Chapter 21, Problem 18SE

An investor wants to select one of seven mutual funds for the coming year. Data showing the percentage annual return for each fund during five typical one-year periods are shown here. The assumption is that one of these five-year periods will occur again during the coming year. Thus, years A, B, C, D, and E are the states of nature for the mutual fund decision.

1. a. Suppose that an experienced financial analyst reviews the five states of nature and provides the following probabilities: .1, .3, .1, .1, and .4. Using the expected value approach, what is the recommended mutual fund? What is the expected annual return? Using this mutual fund, what are the minimum and maximum annual returns?
2. b. A conservative investor notes that the Small-Cap mutual fund is the only fund that does not have the possibility of a loss. In fact, if the Small-Cap fund is chosen, the investor is guranteed a return of at least 6%. What is the expected annual return for this fund?
3. c. Considering the mutual funds recommended in parts (a) and (b), which fund appears to have more risk? Why? Is the expected annual return greater for the mutual fund with more risk?
4. d. What mutual fund would you recommend to the investor? Explain.

To determine

Identify the recommended mutual fund. And compute the expected annual return.

Compute the minimum and maximum annual return Based on the recommended mutual fund.

Answer to Problem 18SE

The technology sector is the recommended mutual fund with expected annual return is 16.97%.

The minimum annual return for the technology sector is –20.1% and maximum annual return for the technology sector is 93.1%.

Explanation of Solution

Calculation:

There are a total of seven mutual funds. The table provides percentage annual return for each fund during five typical one-year periods. There are 5 states of nature for the mutual funds. The probabilities for states of nature are 0.1, 0.3, 0.1, 0.1, and 0.4 respectively.

Expected value approach:

The expected value (EV) of a decision rule $d_i$ is given by:

$\text{EV}\left(d_i\right)={\displaystyle \sum _{j=1}^{N}P\left(s_j\right)}\times V_{ij}$

Where, $V_{ij}$ denote the value of payoff for the decision alternative $d_i$, $P\left(s_j\right)$ denote the probability of the state of nature $s_j$ and total number of nature of states is N.

Since the problem deals with profit, the optimal decision is one with maximum expected value (EV).

The expected value for the Large-Cap Stock mutual fund is as follows:

$\text{EV}\left(\text{Large-Cap Stock}\right)=P\left(s_1\right)\times V_{11}+P\left(s_2\right)\times V_{12}+P\left(s_3\right)\times V_{13}+P\left(s_4\right)\times V_{14}+P\left(s_5\right)\times V_{15}$

Substitute the following values in the expected value approach,

$P\left(s_1\right)=0.1$, $P\left(s_2\right)=0.3$, $P\left(s_3\right)=0.1$, $P\left(s_4\right)=0.1$, $P\left(s_5\right)=0.4$, $V_{11}=35.3$, $V_{12}=20$, $V_{13}=28.3$, $V_{14}=10.4$ and $V_{15}=-9.3$.

Therefore,

$\begin{array}{c}\text{EV}\left(\begin{array}{l}\text{Large-Cap}\\ \text{ Stock}\end{array}\right)=\left(0.1\right)\times \left(35.3\right)+\left(0.3\right)\times \left(20\right)+\left(0.1\right)\times \left(28.3\right)+\left(0.1\right)\times \left(10.4\right)+\left(0.4\right)\times \left(-9.3\right)\\ =3.53+6+2.83+1.04-3.72\\ =9.68\end{array}$

The expected value Large-Cap Stock mutual fund is 9.68%.

Similarly the other expected values are obtained as shown in the table:

Mutual Fund	Expected Annual Return
Large-Cap Stock	9.68
Mid-Cap Stock	$\begin{array}{l}\left[\begin{array}{l}\left(0.1\right)\times \left(32.3\right)+\left(0.3\right)\times \left(23.2\right)+\left(0.1\right)\times \left(-0.9\right)+\\ \left(0.1\right)\times \left(49.3\right)+\left(0.4\right)\times \left(-22.8\right)\end{array}\right]\\ =5.91\end{array}$
Small-Cap Stock	$\begin{array}{l}\left[\begin{array}{l}\left(0.1\right)\times \left(20.8\right)+\left(0.3\right)\times \left(22.5\right)+\left(6\right)\times \left(-0.9\right)+\\ \left(0.1\right)\times \left(33.3\right)+\left(0.4\right)\times \left(6.1\right)\end{array}\right]\\ =15.20\end{array}$
Emergency/Resource Sector	$\begin{array}{l}\left[\begin{array}{l}\left(0.1\right)\times \left(25.3\right)+\left(0.3\right)\times \left(33.9\right)+\left(0.1\right)\times \left(-20.5\right)+\\ \left(0.1\right)\times \left(20.9\right)+\left(0.4\right)\times \left(-2.5\right)\end{array}\right]\\ =11.74\end{array}$
Health Sector	$\begin{array}{l}\left[\begin{array}{l}\left(0.1\right)\times \left(19.1\right)+\left(0.3\right)\times \left(5.5\right)+\left(0.1\right)\times \left(29.7\right)+\\ \left(0.1\right)\times \left(77.7\right)+\left(0.4\right)\times \left(-24.9\right)\end{array}\right]\\ =7.34\end{array}$
Technology Sector	$\begin{array}{l}\left[\begin{array}{l}\left(0.1\right)\times \left(46.2\right)+\left(0.3\right)\times \left(21.7\right)+\left(0.1\right)\times \left(45.7\right)+\\ \left(0.1\right)\times \left(93.1\right)+\left(0.4\right)\times \left(-20.1\right)\end{array}\right]\\ =16.97\end{array}$
Real Estate Sector	$\begin{array}{l}\left[\begin{array}{l}\left(0.1\right)\times \left(20.5\right)+\left(0.3\right)\times \left(44\right)+\left(0.1\right)\times \left(-21.1\right)+\\ \left(0.1\right)\times \left(2.6\right)+\left(0.4\right)\times \left(5.1\right)\end{array}\right]\\ =15.44\end{array}$

From the expected values, it is clear that the technology sector provide maximum expected value. The annual return of technology sector is 16.97%.

From the table of percentage annual return, the minimum annual return for the technology sector is –20.1% and maximum annual return for the technology sector is 93.1%.

To determine

Compute the expected annual return for Small-Cap Stock.

Answer to Problem 18SE

The expected annual return for Small-Cap Stock is 15.20%.

Explanation of Solution

The investor note that the Small-Cap mutual funds are mutual funds that does not have a chance of loss. The Small-Cap mutual funds guaranteed a return of at least 6%.

From part (a), it is clear that expected annual return for Small-Cap Stock is 15.20%. But the recommended mutual fund is technology sector with expected value 16.97%.  The difference between recommended and Small-Cap Stock is 1.77%$\left(=16.97-15.20\right)$.

Therefore, the expected annual return for Small-Cap Stock is 15.20%.

To determine

Identify the mutual fund that explained in parts (a) and (b) is more risky. Explain the reason.

Check whether annual return greater for the mutual fund with more risk.

Answer to Problem 18SE

The technology sector is more risky. Annual return greater for the mutual fund with more risk.

Explanation of Solution

From the table of percentage annual return, the minimum annual return for the technology sector is –20.1% and maximum annual return for the technology sector is 93.1%. That is, the annual return for technology sector varies from –20.1% to 93.1%. The minimum annual return for the Small-Cap Stock is 6% and maximum annual return for the Small-Cap Stock is 33.3%. That is, the annual return for Small-Cap Stock varies from 6% to 33.3%. That is, the range of variation is high for technology sector. Therefore, technology sector is more risky. The technology sector also have a higher expected annual return but only by 1.77%. Hence Annual return greater for the mutual fund with more risk.

To determine

Identify which mutual fund is recommended to the investor. Explain the reason.

Answer to Problem 18SE

The recommended mutual fund is Small-Cap Stock.

Explanation of Solution

The answer will vary. One of the possible answer is given below:

From part (c), it is clear that, risk associated with Small-Cap Stock mutual fund is less compared to technology sector. From part (b), the high risk technology sector only has a 1.77% higher expected annual return. Therefore, mutual fund that recommended to the investor is Small-Cap Stock mutual fund.