Question 13.1: Look again at Tables 13.1 and 13.2. Suppose you think a boom...
Look again at Tables 13.1 and 13.2. Suppose you think a boom will occur only 20 percent of the time instead of 50 percent. What are the expected returns on Stocks U and L in this case?
If the risk-free rate is 10 percent, what are the risk premiums?
| TABLE 13.1 | States of the Economy and Stock Returns | ||
| State of Economy |
Probability of State of Economy |
Rate of Return If State Occurs | |
| Stock L | Stock U | ||
| Recession | .50 | −20% | 30% |
| Boom | \underline{ .50} | 70 | 10 |
| 1.00 | |||
| TABLE 13.2 | Calculation of Expected Return | ||||
| Stock L | Stock U | ||||
|
(1) State of Economy |
(2) Probability of State of Economy | (3) Rate of Return If State Occurs |
(4) Product (2) × (3) |
(5) Rate of Return If State Occurs |
(6) Product (2) × (5) |
| Recession | .50 | −.20 | −.10 | .30 | .15 |
| Boom | \underline{ .50} | .70 | \underline{.35} | .10 | \underline{ .05} |
| 1.00 | E(R_{L}) = .25, or 25% | E(R_{U}) = .20, or 20% | |||
The first thing to notice is that a recession must occur 80 percent of the time (1 − .20 = .80) because there are only two possibilities. With this in mind, we see that Stock U has a 30 percent return in 80 percent of the years and a 10 percent return in 20 percent of the years. To calculate the expected return, we again just multiply the possibilities by the probabilities and add up the results:
E(R_{U}) = .80 × 30% + .20 × 10% = 26%
Table 13.3 summarizes the calculations for both stocks. Notice that the expected return on L is −2 percent.
| TABLE 13.3 | Calculation of Expected Return | ||||
| Stock L | Stock U | ||||
|
(1) State of Economy |
(2) Probability of State of Economy | (3) Rate of Return If State Occurs |
(4) Product (2) × (3) |
(5) Rate of Return If State Occurs |
(6) Product (2) × (5) |
| Recession | .80 | −.20 | −.16 | .30 | .24 |
| Boom | .20 | .70 | \underline{.14} | .10 | \underline{ .02} |
| E(R_{L}) = −.02, or −2% | E(R_{U}) = .26 =, or 26% | ||||
The risk premium for Stock U is 26% − 10 = 16% in this case. The risk premium for Stock L is: −2% − 10 = −12%. This is a little odd; but, for reasons we discuss later, it is not impossible.