This is part 1 of a 2 part series on Gompertz’s law.

After enjoying a podcast interviewing Moshe Milevsky, I got interested in Milevsky’s work and read his book The 7 Most Important Equations for Your Retirement.

It’s a bit of a weird book. I can’t say I didn’t enjoy it. But it gave me the impression Moshe Milevsky didn’t expect his readers to enjoy the book, or at least he didn’t seem to expect them to like the formulas. That seems like a weird proposition for a book with “equations” in its very title. Perhaps the readers were imagined to be part of a captured audience of students, asked to read the book as an assignment? At any rate the author apologizes every time he actually discusses formulae, and having to resort to logarithms seems to make him positively embarrassed.

Then again, Prof. Milevsky wrote many books and this is apparently one of his bestsellers. And the anecdotes of Fibonacci, Gompertz, Halley, Fisher, Huebner, and Kolmogorov1 in the book are lovely, as is Milevsky’s account of his experience of economist Paul Samuelson. Perhaps the professor just knows his audience.

But so do I, being well acquainted with the empty set. So there’s no issue with too much math on this blog.

Let’s therefore discuss what Milevsky doesn’t do in his book and try to explain why his equation #2 might be true: Let’s talk about the Gompertz distribution.

Gompertz’s discovery

Benjamin Gompertz
Image source: Wikipedia

Benjamin Gompertz (1779–1865) was a British mathematician and actuary of German Jewish descent. According to Milevsky, Gompertz was looking for a law of human mortality, comparable to Newton’s laws of mechanics. At his time, statistics of people’s lifespan were already available and formed the basis for Gompertz’s discovery. For instance, one could compile a mortality table of a (hypothetical) group of 45-year-olds as they age. The data might look like this:

Age Alive at Birthday Die in Year
45 98,585 146
46 98,439 161
47 98,278 177
48 98,101 195
49 97,906 214
50 97,692 236

What does this tell us? It doesn’t immediately tell me anything not very obvious: The older one gets, the more likely one dies soon.

But let’s compute the mortality rate: How likely were people to die at a given age in this cohort? To be precise, we compute “Which proportion of people alive at age \(t\) died before age \(t+1\)?”

Age Alive at Birthday Die in Year Mortality Rate
45 98,585 146 0.148%
46 98,439 161 0.164%
47 98,278 177 0.180%
48 98,101 195 0.199%
49 97,906 214 0.219%
50 97,692 236 0.242%

So again, so much so obvious: The probability of dying the next year, conditional on having lived until the current year, increases with age.

But how much does it increase? This is Gompertz’s discovery, now called Gompertz’s law: The mortality rate appears to increase exponentially. One way of seeing this is to take logs and compute their difference:

Age Alive at Birthday Die in Year Mortality Rate Log of Mortality Rate Difference in Log Values
45 98,585 146 0.148% −6.515
46 98,439 161 0.164% −6.416 0.0993
47 98,278 177 0.180% −6.319 0.0964
48 98,101 195 0.199% −6.221 0.0987
49 97,906 214 0.219% −6.126 0.0950
50 97,692 236 0.242% −6.026 0.1000

The difference in log values increases by about 0.1, regardless of the current age. Since \(\exp(0.1) \approx 1.1\), this works out to roughly a 10% increase in the mortality rate per year.

According to Gompertz’s discovery, the likelihood of dying in the next year (conditional on having lived until this year) increases exponentially by a fixed percentage every year.

How do we model this mathematically? And if Gompertz’s law has a rate growing by a rate, how come everything stays a probability, i.e., \(\le 1\)?

Keep reading to learn about hazard functions and find out!

PDFs, CDFs, and Hazard Functions

Some probability density functions
Image source: Wikipedia
Some cumulative distribution functions
Image source: Wikipedia

If you have taken a probability or statistics course, you probably (ha!) know about probability density functions (pdfs). A pdf is a positive function that we use as a density and to make it a probabilty density it needs to integrate to one. If \(f\) is a pdf and \(X\) is a random variable with that distribution then

\[\P(a < X\le b) = \int_a^b f(x) \dx,\]

i.e., the probability of \(X\) landing between \(a\) and \(b\) is the integral from \(a\) to \(b\) of \(f\).2 Let’s say our random variable takes only positive values, then for \(a=0\) and \(b=\infty\) this value will be \(1\), i.e., 100%. For smaller values of \(b\) this will be the so-called cumulative distribution function (cdf):

\[F(b) := \P(X\le b) = \int_0^b f(x)\dx.\]

While pdfs are positive and integrate to \(1\), cdfs are positive, \(0\) at \(0\), monotone (non-decreasing), and \(1\) at the limit to \(\infty\). They are also typically assumed to be right-continuous. To compute the previous expression one can then take

\[\P(a < X \le b) = \int_a^b f(x) \dx = F(b) - F(a).\]

Moreover the fundamental theorem of calculus tells us that (barring a few exceptional points),

\[F'(x) = f(x).\]

What this means is that the cdf determines the pdf and therefore the probability distribution itself: Any function \(F\) with the properties above can serve as a cdf and gives rise to a pdf \(f\), and vice versa.

If you made it this far, you likely knew this in principle. So on to hazard functions now (which I learned about from Wikipedia only recently).

Hazard functions

Some hazard functions
Gnuplot source

If we imagine \(f\) to describe deaths (or failure rates) over time, \(F(t)\) will be the probability to have died by time \(t\). Conversely, \(S(t) := 1 - F(t)\) (the survival function) is the chance to still be alive at time \(t\). If \(f\) is the corresponding pdf and \(X\) is a random variable with this distribution,

\[S(t) = \P(X > t) = \int_t^\infty f(x)\dx = 1 - F(t).\]

Suppose we made it until some time \(t_0 \ge 0\). How will the future look like? We will need to condition on having lived so far and renormalize with \(S(t_0)\) to get the pdf going forward. This is the same as finding the probability of not surviving an additional infinitesimal time:

\[h(t_0) := \frac{f(t_0)}{S(t_0)} = \lim_{t\downarrow 0} \frac{\P(t_0\le X < t_0 + t)}{t S(t_0)} = -\frac{S'(t_0)}{S(t_0)} = - \frac{\d}{\d t}\ln(S(t))\Biggr\rvert_{t=t_0}.\]

This is the hazard function, also known as force of mortality or force of failure. It takes positive values and the last equation indicates that it will not be integrable, i.e., \(\int_0^\infty h(x)\dx = \infty\).

There is also the cumulative hazard function, which is

\[H(t) = \int_0^t h(x)\dx = -\ln S(t).\]

This last equation, which follows from the above, tells us something interesting: We ought to be able to retrieve \(S\), and therefore the cdf \(F\), and therefore the pdf \(f\), from \(h\) itself, because

\[S(t) = e^{-H(t)}.\]

This means, just as any one of a pdf or a cdf determines the other, so does a hazard function: Given either a pdf, a cdf, or a hazard function (or a cumulative hazard function), the probability density and therefore all of these functions and the entire distribution are uniquely determined. The criteria for \(h\) are: (1) being nonnegative and (2) not being integrable.

Back to Gompertz

So how do we model Gompertz’s discovery? We choose the most simple exponentially growing function we know and take it as our hazard function:

\[h(t) := b \eta e^{bt}\]

where \(\eta, b > 0\) are parameters to be determined from the data (see below). This also tells us why our “rate increase of a rate” doesn’t lead to probabilities greater than one over time: It’s a rate increase for the conditional probability at that fixed point in time \(t_0\). The hazard function itself does grow to infinity.

Given this, what is the cdf for this Gompertz distribution? Well,

\[H(t) = \int_0^t h(x)\dx = b \eta\int_0^t e^{bx}\dx = \eta(e^{bt} - 1)\]

and therefore

\[F(t) = 1 - S(t) = 1 - e^{-H(t)} = 1 - \exp\bigl(-\eta(e^{bt} - 1)\bigr)\]


\[f(t) = h(t)S(t) = b\eta\exp\bigl(bt - \eta(e^{bt} - 1)\bigr).\]

So the simple exponential choice for the hazard function yields this not quite so simple double exponential as a pdf for the Gompertz distribution.

pdf cdf hazard
The pdf, cdf and hazard function of the Gompertz distribution for some choices of \(\eta\) and \(b\).
(See above for source.)

We could proceed to compute the mean, variance or moment-generating function (aka Laplace transform) of this distribution, but the math gets somewhat hairy and special functions (mainly the generalized exponential integral, a special function related to the incomplete gamma function) are involved. We will return to this for other reasons in a future blogpost. The infobox on Wikipedia has the data if necessary.

For now, let’s note that \(h(t) = b \eta e^{bt}\) describes a hazard function where the chance of dying in the next moment increases throughout life, but there’s only this time-dependent component. If there are time-independent causes of death or failure (war, some kinds of diseases, voltage surges), one could consider modelling this differently. William Makeham, another British 19th century mathematician, proposed such an addition via the hazard function

\[h(t) = b \eta e^{bt} + \lambda.\]

The result is known as the Gompertz-Makeham distribution.

As simple as this change may seem, it does complicate the resulting integrals quite a bit. So much so that finding closed-form expressions of the quantile function or the moments of this distribution is still an active field of research!

Gompertz from here on out

If a random variable \(X \sim \textrm{Gompertz}(\eta, b)\) describes the expected time of death of a newborn (or, less morbidly, the expected time of failure for a newly built device), how does the world look like at time \(t_0\)? That’s an important question for, e.g., retirement planning.

So let’s compute the pdf after surviving until \(t_0\):

\[\frac{f(t_0 + t)}{S(t_0)} = b\eta\frac{b\eta\exp\bigl(b(t_0+t) - \eta(e^{b(t_0+t)} - 1)\bigr)}{\exp\bigl(-\eta(e^{bt} - 1)\bigr)} = b\eta e^{bt_0}\exp\bigl(-\eta e^{bt_0}(e^{bt}-1) + bt\bigr).\]

This is the pdf of \(\textrm{Gompertz}(\eta e^{bt_0}, b)\). So the future stays Gompertz-distributed with new parameters. This is a useful property of the Gompertz distribution: truncated renormalized versions of it stay within the family, in contrast to, e.g., the Gaussian distribution.

This helps us answer questions like “How long will I spend in retirement?”, which is why Milevsky discusses Gompertz in his book about retirement planning.

Before we can do that, let’s discuss how one could fit the parameters \(\eta\) and \(b\).

How to fit the data: Modal value of human life

The modal value(s) of a distribution is the maximum (or set of maxima) of the pdf. In general, it is distinct from both the mean as well as the median. In the case of Gompertz it’s also an easy computation:

\[\frac{\d}{\d t}f(t) = 0 \ \Rightarrow\ b\eta\exp\bigl(bt - \eta(e^{bt} - 1)\bigr)(b - \eta b e^{bt}) = 0 \ \Rightarrow\ \eta e^{bt} = 1\]

and therefore \(\eta = e^{-bt_m}\) for the modal value \(t_m\).

So once we have \(b\) we can get \(\eta\) from the data by looking for the year (or month) with the most deaths.

The \(b\) parameter we also get from the data as it determines the year-on-year increase in the mortality rate: From the data above we would estimate \(b=0.1\). Milevsky suggests using \(1/b = 9.5\) years and calls this the dispersion coefficient of human life.

As the modal value of human life Milevsky suggests using \(t_m=87.25\) for the general population in North America. Obviously \(t_m\) was lower in Gompertz’s time; for a population of healthy females in a modern society Milevsky suggests using up to \(t_m=90\).

Plugging it all together

For a member of the general population in a developed country we can take \(b = 1/9.5\), \(t_m=87.25\) and therefore \(\eta = e^{- bt_m} = e^{-87.25 / 9.5} \approx 1.026\cdot 10^{-4}\). If that person is \(t_0\ge 0\) years old, the Gompertz pdf for their future looks like

\[b\eta e^{bt_0}\exp\bigl(-\eta e^{bt_0}(e^{bt}-1) + bt\bigr) = b e^{b(t_0 - t_m)} \exp\bigl(-e^{b(t_0 - t_m)}(e^{bt}-1) + bt\bigr)\]

while the cdf is

\[1 - \exp\bigl(- e^{b(t_0-t_m)} (e^{bt} - 1)\bigr).\]

Therefore, the survival function conditioned on having lived until year \(t_0\) is

\[p_{t_0}(t) := \exp\bigl(- e^{b(t_0-t_m)} (e^{bt} - 1)\bigr).\]

This is also the chance of still being alive at year \(t_0 + t\) given being alive at year \(t_0\) and constitutes an answer to the question of “How long will I spend in retirement?”, or more precisely “How likely am I to be alive \(t\) years into my retirement?”. It’s also Milevsky’s equation #2.

In a future blogpost, we will use this to compute values of annuities. For now, let’s wrap up with a quick discussion of how realistic this model is.

Is this model any good?

The first thing to say is that sadly, human babies are more likely to die than the Gompertz distribution would indicate. That’s partially a sad fact of life and partially a consequence of the fact that severe medical problems in unborn babies or during birth are less likely to cause outright death at the point of delivery for the baby in question, but may still cause death after a few days, months, or even years.

On a potentially more positive note, the Gompertz distribution might overestimate the chance of dying next year for very old people. As Wikipedia has it:

At more advanced ages, some studies have found that death rates increase more slowly – a phenomenon known as the late-life mortality deceleration – but more recent studies disagree.

Generally, there seems to be consensus that the Gompertz distribution models mortality really well between the ages of 30 and 80.

A question very dear to “transhumanists” is to what extent the gains in longevity seen in the last centuries can be expected to continue to happen in the future. Obviously a simple two-parameter model won’t tell us a lot about that, and neither will the three-parameter Gompertz-Makeham distribution. Still, I found it interesting if a bit discouraging to read, again in Wikipedia:

The decline in the human mortality rate before the 1950s was mostly due to a decrease in the age-independent (Makeham) mortality component, while the age-dependent (Gompertz) mortality component was surprisingly stable. Since the 1950s, a new mortality trend has started in the form of an unexpected decline in mortality rates at advanced ages and “rectangularization” of the survival curve.

So apparently \(\eta\) and \(b\) have been stable over a long time periods. Whatever that says about human longevity, it also says the Gompertz (or the Gompertz-Makeham) model is not so bad.

That’s it for today. My apologies for not actually having any \(\pi\)s in all of this after all :)

  1. For Kolmogorov, I’d love to have another reference for his supposed War and Peace-inspired nickname “ANK”. The book in question is the only reference I could find. 

  2. This works for probability measures that are absolutely continuous with respect to the Lebesgue measure, meaning such a Lebesgue density exists. All measures on the reals can be decomposed into such a measure, a discrete measure and a so-called singular continuous measure. Let’s just stay with Lebesgue densities here.