Roy's safety-first criterion


Roy's safety-first criterion is a risk management technique that allows an investor to select one portfolio rather than another based on the criterion that the probability of the portfolio's return falling below a minimum desired threshold is minimized.
For example, suppose there are two available investment strategies—portfolio A and portfolio B, and suppose the investor's threshold return level is −1%. then the investor would choose the portfolio that would provide the maximum probability of the portfolio return being at least as high as −1%.
Thus, the problem of an investor using Roy's safety criterion can be summarized symbolically as:
where is the probability of being less than .

Normally distributed return and SFRatio

If the portfolios under consideration have normally distributed returns, Roy's safety-first criterion can be reduced to the maximization of the safety-first ratio, defined by:
where is the expected return of the portfolio, is the standard deviation of the portfolio's return and is the minimum acceptable return.

Example

If Portfolio A has an expected return of 10% and standard deviation of 15%, while portfolio B has a mean return of 8% and a standard deviation of 5%, and the investor is willing to invest in a portfolio that maximizes the probability of a return no lower than 0%:
By Roy's safety-first criterion, the investor would choose portfolio B as the correct investment opportunity.

Similarity to Sharpe ratio

Under normality,
The Sharpe ratio is defined as excess return per unit of risk, or in other words:
The SFRatio has a striking similarity to the Sharpe ratio. Thus for normally distributed returns, Roy's Safety-first criterion—with the minimum acceptable return equal to the risk-free rate—provides the same conclusions about which portfolio to invest in as if we were picking the one with the maximum Sharpe ratio.