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93

due to the fact that A has a much higher volatility than B. On the other hand, if A had variance less than the variance of B, then the VaR invariant cash-flow map would map less to A than is mapped using the linear interpolation method.

9.3.5 Aggregation

Covariance VaR is determined by the volatility of the portfolio P&L, so it behaves just like a volatility. In particular, the rules for adding volatilities apply to the aggregation of covariance VaR measures and, just as one should not simply add volatilities to obtain a total volatility, it is not normally appropriate to add covariance VaR measures. To see why, we now derive a formula for adding the covariance VaR measures of two positions A and B. From the usual rule for the variance of a sum, the variance of the total position is given by

<5a+b = \ + °s + 2 ,

<5a+b = (Pa + ° )2 - 20-.4 - + 2paAaB.

Total VaR = [(VaR,, + VaRB)2 - 2(1 - pVvaRVaRJ1/2. (9.9)

The total VaR is only the same as the sum of the VaRs if we assume the two positions are perfectly correlated, that is, p = 1. Note that the specific VaR in §9.3.3 should be uncorrelated with the VaR due to the market risk factors (see the comments made at the end of §8.1.1). Thus it makes little sense to add the specific VaR to the risk factor VaR. However, to err on the side of conservatism, this is exactly what regulators recommend.

This example has shown that one has to take into account the correlation, when aggregating the VaR for different portfolios. This is particularly important when there is low or negative correlation between the assets or factors, because in that case (9.9) shows that the total VaR will be much less than the sum of the individual VaRs. The interested reader should use the covariance VaR spreadsheet on the CD to see how equity and FX VaR are aggregated to give total VaR.

9.3.6 Advantages and Limitations

The covariance method has one clear advantage: it is very quick and simple to compute, so there is no impediment to intra-day VaR calculations and traders can perform a quick calculation to find the impact of a proposed trade on their VaR limit. However, it also has substantial disadvantages:

The total VaR is only the same as the sum of the VaRs if we assume the two positions are perfectly correlated



It has very limited applicability, being suitable only for linear portfolios or portfolios that are assumed to be linear with respect to the risk factors. It assumes that the portfolio P&L distributions at any point in time are normal. This is certainly true if the returns to individual assets, or risk factors, in the portfolio are normally distributed. But often they are not.15 If there are major differences between a covariance VaR measure and a historical simulation VaR measure for the same portfolio, it is likely that the non-normality of asset or risk factor returns distributions is a main source of error in the covariance VaR measure (Brooks and Persand, 2000). It assumes that all the historic data, including possibly very complex dependencies between risk factors, are captured by the covariance matrix. However, covariance matrices are very limited. Firstly, they are very difficult to estimate and forecast (Chapter 5). Secondly, correlation (and covariance) is only a linear measure of co-dependency (§1.1). And thirdly, unless large GARCH covariance matrices are available (§7.4) it is common to apply the square root of time rule to obtain 10-day covariance matrices. But this assumes that volatility and correlation are constant, which may be a gross simplification for the 10-day horizon.

This last point highlights the main problem with covariance VaR measures - that they are only as accurate as the risk model parameters used in the calculation. We have already noted, many times, how unreliable covariance matrices can be. Moreover, in the covariance equity VaR model, factor sensitivities are also used to compute the VaR, and these too are subject to large and unpredictable forecast errors (§8.2 and §4.5.1). Finally, in the covariance fixed income VaR model, the cash flow map itself can be a source of error - one can obtain quite different VaR measures depending on how the portfolio is mapped to the standard risk factors (§9.3.4).

This last point highlights the main problem with covariance VaR measures - that they are only as accurate as the risk model parameters used in the calculation

9.4 Simulation VaR Models

During the past few years the use of simulation methods for VaR analysis has become standard; simulation methods can overcome some of the problems mentioned in §9.3.6. Historical simulation, in particular, is an extremely popular method for many types of institutions. However, it is not easy to capture the path-dependent behaviour of certain types of complex assets unless Monte Carlo simulation is used.

Monte Carlo VaR measures require a covariance matrix and so their accuracy is limited by the accuracy of this matrix. They also take large amounts of computation time if positions are revalued using complex pricing models. Often full revaluation is simply not possible unless VaR calculations are

15 In that case it is possible to adapt the covariance VaR measure to the assumption that returns are fat-tailed. In §10.3.1 it will be shown how to generalize covariance VaR so that the portfolio P&L is assumed to be generated by a mixture of normal densities. Also see the spreadsheet normal mixture VaR on the CD.



performed overnight, so intra-day VaR measures are often obtained using approximate pricing functions. The trade-off between speed and accuracy in the Monte Carlo VaR methods has been a focus of recent research (Glasserman et al., 2001).

This section outlines the basic concepts for historical VaR and Monte Carlo VaR. It explains when and how they should be applied, and outlines the advantages and limitations of each method.

9.4.1 Historical Simulation

The basic idea behind historical simulation VaR is very straightforward: one simply uses real historical data to build an empirical density for the portfolio P&L. No assumption about the analytic form of this distribution is made at all, nor about the type of co-movements between assets or risk factors. It is also possible to evaluate option prices and other complex positions for various combinations of risk factors, so it is not surprising that many institutions favour this method.

Historical data on the underlying assets and risk factors for the portfolio are collected, usually on a daily basis covering several years. Regulators insist that at least a years data be employed for internal models that are used to compute market risk capital requirements, and recommend using between 3 and 5 years of daily data. These data are used to compute the portfolio value on each day during the historic data period, keeping the current portfolio weights constant. This will include computing the value of any options or other complex positions using the pricing models.

In a linear portfolio we represent the A-day portfolio return AhP,/P, as a weighted sum of the returns R: to assets or risk factors, say,

AhP,/P, = w,Ru + ... + wkRkJ.

The w, are the portfolio weights (so they sum to 1) or the risk factor sensitivities (§8.1.2). Historical data are obtained on each Rh and then the portfolio price changes over h days are simulated as

AhP, = iZ(w,Pt)Ru = ZPiRu,

where the p, are the actual amounts invested in each asset, or, in the case of a factor model, the nominal risk factor sensitivities. This representation allows h-day theoretical (or unrealized) P&Ls for the portfolio to be simulated from historical data on . . ., Rk.

With options portfolios full valuation at each point in time is desirable, but for complex products often a price approximation such as a delta-gamma-vega approximation is used. The current portfolio deltas, gammas and vegas are

The absence of distributional assumptions and the direct estimation of variation without the use of a covariance matrix are the main strengths of the historical VaR model



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