Question

You must make a payment of $1,000 in 10 years. To get the money for this payment, you will make 5 equal deposits, beginning today and for the following 4 quarters, in a bank that pays a nominal interest rate of 8% with quarterly compounding. How large must each of the 5 payments be?

Answer 1

Present value = future value / (1 + (r/n))^n*t

where r = annual rate of interest

n = number of compounding periods per year. This is 4, as the compounding is quarterly.

t = number of years.

First, we calculate the required value of the bank account 1 year from now, by calculating the present value of $1,000 by discounting it for 9 years.

Required value of bank account 1 year from now = $1,000 / (1 + (8%/4)^4*9)

Required value of bank account 1 year from now = $490.22

Future value of annuity = (1 + r) * P * [(1 + r)ⁿ - 1] / r,

where P = periodic payment. We need to calculate this.

r = periodic rate of interest. This is (8%/4) = 2%. We divide by 4 since we need to convert the annual rate into quarterly rate)

n = number of periods. This is 5 (there are 5 deposits, or 5 quarters in the investment period)

$490.22  = (1 + 2%) * P * [(1 + 2%)⁵ - 1] / 2%

P =  $490.22 * 2% / ((1 + 2%) * [(1 + 2%)⁵ - 1])

P = $92.35

Each of the payment must be $92.35