Question

Question: You need $28,974 at the end of 10 years, and your only investment outlet is an 8 percent long-term certificate of deposit (compounded annually). With the certificate of deposit, you make an initial investment at the beginning of the first year.

a. What single payment could be made at the beginning of the first year to achieve this objective?

b. What amount could you pay at the end of each year annually for 10 years to achieve this same objective?

Short answer

Answer

The required amount of annual payment at the beginning and at the end is $1,851.91 and $2,000.06, respectively.

Step-by-step solution

Identification of the required information

Future value (FA) = $28,974

Periods for annual compounding (n) = 10

Interest Rate for annual compounding (i) = 8%

(a) Requisite amount of payment (PMT)

$$\begin{gathered} \mathit{F}\mathit{A}\mathbf{=}\mathit{P}\mathit{M}\mathit{T}\mathbf{\times }\left[\frac{{\left(1+i\right)}^{n+1}-1}{i}-1\right] \\ \mathbf{\$}\mathbf{28}\mathbf{,}\mathbf{974}\mathbf{=}\mathit{P}\mathit{M}\mathit{T}\mathbf{\times }\mathbf{[}\frac{{\left(1+8\%\right)}^{\mathbf{10}\mathbf{+}\mathbf{1}}\mathbf{-}\mathbf{1}}{\mathbf{8}\mathbf{\%}}\mathbf{-}\mathbf{1}\mathbf{]} \\ \mathit{P}\mathit{M}\mathit{T}\mathbf{=}\mathbf{\$}\mathbf{1}\mathbf{,}\mathbf{851}\mathbf{.}\mathbf{91} \end{gathered}$$

(b) Requisite amount of payment (PMT)

$$\begin{gathered} \mathit{F}\mathit{A}\mathbf{=}\mathit{P}\mathit{M}\mathit{T}\mathbf{\times }\mathbf{[}\frac{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{i}\mathbf{)}}^{\mathbf{n}}\mathbf{-}\mathbf{1}}{\mathbf{i}}\mathbf{-}\mathbf{1}\mathbf{]} \\ \mathbf{\$}\mathbf{28}\mathbf{,}\mathbf{974}\mathbf{=}\mathit{P}\mathit{M}\mathit{T}\mathbf{\times }\mathbf{[}\frac{{\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{8}\mathbf{\%}\mathbf{)}}^{\mathbf{10}}\mathbf{-}\mathbf{1}}{\mathbf{8}\mathbf{\%}}\mathbf{-}\mathbf{1}\mathbf{]} \\ \mathit{P}\mathit{M}\mathit{T}\mathbf{=}\mathbf{\$}\mathbf{2}\mathbf{,}\mathbf{000}\mathbf{.}\mathbf{06} \end{gathered}$$