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Question: The change in the value of a portfolio in three months is normally distributed with a mean of $500,000$ and a standard deviation of $3$ million. Calculate the VaR and ES for a confidence level of 99.5% and a time horizon of three months.

My try: We know that $VaR_{99.5} = F^{-1}(0.995)$ where $F^{-1}$ is the quantile function of the distribution. So, for calculate the VaR we only need to take the quantile for a normal distribution with $500,000$ and a standard deviation of $3$ million?

I saw the book solutions and in these they mention the following: The loss has a mean of −500 and a standard deviation of 3000. Also, N−1(0.995) =2.576. The 99.5% VaR in $’000s is −500+3000×2.576) =7,227

I don't understand why they take the negative mean.

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  • $\begingroup$ If you have a gain of X, that is the same thing as saying my loss is -X. $\endgroup$ – noob2 Sep 1 '20 at 18:14
  • $\begingroup$ just to add details to the point made by @noob2, you can calculate the loss percentile as: 500+NormsInv(0.5%)×3000 or 500−NormsInv(99.5%)×3000=-7,227.5, but VaR is traditionally reported with a plus sign (loss is understood), hence the sign reversal: -500+NormsInv(99.5%)×3000=7,227.5 $\endgroup$ – Magic is in the chain Sep 1 '20 at 19:12
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The reason the mean is negative is because the change in portfolio value is seen as a gain rather than a loss, so if the mean is positive, to calculate the respective VaR, we must use the formula of the profit, i.e $\mu - \sigma\Phi(\alpha)$

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