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 $\frac{m^2-1}{12 m^2}$. 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 $-\frac{m^2-1}{12 m^2}(\delta + \mu_x)$.
• 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: $a_{x:\overline{n}|}^{(m)} \approx a_{x:\overline{n}|} - \frac{m-1}{2m}(1 - v^n {}_n p_x) - \frac{m^2-1}{12 m^2}\left[(\delta + \mu_x) - v^n {}_n p_x(\delta + \mu_{x+n})\right]$.

Exam shortcut

Pre-compute $\frac{m-1}{2m}$ and $\frac{m^2-1}{12 m^2}$ 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 $\mu_x^{i\bullet} = \sum_{k \neq i}\mu_x^{ik}$ 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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