Question

What is the future value of $100 deposited in an account for four years paying a 6 percent annual rate of interest, compounded semiannually?

What is the future value of an ordinary annuity of $2,000 each year for 10 years, invested at 12 percent?

Gina has planned to start her college education four years from now. To pay for her college education, she has decided to save $1,000 a quarter for the next four years in an investment account expected to yield 12 percent. How much will she have at the end of the fourth year? (Assume quarterly compounding.)

Answer 1

1)

Future value = PV x (1+r) ⁿ

PV = Present value = $ 100

r = Periodic interest rate = 0.06/2 = 0.03 semi-annually

n = Number of periods = 4 years x 2 = 8

Future value = $ 100 x (1+0.03)⁸

                               = $ 100 x (1.03)⁸

                              = $ 100 x 1.26677008138762

                              = $ 126.677008138762 or $ 126.68

Future value of investment is $ 126.68

2)

Future value of annuity = P x [(1+r) ⁿ – 1/r]

P = Periodic cash flow = $ 2,000

r = Periodic interest rate = 0.12 p.a.

n = Number of periods = 10

Future value = $ 2,000 x [(1+0.12)¹⁰ -1/0.12]

= $ 2,000 x [(1.12)¹⁰ -1/0.12]

                   = $ 2,000 x [(3.10584820834421 -1)/0.12]

                   = $ 2,000 x [(2.10584820834421/0.12)

                  = $ 2,000 x 17.5487350695351

                   = $ 35,097.4701390702 or $ 35,097.47

Future value of the annuity is $ 35,097.47

3)

Future value of annuity = P x [(1+r) ⁿ – 1/r]

P = Periodic cash flow = $ 1,000

r = Periodic interest rate = 0.12/4 = 0.03 p. q.

n = Number of periods = 4 years x 4 q. = 16

Future value = $ 1,000 x [(1+0.03)¹⁶ -1/0.03]

                   = $ 1,000 x [(1.03)¹⁶ -1/0.03]

                   = $ 1,000 x [(1.60470643909879-1)/0.03]

                   = $ 1,000 x [(0.60470643909879/0.03)

                  = $ 1,000 x 20.1568813032929

                   = $ 20,156.8813032929 or $ 20,156.88

Gina will save $ 20,156.88 at the end of the fourth year.