# Income Annuity 101

How are different types of annuities priced? It’s not as hard as you might think, as the basic recipe requires just three ingredients:

1) **Mortality rates** (which vary by age and gender) impact how long payments will be made. For longer projected payout periods, payout rates must be lower.

2) **Interest rates** impact the returns the annuity provider can earn on the underlying annuitized assets. Higher interest rates imply higher payout rates.

3) **Overhead costs** relate to extra charges an annuity provider seeks for business expenses and to manage risks related to the accuracy of their future mortality and interest rate predictions.

The illustration below provides a simple example to illustrate the basic pricing dynamics for an “actuarially fair” annuity. This is an annuity without any overhead costs and it assumes the underlying projections for mortality and returns are correct. For this example, we consider a 65-year old male who is offered $10,000 of income per year as long as he lives. It is a life-annuity income annuity, so payments stop at death.

How much is this offer worth?

*Table 1: Calculating the Cost of a $10,000 Income Stream for a 65-Year Old Male (Life Only)*

Discount Rate: |
2.50% |
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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 | 98.9% | $9,651 |

67 | $10,000 | 95.2% | $9,518 | 97.7% | $9,302 |

68 | $10,000 | 92.9% | $9,286 | 96.5% | $8,957 |

69 | $10,000 | 90.6% | $9,060 | 95.1% | $8,612 |

70 | $10,000 | 88.4% | $8,839 | 93.6% | $8,270 |

71 | $10,000 | 86.2% | $8,623 | 91.9% | $7,928 |

72 | $10,000 | 84.1% | $8,413 | 90.2% | $7,585 |

73 | $10,000 | 82.1% | $8,207 | 88.2% | $7,240 |

74 | $10,000 | 80.1% | $8,007 | 86.1% | $6,893 |

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: |
$148,492 |
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Annuity Payout Rate: |
6.73% |
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*Survival Probabilities are calculated from the IRS Mortality Tables for pension plan valuations. |

Finding how much the offer is worth requires inputs for investment returns, which could be earned on the principal amount financing these payments and the survival probabilities to each subsequent age. However, we want to create a more simplified example, which can illustrate the concept without requiring an entire yield curve of different interest rates for each maturity date. In that event, let’s consider that the underlying principal can earn a fixed return of 2.5%, which is in the neighborhood of current rates for U.S. Treasury bonds with between 10- and 20-year maturities. Our example calculation provides a simplification, because with a typical upward-sloping yield curve, payments coming sooner would earn less interest and later payments would grow at a faster rate.

Additionally, many annuity providers will likely seek higher returns than Treasury bonds offer by including high-quality corporate bonds with higher yields to compensate for slightly higher default risk. But this simplification of one fixed interest rate captures the concept well enough. One other simplification is that I assume the full year’s income arrives at the start of each year, rather than income arriving on a monthly basis, as is the practice with most income annuities.

The 2.5% return acts as a discount rate to reduce the value needing to be set aside today for the future $10,000 payments. The table indicates that at age 75, the discount factor is 78.1%. The interpretation is: if you put $7,812 in the bank today, and it grows at an annual 2.5% compounding interest rate for the next 10 years, you can expect these assets to grow in value to $10,000 by your 75^{th} birthday. And so this is the amount required today to provide the payment at age 75.

The process is the same for the $10,000 payment provided at each age. The later the payments are received, the more time they have to compound and grow, requiring less to be set aside today to fund those payments.

The next part of the table is what differentiates an income annuity from building a bond ladder with maturing bonds providing the desired income. For a bond portfolio, the total cost is the sum of the “discounted value of income” column, which is $257,303. Annuity owners receive an additional discount because the survival probabilities to each subsequent age indicate whether payment will be made. For any particular individual, you are either alive or not. But for a large pool of individuals representing the customer base of the annuity provider, we can rely on the law of large numbers to suggest what percentage of customers will remain alive at each subsequent age.

Our data from the IRS suggests that a 65-year old male has an 83.7% chance of making it to 75. An annuity provider can expect 83.7% of their 65-year old male customers to be alive and receiving income at 75. They don’t know which of their customers will be receiving payments, but they can rely on percentages to tell them how many. Actuaries have estimated mortality rates to the best of their ability for their customer base (note that annuity providers may have more refined mortality data about their customer base than what I am using here). So for each customer, there is an 83.7% chance that the payment must be made at 75.

When we multiply this percentage by the discounted value of the funds needed to provide the $10,000 payment at 75, we see that the annuity company plans $6,539 for the cost of providing this payment at age 75. This is the survival-probability weighted discount factor, and the same process is followed for each age. For another example, a $10,000 payment at age 100 requires $4,214 to be set aside today with a 2.5% interest rate. Given that there is a 1% chance of the 65-year old male reaching 100, the annuity provider further multiplies by the survival probability. In the end, the expected cost of a $10,000 life-contingent payment is only $43. A 65-year old male need only pay $43 today for a guarantee that they will receive $10,000 at age 100, if they are still alive.

When we add survival-weighted costs by age, we see that the total expected cost to provide a $10,000 annual income to a 65-year old male is $148,492. If that dollar amount represented the premium charged, then the payout rate on the annuity would be the $10,000 income it provides divided by this cost. The payout rate is 6.73%. Note, as well, that this is 42% less than the cost of the bond ladder, with the difference being that the bond ladder provides legacy if retirement is less than 35 years, but the bond ladder does not provide additional longevity protection beyond 35 years (this is hypothetical as bonds with maturities beyond 30 years are uncommon).

The annuity provider cannot offer the annuity for the actuarially-fair price and hope to remain in business. Having the provider stay in business is a natural desire for the purchaser. The provider will need to charge more to cover business expenses and profit motives. There is also the issue of adverse selection to consider, as annuity buyers tend to live longer than the average American. Those with terminal illnesses will probably not be in the market to buy income annuities. The Society of Actuaries and annuity providers attempt to correct for this matter by adjusting their mortality numbers, but the experiences of different companies with adverse selection may be broad, and the degree of adverse selection could be misestimated.

Additional financial buffers are needed against misestimating risk as annuity pricing requires long-term projections for interest rates and mortality improvements. If interest rates are lower – or if health improvements accelerate faster and customers live even longer than projected – the cost of providing the guaranteed income will be higher than shown in the table. These costs can be partially offset for an annuity provider who also sells life insurance – systematic increases in longevity mean more claims on annuities but fewer claims on life insurance. As part of risk management, the provider needs an additional buffer of assets. This represents the overhead charge on the annuity and can be viewed as a higher price than what is calculated as the actuarially fair value.

Nevertheless, the 6.73% payout rate was actually quite close to what **www.immediateannuities.com** reported as the best payout rate (6.79%) offered in the market for this type of annuity in late-July 2015. Part of the reason for this similarity may be a lag in updating mortality improvements for annuity providers.

The IRS data I used is based on the Society of Actuaries RP2014 Mortality Tables for retirement plan participants. The Society of Actuaries provided these updates to their previous 2000 Mortality Tables, which show a two-year increase in the life expectancy for men at age 65. In a May 2015 **column at LifeHealthPRO**, Tom Hegna suggests that the revised mortality tables have yet to be incorporated into the calculations for annuity pricing, and when they eventually are, we can expect a reduction in annuity payout rates. As it stands at the time of this writing, with my assumptions of a 2.5% yield to guide the annuity investments and the IRS mortality tables, the overhead charges on income annuities appear to be close to $0, at least for the best rates available in the market.

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

[…] For instance, a 55-year old might purchase a deferred income annuity which will begin income at 65. We have already determined that the cost of a life-only income annuity at 65 is $148,492. If a 55-year male wanted to provide […]