Shortfall risk, safety-first ratio, and Roy's safety-first criterion

CFA level I / Quantitative Methods: Application / Common Probability Distributions / Shortfall risk, safety-first ratio, and Roy's safety-first criterion

The shortfall risk is defined as the risk that portfolio value will fall below some minimum acceptable level over some time horizon. This risk can be easily calculated if the distribution of the return of the portfolio is assumed to be normal.

The safety first ratio is calculated by dividing the difference between the expected return on the portfolio and the minimum acceptable threshold return by the standard deviation of the portfolio.

SFRatio = [E(R_P) - R_L)]/σ_P

Please note that while calculating the z value we subtract the mean return from the given return, but while calculating SFRatio, we are subtracting the threshold return from the mean return. So, to calculate the shortfall risk, we need to find N(-SFRatio) rather than N(SFRatio) as we used to do for z values.

The safety-first ratio equals excess return over the threshold return per unit of risk. The Roy's safety-first criterion is that out of given portfolios, choose the portfolio with the highest safety-first ratio.

Example 7: Using Roy's safety-first criterion and calculating shortfall risk

A portfolio manager has to choose one of the three portfolios for a client. The value of the client's portfolio is $100 million, and the client wants to maximize the probability of his portfolio achieving a value of $104 at the end of the year. The details about the three portfolios are given below in the table:

	Portfolio A	Portfolio B	Portfolio C
Expected annual return	18.00%	14.00%	16.00%
Standard deviation of return	25.00%	12.00%	20.00%

(a) Which portfolio is best suited to the client as per his requirement?
(b) What is the shortfall risk for the portfolio selected in the part (a)?

Solution:

(a) The client wants to maximize the probability of his portfolio having a value of $104 million at the end of the year. It is equivalent to the minimizing the probability that the portfolio value falls below $104 million at the end of the year.

Threshold return required by the client = (104/100) - 1 = 0.04.

SFRatio for Portfolio A= (18-4)/25 = 0.56
SFRatio for Portfolio B= (14-4)/12 = 0.83
SFRatio for Portfolio C= (16-4)/20 = 0.60

Portfolio B has the maximum safety-first ratio. So, it should be chosen by the portfolio manager according to Roy's safety-first criterion.

(b) Shortfall risk for Portfolio B = N(-SFRatio) = N(-0.83) = 0.2024 (from the z table)

The portfolio B has 20.24 percent probability of achieving a return less than the threshold return of 4.00 percent.

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