Q&A: Your porfolio has the following three assets...

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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