Morningstar Magazine - February/March 2015 - 58

Strategies

EXHIBIT 2

p(t; a, m, b) = q(t; a, m ,b) - q(t +1; a, m, b)
In E X H I B IT 2 , I plot the probability distribution function for the age of death for a 65-year-old male
and a 65-year-old female using the parameters I
presented earlier.

When Might a 65-Year-Old Die For a person who reaches age 65, the chart shows
the probability of him or her dying at a certain age.
Male

Female

Probability of Dying (%)
5.0

Man: age 65
Woman: age 65

4.0

To calculate life expectancy, we calculate the mean
age of death:

3.0

LE = a + ∑ tp(t; a, m, b)
t=0

2.0

Using this formula, I find that the life expectancy
of a 65-year-old male is 84.2 years and 87.1 years
for a 65-year-old female.¹⁰
Sometimes, the median age of death is of interest.
To determine this, find the number of years
into the future (t) for which q(t)=0.5. Using the
parameters that I presented earlier, I find
the median age of death for 65-year-old males and
females to be 84.0 and 87.5 years, respectively.
Limits of the Gompertz Formula
The Gompertz formula works well over a wide
range of ages, but not for the very old. As Moshe
Milevsky explains:
"Benjamin Gompertz didn't have access to people
in their 90s and beyond because there weren't
many people around at his time. He had data
on life and death from people between the ages of
20 and 80, where his law works quite well."¹¹
Milevsky goes on to say that more recently
demographers and actuaries have discovered that
"[t]he mortality rate flattens somewhere around
104, and the odds of dying in the next year plateau
at about 50%." So, if you make it to 105, each
year it's a coin toss if you'll make for another year.
This is contrary to Gompertz's formula which
says that each year the probability of living for
another year declines at an increasing rate.
Also, you need to keep in mind that the
Gompertz formula describes a biological law under

1.0
65
70
Future Age

normal circumstances. It does not take
into account events such as freak accidents,
wars, and epidemics.
About Benjamin Gompertz
In late 18th century England, Gompertz, who
was Jewish, was not allowed to enroll in a
university because of his religion. So, instead of
studying mathematics and Newtonian physics
in a formal setting, he taught these subjects
to himself. He worked as an actuary and made his
big discovery by systematically studying records
of death.
Like some earlier mathematicians, Gompertz
never actually wrote down the formula that
I presented here. Rather, he described it in words.
"It was a fellow actuary by the name of William
Makeham, a younger contemporary of Gompertz,
who helped formalize and mathematize
the Gompertz law of mortality," Milevsky says.
Gompertz was eventually recognized for
his contributions and made a Fellow of the Royal
Society. He later became its president. Today,
we are the beneficiaries of his systematic

100

105

110

0.0

approach to analyzing mortality data and
the insights he had while doing so. The formula
based on his work provides an important
ingredient in good financial planning and the
pricing of insurance products. K
Paul D. Kaplan, Ph.D., CFA, is director of research with
Morningstar Canada. He is a member of the editorial board
of Morningstar magazine.
References
Blanchett, David and Paul Kaplan, 2013, "Alpha, Beta, and
Now... Gamma." Journal of Retirement, Fall.
Bernstein, Peter L., 1998, Against the Gods: The
Remarkable Story of Risk. John Wiley & Sons.
Ibbotson, Roger G., Moshe A. Milevsky, Peng Chen,
and Kevin X. Zhu, 2007, Lifetime Financial Advice: Human
Capital, Asset Allocation, and Insurance. Research
Foundation of the CFA Institute.
Milevsky, Moshe A., 2012. The 7 Most Important Equations
for Your Retirement. John Wiley & Sons Canada, Ltd.
Savage, Sam L., 2009, The Flaw of Averages: Why We
Underestimate Risk in the Face of Uncertainty. John Wiley
& Sons.

10 Another application of p(t) is the calculation of the fair value of life insurance. The fair value of one unit of life insurance paid the year the insured dies is L = ∑ p(t)(1+r)-t. The fair value of insurance
t=0
and annuities is related as follows: L = [(1+r)-1-1] A + (1+r)-1.
11 See Milevsky, 2012, chapter 2

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