1. Yejin plans to retire at age 60. She wants to create an annuity fund, which will pay her a monthly allowance of $4000 during her retirement. She wants to save enough money so that the payments lasts for 30 years. A financial advisor has told her the she can expect to earn 5% interest on her funds, compounded annually.

2. Yejin has just turned 28 years old. She currently has no retirement savings. She wants to save part of her salary each month into her annuity fund. Calculate the amount Yejin needs to save each month, to meet her retirement goal.

This question is a bit unusual in that the interest is compounded annually, but payments and withdrawals are made monthly. The effective monthly rate is ...

1.05^(1/12) -1 = 0.407412378% = i

We assume that payments to the annuity are made at the end of the month, and that withdrawals are made at the beginning of the month. (The last payment and the first withdrawal are made on the same day.)

The amount of money required in the fund is ...

A = $4000(1 -(1.00407^-360))/(1 -1.00407^-1) = $757,712.88

The amount of money needed each month to be put into the fund is P, where ...

$757,712.88 = P(1 -1.00407)^(-12(60-28))/(1 -1.00407^-1) = 194.7297P

P = $757,712.88/194.7297 = $3891.10

Yejin needs to save $3891.10 each month to meet her retirement goal.

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Sanity check

Yejin wants payments for 30 years from the fund to which she has contributed for 32 years. The similarity of the time periods means that Yejin's monthly contribution will need to be very similar to the amount she plans to withdraw.

The only ways to reduce the required contribution are to earn a higher interest on deposits, or to adjust the relative time periods (retire later).