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:


Appreciate for anyone clarifying this. Thanks

  • $\begingroup$ What is $g()$ above? $\endgroup$ – Ric 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? $\endgroup$ – Ric Jan 13 '16 at 12:38

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

| improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.