You just turned 50 years old and are planning to retire in 15 years.
You have decided you would like to receive an annuity rather
than managing a lump sum. If you invest $12,500 at the end of
each of the next 10 years, what will your annual payment
be at the beginning of the first year of your retirement?
You can assume an interest rate of 6.5% for the entire period.
At what rate is the interest compounded? Or is 6.5% the total interest rate at the end of the 15 years?
If it's the total interest earned, multiply the $12,500 by 15 years, then multiply by .065. If it's 6.5% per year on the balance, it gets a little more complicated to compute it.
The annual payment from the annuity depends on how long the annuity will last. How many years will annual pay-outs be received? Ten years, 20 years? The question is incomplete. The balance in the account will continue to accrue interest over the life of the annuity, too, and that is not considered here.
To find the annual payment at the beginning of your retirement, we can use the concept of future value of an annuity.
To start, let's calculate the future value of the $12,500 payments over the 10-year period using the formula for future value of an annuity:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = Future Value
P = Periodic Payment (annual investment)
r = Interest Rate per period
n = Number of periods
In this case:
P = $12,500
r = 6.5% = 0.065 (decimal form)
n = 10
Now, let's calculate the future value of the annuity:
FV = $12,500 * [(1 + 0.065)^10 - 1] / 0.065
= $12,500 * [(1.065)^10 - 1] / 0.065
≈ $190,561.46
So, after the 10-year investment period, the accumulated value of your annuity will be approximately $190,561.46.
Next, we'll determine the annual payment at the beginning of the first year of your retirement. Since you want to receive an annuity, you'll need to calculate the present value of the accumulated future value using the formula for present value of an annuity:
PV = FV / (1 + r)^n
Where:
PV = Present Value
FV = Future Value
r = Interest Rate per period
n = Number of periods
In this case:
FV = $190,561.46
r = 6.5% = 0.065 (decimal form)
n = 15
Let's calculate the present value of the accumulated future value:
PV = $190,561.46 / (1 + 0.065)^15
≈ $97,569.06
So, to receive an annuity at the beginning of the first year of your retirement, you would need an annual payment of approximately $97,569.06.