hard · Quantitative Finance microstructure-arb
A risk manager uses Value at Risk (VaR) at the 99% confidence level.
If the returns are normally distributed with mean 0 and standard deviation σ, what is the relationship between VaR and Expected Shortfall (ES)?
- Expected Shortfall is always numerically and unconditionally smaller than the quoted VaR figure.
- The ratio of Expected Shortfall to VaR depends only on the mean of the return distribution, not the tail.
- Expected Shortfall and VaR must be numerically equal whenever the loss distribution is symmetric.
- ES is approximately 1.15 times larger than VaR at high confidence levels for normal distributions.
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