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:

\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 (Return - Risk-Free 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.

See also

References

  1. ↑ Roy, Arthur D. (1952). "Safety First and the Holding of Assets". Econometrica 1952 (July): 431–450.
This article is issued from Wikipedia - version of the Saturday, May 23, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.