# 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.[1]

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:

${\displaystyle {\underset {i}{\min }}\Pr(R_{i}<{\underline {R}})}$

where ${\displaystyle \Pr(R_{i}<{\underline {R}})}$ is the probability of ${\displaystyle R_{i}}$ (the actual return of asset i) being less than ${\displaystyle {\underline {R}}}$ (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:

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

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

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

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

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

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.