5. Life Contingencies and Actuarial Present Values

Introduction and context

Insurance contracts are promises whose financial consequences depend on uncertain future events. In life insurance, the relevant event may be death, survival to a specified date, disability, withdrawal, or the occurrence of one event among several competing events. Actuarial mathematics converts these uncertain future cash flows into values at a common valuation date. The central tool is the actuarial present value (APV): the expected value, at a chosen interest rate, of benefits or premiums payable under a contract.

Life-contingency analysis combines two forms of uncertainty. Financial uncertainty arises because money payable in the future must be discounted. Demographic or event uncertainty arises because the time and nature of the insured event are unknown. A sound valuation therefore requires both a model of interest and a model of survival or failure. This chapter develops the principal one-life, multiple-life, and multiple-decrement techniques used in insurance pricing, reserving, product design, and financial reporting.

The discussion uses both life-table notation and survival-model notation. The life-table approach is especially convenient for operational calculations, while the survival-model approach clarifies assumptions, enables interpolation, and supports advanced analytics.

Learning objectives

After studying this chapter, a student should be able to:

1. Explain survival, death, force of mortality, and life-table functions.

2. Calculate APVs for assurances, endowments, annuities, and varying benefits.

3. Distinguish benefits payable at the moment of death from those payable at the end of a policy year.

4. Derive relationships between insurance and annuity values.

5. Value contracts involving two or more lives.

6. Explain and apply multiple-decrement models.

7. Select suitable assumptions and interpret the managerial significance of results.

8. Recognize modelling, governance, ethical, and implementation risks.

The valuation framework

Let a life aged x have future lifetime T_x, measured in years. The survival function is

\[

{}_tp_x=P(T_x>t),

\]

and the probability of death within t years is

\[

{}_tq_x=1-{}_tp_x.

\]

The one-year death probability is q_x = {}_1q_x, and the number alive at exact age x in a life table is denoted l_x. The associated number dying between ages x and x+1 is d_x=l_x-l_{x+1}. Hence

\[

q_x=\frac{d_x}{l_x}, \qquad p_x=\frac{l_{x+1}}{l_x}.

\]

For an integer n,

\[

{}_np_x=\frac{l_{x+n}}{l_x}, \qquad {}_nq_x=1-{}_np_x.

\]

The conditional probability of dying between durations k and k+1 is

\[

{}_k|q_x={}_kp_x q_{x+k}=\frac{d_{x+k}}{l_x}.

\]

This quantity is fundamental for discrete, end-of-year death benefits.

Let i be the effective annual interest rate, v=(1+i)^{-1}, and δ=ln(1+i) the annual force of interest. A payment of one unit at time t has present value v^t=(1+i)^{-t}, or e^{-δt}. If a benefit B is paid at a random time T, its APV is

\[

E[B(T)v^T],

\]

provided the expectation exists. This expression makes the two components of valuation explicit: B(T) describes the contractual cash flow and v^T discounts it.

Mortality models and life-table interpretation

In a continuous model, the force of mortality at age x+t is

\[

\mu_{x+t}=\lim_{h\to0}\frac{P(t\le T_x<t+h\mid T_x\ge t)}{h}.

\]

The survival function can be written as

\[

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

\]

The density of the future lifetime is

\[

f_{T_x}(t)={}_tp_x\mu_{x+t}.

\]

Consequently, a continuous APV for a death benefit b(t) is

\[

\bar A_x^{(b)}=\int_0^\infty b(t)e^{-δt}{}_tp_x\mu_{x+t}\,dt.

\]

A life table normally records deaths in age intervals rather than exact times. Therefore, practical valuation requires a convention concerning the time of death within each interval. Common assumptions include the uniform distribution of deaths (UDD), constant force of mortality, and assumptions based on fractional-age data. These assumptions can produce slightly different results, particularly when interest rates are high or benefits change rapidly with duration.

Life assurance values

Whole-life assurance

A whole-life assurance pays one unit on death, whenever death occurs. If the benefit is paid at the moment of death, its value is

\[

\bar A_x=E[v^{T_x}]=\int_0^\infty v^t{}_tp_x\mu_{x+t}\,dt.

\]

If the benefit is paid at the end of the year of death, the discrete value is

\[

A_x=\sum_{k=0}^{\infty}v^{k+1}{}_k|q_x.

\]

The distinction is material: a payment at the moment of death is generally made earlier than an end-of-year payment and therefore has a higher value.

Term assurance

An n-year term assurance pays one unit if death occurs within n years and nothing thereafter. With an end-of-year payment,

\[

A_{x:\overline{n}|}=\sum_{k=0}^{n-1}v^{k+1}{}_k|q_x.

\]

For a continuous benefit,

\[

\bar A_{x:\overline{n}|}=\int_0^n v^t{}_tp_x\mu_{x+t}\,dt.

\]

Term assurance is useful for mortgage protection, family-income protection, and temporary business obligations. Its APV is sensitive to mortality during the selected term and to the timing of settlement.

Pure endowment and endowment assurance

A pure endowment pays one unit at time n only if the insured survives n years. Its value is

\[

{}_nE_x=v^n{}_np_x.

\]

An n-year endowment assurance combines an n-year term assurance with a pure endowment. Its end-of-year value is

\[

A_{x:\overline{n}|}^{1}=A_{x:\overline{n}|}+{}_nE_x.

\]

The superscript 1 indicates a unit benefit. If the death benefit is S and the maturity benefit is M, then

\[

APV=S A_{x:\overline{n}|}+M{}_nE_x.

\]

This decomposition is both mathematically efficient and commercially useful because it separates mortality protection from savings or maturity value.

Life annuities

An annuity pays a sequence of benefits while a specified life or lives satisfy the contract condition. A whole-life annuity-due paying one unit at the beginning of each year while a life aged x is alive has value

\[

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

\]

An annuity-immediate pays at the end of each year provided the life is alive at that payment date:

\[

a_x=\sum_{k=1}^{\infty}v^k{}_kp_x.

\]

Thus \(\ddot a_x=1+a_x\). An n-year temporary annuity-due is

\[

\ddot a_{x:\overline{n}|}=\sum_{k=0}^{n-1}v^k{}_kp_x,

\]

and the corresponding annuity-immediate is

\[

a_{x:\overline{n}|}=\sum_{k=1}^{n}v^k{}_kp_x.

\]

A deferred whole-life annuity-due beginning after n years has value

\[

{}_{n|}\ddot a_x=v^n{}_np_x\ddot a_{x+n}.

\]

The fundamental assurance-annuity identity for a whole-life contract is

\[

A_x+d\,\ddot a_x=1,

\]

where d=i/(1+i) is the annual rate of discount. Equivalently,

\[

A_x=1-d\ddot a_x.

\]

For an annuity-immediate, another useful relationship is

\[

A_x+i a_x=1.

\]

These identities provide efficient calculation methods and valuable checks on spreadsheet or software outputs.

Example: a temporary endowment

Suppose a five-year contract pays 100,000 on death during the term and 120,000 on survival to the end. At an interest rate i, its APV is

\[

100{,}000 A_{x:\overline{5}|}+120{,}000v^5{}_5p_x.

\]

If premiums are paid annually in advance while the policy is in force, the APV of a level annual premium P is

\[

P\ddot a_{x:\overline{5}|}.

\]

The equivalence principle sets the APV of premiums equal to the APV of benefits and expenses. Ignoring expenses, the premium is therefore

\[

P=\frac{100{,}000 A_{x:\overline{5}|}+120{,}000{}_5E_x}{\ddot a_{x:\overline{5}|}}.

\]

This is a valuation equation, not necessarily the final market premium: loading, expenses, commissions, profit, capital costs, and risk margins must also be considered.

Assurances with varying benefits

Insurance benefits may increase, decrease, or depend on duration. A linearly increasing end-of-year benefit of k+1 at the end of year k+1 has value

\[

(IA)_x=\sum_{k=0}^{\infty}(k+1)v^{k+1}{}_k|q_x.

\]

A decreasing benefit with amount n-k in year k+1 has value

\[

(DA)_{x:\overline{n}|}=\sum_{k=0}^{n-1}(n-k)v^{k+1}{}_k|q_x.

\]

Increasing and decreasing benefits can be expressed through combinations of term assurances and annuities. For example, a decreasing mortgage protection benefit reflects the declining outstanding loan balance. A family-income benefit may be modelled as a stream of payments rather than a single lump sum. If a death at time t generates an income stream with present value g(t), the assurance value is

\[

\int_0^\infty g(t)e^{-δt}{}_tp_x\mu_{x+t}dt.

\]

The essential discipline is to identify the payment amount and payment date conditional on each possible event time before applying expectation.

Multiple-life contracts

Many products depend on two or more lives: joint-life annuities, last-survivor pensions, spouse benefits, and partnership protection. Let T_x and T_y denote the future lifetimes of two lives.

A joint-life status continues while both lives are alive and ends at the first death. Under independence,

\[

{}_tp_{xy}={}_tp_x{}_tp_y.

\]

The first-death future lifetime is \(T_{xy}=\min(T_x,T_y)\). A joint-life annuity-due therefore has value

\[

\ddot a_{xy}=\sum_{k=0}^{\infty}v^k{}_kp_{xy}.

\]

A last-survivor status continues until both lives have died. Its survival probability is

\[

{}_tp_{\overline{xy}}={}_tp_x+{}_tp_y-{}_tp_{xy},

\]

where the bar denotes last survivor. Thus

\[

\ddot a_{\overline{xy}}=\sum_{k=0}^{\infty}v^k{}_tp_{\overline{xy}}.

\]

The value of a benefit payable at the first death can be obtained by considering the distribution of the minimum lifetime. For a benefit payable at the second death, the distribution of the maximum lifetime is used.

Independence is convenient but not always credible. Spouses may share lifestyle, environment, economic conditions, and exposure to common catastrophes. Dependence can materially affect last-survivor benefits and joint-life protection. Models may therefore use joint-life tables, copulas, common-shock terms, or empirically estimated dependence structures.

Multiple-decrement models

A single-decrement life table treats death as the only exit. Real insurance portfolios often have several possible exits, such as death, disability, surrender, retirement, or lapse. A multiple-decrement model tracks a population exposed to several causes.

Let μ_x^{(j)} be the cause-specific force of decrement j and let

\[

\mu_x=\sum_j\mu_x^{(j)}

\]

be the total force. The probability of remaining in the initial state to duration t is

\[

{}_tp_x^{(\tau)}=\exp\left(-\int_0^t\sum_j\mu_{x+s}^{(j)}ds\right).

\]

The density of exit through cause j at time t is

\[

{}_tp_x^{(\tau)}\mu_{x+t}^{(j)}.

\]

Therefore, the APV of a benefit b_j(t) payable upon cause j is

\[

\sum_j\int_0^\infty b_j(t)e^{-δt}{}_tp_x^{(\tau)}\mu_{x+t}^{(j)}dt.

\]

This framework prevents a common error: adding separate single-decrement probabilities without allowing for the fact that one decrement removes a person from exposure to the others. In a discrete table, the number leaving from cause j during an interval is often denoted d_x^{(j)}, and the total decrement is d_x=\sum_jd_x^{(j)}.

For example, a disability income contract may pay an income on disability, a lump sum on death, and a surrender value on lapse. Each payment is associated with a distinct transition and must be valued using the probability of reaching that transition while still in force.

A unified applied case

Illustrative case: valuation of a family-protection contract

Background. An insurer is designing a ten-year contract for a 40-year-old policyholder. The policy pays 1,000,000 on death during the term, 400,000 if the policyholder becomes permanently disabled, and 250,000 on survival to year ten. Disability and death are competing exits; lapse is also possible. Premiums are paid annually in advance while the policy remains active.

Problem. The product team initially proposes to value death and disability benefits using independent single-decrement probabilities and to value premiums using a survival table that ignores lapse. This approach may overstate both benefits and premium income because it assumes that the policyholder remains exposed to every event simultaneously.

Analysis. The actuary constructs a transition model with three cause-specific forces: death, disability, and lapse. The in-force probability at duration t is based on the sum of the three forces. The APV of benefits is

\[

1{,}000{,}000\int_0^{10}e^{-δt}{}_tp_x^{(\tau)}\mu_{x+t}^{(d)}dt

+400{,}000\int_0^{10}e^{-δt}{}_tp_x^{(\tau)}\mu_{x+t}^{(di)}dt

+250{,}000e^{-10δ}{}_{10}p_x^{(\tau)}.

\]

The premium annuity is based on the probability of being in force at each payment date:

\[

P\sum_{k=0}^{9}v^k{}_kp_x^{(\tau)}.

\]

The equivalence premium is the benefit APV divided by this annuity value, before expenses and risk margins.

Approach and solution. The insurer tests mortality, disability, lapse, interest, and dependence assumptions through sensitivity analysis. It also distinguishes contractual disability definitions, waiting periods, claim settlement timing, and whether lapse benefits are payable.

Outcome. The multiple-decrement valuation produces a lower expected premium income than the single-decrement approach but also avoids overstating event probabilities. Management can then add acquisition expenses, maintenance expenses, commissions, capital costs, and a profit margin transparently.

Learning points. Event timing, competing risks, and policy persistence are as important as the nominal benefit amounts. A mathematically elegant formula is not reliable if its exposure base is wrong.

Professional practice, technology, and governance

APVs are used in product pricing, statutory and financial reporting valuations, embedded-value analysis, liability adequacy testing, reinsurance negotiation, pension valuation, and capital modelling. Modern actuarial systems typically combine life tables, cash-flow engines, policy administration data, scenario generators, and statistical models. Spreadsheet calculations remain useful for education and simple products, but production systems require version control, independent review, audit trails, reproducibility, and controlled assumption changes.

Automation and machine learning may assist in mortality improvement analysis, lapse segmentation, anomaly detection, and claims classification. However, predictive performance does not remove the need for actuarial judgement. Models trained on historical data may reproduce discrimination, fail under changed economic conditions, or confuse correlation with causation. Explainability, fairness testing, data minimisation, cybersecurity, and human oversight are therefore essential.

Governance should clearly document the valuation date, interest basis, mortality and decrement assumptions, payment timing, expenses, benefit definitions, rounding, and treatment of data limitations. Professional standards generally require material assumptions to be reasonable, supportable, and consistently applied, with departures from standard methods justified and disclosed. Policyholders should not be misled by presenting an APV as a guaranteed premium or by concealing the sensitivity of values to assumptions.

Limitations and common errors

Several errors recur in life-contingency work. Confusing q_x with the force of mortality, using survival to age x+n when survival is required only to a payment date, and forgetting whether an annuity is due or immediate can produce substantial differences. Other frequent mistakes include discounting a benefit twice, treating a moment-of-death payment as an end-of-year payment, ignoring expenses, and adding competing-risk probabilities as though each event had exclusive exposure to the original population.

Results are also limited by the quality of mortality and lapse data. Historical experience may not represent the insured population, and future mortality improvement can make period tables inappropriate for long-term contracts. Interest rates may be stochastic rather than constant. Dependence between lives may invalidate simple multiplication of survival probabilities. Finally, APV is an expected value; it does not by itself describe volatility, tail risk, liquidity needs, capital requirements, or the distribution of outcomes.

Best-practice calculation sequence

A reliable valuation normally follows this sequence:

1. Define the contract state, benefit triggers, payment amounts, and payment timing.

2. Specify the valuation date, interest basis, and projection horizon.

3. Select mortality, survival, decrement, and dependence assumptions.

4. Determine whether a life-table or continuous-time representation is appropriate.

5. Write the conditional cash flow for each possible event.

6. Discount each cash flow and weight it by the appropriate probability.

7. Sum benefits and value premiums, expenses, and options separately.

8. Reconcile results using identities and independent calculations.

9. Perform sensitivity, scenario, and reasonableness tests.

10. Document assumptions, limitations, controls, and approval responsibilities.

Chapter Summary

Life-contingency valuation is the expected present value of uncertain insurance cash flows. Survival probabilities describe whether a life remains alive or a policy remains in force; mortality forces and cause-specific hazards describe when and why an exit occurs. Whole-life and term assurances, pure endowments, endowment assurances, and annuities can be valued using life-table sums or continuous survival integrals. Benefit timing is crucial, and assurance-annuity identities provide useful calculation checks.

Varying benefits are valued by assigning the correct amount to each duration. Multiple-life contracts use minimum and maximum future lifetimes for joint-life and last-survivor statuses, while multiple-decrement models recognise competing exits such as death, disability, and lapse. In practice, APVs support pricing, reserving, product management, reporting, and capital decisions, but they depend on data, assumptions, governance, and professional judgement. Sensitivity analysis and transparent documentation are indispensable.

Key Terms

Actuarial present value: Expected discounted value of future contingent cash flows.

Future lifetime: Random time from valuation age until death.

Survival probability: Probability that a life survives a specified future duration.

Mortality rate q_x: Probability that a life aged x dies before age x+1.

Force of mortality: Instantaneous conditional rate of death.

Life table: Table showing numbers surviving and dying at successive ages.

Term assurance: Insurance paying a death benefit during a specified term.

Whole-life assurance: Insurance paying a death benefit whenever death occurs.

Pure endowment: Benefit payable only if the insured survives to a stated date.

Endowment assurance: Combination of term assurance and pure endowment.

Annuity-due: Annuity with payments at the beginning of payment periods.

Annuity-immediate: Annuity with payments at the end of payment periods.

Joint-life status: Status continuing while all specified lives are alive.

Last-survivor status: Status continuing until all specified lives have died.

Multiple-decrement model: Model allowing several competing causes of exit.

Cause-specific force: Instantaneous rate of exit through a specified cause.

UDD: Uniform distribution of deaths within a life-table year.

Equivalence principle: Premium principle equating APVs of premiums and benefits before loadings.

Deferred annuity: Annuity whose payments begin after a specified deferment period.

Mortality improvement: Change over time that reduces or otherwise alters mortality rates.

Review Questions

Short Answer Questions

1. Define actuarial present value and identify its two principal components.

2. Distinguish between q_x and the force of mortality.

3. What is the difference between an annuity-due and an annuity-immediate?

4. Define a pure endowment and give its APV.

5. Why are multiple-decrement probabilities not generally equal to separate single-decrement probabilities?

Long Answer Questions

1. Derive and explain the APV of an n-year endowment assurance.

2. Explain life-table and continuous-time approaches to life-contingency valuation.

3. Discuss joint-life and last-survivor contracts, including the role of independence.

4. Explain how increasing and decreasing benefits are valued and applied in insurance.

5. Describe the construction and professional governance of a multiple-decrement valuation model.

Analytical / Application Questions

1. A policy pays 500,000 at death during five years and 600,000 on survival to year five. Formulate its APV and the level annual premium payable in advance, ignoring expenses.

2. An insurer models death, disability, and lapse as competing exits. Explain why using three separate survival probabilities based on the original population would bias the valuation, and formulate the correct cause-specific APV.

3. A joint-life annuity is valued by multiplying two individual survival probabilities. Identify the assumption involved, explain when it may fail, and suggest an appropriate sensitivity analysis.

22. Multiple Choice Questions with Answers

Question 1

Which expression represents the APV of a pure endowment paying one unit at time n?

A. v^n {}_nq_x

B. v^n {}_np_x

C. v {}_nq_x

D. {}_np_x/v^n

Correct Answer: B. v^n {}_np_x

Question 2

An endowment assurance is most accurately decomposed into:

A. Whole-life assurance and deferred annuity

B. Term assurance and pure endowment

C. Two temporary annuities

D. Lapse benefit and disability benefit

Correct Answer: B. Term assurance and pure endowment

Question 3

Under an independence assumption, the probability that two lives are both alive after t years is:

A. {}_tp_x+{}_tp_y

B. {}_tp_x-{}_tp_y

C. {}_tp_x{}_tp_y

D. 1-{}_tp_x{}_tp_y

Correct Answer: C. {}_tp_x{}_tp_y

Question 4

A whole-life annuity-due pays its benefit:

A. Only at the moment of death

B. At the end of each year while the life is alive

C. At the beginning of each year while the life is alive

D. Only if the life survives a fixed term

Correct Answer: C. At the beginning of each year while the life is alive

Question 5

For a multiple-decrement model, the total force of decrement is:

A. The product of all cause-specific forces

B. The largest cause-specific force

C. The sum of cause-specific forces

D. The difference between mortality and lapse forces

Correct Answer: C. The sum of cause-specific forces

Question 6

A death benefit payable at the moment of death will generally have a higher APV than the same benefit payable at the end of the year of death because it is:

A. Paid later and discounted less

B. Paid earlier and discounted less

C. Paid earlier and discounted more

D. Independent of interest

Correct Answer: B. Paid earlier and discounted less

Question 7

Which identity is a standard relationship between a whole-life assurance and an annuity-due?

A. A_x+d\ddot a_x=1

B. A_x-d\ddot a_x=1

C. A_x+i\ddot a_x=0

D. A_x\ddot a_x=1

Correct Answer: A. A_x+d\ddot a_x=1

Question 8

In a competing-risk model, the probability density of exit through cause j at time t is based on:

A. Total survival multiplied by the cause-specific force

B. Cause-specific survival multiplied by total force only

C. The sum of all discount factors

D. The probability of death alone

Correct Answer: A. Total survival multiplied by the cause-specific force

Question 9

A decreasing death benefit is particularly appropriate for:

A. A fixed maturity endowment only

B. Mortgage protection with a declining loan balance

C. A pure endowment with no death benefit

D. A level immediate annuity only

Correct Answer: B. Mortgage protection with a declining loan balance

Question 10

Which issue most directly threatens the use of a simple joint-life valuation?

A. The existence of an effective interest rate

B. The use of a life table

C. Dependence between the insured lives

D. The definition of a pure endowment

Correct Answer: C. Dependence between the insured lives