Question

Suppose an investor is concerned about a business choice in which there are three prospects—the probability and returns are given below:

PROBABILITY	RETURN
4	\(100
3	\)30
3	-$30

What is the expected value of the uncertain investment? What is the variance.

Short answer

The expected value and the variance of the uncertain investment according to the given probability and return will be $40 and $2940, respectively.

Step-by-step solution

Expected value of the uncertain investment 

The expected value of the uncertain investment can be found out using the given details in the table. In the table, the probability of each return is given. With the probability and its return at each point, the expected value of the uncertain investment will be:

$$\begin{gathered} EV=\left(.4\right)\left(100\right)+\left(.3\right)\left(30\right)+\left(.3\right)\left(-30\right) \\ =40+9-9 \\ =\$40 \end{gathered}$$

The expected value of the uncertain investment is $40.

 Variance of the uncertain investment

The variance of the investment can be obtained from the sum of squared deviations from the mean weighted by their probabilities. The mean is the expected value that is 40.

The variance of the investment will be:

$$\begin{gathered} Variance=\left(.4\right){\left(100-40\right)}^2+\left(.3\right){\left(30-40\right)}^2+\left(.3\right){\left(-30-40\right)}^2 \\ =1440+30+1470 \\ =2940 \end{gathered}$$

The variance of the uncertain investment is 2940.