CFA Level I Exam · Statistical Distributions for Financial Asset Prices and Returns

Roy's Safety-First Ratio and Shortfall Risk Explained

Q: How do you calculate Roy's safety-first ratio?

Subtract the threshold return from the portfolio's expected return, then divide by the portfolio's standard deviation. SFRatio = (E(Rp) − RL) ÷ σp. Do this for each portfolio and choose the highest.

Q: What is the difference between the safety-first ratio and the Sharpe ratio?

Both divide excess return by standard deviation. The Sharpe ratio subtracts the risk-free rate, while the safety-first ratio subtracts the investor's threshold return. If the threshold equals the risk-free rate, they give the same number.

Q: Does a higher safety-first ratio always mean lower shortfall risk?

For normally distributed returns, yes. The shortfall probability is N(−SFRatio), which falls as the ratio rises. For non-normal returns the ranking is not guaranteed to match the true probabilities.

Updated 7 October 2026 · Fact-checked

Shortfall risk is the probability that a portfolio's return falls below a minimum threshold. Roy's safety-first ratio is SFRatio = (E(Rp) − RL) ÷ σp. Calculate it for each portfolio and choose the highest ratio. If returns are normal, the highest ratio gives the lowest shortfall probability.

Understand Safety-First Ratio and Shortfall Risk

Shortfall risk is the chance that a portfolio's return ends up below a target level. That target is called the threshold return (RL). It might be 0% to protect capital, or 4% to meet a spending need. Standard deviation treats gains and losses alike. Shortfall risk looks only at the bad outcome you care about.

Roy's safety-first criterion says: choose the portfolio that minimises the probability that its return falls below RL. You cannot see that probability directly, so you use a ratio that ranks portfolios.

The safety-first ratio (SFRatio) measures how many standard deviations the expected return sits above the threshold. A bigger number means the threshold is further away, so a miss is less likely. If portfolio returns are normally distributed, maximising the SFRatio is exactly the same as minimising the shortfall probability.

To find the probability, treat the SFRatio as a z-score. P(R < RL) = N(−SFRatio), where N( ) is the standard normal cumulative probability. For example, an SFRatio of 1.00 means a shortfall probability of about 15.87%.

The SFRatio looks like the Sharpe ratio. The only difference is what you subtract. Sharpe subtracts the risk-free rate. Safety-first subtracts your minimum acceptable return. If RL equals the risk-free rate, the two ratios are identical.

Key formulas to remember

Roy's safety-first ratio: SFRatio = (E(Rp) − RL) ÷ σp; E(Rp) is expected portfolio return, RL is the threshold return, σp is the portfolio standard deviation. Pick the portfolio with the highest ratio.
Shortfall probability: P(Rp < RL) = N(−SFRatio); Valid when returns are normally distributed. Higher SFRatio means lower shortfall probability.
Sharpe ratio (for comparison): Sharpe = (E(Rp) − Rf) ÷ σp; Same form as SFRatio but uses the risk-free rate Rf in place of RL.

How to solve Safety-First Ratio and Shortfall Risk questions

Use this method for any safety-first question, whether it asks for the ratio, the best portfolio or the shortfall probability.

1Identify the threshold return RL from the question. It may be given as a percentage or as a currency loss that you convert to a return.
2List E(R) and σ for each portfolio. Keep them in the same units, for example all in percent.
3Compute the numerator E(R) − RL for each portfolio.
4Divide each numerator by that portfolio's standard deviation to get the SFRatio.
5Choose the portfolio with the highest SFRatio. It has the lowest shortfall probability if returns are normal.
6If asked for the probability, treat the SFRatio as a z-score and read N(−SFRatio) from the normal table. Use the lower tail.

Quickest way: Compare ratios without the table

When to use it: Use this when the question only asks which portfolio is best or which ratio is highest. You do not need any normal table.

Subtract RL from each expected return.
Divide by each standard deviation, using the calculator.
Pick the biggest result and ignore the rest.
If two ratios are close, recheck the arithmetic before choosing.
On the TI BA II Plus: type E(R), press −, type RL, press ÷, type σ, press =. Repeat for each portfolio.

Common mistakes in Safety-First Ratio and Shortfall Risk

Choosing the portfolio with the highest expected return.
Candidates forget that the criterion is about the chance of missing RL, not about return alone.
Fix: Always compute the SFRatio for every option. A higher return with much higher σ can have a lower ratio.
Subtracting the risk-free rate instead of the threshold return.
The formula looks like the Sharpe ratio, so candidates use Rf out of habit.
Fix: Read the stem for the minimum acceptable or threshold return. Use that as RL. Use Rf only if the question says the threshold is the risk-free rate.
Choosing the lowest SFRatio.
Candidates mix up the ratio with the shortfall probability, where lower is better.
Fix: Remember: highest ratio, lowest shortfall probability.
Reading the shortfall probability as N(SFRatio) instead of N(−SFRatio).
The upper tail and lower tail get confused.
Fix: Shortfall is the lower tail. For a positive SFRatio, the probability is below 50%. Sense-check the answer.
Forgetting to convert a currency threshold into a return.
Questions may say the investor wants to avoid losing more than a given amount on a given portfolio value.
Fix: Compute RL = (minimum amount − portfolio value) ÷ portfolio value, or (minimum amount ÷ portfolio value) − 1. Example: a minimum of 95,000 on a portfolio of 100,000 gives RL = (95,000 − 100,000) ÷ 100,000 = −5%.
Applying the probability result to non-normal returns.
The link between ratio and probability is learned as a rule without its condition.
Fix: State the assumption of normality. The ranking by SFRatio is used regardless, but N(−SFRatio) is exact only for normal returns.

Worked examples

Example 1

An investor requires a minimum return of 2%. Portfolio X has an expected return of 8% and a standard deviation of 12%. Portfolio Y has an expected return of 6% and a standard deviation of 6%. Portfolio Z has an expected return of 10% and a standard deviation of 20%. Which portfolio is preferred under Roy's safety-first criterion? A. Portfolio X, B. Portfolio Y, C. Portfolio Z

Show the solution

RL = 2%.
SFRatio X = (8 − 2) ÷ 12 = 0.50.
SFRatio Y = (6 − 2) ÷ 6 = 0.667.
SFRatio Z = (10 − 2) ÷ 20 = 0.40.
The highest ratio is Y at 0.667.

Answer: B. Portfolio Y has the highest SFRatio, so the lowest shortfall probability. Z has the highest return but the lowest ratio.

Example 2

A fund has an expected return of 9% and a standard deviation of 10%. The client's threshold return is −1%. Assuming returns are normal, the probability of a return below the threshold is closest to: A. 15.9%, B. 30.9%, C. 84.1%. (Use N(−1.00) = 0.1587, N(−0.50) = 0.3085 and N(1.00) = 0.8413.)

Show the solution

SFRatio = (9 − (−1)) ÷ 10 = 10 ÷ 10 = 1.00.
Shortfall probability = N(−1.00).
N(−1.00) = 0.1587, or 15.9%.

Answer: A. About 15.9%. Option B, 30.9%, is N(−0.50), which is the answer you would get if you wrongly used an SFRatio of 0.50, for example by dividing by 20 instead of 10. Option C, 84.1%, is N(1.00), which is the upper tail. It comes from reading N(SFRatio) instead of N(−SFRatio).

Exam tips

Expect a three-portfolio comparison. Do the arithmetic for all three quickly and pick the highest ratio. With 90 seconds per question, skip the normal table unless it is asked for.
Read the threshold carefully. It may be hidden as a minimum acceptable return, a loss limit or a required return to meet a liability.
Know that SFRatio equals the Sharpe ratio when RL = Rf. Questions test this link in conceptual form.
Sense-check any probability. A positive SFRatio means a shortfall probability below 50%. A negative SFRatio means above 50%.

Practice questions from Statistical Distributions for Financial Asset Prices and Returns

Safety-First Ratio and Shortfall Risk: frequently asked questions

What is shortfall risk in CFA Level I?

Shortfall risk is the probability that a portfolio's return falls below a minimum threshold return. It focuses only on the downside you care about, unlike standard deviation, which counts gains and losses alike.

How do you calculate Roy's safety-first ratio?

Subtract the threshold return from the portfolio's expected return, then divide by the portfolio's standard deviation. SFRatio = (E(Rp) − RL) ÷ σp. Do this for each portfolio and choose the highest.

What is the difference between the safety-first ratio and the Sharpe ratio?

Both divide excess return by standard deviation. The Sharpe ratio subtracts the risk-free rate, while the safety-first ratio subtracts the investor's threshold return. If the threshold equals the risk-free rate, they give the same number.

Does a higher safety-first ratio always mean lower shortfall risk?

For normally distributed returns, yes. The shortfall probability is N(−SFRatio), which falls as the ratio rises. For non-normal returns the ranking is not guaranteed to match the true probabilities.