Why Woolhouse's Formula Matters

In practice, annuity payments are made monthly, quarterly, or at other frequencies, but life tables provide data at annual intervals. Computing an mthly annuity value directly would require mthly mortality rates, which are not typically available. Woolhouse's formula bridges this gap by approximating the mthly annuity value using annual annuity values plus correction terms. It is based on the Euler-Maclaurin summation formula, a mathematical technique for approximating sums using integrals and derivative corrections.

This formula is particularly important for pension actuaries, who must compute monthly pension annuity values from annual life tables. The accuracy of the approximation directly affects pension liability calculations, making Woolhouse's formula a practical tool in daily actuarial work.

The Two-Term Woolhouse Approximation

The simplest version of Woolhouse's formula uses two terms: a-ddot_x^(m) approximately equals a-ddot_x - (m-1)/(2m). This approximation subtracts a correction that depends only on the payment frequency m. For monthly payments (m = 12), the correction is (12-1)/(2*12) = 11/24 = 0.4583. For quarterly payments (m = 4), the correction is 3/8 = 0.375.

The two-term formula is simple and easy to compute, but it ignores the effect of mortality on the mthly annuity value. It works reasonably well at younger ages where mortality rates are low, but its accuracy deteriorates at older ages where the force of mortality is significant. For Exam LTAM problems that specify the two-term approximation, simply subtract (m-1)/(2m) from the annual annuity-due value.

The Three-Term Woolhouse Approximation

The three-term version adds a mortality-dependent correction: a-ddot_x^(m) approximately equals a-ddot_x - (m-1)/(2m) - (m^2-1)/(12m^2) * (delta + mu_x). Here delta is the force of interest (delta = ln(1+i)) and mu_x is the force of mortality at age x. For monthly payments, the third-term coefficient is (144-1)/(12*144) = 143/1728 = 0.08275.

The third term captures the interaction between payment frequency and the rate at which the annuitant population is shrinking (through both discounting and mortality). At older ages where mu_x is large, this correction is substantial and significantly improves the approximation. The three-term formula is much more accurate than the two-term version and is the preferred approach in practice.

Approximating the Force of Mortality

The three-term Woolhouse formula requires mu_x, the force of mortality. Since life tables provide q_x rather than mu_x, an approximation is needed. The simplest approach is mu_x approximately equals -ln(p_x) = -ln(1 - q_x), which assumes constant force within the year of age. A more refined approximation averages adjacent values: mu_x approximately equals 0.5 * (-ln(p_{x-1}) - ln(p_x)), which smooths the estimate by using information from both neighboring years.

For example, if q_64 = 0.012 and q_65 = 0.014, then -ln(p_64) = -ln(0.988) = 0.01207 and -ln(p_65) = -ln(0.986) = 0.01410. The averaged estimate is mu_65 approximately equals 0.5 * (0.01207 + 0.01410) = 0.01309. Either approximation method is acceptable on Exam LTAM; the problem will specify which to use or the difference will be negligible.

Temporary Annuity Formula

Woolhouse's formula extends to temporary (term-certain) life annuities. The three-term approximation for a temporary annuity-due is: a-ddot_{x:n}^(m) approximately equals a-ddot_{x:n} - (m-1)/(2m) * (1 - n_E_x) - (m^2-1)/(12m^2) * ((delta + mu_x) - n_E_x * (delta + mu_{x+n})). Here n_E_x = v^n * n_p_x is the pure endowment factor, and mu_{x+n} is the force of mortality at the end of the annuity term.

The temporary annuity formula includes boundary adjustments because the annuity payments stop at duration n. The terms involving n_E_x correct for the fact that the annuity does not continue beyond age x+n. When n approaches infinity, n_E_x approaches 0 and the temporary formula reduces to the whole life formula.

Worked Example

Suppose a-ddot_65 = 12.5, the interest rate is 5% (so delta = ln(1.05) = 0.04879), and mu_65 = 0.0141. To approximate the monthly annuity-due a-ddot_65^(12), apply the three-term formula: a-ddot_65^(12) approximately equals 12.5 - 11/24 - (143/1728) * (0.04879 + 0.0141) = 12.5 - 0.4583 - 0.08275 * 0.06289 = 12.5 - 0.4583 - 0.00520 = 12.0365.

Compare this to the two-term result: 12.5 - 0.4583 = 12.0417. The three-term correction reduces the estimate by about 0.005, which may seem small but can be significant when multiplied by a large benefit amount. For a pension of 50,000 per year, the difference is 50,000 * 0.005 = 250 in present value.

Common Pitfalls

A frequent mistake is forgetting the sign conventions in the formula. All correction terms are subtracted from the annual annuity value. Another common error is using q_x instead of mu_x in the third term. The force of mortality mu_x and the annual mortality rate q_x are related but not equal; mu_x = -ln(1 - q_x). At low mortality rates the difference is small, but at older ages the distinction matters.

Students sometimes apply the whole-life Woolhouse formula to a temporary annuity problem. The temporary annuity version includes additional terms involving n_E_x, and omitting these produces an incorrect result. Always check whether the annuity is whole life or temporary before selecting the appropriate formula.

Study Tips

Exam LTAM frequently tests both two-term and three-term Woolhouse approximations. Memorize both versions and know when each is appropriate (the problem will usually specify). Practice computing mu_x from life table values and substituting into the three-term formula. Be prepared to apply the temporary annuity version, including proper computation of n_E_x and mu_{x+n}. Work through numerical examples to build familiarity with the magnitude of each correction term, and verify that your results are reasonable by checking that the mthly annuity is slightly less than the annual annuity.

Woolhouse's Formula for Life Annuity Approximation