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This PDF is a research paper by Jesús-Adrián Álvar This PDF is a research paper by Jesús-Adrián Álvarez, published by the Society of Actuaries Research Institute (2023). It deeply examines a fundamental and surprisingly unresolved question:
**What does it actually mean for a life to be “long”?
Where does longevity begin?**
The paper argues that traditional definitions—“old age starts at 60 or 70”—are arbitrary, outdated, and disconnected from modern demographic reality. Instead, Álvarez proposes a rigorous, mathematical, population-based definition of when a life becomes “long,” using survivorship ages (s-ages) and concepts from demography, evolutionary biology, and reliability theory.
🧠 1. Purpose of the Paper
The main goal is to develop a formal, scientifically grounded definition of the onset of longevity. The author:
Reviews historical and modern definitions of old age
Shows how chronological-age thresholds fail
Introduces s-ages as a more accurate way to measure longevity
Demonstrates how survival patterns reveal a natural “start” to longevity
Uses mortality mathematics to locate that threshold
Longevity 2023
📜 2. Historical Background: Why Age 60 or 70?
The paper explains how the idea that old age starts at 60–70 came from:
Ancient Greece (age 60 military cut-off)
Medieval Europe (age 70 tax exemption)
Early pension systems (Bismarck’s Germany, Denmark, UK, Australia)
These were social or political definitions—not scientific ones.
Today, many 70-year-olds live healthy, active lives, making old thresholds meaningless.
Longevity 2023
📊 3. The Problem With Traditional Measures of Longevity
Common demographic indicators are examined:
✔ Life Expectancy
Mean lifespan, but ignores lifespan variation.
✔ Modal Age at Death
Most common age at death, but problematic in populations with high infant mortality.
✔ Entropy Threshold
Measures sensitivity of life expectancy to mortality improvements.
All these measures describe aspects of population longevity—but none cleanly answer:
When does a long life begin?
Longevity 2023
🔍 4. The New Solution: Survivorship Ages (s-Ages)
Álvarez and Vaupel propose defining longevity using:
s-age = the age at which a proportion s of the population is still alive.
For example:
x(0.5) = the median age
x(0.1) = age when 10% survive
x(0.37) = the threshold of longevity proposed in this paper
This transforms mortality analysis into a population-relative scale, rather than a fixed chronological one.
Longevity 2023
🚨 5. Breakthrough Finding: Longevity Begins at s = 0.37
Using hazard theory and survival mathematics, the paper shows:
Longevity begins when 37% of the population is still alive.
Mathematically:
Longevity onset occurs at the s-age x(0.37)
This is where cumulative hazard equals 1, meaning:
The population has experienced enough mortality to kill the “average” individual.
This is a universal, population-based threshold, not a fixed age like 60 or 70.
Longevity 2023
🧬 6. Biological Interpretation
From evolutionary biology:
Natural selection pressures drop sharply after reproductive years
After this point, life is governed by “force of failure” (aging processes)
Álvarez connects this transition to the mathematical threshold H = 1, aligning biology with demography
Thus, x(0.37) represents the beginning of “post-Darwinian longevity.”
Longevity 2023
📈 7. Empirical Findings (Denmark, France, USA)
Using mortality data (1950–2020), the paper shows:
🔹 Major longevity indicators (life expectancy, modal age, entropy threshold, s-age 0.37):
All rise dramatically over time
All exceed age 70
All cluster closely around each other
🔹 Key insight:
Longevity begins well after the traditional retirement ages of 60–70.
Longevity 2023
⭐ 8. Main Conclusions
Old age cannot be defined by fixed ages like 60 or 70.
Longevity is population-relative, not chronological.
The onset of longevity should be defined as x(0.37)—the age when 37% of a population remains alive.
This threshold is biologically meaningful, mathematically grounded, and consistent across countries.
Modern populations experience much later onset of old age than historical definitions suggest.
Longevity 2023
🌟 One-Sentence Summary
Longevity begins not at a fixed age like 60 or 70, but at the survivorship age x(0.37), the age at which only 37% of the population remains alive—a dynamic, scientifically derived threshold.... |