After running an optimisation using a quadratic utility (CRRA) function I calculate an CVAR that is equal to the VAR especially for very small risk-aversion levels ($\gamma$=1 and $\gamma$=2 e.g.). What is the intuition behind this finding? I would think that the skewness and kurtosis are approximately normal and these approach one another? Is it even an informative finding?

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    $\begingroup$ Can you give a specific example with some numbers? $\endgroup$ – phdstudent Jul 26 '20 at 20:40

That is incredibly unlikely for a continuous distribution -- though possible for a distribution with a part that is not absolutely continuous, i.e. is atomic.

The way to see this is to remember that the $\alpha$% CVaR/ES/TCE is defined as:

$CVaR(r,\alpha) = E(r|r\leq Var(r,\alpha))$.

Thus getting an $\alpha$-CVaR equal in magnitude to $\alpha$-VaR would imply there are no returns below the $\alpha$-VaR level.


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