Roy's Safety-First Criterion: A Comprehensive Guide

Roy’s safety-first criterion is a risk managementRisk ManagementRisk management encompasses the identification, analysis, and response to risk factors that form part of the life of a business. It is usually done with technique used by investors to compare and choose a portfolio based on the criterion that the probability of a portfolio’s return dropping below a threshold level return is reduced.

In Roy’s safety-first criterion, the optimal portfolio is one that minimizes the probability of the portfolio’s return falling below a threshold return level. The portfolio with the highest Roy’s safety-first criterion has the lowest probability of a portfolio generating a return lower than the threshold level return.

Summary:

• Roy’s safety-first criterion is used by investors to choose a portfolio based on the criterion that the probability of the portfolio’s return dropping below a threshold level is reduced.
• The value given by Roy’s safety-first criterion indicates the number of standard deviations below the mean.
• The formula for Roy’s safety-first criterion is [E(R_P) – R_L] / σ_p

Formula for Roy’s Safety-first Criterion

Where:

• E(Rp) is the expected portfolio return;
• R_L is the threshold level return (the minimum acceptable return); and
• σ_p is the standard deviation, or risk, of the portfolio.

Note that the formula for Roy’s safety-first criterion assumes that portfolio returns are normally distributed.

Visual Representation of Roy’s Safety-first Criterion

Where:

• Expected Return is E(Rp); and
• Threshold Level Return is R_L.

The goal of Roy’s safety-first criterion is to minimize the left tail. The area to the left of the threshold level return is the probability of the portfolio generating a return less than the threshold.

The value given by Roy’s safety-first criterion indicates the number of standard deviations below the mean. For example, a value of 1 indicates one standard deviationStandard DeviationFrom a statistics standpoint, the standard deviation of a data set is a measure of the magnitude of deviations between values of the observations contained below the mean. Therefore, the higher the criterion value, the smaller the left tail and the lower the probability of the portfolio generating a return less than the threshold.

Shortfall Risk and Roy’s Safety-first Criterion

Shortfall risk and Roy’s safety-first criterion goes hand-in-hand. Shortfall risk is the probability of generating a return lower than the threshold level return. In other words, shortfall risk is the area to the left of the threshold level return on a normal distribution graph. It is important to note that:

The higher the safety-first criterion, the lower the shortfall risk.

The lower the safety-first criterion, the higher the shortfall risk.

The shortfall risk can be calculated through a z-table for negative values. Below, we will do a comprehensive example.

For example, consider a portfolio with an expected return of 5%, a standard deviation of 10%, and a threshold return level of 0%. What are the Roy’s safety-first criterion and shortfall risk assuming that the portfolio is normally distributed?

The Roy’s safety-first criterion is calculated as (5% – 0%) / 10% = 0.5.

Illustrated above, the expected return is 5%, the threshold return is 0%, and Roy’s safety-first criterion yields 0.5, which is 0.5 standard deviations below the expected return. Shortfall risk is the area under the curve starting from the left of the threshold return. Using a z-table for negative values, -0.5 corresponds to a z-score of 0.3085 or 30.85%.

Example of Roy’s Safety-first Criterion

Consider three portfolios with the return and risk profiles provided below. Assume that the investor wants to minimize the probability of the portfolio returning less than 0%. In other words, the investor’s minimum acceptable return is 0%. Based on Roy’s safety-first criterion, which portfolio should the investor invest in?

The Roy’s safety-first criterion for Portfolio A is calculated as (5% – 0%) / 5% = 1.

The Roy’s safety-first criterion for Portfolio B is calculated as (10% – 0%) / 12% = 0.83.

The Roy’s safety-first criterion for Portfolio C is calculated as (15% – 0%) / 20% = 0.75.

Based on Roy’s safety-first criterion, the ratio with the largest safety-first criterion has the lowest probability of getting a return less than 0%. In our example, it would be Portfolio A.

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