# 9. Shortfall Risk and Roy's Safety-First Criterion

## n. define shortfall risk, calculate the safety-first ratio, and select an optimal portfolio using Roy's safety-first criterion;

What is shortfall risk?

Shortfall risk is the risk that portfolio value will fall below some minimum acceptable level over some time horizon. The risk that assets in a defined benefit plan will fall below plan liabilities is an example of a shortfall risk. Therefore, shortfall risk is a downside risk.

What kind of risk do safety-first rules focus on?

Shortfall risk

What is Roy's safety-first criterion?

Roy's safety-first criterion states that the optimal portfolio minimizes the probability that portfolio return, `R_P`

(return for the portfolio), falls below the threshold level, `R_L`

(return loss). In symbols, the investor's objective is to choose a portfolio that minimizes `P(R_P < R_L)`

.

According to Roy's safety-first criterion, how can an investor calculate the probability of a portfolio falling below the acceptable rate of loss?

When portfolio returns are normally distributed, the investor can calculate `P(R_P < R_L)`

using the number of standard deviations `R_L`

lies below the expected portfolio return, `E(R_P)`

. The portfolio for which `E(R_P) - R_L`

is largest in terms of units of standard deviations minimizes `P(R_P < R_L)`

. Thus, if returns are normally distributed, the safety-first optimal portfolio maximizes the safety-first ratio (SFRatio)

What is the safety-first ratio formula?

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

- `E(R_P) - R_L`

= the distance from the mean return to the shortfall level.

- `\sigma_P`

= standard deviation of the portfolio's rate of return

How are the safety-first criterion and the Sharpe ratio related? How are they different?

- The only difference between the safety-first and Sharpe ratio is the `R_L`

(acceptable rate of loss) being swapped out for `R_f`

(risk free rate).

- Safety-first $$SFRatio = \frac{E(R_p) - R_L}{\sigma_P}$$

- Sharpe $$SFRatio = \frac{E(R_p) - R_f}{\sigma_P}$$

- The safety-first criterion focuses on the excess return on the threshold return

- The Sharpe ratio focuses on excess return over the risk-free rate

Assuming that returns are normally distributed, the safety-first optimal portfolio can be selected using one of which two criteria?

1. Lowest probability of `R_P < R_L`

2. Highest safety-first ratio.

##### Source:

- CFA