Question

Consider an asset that costs \(120 today. You are going to hold it for 1 year and then sell it. Suppose that there is a 25 percent chance that it will be worth \)100 in a year, a 25 percent chance that it will be worth \(115 in a year, and a 50 percent chance that it will be worth \)140 in a year. What is its average expected rate of return? Next, figure out what the investment’s average expected rate of return would be if its current price were $130 today. Does the increase in the current price increase or decrease the asset’s average expected rate of return? At what price would the asset have a zero average expected rate of return?

Short answer

When the asset cost is $120, the expected rate of return will be 3.125%. When the asset cost changes to $130, the expected rate of return will be -4.808%—the increase in price decreases the asset’s average expected rate of return.

At $123.75, the asset has a zero average expected rate of return.

Step-by-step solution

Step 1. Explanation for the expected rate of return

The expected payoff is calculated by multiplying the probability of each payoff to the actual payoff.

The expected payoff is calculated below:

$$\begin{gathered} ExpectedPayoff=\left(0.25\times \$100\right)+\left(0.25\times \$115\right)+\left(0.50\times \$140\right) \\ =\$25+\$28.75+\$123.75 \\ =\$123.75 \end{gathered}$$

The expected payoff will be $123.75.

The expected rate of return when the price is $120 is calculated below:

$$\begin{gathered} ExpectedRateofReturn=\frac{ExpectedPayoff-Price}{Price}\times 100 \\ =\frac{123.75-120}{120}\times 100 \\ =\frac{3.75}{120}\times 100 \\ =3.125\% \end{gathered}$$

The expected rate of return will be 3.125%.

The expected rate of return when the price is $130 is calculated below:

$$\begin{gathered} ExpectedRateofReturn=\frac{ExpectedPayoff-Price}{Price}\times 100 \\ =\frac{123.75-130}{130}\times 100 \\ =\frac{-6.25}{120}\times 100 \\ =-4.808\% \end{gathered}$$

The expected rate of return will be -4.808%.

The expected rate of return has decreased after the asset price increased from $120 to $130.

Step 2. Calculation of price

The price is calculated below:

It is given that the expected rate of return is zero.

$$\begin{gathered} ExpectedRateofReturn=\frac{ExpectedPayoff-Price}{Price} \\ 0=\frac{123.75-Price}{Price} \\ 0=123.75-Price \\ Price=\$123.75 \end{gathered}$$

The price will be $123.75.