the underlying asset price is virtually unchanged at the same time as its implied volatility increases by 5%. This is very unlikely to happen: the discussion in §2.3 shows that there is often a strong negative correlation between prices and volatilities, particularly in equity markets. However, one could reduce the MRR by hedging the portfolio against this scenario. This would be costly, and of little practical importance, but it might be worth putting on the hedge if the MRR will be reduced considerably. This is one example where regulatory policy does not promote good risk management practice.

This is one example where regulatory policy does not promote good risk management practice

Although it is necessary to follow regulators rules for market risk capital calculations, internal risk management may want to use a more sophisticated approach when assessing the risk characteristics of a portfolio. For one thing, a broader range of movements in both the underlying and the implied volatilities could be applied. Secondly, it would be more informative to compute not the maximum loss, irrespective of how likely it is to occur, but the expected loss of a portfolio, using realistic assumptions about the joint distribution of its risk factors.

It would be more informative to compute the expected loss of a portfolio, using realistic assumptions about the joint distribution of its risk factors

A simple model that employs historic data to generate a joint distribution for movements in an underlying asset and movements in its implied volatility was described in §2.3.4. A joint distribution of risk factor movements, such as those shown in Figures 2.9 and 2.10, can be used to obtain a more realistic assessment of the portfolios risk characteristics. The expected loss of the portfolio can be obtained by multiplying the profit or loss by the relevant probability and then summing over the entire scenario range. In addition, one can compute the loss standard deviation, to indicate the accuracy of the assessment of expected loss.

9.6.3 Stress-Testing Portfolios

The 1996 Amendment also gave guidelines for the rigorous and comprehensive stress-testing of portfolios. This should include stress testing portfolios for:

> the repeat of an historic event, such as the global equity Black Monday crash in 1987;

breakdown in volatility and correlation that is associated with a stress

>- a event;

> changes in liquidity accompanying the stress event

All three types of stress test can be performed using a covariance matrix. In the covariance and Monte Carlo VaR models all that is required is to replace the current covariance matrix by a covariance matrix that reflects the stress test. Even when the VaR model uses historic simulation, it is possible to perform the stress tests using a covariance matrix. Following Duffie and Pan (1997), let r denote the vector of returns that is used in the historical VaR model, and compute its equally weighted covariance matrix V. Suppose that W is a stress

Even when the VaR model uses historic simulation, it is possible to perform the stress tests using a covariance matrix

covariance matrix (of one of the types defined below), and let and D respectively denote the Cholesky decompositions of V and W. Now transform the historic returns series r into another vector of returns r* = DC~r. These returns will reflect the conditions of W rather than V and the stress test is performed using r* in the historical VaR model in place of the actual historic returns r.

What type of covariance matrix should be used in a stress test? If the test is against an extreme event that actually occurred, then it is simply the covariance matrix that pertained at that time. Historic data on all assets and risk factors around the time of extreme events will have to be stored so that historic covariance matrices that include data on these extreme events can be generated.

If the stress test is for a change in liquidity accompanying the stress event then the stressed matrix can be a new covariance matrix that reflects changes in holding periods associated with different asset classes. For example, consider a covariance matrix for positions in Eastern European equity markets. The risk factors that define the covariance matrix will be the relevant equity indices and the relevant exchange rates. Following an extreme event in the markets, suppose it is possible to liquidate the equity positions locally within 5 days without much effect on the bid-ask spread, but that the foreign exchange market becomes very illiquid and that it will take 25 days to hedge the currency exposure. Then an approximate stress covariance matrix can be obtained by multiplying all elements in the equity block and the equity-FX block of the matrix by 5, but the elements in the FX block by 25. Of course, it is not certain that the resulting covariance matrix will be positive semi-definite. The only way to ensure this is to use the same multiplication factor for all elements, which is in effect modelling liquidity deterioration using an / -day VaR measure for reasonably large h.

For a breakdown of volatilities and correlations it is easier to use the decomposition of a covariance matrix V into the product

V = DCD,

where is the correlation matrix and D is the diagonal matrix with standard deviations along the diagonal. Stress tests can therefore be performed by perturbing the volatilities separately from the correlations, and it is only the changes in the correlation matrix that will affect the positive definiteness of V.25 Readers will observe this when using the stress VaR settings in the VaR spreadsheets on the CD. One is therefore free to change any volatilities to any (positive) level during stress tests, and the resulting VaR measures will always be non-negative. But some changes in the correlation matrix would be disallowed, and so it is important to check for positive definiteness (§7.1.3).

25 V is positive semi-definite if and only if xVx 3» 0 for all x 0. But xVx = yCy where = Dx, and, since D is diagonal with positive elements, 0 if and only if x 0. So V is positive semi-definite if and only if is positive semi-definite.

It is worth mentioning that principal components analysis has natural applications to both scenario analysis and stress testing portfolios (Jamshidian and Zhu, 1997). Recall from §6.1 that a principal components representation can be written X = PW where X is a set of standardized returns, P is a matrix of principal components and W is the matrix of factor weights in (6.2). Both X and P have columns that represent time series (the columns of P are the principal components) but W is a matrix of constants, which captures the correlation in the system. Stress-testing correlations can therefore be performed by changing the factor weights matrix W. The advantages of this method include the ability to stress test for correlation breakdown without being required to use a covariance matrix.

Frye (1998) shows how shifts in the first few principal components are useful for the scenario analysis of interest-rate-dependent products. Yield curves lend themselves particularly well to principal component analysis, and the first principal component has the interpretation of a parallel shift in the yield curve (§6.2.1). A scenario that increases the first principal component by the equivalent of 100 basis points, for example, is therefore a computationally efficient method for evaluating the effect of parallel shift scenarios. Similarly, the second component represents a yield curve tilt and the third the curvature, so one can capture very complex scenarios on yield curve movements using just a few scenarios on the first three principal components.

Principal components analysis has natural applications to both scenario analysis and stress testing portfolios