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Assume a market value of $ \$ 10000$.The daily change in the value of a portfolio is linearly dependent on two uncorrelated factors. The delta of a portfolio with respect to the first factor is $5$ and delta with respect to the second factor is $-6$. The standard deviation of the factors are $10$ and $15$ respectively. What is the $10$-day $90 \% $ VaR?

$VaR_A=\sqrt{10} \times 1.2816 \times \sqrt{10} \times 5 = 64.08$

$VaR_B=\sqrt{15} \times 1.2816 \times \sqrt{10} \times -6 = -94.17 $

Since uncorrelated,

$VaR = \sqrt{(-94.17)^2+(64.08)^2} = 113.91$

I want to know if I am on the right track.

Tosh
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1 Answers1

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Denoting the delta coefficients as $\delta_1$ and $\delta_2$, the change in portfolio value $\Delta V$ is related to the factor changes $\Delta f_1$ and $\Delta f_2$ according to

$$\Delta V = \delta_1 \Delta f_1 + \delta_2 \Delta f_2.$$

Subtracting means, squaring and taking the expectation we obtain the following formula for the portfolio variance

$$E\{ \, [\Delta V - E(\Delta V) ]^2 \, \} = \sigma_V^2 = \delta_1^2 \sigma_{f_1}^2 + \delta_2^2 \sigma_{f_2}^2 +2 \rho \delta_1 \delta_2 \sigma_{f_1} \sigma_{f_2}.$$

Since the factors are uncorrelated, we have $\rho = 0$, and

$$\sigma_V = \sqrt{\delta_1^2 \sigma_{f_1}^2 + \delta_2^2 \sigma_{f_2}^2 }.$$

Assuming that $\sigma_{f_1}$ and $\sigma_{f_2}$ are standard deviations for $1$-day factor changes, the $\delta t$-day VaR at the $90 \, \%$ level is given by

$$\text{VaR} = 1.2816 \sqrt{\delta t} \,\sigma_V.$$

Substituting given values,

$$\text{VaR} = 1.2816 \sqrt{10} \, \sqrt{(5)^2(10)^2 + (-6)^2 (15)^2} \approx 417.26 $$

RRL
  • 90,707