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parametric distribution, usually a multivariate normal distribution. It is a partial valuation model because it can only account for linear dependencies (deltas) and ignores non-linear factors, for example, bond convexities or option gammas. This is why it is sometimes called the correlation or ‘‘delta-var’’ variation.

If this frequency distribution is chosen, then VaR is estimated using an equation which specifies portfolio risk as a linear combination of parameters, such as volatility or correlation. It provides an accurate VaR measure if the underlying portfolio is largely linear (e.g. traditional assets and linear derivatives), but is less accurate if non-linear derivatives are present.

Banks that use variance – covariance analysis normally make some allowances for nonlinearities. The Basel Amendment requires that non-linearities arising from option positions be taken into account.

In approaches (a) and (b), a data window must be specified, that is, how far back the historical distribution will go. The Basel Committee requires at least a year’s worth of data. Generally, the longer the data run, the better, but often data do not exist except for a few countries, and it is more likely the distribution will change over the sample period. In approach (b), there is the question of which variances–covariances of the risk factor returns are computed.

(c) Monte Carlo35 approach. Another full valuation approach, involving multiple simulations using random numbers to generate a distribution of returns. Distributional assumptions on the risk factors (e.g. commodity prices, interest rates, equity prices or currency rates) are imposed – these can be normal or other distributions. If a parametric approach is taken, the parameters of the distributions are estimated, then thousands of simulations are run, which produce different outcomes depending on the distributions used. The non-parametric approach uses bootstrapping, where the random realisations of the risk factor returns are obtained through iterations of the historical returns. In either approach, pricing methodology is used to calculate the value of a portfolio.

Unlike (a) and (b), the number of portfolio return realisations is much greater in number, from which the VaR estimates are derived. The Monte Carlo approach is usually rejected because it involves a large number of computations, which present practical problems if traders are computing VaR once, or several times a day. Computation costs are high, too.

3.5.4. VaR, Portfolios and Market Risk

It is possible to show simple applications of VaR for individual trading positions involving two currencies or equities. However, banks compute VaR for large portfolios of equities, bonds, currencies and commodities. Management will want an aggregate number showing the potential value at risk for the bank’s entire trading position. This aggregate VaR is not just a simple sum of the individual positions because they can be positively or negatively correlated with each other, which will raise or reduce the overall VaR. The components of

35 Generally, a Monte Carlo approach models different cash flows of a particular deal, and subjects them to thousands of simulations involving different scenarios to generate the risk parameters. For example, if a bank agrees a loan for a particular venture, detailed cash flow models are subjected to thousands of simulations to assess how changes in the economy will affect the probability of default, exposure at default and loss given default.

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Table 3.7 A Hypothetical Daily Earnings at Risk for a Canadian bank (CDN$m)

DEAR: Daily Earnings at Risk

Table 3.8 Merrill Lynch: value at risk ($m)

Overall VaR is based on a 99% confidence interval and 2-week holding period.VaR for non-trading instruments excludes US banks.

Daily average figures for traded VaR; quarterly average figures for non-traded VaR.

Diversification benefit: the difference between aggregate (firm-wide) VaR and the VaR summed from the four risk categories. The difference arises because the four market risk categories are not perfectly correlated. For example, the simulations of losses at a 99% confidence interval show the losses from each category will occur on different days. There are similar benefits within each category.

( ): negative.

Source: Merrill Lynch (2002), Annual Report, pp. 34–35.

Table 3.8. The daily trading VaR for 2001 is $256 million, which says, with an assumed volatility of $44 million, that average trading losses could exceed $256 million in 1 out of every 100 trading days. Had the confidence interval been 95%, average trading losses could exceed $256 million in 5 out of every 100 trading days. Again, the actual size of the losses on these 5 days is unknown, and could be much higher than $256 million. Table 3.8

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Variations in the model assumptions with respect to the holding period, confidence interval and data window will cause different risk estimates (Beder, 1995). Likewise, Danielsonn (2000) demonstrates the VaR models lack robustness, that is, the VaR forecasts across different assets are unreliable. To illustrate, Danielsonn employs a violation ratio. Violation is defined as the case where the realised loss is greater than the VaR forecast. The violation ratio is the ratio of realised number of VaR violations to the expected number of violations. If the V-ratio >1, the model is under-forecasting the risk; if V-ratio <1, it is over-forecasting. Put another way, over-forecasting means the model is thick tailed relative to the data; under-forecasting means the model is relatively thin tailed. Danielsonn reports disappointing results using this test. Different estimation methods produce different violation ratios, but all vary between, for example, 0.38 and 2.18 (using variations of Riskmetrics ).

For the above reasons, it is necessary to test the actual outcomes with the VaR predictions of losses. However, such tests also have a problem (Kupiec, 1995) because if the period over which the performance of the VaR model is relatively short,38 the tests lack statistical power. It is difficult to evaluate the accuracy of the model on the basis of a year of data. The choice of a 99% confidence interval allows for a loss to occur very 2.5 days in a year. Danielsonn (2000) argues such an allowance is irrelevant in a period of systemic crises, or even for the probability of a bank going bankrupt. If VaR violations occur more than 2.5 times per year under a 99% confidence interval, it does not usually indicate the bank is in any difficulty, and in light of this point, when are VaR breaches relevant? In defence of VaR, it was never meant to indicate that a bank was in difficulty. It is a benchmark number for banks to use to track their market risk exposure.

Both the parametric and non-parametric frequency distributions produce measures which rely on historical data, an implicit assumption is that they are a good predictor of future returns. But historical simulation is sensitive to the sampling period (Danielsonn and de Vries, 1997). For example, in an equity portfolio, the VaR outcomes will be quite different if the October 1987 crash is included than if it is excluded. Or, looking at US share price data from mid-1983 to mid-2000 would suggest sizeable index price falls were the exception, and if they happened, quickly reversed themselves. Agents armed with this information would think the future was like the past (a popular assumption in many models of VaR) and would have found the subsequent share price declines completely mystifying, and outside anything remotely predictable.

However, the non-parametric or historical simulation approaches are superior to the variance – covariance approach (advocated by Riskmetrics ) for two reasons. First, financial market returns do not always follow a normal distribution – large movements in the market (fat tails) occur more often than indicated by the normal distribution. Second, historical simulation allows for non-linearities between the position and risk factor returns, which are important when the VaR being computed includes derivatives, especially options.

There are other criticisms of the use of VaR which relate to the actual 1996 Basel Amendment and ‘‘Basel 2’’ agreement, but these will be discussed in Chapter 4. However,

38 For example, the 250 days specified by the Basel Amendment – see Chapter 4.

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to conclude this section, some quotes from Jorion (defending VaR) and Taleb (rejecting its use) are helpful.

First, comments in favour of VaR by Philippe Jorion:39

‘‘First, the purpose of VaR is not to describe the worst possible outcomes. It is simply to provide an estimate of the range of possible gains and losses. Many derivatives disasters have occurred because senior management did not inquire about the first-order magnitude of the bets being taken. Take the case of Orange County, for instance. There was no regulation that required the portfolio manager, Bob Citron, to report the risk of the $7.5 billion investment pool. As a result, Citron was able to make a big bet on interest rates that came to a head in December 1994, when the county declared bankruptcy and the portfolio was liquidated at a loss of $1.64 billion. Had a VaR requirement been imposed on Citron, he would have been forced to tell investors in the pool: Listen, I am implementing a triple-legged repo strategy that has brought you great returns so far. However, I have to tell you that the risk of the portfolio is such that, over the coming year, we could lose at least $1.1 billion in one case out of 20.’’ (Jorion, 1997, p. 1)

‘‘VaR has other benefits as well. By now, all U.S. commercial banks monitor the VaR of their trading portfolios on a daily basis. Suppose a portfolio VaR suddenly increases by 50 percent. This could happen for a variety of reasons – market volatility could have increased overnight, a trader could be taking inordinate risks, or a number of desks could be positioned on the same side of a looming news announcement. More prosaically, a position could have been entered erroneously. Any of these factors should be cause for further investigation, which can be performed by reverse-engineering the final VAR number. Without it, there is no way an institution can get an estimate of its overall risk profile.’’ (Jorion, 1997, p. 1)

‘‘Still, VaR must be complemented by stress-testing. This involves looking at the effect of extreme scenarios on the portfolio. This is particularly useful in situations of ‘dormant’ risks, such as fixed exchange rates, which are subject to devaluations. Stress-testing is much more subjective than VAR because it poorly accounts for correlations and depends heavily on the choice of scenarios. Nevertheless, I would advocate the use of both methods.’’ (Jorion, 1997, p. 2)

‘‘A second misconception raised in the discussion is that VaR involves a covariance matrix only and does not work with asymmetric payoffs. This is not necessarily the case. A symmetric, normal approximation may be appropriate for large portfolios, in which independent sources of risk, by the law of large numbers, tend to create normal distributions. But the delta-normal implementation is clearly not appropriate for portfolios with heavy option components, or exposed to few sources of risk, such as traders’ desks. Other implementations of VaR do allow asymmetric payoffs. VAR is an essential component of sound risk management systems. VaR gives an estimate of potential losses given market risks. In the end, the greatest benefit of VAR lies in the imposition of a structured methodology for critically thinking about risk. Institutions that go through the process of computing their VAR are forced to confront their

39 Comments from Jorion and Taleb (1997).

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exposure to financial risks and to set up a proper risk management function. Thus the process of getting to VAR may be as important as the number itself. These desirable features explain the widespread view that the ‘quest for a benchmark may be over’.’’ (Jorion, 1997, p. 2)

Nassim Taleb opposes the use of VaR:40

‘‘. . . the professional risk managers I heard recommend a ‘guarded’ use of VaR on the grounds that it generally works or ‘it works on average’ do not share my definition of risk management. The risk management objective function is survival not profits and losses. One trader, according to Chicago legend, made $8 million in eight years and lost $80 million in eight minutes. According to the same standards, he would be ‘in general’ and ‘on average’ a good trader.’’ (Taleb, 1997, p. 3)

‘‘VaR has made us replace about 2500 years of market experience with a co-variance matrix that is still in its infancy.’’ (Taleb, 1997, p. 1)

‘‘[VaR can measure] the risks of common events, perhaps. Those that do not matter, but not the risks of rare events.’’ (Taleb, 1997, p. 2)

‘‘VaR players are all dynamic hedgers and need to revise their portfolios at different levels. As such they can make very uncorrelated markets become correlated. In 1993 hedge funds were long in seemingly independent markets. The first margin call in the bonds led them to liquidate their positions in the Italian, French, and German bond markets. Markets therefore became correlated.’’ (Taleb, 1997, pp. 2 – 3)

3.5.6. Stress Testing and Scenario Analysis to Complement VaR

Given the limitations of VaR, most banks apply scenario analysis and stress testing to complement estimates of market risk produced by VaR. Banks begin by identifying plausible unfavourable scenarios which cause extreme changes to the value of one or more of the four risk factors, i.e. interest, equity, currency or commodity prices. These might include an event which causes most financial agents to act in a similar manner, prompting severe illiquidity, as illustrated by the LTCM case, discussed earlier. Or the unexpected collapse of Enron and WorldCom (two American financial conglomerates in 2002), creating widespread fears about the quality and accuracy of company financial statements which, in turn, contributed to unexpected, dramatic declines in a key or several stock market prices. Other scenarios might be ill-founded rumours which prompt unexpected cash margin calls or changes in collateral obligations.

The stress test, based on a scenario, computes how much a bank’s portfolio could lose. Note the difference from VaR, which estimates the maximum amount that a portfolio, security or business unit (of the bank) could lose over a specified time period. In VaR, the third and fourth moments are assumed to be zero. Some forms of stress testing go beyond the

40 Comments from Taleb (1997) and Taleb and Jorion (1997).

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second moment of the distribution (volatility) and consider the third (skewness) and fourth moment (kurtosis) of the distribution. Skewness will include the worst case outcomes. Kurtosis describes the relative thinness or fatness of the tails of a distribution compared to a normal distribution. For example, in a credit portfolio, if the loss distribution is leptokurtic (i.e. has fat tails), meaning extreme events are more likely than in a normal distribution (e.g. one large credit default results in massive losses). Thin (platykurtosis) tails suggests the opposite.

The size of the change in key risk factors will have to be computed. One group41 recommended changes in volatility such as currency changes of plus or minus 6%, or a 10% change in an equity index. The bank must also choose the frequency with which the stress tests should be conducted.

As a practical example, suppose there are two scenarios: (1) a 40% decline in UK and world equity prices or (2) a 15% decline in residential and commercial property prices. If these are the scenarios, the next step is to decide what stress tests should be conducted. Banks could be asked to identify the potential impact of (1) and (2) on market risk, credit risk and interest rate risk. In addition (or alternatively), building societies could be asked to compute the impact of (1) and (2) on the retail deposit rate, the mortgage rate and the income of building societies. The complexity of the stress tests that must be performed is immediately apparent. Complicated models are required if the banks or building societies are going to produce realistic answers to the questions.

A final task is to decide how to use the results. This presents a very difficult problem for the bank because, by definition, a bank cannot forecast ‘‘surprises’’ or unexpected events. Nor can it judge how frequently they will occur, and therefore, whether or not a special reserve should be created. Furthermore, if such a reserve is kept, how much should the bank be setting aside?

3.6. Management of Credit Risk

3.6.1. Background

Market risk has received an inordinate amount of attention in recent years but managing credit risk is the ‘‘bread and butter’’ of most commercial banks. Every commercial bank, by definition, has a loan portfolio. Increases in credit risk will raise the marginal cost of debt and equity, which in turn increases the cost of funds for the bank. Techniques for credit risk management are well known because the banking sector has had a long history of experience in this area. Nonetheless, loan quality problems are an important cause of bank failure, as will be seen in Chapter 6. For this reason, all bankers, not just those in a credit risk department, should be aware of the key factors affecting the quality of a loan portfolio, and the methods for managing it.

3.6.2. Credit Risk Decisions: Retail versus Corporate

If a bank is looking to minimise its aggregate credit risk, then good risk management of retail and corporate lending is essential. The approaches taken for retail and corporate loans

41 Derivative Policy Group (1995).

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differ considerably, mainly because a corporation is able to produce a variety of financial ratios, which are not available when the suitability of an individual or a small firm for a loan is being assessed. Most bankers concede that lack of information makes retail lending more difficult than corporate lending. On the other hand, loans to corporates which turn out to be bad can be very serious for the bank because of the large sums involved. Countless cases abound: Maxwell and a number of London-based banks; Schroder and Deutsche Bank; and the collapse of Enron and WorldCom. The number of incidents where retail loan defaults have had serious consequences for a bank is very much lower, and usually occurs if a bank is over-exposed in one area such as mortgages, and property prices collapse at the same time as interest rates rise.

The principles used to model credit risk, and the methods used to minimise credit risk for the retail and corporate sectors, are discussed below.

3.6.3. Minimising Credit Risk

There are five key ways a bank can minimise credit risk: through accurate loan pricing, credit rationing, use of collateral, loan diversification and more recently through asset securitisation and/or the use of credit derivatives. The weights applied to each of the methods will vary, depending on whether the loan is commercial or retail.

(1) Pricing the loan: any bank will wish to ensure the ‘‘price’’ of a loan (loan rate) exceeds a risk adjusted rate, and includes any loan administration costs, that is:

where:

RL: interest rate charged on the loan

i: market interest rate, such as LIBOR or an equivalent term42

ip: risk premium, negatively related to the probability of the loan being repaid (ip = 0 if repayment is certain)

In the above equation, ip and i are positively related, for one of two reasons. In the case of a variable rate loan, if the market rate rises, so will the interest rate charged on the loan, and the borrower will find it more difficult to repay the loan, so the probability of default increases. Or, at very high market rates, the loan rate will rise, attracting riskier borrowers (due to adverse selection), so the chances of the loan being repaid fall.

Other factors also influence the loan rate. If there is any collateral or security backing the loan, the rate charged should be lower than in the absence of security. In addition, there are non-price features: the bank may charge a high fee for arranging the loan but the interest rate will be lower. Or, some central banks impose reserve ratios, which means a percentage of a bank’s deposits is held at the central bank, often earning no interest. This is effectively a tax on deposits, which banks will try to make up by imposing higher loan rates.

42 The loan rate set will also depend on the term of the loan. Normally (but not always) the longer the term the higher the loan rate. This will be in the risk premium and will depend on the term structure of interest rates, which is often but not necessarily upward sloping.