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{{Refimprove|date=December 2006}}
'''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.<ref>{{cite journal|last=Roy|first=Arthur D.|year=1952|title=Safety First and the Holding of Assets|journal=Econometrica|volume=1952|issue=July|pages=431–450}}</ref>


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 &minus;1%.


Thus, the problem of an investor using Roy's safety criterion can be summarized symbolically as:
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::<math>\underset{i}{\min}\Pr(R_{i}<\underline{R})</math>
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where <math>\Pr(R_{i}<\underline{R})</math> is the probability of <math>R_{i}</math> (the actual return of asset i) being less than <math>\underline{R}</math> (the minimum acceptable return).
 
==Normally distributed return and SFRatio==
If the portfolios under consideration have [[normal distribution|normally distributed]] returns, Roy's safety-first criterion can be reduced to the maximization of the safety-first ratio, defined by:  
 
::<math>SFRatio_{i}=\frac{E(R_{i})-\underline{R}}{\sqrt{Var(R_{i})}}</math>
 
where <math>E(R_{i})</math> is the [[expected value|expected]] return (the mean return) of the portfolio, <math>\sqrt{Var(R_{i})}</math> is the standard deviation of the portfolio's return and <math>\underline{R}</math> is the minimum acceptable return.
 
===Example===
If [[Portfolio (finance)|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&nbsp;&minus;&nbsp;0]/15 = 0.67,
: SFRatio(B) = [8&nbsp;&minus;&nbsp;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,  
 
: SFRatio = (expected return &minus; 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 &minus; Risk-Free Return]/(standard deviation of [Expected return &minus; Risk-Free Return]).
 
SFRatio has a striking similarity to Sharpe ratio. Thus for normally distributed returns, Roy's Safety-first criterion - with the minimum 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 ==
* [[Value at Risk]]
 
==References==
{{Reflist}}
 
{{DEFAULTSORT:Roy's Safety-First Criterion}}
[[Category:Financial risk]]
[[Category:Portfolio theories]]

Latest revision as of 19:36, 30 November 2014


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