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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1$\begingroup$ Can you give a specific example with some numbers? $\endgroup$– phdstudentJul 26, 2020 at 20:40
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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.
