Question

Question: You wish to retire in 14 years, at which time you want to have accumulated enough money to receive an annual annuity of \(17,000 for 19 years after retirement. During the period before retirement you can earn 8 percent annually, while after retirement you can earn 10 percent on your money. What annual contributions to the retirement fund will allow you to receive the \)17,000 annuity?

Short answer

Answer

The required annual contribution is $5,872.56.

Step-by-step solution

Identification of the required information

Annuity after retirement (PMT) = $17,000

Periods (n) = 19

Interest rate after retirement (i) = 10%

Computation of the value of required annuity at the time of retirement (PV)

$$\begin{gathered} \mathit{P}\mathit{V}\mathbf{=}\mathit{P}\mathit{M}\mathit{T}\mathbf{\times }\left[\frac{1-{\left(1+i\right)}^{-n}}{i}\right] \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{\$}\mathbf{17}\mathbf{,}\mathbf{000}\mathbf{\times }\mathbf{\left[}\frac{\mathbf{1}\mathbf{-}{\left(1+10\%\right)}^{\mathbf{-}\mathbf{19}}}{\mathbf{10}\mathbf{\%}}\mathbf{\right]} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{\$}\mathbf{142}\mathbf{,}\mathbf{203}\mathbf{.}\mathbf{64} \end{gathered}$$

Identification of the required information

Future value (FV) = $142,203.64

Periods (n) = 14

Interest rate (i) = 8%

Computation of the required annual contribution (PMT)

$$\begin{gathered} \mathit{F}\mathit{V}\mathbf{=}\mathit{P}\mathit{M}\mathit{T}\mathbf{\times }\mathbf{\left[}\frac{{\left(1+i\right)}^{\mathbf{-}\mathbf{n}}\mathbf{-}\mathbf{1}}{\mathbf{i}}\mathbf{\right]} \\ \mathbf{\$}\mathbf{142}\mathbf{,}\mathbf{203}\mathbf{.}\mathbf{64}\mathbf{=}\mathit{P}\mathit{M}\mathit{T}\mathbf{\times }\mathbf{\left[}\frac{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{8}\mathbf{\%}\mathbf{)}}^{\mathbf{-}\mathbf{14}}\mathbf{-}\mathbf{1}}{\mathbf{8}\mathbf{\%}}\mathbf{\right]} \\ \mathbf{}\mathbf{}\mathit{P}\mathit{M}\mathit{T}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{\$}\mathbf{5}\mathbf{,}\mathbf{872}\mathbf{.}\mathbf{56} \end{gathered}$$