Updated for the 2026-2027 CFA® Level I curriculum.
Some investors care about more than earning the highest possible return. They may need the portfolio to meet a spending requirement, keep pace with a liability, or earn at least a stated minimum return.
Shortfall risk measures the probability that the portfolio will fall below that threshold. Roy’s safety-first criterion gives you a practical way to compare portfolios based on their expected returns, volatility, and the investor’s minimum acceptable return.
Quick Answer
Roy’s safety-first ratio measures how far a portfolio’s expected return sits above the minimum acceptable return in standard-deviation terms. A higher ratio indicates a lower shortfall probability when returns are normally distributed. Under Roy’s safety-first criterion, choose the portfolio with the highest ratio.
Key Takeaways About the Safety-First Ratio
Shortfall risk is the probability that a portfolio return falls below a required threshold.
The threshold is called the minimum acceptable return, often written as .
Roy’s safety-first ratio compares the return above the threshold with the portfolio’s volatility.
A higher expected return increases the ratio, all else equal.
A higher standard deviation reduces the ratio, all else equal.
The portfolio with the highest safety-first ratio is preferred.
Under normally distributed returns, a higher ratio corresponds to a lower probability of shortfall.
The portfolio with the highest expected return may have a lower safety-first ratio when its volatility is also higher.
What You Need to Know for CFA Level I
When working with shortfall risk and Roy’s safety-first criterion, you should be able to:
Identify the investor’s minimum acceptable return.
Define shortfall risk as a probability.
Calculate Roy’s safety-first ratio.
Rank portfolios using their safety-first ratios.
Select the portfolio with the highest ratio.
Connect the safety-first ratio with a normal-distribution z-score.
Explain how expected return and standard deviation affect the result.
Recognize when the normal-distribution assumption may limit the interpretation.
What Is Shortfall Risk?
Shortfall risk is the probability that a portfolio earns less than the investor’s minimum acceptable return.
The shortfall probability is written as:
Where:
= probability that the portfolio return falls below the threshold
= portfolio return
= minimum acceptable return or threshold return
The minimum acceptable return depends on the investor’s objective. It may represent:
A required spending rate
A liability growth rate
A funding requirement
A capital-preservation target
A minimum acceptable investment result
A portfolio can have a relatively high expected return and still carry meaningful shortfall risk when its return volatility is also high.
What Is Roy’s Safety-First Criterion?
Roy’s safety-first criterion selects the portfolio that minimizes the probability of earning less than the investor’s minimum acceptable return.
The criterion compares each portfolio’s expected return with the threshold and then adjusts that difference for the portfolio’s standard deviation. This produces the safety-first ratio.
The decision rule is straightforward:
Select the portfolio with the highest safety-first ratio
A larger ratio places the minimum acceptable return farther below the portfolio’s expected return in standard-deviation terms.
Roy’s Safety-First Ratio Formula
Roy’s safety-first ratio is:
Formula Breakdown
Where:
= Roy’s safety-first ratio
= expected return of the portfolio
= investor’s minimum acceptable return
= expected return above the minimum acceptable return
= standard deviation of the portfolio’s returns
The numerator measures the portfolio’s expected return cushion above the threshold. The denominator adjusts that cushion for the portfolio’s volatility.
A portfolio with a larger return cushion will generally have a higher ratio. Greater volatility lowers the ratio because the portfolio has more scope to fall below the investor’s threshold.
How Should You Interpret the Safety-First Ratio?
The safety-first ratio expresses the distance between the expected return and the minimum acceptable return in standard-deviation units.
For example, a ratio of 0.50 means the minimum acceptable return lies 0.50 standard deviations below the portfolio’s expected return.
A ratio of 1.00 means the threshold lies one full standard deviation below the expected return. Under a normal distribution, the second portfolio has a lower probability of falling below its threshold.
Safety-First Ratio | General Interpretation |
|---|---|
Higher positive value | Lower shortfall probability under normality |
Smaller positive value | Threshold is closer to the expected return |
Zero | Expected return equals the minimum acceptable return |
Negative value | Expected return is below the minimum acceptable return |
When all candidate portfolios use the same minimum acceptable return, rank them from highest to lowest safety-first ratio.
How Does the Safety-First Ratio Relate to a Z-Score?
When portfolio returns are normally distributed, you can standardize the minimum acceptable return using a z-score:
The numerator appears in the opposite order from the safety-first ratio. Therefore:
The shortfall probability can then be written as:
or:
Where:
= standardized value of the minimum acceptable return
= cumulative probability below the z-score in a standard normal distribution
= threshold z-score
A higher safety-first ratio produces a more negative threshold z-score. Less probability lies below that threshold, so the portfolio has a lower shortfall probability.
You can usually rank portfolios without calculating the exact probability. The portfolio with the highest safety-first ratio will also have the lowest shortfall probability under the normal-return assumption.
Worked Safety-First Example
An investor requires a minimum acceptable return of 4% and is considering two portfolios.
Portfolio | Expected Return | Standard Deviation |
|---|---|---|
A | 10% | 12% |
B | 8% | 6% |
Step 1: Calculate Portfolio A’s Safety-First Ratio
Portfolio A’s safety-first ratio is 0.50.
Step 2: Calculate Portfolio B’s Safety-First Ratio
Portfolio B’s safety-first ratio is approximately 0.67.
Step 3: Select the Portfolio
Portfolio B has the higher safety-first ratio:
Portfolio B is therefore preferred under Roy’s safety-first criterion.
Portfolio A offers a higher expected return of 10%, but it also has twice the standard deviation of Portfolio B. Portfolio B provides the stronger return cushion relative to its volatility and the investor’s 4% threshold.
Calculating the Approximate Shortfall Probabilities
The safety-first ratios can also be converted into threshold z-scores.
Portfolio A
The probability below a z-score of -0.50 is approximately 30.9%.
Portfolio B
The probability below a z-score of approximately -0.67 is about 25.2%.
The probability results confirm the safety-first ranking. Portfolio B has the lower probability of earning less than 4%.
When Is Roy’s Safety-First Criterion Useful?
Roy’s criterion is especially useful when the investor’s main concern is avoiding a return below a specific threshold.
It works well when:
The investor has a clearly defined minimum acceptable return.
The expected return and standard deviation are available for each portfolio.
Returns can reasonably be modeled using a normal distribution.
The investor needs a consistent rule for ranking several portfolios.
For example, a pension fund may want to avoid earning less than the rate at which its liabilities grow. An individual investor may need a portfolio to earn enough to support planned withdrawals.
In both cases, the threshold has a direct financial meaning.
Limitations of Roy’s Safety-First Criterion
The safety-first ratio focuses on expected return, standard deviation, and one minimum acceptable return. Other features of the portfolio may still affect the investment decision.
Consider the following limitations:
The normal-distribution interpretation may be less reliable for skewed or heavy-tailed returns.
Standard deviation treats upside and downside variation in the same way.
Expected returns, standard deviations, and correlations are estimates that may change.
The criterion does not incorporate liquidity, taxes, time horizon, or other investor constraints.
Two portfolios with the same ratio may have different return distributions and different extreme-loss exposure.
The selected portfolio may still have a meaningful probability of falling below the threshold.
Roy’s criterion provides a focused portfolio-ranking rule. A complete investment recommendation also considers the investor’s broader objectives and constraints.
Common Exam Traps
Common mistakes include:
Reversing the numerator and calculating .
Selecting the portfolio with the lowest safety-first ratio.
Choosing the portfolio with the highest expected return without adjusting for volatility.
Using variance instead of standard deviation in the denominator.
Mixing percentages and decimals in the same calculation.
Applying different threshold returns when the investor has one shared minimum acceptable return.
Treating the ratio itself as the exact shortfall probability.
Forgetting that the probability interpretation relies on normally distributed returns.
Assuming a positive safety-first ratio means shortfall cannot occur.
Using the z-score decision rule without recognizing that it has the opposite sign from the safety-first ratio.
Practice Question
An investor requires a minimum return of 3% and is considering the following portfolios:
Portfolio | Expected Return | Standard Deviation |
|---|---|---|
A | 9% | 12% |
B | 7% | 6% |
C | 10% | 15% |
Which portfolio is preferred under Roy’s safety-first criterion?
Portfolio A
Portfolio B
Portfolio C
Correct Answer: B
Calculate the safety-first ratio for each portfolio.
Portfolio A
Portfolio B
Portfolio C
Portfolio B has the highest safety-first ratio and is preferred. Under the normal-return assumption, it also has the lowest probability of earning less than 3%.
Option A has a higher expected return than Portfolio B, but its greater volatility reduces its safety-first ratio.
Option C has the highest expected return, but it also has the highest standard deviation and the lowest ratio among the three portfolios.
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FAQs About Roy’s Safety-First Ratio
What Is Roy’s Safety-First Ratio?
Roy’s safety-first ratio measures how far a portfolio’s expected return is above the investor’s minimum acceptable return after adjusting for portfolio volatility.
A higher ratio indicates a lower probability of falling below the threshold when portfolio returns are normally distributed.
Which Portfolio Should You Choose Using Roy’s Safety-First Criterion?
Choose the portfolio with the highest safety-first ratio.
The highest ratio provides the largest expected-return cushion above the minimum acceptable return for each unit of portfolio standard deviation.
Is a Higher or Lower Safety-First Ratio Better?
A higher safety-first ratio is better under Roy’s criterion. It indicates that the investor’s minimum acceptable return is farther below the portfolio’s expected return in standard-deviation terms.
What Is the Difference Between Shortfall Risk and the Safety-First Ratio?
Shortfall risk is the probability that a portfolio return falls below the investor’s minimum acceptable return.
The safety-first ratio is a ranking measure that compares the return cushion above that threshold with portfolio volatility. Under normally distributed returns, maximizing the ratio minimizes shortfall probability.
Can a Portfolio With a Lower Expected Return Have a Better Safety-First Ratio?
Yes. A portfolio with a lower expected return may still have a higher safety-first ratio when its standard deviation is sufficiently lower.
Roy’s criterion evaluates the expected return above the threshold relative to volatility, so expected return and risk must be considered together.