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IAI Actuarial Core Principles · Economic Modelling · Measures of investment risk

Two portfolios have the same mean return of 10% and are normally distributed. Portfolio P has standard deviation 10% and Portfolio Q has standard deviation 5%. Which statement about the shortfall probability relative to a threshold of 8% is correct?

Q has the lower shortfall probability. With the threshold 8% below the common mean of 10%, Q's z-score is −0.4 against −0.2 for P, so its lower tail probability is smaller, about 0.34 against 0.42.

  1. AP has the lower shortfall probability because its standard deviation is higher
  2. BBoth have equal shortfall probabilities because means are equal
  3. CQ has the lower shortfall probability because the threshold is below the meanCorrect
  4. DP has the lower shortfall probability because the threshold is below the mean
  5. Q has the higher shortfall probability because its z-score is less negative

Explanation

The threshold of 8% is below the mean of 10%. P has z=-0.2 giving about 0.4207 and Q has z=-0.4 giving about 0.3446. Lower volatility gives a smaller shortfall probability when the threshold is below the mean. The option claiming higher risk is better reverses this effect, which holds only when the threshold exceeds the mean.

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