John De Ravin is a member of the Institute’s Retirement Incomes Working Group and in this article he shares a suggestion made by the Convenor of that Working Group, Anthony Asher.

Many actuaries routinely use annuity functions during their working lives. Annuity values are used in many different contexts, which can be anything from calculating annual mortgage repayments to valuing pension liabilities. Of course we all know the formulae for life annuities and term certain annuities, but detailed evaluation calculations involve lengthy summations, commutation functions or (at the very least) in the case of term certain annuities, a (1+i)n calculation.

Wouldn’t it be great if (for the times when we wanted an immediate but approximate figure), an approximation were available that we could quickly and easily use to produce a rough estimate of the annuity value?

Enter Anthony Asher. He has proposed the following approximation for a life annuity:

1/ā(x) = 1/e0(x) + 2/3 * i

where:

ā(x)      is the life annuity value (payable continuously) at age x,
e0(x)    is the remaining expectation of life at age x and
i           is the interest rate used in the calculation of the annuity.

Of course it is true that this approximation requires an estimate of the remaining life expectancy but often an actuary will be able to make a reasonable estimate from general experience. As a measure of the accuracy of the approximation, let’s say that the actuary knows that the male life expectancy at age 65 according to the Australian Life Tables 2010-2012 is 19.22 years. The following are the actual continuous annuity values together with the results of the Asher Approximation, for interest rates varying from zero to 10% per annum:

 i ā(x) Asher Approx. % error 0% 19.22 19.22 1% 17.20 17.04 -1.0 2% 15.50 15.30 -1.3 3% 14.06 13.88 -1.2 4% 12.82 12.71 -0.9 5% 11.76 11.71 -0.4 6% 10.84 10.87 0.2 7% 10.04 10.13 0.9 8% 9.34 9.49 1.7 9% 8.72 8.93 2.4 10% 8.17 8.42 3.1

As may be seen, the approximation is quite accurate over a reasonable range of interest rates. Separate calculations suggest however that the approximation may be less accurate for much older lives, for example lives aged over 80 years.

As an example of the Asher Approximation in use, suppose you are at a dinner party and the conversation turns to investment in a retirement context. Your hostess, a widow, tells you that now that she is 65 and retired, she is considering switching her asset allocation from “balanced” (which her super fund thinks will earn long run average returns of CPI plus 4.5%) to “conservative balanced” (which the fund thinks will earn returns of CPI plus 3%). She knows that the returns will be more stable with the more conservative asset allocation but wonders what reduction in her constant real expenditure she should plan for as a result of the lower investment returns. She has no dependants and no desire to leave any legacy. She turns to you as the actuary and you are in the spotlight: all eyes are on you, awaiting whatever pearls of wisdom you can offer.

“Well, I’d say that you should plan for a reduction in lifestyle of about 14%”, you say. Your host is gobsmacked, as are all the other dinner party guests. How did you do that calculation so quickly and without reference to a calculator or tables? From now on, in their minds, you are a guru (if you weren’t already). You mentally promise yourself you will buy Anthony Asher a drink, the next time you see him.

But how did you do it? Well, constant real sustainable expenditure is 1/ā times financial assets. You know that the female life expectancy at age 65 is about 22 according to the Australian Life Tables, but allowing for future improvements in mortality and your hostess’s apparent good health and socio-economic status, you make a reasonable judgment that her life expectancy is of the order of 25 years. So by the Asher Approximation, the inverse of the annuity value at 4.5% real interest is 0.04 (inverse of life expectancy) plus 0.03 (two thirds of the real interest rate). If the real interest rate fell to 3%, then the inverse of the life annuity would be 0.04 plus 0.02. So her expected real maintainable expenditure falls by one seventh as a consequence of the shift to a lower proportion of growth assets.

The Asher Approximation works for term certain annuities as well, though in that case, my trial calculations using interest rates from 0% to 10% and terms from 5 to 25 years suggest that ideally the constant to be applied to the interest rate in the approximation formula might be slightly lower than the 2/3 factor used above in the context of life annuities. Somewhere between 0.55 and 0.60 seems preferable; for the sake of convenience, you might choose to use 0.60 (three fifths). Thus the approximation for term certain annuities is:

1/ā(N) = 1/N + 3/5 * i

where N is the term of the annuity.

As an example of the application of this amended version of the Asher Approximation, suppose you are at the bank, discussing the terms of a mortgage loan with your bank manager. You would like to purchase an apartment in which to live, and you are planning to borrow $500,000 at 5% over a 25-year term with principal and interest repayments, and the bank manager offers to inform you of the annual repayments. “OK, but I think they will be about$35,000 a year”, you say. The bank manager continues to enter all the data into her PC and after a minute she looks up at you. “How much did you say?” she asks you. “About $35,000 a year”, you reply. “Actually only$34,618”, he says. But you can tell from the awed look on her face that she now knows she will never put one over YOU. How did you do it? You just added 1/N (ie 0.04) to three fifths of the interest rate (0.03) to calculate that your annual repayments would be about 7% of the principal.

(By the way, if you are actually paying 5% on your home mortgage, or anything much over 4%, you really should read strategy 28 of my book Slow and Steady – 100 wealth building strategies for all ages, see www.slowandsteadybook.com.au. You need to get better terms on your mortgage!)

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##### Evaristo Diz says

23 August 2019

excellent aproximation.I think there is a lot of research to do in the actuarial calculations simplications.

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