Multiple-Life, Multiple-Decrement and Multi-State Models

Introduction

Insurance contracts frequently cover more than one person, more than one possible cause of exit, or more than two states of existence. A pension may continue until the last member of a couple dies; a disability income policy may pay while an insured is disabled and terminate on recovery, surrender, or death; a long-term-care contract may move a policyholder among healthy, disabled, recovered, and dead states. These problems cannot be represented adequately by a single-life, two-state survival model.

This chapter develops three connected modelling frameworks:

1. Multiple-life models, for joint-life and last-survivor benefits.
2. Multiple-decrement models, for competing causes of exit from a common state.
3. Multi-state models, for transitions among several states over time.

The frameworks are fundamental to pricing, reserving, experience analysis, product design, solvency assessment, and claims-management analytics. Their usefulness depends on clearly defining states, transitions, timing conventions, and assumptions. A model that is mathematically elegant but poorly specified can produce materially misleading insurance liabilities.

Learning objectives

By the end of this chapter, the reader should be able to:

• distinguish joint-life and last-survivor statuses;
• calculate survival probabilities for two or more lives under stated dependence assumptions;
• explain the relationship between crude and net decrement rates;
• construct and interpret a multiple-decrement table;
• use transition probabilities and transition intensities in a multi-state model;
• derive present values for benefits payable in different states or on transitions;
• apply Markov and semi-Markov ideas appropriately;
• identify modelling, data, governance, and ethical risks in practical applications.

1. Multiple-life models

1.1 Basic notation

Let Tₓ denote the future lifetime of a person aged x. For two lives aged x and y, let Tₓ and Tᵧ denote their future lifetimes. The joint-life status remains in force while both lives are alive. Its future lifetime is

The last-survivor status remains in force while at least one life is alive. Its future lifetime is

The notation varies across actuarial texts; the meaning must always be stated explicitly.

For a single life, the survival function is

and the death probability over the next t years is {}_tq_x = 1 - {}_tp_x. For two lives, the joint survival function is

This is the probability that both are alive at time t. The probability that at least one is alive at t is

The final equality follows from inclusion–exclusion.

1.2 Independence assumption

If the future lifetimes are assumed independent, then

and therefore

Independence is often convenient but is not generally realistic. Spouses may share environment, income, lifestyle, socioeconomic conditions, and exposure to common catastrophes. Dependence is particularly important for joint-life annuities, survivorship benefits, catastrophe-sensitive portfolios, and longevity-risk management.

A more general expression uses the joint distribution function. If Fₓᵧ(t₁,t₂) is the joint probability that both future lifetimes do not exceed their respective arguments, then the joint survival probability is determined by the full joint distribution, not by marginal survival probabilities alone. A copula can be used to join marginal survival distributions to a dependence structure, but the resulting dependence model must be calibrated and stress-tested.

1.3 Joint-life and last-survivor probabilities

Under independence, the probability that the first death occurs by time t is

The probability that the second death occurs by time t is

because both must have died by time t. Consequently, the probability that the second death occurs after t is the last-survivor survival probability given above.

For a continuous-time model, if μₓ(t) and μᵧ(t) are the force of mortality functions and the lives are independent, the joint-life force of mortality is

The first death can occur through either life’s death, so the hazards add. This formula is a special case: it assumes independence and a correctly specified continuous-time hazard framework.

1.4 Annuities and insurance benefits

Let vᵗ denote the discount factor for a payment at time t, where v=(1+i)⁻¹ for an annual effective interest rate i. A continuous joint-life annuity paying 1 per year while both lives survive has expected present value

A last-survivor annuity has expected present value

For discrete annual payments at the beginning of each year, the corresponding forms are sums such as

The exact expression depends on whether payments are made in advance or arrears, whether death benefits are paid immediately or at the end of the year, and whether premiums cease at first death or at second death. These timing conventions must not be mixed.

A useful identity is that a last-survivor annuity can be related to two single-life annuities and a joint-life annuity. Under consistent payment timing,

This follows from the indicator identity: at any time, the indicator that at least one life survives equals the sum of the two individual survival indicators minus the indicator that both survive.

Worked example 1: two-life survival

Suppose the one-year survival probabilities are {}₁p_x=0.98 and {}₁p_y=0.96, and assume independence.

The probability that both survive one year is

{}₁p_xy=0.98×0.96=0.9408.

The probability that at least one survives is

{}₁p_{\overline{xy}}=0.98+0.96-0.9408=0.9992.

Thus, the probability of a first death within one year is 0.0592, while the probability that both have died within one year is (1-0.98)(1-0.96)=0.0008.

2. Multiple-decrement models

2.1 Concept and terminology

A multiple-decrement model begins with a population exposed to several mutually exclusive causes of exit. Examples include:

• death from disease, accident, or other causes;
• withdrawal, retirement, disability, or death from an employment scheme;
• lapse, surrender, maturity, or death in a life-insurance portfolio.

The population is initially in an active state. It may leave through one of several decrements, usually without returning to the active state. The causes are called competing risks because an individual who exits through one cause is no longer exposed to the other causes within that decrement table.

Important quantities include:

• lₓ: number alive and still exposed at age x;
• dₓ^(j): number leaving through decrement j between ages x and x+1;
• qₓ^(j): crude, or multiple-decrement, probability of exit through cause j;
• qₓ: total probability of exit from all causes;
• pₓ: probability of remaining in the active state;
• μₓ^(j): force of decrement j;
• μₓ: total force of decrement.

The one-year accounting identity is

and, if no other exits exist,

The crude probability qₓ^(j) is the probability that the individual exits through cause j before age x+1, with all competing causes operating. It is not generally equal to the probability that would be observed if cause j operated alone.

2.2 Forces and transition probabilities

In continuous time, let μ_x^(j)(t) denote the instantaneous force of decrement j at duration t. The total force is

The probability of surviving all decrements for t years is

The probability of exiting through cause j during the interval (0,t] is the cumulative incidence probability

This formula is central: the individual must first remain exposed to time s, then experience cause j at intensity μ_x^(j)(s). Summing over causes gives {}_tq_x=∑_j{}_tq_x^(j)=1-{}_tp_x.

In a one-year table, if the force is constant over the year, then

The units of each μ are inverse time; q and p are dimensionless probabilities.

2.3 Decrement tables

A decrement table may be constructed from an initial radix, commonly 100,000. For each age or duration, the active number is reduced by the exits attributed to each cause. A simplified table is shown below.

For the first year, the crude probabilities are 600/100,000=0.006 for death and 4,000/100,000=0.040 for surrender. The total exit probability is 0.046 and the active-state probability is 0.954.

A table is descriptive and operationally useful, but it must be interpreted carefully. Observed exits are affected by underwriting, product design, claims administration, economic conditions, and the timing of exposure measurement. A surrender rate is not a pure biological hazard, and a death rate observed among policyholders may reflect selection.

2.4 Cause elimination and the net-risk fallacy

Suppose death and surrender are competing decrements. If surrender is eliminated, the resulting death probability is not generally the original crude death probability. Individuals who would have surrendered remain exposed to death, so the death probability usually increases.

Under constant annual forces μ^(D) and μ^(S), the original death probability is

If surrender is removed, the death probability becomes

The distinction between crude and net rates is essential in pricing and experience studies. Crude rates describe the actual competing environment; net rates describe a hypothetical environment in which other decrements have been eliminated.

Worked example 2: competing risks

Assume constant annual forces μ^(D)=0.006 and μ^(S)=0.040. The total force is 0.046. The probability of remaining active for one year is

p=e^-0.046≈0.9550.

The death probability is

q^(D)=(0.006/0.046)(1-0.9550)≈0.00587.

The surrender probability is

q^(S)=(0.040/0.046)(1-0.9550)≈0.03913.

Their sum is approximately 0.04500, equal to 1−p, subject to rounding. If surrender were impossible, the death probability would instead be 1-e^-0.006≈0.00598, slightly higher because exposure to death is no longer interrupted by surrender.

3. Multi-state models

3.1 From decrements to transitions

A multi-state model generalizes the decrement table by allowing movement among several states and, where appropriate, return to a previous state. Typical states include:

• active or healthy;
• disabled;
• unemployed or deferred;
• surrendered or paid-up;
• dead.

A state is not merely a label. It must have an operational definition linked to contract rights, benefit amounts, premium obligations, and data-recording rules. For example, “disabled” may mean inability to perform one’s own occupation, inability to perform any occupation, or satisfaction of a contractual medical definition.

Let the state space be S={1,2,…,m}. Define P_ij(s,t) as the probability of being in state j at time t given state i at time s. In a time-homogeneous Markov model, transition probabilities depend only on elapsed time:

The Markov property states that, conditional on the current state, the future does not depend on the detailed past history. This is often a useful approximation, but duration in state and prior transitions may matter materially. When they do, a semi-Markov or augmented-state model is more appropriate.

3.2 Transition intensity matrix

For a continuous-time Markov model, let μ_ij(t) be the transition intensity from state i to state j, for i≠j. The generator matrix Q(t) has entries

and diagonal entries chosen so each row sums to zero:

Thus Q has units of inverse time. For a time-homogeneous model, the transition probability matrix is

where e^{Qt} is the matrix exponential. It satisfies the Kolmogorov equations

For time-varying intensities, the transition matrix is obtained from the corresponding differential equations; the simple expression e^{Qt} applies only when Q is constant over time.

3.3 Example state structure

Consider a disability-income contract with states:

1. Healthy/active (A)
2. Disabled (D)
3. Recovered (R)
4. Dead (X)

Possible transitions are A→D, A→X, D→R, D→X, and R→X. Depending on contract design, R may return to A, and a recovered person may become disabled again. Surrender may be represented as an absorbing state S if no later benefits are payable.

A conceptual transition structure can be represented by the following text:

A → D → R → X

A → X and D → X are direct mortality transitions. If recovery to active employment is possible, R → A is added. If surrender terminates all rights, A → S and D → S may be included, subject to contract rules.

The model should distinguish a transition that changes benefit entitlement from an administrative event that does not. Otherwise, the financial interpretation of the state process becomes ambiguous.

3.4 State occupancy and transition benefits

Let p_i(t) be the probability of occupying state i at time t. If p(t) is a row vector, then under a time-homogeneous model

Suppose benefit rate b_i(t) is payable while in state i, and a lump sum c_ij(t) is payable on transition i→j. Under a force of interest δ(t), the expected present value is conceptually

where

The first integral values state-occupancy payments. The second values transition payments: p_i(t) is the probability of being in i at t and μ_ij(t) is the instantaneous rate of moving to j. If benefits are discrete or paid with delays, the expression must be modified to reflect the actual payment timing.

4. Markov versus semi-Markov modelling

The Markov assumption is attractive because it simplifies estimation and computation. However, it may fail when:

• disability recovery depends strongly on time already disabled;
• lapse depends on policy duration and premium history;
• mortality depends on the number and timing of previous disability episodes;
• waiting periods or benefit limits create duration effects;
• prior underwriting information continues to influence risk.

A semi-Markov model allows transition intensities to depend on time since entry into the current state, and possibly on the history of earlier states. An alternative is state augmentation. For example, instead of one disabled state, use Disabled-0–12 months, Disabled-13–24 months, and Disabled-over-24 months. This preserves a Markov representation at the cost of a larger state space.

The choice should be driven by materiality, data volume, interpretability, and governance. Greater complexity is not automatically superior. A poorly estimated semi-Markov model may be less reliable than a transparent Markov model with carefully validated assumptions.

5. Worked application: a disability and surrender model

Background

The following is an illustrative case, not an empirical account of a particular insurer. An insurer offers income protection to employed adults. A policyholder begins in the Active state. During each year, the person may become Disabled, surrender, or die. A disabled policyholder may recover, remain disabled, surrender, or die.

Problem

The product team needs to estimate the expected proportion of a cohort receiving disability benefits after two years and to understand how surrender affects the result. The team proposes a four-state annual transition model:

• A: active;
• D: disabled;
• S: surrendered, absorbing;
• X: dead, absorbing.

Use the following one-year transition matrix, with rows representing the state at the beginning of the year and columns the state at the end:

Each row sums to 1. The matrix is a discrete-time transition matrix, not a continuous-time intensity matrix.

Analysis and solution

Assume the cohort starts entirely active, so

p₀=(1,0,0,0).

After one year,

p₁=p₀P=(0.940,0.030,0.025,0.005).

After two years,

p₂=p₁P.

The active component is

0.940×0.940+0.030×0.250=0.8908.

The disabled component is

0.940×0.030+0.030×0.600=0.0462.

The surrendered component is

0.940×0.025+0.030×0.100+0.025=0.0562.

The dead component is

0.940×0.005+0.030×0.050+0.005=0.0102.

Thus

p₂=(0.8908,0.0462,0.0562,0.0102),

which sums to 1. After two years, 4.62% of the original cohort is disabled. This is an occupancy probability, not the probability of ever having become disabled: some individuals may have entered disability and subsequently recovered to active status, although this particular matrix does not include a D→A transition. If a recovery transition is intended, it must appear explicitly in the matrix.

Outcome and interpretation

The model indicates that surrender materially reduces the active population and therefore reduces future disability exposure. Removing surrender without recalibrating other transitions would generally overstate or understate liabilities depending on how the eliminated individuals differ from continuing policyholders. The insurer should distinguish:

• policyholder exits caused by surrender;
• recovery from disability;
• disability benefit termination;
• death while active or disabled.

Learning points

1. Every row of a transition matrix must sum to one in a discrete-time model.
2. Absorbing states represent terminal outcomes within the model horizon.
3. State occupancy is different from transition incidence.
4. A matrix must reflect contractual definitions, not merely convenient categories.
5. Removing one transition requires reconsideration of the remaining transition probabilities and exposure base.

6. Estimation, validation, and implementation

Data preparation

Multi-state analysis requires reliable event histories. Each record should include entry date, state changes, dates of transitions, censoring date, policy status, and relevant covariates. Duplicate events, delayed reporting, retroactive claims decisions, and inconsistent state definitions can bias estimates.

The analyst must distinguish left truncation, right censoring, competing exits, and missing dates. A policy observed only after underwriting is not necessarily representative of the general population. Exposure should be measured consistently with the rate being estimated.

Estimation approaches

For discrete annual data, transition probabilities can be estimated as counts of transitions from i to j divided by the number exposed in state i, subject to censoring and weighting adjustments. For continuous-time models, cause-specific intensities may be estimated using event-history methods. The Aalen–Johansen estimator is commonly used for nonparametric transition probabilities in competing-risk and multi-state settings. Regression models may introduce age, duration, occupation, health, benefit amount, or economic variables.

The distinction between transition intensity and cumulative incidence must be preserved. A cause-specific hazard describes instantaneous risk among those still eligible for the transition; cumulative incidence describes the resulting probability after accounting for competing events.

Validation

Validation should include:

• reconciliation of transitions and exposure totals;
• comparison of predicted and observed state occupancies;
• back-testing by calendar period and cohort;
• sensitivity tests for dependence and duration effects;
• stress tests for mortality, lapse, recovery, and economic conditions;
• independent review of assumptions and implementation code.

For a transition matrix, numerical checks include non-negative entries, row sums equal to one, and plausible long-term behavior. For an intensity matrix, off-diagonal entries must be non-negative and each diagonal must equal the negative sum of the other entries in its row.

7. Practical and professional applications

Multiple-life models support survivor pensions, joint annuities, reinsurance treaties, mortgage protection, and family income benefits. Multiple-decrement models support lapse-sensitive pricing, employee benefits, pension decrement studies, and underwriting analysis. Multi-state models support disability income, critical illness, long-term care, health insurance, annuity guarantees, and policyholder-behaviour modelling.

For management, the main value is not only a more detailed expected value. Multi-state models reveal where liabilities arise: duration in disability, recovery rates, surrender timing, and death while receiving benefits. They can therefore inform claims intervention, rehabilitation programmes, product redesign, capital allocation, and reinsurance decisions.

8. Limitations, governance, and ethics

Model risk arises from misspecified states, inappropriate independence assumptions, inaccurate transition rates, structural breaks, and overreliance on historical experience. Mortality and morbidity improvements may make old tables obsolete. Economic stress can alter lapse and surrender behaviour. Medical advances may change recovery patterns and claim duration.

Governance should require documented definitions, data lineage, assumption approval, version control, independent validation, and clear limitations. Regulatory reporting may require prescribed mortality, morbidity, discounting, or capital assumptions; a commercially useful internal model must still be reconciled with regulatory bases.

Ethical issues are substantial. Health and disability models use sensitive personal information. Data use must comply with applicable privacy and insurance regulation, including purpose limitation, security, retention controls, and lawful processing. Automated underwriting or claims triage should be tested for disparate impact. A model should not use proxies that unfairly reproduce protected characteristics, and customers should receive appropriate explanations where automated decisions materially affect coverage or benefits.

Artificial intelligence can assist with data cleaning, claim classification, and forecasting, but it does not remove actuarial responsibility. Black-box predictions may be difficult to reconcile with contractual definitions and regulatory expectations. AI systems require audit trails, drift monitoring, human oversight, and controls against leakage of future information into training data.

9. Contemporary developments and best practice

Current practice increasingly combines traditional actuarial tables with granular event histories, generalized linear models, survival analysis, Bayesian methods, and machine learning. The best model is not necessarily the most complex. A robust workflow is:

1. define states and benefits contractually;
2. identify all material transitions;
3. select discrete-time, continuous-time, Markov, or semi-Markov structure;
4. estimate from exposure-consistent data;
5. validate transition probabilities and financial outputs;
6. stress key assumptions;
7. document limitations and decision use;
8. monitor experience and recalibrate responsibly.

Dependence modelling deserves special attention. For multiple lives, common-shock and socioeconomic dependence can materially affect tail outcomes. For multi-state insurance, dependence between morbidity, mortality, lapse, and economic variables may be relevant. Scenario analysis should therefore complement central estimates.

Chapter Summary

Multiple-life models value benefits involving two or more insured lives. The joint-life status ends at the first death, whereas the last-survivor status ends at the second death. Under independence, joint survival is the product of individual survival probabilities, but dependence must be considered where common exposures or shared characteristics are material.

Multiple-decrement models describe competing exits from an active population. Crude decrement probabilities include the effects of all competing causes, while net probabilities describe hypothetical cause-elimination environments. Continuous-time forces add across causes, and cumulative incidence is obtained by weighting a cause-specific force by survival to the event time.

Multi-state models extend decrement tables by representing transitions among active, disabled, recovered, surrendered, dead, and other states. Transition matrices are used in discrete time; intensity or generator matrices are used in continuous time. Markov models condition future behaviour on the current state, while semi-Markov or augmented-state models accommodate duration and history effects.

Applications require disciplined state definitions, reliable event histories, consistent payment timing, validation, sensitivity analysis, and governance. Privacy, fairness, explainability, and regulatory compliance are essential when models use health, behavioural, or automated decision data.

Key Terms

Multiple-life model: An actuarial model involving two or more insured lives.

Joint-life status: A status that continues only while all specified lives are alive.

Last-survivor status: A status that continues while at least one specified life remains alive.

Joint survival probability: The probability that all specified lives survive to a stated time.

Competing risks: Multiple mutually exclusive causes that can terminate exposure.

Multiple-decrement table: A table showing exits from an exposed population by cause.

Crude decrement probability: The probability of exit through a cause while competing causes operate.

Net decrement probability: The probability for a cause in a hypothetical setting where other causes are removed.

Force of decrement: The instantaneous rate of exit through a specified cause.

Cumulative incidence: The probability of experiencing a specified event by a given time in the presence of competing risks.

State: A condition or category determining current rights, exposure, or benefits.

Transition probability: The probability of moving from one state to another over an interval.

Transition intensity: The instantaneous rate of transition between two states.

Generator matrix: A matrix of continuous-time transition intensities with rows summing to zero.

Absorbing state: A state that cannot be left once entered.

Markov property: The assumption that the future depends on the current state, not the detailed past.

Semi-Markov model: A model allowing transition behaviour to depend on time spent in a state or prior history.

State occupancy: The probability or proportion of a cohort in a state at a specified time.

Cause elimination: Hypothetical removal of one competing decrement to derive a net risk.

Review Questions

5 Short Answer Questions

1. Define joint-life and last-survivor statuses.
2. What is the difference between a crude and a net decrement probability?
3. Why do the rows of a discrete-time transition matrix sum to one?
4. State the Markov property in the context of a multi-state insurance model.
5. What is an absorbing state? Give one insurance example.

5 Long Answer Questions

1. Derive the survival probability of a last-survivor status for two independent lives using inclusion–exclusion.
2. Explain how cause-specific forces produce cumulative incidence probabilities in a multiple-decrement model.
3. Compare discrete-time transition matrices with continuous-time generator matrices.
4. Discuss when a semi-Markov model is preferable to a Markov model for disability insurance.
5. Explain the principal governance and ethical risks in health-related multi-state modelling.

3 Analytical / Application Questions

1. A couple has one-year survival probabilities 0.97 and 0.95. Assuming independence, calculate the probabilities that both survive and that at least one survives.
2. Construct a two-year cohort projection for a three-state model with active, disabled, and dead states, given a transition matrix of your choice. Explain the interpretation of each resulting state occupancy.
3. An insurer observes high surrender among policyholders with worsening health. Explain why treating surrender as independent of mortality could bias a disability or mortality liability estimate, and propose two validation tests.

Multiple Choice Questions with Answers

Question 1

For two independent lives, which expression represents the probability that both survive t years?

A. {}_tp_x+{}_tp_y

B. {}_tp_x×{}_tp_y

C. 1-{}_tp_x{}_tp_y

D. (1-{}_tp_x)(1-{}_tp_y)

Correct Answer: B. {}_tp_x×{}_tp_y

Question 2

A joint-life status terminates when:

A. The first specified life dies

B. The second specified life dies

C. Both specified lives survive

D. The policyholder surrenders only

Correct Answer: A. The first specified life dies

Question 3

If two constant decrement forces are μ^(D)=0.006 and μ^(S)=0.040, the total force is:

A. 0.034

B. 0.040

C. 0.046

D. 0.240

Correct Answer: C. 0.046

Question 4

A crude death probability in a multiple-decrement table is measured:

A. With all competing decrements operating

B. After surrender is eliminated

C. Only among people who survive all other causes

D. Without regard to exposure

Correct Answer: A. With all competing decrements operating

Question 5

Which condition must hold for every row of a discrete-time transition matrix?

A. Its entries must all be equal

B. Its diagonal must be zero

C. Its entries must sum to one

D. Its entries must sum to zero

Correct Answer: C. Its entries must sum to one

Question 6

In a continuous-time generator matrix, an off-diagonal entry represents:

A. A discount factor

B. A transition intensity

C. A cumulative probability only

D. A reserve amount

Correct Answer: B. A transition intensity

Question 7

Which is an absorbing state?

A. Active, when recovery is possible

B. Disabled, when recovery is possible

C. Surrendered, when no re-entry is allowed

D. Healthy, when disability is possible

Correct Answer: C. Surrendered, when no re-entry is allowed

Question 8

If p₀=(1,0,0,0) and the first row of a transition matrix is (0.94,0.03,0.025,0.005), the one-year disabled occupancy is:

A. 0.005

B. 0.025

C. 0.030

D. 0.940

Correct Answer: C. 0.030

Question 9

A semi-Markov model is especially useful when transition risk depends on:

A. Only the current calendar year

B. Time already spent in the current state

C. The discount rate alone

D. The matrix row sum

Correct Answer: B. Time already spent in the current state

Question 10

If surrender is eliminated while mortality intensity remains unchanged, the resulting net mortality probability generally differs from the crude mortality probability because:

A. The discount rate changes automatically

B. People who would have surrendered remain exposed to mortality

C. All mortality transitions become impossible

D. The radix must equal zero

Correct Answer: B. People who would have surrendered remain exposed to mortality