Question

You would like to have $9,000,000 in 40 years by making regular deposits at the end of each month in an annuity that pays 4.25% compounded monthly. How much of the $9,000,000 comes from interest? In your calculations, round the monthly payment to the nearest dollar.

Answer 1

In 40 years, there are 40*12=480 months to come, so you make 480 payments of a constant annuity A. Since you pay at the end of each month, the first payment is compounded for 479 months, while the last payment is not compounded, since it is done precisely in 40 years at the end of the contract.

Compounding the annuity A for one month leads to A * (1 + 4.25% / 12 ). Dividing by 12 has to be done because interest rates are (usually) quoted "per annum" (as it is the case here). Compounding for n months leads to A * ( 1 + 4.25% )^n . Let q defined be as (1+4.25%/12). Then after 40 years you own

Sum from n=0 to n=479 ( A * q^n )

which is a geometric series; so it is the same as

A * ( q^480 - 1 ) / ( q - 1 ).

Since this has to be 9,000,000 USD, we can calculate

A = 9,000,000 * ( q - 1 ) / (q^480 - 1) = 7,151 USD (rounded to the nearest dollar).

The sum of all payments is 480 * 7,151 = 3,432,480 USD. So the amount of

9,000,000 USD - 3,432,480 USD = 5,567,520 USD

is due to interest payments.

Answer 2

Looks correct to me =p

Answer 3

Thank you!

Answer 4

My Pleasure.