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

Summary

Schoen and Canudas-Romo ask what the average life expectancy (LE) of cohorts actually alive in a given year is — a question answered by neither period LE (a synthetic cohort) nor cohort LE (one specific cohort) nor cross-sectional average length of life (CAL) (which the paper shows is more sensitive to past mortality than previously recognized). They introduce ACLE (Average Cohort Life Expectancy): a weighted average of all active cohorts' life expectancies at birth, with each cohort's weight equal to its actual probability of survival to its current age. Using both Gompertz population models and Human Mortality Database (HMD) data for England and Wales, Norway, and Switzerland (1880–2000), they show that ACLE is more stable than period LE or CAL under fluctuating mortality, is unaffected by pure timing changes that leave cohort LE unchanged, and under continuing mortality improvement exceeds period LE — meaning published period life expectancies understate average cohort longevity.

Key Claims

• ACLE definition:

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

The denominator equals CAL(t), so ACLE is also interpretable as a weighted integral of cohort LEs against the CAL density $C_{CAL}(a,t-a) = \ell_c(a,t-a)/CAL(t)$.

• Four-measure hierarchy under declining mortality: Cohort LE > ACLE > Period LE > CAL. Under constant mortality all four are equal.

• ACLE is tempo-robust (Figure 5): In a population where cohort mortality timing shifts (earlier or later deaths) without changing completed cohort LE, period LE and CAL each fluctuate by more than 2 years, while ACLE remains flat. This is the paper's key critique of CAL as a tempo correction: CAL is nearly as susceptible to timing distortions as period LE.

• CAL is a weighted lag of past period LEs (Wachter 2004): $CAL(t) = w(t)\,LE(-) + [1-w(t)]\,LE(+)$. After a mortality level shift at $t=0$, the weight $w(t)$ on the old LE declines roughly exponentially but remains at 0.24 even 20 years later. CAL is therefore sensitive to mortality history spanning the full life course, not just recent rates.

• Empirical findings (England/Wales, Norway, Switzerland 1880–2000):

  • In both scenarios (Scenario A: constant future rates; Scenario B: continued decline $c = 0.005$/year), ACLE lies between cohort LE and period LE throughout.
  • Under Scenario B, ACLE exceeds period LE in recent decades in all three countries — current period LE understates average cohort longevity.
  • The rate of ACLE increase is roughly arithmetic mean of cohort and period LE improvement rates; ratio of ACLE to cohort LE $\approx 1 - c/(2b)$ (steady-state, Gompertz model).

• Decomposition (Appendix):

$$\frac{\partial ACLE}{\partial t} = \bar{\dot{e}}_c + Cov(e_c, r_c)$$

Direct component (average change in cohort LEs) dominates; covariance/compositional term is small and consistently negative (~−0.007 to −0.054 per decade in England/Wales), partially offsetting gains.

• ACLE's practical limitation: Computing ACLE requires complete mortality data for all active cohorts, including those not yet extinct — unavailable for at least a century. Must be estimated using future mortality scenarios.

Concepts Introduced or Extended

• Average Cohort Life Expectancy (ACLE)
• Cross-Sectional Average Length of Life (CAL)
• Period vs. Cohort Life Expectancy
• Tempo Effects in Mortality

Entities Mentioned

• Robert Schoen
• Vladimir Canudas-Romo
• James W. Vaupel
• John Bongaarts
• Griffith Feeney
• John R. Wilmoth

Quotes

"ACLE is relatively insensitive to both period and cohort fluctuations in death rates, and is unaffected by changes in the timing of mortality that do not change cohort life expectancy."

"CAL is not only sensitive to tempo effects but generally understates the period level of longevity."

"With mortality steadily declining by more than 0.5% in recent decades, Figure 7 suggests that current life expectancies in many Western countries may understate the average life expectancy of living cohorts."

My Take

This paper clarifies a subtle but important point in the Bongaarts-Feeney (B-F) tempo-effects-in-mortality debate: CAL — the measure B-F (2002) proposed as a tempo correction to period LE — is itself susceptible to timing distortions and always lies below period LE under sustained improvement, making it a downward bias rather than a neutral correction. ACLE is a more defensible measure for the stated purpose (average longevity of living cohorts) but is empirically intractable in real time, requiring projection of future mortality. The practical upshot is that neither period LE nor CAL well represents the lived experience of cohorts alive today; ACLE does, but only with a ~100-year data lag. The paper is primarily a methodological critique and constructive proposal, not an empirical mortality study.