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From Daily Volatility of the Position to Daily VaR

The VaR at confidence level a is the cutoff point of negative P & L variations such that the probability of the P & L being lower than this cutoff point is equal to a. It is the a percentile of the P & L. Let us call Xthe random P & L. The VaR is the cutoff point of X, x(a), such that P[X<x{a)] = a.

Back to the example, the determination of VaR at a requires defining a distribution of value for (AS). The normal distribution is convenient because it is defined by only two parameters, its mean and its standard deviation or volatility. The one-tailed confidence level a, or a percentile, is embedded in the normal distribution. For each value of a, there is unique deviation from mean equal to a multiple of the standard deviation. The process requires defining the distribution of the P & L. The normal distribution is an acceptable distribution for the short-term horizon and for a zero-coupon bond whose sensitivity is approximately constant when interest rate variations are not too big.

An attractive property of the normal distribution is that the a percentiles are easily defined as multiples of the volatility of the distribution, as described in Figure 17.3. Under this framework, defining the VaR is straightforward. Looking for a VaR at 1% one-tailed confidence level, for a daily horizon, all we need to do is to use the 2.33 multiple matching the 1% percentile for the normal distribution.

VaR (1%, daily) = 2.33 x €0.1814 = €0.4227

Normal distribution and one-tailed confidence levels

FIGURE 17.3 Normal distribution and one-tailed confidence levels

VaR at Different Horizons

The daily VaR results from the daily volatility. Under some simple assumptions, the volatility was shown to increase as the square root of time. Following this formula, where His the VaR horizon:

Taking H— 10 days, the horizon selected for the Basel Accord on Market Risk, the 10-day volatility, follows using the "square root of time rule:"

The 10-day VaR is the multiple of the normal distribution matching the confidence level, or 2.33 for 1%.

VaR (1%, 10 days) = 2.33 x €1.337 = €3.115

Note that using the square root of time rule applies only when sensitivities are approximately constant. With options, it cannot be used. Details are in the market risk chapters (Chapter 35 and 36).

Summary and Extensions

The derivation of VaR shows the basic steps used to derive the VaR.

• Map the position to risk factors, here a single interest rate.

• Measure sensitivity to the risk factor.

• Input volatility of risk factor for obtaining volatility of random asset value. When there are multiple risk factors, all volatilities of risk factor and their dependencies need to be included.

• Use the "square root of time rule" for matching the horizon of the VaR measure.

• Use a distribution of random values, here the simple normal distribution.

As calculated, this VaR model relies on some critical assumptions:

• constant sensitivities

• normal distribution of random values

• historical volatility of the risk factor, the interest rate

• square root of time rule.

The main required extensions of this basic scheme are:

• dealing with multiple risk factors, as would be the case for many common assets and for portfolios of assets

• dealing with non-linear positions, whereby sensitivities cannot be used anymore as a proxy of the change of values

• dealing with dependencies across risk factors and individual positions, which drive the diversification effect of individual risks.

The section dedicated to dependencies (Section 9, Chapters 30 to 34) addresses the issue of portfolio and multiple risk factors that are dependent. The chapters on market risk, notably the delta-normal VaR chapter (Chapter 35), use the same method with several factors and portfolios. This requires modeling dependencies between risk factors and the portfolio variance as a pre-requisite (Section 9). Alternate VaR modeling methodologies, such as non-linearity and other techniques not based on the normal distribution are addressed in Chapter 36.

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