Life Contingencies: Study Note for CAS Exam S

By Tom Struppeck · Casualty Actuarial Society · 2014

A short CAS study note on life contingencies — survival models for human lives and the cash flows that start or stop contingent upon survival. Written by Tom Struppeck for CAS Exam S and marked "Revised 9/19/2015", it illustrates terminology and techniques in straightforward contexts for a P&C reader, for whom such calculations are typically intermediate results, and points to where further refinement is possible; exercises follow several chapters, with an appendix of selected solutions.

On the syllabus

Exam MAS-I — objectives A5–A6; the whole note (cited as October 2014, revised September 2015; an online publication).

Introduction

"Life contingencies" is a term for survival models for human lives and the resulting cash flows that start or stop contingent upon survival — a central topic for life insurance actuaries.
In P&C work, life contingency calculations are typically intermediate results used to estimate some other quantity of interest.

1 Life tables and an extended example

A life table presents a family of conditional survival random variables concisely; the chapter builds one from 300 vacuum cleaners that all fail within four years.
The mortality rate (or mortality) is the number expected to fail in the coming year divided by the number still functioning at its start; $l_x$ , $d_x$ and $q_x$ denote the number functioning, the number failing and the mortality.
Actuarial present value (APV) is the present value of a stream of cash flows that depend on a random variable; assuming a single deterministic interest rate, the chapter prices a $100 warranty paid at the end of the year of failure (APV $90.09 at 5%) and the level annual subscription that funds it ($43.30 a year).

2 Comments on Life tables

$(x)$ denotes a life of exact age $x$ ; $d_x = l_x - l_{x+1}$ , $q_x = d_x / l_x$ and $p_x = l_{x+1}/l_x$ .
The radix $l_0$ can be any positive number, usually a round one such as 100,000, since only the mortalities matter.
${}_np_x = l_{x+n}/l_x = p_x\,p_{x+1}\cdots p_{x+n-1}$ , and ${}_nq_x = 1 - {}_np_x$ .
Exercises use the exam's Illustrative Life Table.

3 Contingent Cash flows

Life insurance contracts pay $1 at the end of the year of death: term insurance for a fixed period, whole life with an unlimited term, endowment insurance (term plus $1 at maturity), and the pure endowment (the payment at maturity alone).
The whole life APV satisfies $A_x = v\,q_x + v\,p_x A_{x+1}$ , so $A_x = \sum_{n \ge 0} v^{n+1}\,{}_np_x\,q_{x+n}$ .
A life-annuity-due pays at the start of each year while $(x)$ is alive, $\ddot a_x = 1 + v\,p_x\,\ddot a_{x+1}$ ; a life-annuity-immediate pays at the end of the year, $a_x$ .
By an arbitrage argument $1 = d\,\ddot a_x + A_x$ , so $\ddot a_x = (1 - A_x)/d$ , and the level annual premium $P$ solves $A_x = P\,\ddot a_x$ .
Annuities can be limited to $n$ years or guaranteed for at least $n$ years; the exercises derive the variance of an insurance (its second moment is the first moment at double the force of interest) and of an annuity.

4 An example from workers' compensation insurance

An injured worker's monthly stipend stops on return to work, so "mortality" here is returning to work; the examples assume annual payments and a level 3% discount rate.
An annuity-due on (20) with a 5-year certain period is a five-year annuity-certain plus a life annuity on (25) deliverable in five years: $\ddot a_{\overline{5}|} + (1.03)^{-5}\,\frac{l_{25}}{l_{20}}\,\ddot a_{25}$ .
An annuity on (40) paying until return to work or age 65 is replicated by a life annuity on (40) less one on (65) taken back in 25 years: $\ddot a_{40} - (1.03)^{-25}\,\frac{l_{65}}{l_{40}}\,\ddot a_{65}$ .
Under US regulatory accounting most reserves for future loss payments are not discounted, but certain workers' compensation reserves are.

5 Why P&C companies don't sell level premium products

A level premium buys the first year's coverage plus the option to buy future years at the same price; a purchaser who does not renew has lapsed.
With non-decreasing mortality the early excess premium (policy values, or premium reserves) is positive; with decreasing mortality the rational insured lapses after taking the subsidy and buys cheaper term cover.
P&C risks tend to have decreasing hazard rates over time, because the worst risks have their accidents earlier and only better risks remain in the cohort.

6 What happens mid-year?

Monthly timing needs mortalities between integer ages, typically interpolated from the annual ones: linearly, exponentially or hyperbolically.
In continuous time mortality is described by survival functions and hazard rates, covered elsewhere in the syllabus; a hazard rate describing mortality is called the force of mortality.
A workers' compensation claim in paying status faces competing forces — return to work, reaching 65, death — only one of which stops payment.
Increased mortality is applied, by custom, by multiplying the mortalities (a footnote notes that multiplying the hazard rate by $k$ turns $p_x$ into $p_x^{\,k}$ ).
An annuity can pay based on the survival of more than one person (joint lives): until one of the two dies, $\ddot a_{xy}$ , or until both have died, $\ddot a_{\overline{xy}}$ ; an arbitrage argument relates them to annuities on single lives, $\ddot a_x + \ddot a_y = \ddot a_{xy} + \ddot a_{\overline{xy}}$ .

7 Life expectancy

The curtate life expectancy, the expected number of full years, is $\sum_{n \ge 1} {}_np_x$ from the life table; the complete expectation of life adds 0.5 when deaths are uniformly distributed over the year.
The same calculation applied to claim closure gives the expected number of full years a claim stays open (0.75 in the example; 1.25 complete).

Appendix: Solutions for selected problems from Life Contingencies Study Note for CAS Exam S

Worked solutions to exercises from Chapters 2, 3 and 6.

Sources

Life Contingencies Study Note for CAS Exam S (CAS, revised 2015) — the document, read in full: title, author, revision date, the introduction, Chapters 1–7 and the appendix of solutions
CAS Exam MAS-I Content Outline (2026) — the citation (October 2014, revised September 2015), the link to the CAS copy and the assignment