Aug 09

Value at Risk (VaR). 2007 FRM, Part 8: Trade impact on Value at Risk (VaR)

by David Harper, CFA, FRM, CIPM

FRM | Risk |

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Learning Outcome

  • LO 8.3: Given the cost per dollar of VAR and the relevant betas, expected returns, and correlations, calculate the VAR impact and expected net gain of a project/trade that is not large relative to the firm's portfolio of projects.

Impact of trade = Change to expected return + VaR Impact

This learning outcome is a mouthful but useful building blocks are applied in service of the final answer. Following the assigned reading (Risk Management & Derivatives, Rene Stulz), we start with a three-asset portfolio and then we execute a trade, for example: buy asset #1 and sell asset #3, in amounts equal to 1% of the portfolio.

The question is simply, what is the net impact (gain/loss) of the trade?

There are two parts to the answer:

  1. The change in the portfolio's expected return
  2. The change in portfolio value due to the change in value at risk (VaR). For example, if the trade reduces VaR, the positive value is created as some fraction of the reduced VaR.

The EditGrid spreadsheet below illustrates this analysis. There are two steps in the spreadsheet: 1. Setup portfolio inputs, and 2. Analyze the trade

1. Portfolio inputs

The inputs include asset and portfolio characteristics: expected returns, volatilities, weights to each asset, and value of the portfolio. Note also the 3x3 correlation matrix (we only need three cells).

We need the correlations to compute the covariances (yellow row) and we need the covariances to compute the asset betas (the beta of each asset with respect to the portfolio).

This is a key idea in the analysis: asset betas are used to estimate the impact of the trade on portfolio VaR.

2. Analyze trade

To analyze the trade, we perform two sub-steps

  • 2a. Calculate VaR impact of trade
  • 2b. Calculate "all in" net impact

The VaR impact of the trade uses the betas already calculated. I have a 1% trade illustrated (buy Asset #1 and sell Asset #2). So, under these assumptions, the purchase of Asset #1 (the less risky asset) lowers the portfolio's expected return but also lowers the portfolio's VaR. The increase in VaR is the product of: the asset's beta, the critical value (1.645 @ 95%), portfolio volatility, the portfolio value, and the size of the trade (1%).

But please focus on the critical idea: asset beta multiplied by the portfolio's (scaled) volatility. Since beta is the asset's sensitivity with respect to the portfolio, it links the change in the asset's weight to total portfolio volatility (and the rest of the terms make it a portfolio VaR)!

The final sup-step, the calculation of the "all in" net impact accounts for the benefits of a reduction in VaR. This is is company-specific; under the model assumptions below, a one dollar reduction in VaR corresponds to an $0.11 gain.

Finally, the "net gain from trade" combines the change in the expected return with the dollar gain due to the change in VaR. Under the assumptions below, the trade bought a safer asset and sold a riskier asset. The change in expected return is negative; if we stop there, the trade is bad. But the VaR is reduced, and given the 11% marginal cost of VaR, the benefit a reduced VaR outweighs the cost of a reduced expected return.

EditGrid Spreadsheet by bt/frm2007.

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