Longevity risk

A life annuity is a promise with no end date: whoever grants it pays for as long as the annuitant lives. Its value depends on two things, the interest rate used to discount the payments and the probabilities that the person is still alive at each date. Longevity risk is the possibility that those probabilities turn out higher than assumed when the price or the reserve was set, so that payments last longer than expected. In life insurance the problem is that the insured dies early; here the problem is the opposite.

The risk appears on any balance sheet with payments contingent on survival. In Mexico it is borne by insurers that pay life annuities under the pension insurance arising from the Ley del Seguro Social (Social Security Law), companies with defined benefit plans valued under NIF D-3, the retirement schemes of public bodies and the Federal Government itself, which in 2026 has budgeted 795.8 billion pesos for the pensions that the Twelfth Transitory Article of the Ley del Seguro Social places in its charge, that is, those already in payment in 1997 and those of workers who choose the 1973 law regime (article 5 of the 2026 Federal Expenditure Budget). Every additional year of life for those generations is an additional year of payroll.

The guide to mortality tables covers biometric functions and analytical laws; the guide to pensions covers the mathematics of pension plans. The focus here is uncertainty about the trend: how Mexican mortality has moved, how its evolution is modeled, how it enters reserves and who can absorb the risk when one's own balance sheet is no longer enough.

Even if mortality rates were known exactly, the number of deaths in a portfolio would still be random. That is idiosyncratic risk: with a thousand independent 70-year-old pensioners, some will die sooner and others later than expected. Its relative weight falls with portfolio size, in proportion to \(1/\sqrt{n}\), and an insurer with tens of thousands of annuities diversifies it almost completely.

Systematic or trend risk is uncertainty about the rates themselves. If mortality above age 65 falls faster than projected, every annuity in the portfolio lengthens at once and adding policies does not help: it multiplies the exposure. It is useful to split it into three pieces. Level risk, or base risk, arises when the starting table does not reflect the covered population; annuity buyers tend to live longer than the general population because they decide to buy knowing something about their health. Trend risk concerns the future speed of improvement. Short-term volatility captures shocks such as a pandemic, which in an annuity portfolio works in the payer's favor while it lasts.

The distinction determines the treatment. Idiosyncratic risk is managed through volume and requires little capital; systematic risk calls for prudent assumptions, explicit capital and, when concentration is large, transfer to third parties.

The official series of life expectancy at birth comes from the Demographic Reconciliation for 1950 to 2019 and the Population Projections of Mexico for 2020 to 2070 by the Consejo Nacional de Población (CONAPO, National Population Council), published as open data. Between 2000 and 2015 life expectancy rose from 74.05 to 75.09 years, a modest gain for fifteen years, and between 2015 and 2019 it slipped slightly to 74.84. The gap between the sexes widened over that period: in 2019 women lived on average 6.2 years longer than men.

The COVID-19 pandemic broke the series. INEGI (National Institute of Statistics and Geography) recorded 1,086,743 deaths in 2020 and 1,122,249 in 2021, against 819,672 in 2024. Using the Pan American Health Organization's endemic channel methodology, INEGI estimated that 749,496 deaths were expected from January to December 2020 and 1,076,417 occurred, an excess of 326,921 deaths, equivalent to 43.6%. More than half of the deaths registered in 2020 (55.8%) were among people aged 65 and over. CONAPO estimates that life expectancy fell to 68.91 years in 2020 and 68.76 in 2021, 6.1 years below 2019; the drop was 6.9 years for men and 4.9 for women.

The recovery in the indicator was fast: 75.17 years in 2022 and 75.50 in 2024, already above the pre-pandemic level. CONAPO's projections assume that improvement continues at a pace of about 0.18 years per calendar year until 2050 and somewhat more slowly thereafter, reaching 80.13 years in 2050 and 83.02 in 2070.

For the actuary, the lesson is not that the pandemic reduced longevity risk, but that it made it harder to measure. A two-year shock mixes three effects: direct excess mortality, the early death of frail people who would have died shortly afterward (which can produce somewhat lower mortality in the following years) and possible chronic after-effects. A model calibrated on years that include 2020 and 2021 without special treatment may underestimate the underlying trend; one that discards them entirely ignores real information. Decisions on how to weight those years should be documented in every valuation.

In 1992 Ronald Lee and Lawrence Carter published "Modeling and Forecasting U.S. Mortality" in the Journal of the American Statistical Association (vol. 87, no. 419, pages 659 to 671). Their approach reduces the entire mortality surface, age by year, to a single time index. The logarithm of the central death rate at age x in year t is written as an average age profile, plus each age's response to a general mortality index, plus an error term.

Estimation is straightforward. First, \(a_x\) is the time average of \(\ln m_{x,t}\). Next the centered matrix \(Z_{x,t}=\ln m_{x,t}-a_x\) is formed and a singular value decomposition (SVD) is applied: the first left singular vector, normalized to sum to 1, gives \(b_x\); the first right singular vector, multiplied by the first singular value and by the original sum of \(b_x\), gives \(k_t\). The constraints \(\sum_x b_x = 1\) and \(\sum_t k_t = 0\) resolve the model's identification problem. In the original paper, \(k_t\) is re-estimated in a second stage so that the implied total deaths match those observed in each year.

For projection, \(k_t\) is modeled as a random walk with drift. The estimated drift is simply the average annual change in the index, and forecast uncertainty grows with the square root of the horizon. That simplicity explains its popularity and also its limitations: the age pattern of improvement \(b_x\) is fixed, improvements at all ages are perfectly correlated and the intervals tend to be narrow because they reflect only one risk factor.

The CBD model was designed for pension ages, where the logit of the probability of death is almost linear in age. With two factors, it allows improvement to be faster at some ages than at others and improvements at different ages to be less than perfectly correlated, something Lee-Carter does not capture. Both have extensions with a cohort effect, useful when certain generations show persistently greater improvement. In insurance practice, stochastic models are used more to measure the dispersion of the liability and to calibrate capital than to set the central scenario, which is usually taken from a table with deterministic improvement factors.

An improvement factor is the annual percentage reduction in the probability of death at a given age. With a base table for a reference year and a scale of factors by age (and sometimes by calendar year), one obtains a dynamic table: a matrix of probabilities by age and year. Read across rows, it gives period tables; read along the diagonal, it gives the generational table a cohort will actually experience. Someone aged 65 today will face, at 66, next year's mortality, which is expected to be lower.

In the United States, the Retirement Plans Experience Committee of the Society of Actuaries (SOA) publishes the MP scales, two-dimensional by age and year. Scale MP-2021 reflects the mortality experience of the U.S. population through 2019 and was published together with the MIM-2021 model, a tool that lets each user build their own scale with different long-term assumptions. In the United Kingdom, the Continuous Mortality Investigation (CMI), linked to the Institute and Faculty of Actuaries, maintains the CMI Model: the CMI_2025 version was published in March 2026, calibrated on population data for England and Wales through December 31, 2025, and the projections committee is consulting until November 5, 2026 on the methods for CMI_2026 (Working Paper 216). In both cases the user must choose a long-term rate of improvement: no model estimates it from the data alone.

In Mexico, the CUSF prescribes the projection method for pension insurance arising from the social security laws. Provision 14.2.7 establishes that the base table is projected with improvement factors by sex and age \(TM_x\): for policies offered through the SAOR before November 18, 2022, with base year 2009 and the factors in Annex 14.2.7-a; for those offered from that date onward, with base year 2022 and the factors in Annex 14.2.7-b. The same provision specifies that the present value of an annuity is calculated using the diagonal of the resulting matrix.

The effect of improvement on an annuity grows with the duration of the liability. With the illustrative table used in this guide's example (Gompertz-Makeham with Standard Ultimate Life Table parameters) and a technical interest rate of 5%, the annuity-due at age 65 is worth 13.55 without improvement; with 1% annual improvement it rises 2.8%, with 1.5% it rises 4.35% and with 3% it rises 9.3%. If the technical rate falls to 3%, the same 1.5% improvement makes the annuity 6.0% more expensive: low rates and longevity reinforce each other, because distant payments weigh more in present value.

In pension insurance under the Ley del Seguro Social, the insurer receives the monto constitutivo (the single premium), defined in article 159 of the law as the amount needed to purchase the life annuity and survivor insurance, and undertakes to pay the pension for life. Article 157 provides that the annuity is adjusted every February by the INPC (National Consumer Price Index). It is therefore a real annuity: the insurer bears both longevity risk and real interest rate risk, and an underestimate of improvement translates directly into reserve inadequacy.

In defined benefit plans, IAS 19 requires actuarial assumptions to be unbiased and mutually compatible (paragraph 75), mortality to be determined by reference to the best estimate of the mortality of plan members both during and after employment (paragraph 81) and expected changes to be considered, for example by modifying standard tables with estimates of improvement (paragraph 82). NIF D-3 follows the same best estimate approach. In Mexican practice the debate is rarely whether to include improvement, but which scale to use and how to justify it to the auditor, a subject covered in employee benefits.

In social security the effect is fiscal. Pensions under the 1973 regime are paid from federal funds; the IMSS RJP (the Mexican Social Security Institute's own employees' retirement scheme), a closed plan, depends on its retirees living as long as the valuation assumes; and the cost of the universal pension for older adults grows with the number of survivors. The technical discussion of these issues is in public policy.

When the exposure is large relative to capital, systematic risk is transferred. Longevity transfer markets have developed mainly in countries where corporate defined benefit plans have accumulated very large obligations to retirees. The instruments differ in what is transferred (longevity alone, or longevity together with investment risk) and in who bears the counterparty risk.

There are also internal hedges. A company that sells both life insurance and annuities has a partial natural hedge, because lower mortality reduces claims in one business and increases payments in the other; the offset is never exact, because the ages and populations exposed are different. In product design, programmed withdrawals and annuities with periodic review shift part of the risk to the pensioner. The Ley del Seguro Social provides for exactly that choice between a life annuity and programmed withdrawals in its article 157. Reinsurance and capital are no substitute for a sound central assumption: if the base table is wrong, any transfer will be priced on a mismeasured liability.

The biometric bases for pension insurance are in Chapter 14.2 of the CUSF. For policies issued before the Nuevo Esquema Operativo (New Operating Scheme), Provision 14.2.4 refers to the 1997 demographic experience (EMSSAH-97 and EMSSAM-97 for non-disabled lives, EMSSIH-97 and EMSSIM-97 for disabled lives), without improvement factors. For policies under the New Operating Scheme, Provision 14.2.5 establishes that the policy reserve is calculated with the active-lives mortality tables EMSSAH-09 and EMSSAM-09 when the offer through the SAOR was made before November 18, 2022, and with EMSSAH-15 and EMSSAM-15 from that date onward; in both cases projected with improvements by age and calculation year in accordance with Provision 14.2.7. For disabled and incapacitated lives there are specific IMSS and ISSSTE tables (EMSSI-IMSS-09 and -12, EMSSInc-IMSS-12, among others).

For the capital requirement, Provision 14.2.6 provides specific tables, EMSSAH-RCS and EMSSAM-RCS in their 09 and 15 versions, also projected with improvements, which are used in the calculation under Provision 6.5.5. The regulator thus separates the biometric basis for the reserve, which seeks a central value, from the one that feeds the capital requirement. It is the regulatory expression of the distinction between expected value and systematic risk in section 2.

Outside pension insurance there is no mandatory table for company plans. NIF D-3 leaves the choice to the entity's best estimate, which in practice usually starts from Mexican experience tables adjusted to the plan population. Public sources for testing assumptions are CONAPO's reconciliation and projections, INEGI's registered death statistics and, for pensioner experience, reports from the IMSS and CONSAR (National Retirement Savings Commission). The pension simulator and the pensions and Afores section show how that life expectancy ends up in the amount of an individual account pension.