Aug 05

Diversified VaR of Bond Portfolio – 9 min screencast

by David Harper, CFA, FRM, CIPM

This continues a series of brief tutorials explaining VaR Mapping in Jorion Chapter 11. Previously, I showed the calculation of the un-diversified VaR of the two-asset bond portfolio.

Today I explain Jorion’s Table 11-4 which calculates diversified value at risk (VaR) for the same bond portfolio. The key difference is that diversified VaR should be lower to reflect the benefit of imperfect correlations. The entire calculation implements this formula for portfolio VaR:

As I mention in the screencast, the first VaR above is the classic form: VaR = alpha (e.g., 2.33 @ 99% confidence) multiplied by portfolio volatility. The second VaR above instead uses the correlation matrix; a covariance matrix contains an implicit correlation matrix.

In this case, in implementing the second formula above, we perform two matrix multiplications:

R(xV) is the product of individual VaRs and the correlation matrix. Returns a column vector.
(xV)’ is the transposed individual VaR. Returns a row vector
(xV)’R(xV) multiplies the row vector by the column vector to produce the diversified VaR; i.e., in this case, $2.57 which indeed is less than the undiversified VaR of $2.63. The undiversified VaR is the sum of the individual VaRs.