Woolhouse's Formula for Life Annuity Approximation

Deriving the Approximation

Woolhouse's formula approximates an mthly annuity using the annual annuity plus correction terms based on the Euler-Maclaurin summation formula. The two-term approximation is a-ddot_x^(m) approximately equals a-ddot_x minus (m-1)/(2m). The three-term approximation adds a correction for the slope: a-ddot_x^(m) approximately equals a-ddot_x minus (m-1)/(2m) minus (m^2-1)/(12m^2) * (delta + mu_x). This provides a much better approximation than UDD-based formulas, especially at older ages where mortality is changing rapidly.

Applications and Exam Tips

Woolhouse's formula is particularly important for computing monthly pension annuities from annual life table values. For temporary annuities: a-ddot_{x:n}^(m) approximately equals a-ddot_{x:n} minus (m-1)/(2m)*(1 minus n_E_x) minus (m^2-1)/(12m^2)*((delta + mu_x) minus n_E_x*(delta + mu_{x+n})). The formula requires the force of mortality mu_x, which can be approximated as mu_x approximately equals 0.5*(ln(p_{x-1}) + ln(p_x)) or simply minus ln(p_x). Exam LTAM frequently tests both two-term and three-term Woolhouse approximations.