Interpreting Value at Risk from Portfolio Value Quantiles

Summary

The document clarifies a convention for calculating value at risk from a portfolio’s future value. The apparent discrepancy is between treating VaR as a quantile itself and expressing it as the difference between expected portfolio value and a lower-tail quantile of that value.

Under the convention discussed, the random variable represents future portfolio value rather than loss. VaR at a specified confidence level is therefore the expected value minus the selected quantile, expressing the shortfall from the expectation to that tail outcome. The example uses an expected value of 10 and a quantile of 4, yielding a VaR of 6. If the random variable instead directly represents losses, VaR is commonly expressed as a quantile of that loss distribution. The key is to identify what the variable measures and which sign convention is being used. The brief exchange gives a definition and a simple illustration, but does not address alternative VaR conventions or estimation methods.

Key ideas

• VaR may be expressed as a quantile of losses or as a shortfall from expected portfolio value, depending on the variable used.
• When the random variable is portfolio value, the discussed convention subtracts the selected quantile from expected value.
• Always check whether a distribution represents gains, portfolio values, or losses before interpreting its quantiles.
• The example illustrates the calculation but does not compare VaR estimation methods.

Tags

# VAR interpretation


# VAR interpretation












I definitely struggle to understand the following interpretation of VAR (value at risk) provided by Jorion

$$VAR(c)=E[X]−Q(X,c)$$

where $X$ is a random variable, $E[X]$ its expected value, $Q(X,c)$ the quantile of its distribution such that the associated probability is c.

Isn't VAR the quantile itself? Why should it be viewed as a deviation from the mean?

## Answer by Oscar (score 2)

https://quant.stackexchange.com/a/59458

VaR is the quantile of the loss distribution but $X$ in your post no doubt denotes the the future value of the portfolio. If $E[X]=10$ (expected future value of portfolio) and the quantile $Q(X,c)=4$ for some $c$, then your VaR, that is the lost that represents the outcome at the quantile, is $$VaR(c) = E[X] - Q(X,c) = 10 - 4 = 6$$

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