Question

Suppose that two investments have the same three payoffs, but the probability associated with each payoff differs, as illustrated in the table below:

PAYOFF	PROBABILITY (INVESTMENT A)	PROBABILITY (INVESTMENT B)
\(300	0.10	0.30
\)250	0.80	0.40
$200	0.10	0.30

1. Find the expected return and standard deviation of each investment.

2. Jill has the utility function U = 5I, where I denotes the payoff. Which investment will she choose?

3. Ken has the utility function U = 51I. Which investment will he choose?

4. Laura has the utility function U = 5I 2. Which investment will she choose?

Short answer

1. The expected return and standard deviation of investment A will be $250 and $22.36, respectively; investment B will be $250 and $38.73, respectively.

2. Jill will be indifferent between investments A and B.

3. Ken will choose investment A.

4. Laura will choose investment B.

Step-by-step solution

Explanation for part (a)

The expected return of investment A is calculated below:

$$\begin{gathered} E\left[A\right]=ProbabilityA\times Payoff \\ =\left(0.10\times 300\right)+\left(0.80\times 250\right)+\left(0.10\times 200\right) \\ =30+200+20 \\ =250 \end{gathered}$$

The expected return of investment A will be $250.

The standard deviation of investment A is calculated below:

$$\begin{gathered} \sigma =\sqrt{Probability{(Payoff-E\left[A\right])}^2} \\ =\sqrt{0.10{\left(300-250\right)}^2+0.80{\left(250-250\right)}^2+0.10{\left(200-250\right)}^2} \\ =\sqrt{0.10\left(2500\right)+0+0.10\left(2500\right)} \\ =\sqrt{250+250} \\ =\sqrt{500} \\ =\$22.36 \end{gathered}$$

The standard deviation of investment A will be $22.36.

The expected return of investment B is calculated below:

$$\begin{gathered} E\left[A\right]=ProbabilityA\times Payoff \\ =\left(0.30\times 300\right)+\left(0.40\times 250\right)+\left(0.30\times 200\right) \\ =90+100+60 \\ =\$250 \end{gathered}$$

The expected return of investment b will be $250.

The standard deviation of investment A is calculated below:

$$\begin{gathered} \sigma =\sqrt{Probability{(Payoff-E\left[A\right])}^2} \\ =\sqrt{0.30{\left(300-250\right)}^2+0.40{\left(250-250\right)}^2+0.30{\left(200-250\right)}^2} \\ =\sqrt{0.30\left(2500\right)+0+0.30\left(2500\right)} \\ =\sqrt{750+750} \\ =\sqrt{1500} \\ =\$38.73 \end{gathered}$$

The standard deviation of investment B will be $38.73.

Explanation for part (b)

Jill’s expected utility of investment A will be:

$$\begin{gathered} E\left[U\right]=\left(0.1\right)\left(5\times 300\right)+\left(0.8\right)\left(5\times 250\right)+\left(0.1\right)\left(5\times 300\right) \\ =150+1000+100 \\ =\$1250 \end{gathered}$$

Jill’s expected utility of investment B will be:

$$\begin{gathered} E\left[U\right]=\left(0.3\right)\left(5\times 300\right)+\left(0.4\right)\left(5\times 250\right)+\left(0.3\right)\left(5\times 300\right) \\ =450+500+300 \\ =\$1250 \end{gathered}$$

The expected utility is the same for both investments; thus, Jill will be indifferent between both investments.

Explanation for part (c)

Ken’s expected utility of investment A is calculated below:

$$\begin{gathered} E\left[U\right]=\left(0.1\times \sqrt[5]{300}\right)+\left(0.8\times \sqrt[5]{250}\right)+\left(0.1\times \sqrt[5]{200}\right) \\ =8.66+63.25+7.07 \\ =78.98 \end{gathered}$$

Ken’s expected utility of investment B is calculated below:

$$\begin{gathered} E\left[U\right]=\left(0.3\times \sqrt[5]{300}\right)+\left(0.4\times \sqrt[5]{250}\right)+\left(0.3\times \sqrt[5]{200}\right) \\ =25.98+31.62+21.21 \\ =78.81 \end{gathered}$$

Ken will choose investment A as the expected utility is more in investment A than B.

Explanation for part (d)

Laura’s expected utility of investment A is calculated below:

$$\begin{gathered} E\left[U\right]={\left(0.1\times 5\times 300\right)}^2+{\left(0.8\times 5\times 250\right)}^2+{\left(0.1\times 5\times 200\right)}^2 \\ =45,000+250,000+20,000 \\ =315,000 \end{gathered}$$

Laura’s expected utility of investment B is calculated below:

$$\begin{gathered} E\left[U\right]={\left(0.3\times 5\times 300\right)}^2+{\left(0.4\times 5\times 250\right)}^2+{\left(0.3\times 5\times 200\right)}^2 \\ =135,000+125,000+60,000 \\ =320,000 \end{gathered}$$

Laura will choose investment B as the expected utility is more in investment B than A.