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Interpreting Absolute and Relative Value at Risk from P&L Data

Article Quant Q&A · Author: One Pablo
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Summary

This note explains how to read a historical Value at Risk estimate from a sorted sample of profit and loss outcomes. For a stated confidence level, the relevant tail observation is selected from the loss end of the distribution; gains do not become VaR merely because they fall at a corresponding percentile on the positive side. The example uses a sample of 500 observations and identifies the tail observation based on the chosen confidence level.

It distinguishes absolute VaR, calculated without adding an expected mean, from relative VaR, which accounts for an expected return. The discussion cautions that removing the mean is not necessarily conservative when the mean is negative, and that a historical mean may differ from the future expected return. It frames VaR as satisfying monotonicity: all else equal, a higher-mean P&L distribution represents less risk. The note does not address VaR’s broader limitations or compare it with tail-sensitive alternatives such as expected shortfall.

Key ideas

  • Historical VaR is read from the loss tail of sorted P&L outcomes at the selected confidence level.
  • Positive P&L outcomes do not determine loss-side VaR simply because they occupy a corresponding percentile.
  • Absolute VaR omits the mean, while relative VaR incorporates an expected return.
  • Removing the mean is not always conservative, particularly when the mean is negative.
  • VaR satisfies monotonicity: with otherwise identical distributions, a higher mean corresponds to lower measured risk.

Tags

Risk managementStatisticsBacktesting
Full text
# Absolute and Relative Value at Risk


# Absolute and Relative Value at Risk












Is it correct to calculate the VaR as 99% max between loss and profit. E.g. if 99% VaR on the loss side of the distribution is -100, and on the positive side of the distribution there is a value corresponding to 120 at 99% confidence interval. Is it correct to say that the VaR is then 120?

My thinking is that VaR should be the value corresponding to the 99% worst loss , so if p&l vector has -100, 80, -90, 45, 120 ... until n=500. The worst 5th scenario from the negative numbers is the VaR when the numbers are arranged from smallest to the largest for all 500 observations, with the smallest number starting with a negative value and largest number being positive.

Also, please explain on the concept of Absolute VaR, as my understanding is that absolute VaR is just the VaR relative to 0, whereas relative VaR is the VaR relative to some expected return.

## Answer by AK88 (score 1, accepted)

https://quant.stackexchange.com/a/49882

Terminology around some risk measures can sometimes be very precarious. In your first paragraph, you have got things unnecessarily complicated. If you are looking at a sorted vector of 500 P&L, then your 99% VaR would simply be the P&L corresponding to the 4th smallest number in the vector (e.g. $500 \times (1-0.99) = 5$).

In the second paragraph, you pretty much got this.

Absolute VaR is sometimes referred to as conservative VaR or VaR without the mean. However, conservativeness of it is quite questionable, especially if the mean is negative value. Assuming that you did have a positive mean (historically) you will see a similar distribution to the the following:

On the left side (blue), you see de-meaned P&L (in percentage terms). And if you calculated your VaR from this distribution, you would get a smaller or more conservative number than the VaR from the distribution on the right side (red).

Also, your future expected return may or may not be equal to the mean return that you calculated using some historical data. Therefore, you would de-mean the series first and add back your expected mean to the P&L, if you wanted to see more realistic or as you wrote above, Relative VaR.

Keep in mind that a coherent risk measure should satisfy the monotonicity criteria, which VaR actually does. What this criteria states is that if you have the same P&L distribution, but the means (or expected returns) are different, the you would have less risk for the distribution with higher mean.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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