Thus the 1% 1-day VaR due to the risk factors is $2.33 million x 0.0402 = $0,467 million.

The specific risk of the portfolio also contributes to the P&L variance. In fact from (8.7) we have

P&L variance = pBVBp + pVEp, (9.7)

where VE is the n x n specific risk covariance matrix. There are therefore two components in the covariance VaR:

VaR due to market risk factors = Za(p BVBp)1 2; Specific VaR = Za(pVep)1/2.

Saving the residual covariance matrix from the factor Ve model therefore allows a separate VaR measure for the specific risk of the portfolio. Risk capital charges for specific risk can be substantially reduced by this method.13

Note that the total VaR should not be measured as the sum of these two components.14 The total variance is the sum, not the total volatility, so the total VaR is obtained from (9.7) as

Total VaR = ZpBVB p + pVEp)1/2.

Similar principles are used in the covariance VaR spreadsheet on the CD. It calculates VaR for a dollar investor in an international equity portfolio. Total VaR is disaggregated into equity VaR and FX VaR as explained in the help sheet.

9.3.4 Covariance VaR with Cash-Flow Maps

To compute the covariance VaR of a portfolio of bonds or loans, it may be represented by a cash-flow map p = (PVU PV2, . . ., PVk), where PV, denotes the present value of the income flow at time t for fixed maturity dates t = 1, . . ., k. The present value data should be translated to an absolute scale, so that VaR is measured in terms of P&L, and normally we represent the cash flow amount by its sensitivity to a 1 basis point move in interest rates. Then the covariance matrix must also refer to changes in basis points.

For example, compute the 1% 1-day VaR for a cash flow of $1 million in 3 months time, and $2 million in 6 months time, when both the 3-month and the 6-month interest rates are 6%, the 3-month interest rate has an annualized daily volatility of 10%, the 6-month interest rate has an annualized daily volatility of 9% and their correlation is 0.95.

The present value data should be translated to an absolute scale, so that VaR is measured in terms of P&L. Normally we represent the cash flow amount by its sensitivity to a 1 basis point move in interest rates. Then the covariance matrix must also refer to changes in basis points

13 There is a limit to this reduction: the specific risk charge as measured by (9.7) cannot be less than 50° of the charge that would be made under the standardized rules.

14 In §9.3.5 we see that it is only under very special conditions that one can aggregate covariance VaR measures.

A standard cash-flow mapping method, where cash flows are linearly interpolated between adjacent vertices, is not appropriate for VaR models

First, the cash-flow amounts must be mapped to P&L sensitivities to changes of 1 basis point. The present value of a basis point move (PVBP) is one of the traditional risk measures that were mentioned in §9.2.1. Following footnote 8, to convert the 3-month cash flow to sensitivity terms it should be divided by 4x 104; and to convert this to present value terms it should be discounted at the annual rate of 6% (which means dividing by approximately 1.015); similarly the 6-month cash flow should be divided by 2 x 104 and divided by 1.03 to view it in (approximate) present value sensitivity terms. Thus the 3-month cash flow has the present value of a 1 basis point move of

S(106 x 10"4)/(4 x 1.015) = $24.63;

similarly the 6-month cash flow has a present value, for a 1 basis point move, of

5(2 x 106 x 10~4)/(2 x 1.03) = S97.09.

Secondly, the covariance matrix must refer to rate changes in basis points; if the rates are currently 6% the 10% volatility means that the 3-month rate can vary by ±60 basis points over one year and the 9% volatility means that the 6-month rate can vary by ±54 basis points over one year. In fact, in basis points the 1-day variances are, assuming 250 days per year, (0.1 x 600)2/250 = 14.4 and (0.09 x 600)2/250 = 11.664, respectively. Their 1-day covariance is 0.95(14.4 x 11.664)1/2 = 12.312, so the 1-day covariance matrix in basis point terms is

14.4 12.312

12.312 11.664

Now, recall that the A-day variance of the portfolio P&L is pVp, where V is the A-day x covariance matrix for interest rates at maturity dates t - 1, . . ., , and that the 100a% / -day covariance VaR of this portfolio is simply

VaRa,A = Za(pVp)

Thus, in the example above, we calculate pVp =(24.63 97.09

14.4 12.312 12.312 11.664

24.63 97.09

= $177 570,

so the 1% 1-day VaR is 2.33 x 177 5701/2 = $981.84.

It should be noted that a standard cash-flow mapping method, where cash flows are linearly interpolated between adjacent vertices, is not appropriate for VaR models. In fact the VaR of the original cash flow will not be the same as the VaR of the mapped cash flow if this method is used. To keep VaR constant one has to use quadratic interpolation between adjacent vertices. Consider the cash-flow Xin Figure 9.3 that is between the vertices A and B. Suppose it is tA days from vertex A and tB days from vertex B. The linear interpolation method would be to map a proportion p - tB/{tA + tB) of X to A and a proportion

a x

Figure 9.3 Cash-flow mapping.

1 - p = tA/(tA + tB) to B. But the variance of the resulting mapped cash flow will not be the same as the variance of the original cash flow. A VaR-invariant cash-flow map is to map a proportion p to vertex A and 1 - p to vertex B, where

V(X) = p2 V(A) + (1 -p)2V(B) + 2p(\-p)cov(A, B). (9.8)

The required proportion p is found by solving the quadratic equation (9.8), using estimates for V(A), V(B), V(X) and cov(4, B). The variances of returns at vertices A and and their covariance can be obtained from standard covariance matrices, and the variance of the original cash flow X is obtained by linear interpolation between V(A) and V(B) according to the number of days:

V(X) = [tB/(tA + tB)]V(A) + [tA/(tA + tB)]V(B).

For example, consider a cash flow X - $1 million at 1 year and 200 days. The adjacent vertices are assumed to be A at 1 year and at 2 years, with 365 days between them. Then tA = 200 and tB - 165, so tA/(tA + tB) = 0.548 and tB/ (tA + tB) - 0.452; so with linear interpolation $0.452 million would be mapped to the 1-year maturity and $0.548 million would be mapped to the 2-year maturity. However, this mapping would change the VaR of the cash flow. Now suppose that V(A) = 4 x 10"5 and V(B) = 3.5 x 10"5, and that coy (A, B)= 3.3 x 10"5. By linear interpolation V(X) = 0.452 x 4 x 10"5 + 0.548 x 3.5 x 10-5 = 3.726 x 10~5. So the proportion p of the cash flow that should be mapped to A to keep VaR constant is found by solving

3.72 = 4/+ 3.5(1 -pf+ 6.6/7(1 -p)

0.9p2 - 0.4/7 - 0.22 = 0

giving p= -0.3198 or 0.7643. Taking the positive root shows that a VaR invariant cash flow map is to map $0.7643 million to A and $0.2357 million to B. The fact that much more is mapped to A than under linear interpolation is