Roy's safety-first criterion

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Roy's safety-first criterion is a risk management technique that allows you to select one portfolio over another based on the criteria that the probability of the return of the portfolios falling below a minimum desired threshold is minimized.

In other words, say you have two available investment strategies - portfolio A and portfolio B. Your threshold level return (the minimum return that you are willing to tolerate) is -1%. You would want to pick up the portfolio that would provide you the maximum probability of the net return being higher than (or equal to) −1%.

Thus, Roy's safety criterion can be summarized symbolically as:

where P(Ra < Rm) = probabitily of Ra (The actual return) being less than Rm (The minimum desired return).

[edit] Normally distributed return

If the portfolios under consideration have normally distributed (see normal distribution) returns, Roy's safety-first criterion can be reduced to:

maximize the SFRatio (safety-first ratio).

where SFRatio is defined as [E(Ra) − Rm]/(StdDev of portfolio return) where E(Ra) = expected return of the portfolio (or the mean of the return), Rm = Minimum desired return

[edit] Example

Thus if Portfolio A has a mean return of 10% and standard deviation of 15%, while portfolio B has a mean return of 8% and a standard deviation of 5%, and we are willing to invest in a portfolio that minimizes the probability of a 0% return;

SFRatio(A) = [10 − 0]/15 = 0.67,

SFRatio(B) = [8 − 0]/5 = 1.6

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

[edit] Similarity to excess return

SFRatio = (expected return − minimum return)/(standard deviation of return).

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

Sharpe ratio = [Expected return − Risk-Free Return]/(standard deviation of return).

SFRatio has a striking similarity to Sharpe ratio. Thus for Normally distributed returns, Roy's Safety-first criterion provides the same conclusions (about which portfolio to invest in) as if we were picking the one with the maximum sharpe ratio.