# Roy's safety-first criterion

Roy's safety-first criterion

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:

: $ext\left\{minimize \right\}P\left(mbox\left\{Ra\right\} < mbox\left\{Rm\right\}\right),,$

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

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

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.

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.

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