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)?

  1. Expected Shortfall is always numerically and unconditionally smaller than the quoted VaR figure.
  2. The ratio of Expected Shortfall to VaR depends only on the mean of the return distribution, not the tail.
  3. Expected Shortfall and VaR must be numerically equal whenever the loss distribution is symmetric.
  4. ES is approximately 1.15 times larger than VaR at high confidence levels for normal distributions.

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