Question

Question: Morgan Jennings, a geography professor, invests $50,000 in a parcel of land that is expected to increase in value by 12 percent per year for the next five years. He will take the proceeds and provide himself with a 10-year annuity. Assuming a 12 percent interest rate, how much will this annuity be?

Short answer

Answer

The annuity will be $15,605.95.

Step-by-step solution

Identification of the required information

Present value of land (PV) = $50,000

Periods (n) = 5

Incremental rate (i) = 12%

Computation of the value of land after 5 years (FV)

$$\begin{gathered} \mathit{F}\mathit{V}\mathbf{=}\mathit{P}\mathit{V}\mathbf{\times }{\left(1+i\right)}^{\mathbf{-}\mathbf{n}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{\$}\mathbf{50}\mathbf{,}\mathbf{000}\mathbf{\times }{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{12}\mathbf{\%}\mathbf{)}}^{\mathbf{-}\mathbf{5}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{\$}\mathbf{88}\mathbf{,}\mathbf{117}\mathbf{.}\mathbf{08} \end{gathered}$$

Identification of the required information

Present value (PV) = $88,117.08

Periods (n) = 10

Interest rate (i) = 12%

Computation of the annuity (PMT)

$$\begin{gathered} \mathit{P}\mathit{V}\mathbf{=}\mathit{P}\mathit{M}\mathit{T}\mathbf{\times }\left[\frac{1-{(1+i)}^{-n}}{i}\right] \\ \mathbf{\$}\mathbf{88}\mathbf{,}\mathbf{117}\mathbf{.}\mathbf{08}\mathbf{=}\mathit{P}\mathit{M}\mathit{T}\mathbf{\times }\mathbf{\left[}\frac{\mathbf{1}\mathbf{-}{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{12}\mathbf{\%}\mathbf{)}}^{\mathbf{10}}}{\mathbf{12}\mathbf{\%}}\mathbf{\right]} \\ \mathbf{}\mathit{P}\mathit{M}\mathit{T}\mathbf{=}\mathbf{\$}\mathbf{15}\mathbf{,}\mathbf{605}\mathbf{.}\mathbf{95} \end{gathered}$$