Mortality

Study guides: Exam FAM, Exam ALTAM, Exam ILA-101, Exam ILA-201I, Exam ILA-201U, Exam INV-201

Mortality is the incidence of death in a population, measured as a function of age by the annual mortality rate $q_x$ and the force of mortality $\mu_x$ (the hazard rate of the future lifetime), and varying also with sex, health, duration since underwriting (selection) and calendar year (mortality improvement). Mortality assumptions drive every life insurance and annuity value: higher mortality raises insurance costs and lowers annuity costs.

${}_tp_x = \exp\!\left(-\int_0^t \mu_{x+s}\,ds\right)$

$\mu_x = A + Bc^x$

$\mu^{*}_{x+s} = \mu_{x+s} + \theta$

$q(x, t) = q(x, t-1)\,\big(1 - \phi(x, t)\big)$

The second block is Makeham's law ( $A = 0$ gives Gompertz): an age-independent term plus a term rising exponentially with age. The Standard Ultimate Life Table used on FAM and ALTAM is Makeham with $A = 0.00022$ , $B = 2.7\times10^{-6}$ , $c = 1.124$ , for which ${}_tp_x = \exp\!\big(-At - Bc^x(c^t - 1)/\ln c\big)$
A typical mortality curve falls after high infant mortality, shows an accident hump in the late teens and twenties, then rises roughly exponentially through adult ages and decelerates at the oldest ages; heterogeneity (sex, smoking, socioeconomic group, select versus ultimate) makes a population a mix of such curves
Extra risk for substandard lives is modelled by an age rating (use the table at $x + k$ ), a constant addition $\theta$ to the force (third block, so ${}_tp^*_x = e^{-\theta t}\,{}_tp_x$ ) or a constant multiple of the rate, $q^*_x = k\,q_x$ ; the fourth block is mortality improvement, with $\phi(x, t)$ the proportional fall in $q$ at age $x$ from year $t-1$ to $t$
Lighter mortality than assumed hurts an annuity writer (longevity risk) and helps an insurer, so a company writing both has a partial natural offset; table values and a fractional age assumption fill in between integer ages

Survival under the Standard Ultimate Life Table Using Makeham's law with $A = 0.00022$ , $B = 2.7\times10^{-6}$ , $c = 1.124$ , find $\mu_{60}$ and ${}_{10}p_{60}$ .
Answer
 $\begin{align*} \mu_{60} &= 0.00022 + 2.7\times10^{-6}(1.124)^{60} \\ &= 0.00022 + 0.0030015 \\ &= 0.0032215 \\ \int_0^{10}\mu_{60+s}\,ds &= 0.00022(10) + \frac{0.0030015\,(1.124^{10} - 1)}{\ln 1.124} \\ &= 0.0022 + \frac{0.0030015(2.21857)}{0.116894} \\ &= 0.059167 \\ {}_{10}p_{60} &= e^{-0.059167} \\ &= 0.94255 \end{align*}$ 
A 60-year-old has a $94.3\%$ chance of reaching 70, matching the SULT ratio $\ell_{70}/\ell_{60}$ .

Two Ways to Load for Extra Risk Standard rates are $q_{60} = 0.010$ and $q_{61} = 0.011$ . Find the probability that a substandard life aged 60 dies within two years (a) at $150\%$ of standard mortality and (b) with $0.005$ added to the force of mortality.
Answer Standard: ${}_2p_{60} = (0.990)(0.989) = 0.97911$ .
 $\begin{align*} \text{(a)}\quad {}_2q^*_{60} &= 1 - (1 - 0.015)(1 - 0.0165) \\ &= 1 - 0.968748 \\ &= 0.031253 \\ \text{(b)}\quad {}_2q^*_{60} &= 1 - 0.97911\,e^{-0.005(2)} \\ &= 1 - 0.969368 \\ &= 0.030632 \end{align*}$ 
Both raise the two-year death probability from $0.02089$ to about $0.031$ . The multiple scales with age, so it bites harder at older ages; the addition is the same at every age.

Cohort versus Period Survival with Improvement For calendar year 2026, $q(65) = 0.010$ , $q(66) = 0.011$ , $q(67) = 0.012$ . Mortality improves by $\phi = 2\%$ a year at every age. Find the probability that a life aged 65 in 2026 survives three years, on the period table and on the cohort (diagonal) basis.
Answer The period basis uses the 2026 rates throughout: ${}_3p_{65} = (0.990)(0.989)(0.988) = 0.96736$ . The cohort reaches 66 in 2027 and 67 in 2028:
 $\begin{align*} q(66, 2027) &= 0.011(0.98) \\ &= 0.01078 \\ q(67, 2028) &= 0.012(0.98)^2 \\ &= 0.011525 \\ {}_3p_{65} &= (0.990)(1 - 0.01078)(1 - 0.011525) \\ &= 0.96804 \end{align*}$ 
The cohort death probability is $0.03196$ against $0.03264$ on the period table. Over a whole annuity the gap compounds, which is why ignoring improvement understates annuity values.

How it appears on the exam

FAM (Mortality Models): given a parametric law (Gompertz, Makeham, de Moivre) or a life table, compute ${}_tp_x$ , ${}_{t|}q_x$ , $\mu_x$ and moments of $T_x$ and $K_x$ , with UDD or constant force between integer ages.

FAM ("effect of changes in underlying assumptions"; "modelling extra risk"): recompute a premium or policy value after an age rating, a constant addition to $\mu$ or a multiple of $q$ , often for a term insurance or endowment.

ALTAM ("features of typical mortality curves and heterogeneities"): describe the shape of a curve and the sources of heterogeneity and selection, and work with mortality improvement scales in annuity values, as written and calculation parts.

ILA-101 / ILA-201 (mortality assumption development; "offsets between mortality and longevity risk"): explain how experience studies set the assumption and how insurance and annuity blocks offset, as written answers; INV-201 lists mortality among risks that can only be partially hedged.

Exam traps

Adding to $q$ when the question adds to $\mu$ . A constant addition $\theta$ to the force multiplies survival by $e^{-\theta t}$ ; adding $\theta$ to each $q_x$ gives a slightly different (wrong) answer.

Multiplying the force when the question multiplies the rate. $150\%$ of $q_x$ is not $1.5\mu_x$ : $1 - (1-q)^{1.5} \neq 1.5q$ .

Age rating in the wrong direction. A substandard life aged $x$ rated up $k$ years uses $q_{x+k}$ , and the age in every subsequent factor also shifts by $k$ .

Applying improvement along the period row. A life ages as calendar time passes; use $q(x+t, Y+t)$ , not $q(x+t, Y)$ , or the improvement is ignored.