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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

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  • $\begingroup$ What is $g()$ above? $\endgroup$ – Richard Jan 13 '16 at 10:09
  • $\begingroup$ We have: $F^{-1}_{g(X)}(1-p)=g(F^{-1}(1-p))$ $\endgroup$ – Elekko Jan 13 '16 at 12:33
  • $\begingroup$ Still: how is $g()$ defined? What is it supposed to be. Is $g(x) =x^2$ ? $g$ just pops up in the last term, why? $\endgroup$ – Richard Jan 13 '16 at 12:38
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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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