Lesson 7 of 7 · 12 min

Shortfall probability, the Sharpe ratio and VaR

Under normality, the probability of missing the threshold is N(−SFRatio)N(-\text{SFRatio}); the same idea links the safety-first ratio to the Sharpe ratio and underpins value at risk.

In short

  • Shortfall probability: P(Rp<RL)=N(−SFRatio)=1−N(SFRatio)P(R_p < R_L) = N(-\text{SFRatio}) = 1 - N(\text{SFRatio}).
  • The safety-first optimal portfolio has the highest SFRatio and therefore the lowest shortfall probability.
  • With RLR_L = the risk-free rate, the SFRatio is the Sharpe ratio: the highest-Sharpe portfolio minimises the chance of earning less than the risk-free rate (under normality).
  • If hitting the threshold is a necessity, not a wish, model the required amount as a liability and use fixed-income strategies such as cash-flow matching.
  • Value at risk (VaR): a minimum loss expected at a given probability over a given horizon. Stress testing / scenario analysis estimates losses in extreme scenarios.

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Shortfall probability, the Sharpe ratio and VaR · Portfolio Mathematics