4.1 Mortality tables and life expectancy

Mortality tables are crucial tools in actuarial math, estimating death and survival probabilities at various ages. They're built using data from vital statistics, insurance records, and censuses, with assumptions to simplify the process and ensure consistency.

Key functions in mortality tables include probability of death, survival, and force of mortality. These tables are used to calculate life expectancy, price insurance and annuities, value pension plans, and make population projections. Understanding their limitations is essential for accurate analysis.

Types of mortality tables

• Mortality tables are essential tools in actuarial mathematics used to estimate the probability of death and survival at various ages
• Different types of mortality tables exist to cater to specific populations, time periods, and insurance products
• The choice of mortality table depends on the purpose of the analysis and the characteristics of the group being studied

Construction of mortality tables

Data sources for mortality tables

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• Mortality tables are constructed using data from various sources such as national vital statistics, insurance company records, and population censuses
• The data typically includes the number of deaths and the exposed-to-risk population at each age
• The quality and completeness of the data directly impact the accuracy of the resulting mortality table

Assumptions in mortality table construction

• Several assumptions are made when constructing mortality tables to simplify the process and ensure consistency
• The population is assumed to be homogeneous, meaning that individuals within the same age group have the same probability of death
• The mortality rates are assumed to remain constant over the period for which the table is constructed
• The exposed-to-risk population is assumed to be stationary, with no migrations or changes in the age distribution

Graduation techniques for mortality rates

• Graduation techniques are used to smooth out irregularities in the observed mortality rates and produce a more reliable and consistent set of rates
• Common graduation techniques include moving average methods, parametric models (Gompertz, Makeham), and spline interpolation
• The choice of graduation technique depends on the nature of the data and the desired level of smoothness

Key functions in mortality tables

Probability of death

• The probability of death, denoted as $q_x$, represents the likelihood that an individual aged $x$ will die before reaching age $x+1$
• It is calculated as the ratio of the number of deaths at age $x$ to the number of individuals alive at age $x$
• The probability of death is a fundamental component in the construction of mortality tables and is used to derive other key functions

Probability of survival

• The probability of survival, denoted as $p_x$, represents the likelihood that an individual aged $x$ will survive to age $x+1$
• It is the complement of the probability of death, calculated as $p_x = 1 - q_x$
• The probability of survival is used to calculate the number of individuals expected to be alive at each age in a mortality table

Force of mortality

• The force of mortality, denoted as $\mu_x$, represents the instantaneous rate of mortality at age $x$
• It is defined as the limit of the probability of death over a small interval of time, divided by the length of the interval
• The force of mortality is a continuous function and is related to the probability of death through the equation $q_x = 1 - e^{-\int_x^{x+1} \mu_t dt}$

Curtate expectation of life

• The curtate expectation of life, denoted as $e_x$, represents the average number of complete years a person aged $x$ is expected to live
• It is calculated as the sum of the probabilities of survival from age $x$ to each subsequent age, up to the maximum age in the table
• The curtate expectation of life is a useful measure for estimating the remaining lifetime of an individual at a given age

Complete expectation of life

• The complete expectation of life, denoted as $\stackrel{\circ}{e}_x$, represents the average total future lifetime of a person aged $x$, including fractional years
• It is calculated by adding half a year to the curtate expectation of life, assuming that deaths are uniformly distributed over each year of age
• The complete expectation of life provides a more precise estimate of the remaining lifetime compared to the curtate expectation of life

Interpretation of mortality tables

Cohort vs period tables

• Cohort mortality tables follow a specific group of individuals (a cohort) from birth to death, capturing the actual mortality experience of that group over time
• Period mortality tables represent the mortality rates experienced by different ages in a specific time period, typically a calendar year
• Cohort tables are more suitable for analyzing long-term mortality trends, while period tables are used for short-term projections and pricing insurance products

Select vs ultimate tables

• Select mortality tables account for the initial selection effect, where recently underwritten individuals exhibit lower mortality rates than the general population
• Ultimate mortality tables represent the mortality rates that apply after the initial selection period, typically a few years after underwriting
• Select and ultimate tables are used in combination to price insurance products that have an initial selection process, such as term life insurance

Life expectancy calculations

Life expectancy at birth

• Life expectancy at birth represents the average number of years a newborn is expected to live, based on the mortality rates in a given table
• It is calculated using the complete expectation of life at age 0, denoted as $\stackrel{\circ}{e}_0$
• Life expectancy at birth is a commonly used indicator of the overall health and longevity of a population

Life expectancy at specific ages

• Life expectancy can be calculated for any age using the complete expectation of life function, $\stackrel{\circ}{e}_x$
• For example, life expectancy at age 65 represents the average number of years a 65-year-old is expected to live, based on the mortality rates in the table
• Life expectancy at specific ages is useful for retirement planning, pension calculations, and pricing annuities

Trends in mortality and life expectancy

Historical changes in mortality rates

• Mortality rates have generally declined over time due to improvements in healthcare, living conditions, and technology
• The decline in mortality rates has been more pronounced at younger ages, leading to a compression of mortality at older ages
• Historical changes in mortality rates are studied to understand long-term trends and make projections for future mortality improvements

Factors affecting future mortality improvements

• Several factors can influence future mortality improvements, including medical advancements, lifestyle changes, and socioeconomic conditions
• Advances in medical treatments for chronic diseases (cardiovascular disease, cancer) and the development of new drugs and therapies can lead to further reductions in mortality rates
• Changes in lifestyle factors, such as reduced smoking prevalence and increased awareness of healthy living, can also contribute to mortality improvements
• Socioeconomic factors, such as income inequality and access to healthcare, can impact the distribution of mortality improvements across different population subgroups

Applications of mortality tables

Life insurance pricing

• Mortality tables are used to calculate the probability of death, which is a key factor in determining life insurance premiums
• Insurers use mortality tables to estimate the expected claims they will pay out and set premiums that cover these claims while providing a profit margin
• Different mortality tables may be used for different types of life insurance products (term, whole life) and underwriting classes (standard, substandard)

Annuity pricing

• Mortality tables are used to calculate the probability of survival, which is essential for pricing annuities
• Annuities provide a stream of payments for as long as the annuitant is alive, so the insurer must estimate the expected lifetime of the annuitant to determine the appropriate price
• Annuity pricing also involves considering factors such as interest rates, expenses, and profit margins

Pension plan valuation

• Mortality tables are used to estimate the expected lifetime of pension plan participants and calculate the present value of future pension obligations
• Actuaries use mortality tables to determine the funding requirements for pension plans and ensure that the plans have sufficient assets to meet their liabilities
• Changes in mortality assumptions can have a significant impact on the valuation of pension plans and may require adjustments to funding strategies

Population projections and demographics

• Mortality tables are used in conjunction with fertility and migration data to make population projections and study demographic trends
• Demographers use mortality tables to estimate the future size and age structure of populations, which is important for planning public services, infrastructure, and social programs
• Mortality data can also be used to analyze the impact of demographic changes on various sectors, such as healthcare, housing, and labor markets

Comparing mortality tables

Across time periods

• Comparing mortality tables from different time periods allows actuaries to analyze changes in mortality rates and life expectancy over time
• This comparison helps identify trends in mortality improvements and assess the impact of historical events (wars, pandemics) on population mortality
• Comparing mortality tables across time periods is also useful for updating pricing assumptions and reserving practices for insurance companies

Across populations and regions

• Mortality tables can be compared across different populations and regions to identify disparities in mortality rates and life expectancy
• These comparisons can reveal the impact of socioeconomic factors, healthcare access, and environmental conditions on mortality outcomes
• Comparing mortality tables across populations and regions is important for setting appropriate assumptions for pricing insurance products and managing risk exposure for insurers operating in different markets

Limitations of mortality tables

Data quality and reliability

• The accuracy of mortality tables depends on the quality and completeness of the underlying data used to construct them
• Data issues, such as underreporting of deaths, misclassification of cause of death, and errors in population estimates, can lead to biased or unreliable mortality rates
• Actuaries must assess the quality of the data and make appropriate adjustments or use graduated rates to mitigate the impact of data limitations

Applicability to specific individuals

• Mortality tables represent the average mortality experience of a population and may not accurately reflect the mortality risk of specific individuals
• Individual factors, such as health status, lifestyle, and genetic predispositions, can cause deviations from the average mortality rates in a table
• Actuaries use underwriting and risk classification techniques to assess individual mortality risk and adjust pricing and reserving assumptions accordingly

Impact of socioeconomic factors on mortality

• Socioeconomic factors, such as income, education, and occupation, can have a significant impact on individual mortality risk
• Mortality tables based on general population data may not fully capture the mortality differences across socioeconomic subgroups
• Actuaries and demographers may use specialized mortality tables or adjust standard tables to account for the impact of socioeconomic factors on mortality rates and life expectancy