Life Contingencies Primer

Although life contingencies is primarily SOA territory, P&C actuaries encounter it in workers' compensation lifetime pension claims, structured settlements, and any product with mortality-contingent cash flows. This primer covers survival models, life tables, and the present-value building blocks.

Survival models

Let $X$ be the age at death, with survival function $S_0(x) = P(X>x)$. For a life aged $x$, the future lifetime $T_x = X - x \mid X>x$ has survival function

$\!{}_tp_x = P(T_x > t) = \frac{S_0(x+t)}{S_0(x)}, \qquad {}_tq_x = 1 - {}_tp_x.$

The force of mortality $\mu_x = -\dfrac{d}{dx}\ln S_0(x)$ is the instantaneous death rate, analogous to a hazard rate; it relates to the survival function via $\!{}_tp_x = \exp\left(-\int_0^t \mu_{x+s}\,ds\right)$. Common parametric laws: Gompertz ($\mu_x = Bc^x$), Makeham ($\mu_x = A+Bc^x$, adding an age-independent "accident" component — of particular relevance to WC lifetime claims, which combine an accidental-death-like flat hazard with normal aging mortality).

Life tables

A life table tabulates $l_x$, the expected number of survivors to age $x$ out of an initial cohort $l_0$ (e.g., 100,000), from which $q_x = (l_x - l_{x+1})/l_x$ (one-year mortality rate) and $\!{}_tp_x = l_{x+t}/l_x$ follow directly. Standard tables include the SOA's Annuity 2000/2012 tables and the widely used CSO (Commissioners Standard Ordinary) tables for U.S. life insurance valuation. For WC lifetime pension reserves, actuaries often start from a general population table and adjust for the impairment associated with the disabling injury.

Present values of insurances and annuities

A whole life insurance pays $1 at the moment of death (or end of year of death, for the discrete version); its expected present value is

$A_x = \sum_{k=0}^\infty v^{k+1}\,{}_kp_x\,q_{x+k}.$

A whole life annuity-due pays $1 at the start of each year the insured survives:

$\ddot a_x = \sum_{k=0}^\infty v^k\,{}_kp_x.$

These satisfy the fundamental identity $A_x = 1 - d\,\ddot a_x$ (where $d = iv$), which follows from the fact that a $1 whole life insurance plus the annuity of premiums funding it must reproduce the accumulation of $1. Term and endowment insurances and annuities restrict the payment to a fixed term $n$, e.g. $A^1_{x:\overline{n}\rvert}$ (term insurance, pays only on death within $n$ years) and $A_{x:\overline{n}\rvert}$ (endowment, pays on death within $n$ years or survival to $n$).

Worked example

Using a simplified table: $q_{60}=0.010$, $q_{61}=0.012$, $q_{62}=0.014$, and $i=5\%$ ($v=1/1.05=0.952381$). Compute the EPV of a 3-year term insurance paying $100,000 at the end of the year of death for a life aged 60.

$\!{}_0p_{60}=1$. $\!{}_1p_{60} = 1-0.010 = 0.990$. $\!{}_2p_{60} = 0.990 \times (1-0.012) = 0.990\times0.988 = 0.97812$.

Sum of the deferred death probabilities times discount $\times$ $100,000 gives:

$A^1_{60:\overline{3}\rvert} \times 100{,}000 = 952.38 + 1{,}077.55 + 1{,}183.20 = \$3{,}213.13.$

So the expected present value of the 3-year term benefit is $3,213 per $100,000 of coverage — the building block for pricing structured settlement annuities or reserving lifetime WC pensions with mortality contingency.

Pitfalls

• Applying a generic population mortality table to an impaired-life WC pensioner without an appropriate mortality improvement/impairment adjustment, materially mis-stating reserves.
• Ignoring mortality improvement over time — static tables understate future survival, understating long-duration annuity liabilities.
• Confusing curtate (discrete, end-of-year) and continuous (moment-of-death) formulations, which differ by an approximate factor related to $i/\delta$.
• Treating interest and mortality as independent when they are not in reality (e.g., pandemic years shift both economic and mortality assumptions together) — scenario/stress testing should consider joint shocks.

Exam relevance

This material is the foundation of SOA Exams FAM and ALTAM/ASTAM; P&C actuaries need only the conceptual core for CAS Exam 7/9 discussions of tabular reserves and structured settlements.