Roy's safety-first criterion



Roy's safety-first criterion is a risk management technique, devised by A. D. Roy, 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\Pr(R_<\underline) where is the probability of (the actual return of asset i) being less than (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_=\frac where \text(R_) is the expected return (the mean return) of the portfolio, \sqrt is the standard deviation of the portfolio's return and is the minimum acceptable return.


Portfolio Portfolio may refer to: Objects * Portfolio (briefcase), a type of briefcase Collections * Portfolio (finance), a collection of assets held by an institution or a private individual * Artist's portfolio, a sample of an artist's work or a ...
A has an expected return of 10% and
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
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 Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
of a return no lower than 0%: : SFRatio(A) = = 0.67, : SFRatio(B) = = 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 =\frac. The
Sharpe ratio In finance, the Sharpe ratio (also known as the Sharpe index, the Sharpe measure, and the reward-to-variability ratio) measures the performance of an investment such as a security or portfolio compared to a risk-free asset, after adjusting for it ...
is defined as excess return per unit of risk, or in other words: : \text =\frac. 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.

Asset Pricing

Roy’s work is the foundation of asset pricing under loss aversion. His work was followed by Lester G. Telser’s proposal of maximizing expected return subject to the constraint that the be less than a certain safety level.Telser, L. G., Safety first and hedging, Review of Economic Studies, Vol. 23, 1955, pp. 1-16

Retrieved June 4, 2021.

See also

Omega ratio The Omega ratio is a risk-return performance measure of an investment asset, portfolio, or strategy. It was devised by Con Keating and William F. Shadwick in 2002 and is defined as the probability weighted ratio of gains versus losses for some thres ...
Value at risk Value at risk (VaR) is a measure of the risk of loss for investments. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. VaR is typically used by ...


{{DEFAULTSORT:Roy's Safety-First Criterion Financial risk management Portfolio theories