Question

Andrew Bogut just received a signing bonus of \(1,000,000. His plan is to invest this payment in a fund that will earn 8%, compounded annually. Instructions (a) If Bogut plans to establish the AB Foundation once the fund grows to \)1,999,000, how many years until he can establish the foundation? (b) Instead of investing the entire \(1,000,000, Bogut invests \)300,000 today and plans to make 9 equal annual investments into the fund beginning one year from today. What amount should the payments be if Bogut plans to establish the $1,999,000 foundation at the end of 9 years?

Short answer

The period required will be 9 years, and the annual payment will be $112,056.

Step-by-step solution

Calculation of period

$$\begin{gathered} \mathbf{Future}\mathbf{}\mathbf{Value}\mathbf{}\mathbf{Factor}\mathbf{=}\frac{\mathbf{Future}\mathbf{}\mathbf{Value}}{\mathbf{Amount}\mathbf{}\mathbf{Invested}} \\ \mathbf{=}\frac{\mathbf{1}\mathbf{,}\mathbf{999}\mathbf{,}\mathbf{000}}{\mathbf{1}\mathbf{,}\mathbf{000}\mathbf{,}\mathbf{000}} \\ \mathbf{=}\mathbf{1}\mathbf{.}\mathbf{999} \end{gathered}$$

The future value is 1.999 at 8% in 9 years.

Calculation of annual payment

$$\begin{gathered} \mathbf{Annual}\mathbf{}\mathbf{Payment}\mathbf{=}\frac{\mathbf{Remaining}\mathbf{}\mathbf{Future}\mathbf{}\mathbf{Value}}{\mathbf{present}\mathbf{}\mathbf{Value}\mathbf{}\mathbf{of}\mathbf{}\mathbf{annuity}\mathbf{}\mathbf{factor}} \\ \mathbf{=}\frac{\mathbf{1}\mathbf{,}\mathbf{999}\mathbf{,}\mathbf{000}\mathbf{-}\left(\mathbf{300}\mathbf{,}\mathbf{000}\mathbf{\times }\mathbf{1}\mathbf{.}\mathbf{99900}\right)}{\mathbf{12}\mathbf{.}\mathbf{48756}} \\ \mathbf{=}\mathbf{\$}\mathbf{112}\mathbf{,}\mathbf{056} \end{gathered}$$