Question

Your grandfather has offered you a choice of one of the three following alternatives: \(7,500 now; \)2,200 a year for nine years; or $31,000 at the end of nine years. Assuming you could earn 10 percent annually, which alternative should you choose? If you could earn 11 percent annually, would you still choose the same alternative?

Short answer

• Case 1 - Alternative 3
• Case 2 - Alternative 2

Step-by-step solution

Case 1- Computation of Present value of all alternatives

Present value of first alternative = $7,500

role="math" localid="1656342827200"

Since the present value is higher for third alternative, so alternative third would be preferred.

Case 2- Computation of Present value of all alternatives

Present value of first alternative = $7,500

$\begin{array}{c}\mathbf{Present}\mathbf{}\mathbf{value}\mathbf{}\mathbf{of}\mathbf{}\mathbf{second}\mathbf{}\mathbf{alternative}\mathbf{=}\mathbf{Amount}\mathbf{\times }\left[\frac{1-\frac{1}{{(1+r)}^n}}{r}\right]\\ \mathbf{=}\mathbf{\$}\mathbf{2}\mathbf{,}\mathbf{200}\mathbf{\times }\left[\frac{1-\frac{1}{{(1+0.11)}^9}}{0.11}\right]\\ \mathbf{=}\mathbf{\$}\mathbf{2}\mathbf{,}\mathbf{200}\mathbf{\times }\mathbf{5}\mathbf{.}\mathbf{5370}\\ \mathbf{=}\mathbf{\$}\mathbf{12}\mathbf{,}\mathbf{181}\end{array}$

$\begin{array}{c}\mathbf{Present}\mathbf{}\mathbf{value}\mathbf{}\mathbf{of}\mathbf{}\mathbf{third}\mathbf{}\mathbf{alternative}\mathbf{=}\mathbf{Amount}\mathbf{\times }\frac{\mathbf{1}}{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{r}\mathbf{)}}^{\mathbf{n}}}\\ \mathbf{=}\mathbf{\$}\mathbf{31}\mathbf{,}\mathbf{000}\mathbf{\times }\frac{\mathbf{1}}{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{11}\mathbf{)}}^{\mathbf{9}}}\\ \mathbf{=}\mathbf{\$}\mathbf{31}\mathbf{,}\mathbf{000}\mathbf{\times }\mathbf{0}\mathbf{.}\mathbf{3909}\\ \mathbf{=}\mathbf{\$}\mathbf{12}\mathbf{,}\mathbf{118}\end{array}$

Since the present value under this case is higher for second alternative, so alternative second would be preferred .