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The daily change in the value of a portfolio is, to a good approximation, linearly dependent on two factors (uncorrelated factors). The delta of a portfolio with respect to the first factor is 6 and delta with respect to the second factor is 4. The standard deviation of the factors are 20 and 8,respectively. What is the 5-day 90% VaR? (Ans: 356.21).

Standard deviation of portfolio:

$\sigma_p = \sqrt{20^2+8^2} = 21.54$

5-day 90% VaR

$2.33(\sqrt{5})(21.54)=112.23$

I cannot find how I don't reach the answer.

Tosh
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  • I don't know what the delta of a portfolio is, but you have not used the constants $6$ and $4$ anywhere. – Ross Millikan Dec 25 '17 at 06:30
  • Delta of a portfolio is the change of the value of the portfolio with respect to asset price. The value of the portfolio is $6 \delta x_1 - 4 \delta x_2$ where you use the 6 and 4. But I don't know how to use that in VaR. – Tosh Dec 25 '17 at 06:36

1 Answers1

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Since $\Delta V = 6 \, \delta x_1 - 4 \, \delta x_2$ and the factors are uncorrelated, the correct portfolio standard deviation is

$$\sigma_p = \sqrt{E(\Delta V^2)} = \sqrt{36 E(\delta x_1^2) + 16 E(\delta x_2^2)} = \sqrt{36 \cdot 20^2+ 16 \cdot 8^2} \approx 124.19$$

Also for $90 \%$ VaR the multiplier is approximately $1.282$ not $2.33$. (For a normal distribution the $90th $ percentile corresponds to $1.282$ standard deviations.) The factor $\sqrt{5}$ is correct.

This should give you the desired answer.

RRL
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