Understanding the Mathematics of Personal Finance
LIFE ANNUITIES
The immediate in immediate annuity means that payments start immediately upon paying the premium. Often, the premium is accumulated by the purchaser over time. With a deferred annuity, the premium can be built up with a number of payments that grow with the tax on the interest deferred. This increases the growth of your payments as compared with putting the money into a savings bank. When payments begin, the exclusion ratio is lower than in the case of the immediate annuity, so that more of the payment gets taxed. The IRS is reclaiming some of the tax it deferred during the accumulation period.
Returning to the example above, I need a balance of $413,930 in my account on the day that I want the annuity to start making payments. I’ll assume that I built this balance over the course of 10 years with regular monthly payments. First, I’ll look at doing this in the savings account.
From the Deferred tab in the Ch11Fixedannuities. xls spreadsheet[30] or Table 11.2, I see that my monthly payment for the 10year accumulation period is $2,811.07. I calculated the taxes on the savings and reflected them back to the beginning of the accumulation period. I then took the savings tax PV number from the IAWC (remember you have to enter this yourself), reflected it back from the end of the 10year accumulation period to the beginning of the accumulation period, and added this to the total of the deferral period tax payment present values. The total in this example is $40,381. Summarizing, the present value of all the tax payments for 10 years of putting money into this savings account and then 20 years of pulling the money back
Table 11.2a Adding Deferred Principal Growth to the Table 11.1 Example

out again, at the beginning of this entire 30year “project” is $40,381. Now I want to do the same thing for the annuity.
In the case of an annuity, I don’t pay taxes on the interest accrued during the 10year accumulation period. However, the exclusion ratio is no longer the balance at the beginning of the payout period divided by the sum of the payouts. The numerator is replaced by the sum of the contributions during the accumulation period (the basis). In this example, these numbers are
^ . . 120 ($2,811.07)
Exclusion ratio = = 0.562.
240 ($2,500)
Table 11.2b Adding Deferred Principal Growth to the Table 11.1 Example

While the immediate annuity payments were taxed at a 31% rate, these deferred annuity payments are taxed at a 56% rate. When I reflected these payments back to the beginning of the accumulation period, however, I got a present value of $31,812. The value of deferring taxes is clear. The comments above about inflation also hold here.
A life annuity starts out the same way that an annuity with period certain starts out. You pay a premium—either immediately or accumulated over a period of time (deferred)—and then you start getting regular checks in the mail. There is no pre
determined cutoff date for these checks. They keep coming as long as you’re alive   hence the name “life” annuity.
The structure of life annuities has many variations. Perhaps, the most common is an annuity that funds a married couple. If I buy a life annuity for myself, the payments stop when I die. If I’m one of a couple, I (we) could buy an annuity that keeps funding until both of us are dead. The annuity could be structured so that the payment amounts are constant throughout, or drop by some amount when the first of us dies.
Let’s look at an immediate life annuity for one person—fixed payments starting on day 1 and continuing monthly until that person’s death. The first key question that must be answered before doing any calculations is just how much premium is needed (at a given interest rate) to fund the payments. Unlike the annuity with period certain, we don’t know how many payments there will be.
I downloaded Figure 11.1 from the IRS website in February of 2009 (IRS[31] ). According to the figure’s title, it’s an actuarial table for one life. Looking at the table, you can see ages for males and females and a column labeled “multiples.” Multiples is the number of years that the IRS estimates this annuity will be paying. The name multiples relates to the number that multiplies the annual payout, which in turn is the number in the denominator of the exclusion ratio formula.
How did the IRS come up with this table? Look at Table 10.1, the 2004 Life Table for all men in the United States. Column E in the Life Table is the expected number of years that a man of a given age is most likely to (is “expected to”) live. In other words, column E in the Life Table is the estimate of the number of years that a man of a given age buying an immediate annuity will be collecting payments from that annuity, that is, the IRS multiple.
Looking through these tables, you will see that the 2009 numbers are more optimistic (people are living longer) than the 2004 numbers, and that women continue to outlive men. In the examples to follow, I’m going to stick with my 2004 Life Table numbers just for consistency with the rest of the book. The multiple will be one of the input variables in my spreadsheet, so you can use whichever numbers you wish.
At this point, we can start appreciating why annuities are closely tied to insurance companies. Insurance companies are inherently in the business of generating probability (actuarial) tables for the policies they write, and they understand the importance of averaging over a large number of people and the variabilities involved. Pricing life annuity contracts involves the same kinds of calculations as does pricing a life insurance contract—you just think in terms of how long people will continue to live instead of when they are going to die.
Calculating the relationship between the amount of the payments, the number of payments, and the premium is no different from in the previous examples. The number of monthly payments is 12 times the multiplier. You need to know the person’s age when he or she buys a life annuity—you don’t for a period certain annuity.
If the purchaser of a life annuity dies significantly earlier than the Life Tables predicted he or she would die, the balance in the account is forfeited. This is, in a sense, unfair to the beneficiaries of the purchaser but is a necessary attribute of the
Actuarial tables
Table I (one life) applies to all ages. Tables II—IV apply to males ages 35 to 90 and females age 40 to 95. Table I—Ordinary life annuities—one life—expected return multiples
Age Age Age

Figure 11.1 One person actuarial table from the IRS website (from IRS Publication 939, General Rule for Pensions and Annuities, http://www. irs. gov/publications/p939/ar02.html, accessed February 2009).
annuity if the insurance company is going to have the money to continue making payments to the purchasers who outlive their life expectancies. This issue can be mitigated by adding a periodcertain feature to the life annuity.
With a periodcertain life annuity, a certain number of payments, for example, 10 years, is guaranteed. If the purchaser dies earlier than his or her life expectancy, his or her beneficiaries continue to receive his or her payments until the end of the specified period. What’s happening here is that the insurance company is selling both a life annuity and a decreasing term life insurance policy bundled together. The life insurance policy is written to fund the payments for the period certain of the annuity, if needed. Since life insurance policies cost money, a life annuity with a periodcertain provision must cost more than a simple or “pure” life annuity. Alternatively, for the same cost, a life annuity with a period certain provision will make smaller payments than will a pure life annuity.
Age: 
60 
Multiplier: 
24.2 
Payment: 
$150 
Rate: 
8.9% 
Tax rate: 
25% 
Nominal number of monthly payments 
290.4 
Calculated principal: 
$17,991.52 
Exclusion ratio: 
0.413 
Expected age at death: 
84.2 
Tax fraction before expected death age: 
0.587 
Monthly taxes before expected death age: 
$22.01 
Monthly taxes after expected death age: 
$37.50 
In terms of taxation, the IRS isn’t about to let the purchasers of a life annuity who live longer than the IRS expected them to live not pay taxes on their “extra” income. Starting out, the calculation of the exclusion ratio is the same as for periodcertain annuities. Once enough payments have been made so that the full principal has been excluded, the exclusion ratio goes to 0. From then on, all payments are fully taxed as ordinary income. This switchover occurs at the expected death age.
The Life tab in the spreadsheet Ch11FixedAnnuities. xls is much simpler than the other tabs in this spreadsheet. That’s because there’s no need for a balance sheet or for present value calculations. The nominal number of payments is known once you enter the age and the multiplier (e in the Life Tables). This is the number of annuity payments that will be made if you die exactly on your statistical expected date of death. Enter the monthly payment amount and the interest rate that your balance will grow at and the principal (premium) for the annuity is known, as is the exclusion ratio. Knowing your tax rate (bracket), we then calculate the monthly tax. If you outlive your expected date, the full monthly payment is taxable as ordinary income—at your tax rate (Table 11.3).