Roy's safety-first criterion
Encyclopedia
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.
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, Roy's safety criterion can be summarized symbolically as:
where P(Ra < Rm) is the probability of Ra (the actual return) being less than Rm (the minimum acceptable return).
where SFRatio is defined as [E(Ra) − Rm]/(standard deviation
of portfolio return)
where E(Ra) is the expected
return (the mean return) of the portfolio and Rm is the minimum acceptable return.
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 the investor is willing to invest in a portfolio that maximizes the probability
of a return no lower than 0%:
By Roy's safety-first criterion, the investor would choose portfolio B as the correct investment opportunity.
In contrast, the Sharpe ratio
is defined as excess return per unit of risk, or in other words:
SFRatio has a striking similarity to the Sharpe ratio: for normally distributed returns, Roy's Safety-first criterion with the minimum acceptable return assumed to equal the risk-free rate provides the same conclusion about which portfolio to invest in as does the criterion of maximizing the Sharpe Ratio. However, this partial equivalence no longer holds in the absence of normally distributed returns.
Risk management
Risk management is the identification, assessment, and prioritization of risks followed by coordinated and economical application of resources to minimize, monitor, and control the probability and/or impact of unfortunate events or to maximize the realization of opportunities...
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.
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, Roy's safety criterion can be summarized symbolically as:
- minimize P(Ra< Rm),
where P(Ra < Rm) is the probability of Ra (the actual return) being less than Rm (the minimum acceptable 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]/(standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
of portfolio return)
where E(Ra) is the expected
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
return (the mean return) of the portfolio and Rm is the minimum acceptable return.
Example
Thus if PortfolioPortfolio (finance)
Portfolio is a financial term denoting a collection of investments held by an investment company, hedge fund, financial institution or individual.-Definition:The term portfolio refers to any collection of financial assets such as stocks, bonds and cash...
A has a mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
return of 10% and standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
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
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
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,- SFRatio = (expected return − minimum return) / (standard deviation of return).
In contrast, the Sharpe ratio
Sharpe ratio
The Sharpe ratio or Sharpe index or Sharpe measure or reward-to-variability ratio is a measure of the excess return per unit of deviation in an investment asset or a trading strategy, typically referred to as risk , named after William Forsyth Sharpe...
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 the Sharpe ratio: for normally distributed returns, Roy's Safety-first criterion with the minimum acceptable return assumed to equal the risk-free rate provides the same conclusion about which portfolio to invest in as does the criterion of maximizing the Sharpe Ratio. However, this partial equivalence no longer holds in the absence of normally distributed returns.