Question

Suppose that a risk-free investment will make three future payments of \(100 in one year, \)100 in two years, and $100 in three years. If the Federal Reserve has set the risk-free interest rate at 8 percent, what is the proper current price of this investment? What is the price of this investment if the Federal Reserve raises the risk-free interest rate to 10 percent?

Short answer

The current price of the investment will be $257.42. After the Federal Reserve raises the risk-free interest rate to 10%, the price of an investment will be $248.68.

Step-by-step solution

Step 1. Explanation for the present value when the interest rate is 8%

When the future payment is divided into different years, the total present value is calculated by adding all the present values of different years.

The total present value is calculated in the following manner:

$$\begin{gathered} PresentValue=\frac{A}{{\left(1+i\right)}^n} \\ i=RateofInterest \\ i=\frac{8}{100} \\ i=0.08 \\ n=NumberofYearofInvestment \\ n=1,2,3 \\ A=Amountafterone,twoandthreeyear \\ A=\$100 \\ PresentValue=\frac{100}{\left(1+0.08\right)}+\frac{100}{{\left(1+0.08\right)}^2}+\frac{100}{{\left(1+0.08\right)}^3} \\ =\frac{100}{\left(1.08\right)}+\frac{100}{{\left(1.08\right)}^2}+\frac{100}{{\left(1.08\right)}^3} \\ =\frac{100}{1.08}+\frac{100}{1.17}+\frac{100}{1.26} \\ =92.59+85.47+79.36 \\ =\$257.42 \end{gathered}$$

The total present value will be $257.42.

Step 2. Explanation for present value when interest rate changes to 10%

The total present value with the rate of interest as 10% is calculated in the following manner:

$$\begin{gathered} i=\frac{10}{100} \\ i=0.1 \\ TotalPresentValue=\frac{100}{\left(1+0.1\right)}+\frac{100}{{\left(1+0.1\right)}^2}+\frac{100}{{\left(1+0.1\right)}^3} \\ =\frac{100}{\left(1.1\right)}+\frac{100}{{\left(1.1\right)}^2}+\frac{100}{{\left(1.1\right)}^3} \\ =\frac{100}{1.1}+\frac{100}{1.21}+\frac{100}{1.331} \\ =90.91+82.64+75.13 \\ =\$248.68 \end{gathered}$$

The total present value will be $248.68.