4G Payout Annuities
A payout annuity works like a home mortgage, except that you turn the tables on the bank - you become the lender and the bank is the borrower. You give the bank a sum of money, and the bank pays you back with regular and equal payments over a specified time period. The bank pays simple interest on the outstanding balance, just as you would do with a home mortgage.
Payout annuities are popular with retired people who have saved some money over their working years, possibly in a retirement plan or through a savings annuity. Instead of taking possession of all their savings at once, they deposit them in a payout annuity earning interest, while taking out regular payments for living expenses.
The relevant formulas for a payout annuity are the same as those for a simple interest amortized loan; for convenience we repeat those formulas here. The notation is
|p = payment amount||r = annual interest rate|
|n = # of payments per year||R = r/n = periodic interest rate|
|t = # of years||P = original principal|
|N = n · t = total # of payments.|
The formula specifying the payment amount p is
Solving this formula for the principal P gives
Mr. Harrison is ready to retire, having already bought his golf clubs. He has accumulated $300,000 in savings, and he will deposit this amount in a 20-year payout annuity with an annual interest rate of 6.6%. We calculate
The principal is P = $300,000, the annual interest rate is r = .066, and the periodic rate is R = r/12 = .066/12 = .0055. Mr. Harrison will make 12 payments a year for 20 years, for a total of N = 20 · 12 = 240 payments. The formula gives his monthly payment from the annuity as
Over 20 years, receiving 240 payments, Mr. Harrison will receive from the annuity the total amount
240 · $2254.42 = $541,060.80 .
The amount of interest he will earn is the difference between the amount he takes out and his initial principal, or
$541,060.80 − $300,000.00 = $241,060.80 .
Elizabeth is contemplating retirement in a few years. From the Bank of England she can get a 25-year payout annuity at a 6% annual interest rate. If she wants her monthly payments to be $5000, how much should she deposit into the annuity?
We use the formula giving the principal P in terms of the other variables. The annual interest rate is r = .06, and the periodic rate is R = r/12 = .06/12 = .005. The number of monthly payments over 25 years will be 12 · 25 = 300. As Elizabeth wants the monthly payment to be p = $5000, we find that her required principal is
A young newly married couple has just opened a 40-year savings annuity, making monthly payments of $100 at a 5.4% annual interest rate. Their plan is that, after expiration of the annuity in 40 years, they will deposit the balance into a 25-year payout annuity at the same interest rate. We calculate
The number of payments the couple makes into the savings annuity will be 12 · 40 = 480, and so the amount they will pay into that annuity is
$100 · 480 = $48,000 .
For each annuity the annual interest rate is r = 5.4% = .054, and the monthly rate is R = r/12 = .0045. In the savings annuity we have p = $100 and N = 480, and so the formula for the expiration amount A in the first annuity gives
The expiration amount in the first annuity becomes the principal in the second annuity. The number of payments in the second annuity is N = 12 · 25 = 300, and the monthly interest rate is R = .0045; thus the formula for the payment in the payout annuity gives the monthly payment
We multiply this payment amount by the number of payments 300, and find that the total amount that the couple will receive from the payout annuity is
300 · $1031.01 = $309,303 .
Note that the difference in what the couple receives in the payout annuity and what they pay into the savings annuity is
$309,303 − $48,000 = $261,303 .
This striking difference highlights the advantage of beginning a savings plan early in life.
|(A) 15-year annuity, 4.8% annual interest,|
|(B) 20-year annuity, 5.4% annual interest,|
|(C) 25-year annuity, 6% annual interest.|