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 (the minimum return that the investor is willing to tolerate) 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:

${\underset {i}{\min }}\Pr(R_{i}<{\underline {R}})$ where $\Pr(R_{i}<{\underline {R}})$ is the probability of $R_{i}$ (the actual return of asset i) being less than ${\underline {R}}$ (the minimum acceptable return).

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:

${\text{SFRatio}}_{i}={\frac {{\text{E}}(R_{i})-{\underline {R}}}{\sqrt {{\text{Var}}(R_{i})}}}$

where ${\text{E}}(R_{i})$  is the expected return (the mean return) of the portfolio, ${\sqrt {{\text{Var}}(R_{i})}}$  is the standard deviation of the portfolio's return and ${\underline {R}}$  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%:

SFRatio(A) = [10 − 0]/15 = 0.67,
SFRatio(B) = [8 − 0]/5 = 1.6

By Roy's safety-first criterion, the investor would choose portfolio B as the correct investment opportunity.

Similarity to Sharpe ratio

Under normality,

${\text{SFRatio}}={\frac {\text{ Expected Return - Minimum Return}}{\text{standard deviation of Return}}}.$

The Sharpe ratio is defined as excess return per unit of risk, or in other words:

${\text{Sharpe ratio}}={\frac {\text{ Expected Return - Risk-Free Return}}{\text{standard deviation of Portfolio Return}}}$ .

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.