Mortality Table

Title: Mortality Tables — What They Are, How They Work, and Practical Steps for Using and Building Them

Key takeaways

– A mortality table (life table or actuarial table) gives the probability that members of a defined population will die during a specific time interval, usually expressed by age.
– Insurers, pension programs, and governments use mortality tables to price products, set reserves, and project liabilities.
– There are two main types: period (specific year) and cohort/generation (lifetime of a birth cohort). Cohort tables are usually more relevant for long-term actuarial work.
– Building and using mortality tables requires good data, appropriate segmentation (sex, smoking, occupation), smoothing/graduation, and ongoing updates to reflect mortality improvements or shocks (e.g., pandemics).

Source note: Core definitions and context summarized from Investopedia (Matthew Collins). https://www.investopedia.com/terms/m/mortality_table.asp

1) What is a mortality table?

A mortality table is a grid of numbers showing, for each age (typically in one‑year increments from birth to age 100 or more), the probability of dying during the next year and related quantities (number alive at each exact age, number dying during the interval, and life expectancy remaining at each age). Actuaries use these tables to estimate expected deaths, survival probabilities, life expectancy, insurance premiums, pension liabilities, and reserve requirements.

2) Core components and common notation

– l_x: number of survivors at exact age x in a hypothetical initial cohort (often per 100,000 or per 1,000).
– d_x: number dying between ages x and x+1 = l_x − l_{x+1}.
– q_x: probability of dying between exact ages x and x+1 (often given per 1,000 or as a decimal). q_x = d_x / l_x.
– p_x: probability of surviving from age x to x+1 = 1 − q_x.
– e_x: remaining life expectancy at age x (typically derived from the total future person‑years lived by the cohort from age x onward divided by l_x).

3) Types of mortality tables

– Period life table: based on mortality rates observed in a single calendar year (or other short period). It represents mortality if current rates applied throughout life.
– Cohort (generation) life table: follows a birth cohort, incorporating projected future changes in mortality rates as the cohort ages. More realistic for long-term liability modelling.

4) How a mortality table works (intuitively)

– Start with an initial cohort (e.g., 100,000 births). For each age x, the table gives q_x — the fraction of those alive at age x who are expected to die before x+1.
– By applying q_x sequentially you obtain l_x for every age and can compute cumulative survival probabilities, expected number of deaths in any interval, and life expectancy.

5) Simple example (how to read and calculate)

Practical example using a per-1,000 basis:
– Suppose l_40 = 1,000 (i.e., 1,000 people aged 40 in the cohort) and q_40 = 0.002 (0.2%).
– Expected deaths between age 40 and 41: d_40 = q_40 * l_40 = 0.002 × 1,000 = 2 people.
– Survivors at age 41: l_41 = l_40 − d_40 = 1,000 − 2 = 998.
– Probability that a 40‑year‑old survives to age 45 = p_40 × p_41 × p_42 × p_43 × p_44, where each p_k = 1 − q_k. Multiply the one‑year survival probabilities across the ages of interest.

6) Practical steps: reading a mortality table (for non‑actuaries)

1. Identify the table’s basis (period vs cohort), initial cohort size (e.g., per 1,000 or 100,000), and the population segment (sex, smoker status).
2. Locate the row for the person’s exact age x. Note q_x (probability of death this year) and p_x = 1 − q_x.
3. To estimate survival to a future age y, multiply the year‑by‑year p_k values from x to y−1.
4. To estimate remaining life expectancy, read e_x from the table or compute total future person‑years divided by l_x.

7) Practical steps: building a mortality table (for actuaries/researchers)

1. Define scope and segmentation: choose cohort vs period, population, and risk strata (sex, smoking, occupation, socio‑economic class).
2. Gather data: obtain accurate death counts and exposures (population at risk) by single year of age and calendar year. Ensure sample size is adequate.
3. Calculate crude age‑specific death rates (deaths / exposure) and convert to q_x or other chosen metric.
4. Smooth/graduation: apply statistical smoothing to remove random variation (e.g., splines, parametric models, graduated tables).
5. Test and validate: back‑test against historical data and perform goodness‑of‑fit checks.
6. Make mortality improvement assumptions: project future mortality changes if creating cohort tables or reserves.
7. Document methods, assumptions, and limitations; obtain peer or regulatory review.

8) Practical steps: using mortality tables in insurance and pensions

1. Select the appropriate table(s) for the product and population (e.g., separate smoker/non‑smoker tables).
2. For pricing: compute expected present value of future benefits by multiplying benefit amounts by survival or death probabilities and discounting at an appropriate interest rate.
3. For reserving: compute expected future payouts under current policies using the table and discount appropriately; apply margins and regulatory requirements.
4. For risk management: run sensitivity tests (mortality improvement, shock scenarios) and capital impact analyses.

9) Practical steps for consumers and personal planning

1. Use published life tables or calculators (look for cohort tables where possible) to obtain a baseline life expectancy for your sex and age.
2. Adjust for known personal factors (smoking, occupation, severe health conditions) qualitatively — or use tables that incorporate those factors if available.
3. Use the survival probabilities to inform insurance amounts and retirement planning: e.g., probability you reach retirement age or the expected number of retirement years.
4. Remember that tables show population averages; individual outcomes vary.

10) Requirements, standards, and common practice

– Tables are usually constructed separately for men and women and often segmented by other risk factors.
– Regulatory bodies and large institutions (e.g., Social Security Administration, insurers) publish standard or required tables for specific regulatory uses.
– Good practice: use large, representative datasets, apply graduation/smoothing, and update tables regularly to reflect mortality trends and new data.

11) Limitations and important considerations

– Heterogeneity: population averages hide wide variation by health, lifestyle, and socio‑economic status.
– Period vs cohort differences: period tables may understate future survival improvements; cohort tables require assumptions about future trends.
– Data quality: small sample sizes, reporting errors, and short time series reduce table reliability.
– Shocks and non‑stationarity: pandemics, medical breakthroughs, or sudden mortality shifts can rapidly invalidate assumptions.
– Legal/regulatory constraints: insurers must comply with local actuarial standards and solvency rules.

12) Conclusion

Mortality tables are essential actuarial tools that quantify death and survival probabilities by age and other factors. Whether you are an insurer pricing a life product, a pension plan projecting liabilities, a policymaker assessing future benefit needs, or an individual planning for retirement, understanding the structure, uses, limitations, and practical steps for applying mortality tables helps ensure more reliable decisions.

Primary source

– Investopedia — “Mortality Table,” Matthew Collins. https://www.investopedia.com/terms/m/mortality_table.asp