22.6 22.5 Annuities
We encountered in Section 21.5 (page 885) the concept of an annuity.
Annuity DEFINITION
An annuity is a sequence of periodic payments.
An annuity is much like a mortgage but from the opposite point of view: Instead of making a payment every month, you receive a check every month! In purchasing an annuity, you make a loan to a company in exchange for periodic “payback” payments.
We restrict our discussion to annuities for which payments are made at the end of each period and for which the compounding period is the same as the payment period.
The basis for the annuity payments can be a lump sum or even the winnings from a lottery. For example, winners of lotteries are often offered the choice of receiving either the jackpot amount paid as an annuity over a number of years or else a smaller lump sum to be paid immediately. The cost to the lottery administration is the same. If the winner wants an annuity, the administration uses the lump sum to buy either government bonds that pay at the specified rate or else an equivalent annuity from an insurance company.
You can think of the government or the insurance company as borrowing the lump sum and paying it back via the annuity payments. In effect, the government or the insurance company amortizes the lump sum over the term of the annuity. Annuities can vary in the amount of the payments (level or graduated) and the term (how long payments continue).
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EXAMPLE 13 Winning a Lottery
The world record lottery jackpot with a single winner was $370.9 million in May 2013. The winner had the option of receiving the “annuity value” of the prize, $590.5 million, in 30 graduated annual installments, the first payment being right away. To take into account inflation, each installment was to be 4% larger than the previous one. However, the winner chose instead the instant lump sum of $370.9 million. Well, she was 84 years old at the time … (but had she chosen the annuity, after her death any remaining payments would have been made to her estate). What was the interest rate of the annuity?
We don’t offer a solution here. In the past, lotteries used to offer annuities with level payments over 25 to 30 years. For such an annuity, we could use the amortization formula and software to determine the effective interest rate on which the payments are based. Now the annuities offered by lotteries usually feature graduated payments (4% or 5% per year), a situation that offers more complications than we want to go into here. Such calculations are commonly handled by actuaries (see Spotlight 22.4).
What Is an Actuary? Spotlight 22.4
The Truth in Savings Act and the Truth in Lending Act specify that the APY for savings and the APR for loans must be calculated “according to the actuarial method.”
Actuaries are financial experts who manage risks. They assess the costs and likelihood of risks such as tornadoes, floods, auto accidents, and deaths. Actuaries are crucially involved in setting the premiums for insurance against risks. Their calculations take into account historical rates—such as the percentage of female 85-year-olds who live to be 86, or the percentage of unmarried male drivers under age 25 who have auto accidents—and project those rates and the accompanying costs into the future.
Other actuaries concentrate on setting up and evaluating healthcare plans or pension and fringe benefit plans. For example, the city of Beloit, Wisconsin, hired a consulting actuary to estimate the current and future costs of free lifetime medical benefits to families of police and firefighters.
Another major activity of actuaries is managing return on investment. Contrary to popular belief, insurance companies (particularly life insurance companies) do not earn all their money from premiums paid. In fact, a substantial portion of their income comes from return on investment of financial reserves, funds that they are required to have to meet current and future insurance obligations.
Becoming an actuary requires training in mathematics, statistics, economics, and finance and includes a sequence of professional exams taken over several years.
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Self Check 12
Why would lotteries offer graduated payments?
- One reason is that because of inflation, the value (purchasing power) of a level payment would decline over the 25 or 30 years of the term of the annuity.
A more common situation than winning a jackpot is saving toward retirement and then at retirement purchasing an annuity. One possibility is an annuity for a fixed number of years (perhaps as long as you expect to live), called an annuity-certain.
EXAMPLE 14 How Much Do You Need to Retire?
Suppose that your father is ready to retire at 65 and wants to purchase an annuity that pays $5000 per month for 25 years. The insurance company offering the annuity is willing to assume that the long-range steady interest rate will be 3% per year compounded monthly (and that rate also takes into account its costs and profit). What should be the cost of the annuity—that is, how much should your father expect to pay for such a stream of income?
We apply the amortization formula “in reverse.” We know the amount of the monthly payment and we need to find the principal .
We apply the amortization formula with , , , and , to find the amount :
So such an annuity would cost about a million dollars in retirement savings.
Self Check 13
How much would such an annuity cost if the insurance company expects the long-range steady interest rate to be 4%?
- $947,262.41
However, if your father retires at 65 and purchases a 25-year annuity, he might be in trouble if he lives longer than the term of the annuity (past 90), because the payments would stop and he would have no further income from the annuity. (About 2% of U.S. children born since 2000 can expect to live to age 100.) Similarly, if your father were to die sooner, his designated beneficiaries would still get the payments due after his death, but that money wouldn’t have helped him meet living expenses while he was alive.
Another drawback to such an annuity is that the purchasing power of the level payments will decline with inflation.
An approach that avoids some of these disadvantages is a life annuity: You receive payments for as long as you live. How much you receive is based on the life expectancy of people your age, as determined from population data. For example, Social Security is in effect a nonlevel life annuity:
- You get payments until you die.
- How much you get depends on your age when you choose to start taking payments (as well as on your pre-retirement earnings).
- The amount of payments in a given year is determined by the consumer price index (CPI) (see Chapter 21, pages 888–893).
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There are many variations on life annuities, such as payments that increase with anticipated cost-of-living increases, or payments that last until both you and your life partner die (see Spotlight 22.4). We focus on a simple one-life annuity.
The insurance company that sells you the annuity makes money on your policy if you die younger than average and loses money if you die older than average. As in any kind of insurance, over a large number of people, the company expects gains to balance losses (actually, overbalance losses, to account for expenses and profit). This is a manifestation of the law of large numbers (discussed in Section 8.5 on page 378). The company invests the annuity funds, and its profits depend on the rate that its investments earn compared with the rate that it pays on the annuity.
A drawback of an annuity-certain is that you may outlive your annuity (and then have no income). A drawback of a life annuity is that you may die before receiving many (or even any) payments, thus in effect “losing” your retirement savings. On the other hand, if you live a long time, a life annuity may turn out to have been a better deal.
The amount of the payments for each annuity variation depends on prevailing and predicted rates of interest and (in the case of life annuities) on your age and sex. How much the annuitant (the purchaser of the annuity) receives per month depends on gender. Because women on average live longer than men, the monthly payment to a woman may be lower.
EXAMPLE 15 Life Income Annuity
Suppose that your 65-year-old father retires and purchases for $250,000 a life income annuity. According to the table from one particular insurance company, he would receive $6.3448 per month for every $1000, so his monthly income would be $1586. According to the Social Security Administration actuarial life table, his life expectancy at age 65 is about 17 years = 204 months. If he lived exactly that long, he would receive a total of .
However, simple algebra cannot be used to find the rate of interest that the annuity would need to earn to last that long. We use the RATE function in a spreadsheet (for more details, see Spotlight 21.3 (page 888); entering gives a monthly rate of 0.2636%, for an effective annual rate of .
Now let’s consider instead the case of your mother retiring now, also at age 65 and also with a $250,000 life income annuity; she would receive $5.9010 per month for every $1000, or $1475 per month. Her life expectancy would be about 19.72 years = 237 months. If she lived exactly that long, she would receive a total of . The rate of interest that her annuity would need to earn to last that long can be calculated from the amortization formula; using gives a monthly rate of 0.2997%, for an effective annual rate of 3.66%. The difference between this rate and the one for your father probably reflects the fact that the company uses life expectancies (which vary by region of the country) that differ from those for the nation as a whole.
Notice that a man and a woman who save the same amount receive different monthly incomes at retirement: The woman receives less per month but for longer—93% as much for 16% longer. Yet their living expenses are likely to be the same. That consideration has resulted in some companies offering “merged gender” rate schedules for annuity payments, so that the individual receives the same monthly payment regardless of gender.
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