price the risks of swaps. The CME-QBT contract is discussed in Arditti and Curran (1998). Longstaff and Schwartz (1995) show how to value credit derivatives.

From From Risk Risk Management Management andand Derivatives, Derivatives, 1st1st edition, edition, by by René René M.M. Stulz. Stulz. © 2003. © 2003. Reprinted Reproduced with with permission permission of South-Western, of South-Western, a division a division of Thomson of Thomson Learning: Learning: www.thomsonrights.com. www.thomsonrights.com. Fax 800 Fax730-2215. 800 730-2215.

This chapter has permission been reproduced with permission from Market, Understanding Market, Credit and Operational Risk: The At Reproduced with from Understanding Credit and Operational Risk: The Value Value at Risk Approach, by Linda Allen, Jacob Boudoukh and Anthony Saunders, published by Blackwell Risk Approach, by Linda Allen, Jacob Boudoukh, Anthony Saunders. © 2004 Blackwell Publishing. Publishing, Oxford 2004 (© Blackwell Publishing, all rights reserved).

EXTENDING THE VaR APPROACH TO OPERATIONAL RISKS

5.1 Top-Down Approaches to Operational Risk Measurement 5.1.1 Top-down vs. bottom-up models 5.1.2 Data requirements 5.1.3 Top-down models 5.2 Bottom-Up Approaches to Operational Risk Measurement 5.2.1 Process approaches 5.2.2 Actuarial approaches 5.2.3 Proprietary operational risk models 5.3 Hedging Operational Risk 5.3.1 Insurance 5.3.2 Self -insurance 5.3.3 Hedging using derivatives 5.3.4 Limitations to operational risk hedging 5.4 Summary Appendix 5.1 Copula Functions

All business enterprises, but financial institutions in particular, are vulnerable to losses resulting from operational failures that undermine the public's trust and erode customer confidence. The list of cases involving catastrophic consequences of procedural and operational lapses is long and unfortunately growing. To see the implications of operational risk events one need only look at the devastating loss of reputation of Arthur Andersen in the wake of the Enron scandal, the loss of independence of Barings Bank as a result of Nick Leesonrs rogue trading operation, or UBS' loss of US$100 million due to a trader's

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159

error, just to name a few examples.' One highly visible operational risk event can suddenly end the life of an institution. Moreover, many, almost invisible individual pinpricks of recurring operational risk events over a period of time can drain the resources of the firm. Whereas a fundamentally strong institution can often recover from market risk and credit risk events, it may be almost impossible to recover from certain operational risk events. Marshall (2001) reports that the aggregate operational losses over the past 20 years in the financial services industry total approximately US$200 billion, with individual institutions losing more than US$500 million each in over 50 instances and over US$1 billion in each of over 30 cases of operational f a i l ~ r e sI.f~anything, the magnitude of potential operational risk losses will increase in the future as global financial institutions specialize in volatile new products that are heavily dependent on technology. Icingsley et al. (1998) define operational risk to be the "risk of loss caused by failures in operational processes or the systems that support them, including those adversely affecting reputation, legal enforcement of contracts and claims" (p. 3). Often this definition includes both strategic risk and business risk. That is, operational risk arises from breakdowns of people, processes, and systems (usually, but not limited to technology) within the organization. Strategic and business risk originate outside of the firm and emanate from external causes such as political upheavals, changes in regulatory or government policy, tax regime changes, mergers and acquisitions, changes in market conditions, etc. Table 5.1 presents a list of operational risks found in retail banking. Operational risk events can be divided into high frequencyllow severity (HFLS) events that occur regularly, in which each event individually exposes the firm to low levels of losses. In contrast, low frequencylhigh severity (LFHS) operational risk events are quite rare, but the losses to the organization are enormous upon occurrence. An operational risk measurement model must incorporate both HFLS and LFHS risk events. As shown in figure 5.1, there is an inverse relationship between frequency and severity so that high severity risk events are quite rare, whereas low severity risk events occur rather frequently. In order to calculate expected operational losses (EL),one must have data on the likelihood of occurrence of operational loss events (PE) and the loss severity (loss given event, LGE), such that EL = PE x LGE. Expected losses measure the anticipated operational losses from HFLS events. VaR techniques can be used to measure unexpected losses.

Reproduced with permission from Understanding Market, Credit and Operational Risk: The Value At Risk Approach, by Linda Allen, Jacob Boudoukh, Anthony Saunders. © 2004 Blackwell Publishing.

Table 5.1 Operational risk categories Process risk Pre-transaction: marketing risks, selling risks, new connection, model risk Transaction: error, fraud, contract risk, product complexity, capacity risk Management information Erroneous disclosure risk People risk Integrity: fraud, collusion, malice, unauthorized use of information, rogue trading Competency Management Key personnel Health and safety Systems risk Data corruption Programming errorslfraud Security breach Capacity risks System suitability Compatibility risks System failure Strategic risks (platformlsupplier) Business strategy risk Change management Project management Strategy Political External environmental risk Outsourcing/external supplier risk Physical security Money laundering Compliance Financial reporting Tax Legal (litigation) Natural disaster Terrorist threat Strike risk Source: Rachlin (1998), p. 127.

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161

123 276 623 1,406 3,885 10,73736,365 123,163 51 1,184 2,600,000

.c

I

4

Empirical distribution

I

+

Fit to model distribution

I

+

Fit using EVT

Severity of loss events ($'000)in logorithrnic scale

Figure 5.1 Frequency and severity of loss events Source: Ceske and Hernandez (1999), p. 19.

However, LFHS events typically fall in the area of the extreme tail (the area fit using extreme value theory (EVT) shown in figure 5.1). Analysis of operational risk requires all three measures. The typical risk assessment period in these operational risk measurement models is assumed to be one year.

5.1 TOP-DOWN APPROACHES TO OPERATIONAL RISK MEASUREMENT Financial institutions have long articulated the truism that "reputation is e~erything."~ Particularly in businesses that deal with intangibles that require public trust and customer confidence, such as banking, loss of reputation may spell the end of the institution. Despite this recognition (unfortunately often limited to the firm's advertising campaign), banks and other financial institutions have been slow at internalizing operational risk measurement and management tools to protect their reputational capital. As backward as financial firms have been in this area, nonfinancial firms are often even less sophisticated in assessing potential operational weaknesses.

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UNDERSTANDING MARKET, CREDIT, AND OPERATIONAL RISK

5.1 .I

Top-down vs. bottom-up models

Historically, operational risk techniques, when they existed, utilized a "top-down" approach. The top-down approach levies an overall cost of operational risk to the entire firm (or to particular business lines within the firm). This overall cost may be determined using past data on internal operational failures and the costs involved. Alternatively, industry data may be used to assess the overall severity of operational risk events for similar-sized firms as well as the likelihood that the events will occur. The top-down approach aggregates across different risk events and does not distinguish between HFLS and LFHS operational risk events. In a top-down model, operational risk exposure is usually calculated as the variance in a target variable (such as revenues or costs) that is unexplained by external market and credit risk factors. The primary advantage of the top-down approach is its simplicity and low data input requirements. However, it is a rather unsophisticated way to determine a capital cushion for aggregate operational losses that may not be covered by insurance. Nonetheless, top-down operational risk measurement techniques may be appropriate for the determination of overall economic capital levels for the firm. However, top-down operational risk techniques tend to be of little use in designing procedures to reduce operational risk in any particularly vulnerable area of the firm. That is, they do not incorporate any adjustment for the implementation of operational risk controls, nor can they advise management about specific weak points in the production process. They over-aggregate the firm's processes and procedures and are thus poor diagnostic tools. Top-down techniques are also backward looking and cannot incorporate changes in the risk environment that might affect the operational loss distribution over time. In contrast to top-down operational risk methodologies, more modern techniques employ a "bottom-up" approach. As the name implies, the bottom-up approach analyzes operational risk from the perspective of the individual business activities that make up the bank's or firm's "output." That is, individual processes and procedures are mapped to a combination of risk factors and loss events that are used to generate probabilities of future scenariosr o c ~ u r r e n c e HFLS .~ risk events are distinguished from LFHS risk events. Potential changes in risk factors and events are simulated, so as to generate a loss distribution that incorporates correlations between events and processes. Standard

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Table 5.2 Top -down and bottom-up operational risk measurement models Operational risk problem

Primarily use top-down or bottom-up model

Operational risk model recommended

Control Mitigation Prevention Economic capital

Bottom-up Bottom-up Bottom-up Top-down

Regulatory capital Efficiency optimization

Top-down Top-down and bottom-up

Process approach Process approach Process approach Multi-factor, scenario analysis Risk profiling Risk profiling and process approach

Source: Adapted from Doerig (2000), p. 95.

VaR and extreme value theory are then used to represent the expected and unexpected losses from operational risk exposure. Bottom-up models are useful to many constituencies within the firm - from the internal risk auditor to the line area middle managers to the operations staff. Results of the analysis may be utilized to correct weaknesses in the organization's operational procedures. Thus, bottomup models are forward looking in contrast to the more backward looking top-down models. The primary disadvantages of bottom-up models are their complexity and data requirements. Detailed data about specific losses in all areas of the institution must be collected so as to perform the analysis. Industry data are required to assess frequencies both for LFHS and HFLS events. Moreover, by overly disaggregating the firm's operations, bottom-up models may lose sight of some of the interdependencies across business lines and processes. Therefore, neglecting correlations may lead to inaccurate results since many of the operational risk factors have a systematic component. Most firms that have operational risk measurement programs use both top-down and bottom-up operational risk measurement model^.^ Table 5.2 shows how both top-down and bottom-up models can be used to address different operational risk problems.

5.1.2

Data requirements

The operational risk measurement methodology that is chosen is often determined by data availability. Senior (1999) interviewed top

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UNDERSTANDING MARKET, CREDIT, AND OPERATIONAL RISK

managers at financial firms and found that the biggest impediment to the implementation of precise operational risk measurement models is the absence of accurate data on operational risk events. Ceske and Hernandez (1999) present four choices for obtaining data inputs: internal collection of data, external data, simulating data using educated estimates, and extrapolating data based on limited samples. Internal data are most applicable to the individual institution and are therefore the most useful in determining the firm's operational loss distribution. However, internal data are biased toward HFLS events. It is likely that there will be no LFHS events at all in the internal database, simply because many firms do not survive the catastrophic losses associated with this type of operational risk events. Moreover, it is extremely costly and time-consuming to develop a historical internal database on operational risk events. Thus, internal data should be supplemented with external data obtained from other institutions. This expands the database to include more LFHS events, particularly if the scope of the external database is industry-wide. However, external data must be scaled and adjusted to reflect institutional differences in business unit mix, activity level, geography and risk control mechanisms across firms. Moreover, competing firms are reluctant to release sensitive and detailed information about their internal processes and procedures to competitors. Ceske and Hernandez (1999) advocate the creation of a data consortium for financial insti~ tutions along the lines of the insurance and energy i n d ~ s t r i e s ."The database would contain information on non-public, internal, operational loss events, with the sources of the losses concealed. This would help financial institutions to learn the lessons from operational risk failures at other institutions" (Ceske and Hernandez, 1999, p. 18). Thus, individual firm confidentiality would be preserved while minimizing the cost of developing a comprehensive database on operational risk events for financial institution^.^ However, Ong (1998) argues against this emphasis on data collection because it would only encourage "follow the pack" decision making that would not necessarily improve risk management. Another source of data is obtained from management-generated loss scenarios. These scenarios emanate from either educated estimates by operational line managers or from extrapolation from smaller databases. Using either of these methods, management must construct frequency and severity estimates from individual operational risk events across individual business lines using bootstrapping and

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,

165

jackknife methodologies in order to construct "synthetic data points."' The operational risk loss distribution is then obtained by considering all possible imaginable scenarios. The distribution can be specified using either parametric models or may be based on non-parametric, empirical distributions. Empirical distributions may not be representative and the results may be driven by outliers. In practice, loss severity is typically modeled using lognormal, gamma or Pareto distributions, although the uniform, exponential, Weibull, binomial and beta distributions are sometimes used.9 For catastrophic losses (in the fat tails of the distribution), extreme value theory is used. Loss frequency parametric distributions such as Poisson, beta, binomial, and negative binomial are most often used (see discussion in section 5.2.2.2). However, the current state of data availability still does not permit long run backtesting and validation of most operational risk measurement models.

5.1.3 Top-down models The data requirements of top-down models are less onerous than for bottom-up models.1° Top-down models first identify a target variable, such as earnings, profitability or expenses. Then the external risk (e.g., market and credit risk) factors that impact the target variable are modeled, most commonly using a linear regression model in which the target variable is the dependent variable and the market and credit risk factors are the independent variable. Operational risk is then calculated as the variance in value of the target variable that is unexplained by the market and credit risk factors (i.e., the variance in the residual of the regression that is unexplained by the independent variables). Sometimes operational risk factors are directly modeled in the regression analysis. Then operational risk is calculated as the portion of the target variable's variance explained by the operational risk independent variable.

5.1.3.1 Multi-factor models One top-down model that can be estimated for publicly traded firms is the multi-factor model. A multi-factor stock return generating function is estimated as follows:

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UNDERSTANDING MARKET, CREDIT, AND OPERATIONAL RISK

where R, is the rate of return on firm i's equity; I,,, I,,, and I,, are the external risk factor indices (i.e., the change in each market and credit risk factor at time t); PIi, P2i and p,, are firm its sensitivity to changes in each external risk factor; and is the residual term. The risk factors are external to the firm and include as many market and credit risk factors as possible (i.e., interest rate fluctuations, stock price movements, macroeconomic effects, etc.). The multi-factor model measures operational risk as 0;= (1 - R2)c: where of is the variance of firm irs equity return from equation (5.1) and R, is the regression's explanatory power. The multi-factor model is easy and inexpensive to estimate for publicly traded firms. However, as in most top-down models, it cannot be used as a diagnostic tool because it does not identify specific risk exposures. More importantly, however, the multi-factor model is useful in estimating the firm's stock price reaction to HFLS operational risk events only. In contrast, LFHS events often have a catastrophic impact on the firm (often leading to bankruptcy or forced merger) as opposed to the marginal decline in equity returns resulting from the HFLS operational risk events that are measured by equation (5.1). Thus, the multi-factor model does not perform well when large scale events (such as mergers or catastrophic operational risk events) break the continuity of equity returns.', 5.1.3.2

Income-based models

Also known as Earnings at Risk models, income-based models extract market and credit risk from historical income volatility, leaving the residual volatility as the measure of operational risk. A regression model similar to equation (5.1) is constructed in which the dependent variable is historical earnings or revenues. Since long time series of historical data are often unavailable, income-based models can be estimated using monthly earnings data, in which annualized earnings are inferred under that assumption that earnings follow a Wiener process. Thus, monthly earnings volatility can be annualized by multiplying the monthly result by 'lt where t = 12. Since earnings for individual business lines can be used in the incomebased model, this methodology permits some diagnosis of concentrations of operational risk exposure. Diversification across business lines can also be incorporated. However, there is no measure of opportunity cost or reputation risk effects. Moreover, this methodology is sensitive to HFLS operational risk events, but cannot measure LFHS risk events that do not show up in historical data.

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5.1.3.3

167

Expense-based models

The simplest models are expense-based approaches that measure operational risk as fluctuations in historical expenses. Historical expense data are normalized to account for any structural changes in the organization.13Unexpected operational losses are calculated as the volatility of adjusted expenses. The primary disadvantage of expensebased models is that they ignore all operational risk events that do not involve expenses, e.g., reputational risk, opportunity costs, or risks that reduce revenues. Moreover, improving the operational risk control environment may entail increased expenses. Thus, expense-based models would consider the implementation of costly risk control mechanisms as an increase, rather than a decrease in operational risk exposure. Finally, since organizational changes are factored out of the analysis, expense-based models do not consider structural operational risk exposure (e.g., the operational risks of new business ventures).

5.1.3.4

Operating leverage models

A class of models that joins both the income-based and expense-based

approaches is the operating leverage model. Operating leverage measures the relationship between operating expenses (variable costs) and total assets. Marshall (2001) reports that one bank estimated its operating leverage to be 10 percent multiplied by the fixed assets plus 25 percent multiplied by three months of operating expenses. Another bank calculated its operating leverage to be 2.5 times the monthly fixed expenses for each line of business. Operating leverage risk results from fluctuations from these steady state levels of operating leverage because of increases in operating expenses that are relatively larger than the size of the asset base. Data are readily available and thus the model is easy to estimate. However, as is the case with income-based and expense-based models, the operational risk measure does not measure nonpecuniary risk effects, such as the loss of reputation or opportunity costs.

5.1.3.5

Scenario analysis

Scenario analysis requires management to imagine catastrophic operational shocks and estimate the impact on firm value. These scenarios focus on internal operations and try to estimate the impact of LFHS operational risk events, such as a critical systems failure, major

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UNDERSTAND!NG MARKET, CREDIT, AND OPERATIONAL RISK

regulatory changes, losses of key personnel, or legal action. Marshall (2001) enumerates some possible scenarios: (i) the bank's inability to reconcile a new settlement system with the original system, thereby preventing its implementation (such as in the case of the TAURUS system cancellation by the London Stock Exchange in 1993 resulting in a US$700 million loss); (ii) a class action suit alleging incomplete disclosure (such as in Merrill Lynch's exposure to allegations about conflicts of interest affecting the accuracy of its stock recommendations resulting in a US$100 million fine plus pending legal action); (iii) a significant political event (such as the overthrow and reinstatement of Venezuela's president); (iv) massive technology failure (such as eBayrs internet auction failure that reduced market value by US$5 billion in 1999); (v) non-authorized trading (such as Barings Bank's losses of US$1.6 billion in 1995); and many others. The enumeration of scenarios is only limited by management's imagination.14 The primary advantage of scenario analysis is its incorporation of LFHS operational risk events that may not have transpired as of yet. This is also the model's primary disadvantage, however. Scenario analysis is by its very nature subjective and highly dependent on management's subjective assessment of loss severity for each operational risk scenario. Moreover, it comprises a laundry list of operational risk events without attaching a likelihood estimate to each event. Thus, scenario analysis is often used to sensitize management to risk possibilities, rather than strictly as an operational risk measure.

5.1.3.6

Risk profiling models

Risk profiling models directly track operational risk indicators. Thus, they do not use income or expenses as proxies for operational risk, but rather measure the incidence of risk events directly. For example, commonly used operational risk indicators are: trading volume, the number of mishandling errors or losses, the number of transaction fails or cancellations, the staff turnover rate, the percentage of staff vacancies, the number of incident reports, the amount of overtime, the ratio of supervisors to staff, the pass-fail rate in licensing exams for the staff, the number of limit violations, the number of process "fails," the number of personnel errors, the average years of staff experience, backlog levels, etc. Risk indicators can be divided into two categories: performance indicators and control indicators. Performance indicators (such as the number of failed trades, staff turnover rates, volume and systems downtime) monitor operational efficiency. Control indicators

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Training expenditure Employee errors as a % of trade volume

p

.-

150

-; roo I-

1

Customer complaints asasof trade volume Mar

Jun

Sep

Dec

Mar

Jun

Sep

98

98

98

98

99

99

99

Quarter

Figure 5.2 Composite risk indicators: training dollars vs. employee error rate vs. customer complaints Source: Taylor and Hoffman (1999), p. 15.

measure the effectiveness of controls, e.g., the number of audit exceptions and the number of outstanding confirmations. Risk profiling models can track operational risk changes over time. The results can be used as a diagnostic tool to target operational risk weaknesses. The results can be incorporated into an operational risk scorecard (see discussion in section 5.2.1.1 ) .I5 However, risk profiling models assume that there is a direct relationship between operational risk indicators and target variables such as staff turnover rate. If this is not true, then the risk indicators may not be relevant measures of operational risk. Moreover, risk profiling may concentrate on the symptom (say, increased overtime), not the root cause of the operational risk problem. Finally, risk profiling models should analyze the relationships among different indicator variables to test for cross correlations that might yield confounding results. For example, figure 5.2 shows the inverse relationship between training expenditures and employee errors and employee complaints. A composite risk indicator can be determined using, say, the average expenditure required to reduce errors or customer complaints by 1 percent. Thus, a risk profiling model will examine several different risk indicators in order to obtain a risk profile for the company. Doerig (2000) states that each business unit uses approximately 10-1 5 risk indicators to assess its operational risk exposure.

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It is a matter of judgment, however, which risk indicators are most relevant to the overall operational risk exposure of the firm.16

5.2 BOTTOM-UP APPROACHES TO OPERATIONAL RlSK MEASUREMENT Top-down models use various statistical techniques (e.g., regression analysis) to take a "bird's eye view" of the firm's operational risk. Bottom-up models may use the same techniques, but instead apply them to the nuts and bolts of the firm's operational processes and procedures. Thus, bottom-up models are more precise and targeted to the measurement of specific operational risk problems, but at the same time, are more complicated and difficult to estimate than are top-down models. Bottom-up models use two different approaches to estimate the operational risk of a particular business line or activity: (i) the process approach and (ii) the actuarial approach.17 The process approach focuses on a step-by-step analysis of the procedures used in any activity. This can be used to identify operational risk exposures at critical stages of the process. In contrast, the actuarial approach concentrates on the entire distribution of operational losses, comprised of the severity of loss events and their frequency. Thus, the actuarial approach does not identify specific operational risk sources, but rather identifies the entire range of possible operational losses taking into account correlations across risk events.

5.2.1

Process approaches

The process approach maps the firm's processes to each of the component operational activities,'' Thus, resources are allocated to causes of operational losses, rather than to where the loss is realized, thereby emphasizing risk prevention. There are three process models: causal networks or scorecards, connectivity and reliability analysis. 5.2.1.1

Causalnetworks orscorecards

Causal networks, also known as scorecards, break down complex systems into simple component parts to evaluate their operational risk exposure. Then data are matched with each step of the process map

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Figure 5.3 Process map for a transaction settlement Source: Srnithson (2000), p. 58.

to identify possible behavioral lapses. Data are obtained using incident reports, direct observation and empirical proxies. For example, figure 5.3 shows a process map for a transaction settlement. The transaction is broken into four steps. Then data regarding the number of days needed to complete the step is integrated into the process map to identify potential weak points in the operational cycle. Scorecards require a great deal of knowledge about the nuts and bolts of each activity. However, the level of detail in the process map is a matter of judgment. If the process map contains too much detail, it may become unwieldy and provide extraneous data, detracting from the main focus of the analysis. Thus, the process map should identify the high risk steps of the operational process that are the focus of managerial concern. Then all events and factors that impact each high risk step are identified through interviews with employees and observation. For example, the high risk steps in the transaction settlement process map shown in figure 5.3 relate to customer interaction and communication. Thus, the process map focuses on the customer-directed steps, i.e., detailing the steps required to get customer confirmation, settlement instructions and payment notification. In contrast, the steps required to verify the price and position are not viewed by management as particularly high in operational risk and thus are summarized in the first box of the process map shown in figure 5.3. Mapping the procedures is only the first step in the causal network model. Data on the relationship between high risk steps and component risk factors must be integrated into the process map. In the process map shown in figure 5.3, the major operational risk factor is

-

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UNDERSTANDING MARKET, CREDIT, AND OPERATIONAL RISK

Time progression Event Event occurrence detection

External event occurs

Response implementation Response identification

Staff detects Staff correctly Staff implements appropriate Success diagnoses response event response Staff implements inappropriate Failure response Staff misdiagnoses response Staff fails to detect event.

Figure 5.4

Outcome

Failure Failure

Generic event tree

Source: Marsha11 (2001), p. 259.

assumed to be time to completion. Thus, data on completion times for each stage of the process are collected and input into the process map in figure 5.3. In terms of the number of days required to complete each task, figure 5.3 shows that most of the operational risk is contained in the last two steps of the process - settlement instructions and payment notification. However, there may be several different component risk factors for any particular process. If another operational risk factor were used, say the number of fails and errors at each stage of the process, then the major source of operational risk would be at another point of the process, say the position reconciliation stage. Another technique used in causal networks is the event tree. The event tree evaluates each risk events' direct and indirect impacts to determine a sequence of actions that may lead to an undesirable outcome. For example, figure 5.4 shows a generic event tree triggered by some external event. As an example, we can apply the generic event tree to Arthur Andersen's operational risk in the wake of the external event of Enron's bankruptcy declaration and the resulting SEC investigation into Enron's financial reporting. One can argue that Arthur Andersen employees, while detecting the event, failed to correctly interpret its significance for Arthur Andersen's reputation as Enron's auditor. In directing employees to shred documents, the staff misdiagnosed the appropriate response, resulting in a failed outcome. Event trees are particularly useful when there are long time lags between an event's occurrence and the ultimate outcome. They help identify chronological dependencies within complex processes. However,

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both event trees and process maps are somewhat subjective. Management has to identify the critical risk factors, break down the process into the appropriate level of detail and apply the correct data proxies. Moreover, by focusing on individual processes at the microlevel, the analysis omits macrolevel interdependencies that may result from a single failed activity that produces many failed processes. Moreover, there is no analysis of the likelihood of each external risk event.19 5.2.1.2

Connectivity models

Connectivity models are similar to causal networks, but they focus on cause rather than effect. That is, they identify the connections between the components in a process with an emphasis on finding where failure in a critical step may spread throughout the procedure. Marshall (2001) shows that one technique used in connectivity models is fishbone analysis. Each potential problem in a process map is represented as an arrow. Each problem is then broken down into contributing problems. An example of fishbone analysis for errors in a settlement instruction is shown in figure 5.5. The root cause of

Safekeeping error

Broker error

Error in message content Incorrect lSlN

Free-text error

Security error

Figure 5.5 Example of a fishbone diagram for errors in a settlement instruction Source: Marshall (2001),p. 252.

w

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UNDERSTANDING MARKET, CREDIT, AND OPERATIONAL RISK

( Late settlement

E lw Staff error

Missing

System failure

confirmation

Telecom failure

Human error

Booking error

\

/

<

)/I

R

Counterparty 1

complexity

Figure 5.6 Causal structure of late settlement losses Source: Marshall (2001),p. 95.

the error message is traced to either a safekeeping error, a broker error, a free-text error, or a security error. Within each of these possible problems, the specific cause of the error is identified. Another technique used in connectivity models is fault tree analysis. A fault tree integrates an event tree with fishbone analysis in that it links errors to individual steps in the production process. Management specifies an operational risk event to trigger the analysis. Then errors are identified at each stage of the process. In both fishbone and fault tree analysis, as well as for causal networks, care should be taken to avoid over-disaggregation which will make the analysis unnecessarily complex, thereby losing its focus. Connectivity models suffer from some of the same disadvantages as do causal networks. They are subjective and do not assess probabilities for each risk event. However, when combined with a scorecard to assess subjective probabilities, one obtains the fault tree shown in figure 5.6. This is taken from Marshall's (2001) example of the analysis of late

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settlement losses for a financial institution. As shown in figure 5.6, late settlement occurs because of late confirmation (with a 40 percent probability), staff error (5 percent probability) or telecom failure (5 percent probability); the remainder of the cause of the late settlement operational risk event is the result of unknown factors (occurring with However, late confirmations themselves can a 50 percent pr~bability).~' be the result of several errors: missing trades, system failures, human errors, booking errors, or counterparty errors. Each of these operational risk events is assigned a probability in figure 5.6. Finally, the booking error cause can be the result of product complexity or product volume. Thus, the fault tree measures the extent of interdependencies across steps that make up complex processes.

5.2.1.3

Reliability models

Reliability models use statistical quality control techniques to control for both the impact and the likelihood of operational risk events. They differ from causal networks and connectivity models in that they focus on the likelihood that a risk event will occur. Reliability models estimate the times between events rather than their frequency (the event failure rate).21This methodology is similar to intensity-based models of credit risk measurement (see chapter 4, section 4.2.2). If p(t) is the probability that a particular operational risk event will occur at time t, then the time between events, denoted h(t), can be calculated as follows:

Thus, the reliability of a system is the probability that it will function without failure over a period of time t, which can be expressed as:

External as well as internal data are needed to estimate the reliability function R(t). Thus, the data requirements may be daunting. Moreover, the model must be estimated separately for LFHS events in contrast to HFLS events. However, by focusing only on frequency

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and not on impact, reliability models do not measure the severity of the risk event.

5.2.2 Actuarial approaches The actuarial approach combines estimation of loss severity and frequency in order to construct operational loss distributions. Thus, the actuarial approach is closest to the VaR models discussed in the remainder of this book. There are three actuarial approaches: empirical loss distributions, explicit parametric loss distributions and extreme value theory.

5.2.2.1 Empirical loss distributions Both internal and external data on operational losses are plotted in a histogram in order to draw the empirical loss distribution. External industry-wide data are important so as to include both LFHS and HFLS operational risk events. The relationship shown in figure 5.1 represents an empirical loss distribution. This model assumes that the historical operational loss distribution is a good proxy for the future loss distribution. Gaps in the data can be filled in using Monte Carlo simulation techniques. Empirical loss distribution models do not require the specification of a particular distributional form, thereby avoiding potential errors that impact models that make parametric distributional assumptions. However, they tend to understate tail events and overstate the importance of each firm's idiosyncratic operational loss history. Moreover, there is still insufficient data available to backtest and validate empirical loss distributions.

5.2.2.2 Parametric loss distributions Examining the empirical loss distribution in figure 5.1 shows that in certain ranges of the histogram, the model can be fit to a parametric loss distribution such as the exponential, Weibull or the beta distribution. In contrast to the methodology used in market risk rnea~urement,~~ parametric operational loss distributions are often obtained using different assumptions of functional form for the frequency of losses and for the severity of operational losses. Typically, the frequency of operational risk events is assumed to follow a Poisson distribution. The distribution of operational loss severity is assumed to be either

1

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Events

Figure 5.7(a) Distribution of mishandling events per day Source: Laycock (1998),p. 138.

lognormal or Weibull in most studies. The two distributions are then combined into a single parametric operational loss distribution using a process called convolution. An example of the procedure used to fit actual data2' to empirical distributions is given by Laycock (1998), who analyzes mishandling losses and processing errors that occur because of late settlement of cash or securities in financial transactions. Figure 5.7 (a) shows that the likelihood of daily mishandling events can be modeled as a Poisson distribution, with the caveat that actual events are more likely to be correlated than those represented by the theoretical distribution. That is, when it is a bad day, many mishandling events will be bunched together (as shown i n the extreme right tail region of the data observations which lies above the Poisson distribution values). Moreover, there are more no-event days than would be expected using the Poisson distribution (as shown by the higher probability density for the observed data in the extreme low-event section of the distributions). Laycock (1998) then plots the loss severity distribution for mishandling events and finds that the Weibull distribution is a "good" fit, as shown in figure 5.7(b). Finally, the likelihood and severity distributions are brought together to obtain the distribution of daily losses shown in figure 5.7(c). Separating the data into likelihood and severity distributions allows risk managers to ascertain whether operational losses from mishandling stem from infrequent, large value losses

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Size of losses

Figure 5.7(b) Mishandling loss severity distribution Source: Laycock (1998), p. 139.

Size of losses

Figure 5.7(c) Distribution of daily mishandling operational losses Source: Laycock (1998), p. 140.

or from frequent, small value losses. However, the data required to conduct this exercise are quite difficult to obtain. Moreover, this must be repeated for every process within the firm. Even if the operational loss distribution can be estimated for a specific business unit or risk variable, there may be interdependencies across risks within the firm. Therefore, operational losses cannot be simply

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aggregated in bottom-up models across the entire firm. For example, Ceske and Hernandez (1999) offer the simplified example of measuring the operational risk on a trading desk comprised of operational losses on foreign exchange (denoted X) and operational losses on precious ~ and Y are independent, then S = X + Y can metals (denoted Y ) . If~ X be represented as:

where F denotes distribution functions and f,(Y) is the probability density function for the random variable Y. However, X and Y are generally not independent. Thus, one must specify the interdependencies between the two random variables in order to specify the (joint) operational loss distribution. This requires a large amount of information that is generally unavailable. Ceske and Hernandez (1999) suggest the use of a copula function that represents the joint distribution as a function of a set of marginal distributions. The copula function can be traced out using Monte Carlo simulation to aggregate correlated losses. (See Appendix 5.1 for a discussion of copula functions.)

5.2.2.3

Extreme value theory

As shown in figure 5.1, it is often the case that the area in the extreme tail of the operational loss distribution tends to be greater than would be expected using standard distributional assumptions (e.g., lognormal or Weibull). However, if management is concerned about catastrophic operational risks, then additional analysis must be performed on the tails of loss distributions (whether parametric or empirical) comprised almost entirely of LFHS operational risk events. Put another way, the distribution of losses on LFHS operational risk events tends to be quite different from the distribution of losses on HFLS events. The Generalized Pareto Distribution (GPD) is most often used to represent the distribution of losses on LFHS operational risk events.25As will be shown below, using the same distributional assumptions for LFHS events as for HFLS events results in understating operational risk exposure. The Generalized Pareto Distribution (GPD) is a two parameter distribution with the following functional form:

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UNDERSTANDING MARKET, CREDIT, AND OPERATIONAL RISK

r

0 Mean

$4.93 $6.97 $22.23 $53.53

t

95% VaR normal dist.

t

f

E

t

S

99% 99% VaR VaR Mean of normal GPD ~?xtmne dist. losses beyond the 99th percentile VaR under the GPD

Figure 5.8 Estimating unexpected losses using Extreme Value Theory (ES = the expected shortfall assuming a Generalized Pareto Distribution (GPD) with fat tails)

The two parameters that describe the GPD are 5 (the shape parameter) and p (the scaling parameter). If 5 > 0,then the GPD is characterized by fat tails.26 Figure 5.8 depicts the size of losses when catastrophic events Suppose that the GPD describes the distribution of LFHS operational losses that exceed the 95th percentile VaR, whereas a normal distribution best describes the distribution of values for the HFLS operational risk events up to the 95th percentile, denoted as the "threshold value" u, shown to be equal to US$4.93 million in the example presented in figure 5.8.28The threshold value is obtained using the assumption that losses are normally distributed. In practice, we observe that loss distributions are skewed and have fat tails that are

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inconsistent with the assumptions of normality. That is, even if the HFLS operational losses that make up 95 percent of the loss distribution are normally distributed, it is unlikely that the LFHS events in the tail of the operational loss distribution will be normally distributed. To examine this region, we use extreme value theory. Suppose we had 10,000 data observations of operational losses, denoted n = 10,000. The 95th percentile threshold is set by the 500 observations with the largest operational losses; that is (10,000 500)/10,000 = 95%; denoted as Nu= 500. Suppose that fitting the GPD parameters to the data yields 6 = 0.5 and P = 7.29McNeil (1999) shows that the estimate of a VaR beyond the 95th percentile,taking into account the heaviness of the tails in the GPD (denoted VaR,) can be calcillated as follows:

Substituting in the parameters of this example for the 99th percentile yields: VaR, or

m,.,,,

That is, in this example, the 99th percentile VaR for the GPD, denoted -

VaR,,,, is US$22.23 million. However, VaR,,, does not measure the severity of catastrophic losses beyond the 99th percentile; that is, in the bottom 1 percent tail of the loss distribution. This is the primary area of concern, however, when measuring the impact of LFHS operational risk events. Thus, extreme value theory can be used to calculate the Expected Shortfall to further evaluate the potential for losses in the extreme tail of the loss distribution. The Expected Shortfall, denoted mO.,,, is calculated as the mean of the excess distribution of unexpected losses beyond the threshold $22.23 million VaR,.,,. McNeil (1999) shows that the expected shortfall (i.e., the mean of the LFHS operational losses exceeding VaR,.,,) can be estimated as follows: -

ES, =

(VaR,/(l - 5)) + (B - c ~ ) l (-l 5).

where q is set equal to the 99th percentile. Thus, in our example, -

ES, = (($22.23)/0.5)+ (7 - 0.5(4.93))/0.5= US$53.53 million to obtain the values shown in figure 5.8. As can be seen, the ratio of the extreme (shortfall) loss to the 99th percentile loss is quite high:

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This means that nearly 2'12 times more capital would be needed to secure the bank against catastrophic operational risk losses compared to (unexpected) losses occurring up to the 99th percentile level, even when allowing for fat tails in the VaR,.,, measure. Put another way, coverage for catastrophic operational risk would be considerably underestimated using standard VaR methodologies. The Expected Shortfall would be the capital charge to cover the mean of the most extreme LFHS operational risk events (i.e., those in the 1 percent tail of the distribution). As such, the amount can be viewed as the capital charge that would incorporate risks posed by extreme or catastrophic operational risk events, or alternatively, a capital charge that internally incorporates an extreme, catastrophic stress-test multiplier. Since the GPD is fat tailed, the increase in losses is quite large at high confidence levels; that is, the extreme values of E S , (i.e., for high values of q, where q is a risk percentile) correspond to extremely rare catastrophic events that result in enormous losses.30

mO.,,

5.2.3

Proprietary operational risk models31

The leading32proprietary operational risk model currently available is OpVar, offered by OpVantage which was formed in April 2001 by a strategic alliance between NetRisk, Inc. and PricewaterhouseCoopers (PwC). OpVar integrates NetRisk's Risk Ops product with an operational risk event database originally developed by PwC to support its in-house operational risk measurement product. The operational loss database currently contains more than 7,000 publicly disclosed operational risk events, each amounting to a loss of over US$1 million for a total of US$272 billion in operational losses. In addition, the database contains over 2,000 smaller operational risk events amounting to less than US$1 million each. The data cover a period exceeding 10 years, with semiannual updates that add approximately 500 new large operational risk events to the database each half year. Figure 5.9(a) shows the distribution of operational risk events by cause. Clients, products and business practices overwhelmingly account for the majority (71 percent) of all operational losses in the OpVar database. However, this database may not be relevant for a particular financial institution with distinctive characteristics and thus OpVar's accuracy hinges on its ability to scale the external data and create a customized database for each financial firm. OpVar is currently installed at more than 20 financial institutions throughout the world,

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Clients, products and business practices 71% H External fraud 7% El Execution, delivery and process management 3% @ Internal I fraud 10% Damage to physical assets 5% H Employment practices and workplace safety 2% H Business disruption and system failures 2%

Figure 5.9(a) Total operational losses by cause amount Source: www.opvantage.com

Commercial banking 16% fl Retail brokerage 8% Ed Trading and sales 21% B4. Asset management 4% H Institutional brokerage 3% H Corporate finance 3% H Insurance 18% Retail banking 23% Agency services 3% Other 1%

. I .

t.l

.

' ,,"

&me5.9(b)

Total operation losses by business uriit type

Source: www.opvantage.com

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including Bank of America, Banco Sabadell, CIBC, ING, Sanwa, Societe Generale and Swiss Re. Figure 5.9(b) shows that most operational losses originate in the banking sector (16 percent in commercial banking and 22 percent in retail banking). OpVar is a bottom-up model that uses several different methodologies. It features multiple curve fitting techniques employing both parametric (e.g., lognormal) and empirical models of severity distributions, frequencies, and operational losses. Moreover, OpVar uses actuarial methods and Monte Carlo simulation to fill in gaps in the data. Graphical displays of causes and effects of operational risk events incorporate the process approach through the analysis of fault trees, causal networks, and risk profiles. Another major proprietary operational risk measurement model, Algorithmics Algo OpRisk, consists of three components: Algo Watchdog, Algo OpData, and Algo 0pCapita1.~~ Algo Watchdog is a bottom-up factor model that uses simulations and Bayesian analysis to predict the sensitivity of operational losses to risk events. AIgo OpData provides a flexible framework to store internal data on operational losses. The database is two-dimensional in that each operational risk event (or near miss) is sorted by organizational unit and by risk category. For financial firms, there are nine organizational units (corporate finance, merchant banking, Treasury, sales, market making, retail banking, card services, custody, and corporate agency services)34and five risk categories (two categories of employee fraud: collusion and embezzlement and three categories of systems failure: network, software, and hardware). Finally, OpCapital calculates operational risk capital on both an economic and a regulatory basis (following BIS I1 proposals; see discussion in section 6.3) using an actuarial approach that estimates loss frequency and severity distributions ~ e p a r a t e l y . ~ ~ Another proprietary model called 6 Sigma, developed by General Electric for the measurement of manufacturing firms' operational risk, has been adapted and applied to the operational risk of financial firms by Citigroup and GE Capital. This model primarily utilizes a topdown approach, focusing on variability in outcomes of risk indicator variables, such as the total number of customer complaints, reconciliations, earnings volatility, etc. However, because of the shortcomings of top-down models, 6 Sigma has added a bottom-up component to the model that constructs process maps, fault trees, and causal networks. Several companies offer automated operational risk scorecard models that assist middle managers in using bottom-up process approaches

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to create causal networks or fault trees. JP Morgan Chase's Horizon (marketed jointly with Ernst 6 Young) and Accenture's operational risk management framework focus on key risk indicators identified by the financial institution. These models essentially massage manual data inputs into graphs and charts that assist the manager in visualizing each process's operational risk exposure. Capital requirements (either economic or regulatory) are also computed using the data input into the model. However, if the data inputs are subjective and inaccurate, then the outputs will yield flawed operational risk measures. The pressure to develop more proprietary models of operational risk measurement has been increased by the BIS I1 consideration of an operational risk component in international bank capital requirements (see section 6.3). Moreover, in June 2005, the US is scheduled to move to a T + 1 settlement standard, such that all securities transactions will be cleared by one day after the trade. The Securities Industry Association conducted a survey and found that only 61 percent of equity transactions at US asset management firrns and 87 percent of equity transactions at US brokerage houses comply with the straightthrough-processing standards required to meet the T + 1 requirement. The compliance levels in the fixed income markets were considerably lower: only 34 percent of asset managers and 63 percent of brokerage houses in the US were capable of straight-through-processing(see Bravard and David, 2001). Compliance with T + 1 standards will require an estimated investment of US$8 billion with an annual cost saving of approximately US$2.7 billion. Failure to meet the standard would therefore put firms at a considerable competitive disadvantage. Thus, the opportunity for operational losses, as well as gains through better control of operational risk, will expand considerably.

5.3 HEDGING OPERATIONAL RISK Catastrophic losses, particularly resulting from LFHS operational risk events, can mean the end of the life of a firm. The greater the degree of financial leverage (or conversely, the lower its capital), the smaller the level of operational losses that the firm can withstand before it becomes insolvent. Thus, many highly levered firms utilize external institutions, markets, and/or internal insurance techniques to better manage their operational risk exposures. Such risk management can take the form of the purchase of insurance, the use of self-insurance, or hedging using derivatives.

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5.3.1

Insurance

Insurance contracts can be purchased to transfer some of the firm's operational risk to an insurance company. The insurance company can profitably sell these policies and absorb firm-specific risk because of its ability to diversify the firm's idiosyncratic risk across the policies sold to many other companies. The most common forms of insurance contract sold to financial firms are: fidelity insurance, electronic computer crime insurance, professional indemnity, directors' and officers' insurance, legal expense insurance, and stockbrokers' indemnity. Fidelity insurance covers the firm against dishonest or fraudulent acts committed by employees. Electronic computer crime insurance covers both intentional and unintentional errors involving computer operations, communications, and transmissions. Professional indemnity insurance covers liabilities to third parties for claims arising out of employee negligence. Directors' and officers' insurance covers any legal expenses associated with lawsuits involving the discharge of directors' and officers' fiduciary responsibilities to the firm's stakeholders. Stockbrokersfindemnity insurance protects against stockbrokersf losses resulting from the regular course of operations such as the loss of securities and/or cash, forgery by employees, and any legal liability arising out of permissible transactions. All insurance contracts suffer from the problem of moral hazard; that is, the mere presence of an insurance policy may induce the insured to engage in risky behavior because the insured does not have to bear the financial consequences of that risky behavior. For example, the existence of directors' insurance limiting the directors' personal liability may cause directors to invest less effort in monitoring the firm's activities, thereby undermining their responsibility in controlling the firm's risk taking and questionable activities. Thus, insurance contracts are not written to fully cover all operational losses. There is a deductible, or co-insurance feature which gives the firm some incentive to control its own risk taking activities because it bears some of the costs of operational failures. The impact of insurance, therefore, is to protect the firm from catastrophic losses that would cause the firm to become insolvent, not to protect the firm from all operational risk. To better align the interests of the insured and the insurer, losses are borne by both parties in the case of an operational risk event. Figure 5.10 shows how operational losses are typically distributed. Small losses fall entirely under the size of the deductible and are thus completely absorbed by the firm (together with the cost of the insurance prem i ~ m )Once . ~ ~the deductible is met, any further operational losses

Reproduced with permission from Understanding Market, Credit and Operational Risk: The Value At Risk Approach, by Linda Allen, Jacob Boudoukh, Anthony Saunders. © 2004 Blackwell Publishing. EXTENDING THE VaR APPROACH TO OPERATIONAL RISKS

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9 10 11 12 13 14 15 16 17 18 19 20 Operational losses

Figure 5.10 The typical payout structure on an operational risk insurance policy

are covered by the policy up until the policy limit is met. The firm is entirely responsible for operational losses beyond the policy limit. The higher (lower) the deductible and the lower (higher) the policy limit, the lower (higher) the cost of the insurance premium and the lower (higher) the insurance coverage area on the policy. Thus, the firm can choose its desired level of risk reduction by varying the deductible and policy limit of each operational risk insurance policy. Despite their role as outsiders to the inner workings of insured firms, -insurance companies have a comparative advantage in absorbing risks. Insurance companies diversify risks by holding large portfolios of.policies. Moreover, insurance companies have access to actuarial information and data obtained from past loss experience to better assess operational risk exposure. This expertise can also be used to advise their clients about internal risk management procedures to prevent operational losses. Finally, insurance companies spread risk among themselves using the wholesale reinsurance market.37 The primary disadvantage of insurance as a risk management tool is the limitation of policy coverage. Hoffman (1998) estimates that insurance policies cover only 10-30 percent of possible operational losses. Large potential losses may be uninsurable. Moreover, there may be ambiguity in the degree of coverage that results in delays in settling claims, with potentially disastrous impacts on firm solvency.38Large claims may threaten the solvency of the insurance companies themselves, as

Reproduced with permission from Understanding Market, Credit and Operational Risk: The Value At Risk Approach, by Linda Allen, Jacob Boudoukh, Anthony Saunders. © 2004 Blackwell Publishing. 188

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evidenced by the problems suffered after Hurricane Andrew in 1992 (which resulted in insured losses of US$19.6 billion) and the terrorist attacks on the World Trade Center on September 11, 2001. Although estimates of losses to the insurance industry resulting from the September 11 attacks range from US$30 billion to US$70 billion, it is clear that this will be the most expensive catastrophic loss event ever recorded in the history of the insurance industry. Insurance premium costs have gone up and policy coverage narrowed in the wake of the terrorist attacks. Moreover, US property-liability insurers responded to large losses (such as Hurricane Andrew and the Northridge earthquake) by significantly increasing their capital from US$0.88 in equity per dollar of incurred losses in 1991 to US$1.56 in 1997. Thus, Cummins, Doherty, and Lo (2002) find that 92.8 percent of the US property-liability insurance industry could cover a catastrophe of US$100 billion. However, Niehaus (2002) contends that a major disaster would seriously disrupt the insurance industry, particularly since many propertylcasualty insurers lost money in 2002. Even without considering the cost of major catastrophes, insurance coverage is expensive. The Surety Association of America estimates that less than 65 percent of all bank insurance policy premiums have been paid out in the form of settlements (see Hoffman, 1998, and Marshall, 2001). Thus, the firm's overall insurance program must be carefully monitored to target the areas in which the firm is most exposed to operational risk so as to economize on insurance premium payments. The firm may obtain economies of scope in its operational risk insurance coverage by using integrated, combined or basket policies. These policies are similar in that they aggregate several sources of risk under a single contract. For example, Swiss Re's Financial Institutions Operational Risk Insurance product provides immediate payout in the event of a wide variety of operational risk incidents. To price such a comprehensive policy, the insurance company often sets very high deductibles, often as high as US$100 million. In exchange for this, the firm receives a wide range of insurance coverage at a relatively low premium cost.

5.3.2 Self-insurance The firm can reduce the cost of insurance coverage by self-insuring. Indeed, the presence of a deductible and a policy limit amounts to a form of self-insurance. The most common form of self-insurance is

Reproduced with permission from Understanding Market, Credit and Operational Risk: The Value At Risk Approach, by Linda Allen, Jacob Boudoukh, Anthony Saunders. © 2004 Blackwell Publishing. EXTENDING THE VaR APPROACH TO OPERATIONAL RISKS

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the capital provision held as a cushion against operational losses. Regulatory capital requirements set minimum levels of equity capital using some measure of the firm's operational risk exposure (see discussion in section 6.3 for the BIS proposals on operational risk capital req~irernents).~~ However, capital requirements may be an exceedingly costly form of self-insurance to protect the firm against operational risk losses because equity capital is the most expensive source of funds available to the financial institution. Indeed, Leyden (2002) suggests that internal market risk models that economize on capital requirements have a return on investment of up to 50 percent. Alternatively, the firm could set aside a portfolio of liquid assets, such as marketable securities, as a cushion against operational losses. Moreover, the firm could obtain a line of credit that precommits external financing to be available in the event of losses. Thus, the firm allocates some of its debt capacity to covering losses resulting from operational risk events. Finally, some firms self-insure through a wholly owned insurance subsidiary, often incorporated in an offshore location such as Bermuda or the Cayman Islands, known as a captive insurer.40This allows the firm to obtain the preferential tax treatment accorded to insurance companies. That is, the insurance company can deduct the discounted value of incurred losses, whereas the firm would only be able to deduct the actual losses that were paid out during the year. Suppose that the firm experiences a catastrophic operational risk event that results in a loss of reputation that will take an estimated three years to recover. Under current US tax law, the firm can reduce its tax liabilities (thereby regaining some of the operational losses through tax savings) only by the amount of out-of-pocket expenses actually incurred during the tax year. Operational losses realized in subsequent tax years are deductible in those years, assuming that the firm survives until then. In contrast, a captive insurer can deduct the present value of all future operational losses covered by the policy immediately in the current tax year. Thus, the formation of a captive insurer allows the firm to co-insure with the relevant tax authorities. Risk prevention and control can be viewed as a form of selfinsurance. The firm invests resources to construct risk mitigation techniques in the form of risk identification, monitoring, reporting requirements, external validation, and incentives to promote activities that control operational risk. Of course, these techniques must themselves be credible since operational risk problems may be pervasive and may even infect the risk monitoring and management apparatus.

Reproduced with permission from Understanding Market, Credit and Operational Risk: The Value At Risk Approach, by Linda Allen, Jacob Boudoukh, Anthony Saunders. © 2004 Blackwell Publishing. 190

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Self-insurance tends to be less costly than external insurance when the firm has control over its risk exposure. Thus, routine, predictable losses that can be controlled using internal management and monitoring techniques are most often self-insured. If the risk is unique to a particular firm, and thus cannot be diversif