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

# 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.

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