Mortality tables and longevity

A mortality table summarizes, age by age, the probability that a person dies before reaching one more year of age. Every other biometric function derives from that column of numbers: survivors, deaths, force of mortality and life expectancy. With them we calculate life insurance premiums, the single premiums that fund life annuities (montos constitutivos), defined benefit obligations under NIF D-3 and capital requirements.

The technical problem is not applying the table but choosing it. A table built on the general population does not describe insured lives, who go through medical underwriting and belong to a different socioeconomic level; a period table, fixed in the calendar, underestimates the future lifetime of someone who is 65 today because it ignores the fact that mortality will keep falling. That gap is longevity risk, and in a life annuity portfolio it can be worth several percentage points of the reserve.

This page complements the mortality summaries in Actuarial mathematics and the Formula sheet: here the emphasis is on how a table is built, projected and regulated, with a complete numerical example and a map of the tables in force under the Circular Única de Seguros y Fianzas (CUSF, Single Insurance and Surety Circular).

The table starts from a hypothetical cohort of l₀ newborns (the radix, typically 100,000) subjected to observed mortality rates. Each column answers a different question: how many are still alive at age x, how many die between x and x+1, what fraction of the living die and how many years they have left on average. All of them are transformations of a single one: the q_x series.

The force of mortality μ_x is the instantaneous version of q_x: the death rate per unit of time for someone exactly age x. It is the natural function for modeling, because survival is the exponential of its integral and because the analytical laws (Gompertz, Makeham) are written in terms of μ_x. For adult ages and small values, μ_x+1/2 is approximately q_x, a handy approximation for checking calculations by hand.

Life expectancy comes in two versions. The complete one, e̊_x, integrates the survival curve in continuous time; the curtate one, e_x, counts only complete years. If deaths are spread uniformly within each year of age, e̊_x is approximately e_x + 1/2. Mixing them up in a valuation introduces a half-year bias for every life.

In 1825 Benjamin Gompertz observed that, in adulthood, the force of mortality grows roughly geometrically: each year of age multiplies it by a constant factor c. In 1860 William Makeham added a constant term A that does not depend on age and represents accidental causes. With three parameters, Makeham's law reproduces mortality between ages 30 and 90 quite accurately, which makes it useful for smoothing noisy data, extrapolating to ages with little exposure and building calculators.

The parameter c has a direct interpretation: the doubling time of the senescent part of mortality is ln 2 / ln c. With c = 1.095, mortality doubles every 7.6 years, a typical value for modern populations. B sets the level of the curve and A the plateau of accidental mortality that dominates at young ages. Since the integral of B c^x has a closed form, survival has an explicit formula, with no need for numerical integration.

In practice the parameters are estimated by maximum likelihood, assuming that observed deaths D_x follow a Poisson distribution with mean E_x·μ_x+1/2, where E_x is the central exposure. Above age 95 the law tends to overstate mortality because the data show a deceleration; to close the table, logistic models such as Perks or Kannisto are used, or a limiting age is set with q_ω = 1. When no parametric law is sought, graduation is done with Whittaker-Henderson or penalized splines.

Building a table is a statistical exercise in four stages. First, the database is assembled: each insured life contributes exposure from entry into the study until death, lapse or the end of observation, and deaths are recorded by age and sex. Second, crude rates m_x = D_x/E_x are calculated and converted to q_x. Third, they are graduated to remove random noise without erasing the true shape of the curve. Fourth, the graduated table is validated against the data with goodness-of-fit tests (chi-square, runs of signs, standardized deviations by age).

Insured experience differs from population experience because of three effects. Medical selection means newly insured lives die less during the first policy years; that is why select and ultimate tables exist. The self-selection of those who buy life annuities makes them longer lived than the population. And the socioeconomic level of insured lives sets them apart from the national average. That is why a population table is almost never applied to an insured portfolio without adjustment.

The prudence margin depends on the use. For death benefits, a table with high mortality is conservative; for a life annuity, low mortality is the conservative choice. The same insurer may need two different tables for two products, and the Mexican regulator reflects that asymmetry by using tables with lower mortality to calculate annuity capital than to calculate the annuity reserve.

A period table describes the mortality of a single calendar year: it takes q₆₅ for 2026, q₆₆ for 2026, q₆₇ for 2026, and so on. Nobody lives that way, because someone who is 65 today will be 66 in 2027 and will face 2027 mortality, which will be lower. A generational table follows the cohort along the diagonal of an age by year matrix: q₆₅ for 2026, q₆₆ for 2027, q₆₇ for 2028. With the EMSSAH-15 table and its CUSF improvement factors, the life expectancy of a 65-year-old man in 2026 is 20.0 years under the period table and 20.7 under the generational table; for women, with EMSSAM-15, it is 22.5 and 23.4 years.

The practical way to move from a base table to a generational one is to apply improvement factors by age. If mortality at age x falls at a constant annual rate TM_x, the projected probability for base year + t is q_x multiplied by (1 - TM_x)^t. The CUSF uses exactly this expression for pension insurance arising from the social security laws, with factors by sex published in an annex. In the United States, the Society of Actuaries publishes two-dimensional (age and year) scales in the MP family, in which improvement gradually converges to a long-term rate.

In reality, improvement factors are not constant. There are cohort effects (generations that improved faster than their neighbors), transitory shocks such as the COVID-19 pandemic, which raised mortality in 2020 and 2021 in Mexico and almost everywhere else, and structural changes driven by chronic diseases. That is why the choice of long-term improvement rate is an assumption that must be documented and stress tested.

In 1992 Ronald Lee and Lawrence Carter published a model that became the standard for projecting mortality. It decomposes the log of the central death rate into an average age profile a_x, a time index k_t that summarizes the overall level of mortality in year t, and a sensitivity b_x that indicates how much each age responds to movements in k_t. To identify the model, the b_x are constrained to sum to one and the k_t to sum to zero.

Classic estimation uses the singular value decomposition of the centered matrix ln m_x,t - a_x, followed by a refit of k_t so that total fitted deaths match observed deaths. Then k_t is projected as a random walk with drift: the estimated drift gives the central trend and the variance of the innovations gives prediction intervals. Those intervals are used to generate stochastic longevity scenarios, which are useful for economic capital.

Its limitations are well known: a single time factor forces all ages to improve in perfectly correlated fashion, and it does not capture cohort effects. The most widely used extensions are Renshaw-Haberman (which adds a cohort term), Cairns-Blake-Dowd (which models logit q_x directly at older ages with two factors) and age-period-cohort models. Modern estimation is done by Poisson maximum likelihood with packages such as StMoMo in R.

Longevity risk has two components. The idiosyncratic component is the random variation in when each pensioner dies; it diversifies away with portfolio size, because its relative deviation falls with the square root of the number of lives. The systematic component is trend error: if the whole population lives longer than projected, no portfolio, however large, can diversify it. This is the component that requires capital.

Sensitivity is measured by recalculating the liability under a mortality shock. A permanent -20% shock to μ_x at all ages is a common benchmark in solvency frameworks; for a life annuity at age 65 with a 3.5% real technical interest rate, it raises the present value by about 7%, as the example on this page shows. The effect grows the lower the discount rate and the younger the annuitant, because distant cash flows carry more weight.

The management tools are reserving with a generational table, the special mathematical reserve the LISF (Insurance and Surety Institutions Law) requires for social security pension insurance, longevity reinsurance, longevity swaps (actual annuity payments are exchanged for fixed payments based on an agreed table) and bulk portfolio purchases by specialized reinsurers. In corporate defined benefit plans, the same risk shows up in the retirement DBO (defined benefit obligation) under NIF D-3.

Articles 216 and 217 of the LISF require insurers that write social security pensions to set up a special mathematical reserve whose purpose is to meet possible improvements in the life expectancy of the insured population. The biometric bases for that line are set by the CUSF in Chapter 14.2. Policies issued before the new operating scheme are valued with the EMSSAH-97 and EMSSAM-97 tables; those under the new scheme, with EMSSAH-09 and EMSSAM-09 projected with 2009 base improvement factors when the offer in the SAOR (the system that administers annuity offers and resolutions) was made before November 18, 2022, and with EMSSAH-15 and EMSSAM-15 projected with 2022 base factors when the offer was made later. Provision 14.2.7 sets out the projection q_x^2022+t = q_x²⁰²²(1 - TM_x)^t and requires using the diagonal of the resulting matrix: it is a generational table by mandate.

For the pension solvency capital requirement, the CUSF uses tables with deliberately lower mortality: EMSSAH-RCS-09 gives q₆₅ = 0.00808 compared with 0.01512 under EMSSAH-09, and EMSSAH-RCS-15 gives 0.01328 compared with 0.01807 under EMSSAH-15. The difference between the two valuations works as the capital's longevity shock. For life insurance with terms longer than one year valued under the statutory method, Annex 5.3.3-a sets the CNSF M 2013 table (unisex, with no distinction by sex), together with disability, accidental death and other benefit experience.

Outside the insurance sector, the official demographic reference is the Conciliación Demográfica de México 1950-2019 (Demographic Reconciliation of Mexico) and the Proyecciones de la Población de México y de las Entidades Federativas 2020-2070 (Population Projections for Mexico and its States), published by the Consejo Nacional de Población (CONAPO, National Population Council). They are used for social security valuations, studies of state pension systems and as a point of comparison in employee benefit valuations, where the actuary may adopt market tables or the plan's own experience provided the choice is justified.