Question

If you make 30 semiannual deposits of $2000 into a fund that earns 10% compounded quarterly, how much money will be in the fund two years after the last deposit?

Answer 1

No. of semiannual deposits = 30

deposit amount = $2000

interest rate = 10%; Compounded quarterly

semi annual interest rate = 10%/2 = 5%

effective interest rate = (1+ i/n )ⁿ - 1

Where, i = interest rate; and n = no. of compounding period in half year (since deposit is for semiannual)

effective interest rate for semi annual = (1+ 5%/2 )² - 1

= (1+ 2.5% )² - 1 = (1+ 0.025 )² - 1 = (1.025)² - 1 = 1.050625 - 1 = 0.050625

effective interest rate for semi annual = 5.0625%

Future value of Annuity (FV)= P * [ { ( 1 + r ) ⁿ - 1 } / r ]

P = periodic payment = $2000

r = effective interest rate for semi annual = 5.0625% = 0.050625

n = no. of periodic payments = 30

FV = 2000 * [ { ( 1 + 0.050625 ) ³⁰ - 1 } / 0.050625 ]

FV = 2000 * [ { ( 1.050625 ) ³⁰ - 1 } / 0.050625 ]

Note: Use financial calculator to solve above equation

FV of the fund after the last deposit = $134312.7

Calculation of value of the fund two years after the last deposit:

Value of the fund after the last deposit (P) = $134312.7

No. of years (t) = 2

no. of compounding in a year (n) = 4

interest rate (r) = 10% = 0.10

value of the fund two years after the last deposit (VF) = P * (1 + r/n )^{(n * t)}

VF = 134312.7 * (1 + 0.10/4 )^{(4 * 2)}

VF = 134312.7 * (1 + 0.025)⁽⁸⁾

VF = 134312.7 * (1.025)⁸

VF = 134312.7 * (1.2184)

value of the fund two years after the last deposit = 163647