Safety–First Rules, Shortfall Risk, and VaR
An investor has a CAD 750,000 growth portfolio. After a year, the investor plans to liquidate CAD 30,000 without invading the initial capital of CAD 750,000. Three alternative allocations with normally distributed returns are shown below.
| Measures | A | B | C |
|----|----|----|----|
| Expected annual return | 22% | 11% | 16% |
| Standard deviation of return | 23% | 10% | 16% |
The probability that the return on the safety-first optimal portfolio is less than the shortfall level is _closest_ to:
Correct!
Allocation A has the highest safety-first ratio. The CAD 30,000 liquidation is 4% of the portfolio, so 4% is used as the minimum return:
$$\displaystyle \text{SFRatio} = \frac{E(R_P)-R_L}{\sigma_P} = \frac{22 \% - 4 \%}{23 \%} \approx 0.78$$
The probability that its return will be less than _RL_ is then:
$$\displaystyle N(-\text{SFRatio}) = N(-0.78) $$
Use a standard normal table.
$$\displaystyle P(R_A) < 4 \% = N(-0.78) $$
$$\displaystyle = 1 - N(0.78) = 1 - 0.7823 = 0.2177$$
The safety-first optimal portfolio has roughly a 22% chance of not meeting a 4.00 percent return threshold.
Incorrect.
This is the probability for a portfolio that is not the optimal safety-first allocation.
$$\displaystyle \text{SFRatio} = \frac{E(R_P)-R_L}{\sigma_P} = \frac{11-4}{10} = 0.700$$
This is the probability that its return will be less than _RL_.
$$\displaystyle = N(-\text{SFRatio}) = -0.700$$
Use a standard normal table.
$$\displaystyle P(R_A) < 4.00 = N(-0.70) $$
$$\displaystyle = 1 - N(0.70) = 1 - 0.75804 = 0.24196$$
This portfolio has roughly a 24% chance of not meeting a 4.00 percent return threshold.
Incorrect.
This is the safety-first ratio used in calculating the most likely probability the return on the safety-first optimal portfolio is less than the shortfall level.
0.22.
0.24.
0.78.
