Survival Models and Life Tables

Introduction and context

Actuarial mathematics studies uncertain future cash flows whose timing or occurrence depends on human life. Mortality is therefore not merely a demographic statistic: it is a primary risk driver for life insurance, annuities, pensions, health insurance, longevity swaps and solvency assessment. This chapter develops the mathematical language used to describe mortality, beginning with the future lifetime random variable and progressing to life tables, selection effects and mortality improvement.

A survival model must distinguish carefully between:

• the probability that a person survives to a specified age;
• the probability of dying during an interval, conditional on survival to its beginning;
• the instantaneous rate at which mortality is occurring; and
• the way empirical mortality observations are arranged in a life table.

These quantities are related, but they are not interchangeable. Sound actuarial analysis depends on knowing which denominator and time scale each measure uses.

Learning objectives

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

1. Define and interpret the survival function, distribution function, density and hazard rate.
2. Relate mortality probabilities, force of mortality and survival probabilities.
3. Distinguish curtate from complete future lifetime models.
4. Construct and interpret the principal columns of a life table.
5. Explain select and ultimate mortality and the effects of underwriting selection.
6. Assess mortality improvement assumptions and their consequences for insurance and pension liabilities.
7. Identify important data, governance and modelling limitations.

1. The future lifetime random variable

Let T_x denote the future lifetime of a life currently aged x. It is measured from age x until death. Thus, if death occurs at exact age x+t, then T_x = t. The variable is non-negative and may be continuous, discrete, or represented by a mixed model depending on the actuarial application.

The complete future lifetime is the exact remaining lifetime. The curtate future lifetime is the completed number of future years lived:

where ⌊·⌋ denotes the floor function. If a person aged 40 dies at age 73.6, then T₄₀=33.6 and K₄₀=33. The curtate variable is especially useful when benefits are paid at integer policy anniversaries.

The age-at-death random variable is X, with X=x+T_x for a life currently aged x.

2. Survival and mortality functions

2.1 Distribution and survival functions

The cumulative distribution function of T_x is

It represents the probability that a life aged x dies within the next t years. The survival function is

where {}_t p_x is the standard actuarial notation for survival from age x to age x+t. For a continuous lifetime distribution, replacing “>” with “≥” does not change the probability. In a model with a positive probability of death at an exact time, the distinction must be retained.

The probability of death within the next t years is

The familiar one-year mortality probability is q_x = {}₁q_x, while p_x = {}₁p_x and therefore p_x+q_x=1.

For ages x and x+t, the conditional survival probability is

Under the usual continuous-life interpretation,

Consequently, survival probabilities satisfy the composition rule

This identity is fundamental: survival over a longer period is the product of survival over successive conditional periods.

2.2 Density and hazard rate

If T_x has a differentiable distribution, its density is

The density is a probability per unit of time, not a probability at an exact point. For a continuous variable, Pr(T_x=t)=0.

The force of mortality, or hazard rate, at duration t is

provided S_x(t)>0. It measures the instantaneous rate of death at age x+t, conditional on survival to that age. Its unit is inverse time, such as deaths per person-year in an instantaneous limiting sense.

Since f_x(t)=μ_x+tS_x(t), integration gives

and

These formulas assume an absolutely continuous lifetime distribution and a measurable, non-negative force of mortality. They show that the force of mortality is not itself a probability: it may exceed 1 per year, whereas a probability must lie between 0 and 1.

2.3 Conditional mortality within an interval

The probability of death between ages x+t and x+t+s, given survival to age x+t, is

Using the hazard representation,

If the force is constant over the interval, μ_x+t+r=μ, then

The approximation q≈μs is reasonable only when μs is small. It is not an identity.

3. Important mortality models

3.1 Constant-force model

The constant-force model assumes mortality does not vary within the age or duration interval under consideration. If μ_x=μ for a specified range, then

For example, if μ=0.02 per year, the five-year survival probability is e^−0.1≈0.9048, and the five-year death probability is approximately 0.0952. The assumption is convenient for interpolation but may be unrealistic over broad adult age ranges.

3.2 Uniform distribution of deaths (UDD)

Life tables commonly provide annual probabilities q_x rather than a continuous hazard. Under UDD, deaths are assumed uniformly distributed throughout each year of age. For 0≤t≤1,

and, for the corresponding within-year force,

UDD is an interpolation assumption. It is not a claim that every population experiences a uniform pattern of deaths within each year.

3.3 Constant force within each year

Another interpolation method assumes that the force is constant over the year of age. If the annual mortality probability is q_x, then

where p_x=1−q_x. Hence

This differs from UDD except at the endpoints. The chosen interpolation can affect monthly benefits, reserves and premium calculations, particularly at high ages or when cash flows are frequent.

4. Curtate and complete future lifetime

4.1 Curtate distribution

For the curtate future lifetime K_x,

This means the life survives k complete years and dies during the following year. The tail probability is

Therefore, for a non-negative integer-valued random variable,

provided the expectation is finite.

4.2 Complete distribution and expectation

For a continuous future lifetime,

This follows from the tail-integral identity for non-negative random variables. More generally, for a measurable function g,

when the integral exists.

Curtate and complete expectations are not generally equal. Under UDD, a common relationship is

when the lifetime is modelled with deaths uniformly distributed in each year and there is no terminal-age complication. This is a model-dependent result, not a universal correction.

Worked example 1: survival and curtate probability

Suppose a life aged 60 has annual survival probabilities

p₆₀=0.96, p₆₁=0.95 and p₆₂=0.94. Then

Thus the probability of surviving three complete years is 0.85728, or 85.728%. The probability that the curtate future lifetime equals 2 is

The result concerns death between ages 62 and 63, conditional on being age 60 at the outset.

5. Life tables

A life table converts mortality probabilities into a standardized population schedule. It may be based on a real population, an insured portfolio, a pension scheme or a regulatory standard. A table is not necessarily a forecast; it is a statistical model or summary of mortality experience.

Typical columns are shown below.

The radix l₀ is an arbitrary starting number, often 100,000. Because the table is usually scaled, changing the radix does not change the probabilities.

The basic one-year relationships are

For an integer period n,

and

Under a standard one-year table with deaths spread within the year, L_x is often approximated by

This midpoint expression assumes approximately uniform deaths over the year. A more general definition is the expected number of person-years lived by the cohort between ages x and x+1.

The total future person-years above age x are

where ω is the limiting age of the table. The complete expectation of life is

This is a cohort expectation under the mortality schedule, not necessarily the observed average lifespan of a real contemporary population.

Constructing a life table: a compact example

Assume a radix of 100,000 and annual mortality probabilities as follows.

At age 60, d₆₀=100000×0.04=4000. The next row begins with l₆₁=96000. The three-year survival probability is therefore 85728÷100000=0.85728, agreeing with the previous example.

6. Estimating mortality from data

Mortality data can be arranged by calendar year, age, sex, duration since underwriting, policy type and socioeconomic or medical characteristics. The central empirical quantities are deaths and exposure to risk. A crude central death rate is

where D_x is the number of deaths and E_x is central exposure, usually person-years lived or exposed during the period. Unlike q_x, m_x is a rate and need not be bounded by 1.

The relationship between a central rate and a one-year probability requires an assumption about the timing of deaths. Under a constant force approximation,

If deaths are approximately uniform and exposure is measured appropriately, other conversions may be more suitable. Analysts must avoid treating rates, probabilities and hazards as synonymous.

Practical estimation issues include incomplete death registration, delayed reporting, left truncation, right censoring, withdrawals, changing exposure and small-cell volatility. Credibility methods, graduation and smoothing are used to produce a stable actuarial basis. Graduation should preserve broad mortality patterns while avoiding excessive reaction to random fluctuations.

7. Selection and duration effects

7.1 Select and ultimate mortality

Underwriting creates selection because insured lives are not a random sample of the general population. Applicants may be screened by health, occupation, income, lifestyle and medical history. Immediately after underwriting, mortality may be lower than that of otherwise comparable lives without recent selection.

A select-and-ultimate table represents mortality by age at selection and duration since selection. It may use notation such as

where [x] denotes age at selection and t denotes duration since selection. After a specified select period, mortality becomes ultimate and depends primarily on attained age:

for durations beyond the assumed selection period. This equality is a modelling assumption, not a biological law.

Selection can be adverse as well as favourable. For example, guaranteed-issue products may attract applicants with poorer health, while lapse or anti-selection may cause healthier policyholders to leave when premiums are not competitive. Re-entry, policy replacement and underwriting changes can create additional duration effects.

7.2 Policyholder behaviour and mortality

Mortality assumptions may interact with lapses. If unhealthy policyholders are more likely to retain cover and healthy policyholders more likely to lapse, the surviving portfolio can deteriorate over time. Therefore, mortality and lapse assumptions should be assessed jointly rather than independently whenever behaviour is material.

8. Mortality improvement and longevity risk

A static period life table describes mortality rates observed during a particular period. A projection model introduces calendar time. A common multiplicative framework is

where μ_x⁽⁰⁾ is a base force of mortality, R_x,t is an improvement factor, x is age and t is calendar time. If improvement is represented by an annual reduction rate r_x,t, then

with 0≤r_x,t≤1 under this interpretation. A 2% annual reduction applied for 10 years gives a factor of 0.98¹⁰≈0.8171, meaning mortality is about 81.71% of its starting level, before allowing for changes in age or other effects.

The Lee–Carter model is a widely used statistical framework. In one common form,

where m_x,t is a central death rate, a_x describes the average age pattern, b_x measures age sensitivity to the mortality index, κ_t is a time-varying mortality index and ε_x,t is a residual. Identification constraints are required because different combinations of parameters can produce the same fitted rates. Extensions may include cohort effects, multiple populations, non-linear trends and stochastic period models.

Improvement assumptions affect annuity liabilities strongly. If pensioners live longer than expected, annuity payments continue for more years. Insurers therefore face longevity risk, while life insurers selling mortality protection may benefit financially from improvement, subject to product mix and other risks.

9. Worked application: annuity sensitivity to mortality improvement

Consider an illustrative, notional pension portfolio with 10,000 lives aged 65. Suppose a simplified model assigns a one-year survival probability of 0.98 in each of the next five years, ignoring discounting and payments after year five. The expected number alive at each year-end is:

for year k. The expected total of year-end payment units over five years is

Now suppose mortality improvement raises the annual survival probability to 0.985. The corresponding total is

The simplified expected payment units increase by approximately 696, or 1.48%. This is not a complete annuity valuation because it excludes discounting, exact payment timing, age-varying mortality, expenses, withdrawals and payments beyond five years. It nevertheless demonstrates why modest annual changes compound into material liability effects.

10. Illustrative case study: pricing a selected life portfolio

Background

An insurer issues medically underwritten term assurance to applicants aged 45–65. Historical pricing used an ultimate population table, even though underwriting was intensive and medical evidence was collected at application.

Problem

Actual claims during the first three policy durations were materially below the pricing basis, while claims after duration ten were close to the ultimate basis. Management must decide whether to reduce premiums, strengthen reserves, or revise the mortality basis.

Analysis

An actuary separates experience by age at entry and duration since underwriting. The analysis identifies three features:

1. First-duration mortality is low because severe risks are declined and accepted lives are selected.
2. Mortality increases gradually toward the ultimate level as the selection effect wears off.
3. A recent medical-treatment trend suggests that ultimate mortality may also be improving.

The actuary checks exposure definitions, claims delays, policy withdrawals, claim coding and credibility. A simple comparison of deaths divided by the number of policies is rejected because policies have different durations and exposure times.

Approach and solution

The insurer adopts a select-and-ultimate table for new business, with credibility-weighted experience for early durations and an external industry table for later durations. A separate improvement scale is applied to future calendar years. Sensitivity tests examine:

• no mortality improvement;
• central improvement;
• accelerated improvement;
• weaker underwriting selection;
• adverse anti-selection after premium increases.

Pricing uses the selected basis, while valuation uses prudent margins and governance-approved stresses. Experience is monitored quarterly, but changes to the table require sufficient credibility and documented expert review.

Outcome

The revised basis produces lower expected early claims but does not automatically justify equivalent premium reductions. Acquisition expenses, capital costs, lapse risk, guarantees and profit requirements remain relevant. The insurer instead retains part of the margin, improves risk classification and establishes a formal mortality-monitoring dashboard.

Learning points

The case demonstrates that mortality is multidimensional. Age alone may be inadequate; duration, selection, calendar year, product design and behaviour can all matter. It also illustrates the distinction between best-estimate pricing assumptions and prudent valuation assumptions.

11. Limitations, governance and professional judgement

Mortality models are exposed to both parameter risk and model risk. Parameter risk concerns uncertainty in estimated rates. Model risk concerns an unsuitable structure, such as assuming a constant improvement rate when improvement varies by age or cohort.

Important controls include:

• clear definitions of age, exposure, death and policy duration;
• reconciliation between source data and model inputs;
• documented interpolation and extrapolation methods;
• credibility analysis and statistical diagnostics;
• sensitivity, scenario and reverse-stress testing;
• independent model validation;
• version control and reproducibility;
• monitoring of actual-to-expected mortality;
• explicit treatment of uncertainty and margins.

Ethical issues arise when mortality models use sensitive health or socioeconomic information. Risk classification should comply with applicable insurance law, privacy requirements and anti-discrimination standards. Machine-learning models may detect predictive patterns, but opacity, proxy discrimination, data drift and lack of explainability can make them unsuitable for ungoverned underwriting decisions. Human oversight, fairness testing and audit trails are essential.

Contemporary developments include high-frequency mortality monitoring, medical and wearable data, pandemic scenario analysis, cause-of-death modelling, stochastic longevity models and longevity-risk transfer through reinsurance or capital-market instruments. These tools improve responsiveness but do not remove uncertainty. Greater data volume can amplify bias or produce false precision if definitions and exposures are poor.

Best-practice checklist

Before approving a mortality basis, an actuary should ask:

1. What population and product does the table represent?
2. Are deaths, exposures and durations measured consistently?
3. Is the basis period, cohort or projected, and is that choice appropriate?
4. Have selection, underwriting and lapse interactions been considered?
5. Are interpolation, terminal-age and extrapolation assumptions documented?
6. Are improvement assumptions supported by evidence and sensitivity-tested?
7. Is uncertainty reflected in pricing, reserving and capital decisions?
8. Can another qualified reviewer reproduce the results?

Chapter Summary

The future lifetime T_x provides the foundation for survival modelling. Its distribution function describes death by a specified time, while the survival function {}_t p_x describes survival to that time. The force of mortality μ_x+t is an instantaneous conditional rate and determines survival through an integrated hazard.

Curtate future lifetime is appropriate when time is measured in completed years; complete future lifetime retains the exact time of death. Life tables translate mortality probabilities into survivors, deaths, exposures and expectations of life. UDD and constant-force assumptions provide alternative ways to interpolate annual data.

Mortality varies by age, selection duration, calendar year, cohort and portfolio characteristics. Select-and-ultimate tables capture underwriting effects, while mortality-improvement models project future changes. These assumptions have direct consequences for premiums, reserves, annuity values and solvency. Reliable practice requires sound data definitions, transparent assumptions, validation, sensitivity testing and ethical governance.

Key Terms

Survival function: Probability that a future lifetime exceeds a specified duration.

Future lifetime: Remaining lifetime of a person currently aged x.

Complete future lifetime: Exact remaining lifetime measured continuously.

Curtate future lifetime: Completed number of future years lived.

Mortality probability: Probability of death within a specified interval, conditional on survival to its beginning.

Survival probability: Probability of surviving a specified interval.

Density: Rate at which probability is distributed across possible death times.

Force of mortality: Instantaneous conditional rate of death.

Hazard rate: General statistical term for the force of mortality.

Life table: Structured schedule summarising survival, death and exposure by age.

Radix: Arbitrary initial number of lives in a life table.

Select mortality: Mortality dependent on age at selection and duration since selection.

Ultimate mortality: Mortality after the selection effect is assumed to have worn off.

UDD: Uniform distribution of deaths within each age year.

Central death rate: Deaths divided by an appropriate person-time exposure.

Mortality improvement: Reduction in mortality rates over calendar time.

Longevity risk: Risk that people live longer than assumed.

Graduation: Smoothing mortality observations to obtain a stable actuarial schedule.

Anti-selection: Behaviour that causes risks to differ systematically from the assumed portfolio.

Review Questions

Short Answer Questions

1. Define the survival function and explain its relationship with the distribution function.
2. Why is the force of mortality not itself a probability?
3. Distinguish complete future lifetime from curtate future lifetime.
4. What is the purpose of the radix in a life table?
5. Explain the difference between select and ultimate mortality.

Long Answer Questions

1. Derive the relationship between the force of mortality and the survival function, stating the required assumptions.
2. Explain the principal columns of a life table and show how they are connected.
3. Compare UDD and constant-force interpolation, including their practical implications.
4. Discuss the statistical and actuarial challenges in estimating mortality from portfolio data.
5. Explain how mortality improvement affects life insurance and pension liabilities.

Analytical / Application Questions

1. A portfolio has annual survival probabilities of 0.97, 0.965 and 0.96 over three successive years. Calculate the three-year survival probability and the probability of dying during the third year, conditional on survival to the start of the first year.
2. An insurer observes lower mortality in durations one to three and mortality close to an industry table thereafter. Recommend an appropriate mortality structure and explain the data and governance checks required.
3. Compare the liability implications of two five-year survival assumptions, 0.98 and 0.985 per year, for a pension portfolio. Explain why the difference should not be interpreted as a complete valuation result.

Multiple Choice Questions with Answers

Question 1

Which quantity is defined as Pr(T_x > t)?

A. The density function

B. The survival function

C. The force of mortality

D. The central death rate

Correct Answer: B. The survival function

Question 2

If q_x=0.04, what is p_x?

A. 0.04

B. 0.96

C. 1.04

D. 25

Correct Answer: B. 0.96

Question 3

Under a constant force μ, which expression gives survival for t years?

A. 1−μt

B. μ^t

C. e^−μt

D. e^μt

Correct Answer: C. e^−μt

Question 4

A life aged 40 dies at age 73.6. What is the curtate future lifetime?

A. 33

B. 33.6

C. 40

D. 73.6

Correct Answer: A. 33

Question 5

In a life table, which relationship is correct?

A. d_x=l_x+p_x

B. l_x+1=l_xq_x

C. d_x=l_xq_x

D. q_x=l_xd_x

Correct Answer: C. d_x=l_xq_x

Question 6

A life table has l₆₀=100000 and l₆₃=85728. What is {}₃p₆₀?

A. 0.14272

B. 0.85728

C. 1.1677

D. 85,728

Correct Answer: B. 0.85728

Question 7

Which statement about the force of mortality is correct?

A. It must always be between 0 and 1

B. It is an instantaneous conditional rate

C. It is identical to annual q_x in every model

D. It is measured as a number of survivors

Correct Answer: B. It is an instantaneous conditional rate

Question 8

What does UDD assume within a one-year age interval?

A. The force is zero

B. Deaths are uniformly distributed

C. All lives die at the midpoint

D. Mortality improves exponentially

Correct Answer: B. Deaths are uniformly distributed

Question 9

Which notation most directly represents mortality depending on age at selection and duration?

A. q_[x]+t

B. q_x+t only

C. l₀

D. T_x

Correct Answer: A. q_[x]+t

Question 10

If annual survival is 0.985 for five years, which expression gives the probability of surviving all five years under constant annual probabilities?

A. 5×0.985

B. 1−0.985⁵

C. 0.985⁵

D. 0.985÷5

Correct Answer: C. 0.985⁵