4. Survival Models, Mortality and Life Tables

Introduction and Context

Life insurance, annuities, pensions, health insurance and long-term care products all depend on assumptions about how long individuals survive. The actuarial problem is not simply to predict whether a person will die during a particular year. It is to model the timing of death, quantify uncertainty, compare mortality across populations, and translate mortality patterns into premiums, reserves, capital requirements and strategic decisions.

Survival models provide the mathematical language for this purpose. They describe the distribution of future lifetime through survival probabilities, death probabilities, hazard rates and expected lifetimes. Life tables provide an operational summary of mortality experience, while graduation and mortality-improvement methods convert observed data into smooth and credible assumptions for pricing and valuation.

This chapter develops the principal survival models used in actuarial work. It also examines exposure measurement, population heterogeneity, longevity risk, professional judgement and the growing role of statistical and digital methods in mortality analysis.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Define and interpret survival functions, distribution functions, density functions and force of mortality.

2. Distinguish between curtate and complete future lifetime.

3. Derive and apply relationships among survival probabilities, death probabilities and hazard rates.

4. Explain the structure and construction of life tables.

5. Measure mortality exposure and understand the effect of different exposure bases.

6. Describe graduation methods and mortality-improvement models.

7. Analyse longevity risk and the effects of population heterogeneity.

8. Apply survival concepts to insurance, pensions and risk-management decisions.

9. Identify limitations, governance requirements and ethical issues in mortality modelling.

The Random Future Lifetime

Let \(T_x\) denote the future lifetime of a person aged \(x\), measured from exact age \(x\) until death. It is a non-negative continuous random variable. Its survival function is

\[

{}_tp_x=P(T_x>t), \quad t\geq 0.

\]

The notation \({}_tp_x\) means the probability that a life aged \(x\) survives at least \(t\) further years. The corresponding distribution function is

\[

{}_tq_x=P(T_x\leq t)=1-{}_tp_x.

\]

Thus, \({}_tq_x\) is the probability of death within \(t\) years. For one year, the familiar actuarial notation is \(p_x={}_1p_x\) and \(q_x={}_1q_x\), with \(p_x+q_x=1\).

If the distribution is continuous and has density \(f_x(t)\), then

\[

f_x(t)=\frac{d}{dt}{}_tq_x=-\frac{d}{dt}{}_tp_x.

\]

The density describes the relative likelihood of death at a particular future duration, although a density is not itself a probability at a single point. Probabilities are obtained by integrating the density over an interval.

A fundamental relationship is

\[

{}_tp_x=\frac{S(x+t)}{S(x)},

\]

where \(S(a)\) is the probability of surviving from birth to age \(a\), or any proportional survival function defined on attained age. Consequently,

\[

{}_{s+t}p_x={}_sp_x\,{}_tp_{x+s}.

\]

This identity states that survival for \(s+t\) years equals survival for the first \(s\) years multiplied by conditional survival for the following \(t\) years. It is essential in actuarial calculations and reminds us that survival probabilities are conditional on reaching the starting age.

Force of Mortality and Hazard Rates

The force of mortality, also called the hazard rate, at age \(x+t\) is defined by

\[

\mu_{x+t}=\lim_{h\to 0}\frac{P(t\leq T_x<t+h\mid T_x\geq t)}{h}.

\]

It measures the instantaneous rate of death conditional on survival to the relevant age. The force of mortality is not a probability and may exceed one when expressed per year. Its interpretation is local: it describes the intensity of mortality over a very small interval.

The survival function and force of mortality are linked by

\[

\mu_{x+t}=\frac{f_x(t)}{{}_tp_x}=-\frac{d}{dt}\log({}_tp_x),

\]

and therefore

\[

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

\]

The integral

\[

\Lambda_x(t)=\int_0^t\mu_{x+s}\,ds

\]

is the cumulative hazard. Survival is \(e^{-\Lambda_x(t)}\). This formulation is useful because mortality assumptions can be specified either directly through survival probabilities or through a hazard function.

The hazard interpretation is particularly valuable in statistical modelling. For example, a proportional-hazards model may write an individual’s hazard as a baseline hazard multiplied by a factor reflecting health, lifestyle or socioeconomic characteristics. In insurance practice, however, care is needed: underwriting selection, policy duration and behavioural changes can make observed policyholder hazards differ substantially from population hazards.

Curtate and Complete Future Lifetimes

The complete future lifetime \(T_x\) measures the exact remaining lifetime. The curtate future lifetime, denoted by \(K_x\), counts only the number of complete years lived after age \(x\). It is defined as

\[

K_x=\lfloor T_x\rfloor.

\]

If a person aged 40 dies at age 73.6, the complete future lifetime is 33.6 years and the curtate future lifetime is 33 years.

For integer \(k\geq 0\),

\[

P(K_x=k)={}_kp_xq_{x+k}.

\]

The person must survive \(k\) complete years and then die during the next year. The expected curtate future lifetime is

\[

\mathring{e}_x=E[K_x]=\sum_{k=1}^{\infty}{}_kp_x.

\]

The complete expectation of life is

\[

e_x=E[T_x]=\int_0^{\infty}{}_tp_x\,dt.

\]

Under a uniform distribution of deaths between integer ages, the relationship is approximately

\[

e_x\approx \mathring{e}_x+\frac{1}{2}.

\]

This approximation is not universal. It depends on the within-year distribution of deaths and may be inaccurate where mortality changes rapidly, data are sparse or ages are high. Complete lifetime measures are natural for continuous-time annuity theory, whereas curtate lifetime measures commonly arise in discrete annual insurance benefits.

Life-Table Functions and Construction

A life table converts mortality assumptions or observed deaths into a systematic age-by-age framework. A standard table begins with a radix, commonly \(l_0=100,000\), representing an arbitrary number of births or lives entering the table. The radix has no effect on probabilities; it provides a convenient scale.

The main functions are:

• \(l_x\): number surviving to exact age \(x\);
• \(d_x=l_x-l_{x+1}\): number dying between ages \(x\) and \(x+1\);
• \(q_x=d_x/l_x\): probability of death between ages \(x\) and \(x+1\), conditional on survival to age \(x\);
• \(p_x=l_{x+1}/l_x=1-q_x\): probability of surviving one year;
• \(L_x\): total years lived by the table population between ages \(x\) and \(x+1\);
• \(T_x=\sum_{y\geq x}L_y\): total future years lived above age \(x\);
• \(e_x=T_x/l_x\): complete expectation of life at age \(x\), under the table’s conventions.

For a one-year interval, \(L_x\) is often approximated by the average of \(l_x\) and \(l_{x+1}\), giving \(L_x\approx(l_x+l_{x+1})/2\). More refined adjustments are required when deaths are not uniformly distributed throughout the year, particularly for infancy and advanced ages.

A life table may be complete, providing values for every single year of age, or abridged, providing grouped ages such as 0, 1–4, 5–9 and so on. A period life table applies mortality rates observed during a specified calendar period to a hypothetical cohort. A cohort life table follows an actual birth cohort through time and therefore reflects the mortality improvements experienced by that cohort. The distinction is important: period life expectancy is not a forecast of the actual lifetime of a current newborn unless future mortality remains unchanged.

From Observations to a Life Table

Suppose deaths and exposure are available for each age. A central crude mortality rate is

\[

m_x=\frac{D_x}{E_x},

\]

where \(D_x\) is observed deaths and \(E_x\) is exposure to risk. To obtain a probability of death, an assumption is required concerning the timing of deaths within the interval. A common relationship is

\[

q_x=\frac{m_x}{1+(1-a_x)m_x},

\]

where \(a_x\) is the average fraction of the year lived by those dying in the interval. If deaths are uniformly distributed, \(a_x\approx 0.5\), producing \(q_x\approx m_x/(1+0.5m_x)\). At high mortality rates, the distinction between rates and probabilities is material.

Beginning with a radix, the table is then built recursively:

\[

d_x=l_xq_x, \qquad l_{x+1}=l_x-d_x.

\]

This process appears simple, but the credibility of the resulting table depends on data quality, exposure definitions, age accuracy, treatment of withdrawals, late reporting, medical selection and the treatment of open-ended final age groups.

Exposure Measurement and Mortality Investigation

A mortality rate has meaning only when its denominator is properly defined. Exposure to risk represents the amount of time, or number of lives, subject to the possibility of death. Common measures include beginning-of-year lives, mid-year population, central exposure measured in person-years, and policy-year exposure in insurance portfolios.

Central exposure is generally preferable when deaths occur throughout an interval. If a portfolio contains 10,000 lives observed for an average of 0.8 years, its exposure is approximately 8,000 life-years. A rate based on 120 deaths is then 120/8,000 = 0.015 deaths per life-year.

Exposure may be split by attained age, duration since entry, calendar year, sex, underwriting class, product, geography and socioeconomic group. These classifications are not interchangeable. For example, a mortality table by attained age may conceal strong duration effects caused by selection at policy issue. A table by policy duration may reveal high early mortality followed by lower mortality among survivors.

Important sources of error include inaccurate dates of birth, duplicate records, deaths reported after termination, population migration, policy lapses and the use of census counts rather than person-years. A professional investigation reconciles deaths and exposures, documents inclusion rules and performs sensitivity tests.

Graduation: Turning Experience into an Actuarial Basis

Crude age-specific rates often fluctuate because of random variation, especially at young ages, advanced ages or in small portfolios. Graduation is the process of adjusting observed rates to produce a smooth series that reflects the underlying mortality pattern while retaining material features of the data.

The principal approaches are:

1. Parametric graduation. A functional form, such as a Makeham or Gompertz model, is fitted to the data. Parametric methods are interpretable but may be too restrictive if mortality has unusual shapes.

2. Whittaker–Henderson methods. A balance is struck between fidelity to observed rates and smoothness, usually by minimizing a weighted sum of squared deviations and a roughness penalty.

3. Kernel and local smoothing. Nearby ages receive weighted influence, allowing flexible patterns but requiring boundary and bandwidth decisions.

4. Generalized linear models. Death counts may be modelled using Poisson or binomial likelihoods with exposure offsets and selected age, period or cohort effects.

5. Credibility-based approaches. Large and reliable experience may receive greater weight than standard tables, while sparse experience is blended with a broader reference population.

Graduation is not merely cosmetic. An over-smoothed table may remove genuine mortality features; an under-smoothed table may create unstable premiums and reserves. Validation should examine residuals, goodness of fit, out-of-sample performance, parameter stability and the practical consequences for liabilities.

Mortality Laws and Improvement Models

Several classical laws provide useful conceptual benchmarks. The Gompertz law assumes that the force of mortality increases exponentially with age, often represented as \(\mu_x=Bc^x\). The Makeham law adds an age-independent component: \(\mu_x=A+Bc^x\). These models can describe adult mortality over selected age ranges but are not universally appropriate, particularly at childhood ages, extreme old ages or during periods of rapid medical change.

Mortality improvement means that mortality at a given age declines over time. A simple age-period model may express future mortality as

\[

m_{x,t}=m_{x,t_0}(1-r_x)^{t-t_0},

\]

where \(r_x\) is an annual improvement rate. More advanced approaches model the logarithm of mortality as a combination of age, period and cohort effects. The Lee–Carter framework, for example, represents log mortality approximately as an age pattern plus a time-varying mortality index, with parameters estimated from historical data. Extensions may include cohort effects, multiple populations, stochastic period dynamics and constraints to ensure coherent forecasts.

Mortality improvement assumptions must be distinguished from mortality experience. Recent improvement may reflect medical advances, smoking changes or temporary factors. Forecasts must also consider future medical technology, obesity, infectious disease, climate effects, social inequality and behavioural change. A single deterministic improvement scale can conceal substantial uncertainty; scenario analysis and stochastic modelling are therefore important for long-term liabilities.

Longevity Risk and Population Heterogeneity

Longevity risk is the risk that people live longer than expected, increasing the cost of annuities, pensions and long-term care. It affects insurers, pension trustees, governments and households. Systematic longevity risk cannot be diversified away merely by increasing the number of policyholders because the same improvement may affect an entire population.

Risk-management responses include prudent improvement assumptions, reserves, capital margins, longevity swaps, reinsurance, pension buy-ins and buy-outs, delayed retirement incentives and product designs that share longevity outcomes. Basis risk arises when the mortality of a hedging index differs from the mortality of the insured portfolio.

Population heterogeneity is another central issue. Individuals differ in health, income, occupation, genetics, access to care and behaviour. A population mortality rate is an average over groups with different hazards. If high-risk and low-risk lives are mixed, the aggregate hazard may change over time because frailer lives die earlier, leaving a progressively selected survivor population. This is one reason why aggregate hazards need not follow the same pattern as subgroup hazards.

Selection effects are particularly important in life insurance. Medical underwriting may initially produce mortality below that of the general population. Over time, the advantage may reduce as underwriting information becomes less current. Anti-selection can arise when applicants with private knowledge of poor health are more likely to purchase or increase cover. Product design, underwriting, claims investigation and monitoring must therefore be considered alongside statistical modelling.

Applications in Insurance and Pensions

Mortality assumptions enter the calculation of net premiums, reserves and capital. For a discrete whole-life insurance benefit of one unit payable at the end of the year of death, the expected present value is based on terms such as

\[

A_x=\sum_{k=0}^{\infty}v^{k+1}{}_kp_xq_{x+k},

\]

where \(v\) is the discount factor. For a life annuity-due paying one unit at the beginning of each year while the insured is alive,

\[

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

\]

These expressions show why mortality assumptions and interest assumptions interact. Lower mortality increases the value of life annuities but may reduce the value of death benefits. In pension valuation, the choice between period and cohort tables, improvement scales, retirement-age assumptions and survivor benefits can materially change liabilities.

Health and long-term care products may require multi-state models rather than a single death state. A policyholder may move between healthy, disabled, recovered and dead states. Nevertheless, the survival concepts developed here remain foundational because transition intensities and state-occupancy probabilities are built from the same hazard-based logic.

Illustrative Applied Case: Longevity Review for a Pension Portfolio

Background

A fictional occupational pension scheme has 25,000 retired members. Its previous valuation used a mortality table based on historical population data with a fixed annual improvement scale. The trustees are concerned that pensioners may live longer than assumed and that the portfolio differs from the national population in socioeconomic composition.

Situation and Problem

The scheme’s actuary observes that recent experience is broadly consistent with the existing table but that members aged 80 and above have survived at higher rates than projected. The portfolio also contains a large proportion of former professional employees, whose mortality may be lower than the population average. The problem is to decide whether to change the central assumption and how much capital to hold for adverse longevity outcomes.

Analysis

The actuary first verifies exposure and death records, separating attained age, calendar year and pension duration. Crude rates are compared with the current table and graduated using a weighted smoothing method. Residual analysis indicates that the difference is not explained by random fluctuation alone. The actuary then fits an age-period-cohort model and tests alternative improvement scenarios, including lower, central and higher future improvement.

The analysis distinguishes basis risk from model risk. Basis risk concerns whether a population improvement index represents the scheme’s members. Model risk concerns the chosen forecasting form, parameter uncertainty and possible structural change. Sensitivity tests show that a one-year increase in projected life expectancy has a material effect on liabilities, especially for pensions with inflation protection and contingent spouse benefits.

Approach and Solution

The trustees adopt a central assumption based on a graduated member-specific table with an explicit improvement model. They also approve a range of adverse scenarios rather than relying on one deterministic forecast. The valuation report documents data limitations, expert judgements, governance responsibilities and the circumstances that would trigger an assumption review. A longevity hedge is investigated, but its index basis is found to create significant mismatch; consequently, only partial hedging is considered.

Outcome

The scheme reports a higher best-estimate liability and establishes a longevity risk margin. Monitoring is scheduled annually, with a more comprehensive experience investigation every three years or sooner if mortality deviates materially from expectations.

Learning Points

The case demonstrates that mortality modelling is not a one-time selection of a table. It requires data validation, graduation, improvement forecasting, scenario analysis, governance and an explicit assessment of basis risk. The most sophisticated model cannot compensate for poorly defined exposure or unexamined population differences.

Challenges, Limitations and Professional Judgement

Mortality data are subject to sampling variation, recording error, delayed reporting and changing population composition. Small portfolios may produce unstable results, while large populations may still be unrepresentative of a particular insurance book. Extrapolation to extreme ages is especially uncertain because observations are limited and institutional conditions may change.

Common mistakes include treating a mortality rate as a probability, confusing period and cohort life expectancy, ignoring selection by underwriting or duration, using an unadjusted population table for a specialised portfolio, and presenting a single forecast as if it were certain. Another error is to focus on statistical fit while neglecting the financial consequences of tail outcomes.

Actuarial judgement should be transparent rather than hidden. Reports should distinguish observed data, model assumptions, expert adjustments and uncertainty margins. Independent review, sensitivity testing, version control and documented approval are essential elements of model governance.

Ethics, Regulation and Data Governance

Mortality modelling affects pricing, access to insurance, pension security and intergenerational fairness. The use of medical, genetic, lifestyle or socioeconomic data raises questions of privacy, consent, discrimination and proportionality. Data should be relevant, lawfully obtained, securely stored and used consistently with applicable regulation.

Regulatory frameworks commonly require insurers and pension institutions to maintain adequate technical provisions, conduct prudent assumption-setting and demonstrate governance over actuarial models. Professional standards generally require integrity, competence, objectivity, communication of uncertainty and appropriate documentation. Automated or machine-learning models do not remove these duties. If an algorithm produces materially different outcomes for groups, the actuary must investigate whether the difference is justified by legitimate risk factors or reflects data bias.

Technology, Analytics and Emerging Developments

Modern mortality investigations use relational databases, reproducible statistical code, cloud computing and automated data-quality checks. Generalized linear models, survival-analysis software and stochastic simulation support rapid scenario testing. Machine-learning methods can identify nonlinear relationships and interactions, but predictive performance must be balanced against interpretability, stability and fairness.

Emerging work includes real-time experience monitoring, alternative data sources, dynamic underwriting and models integrating medical advances. These developments may improve segmentation but can also magnify privacy and governance risks. A sound actuarial process therefore combines computational capability with explainable assumptions, human review and clear accountability.

Best Practices in Mortality Work

A robust mortality exercise should:

• define the population, observation period, age basis and exposure measure clearly;
• reconcile deaths, entries, exits and late reports;
• separate experience analysis from future improvement assumptions;
• investigate selection, duration, cohort and socioeconomic effects;
• use graduation appropriate to the volume and quality of data;
• validate models through residual, sensitivity and out-of-sample analysis;
• present central, adverse and alternative scenarios;
• document judgement, limitations, approvals and review triggers; and
• communicate financial implications in language understood by decision-makers.

Chapter Summary

Survival models describe the uncertain timing of death through the survival function, distribution function, density and force of mortality. Curtate future lifetime counts completed years, while complete future lifetime measures exact remaining time. Life tables organise mortality into survivorship, deaths, probabilities, exposures and expectations of life. Their construction requires careful treatment of rates, probabilities, within-year timing and open age intervals.

Observed mortality must be interpreted through appropriate exposure measures and often graduated to reduce random fluctuation. Mortality improvement models project future change but are uncertain and should be tested through scenarios. Longevity risk is systematic and particularly important for annuities and pensions. Population heterogeneity, underwriting selection and anti-selection mean that aggregate mortality assumptions may not represent a particular portfolio. Professional practice therefore combines mathematical models, data governance, expert judgement, ethical safeguards and continuous monitoring.

Key Terms

Survival function: The probability that a life survives beyond a specified future duration.

Future lifetime: The random time remaining until death for a person currently aged x.

Distribution function: The probability that future lifetime is less than or equal to a specified duration.

Density function: The derivative of the future-lifetime distribution function for a continuous model.

Force of mortality: The instantaneous conditional rate of death at a specified age or duration.

Hazard rate: Another name for the force of mortality, especially in statistical survival analysis.

Cumulative hazard: The integral of the hazard rate over a specified interval.

Curtate future lifetime: The number of complete future years lived after the current age.

Complete future lifetime: The exact remaining lifetime, including fractional years.

Life table: A structured tabulation of survival, deaths, mortality probabilities and related functions by age.

Radix: The initial arbitrary number of lives from which a life table is constructed.

Exposure to risk: The amount of time or number of lives exposed to the possibility of death.

Central mortality rate: Observed deaths divided by person-years of exposure.

Graduation: Smoothing or adjusting crude mortality rates to estimate an underlying mortality pattern.

Period life table: A table applying mortality observed in a specified period to a hypothetical population.

Cohort life table: A table following an actual birth cohort through successive periods.

Mortality improvement: The reduction in mortality rates over calendar time.

Longevity risk: The risk that people live longer than assumed, increasing long-term liabilities.

Population heterogeneity: Variation in mortality risks among individuals or subgroups in a population.

Selection effect: A difference in mortality arising because entry, underwriting or survival changes the composition of a group.

Review Questions

Short Answer Questions

1. Define the survival function and explain its relationship with the probability of death within a specified period.

2. What is the difference between a mortality rate and a probability of death?

3. Distinguish between curtate and complete future lifetime.

4. Explain the purpose of the radix in a life table.

5. Why is exposure to risk important when calculating mortality rates?

Long Answer Questions

1. Derive the relationship between the force of mortality and the survival function, and interpret the cumulative hazard.

2. Explain the principal functions in a complete life table and describe how they are constructed.

3. Discuss the distinction between period and cohort life tables and its importance in pension valuation.

4. Compare parametric graduation, Whittaker–Henderson graduation and generalized linear modelling.

5. Explain how mortality improvement assumptions should be developed and governed in long-term insurance work.

Analytical / Application Questions

1. An insurer observes 150 deaths over 10,000 life-years of exposure. Explain why the crude rate cannot automatically be treated as the one-year probability of death, and identify the additional assumption required.

2. A pension scheme’s members live longer than the population underlying its standard table. Analyse the possible sources of the difference and recommend a modelling and governance response.

3. An underwriting portfolio has low mortality during the first five policy years but mortality converges toward population levels thereafter. Explain the likely selection mechanism and discuss how the insurer should reflect it in experience analysis and pricing.

22. Multiple Choice Questions with Answers

Question 1

Which expression gives the probability that a life aged x survives for t years?

A. {}_tq_x

B. {}_tp_x

C. q_{x+t}

D. \mu_{x+t}

Correct Answer: B. {}_tp_x

Question 2

If the force of mortality is known over an interval, which formula gives the corresponding survival probability?

A. The integral of the force of mortality

B. The force of mortality divided by exposure

C. The exponential of the negative integrated force of mortality

D. The reciprocal of the force of mortality

Correct Answer: C. The exponential of the negative integrated force of mortality

Question 3

A person aged 50 dies at age 78.4. What are the complete and curtate future lifetimes?

A. 28 years and 28.4 years

B. 28.4 years and 28 years

C. 78.4 years and 78 years

D. 50.4 years and 50 years

Correct Answer: B. 28.4 years and 28 years

Question 4

In a life table, what does d_x represent?

A. The number surviving to age x

B. The probability of surviving one year from age x

C. The number dying between ages x and x+1

D. The total future years lived above age x

Correct Answer: C. The number dying between ages x and x+1

Question 5

Why is a mortality rate based on person-years of exposure generally useful?

A. It ignores the timing of entry and exit

B. It measures deaths relative to the time actually exposed to risk

C. It is always equal to the probability of death

D. It eliminates all sampling variation

Correct Answer: B. It measures deaths relative to the time actually exposed to risk

Question 6

What is the main purpose of graduation?

A. To increase every observed mortality rate

B. To replace exposure data with population counts

C. To produce a stable and credible estimate of the underlying mortality pattern

D. To guarantee that future mortality will be predicted accurately

Correct Answer: C. To produce a stable and credible estimate of the underlying mortality pattern

Question 7

Which statement best distinguishes a period life table from a cohort life table?

A. A period table follows one actual birth cohort throughout life

B. A cohort table uses only mortality from one calendar year

C. A period table applies mortality from a specified period to a hypothetical population

D. There is no conceptual difference between them

Correct Answer: C. A period table applies mortality from a specified period to a hypothetical population

Question 8

Why is longevity risk not fully diversified by increasing the number of annuitants?

A. All annuitants necessarily have identical mortality

B. Future mortality improvement can affect the whole population simultaneously

C. Larger portfolios always have higher mortality

D. Diversification applies only to death benefits

Correct Answer: B. Future mortality improvement can affect the whole population simultaneously

Question 9

A portfolio has mortality below the general population immediately after medical underwriting, followed by convergence over time. What is the most plausible explanation?

A. Random discounting

B. Initial underwriting selection weakening with duration

C. A change from complete to curtate lifetime

D. A life-table radix error

Correct Answer: B. Initial underwriting selection weakening with duration

Question 10

Which issue is an example of model risk in a longevity assessment?

A. A death was recorded twice in the database

B. The chosen mortality-improvement forecasting model may be structurally inappropriate

C. A policyholder paid a premium late

D. The radix was set at 100,000 instead of 10,000

Correct Answer: B. The chosen mortality-improvement forecasting model may be structurally inappropriate