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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 ...



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