Mortality tables

Mortality tables are a cornerstone of actuarial science, providing a statistical framework to assess the probability of death at various ages within a given population. These tables are essential tools within the broader field of financial risk management, helping actuaries, insurers, and pension fund managers quantify longevity risk and price products accurately. Mortality tables essentially present a snapshot of mortality rates, indicating how many people from a starting group are expected to survive to each successive age. The data within a mortality table is crucial for various financial calculations, influencing everything from life insurance premiums to pension plan liabilities. They are dynamic instruments, regularly updated to reflect changes in public health, medical advancements, and lifestyle trends.

History and Origin

The genesis of mortality tables can be traced back to the 17th century, driven by the burgeoning popularity of life annuities and the need for a more scientific approach to their valuation. Early attempts at constructing these tables were based on limited data. John Graunt, a pioneer in statistics and demography, is widely credited with devising a rudimentary life table in 1662 based on his analysis of the London Bills of Mortality. His work provided early insights into age-specific death rates and the phenomenon of "excess deaths" during epidemics, laying foundational concepts for numerical analysis of health data.³⁴, ³⁵

Building on Graunt's work, the astronomer Edmond Halley, also a Fellow of the Royal Society, constructed one of the earliest known mortality tables in 1693, using data from the city of Breslau (now Wrocław). ³¹, ³², ³³Halley's table was arguably the first based on population data and was specifically used to advise bankers on annuity rates, demonstrating the practical application of these statistical instruments in financial services. ²⁹, ³⁰This methodology became a standard, solidifying the role of mortality tables in the nascent life insurance industry and for calculating annuities. ²⁸The scientific community's interest in measuring the distribution of lifetimes grew, leading to significant innovations in the 19th century with large-scale national censuses and the theoretical formalization of life trajectories by statisticians like Lexis.
²⁷

Key Takeaways

• Mortality tables are statistical tools used in actuarial science to project the probability of death at different ages.
• They are fundamental for pricing life insurance policies, valuing annuities, and calculating pension liabilities.
• The tables track a hypothetical group of individuals from birth, showing how many are expected to survive to each subsequent age.
• Data quality and the ability to capture future trends are critical for the accuracy and reliability of mortality tables.
• Mortality tables are periodically updated to reflect changes in population health, medical advancements, and other demographic shifts.

Formula and Calculation

A mortality table is built upon several key components and their interrelationships. While there isn't a single overarching "formula" for the entire table, individual probabilities and functions are derived. The core element is the mortality rate for each age.

Let:

• (x) = Exact age
• (l_x) = Number of lives surviving to exact age (x) from a starting cohort (e.g., 100,000 births)
• (d_x) = Number of lives dying between age (x) and (x+1)
• (q_x) = Probability of death between age (x) and (x+1) (i.e., the mortality rate)
• (p_x) = Probability of survival from age (x) to (x+1)

The fundamental relationships are:

The data used to construct mortality tables typically comes from national vital statistics, population census data, and the experience of large insurance companies. Actuaries use various statistical techniques, including data smoothing and projection methods, to address challenges like data quality issues and the need to forecast future mortality rates.
²⁶

Interpreting the Mortality Table

Interpreting a mortality table involves understanding the probabilities of survival and death at different ages. The (l_x) column (number of survivors) provides insight into how a cohort diminishes over time, while the (q_x) column (mortality rate) highlights the likelihood of death at a specific age. For instance, a high (q_x) at very young ages reflects infant mortality, and an increasing (q_x) at older ages indicates the natural progression toward higher death rates as individuals age.

These tables do not predict the lifespan of any single individual but rather provide a statistical average for a large group. They are crucial for assessing longevity risk and calculating life expectancy, which is the average number of additional years a person is expected to live at a given age. Understanding the trends within a mortality table, such as improving mortality rates over time due to medical advancements, is vital for long-term financial planning and product design.

Hypothetical Example

Consider a simplified mortality table for a hypothetical population, starting with 100,000 births ((l_0 = 100,000)):

In this example:

• At age 0, out of 100,000 births, 500 are expected to die before reaching age 1, resulting in a mortality rate of 0.00500.
• By age 65, 85,000 individuals are still alive from the original cohort, and 850 of them are expected to die before reaching age 66.
• At age 100, only 1,000 individuals remain, with a significantly higher mortality rate of 0.50000, meaning 500 are expected to die before age 101.

This table allows an actuary to calculate the probability of survival from one age to another, or the probability of death within a specific age range. This information is directly applicable to understanding the time value of money in long-term investments that depend on human lifespan.

Practical Applications

Mortality tables are indispensable tools with wide-ranging practical applications across the financial industry, particularly in insurance and retirement planning.

In the insurance sector, mortality tables are fundamental for:

• Pricing Life Insurance Policies: Insurers use mortality tables to calculate the likelihood of a policyholder dying at each age, which directly informs the premiums charged. Accurate mortality forecasts are essential to ensure the solvency of an insurer and its ability to meet future claims.
  ²⁵* Valuing Annuities: For annuities, which provide a stream of income payments for life, mortality tables help determine how long payments are expected to last, impacting the initial lump sum or premium required.
• Setting Reserves: Insurance companies are legally required to hold sufficient financial reserves to cover future policy obligations. Mortality tables are key to calculating these reserves, ensuring the company has enough assets to pay out benefits when due. Regulatory bodies often specify which mortality tables must be used for valuation purposes. For example, in the U.S., state financial regulations often adopt standard tables like the Commissioners Standard Ordinary (CSO) Mortality Table for determining reserve liabilities.
  ²², ²³, ²⁴
  Beyond insurance, mortality tables are crucial in pension fund management. Actuaries use them to estimate the expected lifetimes of pension plan participants, which directly influences the calculation of the present value of future pension obligations. ²¹This helps determine the funding requirements for pension plans, ensuring they have sufficient assets to meet their liabilities.
  ²⁰
  Furthermore, government agencies like the U.S. Social Security Administration (SSA) utilize mortality projections in their long-term actuarial estimates for programs like Old-Age and Survivors Insurance (OASI) and Disability Insurance (DI). These projections inform assessments of the programs' financial health and potential future costs. ¹⁶, ¹⁷, ¹⁸, ¹⁹The Internal Revenue Service (IRS) also specifies the use of actuarial tables, which incorporate mortality data, for valuing annuities, life estates, remainders, and reversions for tax purposes.
  ¹⁴, ¹⁵

Limitations and Criticisms

While mortality tables are powerful tools, they are not without limitations and criticisms. Their accuracy heavily relies on the quality and completeness of the underlying data. Issues such as underreporting of deaths, misclassification of causes of death, or errors in population estimates can lead to biased or unreliable mortality rates. ¹¹, ¹², ¹³Actuaries must carefully assess data quality and make adjustments to mitigate these issues.

A significant limitation is that mortality tables represent the average mortality experience of a population and may not accurately reflect the specific mortality risk of individuals. Personal factors such as health status, lifestyle choices, and genetic predispositions can cause deviations from average rates. ¹⁰Actuaries often address this through underwriting and risk classification techniques to assess individual risk and adjust pricing accordingly.

Moreover, mortality tables are typically based on historical data and may not fully capture future changes in mortality rates. Advances in medicine, changes in lifestyle, or unforeseen events like pandemics can lead to shifts in mortality trends that historical data might not predict. ⁸, ⁹This challenge necessitates continuous monitoring and updating of tables, as well as the use of sophisticated mortality projection models like the Lee-Carter model, though even these models can be sensitive to outliers and require robust statistical approaches. ⁶, ⁷The Social Security Administration, for example, has had to adjust its mortality assumptions to account for the effects of the COVID-19 pandemic.
⁴, ⁵
Another criticism is the potential for "false cohort effects" in some historical mortality tables derived from periodic census observations. These can lead to misinterpretations of longevity improvements across generations. ³Ongoing research and methodological improvements aim to refine the construction of mortality tables to address these complexities and ensure their continued relevance and accuracy in dynamic demographic environments.

Mortality Tables vs. Life Expectancy

Mortality tables and life expectancy are closely related but represent distinct concepts in actuarial science. A mortality table is a comprehensive statistical summary that presents the probability of death and survival at each specific age within a given population. It is a detailed dataset from which various other actuarial measures are derived. The table illustrates the entire "dying out" process of a hypothetical cohort from birth, showing the number of survivors at each age and the corresponding death rates.

Life expectancy, on the other hand, is a single, summarized metric derived from a mortality table. It represents the average number of additional years a person is expected to live at a given age, assuming that the current age-specific mortality rates remain constant throughout their remaining lifetime. For instance, "life expectancy at birth" is the average number of years a newborn is expected to live, while "life expectancy at age 65" is the average number of additional years a 65-year-old is expected to live. While a mortality table provides the granular data on age-by-age survival and death probabilities, life expectancy offers a concise summary measure of overall longevity. Both are critical for financial planning, but the mortality table is the foundational data set, and life expectancy is a key output.

FAQs

How often are mortality tables updated?

Mortality tables are updated periodically to reflect changes in population mortality. In the U.S., major updates to standard tables, such as those used by the IRS for tax purposes, occur approximately every 10 years to incorporate the most recent mortality experience. ²Insurance regulators also mandate updates, with tables like the Commissioners Standard Ordinary (CSO) Mortality Table being revised over time. ¹However, actuaries continuously monitor emerging trends and may make adjustments to their models more frequently.

Who creates mortality tables?

Mortality tables are typically created by actuaries and demographers who specialize in statistical analysis of population data. Professional actuarial organizations, such as the Society of Actuaries (SOA) in the U.S., play a significant role in developing and recommending new mortality bases for various applications, often in collaboration with regulatory bodies. Government agencies, like the Social Security Administration, also develop their own mortality projections for long-range financial projections.

Why are mortality tables important for financial planning?

Mortality tables are crucial for financial planning because they enable the accurate assessment of longevity risk. For individuals, understanding life expectancy derived from these tables can inform decisions about retirement savings, the timing of taking Social Security benefits, and the purchase of annuity products. For institutions, they are fundamental for ensuring the financial stability of pension plans and life insurance companies by allowing precise calculation of future liabilities and appropriate pricing of products. Without reliable mortality data, long-term financial commitments would be based on mere speculation, posing significant risks.