Average Cohort Life Expectancy (ACLE)

Definition

ACLE (Average Cohort Life Expectancy) is the survivorship-weighted average of the life expectancies at birth of all cohorts alive at a given time $t$ . Proposed by Schoen and Canudas-Romo (2005), it answers the question "what is the average life expectancy of the people alive right now?" — a question answered by neither period life expectancy (LE) (a synthetic-cohort construct) nor cohort LE (only one birth cohort) nor the Cross-Sectional Average Length of Life (CAL) (which is sensitive to past mortality fluctuations). The weight given to age $a$ is the cohort's actual probability of surviving to that age, $\ell_c(a, t-a)$ .

$ACLE(t) = \frac{\int_0^\omega e_c(0,\,t-a)\,\ell_c(a,\,t-a)\,da}{\int_0^\omega \ell_c(a,\,t-a)\,da} = \frac{\int_0^\omega e_c(0,\,t-a)\,\ell_c(a,\,t-a)\,da}{CAL(t)}$

The denominator equals CAL(t) (the cross-sectional average length of life), so ACLE is also the expectation of cohort LEs under the CAL density $C_{CAL}(a,t-a) = \ell_c(a,t-a)/CAL(t)$ .

Key Ideas

Four-measure hierarchy under declining mortality:

Cohort LE > ACLE > Period LE > CAL

Under constant mortality all four are equal. ACLE is always intermediate between cohort and period LE, and above CAL under declining mortality. The ratio of ACLE to cohort LE is approximately $1 - c/(2b)$ in a Gompertz model with constant improvement rate $c$ and age-gradient $b$ .
ACLE vs. CAL — the tempo sensitivity problem: CAL was proposed by Bongaarts and Feeney (2002) as a tempo-robust alternative to period LE. Schoen and Canudas-Romo (2005) show this claim fails. In a pure-tempo scenario (timing shifts that leave all cohort LEs fixed), period LE and CAL each fluctuate by more than 2 years, while ACLE remains exactly flat (since it is determined solely by cohort LEs). CAL is sensitive to timing changes because it integrates past cohort survival probabilities, which encode the history of mortality at all ages — including past timing shifts.
CAL as a lagged average of period LEs: After a mortality level shift at $t=0$ , $CAL(t) = w(t)\,LE(-) + [1-w(t)]\,LE(+)$ , where $w(t)$ declines roughly exponentially. Even 20 years after the shift, 24% of the weight still attaches to the pre-shift level. CAL is thus history-sensitive in a way ACLE is not.
ACLE is insensitive to timing changes that don't affect cohort LE: This follows directly from the definition. ACLE depends only on the completed life expectancies of active cohorts, not on when within those cohorts deaths occur. If a timing shift compresses deaths into younger ages for a cohort (raising period LE) but does not change that cohort's completed LE, ACLE is unchanged.
Empirical relationship to period LE: Under Scenario A (constant future mortality), ACLE lies between period LE and cohort LE throughout the 1880–2000 histories of England/Wales, Norway, and Switzerland. Under Scenario B (continuing improvement at $c = 0.005$ per year), ACLE exceeds period LE in recent decades — suggesting that published period life expectancies currently understate the average longevity of living cohorts.
ACLE decomposition: $\partial ACLE/\partial t = \bar{\dot{e}}_c + Cov(e_c, r_c)$ . The first term is the cohort-survivorship-weighted average of cohort LE improvements (the direct component); the second is a covariance between cohort LE levels and cohort growth rates. In all three countries, the direct term dominates; the covariance term is small and negative (~10% of the direct term), partially offsetting improvement.
Slope relationships: In continually declining mortality models, ACLE improvement rate ≈ arithmetic mean of cohort and period LE improvement rates. Cohort LE diverges from period LE and CAL over time; ACLE stays roughly parallel to both.

How It Works

At time $t$ , the persons alive at age $a$ belong to the cohort born at $t-a$ . That cohort's completed life expectancy (from birth) is $e_c(0,t-a)$ . The number of people of that cohort alive at time $t$ per unit birth is $\ell_c(a,t-a)$ . ACLE sums the product $e_c(0,t-a)\cdot\ell_c(a,t-a)$ over all ages and divides by the total population count — which equals CAL(t) under constant births. The result is the cohort-survivorship-weighted mean of all active cohorts' life expectancies.

Practical computation requires:

Complete cohort life tables for all cohorts currently alive — ages 0 through ω.
For cohorts not yet extinct (typically ages 0–85+ at the reference year), future mortality must be projected under an assumed scenario (e.g., constant last-period rates; or continued decline at $c = 0.005$ per year).
The measure is therefore not a current real-time statistic; it is retrospectively computable for cohorts that have fully died out, and scenario-dependent for living cohorts.

Why It Matters

Resolves what period LE and CAL cannot: Period LE answers a synthetic-cohort question; CAL is sensitive to mortality history; cohort LE covers only one birth year. ACLE uniquely gives the average actual LE of all people alive today — the question most intuitively asked.
Critiques the CAL-as-tempo-correction claim: The paper's finding that CAL is susceptible to timing distortions (almost as much as period LE) undermines the Bongaarts-Feeney (2002) argument that CAL corrects for a tempo bias in period LE. If the proposed correction is itself sensitive to timing, it does not resolve the underlying concern.
Forecasting implication: Under continuing improvement, ACLE > period LE, suggesting that actuarial projections based on period life tables may systematically underestimate the longevity of living cohorts, with implications for pension solvency and Disability Insurance (DI) roll duration.

Open Questions

ACLE requires future mortality projections for living cohorts; results depend on the scenario chosen. It is unclear which scenario assumption is most appropriate for applied actuarial use.
Is there a tractable approximation to ACLE that can be computed from current and recent period data, analogous to the Bongaarts-Feeney (BF) approximation for CAL?
Under accelerating mortality improvement ( $r$ increasing), the ACLE–period LE gap grows. How large is this gap in contemporary high-income populations?

Sources

Schoen and Canudas-Romo 2005 — Changing Mortality and Average Cohort Life Expectancy