From the February 2012 issue of Research Magazine • Subscribe!

Gompertz’ Law of Mortality: How Long Must Your Money Last?

For a man who never formally studied in or attended university, Benjamin Gompertz achieved a level of scientific immortality that even the most senior and tenured professors could only dream of today.

While most academics toil in obscurity, and their names eventually perish despite their frenetic publishing, Benjamin Gompertz has joined a very small distinctive group of scholars with an actual equation named after him. The equation has been admired and used by researchers in demographic studies, the world over, for almost two centuries now. His equation might not be as famous as Albert Einstein’s ubiquitous E = MC2, but it sure is a lot more useful for retirement income planning – even if you are a nuclear engineer.

Now, just to be clear, Benjamin Gompertz was not a college drop-out, who decided to tinker in his parents’ garage, instead of staying in school. I surmise that Benjamin would have loved nothing more than to enroll in university as an eager teenager. He just wasn’t allowed. You see, back in 1795 England, there were strict quotas on these sorts of things, and he was Jewish; hence, no admission. He had to make do with an informal “street” education.

But, despite the handicap, this self-starter taught himself everything there was to know about mathematics – becoming a virtual expert and devotee of Newtonian physics along the way – and made it all the way to the top of the British scientific aristocracy. He was made a fellow and eventually was elected president of the Royal Society. This honor would certainly be inconceivable in today’s hierarchical scientific world.

In fact, I’m not sure what happened to his childhood classmates, who did make it into university, but I doubt any of them have an equation named in their honor, used daily almost two centuries later.

Simplistic Retirement Planning

I have observed that when financial advisors discuss retirement income planning with their clients, they start by asking questions about how long they would like to plan for, or the age to which they expect to live, for example age 85 or 90. Consistent with the pick-your-timeline philosophy, many of the popular financial planning software tools and web-based retirement calculators force users to select a lifetime horizon in advance.

Perhaps you too have played with these tools, using various lifetime horizons. I can just hear the discussions “Aunt Gemma lived to 97, but Uncle Bob only made it to 82, so maybe we should use age 90?” or “Oh dear, we can only spend $60,000 per year if we plan to 90” which then leads to the inevitable reductio ad absurdum “Ok, lets plan to 85, because we really need $75,000 per year”.

The problem with this approach is that you really shouldn’t be picking your life horizon in advance. Life is random, and you know it. In my opinion, the next step in a scientific approach to retirement income planning is to understand how random your remaining lifespan really can be. To make an informed decision, you need to know the odds of living to various ages. Then, you can decide how long you want to plan for ­— and more importantly how you plan to adjust your spending if you live to a very old age.

This is precisely where Benjamin Gompertz’s handy little equation comes in.

Gompertz’s Big Discovery

Benjamin Gompertz, like other demographers and actuaries in the 19th century, spent much of his life examining records of death – and specifically the exact ages at which people died. Until Gompertz, scientists and researchers would compile or collect these records, but had never given much thought to extracting any forward-looking patterns or formal laws of mortality. They knew how many people had died in Carlisle or Northampton in the past, and could predict how many might die in the next few years – which was very important for insurance pricing – but the entire activity was rather ad hoc in the early 19th century.

The mortality tables complied by statisticians was a single frame from a snapshot of the past. Benjamin Gompertz figured out how to convert these individual frames into a movie.

The first two columns of Table #1 are an example of the snapshot from a hypothetical life (a.k.a mortality) table. You will see that there were 98,585 people who were 45 years of age and alive in a given (hypothetical) year. Then, of that large group, 146 people died between the age of 45 and 46, then 161 people died between the age of 46 and 47, then 177 people who died between the age of 47 and 48, etc.

The fact that these annual mortality rates increased with age was well documented and understood well before the time of Benjamin Gompertz.

But, Gompertz went one step further with these numbers in search of a pattern or a natural law. He wanted something like the laws of gravity, which were put forth by his British hero Sir Isaac Newton. So, he tinkered, played with and manipulated the mortality rates from many different mortality tables. Along the way, he decided to compute the natural logarithm of these numbers.

Benjamin Gompertz looked at the differences in values between two adjacent ages, and that is exactly when the light bulb went on!

When he subtracted subsequent values from each other, displayed in the sixth and final column of the table, he got numbers that were extremely close. As you can see, they are between 9% and 10% regardless of age! He did this process for many different age blocks, different mortality tables with populations from different cities and countries. Sure, the mortality rates were quite different depending on age, gender, country of origin and city, but the difference in the logs was the same.

To Benjamin Gompertz this was a very odd coincidence, and an indication that perhaps something deeper was at work. Why should the difference in the natural logarithm of death rates be constant with age? In fact, if you plot the values themselves (column 5) they fall on a straight line, with a slope that is approximately 0.0975. Mathematically speaking, if the difference between the natural logarithm of mortality rate is constant over time, then the mortality rate itself is growing exponentially at the rate of 9.75% per year.

Benjamin Gompertz deduced that there was a law of nature at work. Death was no longer just a random event whose likelihood increased with age. There was an underlying force of mortality that led to these values.

Benjamin Gompertz discovered that your probability of dying in the next year increases by approximately 9% to 10% per year, from adulthood until old age. Subsequent bio-demographic research has shown that every species on earth has its own rate.

From a mathematical point of view, assuming this line and working backwards, if you start with a species whose “chances of dying” increases by (say) 9% per year, you can invert the relationship and obtain the probability you will survive to any age. That is Gompertz’s equation, and why it is named after him.

According to his obituary, he was born on March 5th, 1779 in London and died on July 14th, 1865 in London, at the age of 86, which is exactly what one might expect under the Gompertz law of mortality.

Reprinted by permission of the publisher, John Wiley & Sons Canada, Ltd., from The 7 Most Important Equations for Your Retirement, by Moshe A. Milevsky.  Copyright © 2012 by Moshe A. Milevsky.

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