Morningstar - Q2 2023 - 51

EXHIBIT 1
Equations Discussed in Life Annuities and The 7 Most Important Equations Quant U will focus on the shaded concepts
and equations.
Equation Solves For
How Long Will a Nest Egg Last?
Biological Mortality Rate
Survival Probabilities
Benjamin Gompertz's Law of Mortality
Basic Annuity Pricing Formula
Gompertz's Annuity Pricing Model
Duration and Interest Rate Sensitivity of a Life Annuity
Money's Worth Ratio of a Life Annuity
The Implied Longevity Yield
Lifetime Ruin Probability from Self-Annuitizing
Lifetime Ruin Probability
Payout of an Immediate Variable Annuity
Cost of a Tontine
Value of a Financial Legacy
Proper Spending Rate
Stocks vs. Cash Allocation
Source: Author.
t =
r�
r� = ln(1 + r)
Hence, as expected, to make a nest egg last longer,
you need to lower your withdrawal rate, obtain a
higher rate of return, or both.
The main limitation of this equation is that it
assumes a constant rate of return, which
is highly unlikely to be available. A more realistic
approach is to model each period's rate of
return as a random variable. I will discuss this
approach in a future issue of Quant U.
Benjamin Gompertz's Law of Mortality
This equation is so important that Milevsky
discusses it in both books (as Equation 2
in The 7 Most Important Equations and Equation 3
in Life Annuities).
In fact, I have considered it important for a long
time now, having already discussed it in
Quant U about eight years ago ( " When Will I Go? "
February/March 2015), based on Chapter 2 of
The 7 Most Important Equations.
While my notation differs a bit from Milevsky's, it
conveys the same relationship. I write the
Benjamin Gompertz Law of Mortality as follows:
t =
1
r�
ln
c
c - Wr
q(t; a, m, b) = exp 1 - exp
q( ; a, m, b) =
t
b
exp
a - m
b
The left-hand side of this equation is the
probability that a person of age a will live for at
least t more years, given the following parameters:
q(t; a, m, b) =
q(t; a, m, b) - q( ; a, m, b)
1 - q( ; a, m, b)
m = the mode of the distribution of the
age of death.
p(t; a, m, b) = q(t-1; a, m, b) -
T
LE (a, m, b) = a + t
t =1
∑
b = the dispersion of the age of death
around the mode.
p(t; a, m, b)
q(t; a, m, b)
The b parameter is like the standard deviation
parameter of a normal distribution.
LE (a, m + m, b) =
At Morningstar, we frequently use the
Gompertz formula to model longevity. We use
Parameter
m
b
Males
88.0
10.6
o
Females
91.0
8.9
The beauty of the Gompertz formula is that it
gives a very accurate estimate of a person's
probability of survival with only two parameters,
thus making it easy to customize. To model
the probability of survival of a person whose life
expectancy differs from the average life
expectancy for their sex, we just need to adjust
the m parameter, as I demonstrate below.
The Truncated Gompertz Formula
The Gompertz formula will give a survival
probability for any non-negative value of t, with
the value of q(t;a,m,b) decreasing as t increases
and tending toward 0. In practice, we need
morningstar.com/products/magazine
51
the following default values for m and b for males
and females generically:
Life Annuities
-
Equation 1
Equation 2
Equation 3
Equation 4
Equation 5
Equations 6, 7
Equation 8
Equation 9
Equation 10
-
The 7 Most Important
Equations
Equation 1
-
-
Equation 2
Equation 3
-
-
-
-
-
Equation 7
Equations 11a, 11b -
Equation 12b
-
-
-
1 -
ln
c
c - Wr�
Equation 6
Equation 4
Equation 5
Originators
Leonardo Fibonacci (1170-1250)
Not Indicated
Not Indicated
Benjamin Gompertz (1779-1865),
William Makeham (1826-1891)
Edmond Halley (1656-1742)
Moshe A. Milevsky
Moshe A. Milevsky
Not Indicated
Moshe A. Milevsky
Not Indicated
Andrei N. Kolmogorov (1903-1987)
Not Indicated
Moshe A. Milevsky
Solomon S. Huebner (1882-1964)
Irving Fisher (1867-1947), Menahem Yaari (b. 1935)
Paul Samuelson (1915-2009)
http://www.morningstar.com/products/magazine
Morningstar - Q2 2023

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