Shortfall risk and Roy's safety-first ratio

What is shortfall risk?

Shortfall risk is the risk that a portfolio’s value or return falls below some minimum acceptable level over a given time horizon. A pension fund whose assets might drop below its liabilities faces shortfall risk. So does a charity that must pay out a fixed sum each year.

Standard deviation does not capture this well. It treats a surprise above the mean exactly like a surprise below it. An investor with a hard floor cares only about the left side, so the question becomes: which portfolio is least likely to finish under the floor?

How do you set the threshold return?

The floor is the threshold return , written R_L. When the floor comes from a money need, divide the money you must earn by the portfolio’s value today.

$$R_L = \frac{\text{money that must be earned}}{\text{portfolio value}}$$

How does Roy’s criterion choose a portfolio?

Roy’s safety-first criterion says to choose the portfolio that minimizes the probability that its return falls below R_L. If returns are normally distributed, that portfolio is the one with the highest safety-first ratio : the distance from the expected return down to the threshold, measured in standard deviations.

$$\mathrm{SFRatio} = \frac{E(R_p) - R_L}{\sigma_p}$$

The charity is choosing among three portfolios. Under a normal distribution, the probability of finishing below 4.2% is N(−SFRatio).

Portfolio	E(R)	σ	SFRatio	P(R below 4.2%)
Cautious	5.5%	7%	0.186	42.6%
Balanced	8.5%	12%	0.358	36.0%
Growth	11.0%	20%	0.340	36.7%

Roy's safety-first criterionminimize P(return below R_L)

• 1. Set the threshold R_L Use it payout ÷ portfolio value, or the minimum return you can accept
• 2. Compute each SFRatio Memorize (E − R_L) ÷ σ, one per portfolio
• 3. Choose the highest ratio Memorize with normal returns it has the lowest shortfall probability
• 4. Shortfall probability Memorize N(−SFRatio), read from the z-table

Exam tip Percent units cancel in the ratio, so type 8.5 and 12 as they are. Only the final probability needs a table.

How do you find the shortfall probability?

A ratio of 0.358 means the threshold lies 0.358 standard deviations below the expected return. For a normal distribution, the probability of landing below a point z standard deviations under the mean is N(−z).

$$P(R_p < R_L) = N(-\mathrm{SFRatio})$$

The exam supplies a table of N(z) for positive z, and N(−z) = 1 − N(z).

z	0.34	0.35	0.36	0.37	0.38
N(z)	0.6331	0.6368	0.6406	0.6443	0.6480

Because N(−z) falls as z rises, the portfolio with the highest ratio always has the smallest left tail. That is why ranking by ratio and ranking by shortfall probability give the same order, as the table shows: 0.358 beats 0.340 beats 0.186.

How is it different from the Sharpe ratio?

The formula is the same, but the thing you subtract differs.

	Safety-first ratio	Sharpe ratio
Subtracts	the investor’s threshold R_L	the risk-free rate
Answers	which portfolio is least likely to miss the floor	how much return per unit of risk above a risk-free investment
Same number when	R_L is set equal to the risk-free rate	(always)

The choice can differ. With a risk-free rate of 1%, the Sharpe ratios of the charity’s portfolios are 0.643 (Cautious), 0.625 (Balanced) and 0.500 (Growth), so Sharpe prefers Cautious. Roy’s criterion, using the charity’s 4.2% floor, prefers Balanced. When a question gives a threshold, the threshold goes in the ratio.

What are the other measures of downside risk?

Portfolio risk measuresstandard deviation treats gains and losses alike; these look at the bad side

• Standard deviation Memorize spread both above and below the mean
• Shortfall risk Memorize chance of finishing below a threshold return R_L
  • Safety-first ratio standard deviations between the mean and R_L
• Value at risk Know the idea money loss exceeded only with a small stated probability
• Stress testing Know the idea loss under a specified extreme event

Two other measures express bad outcomes in money and are developed in risk management. Value at risk states a loss that will be exceeded only with a small stated probability over a stated period, for example a 5% one-day value at risk of EUR 2 million. Stress testing asks what the portfolio would lose under a specified extreme event, such as a sharp rate rise or a market crash. Both complement standard deviation instead of replacing it.

Go deeper Why the result depends on normality

The link between the ratio and the probability, N(−SFRatio), holds only if returns are normally distributed. Real returns often have fat tails and negative skewness, which puts more probability in the left tail than a normal curve implies. The ranking by ratio is then less reliable, and the true shortfall probability is higher than the table value. This is why the lesson on skewness and kurtosis matters for risk measurement.

Go deeper Turning a need into a threshold

A fixed payout gives a simple threshold, as in the charity example. In practice the need may grow with inflation or follow a spending rule. If the charity’s payout is expected to rise 2% to EUR 64,260, its threshold rises to 64,260 ÷ 1,500,000 = 4.284%. Portfolio construction uses thresholds like this when it turns objectives into constraints.