Asset | X | Y | Z |

Expected returns | 5% | 10% | 15% |

standard deviations | 10% | 20% | 35% |

You views any porfolio return below a level of 3% as unacceptable. Find the asset that minimizes the probability that the portfolio will fall below 3% annual return, and what is the probability?

**CppQuant Answer**

make use of Roy's Safety-First criterion.

- Compute the SFRatio for each asset: SFRatio = (R(e)-R(l)) / stderr
- The probability of shortfall will be N(-SFRatio) = 1 - N(SFRatio). Or in Excel, NORMSDIST(-SFRatio)

Asset X Y Z SFRation (5%-3%)/10% = 0.2 (10% - 3%)/20% = 0.35 (15% - 3%)/35% = 0.34 N(-SFRation) NORMSDIST(-0.2)=0.4207 NORMSDIST(-0.35)=0.3632 NORMSDIST(-0.35)=0.3669

- Y<Z<X: Thus, asset Y minimizes the probability that the return will fall short of 3%, with a probability of approximately 36%.

*Bonus Points*

- Safety-first rules are concerned with shortfall risk. According to Roy's safety-first criterion, an optimal portfolio minimizes the probability that the porfolio return (
**NOT**the returns on all the securities in the portfolio) fall below the threshold level. - If returns are normally distributed, the safety-first optimal portfolio maximizes the safety-first ratio (SFRatio).
- SFRatio gives the distance (from the mean return to the shortfall level) in units of standard deviation: it measures the excess return over the threshold level per unit of risk.
- The SFRatio becomes the Sharpe ratio if we substitute the risk free rate for the threshold return R(l): The safety-ratio focuses on the excess return over the threshold return, while the Sharpe ratio focuses on the excess return over the risk-free rate.

*Category: Quantitative Analysis > Probability*

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