P&L is pVRp at the individual asset level, or pBVxBp at the risk factor level.7 The variance of P&L is the crucial determinant of the portfolio VaR when it is measured using the covariance method (§9.3).

7.1.2 Simulating Correlated Risk Factor Movements in Derivatives Portfolios

Independent Monte Carlo simulation on each of these returns will produce very odd yield curves

Another application of covariance matrices is to generate correlated paths for the underlying risk factors in an options portfolio. As a simple example, consider a portfolio of domestic swaps where the risk factor is an AA zero-coupon yield curve. Suppose that the cash flow map is to different maturities along this curve, so the risk factor returns are summarized by a vector of returns to zero-coupon AA bonds of different maturities denoted r - (Ru . . ., Rk). Independent Monte Carlo simulation on each of these returns will produce very odd yield curves and we need to use the covariance matrix of r, denoted V, to produce simulations that accurately reflect the correlation in the system.

To see how this is done, let z = (z,, . . ., zk) be the vector of observations on independent standard normal variates that is simulated by Monte Carlo. Then the covariance matrix V(z) = I, the identity matrix. Denote by the Cholesky decomposition of V. The Cholesky decomposition of V is a triangular matrix such that V = CC. Transform z into r = Cz. Since V(r) - CV(z)C = = V, the random sample z has been converted to a set of normal returns r that reflects the appropriate covariance structure.

Figure 7.1 illustrates a set of 10 simulations on a yield curve. In Figure 7.1a each of the simulations is uncorrelated and in Figure 7.1b each of the simulations is correlated. That is, on each curve in Figure 7.1b the six maturities have been moved in a correlated fashion, rather than independently as in Figure 7.1a. The correlated simulations were achieved by pre-multiplying the simulation vector by the Cholesky decomposition of the covariance matrix of the yield curve, as described above.

Vanilla options will have the underlying price, the implied volatility and the interest rate as risk factors

Most derivative portfolios have many risk factors other than a yield curve. Even vanilla options will have the underlying price, the implied volatility and the interest rate as risk factors, so the simulation of a correlated price path requires specification of a 3x3 covariance matrix for the underlying asset return, the implied volatility and the interest rate. This type of correlated simulation for option risk factors is illustrated in the Monte Carlo VaR spreadsheet on the CD.8 A certain amount of historic data will be required to obtain statistical estimates of the variances of these risk factors, and of the covariance between the underlying asset and the implied volatility (unless the

7 At the risk factor level this is only the P&L variance due to market risk factors. There will be another term for P&L variance due to specific risk factors (§8.1.2 and §9.3.3).

8 However, it should be noted that the dependency between price changes and volatility changes is highly nonlinear, as we have seen in §2.3. Therefore it is not well captured by the linear covariance matrix.

underlying is a bond, it is often assumed that the correlation between the interest rate and the other two risk factors is small, if not zero).

7.1.3 The Need for Positive Semi-definite Covariance Matrices

In §7.1.1 it was shown that the variance of a linear portfolio is a quadratic form wVw, where w contains the portfolio weights or, in a factor model representation, their net sensitivities. A symmetric matrix A is positive semi-definite if and only if xAx 0 for all non-zero x. Thus if the covariance matrix V were not positive semi-definite there would be some non-trivial portfolios that have a negative variance. This is not possible, since variance must always be non-negative. Another reason why covariance matrices should be positive definite is that in order to generate correlated scenarios as described above, either the Cholesky or the singular value decomposition of the covariance

matrix is usually employed (§7.1.2). The Cholesky decomposition is like the square root of a matrix: it only exists if the matrix is positive definite.9

A symmetric matrix A is positive semi-definite if and only if none of its eigenvalues are negative.10 Thus tests for positive semi-definiteness are based on calculating the eigenvalues and ensuring that they are all non-negative. It is always prudent to pass a covariance matrix through a simple eigenvalue check for positive semi-definiteness.

The covariance matrix is commonly estimated by (XX)/T, where T is the number of data points and X is the T x k matrix of data on the asset (or risk factor) returns. This matrix, which corresponds to using equally weighted averages of squared risk factor returns, will always be positive definite when T>k.11 But if other covariances and variances are used in a covariance matrix, such as EWMA or GARCH estimates, positive semi-definiteness is not always guaranteed (§7.4 and §4.5).

In some cases a covariance matrix that is not positive definite is a blatant case of misspecification.12 In some cases non-positive semi-definiteness is simply a result of rounding error (this will happen if, for example, linear interpolation is used between risk factors); in that case it may happen that some eigenvalues are negative, but then they will be extremely small. Then it may not alter the original covariance greatly if one simply gives these negative eigenvalues a value of zero. Whenever negative eigenvalues are made zero, it is of course necessary to check the new variance and covariances to ensure that they accurately reflect the volatilities and correlations in the system.

7.1.4 Stress Testing Portfolios Using the Covariance Matrix

Sections 7.1.1 and 7.1.2 explained the role of a covariance matrix for the calculation of portfolio risk. To obtain an everyday or normal markets risk measure for a portfolio it is appropriate to use a covariance matrix that is currently relevant for the risk horizon of the measure. This covariance matrix will often be a forecast of variances and covariances that are obtained using the statistical methods described in Chapter 3 or 4. But in addition to an everyday risk measure, risk management will require estimates of the portfolio risk under extreme market conditions. A stress covariance matrix is applied in the

9 In fact if V has eigenvalues . . ., Xk and V = MAM, where M is the matrix of eigenvectors and is the diagonal matrix of eigenvalues, then C = MA2, where "2 is the diagonal matrix with the square roots of the eigenvalues along the diagonal. If V is not positive semi-definite some of these eigenvalues will be negative, so does not exist. If V is not positive definite because some eigenvalues are zero, then is not defined uniquely.

"This can be seen by writing xAx = uAu, where A is the diagonal matrix of eigenvalues of A.

11 A matrix of the form AA is always positive definite, since xAAx is the sum of the squares of the elements of Ax, so it will always be non-negative for non-zero x.

12 Even if the result is positive semi-definite, some methods of estimating the covariance matrix may still be quite ridiculous.

Thus tests for positive semi-definiteness are based on calculating the eigenvalues and ensuring that they are all non-negative

A stress covariance matrix is applied in the risk model, keeping the other parameters such as the portfolio weights and the estimates of pricing model parameters constant