Aug 03

Value at Risk (VaR): 2007 FRM. Part 4 - Illustrated VaR

by David Harper, CFA, FRM, CIPM

FRM | Risk |

var_calculations

I previously explained the elements of basic value at risk (VaR). By basic, I mean parametric instead of historical simulation or Monte Carlo simulation. And the easiest parametric approach assumes portfolio returns are normally (Gaussian) distributed. This post simply applies these formulas with the spreadsheet below.

Inputs

The EditGrid spreadsheet below (which can be downloaded into Excel) contains two sections: inputs and outputs. For inputs we need:

  • Trading days per year. So that we can take annual asset returns and volatilities and convert (scale) them into a n-day periodic value at risk (VaR).
  • The initial portfolio value
  • The two key design choices: time horizon and confidence interval.
  • Because we illustrate a two-asset portfolio, we need the asset parameters: volatility and expected return for each asset, the weight of Asset A (we can figure Asset B's weight as the remainder), and, importantly, we need the correlation between the Assets .The correlation is bounded by -1.0 (perfect negative correlation) and +1.0 (perfect positive correlation).

Output

The solution divides into two parts:

  1. Solve for portfolio volatility
  2. Solve for portfolio value at risk

1. Solve for portfolio volatility

For FRM candidates, please note our calculation of portfolio variance. The third term in the formula is two (2)(weight of A)(weight of B)(Covariance A,B). This term incorporates the benefits of diversification. Note also, the covariance is the product of (correlation)(volatility A)(volatility B).

portfolioVARVariance

To illustrate the impact of this term, consider: if you change the correlation to 1.0 (perfect correlation), then the portfolio volatility will be the weighted average of the asset volatilities. In others, no reduction in portfolio volatility because there are no diversification benefits.

2. Solve for VaR

Once we solve for the portfolio volatility (per year), we can calculate the VaR. Note the standard deviation is multiplied by the square root of time. Then the dollar VaR is simply the product of the portfolio value, the critical value, and the scaled volatility:

$var

The above is technically relative VaR. By relative, we mean relative to the portfolio's final expected value. Because there is an expected gain, the number of dollars we could lose in absolute terms is given instead by absolute VaR.

The illustrated example is shown in the EditGrid spreadsheet below. The spreadsheet contains additional explanatory notes. You can open your own write-able copy here.

EditGrid Spreadsheet by bt/admin.

Comments

  1. Naren 03 Aug 2007

    F14: should read as Expected return
    f15:should read as portfolio weight of Asset B

    Am I correct?

  2. David Harper, CFA, FRM 04 Aug 2007

    Naren: Yes, very correct, thanks for spotting those bad labels. Corrected as above

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