Roy's safety-first criterion is a
risk management
Risk management is the identification, evaluation, and prioritization of risks, followed by the minimization, monitoring, and control of the impact or probability of those risks occurring. Risks can come from various sources (i.e, Threat (sec ...
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:
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
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is
f(x ...
returns, Roy's safety-first criterion can be reduced to the maximization of the safety-first ratio, defined by:
where
is the
expected return (the mean return) of the portfolio,
is the standard deviation of the portfolio's return and is the minimum acceptable return.
Example
If
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 of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
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 a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
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,
:
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 ...
is defined as excess return per unit of risk, or in other words:
:
.
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
In cognitive science and behavioral economics, loss aversion refers to a cognitive bias in which the same situation is perceived as worse if it is framed as a loss, rather than a gain. It should not be confused with risk aversion, which descri ...
. 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
Chance-constrained portfolio selection Chance-constrained portfolio selection is an approach to portfolio selection under loss aversion.
The formulation assumes that (i) investor's preferences are representable by the expected utility of final wealth, and that (ii) they require that th ...
.
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 of investment/capital. 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 us ...
References
{{DEFAULTSORT:Roy's Safety-First Criterion
Financial risk management
Portfolio theories