A company is considering investing in a particular project. The project requires an investment of X today. Additional investments are required at the beginning of each of the next five years, with each year’s investment 5% greater than the previous year’s investment. The investment is expected to produce an income of 100 per year at the end of each year forever, with the first payment expected at the end of the first year. At an annual effective interest rate of 10.25%, the project has a net present value of zero.

Calculate X.

• 183
• 192
• 205
• 215
• 225

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: A

The present value of the income is

[[math]] 100a_{\overline{\infty}|0.1025} = 100/0.1025 = 975.61. [[/math]]

The present value of the investment is

[[math]] \begin{array}{l}{{X\biggl[1+1.05/1.1025+(1.05/1.1025)^{2}+(1.05/1.1025)^{3}+(1.05/1.1025)^{4}+(1.05/1.1025)^{5}\biggr]}}\\ {{=X[1+1.05^{-1}+1.05^{-2}+1.05^{-3}+1.05^{-5}]=X\frac{1-1.05^{-6}}{1-1.05^{-1}}=5.3295X.}}\end{array} [[/math]]

Then 975.61=5.3295X for X = 183.06.

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.