Derive and apply two-term and three-term Woolhouse approximations for calculating expected present values of state-dependent cash flows.

Mortality tables are tabulated annually, but real annuities pay monthly. Woolhouse bridges the gap with a correction that costs you one extra mortality lookup and earns you four-decimal accuracy.

Standard life tables give integer-age survival probabilities. A life annuity that pays $1/m$ at the start of every $1/m$ period (monthly: $m=12$) has EPV

$${\ddot{a}}_x^{(m)}=\frac{1}{m}\sum _{k=0}^{\infty }{v}^{k/m}{}_{k/m}{p}_x$$

Evaluating this directly requires fractional-age survival probabilities. Woolhouse converts the fine-grid sum into the coarse annual annuity ${\ddot{a}}_x$ plus a small correction, using the Euler-Maclaurin summation formula.

Euler-Maclaurin links a sum to an integral plus boundary corrections involving derivatives of the integrand. Applied to $g(t)=v^t{}_t{p}_x$, it yields the Woolhouse series. The leading derivative term evaluates to $-(\delta +{\mu }_x)$, because

$$\frac{d}{dt}{\left[v^t{}_t{p}_x\right]}_{t=0}=-(\delta +{\mu }_x)$$

That single identity drives the three-term coefficient. You do not need to re-derive Euler-Maclaurin on exam day; you need to recognize where $(\delta +{\mu }_x)$ comes from and apply the formula.

Common mistakes

• Using ${\mu }_x$ instead of ${\mu }_x^{i\bullet }$ in multi-state problems. Plugging single-decrement mortality into a disability annuity ignores the recovery and disability-death exits. Correct is the sum of ALL forces out of the paying state at the starting age.
• Forgetting the endpoint correction on temporary annuities. Treating ${\ddot{a}}_{x:\overline{n}|}^{(m)}$ as if it ran to infinity drops the $v^n{}_n{p}_x$ factor and over-corrects by roughly $\frac{m-1}{2m}\cdot v^n{}_n{p}_x$.
• Swapping the coefficients. Two-term uses $\frac{m-1}{2m}$; curvature uses $\frac{m^2-1}{12m^2}$. For $m=12$ the values are $11/24$ and $143/1728$, NOT $11/24$ and $11/144$.

Bottom line

• Two-term Woolhouse: $a_x^{(m)}\approx a_x-\frac{m-1}{2m}$. Three-term adds $-\frac{m^2-1}{12m^2}(\delta +{\mu }_x)$.
• Three-term needs ${\mu }_x$ at the starting age, plus the force of interest $\delta =\ln (1+i)$.
• For $m=12$: the coefficients are $\frac{11}{24}=0.4583$ and $\frac{143}{1728}=0.0828$.
• Temporary annuity rule: $a_{x:\overline{n}|}^{(m)}\approx a_{x:\overline{n}|}-\frac{m-1}{2m}(1-v^n{}_n{p}_x)-\frac{m^2-1}{12m^2}\left[(\delta +{\mu }_x)-v^n{}_n{p}_x(\delta +{\mu }_{x+n})\right]$.

Exam shortcut

Pre-compute $\frac{m-1}{2m}$ and $\frac{m^2-1}{12m^2}$ at the top of your scratch sheet for $m=2,4,12$. On a 3-hour exam you will use them repeatedly. DECISION: Question asks for three significant figures and $m=12$ → use three-term. Asks for two figures or gives only ${\ddot{a}}_x$ without ${\mu }_x$ → two-term is intended. For multi-state premiums, write ${\mu }_x^{i\bullet }={\sum }_{k\ne i}{\mu }_x^{ik}$ FIRST, then assemble. Compute the force sum before touching the Woolhouse formula to prevent mid-formula errors.

The full lesson (about 2,143 words, 14 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 2b

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