Gompertz Law of Mortality, or, your body wasn’t built to last

by on August 2, 2009 at 8:44 am in Science | Permalink

I thought this was one of the more interesting blog posts I've read in some time:

What do you think are the odds that you will die during the next
year?  Try to put a number to it – 1 in 100?  1 in 10,000?  Whatever it
is, it will be twice as large 8 years from now.

This startling fact was first noticed by the British actuary
Benjamin Gompertz in 1825 and is now called the “Gompertz Law of human
mortality.”  Your probability of dying during a given year doubles
every 8 years.  For me, a 25-year-old American, the probability of
dying during the next year is a fairly miniscule 0.03% – about 1 in
3,000.  When I’m 33 it will be about 1 in 1,500, when I’m 42 it will be
about 1 in 750, and so on.  By the time I reach age 100 (and I do
plan on it) the probability of living to 101 will only be about 50%. 
This is seriously fast growth – my mortality rate is increasing
exponentially with age.

And if my mortality rate (the probability of dying during the next
year, or during the next second, however you want to phrase it) is
rising exponentially, that means that the probability of me surviving to a particular age is falling super-exponentially.

The post has much more, including excellent visuals.  Here is Wikipedia on the law.  Here is an attempted derivation of the law

Three thought questions: a) what does the law imply about systematic risk and asset pricing? b) what does the law imply for regulatory structures and Arnold Kling's chess game analogy?  c) what does the law imply for the Fermi paradox?

DanC August 2, 2009 at 9:49 am

The following comments are from Explanation of the Decline in Mortality among the Oldest-old: a Demographic point of view
Graziella Caselli , Department of Demography, university of Rome “La Sapienza”
James W. Vaupel, Anatoli I. Yashin, Max Planck Institute for Demographic Research, Rostock

“In many highly developed countries, remarkable progress has been made in recent decades in reducing death rates, especially at older ages. New statistical data on mortality over time and up to the highest ages have revealed the time and age pattern of these improvements. These data have permitted reliable estimation of the age-trajectory of mortality, which turns out to follow a logistic pattern with deceleration at advanced ages. Individuals are heterogeneous with regard to their chances of death, and the frail tend to die first. Deeper understanding of the age-trajectory of mortality and the pattern of mortality improvements hinges on the development of statistical models that incorporate such mortality selection. ”

You can see a series of articles on the topic at http://www.iussp.org/Activities/scc-lon/lon-prog00.php

Brian August 2, 2009 at 11:07 am

I’m not sure I see how the existence of a sharp upper limit on human life spans affects financial regulatory structures. What did you mean by point b)?

pedroj August 2, 2009 at 11:33 am

This is another interesting theory relating machines failure with biological life-span
Reliability theory of aging and longevity
http://en.wikipedia.org/wiki/Reliability_theory_of_aging_and_longevity
and an interesting book chapter
http://longevity-science.org/Aging-Theory-2006.pdf
You can see there one graph comparing drosophila and human life-span. The curious thing is Gompertz law is apparently not valid above some age where life expectancy becomes higher, not lower!

Richard S August 2, 2009 at 12:16 pm

“And if my mortality rate (the probability of dying during the next year, or during the next second, however you want to phrase it) is rising exponentially, that means that the probability of me surviving to a particular age is falling super-exponentially.”

Wrong! Every year you survive obviously increases (not decreases) your chances of surviving to a particular age. A 100 year-old man has a much better chance of living to 101 than a 30 year-old does. Duh.

Parke August 2, 2009 at 12:40 pm

That was interesting! But the stats teacher in me kept getting distracted trying to figure out if the author meant “odds” or “probability,” since the terms (incorrectly) seemed to be used interchangeably in the post. The odds equal p/(1-p), so they are similar in purpose, but they are on different scales. I tried to check Wikipedia to see whether Gompertz Law refers to odds or probabilities, but I found the following description instead, which is more complicated, more interesting, and, in the end, more than I wanted to tackle on a Sunday.

http://en.wikipedia.org/wiki/Gompertz-Makeham_law_of_mortality

Ryan August 2, 2009 at 3:50 pm

Why do we assume that death is probabilistic at all?

Whatever the risks of bungee jumping or walking down a dark alley at night, there is no PROBABILITY that these things will kill you. It is your choice whether you bungee jump or walk down dark alleys. It has nothing to do with probability.

There are infinitely many ways to die, but just because X% of us die as the result of heart disease doesn’t mean that any one individual has an X% chance of dying that way. Not every ratio is a probability.

timn2bama August 2, 2009 at 6:54 pm

Why all the math fuss! Just live life to the fullest and don’t worry about when you going to die!

Michael F. Martin August 2, 2009 at 11:17 pm

a) what does the law imply about systematic risk and asset pricing? b) what does the law imply for regulatory structures and Arnold Kling’s chess game analogy? c) what does the law imply for the Fermi paradox?

a) If you’re assuming that Gompertz puts a sharp cut-off on the lowest “rational” discount rate, then you’re not adjusting for individual preferences that take family or friends in their argument. Dead hand control has been around for a long time, and doesn’t seem to be on its way out. So not sure how this would matter to asset pricing. But maybe there’s less concern about very low-frequency systematic risks because the size of the group that can interact in the economy isn’t growing as fast as it might be were there not a double-exponential cut-off on survivorship?

b) If the term of 99.9% of loans is mismatched with the productive lifespan of 99.9% of borrowers, then an oscillation in credit on the order of lifespan might show up?

c) You got me

Jimbino August 3, 2009 at 12:20 am

“…that means that the probability of me surviving to a particular age…” should read, “…that means that the probability of my surviving to a particular age…” in the interest of English grammar.

Anderson August 3, 2009 at 6:26 pm

What do you think are the odds that you will die during the next year? Try to put a number to it — 1 in 100? 1 in 10,000? Whatever it is, it will be twice as large 8 years from now.

I categorically disagree.

Whatever my chance of dying in 2010, it will be zero in 2018, at least if I’m the one wondering about it.

anonymous August 3, 2009 at 9:36 pm

Gompertz’s law only applies to human mortality because our own bodies kill us. External threats (famines, wars, plagues, car crashes, murders) may be constant or they may vary slightly over time, but in any case do not grow exponentially.

Similarly, external threats to our civilization’s existence (gamma-ray bursts, supernovas, asteroid strikes) might vary over time but are not observed to grow exponentially. If Gompertz’s law were to apply to the Fermi paradox, it could only be if an advanced technological civilization creates lethal threats to itself that grow exponentially over time.

We might readily posit that an advanced technological civilization’s capability to kill itself off is likely to grow over time. And to the extent that scientific progress seems to be advancing exponentially (the rate of new discoveries seems to be proportional to what we already know), this self-threat might indeed grow exponentially. However there is unfortunately no empirical evidence whatsoever to support a Gompertz law here, given that we are working with a null dataset of extinct interstellar-contact-capable civilizations.

anonymous August 3, 2009 at 10:54 pm

If genetic replication of biological cells occasionally goes awry and metastatizes into a cancer (giving rise to mortality rates governed by Gompertz’s law), then perhaps the replication of memes (or Susan Blackmore’s “temes”) at the core of an advanced technological civilization may be susceptible to similar “cancers”.

Some biological cancers are known to have external causes, ie viral infection. Could the same apply to “meme cancers”? We could even hypothesize the existence of a Toxoplasma-like meme parasite that spreads by interstellar panspermia and induces self-destructive behavior in advanced civilizations. A Gompertz law might apply if the willingness of an advanced civilization to reject alluring new memes declines linearly over time.

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