Calculating VaR and Expected Shortfall with Gain-Loss Sign Conventions

Summary

The document explains how to interpret a portfolio change with a positive mean when calculating loss-based risk measures. A gain in portfolio value corresponds to a negative loss, so the loss distribution has the opposite mean while keeping the same standard deviation. This sign convention lets the upper tail of the loss distribution represent adverse outcomes.

For the stated normal model, the response applies the 99.5% standard normal quantile to the loss mean and volatility to obtain VaR. It also points toward a profit-based expression as an equivalent way to handle the sign. The example reports a VaR, but it does not actually calculate expected shortfall, despite the question asking for both. The discussion is brief and assumes normally distributed changes; it gives no treatment of non-normal tails or alternative conventions for reporting VaR.

Key ideas

• A positive mean portfolio change is a gain, so the corresponding mean loss is negative.
• VaR can be computed from the loss distribution or by consistently translating a profit-based convention.
• The example uses a normal quantile and the portfolio change standard deviation.
• The response does not provide an expected shortfall calculation.

Tags

# Calculate the VaR and ES for a confidence level of 99.5%


# Calculate the VaR and ES for a confidence level of 99.5%












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.

## Answer by Rodrigo Palacios (score 0, accepted)

https://quant.stackexchange.com/a/57817

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.