Translation Invariance of Value at Risk

Summary

The document explains translation invariance for a risk measure: adding a constant amount to a profit-and-loss position reduces its VaR by that amount. For a continuous P&L distribution, VaR is expressed as the negative of a lower-tail quantile, so shifting the distribution by a constant shifts that quantile by the same amount. The answer supports the property by showing that the probability defining the VaR threshold is unchanged when the same constant is added to both the position and its threshold.

The original question applies this idea to a future value adjusted by an interest-rate term. However, the response does not work through that specific expression or clarify the sign and units of its adjustment. Its final statement that the result is true only for elliptical distributions conflicts with the preceding quantile argument, which establishes translation invariance for continuous distributions under the stated P&L convention. The distribution caveat is therefore not explained and should not be treated as a general requirement for translation invariance.

Key ideas

• Translation invariance means a constant gain added to P&L reduces VaR by that constant.
• For continuous P&L, VaR can be represented using a lower-tail quantile with a negative sign.
• Shifting both the P&L and the VaR threshold leaves the defining probability unchanged.
• The answer’s elliptical-distribution caveat is unsupported by its preceding derivation.

Tags

# Is Value-at-Risk translation invariant?


# Is Value-at-Risk translation invariant?












Let: $X=V_1-V_0R_0$ where $R_0$ is the interest rate. Then, is it so that this risk measure is Translation Invariant as:

$\textit{VaR}_{\alpha}(X)=\textit{VaR}_{\alpha}(V_1-V_0R_0)=V_0+\textit{VaR}_{\alpha}(V_1)=V_0+g(F^{-1}_{V_{1}}(\alpha))$?

Appreciate for anyone clarifying this. Thanks

## Answer by Richi Wa (score 1, accepted)

https://quant.stackexchange.com/a/22730

Translation invariance of a risk measure $\rho$ is defined as $$ \rho(X+k) = \rho(X)-k, $$ where $X$ is a random variable such that $\rho(X)$ exists and $k$ is a constant. The meaning is that if I add an amount $k$ to my risky positions then the risk is reduced by this amount.

For VaR we consider the case that $X$ has a continuous distribution and that it is a profit and loss random varibale. Then $$ VaR_\alpha(X) = -F^{-1}_{1-\alpha}(X) $$ and $$ P[-VaR_\alpha(X) \le X] = 1-\alpha. $$ Note that $VaR$ is a posive number and e.g. for $\alpha=99\%$ the quantile $F^{-1}_{1-\alpha}(X)$ is a negative number.

It also holds that $$ P[-VaR_\alpha(X)+k \le X+k] = P[-VaR_\alpha(X) \le X] = 1-\alpha, $$ and thus $VaR_\alpha(X+k) = VaR_\alpha(X)-k$.

This is only true for elliptical distributions

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