Period vs. Cohort Life Expectancy

Definition

Period life expectancy ( $e_0$ ) is the expected mean age at death for a synthetic cohort — a hypothetical group experiencing the age-specific death rates observed in a single calendar period throughout its entire life. Cohort life expectancy is the observed (or projected) mean age at death for an actual birth cohort followed from birth. The two measures answer different questions and differ systematically under sustained mortality decline.

Key Ideas

Observation design, not bias. The period/cohort distinction is not a question of which measure is "correct." Period $e_0$ answers: "what is the typical age at death for someone born into today's mortality environment?" Cohort $e_0$ answers: "what is the typical age at death for people actually born in year $\tau$ ?" Both are valid; they are simply different aggregations.
Period $e_0$ > cohort $e_0$ under declining mortality. Period survival $\ell_p(x,t)$ uses the favorable mortality rates current at time $t$ . Cohort survival $\ell_c(x,t-x)$ uses rates from $(t-x)$ years ago, when mortality was higher. Under improvement, the cohort-based measure is necessarily lower. This is not a defect of either measure; it is a mathematical consequence of using different mortality schedules.
Six-measure taxonomy (Wilmoth 2005, Table 1). Wilmoth formalizes six distinct mean-lifespan measures to clarify which question each answers:
1. $e_0$ — pure period measure; depends only on period death rates at $t$
2. $e_0^* = \text{CAL}$ — cross-sectional average length of life; relative size of constant-birth population; depends on cohort survival probabilities
3. $e_0'$ (MAD) — mean age at death in constant-birth population; a population-dynamics measure
4. $e_0^{BF}$ — Bongaarts-Feeney approximation; no clear standalone interpretation except in special cases; "least interesting"
5. Expected cohort life expectancy (LE) — linear-shift projection for the cohort born at time $t$
6. Achieved cohort LE — linear-shift projection for the cohort whose mean age at death is attained at time $t$
Period-to-cohort speed conversion. If period $e_0$ improves at rate $r$ per calendar year, cohort $e_0$ improves at $s = r/(1-r)$ per birth year. Under $r = 0.2$ yrs/yr, $s = 0.25$ yrs/birth-year — a $25\%$ steeper improvement pace. This is an exact result under the linear shift model and an approximation under more general mortality trend assumptions.
ACLE as a fifth measure (Schoen and Canudas-Romo 2005): Average Cohort Life Expectancy (ACLE) extends Wilmoth's taxonomy with a measure that directly answers "what is the average LE of people alive today?" Under declining mortality the hierarchy is: Cohort LE $>$ ACLE $>$ Period LE $>$ CAL. ACLE is approximately the arithmetic mean of cohort and period LE, with improvement rate averaging theirs. Under continuing mortality improvement ( $c > 0$ ), published period LE understates ACLE — i.e., understates the actual average longevity of living cohorts. The practical limitation is that ACLE requires complete mortality data for all active cohorts, unavailable for a century. See Average Cohort Life Expectancy (ACLE).
Hybrid construction of observed cohort LE. In practice, cohort LE is never purely observed. For recent cohorts, deaths at older ages have not yet occurred and must be projected. For a birth cohort born in 1941, only mortality through roughly age 60–70 is directly observed; the rest is model-dependent projection. Cohort LE estimates for recent cohorts are therefore more sensitive to model assumptions than period LE. See Period Mortality.

How It Works

The period-cohort transformation is a coordinate change: the cohort birth year $\tau = t - x$ , where $t$ is calendar time and $x$ is age. A period life table conditions on $t$ , holding $x$ variable. A cohort life table conditions on $\tau = t - x$ , holding $\tau$ fixed. Period survival probabilities are computed as:

$\ell_p(x,t) = \exp\!\left(-\int_0^x \mu(a,t)\, da\right)$

using current (period) death rates $\mu(a,t)$ at all ages $a$ . Cohort survival is:

$\ell_c(x,\tau) = \exp\!\left(-\int_0^x \mu(a,\, \tau+a)\, da\right)$

using the death rates that actually prevailed when the cohort of birth year $\tau$ was at each age $a$ .

Why It Matters

Social Security solvency. Social Security Administration (SSA) uses period life tables for cost projections. Cohort life tables — which better reflect actual beneficiary longevity — generally show longer expected lives, implying higher future benefit costs. The choice of measure directly affects actuarial balance estimates.
DI program design. Disability Insurance (DI) beneficiaries' expected time on the rolls depends on cohort survival. Period LE understates this duration under improving mortality, biasing cost estimates downward.
Mortality inequality research. Period LE comparisons can mask cohort-specific divergence. A low socioeconomic status (SES) cohort may have worse cohort LE than period LE implies if low-SES workers benefited less from recent medical improvements. See Income-Mortality Gradient.
NDC pension design (Italy). Italy's 1995 Notional Defined Contribution (NDC) reform computed transformation coefficients from 1990 Istituto Nazionale di Statistica (ISTAT) period life tables, updated only every 10 years. Because cohort LE exceeds period LE under improving mortality, the expected annuity is systematically underestimated $\to$ transformation coefficients are too high $\to$ pensions are systematically overpaid. Belloni and Maccheroni (2006) estimate this overpayment at $\approx 6$ percentage points (pp) of net present value ratio (NPVR), with roughly two-thirds from the cross-sectional vs. longitudinal table divergence and one-third from the decennial revision lag. The fix is to use prospective cohort tables or to update coefficients continuously. See Notional Defined Contribution and Actuarial Neutrality.

Application: COVID-19 and Time-Limited Events (Parra et al. 2022)

The period vs. cohort distinction has acute policy relevance for pandemics. Published COVID-19 LE loss (LEL) estimates (Murphy et al. 2021, Andrasfay and Goldman 2021) use the period approach, which assumes 2020 COVID mortality rates persist indefinitely. For a time-limited event expected to decline, this produces dramatic overestimates: period-based LEL exceeds cohort-based LEL by a factor of >1,000 for younger ages. Parra et al. (2022) apply the cohort approach to individual-level death certificate data (three Midwest areas plus national Centers for Disease Control and Prevention (CDC) data, Mar 2020–Sep 2021) and find:

Cohort-based national LEL = 11 days (vs. 1–2 years in period estimates)
The period approach inflates LEL for the young because it assumes decades of continued exposure; the cohort approach inflates LEL for the old because elderly bear the actual population fatality rate (PFR)
The cohort approach is generally more appropriate for a time-limited event

More generally: period-based LE loss is the right measure only if the cause of death is permanent. Cancer or heart disease are (quasi-)permanent features of the mortality landscape, so period estimates are appropriate. A pandemic — or a war, a natural disaster — is time-limited; the cohort approach, which integrates actual cumulative mortality over the event window, is methodologically superior. See Years of Life Lost for the YLL framework that underpins this calculation.

Open Questions

Under accelerating improvement ( $r$ increasing over time), the period-cohort gap grows. It is empirically unclear whether high-income country improvement rates are accelerating, stable, or decelerating, which affects the gap magnitude.
The linear shift model yields $s = r/(1-r)$ exactly. Under non-linear trend models (e.g., Lee-Carter), the exact relationship is more complex; the $s = r/(1-r)$ formula is an approximation.