# After graduating from college with a finance degree, you begin an ambitious plan to retire in 25...

## Question:

After graduating from college with a finance degree, you begin an ambitious plan to retire in {eq}25 {/eq} years. To build up your retirement fund, you will make quarterly payments into a mutual fund that on average will pay {eq}12 \% {/eq} APR compounded quarterly. To get you started, a relative gives you a graduation gift of {eq}\$5,000 {/eq}. Once retired, you plan on moving your investment to a money market fund that will pay {eq}6 \% {/eq} APR with monthly compounding. As a young retiree, you believe you will live for {eq}30 {/eq} more years and will make monthly withdrawals of {eq}\$10,000 {/eq}. To meet your retirement needs, what quarterly payment should you make?

## Future and present value of annuity:

Future value of annuity is the value of a series of equal periodic payments or receipts at the end of the series, whereas, the present value of annuity is the value of the series of payments or receipts at the beginning of the series.

Let us first find out the total money required at retirement, which will be the present value of monthly series of $10,000.00 withdrawals Now, Present value of annuity = Annuity x PVAF(r,n) PVAF or present value annuity factor is the sum of discounting factors at a given rate r for n number of years It can be calculated using the following formula: Here, Annuity =$10,000.00

r = 6 / 12 = 0.5% = 0.005

n = 30 x 12 = 360

So,

Money required at retirement
= 10,000 x PVAF(0.5%,360 months)
= 10,000 x 166.7916
= $1,667,916.00 Now, Deposits will be made to accumulate the given amount. So, Future value of annuity =$1,667,916.00

Also,

Future value of annuity
= Annuity x ((1 + Periodic rate)^Number of periods - 1) / Periodic rate

Here,

Future value of annuity = $1,667,916.00 Annuity = Quarterly deposits Periodic rate = 12 / 4 = 3% = 0.03 Number of periods = 25 x 4 = 60 So, 1,667,916.00 = Quarterly deposits x (1.03^60 - 1) / 0.03 or, Quarterly deposits required =$10,229.26 