# How does an annuity work?

An annuity is an insurance contract which a guaranteed income as long as the buyer is alive. While we buy insurance to protect against a terrible event, like a fire burning down your house, or for premature death, longevity, while considered a blessing in many cultures, can be a terrible event, if it means running out of money or income! True annuities last for life, paying out as long as you are alive, and in general, pay a much higher return than a bank deposit or bond. However, you don’t get to keep any principal value of the annuity – once you pass away, the money used to purchase the annuity is forfeit. CPF Life, which we have described elsewhere, is an example of such an annuity.

Longevity can be terrible,if it means

running outof money or income

So, how do annuities work? And how can an annuity promise to pay however long you may live, which means that it needs a really, really high rate of return? A little fable may help here:

Imagine a 99 year old grandmother who plays mahjong with her 3 best friends every week. All four of them are exactly 99 years old, quite healthy, and have been retired for 34 years. Recently, the game has become a little more tiresome, so this grandmother decided to try something else for fun. The last time they played, she proposed that they each take $100 and put it on the kitchen table. “Whoever survives to 31st December gets to split the $400” she said. “If you don’t make it, you forfeit the money”.

Everyone thought it was an interesting idea, and agreed, but decided that it was too risky to keep $400 on the kitchen table (the children may come to visit, and the grandkids may just pilfer it), so they put it in a bank which was paid 4% interest on 1-year fixed deposits.

So what happens next year? There is a roughly 25% chance that any one of them will pass away before 31st December, and hence the odds imply that on average, the three survivors will not only make it to the exclusive centenarian club, but also get to split the $416 dollars (remember the bank pays 4% in interest, or $16) between themselves.

This in turn, means that each survivor will get $138.67, which is a 38.67% return on their money. This return is from 4% of bank interest, and an incredible 34.67% return from longevity! While the heirs of the non-survivor may rue the loss of $100 from their inheritance, the survivors get a superior investment return.

This little fable shows the power of life annuities – by pooling the longevities of everyone together, the annuity fund is able to generate very high returns for the survivors, making it possible to sustain payments to the survivors for the rest of their lives. No other financial product can guarantee such a high rate of return.

So far, we have only showed you how an annuity works on a year-by-year basis. But real-life annuities make payments for the rest of your life, not just for one year. So how does this work out? We start by looking at the life table once again. Here is the life table for Singapore across genders, where we are only showing the statistics for every 5 years in age to save space:

Age x (Years) | Prob of dying between age x and x+1 | Number of survivors at age x | Chance of survival to this age |
---|---|---|---|

q(x) | l(x) | l(x) / l(65) | |

65 | 0.00841 | 91,508 | 100% |

70 | 0.01387 | 86,888 | 95.0% |

75 | 0.02391 | 79,605 | 87.0% |

80 | 0.04269 | 68234 | 74.6% |

85 | 0.07280 | 51,883 | 56.7% |

90 | 0.12151 | 32,255 | 35.2% |

95 | 0.19462 | 14,415 | 15.8% |

100 | 0.33333 | 3,807 | 4.16% |

The life table shows the chance of dying, and also the number of survivors for every single age. To work out the chance of surviving to any particular age, say age 90, we just divide the number of survivors at age 90 by the number at age 65, and this gives us the chance, which is 35.2%. Now suppose 1,000 people put $1 on the kitchen table at the age of 65, so that there is a total of $1,000. At the age of 90, this $1,000 only needs to be split amongst 352 people who survive till then (not counting any interest), which gives each of the survivors $2.84. If we add back the interest over the 25 years, this is going to be an even larger amount.

Let’s work out the numbers in the next table, where we assume that a bank will pay 4% interest every year. For a large group of people, how much is needed to be set aside today, in order to pay them $1,000 every year in the future as long as they live?

Age | Chance of survival to this age | Longevity Return | Total Interest Earned | Total Return (Longevity & Interest) | Amount at 65 needed for $1,000 a month |
---|---|---|---|---|---|

65 | 100% | 0.0% | 0.0% | 0.0% | $12,000 |

70 | 95.0% | 5.3% | 21.7% | 28.1% | $9,365 |

75 | 87.0% | 15.0% | 48.0% | 70.2% | $7,052 |

80 | 74.6% | 34.1% | 80.1% | 141.5% | $4,968 |

85 | 56.7% | 76.4% | 119.1% | 286.5% | $3,105 |

90 | 35.2% | 183.7% | 166.6% | 656.3% | $1,587 |

95 | 15.8% | 534.8% | 224.3% | 1959% | $583 |

100 | 4.16% | 2303% | 294.6% | 9385% | $127 |

Accounting for the other years not shown, an annuity premium of $168,646 is needed at age 65 to provide an income of $1,000 a month forever. Of this $168,646, only $16,841, or about 10% of the total, is required to provide for the payments after the age of 85! In comparison, $12,000 is needed to provide for the annuity payments in the immediate next year, which is why it may sometimes be cheaper to buy a deferred annuity. Of course, a commercial annuity from an insurer will cost more, as they have expenses, and adverse selection (people who buy annuities tend to live longer than average) to account for.

What we see is that the returns to longevity really only kick in after the average life expectancy of 85 years. From that point on, it overtakes the interest earned as the bigger contributor to returns. It also shows how cheap it is for an insurer to provide payments for the very long-lived once the longevity returns kick in.

No other financial product can

guarantee such a high rate of return

Finally, it is clear that putting some money into an annuity gives you the chance to enjoy a very high rate of return (depending on longevity, of course). Even if the idea of investing in “longevity futures” is not appealing, the being able to diversify risks and sources of stable returns, especially in retirement should for some investment in annuities.

Other perspectives on annuities which could be of interest can be found at About Annuity Benefits, Rate of Return by Investment Moats.

## 3 thoughts on “How does an annuity work?”