Theory of Survival and Demographic Tables

Life insurers enter contracts 30, 50, 70 years into the future: calculating premiums and reserves requires a quantitative model of mortality. Life actuarial mathematics is built on the analysis of the random variable "time of life" $T_x$ — the residual life expectancy of a person aged $x$. Mortality tables are the main statistical tool: they compile data on mortality by age and sex and are used to calculate premiums, reserves, and pension obligations. Without them, neither classical life insurance nor pension systems are possible.

Survival Functions

Let $X$ be the random age at death of a newborn (age 0). Define:

Survival function: $S(x) = P(X > x)$ — probability of surviving to age $x$. $S(0) = 1$, $S(\infty) = 0$, $S$ is decreasing.

Related functions:

• $F(x) = P(X \leq x) = 1 − S(x)$ — distribution function.
• $f(x) = F'(x) = −S'(x)$ — density.
• Force of mortality: $\mu(x) = f(x)/S(x) = −d, \ln S(x)/dx$.

Decoding $\mu(x)$: the instantaneous probability of death per unit time conditional on surviving to $x$. Analog of the hazard rate in survival analysis.

Restoring $S$ from $\mu$: $S(x) = \exp(−\int_0^x \mu(s),ds)$.

Conditional probabilities (actuarial notation):

• $_{t}p_x = P(X > x + t,|,X > x) = S(x+t)/S(x)$ — probability of surviving $t$ more years from $x$.
• $_{t}q_x = 1 − _{t}p_x$ — probability of dying in $(x, x+t]$.
• $p_x = _{1}p_x$, $q_x = _{1}q_x$ — annual probability of surviving/dying.

Expected residual life: $\bar{e}x = \int_0^\infty,{t}p_x,dt = \int_0^\infty S(x+t)/S(x),dt$.

Parametric Models

1. Exponential (constant mortality): $\mu(x) = \lambda$. $S(x) = e^{−\lambda \cdot x}$, $\bar{e}_x = 1/\lambda$. Too simplified: mortality actually depends on age.

2. Gompertz (1825): $\mu(x) = B \cdot c^x$, $c > 1$ — exponential growth. Well approximates adult mortality (30–80 years). For typical people: $c \approx 1.085$, i.e., mortality doubles every 8–9 years.

3. Gompertz-Makeham (1860): $\mu(x) = A + B \cdot c^x$. Constant $A$ — accidents, independent of age. Standard for actuarial tables of the 20th century.

4. Lee-Carter model (1992): $\ln \mu(x, t) = \alpha(x) + \beta(x)\cdot\kappa(t)$. $\alpha$ — average logarithm of mortality by age, $\beta$ — sensitivity to time trend, $\kappa$ — general trend of decreasing mortality. Standard for mortality forecasts for 30–50 years.

5. Cairns-Blake-Dowd (CBD, 2006): $\mathrm{logit}, q(x, t) = \kappa_1(t) + (x−\bar{x})\cdot\kappa_2(t)$. Two time trends: level and slope of the age profile.

Mortality Tables

Structure. For each age $x = 0, 1, ..., 110$:

• $l_x$ — number surviving to $x$ from $l_0 = 100,000$ (or $1,000,000$) in a hypothetical cohort.
• $d_x = l_x − l_{x+1}$ — number dying during year $x$.
• $q_x = d_x / l_x$ — annual probability of death.
• $p_x = 1 − q_x$.
• $\bar{e}x = (l{x+1} + l_{x+2} + ... + l_{110}) / l_x + 0.5$ (mid-year adjustment) — expected residual life.

Data sources: Census + death registry (Rosstat, US Social Security Administration, Office for National Statistics UK). Human Mortality Database (HMD) collects data for 40+ countries.

Numerical Example

Russia 2019 (men), simplified: $q_{45} = 0.0085$, $q_{55} = 0.0175$, $q_{65} = 0.0345$, $q_{70} = 0.0510$, $q_{80} = 0.108$.

Probability of surviving from 45 to 65: ${20}p{45} = \prod_{k=45}^{64} p_k \approx \exp(−\sum q_k) \approx \exp(−0.305) \approx 0.737$.

That is, a 45-year-old man has a $\sim 74%$ chance of surviving to 65 — noticeably lower than for women ($\sim 89%$).

Expected residual life at age 65, $\bar{e}_{65} \approx 14.0$ years (men), $19.5$ years (women). The gap in life expectancy is one of the highest in the world.

Selection Effect and Underwriting

Newly insured individuals are usually healthier than the general population (medical underwriting). Select-ultimate tables are used: $q_{[x]+t}$ — mortality at age $x + t$ for a person insured at age $x$. After 5–15 years the "selective effect" disappears, and $q$ approaches ultimate $q_{x+t}$.

Real-World Applications

• Life insurance. Calculating net premiums, reserves, dividends for participating policies. Actual tables at Russian insurers — Russia-mortality with adjustments for selection.
• Pension funds. Assessing obligations for lifetime pension payments. An error in $e_{65}$ by 1 year → $\sim$5% additional obligations → billions for large funds (UK USS, Dutch ABP).
• State pension systems. Russian Social Fund, US Social Security project long-term deficits based on projected mortality.
• Medical insurance. Age-based premiums, calculation of expected treatment costs for cohorts.
• Annuities (lifetime income). Pricing — reverse problem: premium for guaranteed lifetime payout of $X$ rub./month based on $e_x$ and rate.

Assignment. Using a mortality table (e.g., Human Mortality Database, Russia men 2019): (a) Plot the curves $\mu(x)$ (logarithmic) and $S(x)$ for ages 0–100. (b) Fit Gompertz-Makeham parameters $A$, $B$, $c$ by least squares on the interval 30–80 years. (c) Compute $P(\text{survive to 65} \mid 45,\text{years})$, $\bar{e}{65}$ for men and women. (d) Compare Russia 2019 data with Japan 2019 — what is the difference in $\bar{e}{65}$? Plot both $\mu(x)$ curves on a single graph.