Question

Betty DeRose is going to deposit $16,000 into a savings account at the beginning
of every six months for the next eight years. Assume the savings account will
earn 10% interest compounded semi-annually.

Calculate the amount of interest Betty DeRose will earn over the eight year
period.

Answer 1

In order to calculate the amount of interest earned over the 8 year period, we will compute the total amount (total annuity deposits + interest) that Betty will receive at the end of 8 year period and subtract it with total annuity deposits of $16000 per half year for a 8 year time period that Betty will make.

To calculate the future value of annuity payments with interest, we will use the following formula:

FV = {A [(1+i)^n -1] / i} * (1+i)

where

FV - future value

A - annuity of $16000 per 6 month period

i - interest rate for every conversion period (since in our case it is 10% interest compounded semi annually, i will be determined as 0.1/2 i.e. 0.05)

n - number of conversion period (in our case it is 8 year time frame so 16 conversion period of 6 months each)

Substituting in the above mentioned formula,

FV = { 16000[(1+0.05)^16 - 1] /0.05 } * (1+0.05)

= { 16000[(2.1829 - 1]/0.05} * 1.05

= $397445.86 (rounded off to 2 decimal places)

Total 6 monthly deposits made = $16000 * 16 = $256000

Interest earned over 8 year period = Future value - Total 6 monthly deposits = 397445.86-256000 = $141445.86.

Formula explained in detail:

FV = {A [(1+i)^n -1] / i} * (1+i)

1 2

Part 1 computes the future value of annuity deposits made at the end of a period (say at the end of year/quarter/month) [ It has been derived using the basic formula of compound interest ]

When Part 2 in multipled to part 1, we get the future value of annuity deposits made at the beginning of a period (say at the beginning of the year/quarter/month)

Further it would be useful to refer the basic formulae and concepts of compound interest first and then refer to the above formula to comprehend things better.