## Incremental VAR

AIM 74.4: Compute incremental VAR, explain why calculating incremental VAR may be difficult, and give a useful approximation.

Incremental VAR is the change in VAR from the addition of a new position in a portfolio. Since it applies to an entire position, it is generally larger than marginal VAR and may include nonlinear relationships, which marginal VAR generally assumes away. The problem with measuring incremental VAR is that, in order to be accurate, a full revaluation of the portfolio after the addition of the new position would be necessary. The incremental VAR is the difference between the new VAR from the revaluation minus the VAR before the addition. The revaluation requires not only measuring the risk of the position itself, but it also requires measuring the change in the risk of the other positions that are already in the portfolio. For a portfolio with hundreds or thousands of positions, this would be time consuming.

For small additions to a portfolio, we can estimate the incremental VAR with the following steps:

Step 1: Estimate the risk factors of the new position and include them in a vector [r|]. Step 2: For the portfolio, estimate the vector of marginal VARs for the risk factors

[MVARp. Step 3: Take the cross product.

This probably requires less work and is faster to implement because it is likely the managers already have estimates of the vector of MVAR^ values in Step 2.

Example: Computing VAR using matrix notation

A portfolio consists of assets A andB. The volatilities are 6% and 14%, respectively.. ■■ There are $4 million and $2 million invested in them, respectively. If-we assume .they are uncorrected with each other, compute the VAR of the portfolio using a confidence parameter, Z, of 1.65. "'

We can use matrix notation to derive the dollar variance of the portfolio: ap2V2 = [$4 $2

This value is in ($ millions)2. VAR is then the square root of this value, times 1.65: VAR = (1.65)($368,782) = $608,490 :

Professor's Note: Matrix multiplication consists of multiplying each row by each lliiP column. For example: (4 x 0.062) + (2 x 0) = 0.0144; 0.0144,x 4 = 0.0576.

Example: Computing incremental VAR

A portfolio .consists of assets A and B, The volatilities "are 6% and 14%, respectively. " : There are $4 million and $2- million invested in them respectively/ If we assume they are ^ uncorrelated with each other, compute the incremental VAR for an increase of $ 10,000 in Asset A. Assume a Z-score of 1.65.

Answer:

To find incremental VAR, we compute the covariances of each position: .

cov(RA,Rpj |
0.062 ■ 0 |
$4 |
'0.0144 | ||

cov(Rb,Rp) |
0 0.142 |
$2 |
0.0392 |

The marginal VARs of the two positions are:

MVARa = Zc XÍ^X^Rp) = 165x 0ß44 = 0-064428; aP VO.136

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