Aug 15

Value at Risk (VaR). 2007 FRM, Part 11. Incremental & Component VaR

by David Harper, CFA, FRM, CIPM

FRM | Risk |

portfolioVar

Learning Outcomes

  • LO 9.5: With respect to portfolio risk analysis, define marginal VAR.
  • LO 9.6: With respect to portfolio risk analysis, define and calculate incremental VAR, explain why calculating incremental VAR may be difficult, and give a useful approximation.
  • LO 9.8: Estimate component VAR in a portfolio with a large number of positions and use it to decompose VAR.

We continue the previous post in order to define marginal, incremental and component VaR (based on Jorion's 3rd edition Value at Risk). All three are calculated in the EditGrid spreadsheet below, which can be uploaded to Excel (File > Export As > Excel).

We assume a two-asset portfolio: a $200 position in Canadian dollars and a $100 position in Euors. Previously, we defined...

  • Diversified portfolio VaR (cell D17 in the spreadsheet): Portfolio volatility x Portfolio value x critical value (e.g., 1.645 @ 95%). As such, diversified portfolio value incorporates asset/position correlations into the portfolio volatility
  • Individual VaR (cells D23 and E23): Position (e.g., CAD, EUR) volatility x Position value x critical-z

...now let's define marginal, incremental and component VaR:

image

9.6 Incremental VaR

Incremental VaR (cell D30) is the change in portfolio VaR implied by a change in a position; e.g., what happens to portfolio VaR if we add to the Canadian dollar position? The full revaluation approach to incremental VaR requires "re-pricing" the portfolio VaR before and after the change in position; the difference is the incremental VaR. On the chart above, it represents the vertical distance on the y-axis (additional portfolio VaR) as we move to the right on the x-axis (additional position). 

But the spreadsheet below does not do this. Instead, is uses an approximation to estimate the incremental VaR: marginal VaR multiplied by the new (additional) position.

9.5 Marginal VaR

Marginal VaR (cells D25 and E25) is a first derivative. FRM candidates surely know what that means by now! It's the slope of a tangent line. As Jorion says, marginal VaR is the "change in portfolio VaR resulting from an additional dollar of exposure to a given component." Notice that each position contributes a marginal VaR, which is a function of the variance-covariance matrix.

9.8 Component VaR

Component VaR (cells D32 and E32) is a position's contribution to the portfolio VaR. If the position were eliminated, portfolio VaR would drop by the component VaR. Two key things to remember about component VaR:

  • Component VaRs (i.e., the component VaR for each position) sum to portfolio VaR
  • Component VaR is estimated with marginal VaR. Therefore, it is an estimate (based on an underlying linear relationship, itself an approximation). As an estimate, it approximates incremental VaR. Note the gap between the incremental and component VaR in the chart above. 

Here is the EditGrid spreadsheet:

 

EditGrid Spreadsheet by bt/frm2007.

Comments

  1. jay park 05 Oct 2007

    ...

Leave a Comment