When Value at Risk and Conditional Value at Risk Can Coincide

Summary

The document asks why an optimized portfolio can produce equal Value at Risk (VaR) and Conditional Value at Risk (CVaR), particularly under low risk aversion, and whether this equality is informative. The answer defines CVaR, also called expected shortfall or tail conditional expectation, as the conditional mean of returns at or below the VaR threshold.

For a continuous return distribution, equality in magnitude would require the tail average to sit exactly at the threshold, which is described as extraordinarily unlikely. It can occur in distributions with an atomic or discrete component. The answer therefore cautions against treating the observed equality as evidence about near-normal skewness or kurtosis. The exchange is brief and gives a conceptual condition rather than investigating the user’s optimization, return convention, or numerical implementation, so those details would need checking before interpreting an actual result.

Key ideas

• CVaR is the conditional mean of returns at or below the VaR cutoff.
• For continuous distributions, equality between VaR and CVaR is described as highly unlikely.
• Equality may occur when a distribution has an atomic or discrete component.
• An observed equality should be checked against the distribution and calculation conventions before drawing conclusions.

Tags

# When is the VaR equal to the CVaR?


# When is the VaR equal to the CVaR?












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 (e.g. $\gamma=1$ and $\gamma=2$). 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?

## Answer by kurtosis (score 2, accepted)

https://quant.stackexchange.com/a/55933

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.