Question

After graduating from college with a bachelor of business administration, you begin an ambitious plan to retire in 27.00 years. To build up your retirement fund, you will make quarterly payments into a mutual fund that on average will pay 11.92% APR compounded quarterly. To get you started, a relative gives you a graduation gift of $3,584.00. Once retired, you plan on moving your investment to a money market fund that will pay 5.76% APR with monthly compounding. As a young retiree, you believe you will live for 34.00 more years and will make monthly withdrawals of $10,523.00. (YOUR WITHDRAWALS ARE AT THE BEGINNING OF THE MONTH!!!!) To meet your retirement needs, what quarterly payment should you make?

Answer 1

Step 1

In first step , we will find out the value of fund at the time of retirement

We can use the present value of annuity due formula to find out this value.

Present value of annuity due = P + P x {[1 - (1+r)^-(n-1)]/r}

Present value of annuity due = value of fund at the time of retirement = ?

P = monthly withdrawal from fund = $10523

r = monthly APR after retirement = 5.76%/12 = 0.0048

n = number of monthly withdrawals = 34 years * 12 = 408

Present value of annuity due = 10523 + 10523 x {[1 - (1+0.0048)^-(408-1)]/0.0048}

Present value of annuity due = 10523 + 10523 x 178.66

Present value of annuity due = $10523 + $18,80,050.33

Value of fund at the time of retirement = $18,90,573.33

Step 2

In second step , we will find out the value of graduation gift at the time of retirement.

We can use the future value of sum formula to find out this value.

Future value of sum = P x (1+r)^n

Future value of sum = Value of graduation gift at the time of retirement = ?

P = graduation gift = $3584

r = quarter APR before retirement = 11.92% /4 = 0.0298

n = number of quarterly compounding = 27 years * 4 = 108

Future value of sum = 3584 x (1+0.0298)^108

Future value of sum = 3584 x 23.84031

Future value of sum = 85443.66

Value of graduation gift at the time of retirement = $85,443.66

Step 3

In step 3 , we will find out the quarterly payment required to the fund till retirement to arrive at the fund value of $18,05,129.67 at retirement (Step 1 Value - Step 2 Value)

We can use the future value of annuity formula to know the quarterly payment required to the fund.

Future value of annuity = P x {[(1+r)^n -1]/r}

Future value of annuity = fund value at the time retirement of quarterly payments till retirement = $18,05,129.67

P = Quarterly payment to the fund = ?

r = quarter APR before retirement = 11.92% /4 = 0.0298

n = number of quarterly compounding = 27 years * 4 = 108

1805129.67 = P x {[(1+0.0298)^108 -1]/0.0298}

1805129.67 = P x 766.453265

P = 2355.17

You should make quarterly payment of $2355.17 to the fund to meet your retirement needs.