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distribution. Conditional VaR is the expected loss given that the loss has exceeded the VaR threshold. That is:11

Conditional VaR = E(X\X > VaR). (9.4)

A simple example will illustrate the difference between VaR and conditional VaR. Suppose that 100000 simulations of the risk factors at the risk horizon are used to generate a P&L distribution such as that in Figure 9.1, and consider the 1000 largest losses that are simulated. Then the 1% VaR is the smallest of these 1000 losses and the 1% conditional VaR is the average of these 1000 losses. So conditional VaR will always be at least as great as VaR, and usually it will be greater.

The conditional VaR risk measure will be encountered again in §10.2.1, when it will be related to the mean excess loss in the peaks over threshold model. An estimate of conditional VaR may be obtained by fitting the standard parametric distribution for excesses over a threshold, that is, the generalized Pareto distribution that is described in §10.2.1.

9.3 Covariance VaR Models

The covariance VaR methodology was introduced in October 1994 by J.P. Morgan; in fact it is the methodology that underlies the RiskMetrics daily data that are available at www.riskmetrics.com. A detailed discussion of these data, which consist of three very large covariance matrices of the major risk factors in global financial markets, has been given in §7.3.

9.3.1 Basic Assumptions

In the covariance VaR methodology the only data necessary to compute the VaR of a linear portfolio is a covariance matrix of all the assets in the portfolio. One does of course need to know the portfolio composition, but the only other data necessary are the variances and covariances of the asset returns. These can be measured using any of the standard methods - usually a moving average or GARCH methodology will be employed, as explained in Part 1 of the book - and regulators recommend that at least one year of historic data be used in their construction.

The fundamental assumption is that the portfolio P&L is normally distributed. That is, if AhP, - Pt+h - P, denotes the A-day unrealized P&L, we assume12

11 Of course conditional VaR depends on the same parameters a, the significance level, and h, the holding period, as the corresponding VaR.

12 The validity of such an assumption is questionable, and should be investigated using some of the tests that are explained in §10.1.



AhP, ~ N(\i„ G2). (9.5)

The 100a% A-day Value-at-Risk is that number VaR„ h such that Prob(A/jf>( < - VaR„ h) = a. Now, applying the standard normal transformation:

Prob([AAP, - p,]/c, < [-VaRa>A - ,]/ ,) = a;

or, since [A„P, - \i,]/at ~ jV(0, 1), and denoting [AAP, - \i,]/o, by the standard normal variate Z„

Prob(Z, < [-VaRaA - uj/a,) = a. But for a standard normal variate Z„

Prob(Z, < -Za) = a, where Za is the lOOath percentile of the standard normal density. Therefore,

[-VaRa>A - u,]/c, = -Za; written another way, we have the formula for covariance VaR,

VaRM = Zaa, - p,. (9.6)

It has already been mentioned that VaR as a risk measure is only suitable for

short-term risks, so it is normal to assume that u, = 0. Now, Za is simply a efmn she

constant given in the standard normal tables (1.645 for a = 0.05, 1.96 for ortfolio VaR in the

a = 0.025, 2.33 for a = 0.01, and other values may be found from the tables in ° ° ? f

J covariance method

the back of the book). Thus it is the volatility of the P&L, that is, o,, that determines the portfolio VaR.

9.3.2 Simple Cash Portfolios

The P&L volatility is easily computed for a simple cash portfolio. Recall from §7.1.1 that the variance of a linear portfolio is a quadratic form: that is, the portfolio return has a variance wVw, where w is the vector of portfolio weights and V denotes the covariance matrix of asset returns. Similarly, the portfolio P&L has a variance pVp, where p is the vector of nominal amounts invested in each asset. Thus if the portfolio is represented as a linear sum of its constituent assets, the covariance method gives the portfolio VaR as (9.6) with

c, = (pVp)1/2.

For example, consider a portfolio with two assets with $ 1 million invested in asset 1 and $2 million invested in asset 2. If asset 1 has a 10-day variance of 0.01, asset 2 has a 10-day variance of 0.005 and their 10-day covariance is 0.002, then

™-o 2)(o°;2 S)GH38-

and the P&L volatility is (pVp)I/2 = 0.0381/2 = $0.195 million, so the 5% 10-day VaR is 1.645 x 0.195 =$0.32 million.



Often the historic asset information is given in terms of annualized volatilities and correlations, instead of / -day variances and covariances. But the formula pVp for the P&L variance can be written equally well in terms of volatilities and correlations. Since the covariance is the correlation multiplied by the product of the square roots of the variances we have

where is the correlation matrix and v is the vector of positions multiplied by the square root of the corresponding asset variance. For example, the data in the above example could have been expressed in the following form: the correlation between two assets is 0.2828, asset 1 has annualized volatility of 50% and asset 2 has an annualized volatility of 35.355%. Then one may calculate the asset 10-day standard deviations as 0.5/(250/10)1/2 = 0.5/5 = 0.1 and 0.35355/(250/10)1/2 = 0.35355/5 = 0.0707, respectively (assuming 250 days per year). Then

which is the same as pVp in the previous example. 9.3.3 Covariance VaR with Factor Models

Many linear portfolios are too large to be represented at the asset level and are instead represented by a mapping. Commonly a factor model is used for large equity portfolios or a cash-flow map is used for fixed income portfolios. In this subsection we suppose that a large equity portfolio, with n assets, has been represented by a factor model with risk factors. Now from §7.1.1 we know that the variance of the portfolio P&L that is due to the risk factors is pBVBp, where p denotes the n x 1 vector of nominal amounts invested in each asset, is the n x matrix of factor sensitivities (the (z*,y)th element of is the beta of the rth asset with respect to the yth risk factor) and \x is the x risk factor covariance matrix appropriate to the risk horizon.

For a simple example, consider an equity portfolio with $2 million invested in US and UK stocks. Suppose the net portfolio beta with respect to the FTSE 100 is 1.5 and the net portfolio beta with respect to the S&P 500 is 2. Then the vector pB = (3, 4). Suppose the 1-day risk factor covariance matrix for the FTSE 100 and the S&P 500 is

pVp = vCv,

Then the variance of the P&L due to the risk factors is



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