Wade Pfau, Ph.D., CFA, RICP®

# Annuity Pricing 102

In today’s post, I continue the discussion on annuity pricing which I began recently.

In today’s scenario, we’ll see what happens to the price of an annuity if:

1. We guarantee income will be provided for at least 10 years, regardless of whether the annuitant lives.
2. We provide a cash refund if the annuitant dies before at least receiving their full principal payment.
3. We include a cost-of-living adjustment for annuity payments.

Pricing an Income Annuity with Period-Certain Payments

The first two types of provision are popular if you worry about the “hit by a bus” factor.

First, practically speaking, to provide 10 years of certain income, the “survival probabilities” become 100% for the first 10 years of payments (see Table 2). The annuitant does not become immortal, but from the perspective of the actuaries who are pricing this annuity, payments must be made regardless of survival status.

Mathematically, there is a 100% chance the first 10 payments will be made. The 10-year period certainty provision raises our hypothetical annuity’s cost from \$148,492 to \$153,761. The increased cost lowers the payout rate (which is the initial \$10,000 income divided by the cost) from 6.73% to 6.50%.

Switching from a life-only annuity to one that guarantees payments for at least 10 years lowers the payout rate by 0.23%.

Table 2 Calculating the Cost of a \$10,000 Income Stream for a 65-Year Old Male (10-Year Period Certain)

 Discount Rate: 2.50% Age Income Discount Factor Discounted Value of Income Survival Probabilities* Survival-Weighted Discounted Value 65 \$10,000 100.0% \$10,000 100.0% \$10,000 66 \$10,000 97.6% \$9,756 100.0% \$9,756 67 \$10,000 95.2% \$9,518 100.0% \$9,518 68 \$10,000 92.9% \$9,286 100.0% \$9,286 69 \$10,000 90.6% \$9,060 100.0% \$9,060 70 \$10,000 88.4% \$8,839 100.0% \$8,839 71 \$10,000 86.2% \$8,623 100.0% \$8,623 72 \$10,000 84.1% \$8,413 100.0% \$8,413 73 \$10,000 82.1% \$8,207 100.0% \$8,207 74 \$10,000 80.1% \$8,007 100.0% \$8,007 75 \$10,000 78.1% \$7,812 83.7% \$6,539 76 \$10,000 76.2% \$7,621 81.1% \$6,183 77 \$10,000 74.4% \$7,436 78.3% \$5,820 78 \$10,000 72.5% \$7,254 75.1% \$5,451 79 \$10,000 70.8% \$7,077 71.7% \$5,077 80 \$10,000 69.0% \$6,905 68.0% \$4,698 … … … … … … 95 \$10,000 47.7% \$4,767 6.4% \$305 96 \$10,000 46.5% \$4,651 4.7% \$217 97 \$10,000 45.4% \$4,538 3.3% \$151 98 \$10,000 44.3% \$4,427 2.3% \$102 99 \$10,000 43.2% \$4,319 1.6% \$67 100 \$10,000 42.1% \$4,214 1.0% \$43 101 \$10,000 41.1% \$4,111 0.7% \$27 102 \$10,000 40.1% \$4,011 0.4% \$17 103 \$10,000 39.1% \$3,913 0.3% \$10 104 \$10,000 38.2% \$3,817 0.2% \$6 Present Value of the Annuity = Sum of Survival-Weighted Discounted Values: \$153,761 Annuity Payout Rate: 6.50% *Survival Probabilities are calculated from the IRS Mortality Tables for pension plan valuations.

An income annuity with a cash refund could be priced in a similar way, though the math becomes more complex than what can be shown in simple tables. For each age, we consider the probability it represents the age of death, and then also include the remaining refund amount from the initial principal to be provided to beneficiaries. The refund amount is discounted by the factor for that age to know the expected cost today to provide the refund.

I estimate with a more complex table (not shown) that a cash refund provision on the original life-only annuity would raise the cost from \$148,492 to \$172,296. The cost for the cash refund rider is \$23,804, while the increased cost lowers the payout rate from 6.73% to 5.8%.

The tradeoff is that while you could receive the highest income while alive with a life-only income annuity, you can alleviate fears about not living long enough to fully collect on annuitized assets by adding provisions which will create more opportunities for your beneficiaries to receive something in the event of an early death. Academics who study income annuities generally suggest a life-only income to fully maximize the income-producing power, with legacy goals covered through other means, but these sorts of provisions are quite popular in practice.

Pricing for an Income Annuity with Cost-of-Living Increases

Another income annuity option we can price is an annuity with a cost-of-living adjustment (COLA). Using a COLA requires a minor adjustment to our table. If the COLA is 2%, then rather than having a fixed income of \$10,000 each year, the annuity instead provides an income which grows by 2% each year. Naturally, the ability to receive more income over time raises the cost.

In this case, the cost for a 2% COLA increases the life-only annuity price from \$148,492 to \$180,536. (calculations in Table 3). While we must remember that income increases over time with this provision, the payout rate for the initial income amount falls from 6.73% to 5.54%. With a lower payout rate, but 2% annual income growth, the income payment in year 10 would finally be larger than the initial income payment for the no-COLA version if the same premium amount was applied to each contract. The tradeoff is whether to accept a lower initial income, but gain the ability to have income grow over time.

Table 3: Calculating the Cost of a \$10,000 Income Stream for a 65-Year Old Male (Life Only, with 2% COLA)

 Discount Rate: 2.50% Age Income Discount Factor Discounted Value of Income Survival Probabilities* Survival-Weighted Discounted Value 65 \$10,000 100.0% \$10,000 100.0% \$10,000 66 \$10,200 97.6% \$9,951 98.9% \$9,844 67 \$10,404 95.2% \$9,903 97.7% \$9,678 68 \$10,612 92.9% \$9,854 96.5% \$9,505 69 \$10,824 90.6% \$9,806 95.1% \$9,322 70 \$11,041 88.4% \$9,758 93.6% \$9,131 71 \$11,262 86.2% \$9,711 91.9% \$8,928 72 \$11,487 84.1% \$9,663 90.2% \$8,713 73 \$11,717 82.1% \$9,616 88.2% \$8,483 74 \$11,951 80.1% \$9,569 86.1% \$8,238 75 \$12,190 78.1% \$9,523 83.7% \$7,971 76 \$12,434 76.2% \$9,476 81.1% \$7,688 77 \$12,682 74.4% \$9,430 78.3% \$7,381 78 \$12,936 72.5% \$9,384 75.1% \$7,052 79 \$13,195 70.8% \$9,338 71.7% \$6,699 80 \$13,459 69.0% \$9,293 68.0% \$6,322 … … … … … … 95 \$18,114 47.7% \$8,636 6.4% \$552 96 \$18,476 46.5% \$8,593 4.7% \$400 97 \$18,845 45.4% \$8,551 3.3% \$284 98 \$19,222 44.3% \$8,510 2.3% \$195 99 \$19,607 43.2% \$8,468 1.6% \$132 100 \$19,999 42.1% \$8,427 1.0% \$87 101 \$20,399 41.1% \$8,386 0.7% \$55 102 \$20,807 40.1% \$8,345 0.4% \$35 103 \$21,223 39.1% \$8,304 0.3% \$21 104 \$21,647 38.2% \$8,264 0.2% \$13 Present Value of the Annuity = Sum of Survival-Weighted Discounted Values: \$180,536 Annuity Payout Rate: 5.54% *Survival Probabilities are calculated from the IRS Mortality Tables for pension plan valuations.

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