How does Roy's safety-first criterion work, and how is it different from Sharpe?

Roy's safety-first criterion is a portfolio selection rule that minimizes the probability of the portfolio return falling below a threshold level (RL) that the investor cannot afford to breach.

The Formula

SFRatio = (E(Rp) - RL) / σp

Choose the portfolio with the highest SFRatio.

Notice this is identical in structure to the Sharpe ratio — the only difference is that Rf is replaced by RL (the investor's minimum acceptable return or "disaster level").

When Does It Differ from Sharpe?

If RL = Rf, the SFRatio equals the Sharpe ratio and you get the same ranking. But when RL differs from Rf, the rankings can diverge.

Worked Example — Brookstone Pension Fund

Brookstone Pension has $200 million in assets and $186 million in liabilities. The board mandates that assets must not fall below liabilities, so the minimum return is:

RL = (186 - 200) / 200 = -7.0%

The risk-free rate is 3.5%. Consider three portfolios:

Sharpe ratios (using Rf = 3.5%):

• Ironwood: (8.5 - 3.5) / 10 = 0.500
• Crestview: (12 - 3.5) / 18 = 0.472
• Harborline: (6 - 3.5) / 5 = 0.500

Sharpe says: Ironwood and Harborline are tied; Crestview is worst.

SFRatios (using RL = -7.0%):

• Ironwood: (8.5 - (-7)) / 10 = 1.550
• Crestview: (12 - (-7)) / 18 = 1.056
• Harborline: (6 - (-7)) / 5 = 2.600

Safety-first says: Harborline is best because it has the lowest probability of breaching the -7% floor.

Interpretation Using Z-Scores

Assuming normality, the SFRatio is the number of standard deviations between the expected return and the threshold. A higher SFRatio means a lower probability of disaster.

For Harborline: P(R < -7%) = P(Z < -2.60) = 0.47% — less than 1 in 200.

Exam tip: Safety-first questions almost always give you a threshold return that differs from Rf. Compute the SFRatio for each portfolio and pick the highest.

Practice more portfolio selection criteria in our CFA Level I question bank.