Understand and explain features of typical mortality curves and heterogeneities in mortality.

Plot human mortality rates on a log scale and three landmarks pop out: an infant spike, a young-adult accident hump, and a near-straight senescent line. Pricing any contingent cash flow starts with knowing where on this curve your insured sits.

$q_x$ is the probability a life age $x$ dies within one year. ${\mu }_x$ is the instantaneous force of mortality. On a log scale, human ${\mu }_x$ traces a U with a bump:

• Infant mortality. ${\mu }_0$ is high (congenital defects, birth trauma), then drops sharply through age 1 to a low plateau near age 10.
• Accident hump. A local bump from roughly age 18 to 25, driven by motor vehicle accidents, violence, and suicide. More pronounced for males.
• Senescent mortality. From about age 40 onward, $\log {\mu }_x$ is nearly linear in $x$. Doubling time roughly 8 years.
• Late-life deceleration. Past age 95, mortality rates increase slower than Gompertz predicts. Empirically debated but exam-relevant for very old ages.

Gompertz (1825) modeled senescent mortality as exponential in age. Think of $B$ as the baseline hazard level anchored at age 0 and $c$ as the per-year multiplicative growth factor: $B$...

Common mistakes

• Using Gompertz at infant ages. Gompertz fits ages 40 plus. Below 40 the accident hump and infant spike break the log-linear shape.
• Confusing $q_x$ and ${\mu }_x$. $q_x$ is annual probability; ${\mu }_x$ is instantaneous rate. For small rates $q_x\approx 1-e^{-{\mu }_x}\approx {\mu }_x$, but the approximation breaks at high ages.
• Reading the select table by attained age. Select rows index age at selection $[x]$, not current age. A life selected at 60 in policy year 3 uses row $[60]$, column 2, not row 62.

Bottom line

• Three landmarks: infant peak, accident hump near ages 18 to 25, senescent log-linear rise past age 40.
• Gompertz ${\mu }_x=B{c}^x$: force of mortality grows exponentially with age. Makeham adds a constant background term $A$ for accident and infectious-disease deaths.
• Doubling time of adult force of mortality is $\ln 2/\ln c$, roughly 7 to 8 years.
• Heterogeneity sources: sex, smoking, socioeconomic class, occupation, underwriting class, ethnicity, geography.

Exam shortcut

Plot $\log q_x$ versus $x$. A straight line past 40 with intercept and slope gives Gompertz $B$ and $c$ directly: slope $=\ln c$, intercept $=\ln B$. DECISION: Select period not yet over → read $q_{[x]+t}$ from the select column. Select period ended → read ultimate $q_{x+t}$ at attained age. "SAGES" for heterogeneity drivers: Sex, Age, Geography, Economics, Smoking. Check each before pricing.

The full lesson (about 1,708 words, 11 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 1a

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