Vol I, Issue 21. Quarter 3 – 2025.

Around 1800 a Jewish student in England names Benjamin Gompertz wanted to enter a University in England, but this was almost impossible due to the political norms of the times.  Given the social constraints in place, Benjamin decided to educate himself in a wide variety of areas of applied mathematics and economics. One of his projects involved the analysis of death records for a single village. By counting the population and deaths in the area over a number of years he began to create actuarial tables. A sample of the data is repeated in the book “The 7 Most Important Equations for Your Retirement” by Moshe Milevsky.  Consider the slice of data presented below.

Age	Alive at Birthday	Die in Year	Mortality Rate %	Nat. Log of Mortality Rate	Change in Value with Age
45	98,585	146	0.1481%	-6.5151
46	98,439	161	0.1636%	-6.4158	9.9280%
47	98,278	177	0.1801%	-6.3194	9.6382%
48	98,101	195	0.1988%	-6.2208	9.8652%
49	97,906	214	0.2186%	-6.1258	9.4966%
50	97,692	236	0.2416%	-6.0257	10.0044%
51	97,456	259	0.2658%	-5.9303	9.5415%
52	97,197	285	0.2932%	-5.8320	9.8322%
53	96,912	313	0.3230%	-5.7354	9.6651%
54	96,599	345	0.3571%	-5.6348	10.0576%

The first two columns are probably self-explanatory, as they refer to the number of people alive in the town at a given age. Note, that this does not say that there are 98,585 people of an age of 45 in a given year. Rather it is that of the people tracked, 98,585 lived to be at least 45. We then see that 146 such people died during the calendar year that included their 45^th birthday. Dividing these two terms yields a mortality rate. For example, 146/98,585 = 0.001481, or 0.1481% of the people who live to the age of 45 die during the year in which they turn 45. The next column shows the natural logarithm of this value. All of these values are negative because the natural log of any value between 0 and 1 is negative. Stated differently ln(0.001481) = -6.5151. Or equivalently e^(-6.5151) = 0.001481.

The last column in the table (Column 6) shows the change in the values listed in Column 5. The most obvious take-away is that the mortality rate increases by between 9 and 10% with each year of age.

Let’s make all of this as plain as possible. What the chart ultimately shows is that the probability of dying in each year is about 9.5% higher than it was last year. Here is a simple way to view what that means. A man who is alive at his 59^th birthday has about a 1% probability of dying over the next 365 days. If he makes it to the age of 60, the likelihood of death in the next 365 days is roughly 1% * (1 + 9%) = 1.09%. The corresponding values for the ages of 61, 62, and 63 become 1.09% * 1.09 which is roughly 1.188%, followed by 1.295%, and 1.412% respectively. It should come as no surprise that the likelihood of dying increases with age.

This is a good point to step back and think about the nature of these results. Gompertz expanded his study to consider different age ranges, both sexes, many countries, and even different species. He was shocked to learn that the same type of pattern applied to almost any population that he studied. This result that the mortality rate of humans rises annually at this 9 or 10% rate seems to be almost universal. Of course, the starting point differs for each of the populations considered but the rate of the increase in the mortality rate seems to be a simple law of nature.

The end result is that when you build your spreadsheet model to evaluate things that depend on you still being alive at some future age, it becomes very simple to include mortality into the calculations. To make it not sound so bleak, instead of thinking of this as the year of your death, we can say that a given year becomes the end of your planning horizon with a known probability. I understand that this may be a bit depressing, but you are going to have to get over it, because it’s going to be true, whether you like it or not, so you may as well account for it.

Fortunately, you can make use of this observation to help you compute a host of important values such as:

• The remaining life expectancy of a decision maker of any age,
• The present value of a stream of guaranteed payments that end at death,
• The probability of a pool of assets surviving the investor,
• The expected value of lottery winnings, and
• The break-even point for applying for Social Security benefits.

Being able to estimate a mortality rate for any age has lots of applications and can be useful in building your own models as you consider major decisions about your money. I promise that we will talk about more pleasant things in other posts, but this had to be done at some point.

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