Question

Evaluating Investments. An investor plans to put $50.000 in one of four investments. The return on each investment depends on whether next year's economy is strong or weak. The following table summarizes the possible payoffs. in dollars for the four investments.

Let V, W, X and Y denotes the payoffs for the certificate or deposit office complex, land speculation. and technical school, respectively the V, W, X and Y are random variables . assume that nest year's economy has a 40% chance of being strong and a 60% chance of being weak.

Part(a) Find the probability distribution of each random variable V, W, X, and Y

Part (b) Determine the expected value of each random variable.

Part (c) Which investment has the best expected payoffs? the worst?

Part (d) Which investment would you select? Explain

Short answer

Part (a)

	Strong	Weak
v	6000	6000
P(V=v)	0.40	0.60

	Strong	Weak
w	15000	5000
P(W = w)	0.40	0.60

	Strong	Weak
x	33000	-17000
P(X = x)	0.400	0.60

	Strong	Weak
y	5500	10000
P(Y = y)	0.40	0.60

Part (b)

E(V) = 6000

E(W) = 9000

E(X) = 3000

E(Y) = 8200

Part (c)

The investment in an office complex has the highest expected return.

The worst-case scenario for a land speculation investment is a loss.

Part (d) The investment in the Office Complex must be chosen.

Step-by-step solution

Part (a) Step 1. Given information.

The potential returns on the four investments are given as

Consider the payoffs for the certificate of deposit, office complex, land speculation, and technical school as V, W, X, and V, respectively. It is also assumed that the economy will be strong 40% of the time and weak 60% of the time next year.

 Part (a) Step 2. Each random variable's probability distribution

The payoff probability distribution for the certificate of deposit is defined as follows:

	Strong	Weak
v	6000	6000
P(V=v)	0.40	0.60

The payoff probability distribution for the office complex is defined as follows:

	Strong	Weak
w	15000	5000
P(W = w)	0.40	0.60

The payoff probability distribution for land speculation is defined as follows:

	Strong	Weak
x	33000	-17000
P(X = x)	0.400	0.60

The payoff probability distribution for technical school is defined as follows:

	Strong	Weak
y	5500	10000
P(Y = y)	0.40	0.60

Part (b) Step 1. Each random variable's expected value.

V's expected value can be calculated as

$$\begin{gathered} E\left(V\right)=\sum _{}P(V=v) \\ E\left(V\right)=\left[\right(6000\times 0.40)+(6000\times 0.60\left)\right] \\ E\left(V\right)=(2400+3600) \\ E\left(V\right)=6000 \end{gathered}$$

W's expected value can be calculated as

$$\begin{gathered} E\left(W\right)=\sum _{}P(W=w) \\ E\left(W\right)=\left[\right(15000\times 0.40)+(5000\times 0.60\left)\right] \\ E\left(W\right)=(6000+3000) \\ E\left(W\right)=9000 \end{gathered}$$

X's expected value can be calculated as

$$\begin{gathered} E\left(X\right)=\sum _{}P(X=x) \\ E\left(X\right)=\left[\right(33000\times 0.40)+(-17000\times 0.60\left)\right] \\ E\left(X\right)=(13200-102000) \\ E\left(X\right)=3000 \end{gathered}$$

Y's expected value can be calculated:

$$\begin{gathered} E\left(Y\right)=\sum _{}P(Y=y) \\ E\left(Y\right)=\left[\right(5500\times 0.40)+(-10000\times 0.60\left)\right] \\ E\left(Y\right)=(2200-6000) \\ E\left(Y\right)=8200 \end{gathered}$$

Part (c) Step 1. The investment with the highest and lowest payoff

The expected values of the random variables V, W, X, and Y are as follows:

E(V) = 6000

E(W) = 9000

E(X) = 3000

E(Y) = 8200

The Office Complex investment has the best expected payoff, while the Land Speculation investment has the worst expected payoff.

Part (d) Step 1. The investment which should be selected

The expected values of the random variables V, W, X, and Y are as follows:

E(V) = 6000

E(W) = 9000

E(X) = 3000

E(Y) = 8200

The Office Complex investment has the best expected payoff, while the Land Speculation investment has the worst expected payoff.

As a result, the Office Complex investment must be chosen.