Woolhouse's Formula

Study guide: Exam ALTAM

Woolhouse's Formula approximates the expected present value of an annuity paid $m$ times a year from the value of the same annuity paid once a year, using the Euler–Maclaurin formula. It turns annual life annuity values from a life table into monthly or continuous ones, and in a multiple state model it does the same for payments made while the life is in a given state.

$\ddot{a}^{(m)}_x \approx \ddot{a}_x - \frac{m-1}{2m} - \frac{m^2-1}{12m^2}\left(\delta + \mu_x\right)$

$\begin{aligned} \ddot{a}^{(m)}_{x:\overline{n}|} \approx {} & \ddot{a}_{x:\overline{n}|} - \frac{m-1}{2m}\left(1 - {}_nE_x\right) \\ & - \frac{m^2-1}{12m^2}\left(\delta + \mu_x - {}_nE_x\left(\delta + \mu_{x+n}\right)\right) \end{aligned}$

$\begin{aligned} \frac{1}{m}\sum_{k=0}^{\infty} g\!\left(\tfrac{k}{m}\right) \approx {} & \sum_{k=0}^{\infty} g(k) - \frac{m-1}{2m}\,g(0) \\ & + \frac{m^2-1}{12m^2}\,g'(0) \end{aligned}$

Symbols: Following Dickson, Hardy & Waters (chapter 5), $\ddot{a}^{(m)}_x$ pays $1/m$ at the start of each $m$ -th of a year while $(x)$ is alive, $\delta$ is the force of interest, $\mu_x$ is the force of mortality, and ${}_nE_x = v^n\,{}_np_x$ . The first block is the three-term formula, and dropping its last term gives the two-term formula $\ddot{a}^{(m)}_x \approx \ddot{a}_x - \frac{m-1}{2m}$ .
Where it comes from: The third block is the Euler–Maclaurin formula written once with step $1/m$ and once with step 1, with the integral of $g$ eliminated between them. For a single life $g(t) = v^t\,{}_tp_x$ , so $g(0) = 1$ and $g'(0) = -(\delta + \mu_x)$ , which gives the first block. A term annuity also has terms at $t = n$ , which gives the second block.
Continuous limit: Letting $m \to \infty$ gives $\bar{a}_x \approx \ddot{a}_x - \frac{1}{2} - \frac{1}{12}\left(\delta + \mu_x\right)$ . If the force of mortality is not given, it can be estimated from the table as $\mu_x \approx -\frac{1}{2}\left(\ln p_{x-1} + \ln p_x\right)$ .
State-dependent cash flows: For payments made while the life is in state $j$ , starting from state $i$ , take $g(t) = v^t\,{}_tp^{ij}_x$ . If $j = i$ , then $g(0) = 1$ and $g'(0) = -\left(\delta + \sum_{k \neq i}\mu^{ik}_x\right)$ . If $j \neq i$ , then $g(0) = 0$ and $g'(0) = \mu^{ij}_x$ , so the two-term formula makes no adjustment at all and only the third term changes the value.

Monthly Life Annuity on the SULT A pension of $1,000 a month is paid in advance for life to a retiree aged 65. On the SULT at $5\%$ , $\ddot{a}_{65} = 13.5498$ and $\mu_{65} = 0.005605$ , and $\delta = \ln 1.05 = 0.048790$ . Estimate the EPV with the two-term and the three-term Woolhouse formulas.
Answer With $m = 12$ , the coefficients are $\frac{11}{24} = 0.458333$ and $\frac{143}{1{,}728} = 0.082755$ , and $\delta + \mu_{65} = 0.054395$ .
 $\begin{align*} \ddot{a}^{(12)}_{65} &\approx 13.5498 - 0.458333 \\ &= 13.0915 \\ \ddot{a}^{(12)}_{65} &\approx 13.091467 - 0.082755(0.054395) \\ &= 13.0870 \end{align*}$ 
The EPVs are $12{,}000(13.0915) = 157{,}098$ and $12{,}000(13.0870) = 157{,}044$ . Summing the monthly payments directly under the Makeham law gives $13.08696$ , so the three-term value is exact to four decimals, while the two-term value overstates the EPV by about $54.

Monthly Disability Income in a Multiple State Model A disability income policy on a healthy life uses the healthy–sick–dead model with constant intensities $\mu^{01} = 0.02$ , $\mu^{02} = 0.01$ , $\mu^{10} = 0.10$ and $\mu^{12} = 0.05$ , and $\delta = 0.05$ . The annual values are $\ddot{a}^{00}_x = 14.7924$ and $\ddot{a}^{01}_x = 1.42691$ . The policy pays $2,000 a month in advance while the insured is sick, and premiums are paid monthly in advance while the insured is healthy. Use the three-term formula to find the annual premium.
Answer For the healthy-state annuity, $g'(0) = -(0.05 + 0.02 + 0.01) = -0.08$ . For the sick-state annuity, $g(0) = 0$ and $g'(0) = \mu^{01} = 0.02$ .
 $\begin{align*} \ddot{a}^{(12)00}_x &\approx 14.7924 - 0.458333 - 0.082755(0.08) \\ &= 14.32745 \\ \ddot{a}^{(12)01}_x &\approx 1.42691 + 0.082755(0.02) \\ &= 1.42857 \\ P &= \frac{24{,}000(1.42857)}{14.32745} \\ &= 2{,}393.01 \end{align*}$ 
The premium is $2,393.01 a year, or $199.42 a month. The exact monthly values from the model are $14.32743$ and $1.42856$ . Using the two-term formula, which leaves $\ddot{a}^{01}_x$ unchanged, gives $2,389.12 instead.

How it appears on the exam

ALTAM asks candidates to derive the two-term and three-term formulas and to apply them to the EPVs of state-dependent cash flows. The inputs can be annual annuity values from a multiple state model, such as disability income or a CCRC, together with the transition intensities at the starting age.

Premiums and policy values: The converted annuities then feed a premium by the equivalence principle or a policy value for a contract with monthly premiums or benefits.

Exam traps

Force of interest, not the rate: The third term uses $\delta$ , not $i$ . With $i = 5\%$ , writing $0.05$ in place of $\delta = 0.04879$ makes the third term too large.

Annuity-due versus immediate: The formulas are for annuities-due. An $m$ -thly annuity-immediate is $a^{(m)}_x = \ddot{a}^{(m)}_x - \frac{1}{m}$ , and applying the formula to $a_x$ directly gives the wrong constant term.

The end of a term annuity: A term annuity needs the correction at $t = n$ as well. Leaving out the ${}_nE_x$ terms subtracts a full correction for payments that are never made.

The starting state: For an annuity paid in a state other than the starting state, $g(0) = 0$ , so there is no $\frac{m-1}{2m}$ term. Subtracting it anyway removes about $0.46$ from a value that may be close to 1.