Question

You would like to have ​$75,000 in 15 years. To accumulate this​ amount, you plan to deposit an equal sum in the bank each year that 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 this​ amount?
b. If you decide to make a large​ lump-sum deposit today instead of the annual​ deposits, how large should the​ lump-sum deposit​ be? ​ (Assume you can earn 8 percent on this​ deposit.)
c. At the end of year​ 5, you will receive ​$20,000 and deposit it in the bank in an effort to reach your goal of ​$75,000 at the end of year 15. In addition to the​ lump-sum deposit, how much must you invest in 15 equal annual deposits to reach your​ goal? ​ (Again, assume you can earn 8 percent on this​ deposit.)

Answer 1

a.

Annual deposits can be computed using formula for future value of annuity as:

FV = C x [(1+r) ⁿ- 1 /r]

FV = Future value of annuity = $ 75,000

C = Periodic Cash flows

r = Rate per period = 8 % or 0.08 p.a.

n = Numbers of periods = 15

$ 75,000 = C x [(1 + 0.08)¹⁵ – 1/0.08]

                = C x [(1.08)¹⁵ – 1/0.08]

               = C x [(3.172169 – 1)/0.08]

               = C x (2.172169/0.08)

              = C x 27.15211

    C = $ 75,000/27.15211 = $ 2,762.22

$ 2,762.22 need to deposit annually to get the desired maturity amount.

b.

Formula for compound interest can be used to compute the lump-sum deposit as:

A = P x (1+i/n)^nxt

A = Amount on maturity = $ 75,000

i = Annual interest rate = 8% or 0.08

n = Compounding frequency in a year = 1

t = No. of years = 15

$ 75,000 = P x (1+0.08/1)^1x15

$ 75,000 = P x (1.08)¹⁵

$ 75,000 = P x 3.172169114

P = $ 75,000 / 3.172169114 = $ 23,643.16

$ 23,643.16 needs to deposit today.

c.

Future value of $ 20,000 at the end of 15 years deposited after 5 year computed using compound interest formula as:

A = P x (1+i/n)^nxt

A = Amount on maturity

P = $ 20,000

i = Annual interest rate = 8% or 0.08

n = Compounding frequency in a year = 1

t = No. of years = 15 – 5 = 10

A = $ 20,000 x (1+0.08)15

A = $ 20,000 x (1.08)¹⁵

A = $ 20,000 x 2.158925

A = $ 43,178.50

Future amount needs to deposit annually = $ 75,000 - $ 43,178.50 = $ 31,821.61

Applying formula for FV of annuity, we get:

FV = C x [(1+r) ⁿ- 1 /r]

FV = Future value of annuity = $ 31,821.61

C = Periodic Cash flows

r = Rate per period = 8 % or 0.08 p.a.

n = Numbers of periods = 15

$ 31,821.61 = C x [(1 + 0.08)¹⁵ – 1/0.08]

                     = C x [(1.08)¹⁵ – 1/0.08]

                    = C x [(3.172169 – 1)/0.08]

                   = C x (2.172169/0.08)

                   = C x 27.15211

    C = $ 31,821.61 /27.15211 = $ 1,171.98

$ 1,171.98 needs to deposit annually along with lump-sum to get the desired maturity amount.