Aug 14

Value at Risk (VaR). 2007 FRM, Part 10: Diversified Portfolio VaR

by David Harper, CFA, FRM, CIPM

FRM | Risk |

portfolioVaRintro

Learning Outcomes

  • LO 9.1: With respect to portfolio risk analysis, define diversified portfolio VAR.
  • LO 9.2: With respect to portfolio risk analysis, define individual VAR.
  • LO 9.3: For a two-asset portfolio, calculate the portfolio VAR when the returns of the two assets have a zero correlation and when the correlation is one (perfectly correlated).

 

LO 9.1 Diversified Portfolio VaR

The portfolio VaR learning outcomes are based on Chapter seven of Jorion's Value at Risk. The term 'diversified portfolio VaR' refers to the plain-old value at risk (VaR) that we typically mean: portfolio volatility x portfolio value x critical-z (e.g., 1.645 @ 95% confidence). Where portfolio volatility incorporates correlations between assets (diversification).

Jorion introduces matrix notion: necessary with many positions, but still instructive with a two-asset portfolio. Here is the very important formula for "diversified" portfolio VaR:

portfoliovarmatrix

The EditGrid spreadsheet below (to import into Excel select File > Export As > Excel) performs this calculation. The blue highlight contains the matrix math. Note that portfolio variance is the product of three matrices: position weights transposed (x'), the covariance-variance matrix, and the position weights. (The weight can be percentages or dollars. If percentages, the resulting volatility is percentage; if dollars, the resulting volatility is dollars).

In the example below, we use dollar positions. The matrix product then produces a variance. Take the square root, get the portfolio (dollar) volatility. Multiply by 1.645 (@ 95%) and we get the "diversified" portfolio VaR.

 

LO 9.2 Individual VaR

This is just the VaR of the position. So, as shown below, it equals the product of: position x position volatility x 1.645 (@95%). If the assets are imperfectly correlated (correlation < 1), the sum of individual VaRs must be greater than diversified VaR.

 

LO 9.3 Portfolio VaR with 0/1 correlation

If two assets have a zero correlation, the VaR simplifies to (see column G in EditGrid spreadsheet below):

portfoliovar_zerocorrelate

 

If two assets are perfectly correlated, the VaR simplifies to the sum of the individual VaRs (see column G in EditGrid spreadsheet below):

portfoliovar_perfectcorrelate  

 

Here is the EditGrid spreadsheet:

EditGrid Spreadsheet by bt/frm2007.

Comments

  1. vicky 23 Jun 2008

    thanks for your explanation about VAR (all about VAR).
    it really help me to understand what is VAR all about, because you give a ‘easy to understand’ excel example in this website, not a ‘complicated and boring’ math equation all around.

    many thanks for you to create this useful website.

    best regard.

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