Question

Prove that CVar is a coherent risk measure

Answer 1

CVar or Conditional Value at Risk is a coherent risk measure.

CVar is derived on the basis of taking a weighted average of the “extreme” losses in the tail of the distribution of possible returns. This is beyond the VaR cutoff point.

The formula for CVar is: CVar = min {v + (1/1-α)*E*[R-v]+}

As such we can see that Cvar satisfies the sub-additivity property. This property is not valid in case of a normal VaR. Thus CVar is coherent for general loss distributions, including discrete distributions.