Question

Complex present value​) You would like to have ​$56 comma 000 in 16 years. To accumulate this amount you plan to deposit each year an equal sum in the​ bank, which will earn 6 percent interest compounded annually. Your first payment will be made at the end of the year. a. How much must you deposit annually to accumulate this​ amount? b. If you decide to make a large​ lump-sum deposit today instead of the annual​ deposits, how large should this​ lump-sum deposit​ be? (Assume you can earn 6 percent on this​ deposit.) c. At the end of 6 years you will receive ​$12 comma 000 and deposit this in the bank toward your goal of ​$56 comma 000 at the end of 16 years. In addition to this​ deposit, how much must you deposit in equal annual deposits to reach your​ goal? (Again assume you can earn 6 percent on this​ deposit.) a. How much must you deposit annually to accumulate ​$56 comma 000 in 16 ​years?

Answer 1

a). Annuity = [FVA x r] / [(1 + r)ⁿ - 1]

= [$56,000 x 0.06] / [1.06¹⁶ - 1]

= $3,360 / 1.5404 = $2,181.32

b). PV = FV / (1 + r)ⁿ

= $56,000 / 1.06¹⁶ = $56,000 / 2.5404 = $22,044.19

c). First, we need to find the future value of $12,000

FV = PV x (1 + r)ⁿ

= $12,000 x 1.06^{(16 - 6)} = $12,000 x 1.7908 = $21,490.17

Remaining Fund = $56,000 - $21,490.17 = $34,509.83

Now, we can find the annuity for this future value

Annuity = [FVA x r] / [(1 + r)ⁿ - 1]

= [$34,509.83 x 0.06] / [1.06¹⁶ - 1]

   = $2,070.59 / 1.5404 = $1,344.23