Question

You would like to have $47,000in 9 years. To accumulate this amount you plan to deposit each year an equal sum in the bank, which will earn 8 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 $47,000 in 9 years?

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 8 percent on this deposit.)

c. At the end of 6 years, you will receive $11,000 and deposit this in the bank toward your goal of $47,000 at the end of 9 years. In addition to this deposit, how much must you deposit in equal annual deposits to reach your goal? (Again assume you can earn 8 percent on this deposit.)

Answer 1

(a) Future Value required = FV = $47000

Number of years = n = 9

Let deposits made each year be P

Interest Rate = r = 8%

Hence, FV = P(1+r)^n-1 +....+ P(1+r)² + P(1+r) + P = P[(1+r)ⁿ -1]/r = P[(1+0.08)⁹ -1]/0.08

=> P[(1+0.08)⁹ -1]/0.08 = 47000

=> P = 3763.75

(b) Let the lumpsum be PV

Given, FV = 47000

r = 8%

n = 9

=> FV = PV(1+r)ⁿ

=> 47000 = PV(1+0.08)⁹

=> PV = 47000/(1+0.08)⁹

=> PV = $23511.70

(c)

Future Value required = FV = $47000

Number of years = n = 9

Let deposits made each year be P

Interest Rate = r = 8%

Lumpsum Received after 6 years = X = 11000

Hence, FV = P(1+r)^n-1 +....+ P(1+r)² + P(1+r) + P + X(1+r)³ = P[(1+r)ⁿ -1]/r + X(1+r)³ = P[(1+0.08)⁹ -1]/0.08 + 11000(1+0.08)³

=> 47000 = P[(1+0.08)⁹ -1]/0.08 + 11000(1+0.08)³

=> P[(1+0.08)⁹ -1]/0.08 = 33143.168

=> P = $2654.10