Question

Mike plans to make contributions to his retirement account for 15 years. After the last contribution, he will start withdrawing $10,000 a quarter for 10 years. Assuming Mike's account earns 8% compounded quarterly, how large must his quarterly contributions be during the first 15 years, in order to accomplish his goal

Answer 1

$2398.52

Explanation:

The amortization formula can be used to find the principal amount that is required to support payments of $10,000 per quarter.

A = P(r/n)/(1 -(1 +r/n)^(-nt))

A is the periodic payment, P is the principal amount, r is the annual interest rate, n is the number of times per year interest is compounded, and t is the number of years.

10,000 = P(.08/4)/(1 -(1 +.08/4)^(-4·10)) = .02P/(1 -1.02^-40)

P = 10000(1 -1.02^-40)/.02

P = $273,554.79 . . . . . balance required when payouts start

__

The annuity formula can be used to find the periodic payment required to achieve this balance.

A = P((1 +r/n)^(nt) -1)/(r/n)

A is the sum of the payments P made n times per year. Interest rate r is also compounded n times per year for t years.

273,554.79 = P((1 +.08/4)^(4·15) -1)/(.08/4) = P(1.02^60 -1)/.02 = 114.05154P

273,554.79/114.05154 = P = 2398.52

Mike must deposit $2398.52 quarterly to accomplish his goal.