VaR DEFINITIONS AND APPLICATIONS

10.2 VaR DEFINITIONS AND APPLICATIONS

VaR defines the loss in market value of say, a portfolio, over the time horizon T that is exceeded with probability 1 − P VaR . In other words, it is the probability

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that returns (losses), say ξ , are smaller than −VaR over a period of time (horizon) T , or:

− VaR

P VaR = P(ξ < −VaR) = P T (ξ ) dξ

where P T (.) is the probability distribution of returns over the time period (0, T ). When the returns are normal, VaR is equivalent to using the variance as a risk measure. When risk is sensitive to rare events and extreme losses, we can fit ex- treme distributions (such as the Weibull and Frechet distributions) or build models based on simulation of VaR. When risks are recurrent, VaR can be estimated by using historical time series while for new situations, scenarios simulation or the construction of theoretical models are needed.

The application of VaR in practice is not without pitfalls, however. In the financial crises of 1998, a bias had made it possible for banks to lend more than they ought to, and still seem to meet VaR regulations. If market volatility increases, capital has to be put up to meet CAR requirements or shrink the volume of business to remain within regulatory requirements. As a result, reduction led to

a balance sheet leverage that led to a vicious twist in credit squeeze. In addition, it can be shown, using theoretical arguments, that in nonlinear models (such long-run memory and leptokurtic distributions), the exclusive use of the variance– covariance of return processes (assuming the Normal distribution) can lead to VaR understating risk. This may be evident since protection is sought from unexpected events, while the use of the Normal distribution presumes that risk sources are stable. This has led to the use of scenario and simulation techniques in calculating the VaR. But here again, there are some problems. It is not always possible to compare the VaR calculated using simulation and historical data. The two can differ drastically and therefore, they can, in some cases, be hardly comparable. Nonstationarities of the underlying return processes are also a source of risk since they imply that return distributions can change over time. Thus, using stationary parameters in calculating the VaR can be misleading again. Finally, one should be aware that VaR is only one aspect of financial risk management and therefore over- focusing on it, may lead to other aspects of risk management being neglected. Nonetheless, VaR practice and diffusion and the convenient properties it has, justifies that we devote particular attention to it. For example, the following VaR properties are useful in financial risk management:

r Risks can be aggregated over various instruments and assets. r VaR integrates diversification effects of portfolios, integrating a portfolio risk

properties. This allows the design of VaR-efficient portfolios as we shall see later on.

r VaR provides a common language for risk management, applicable to portfolio management, trading, investment and internal risk management.

r VaR is a simple tool for the selection of strategic risk preferences by top management that can be decomposed in basic components and applied at

313 Table 10.1 VaR methods and parameters in a number of banks.

VAR DEFINITIONS AND APPLICATIONS

Bank Technique applied Confidence interval Holding period Abbey National

Variance/covariance 95 % 1 day Bank Paribas

Monte Carlo simulations 99 % 1 day BNP

Variance/covariance 99 % 1 day Deutsche Bank Paris

Variance/covariance 99 % 1 day Soci´et´e G´en´erale

Historical simulation 99 % 10 days Basle Committee

Variance/covariance, 99 % 10 days or 1 historical simulation,

√ (days∗ 10)

Monte Carlo

various levels of a portfolio, a firm or organization. Thus, risk constraints at all levels of a hierarchical organization can be used coherently.

r Finally, VaR can be used as a mechanism to control the over-eagerness of some traders who may assume unwarranted risks as this was the case with Barings

(see also Chapter 7).

Example: VaR in banking practice

Generally, the VaR used by banks converges to the Basle Committee specifications as shown in Table 10.1. Most banks are only beginning to apply VaR methodolo- gies, however. At the Soci´et´e G´en´erale, for example, market risk measurement consists in measuring the potential loss due to an accident that may occur once in

10 years (with a confidence interval of 99.96 %). Correlations are ignored, lead- ing thereby to an underestimation of this risk. This too, led to the introduction of the VaR approach including correlations, with a probability of 99 %. The first technique was maintained, however, in order to perform stress tests required by regulators. In most banks the time period covered was one day, while the Basle Committee requires 10 days. This means that VaR measures must be aggregated. If risks are not correlated over time, then aggregation is simple, summarized by their sum (and thus leading to a linear growth of variance). In this case, to move from a one day to a 10-day VaR we calculate:

√ VaR 10 days = VaR 1 day ∗ 10

When there is an inter-temporal correlation, or long-run memory, the calcula- tion of the VaR is far more difficult (since the variance is essentially extremely large and the measurement of VaR not realistic). Regulatory institutions, operat- ing at international, EU and national levels have pressured banks to standardize their measurements of risks and the application of controls, however. In particular, the CRB operating by a regulation law of 21 July 1995 defines four market risks that require VaR specification:

(1) Interest-rate risk: relating to obligations, titles, negotiable debts and related instruments.

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(2) Price risk of titles and related instruments. (3) Regulated credit risk (4) Exchange risk.

Financial institutions have to determine market positions commensurate with Basle (VaR) regulation for these risks.

Numerical examples

(1) Say that a firm buys in a position in DM that produces a loss if the dollar appreciates (and a profit if it depreciates). We seek to determine the VaR cor- responding to the maximum loss that can be sustained in 24 hours with a 5 % probability. Assume that the returns distributions are stable over time. Based on data covering the period of 2000 for the DM/USD exchange rate, a histogram can be constructed and used to calculate the probability of an appreciation larger than 5 %. The critical value for such dollar appreciation turns out to be 1.44 %; which is applied to a market value of $1 million, and thereby leading to a VaR of $14 400.

(2) A US investor assumes a long position of DM 140 millions. Volatility in DM/USD FX is: 0.932 while the exchange rate is 1.40 DM/USD. As a result, the VaR for the US investor is calculated by:

VaR USD = 140 millions DM * 0.932 %/1.40DM/USD = 932 000 USD

(3) We define next two positions, each with it own VaR calculation, VaR 1 , VaR 2 respectively. The VaR of both positions is thus

VaR = VaR 2 1 + VaR 1 + 2ρ 12 VaR 1 VaR 2 where ρ 1,2 is the correlation of the positions. The US investor assumes then a long

position of DM 140 million in a German bond for 10 years (thereby maintaining

a long position in DM). The bond volatility is calculated to be: 0.999 % while the DM/USD volatility is: 0.932 % with correlation −0.27. The interest-rate and

exchange-rate risks are thus:

Interest-rate risk: 140 million DM * 0.999 % / 1.40 = 999 000 USD Exchange-rate risk: 140 million DM * 0.932 %/1.40 = 932 000 USD As a result, the VaR in both positions is:

VaR USD = (999 000) 2 + (932 000) 2 + 2 * (−0.27) * 999 000 * 932 000 = 1.17 million USD

Non-diversifiable risks that have perfect correlation have a risk which is simply their sum. A correlation 1 would thus lead to a risk of 999 000 USD + 932 000

USD = 1.93 million USD. When risk is diversifiable the risk can be reduced with

315 the difference benefiting the investor. In our case, this is equal to: ($1.93 million

VAR STATISTICS

−$1.17 million = $760 000).