Morningstar - Q2 2023 - 50

Strategies
t =
1
r�
ln
r� = ln(1 + r)
Solving the Retirement Equations, Part I
Moshe Milevsky provides an
important foundation.
QUANT U
Paul D. Kaplan
Moshe A. Milevsky, professor of finance at York
University's Schulich School of Business in
Toronto, is one of my favorite authors in financial
economics. His website describes him as
" a leading authority on the intersection of wealth
management, financial mathematics, and
insurance. " 1
He is a prolific writer on the
economics of annuities and insurance. While most
of his articles and papers are for his fellow
academics, he also writes for a broader audience,
especially in his books.
Two of his books stand out to me-one for
the way it combines mathematics with the stories
of the people behind the math (The 7 Most
Important Equations for Your Retirement, Wiley,
2012) and the other for its history of annuities (Life
Annuities: An Optimal Product for Retirement
Income, CFA Institute Research Foundation, 2013).
Both books present equations for answering
specific questions in retirement-income planning
and the pricing of insurance instruments. Along
with co-author Thomas Idzorek, I incorporate some
of these equations into a comprehensive financial
planning methodology in a forthcoming CFA
Institute Research Foundation publication,
Lifetime Financial Advice: A Personalized Optimal
Multi-Level Approach.
EXHIBIT 1 lists the equations that these books
discuss, and in this and subsequent issues of
Quant U, I will review the highlighted ones. Going
through these equations should provide a solid
1 moshemilevsky.com
50
Morningstar Q2 2023
foundation in retirement-income planning and the
uses of annuities and life insurance.
How Long Will a Nest Egg Last?
Suppose that you have accumulated a nest egg of
W dollars and that you plan to continuously
withdraw c dollars per year. How many years
could you make these withdrawals before
running out of money? That depends in part on
the rate of return on the nest egg while it
is being depleted. Here we assume that it is
a constant, continuous rate, which we denote
as . Equation 1 in The 7 Most Important
Equations gives the number of years until the nest
egg runs out. We can write this equation
as follows:
t =
1
r�
ln
r� = ln(1 + r)
t =
1
r�
ln
c
c - Wr�
Although Milevsky wrote this equation in
continuous time, which assumes that the
withdrawals take place continually, we can easily
rewrite it in discrete time so that withdrawals
take place once a year (or at whatever frequency).
Let r denote the conventional discrete time
rate of return in the desired frequency,
such as 3% per year. The continuous rate of return,
, is related to the discrete rate of return, r,
as follows:
c
c - Wr
q(t; a, m, b) = exp 1 - exp
t =q( ; a, m, b) c - Wr�
1
r�
ln
=
r� = ln(1 + r)
The number of years (or whatever frequency) is
given by the following equation:
t =
p(t; a, m, b) = q(t-1; a, m, b) -
T
1
r�
c - Wr
LE (a, m, b) = a + t
t =1
q(t; a, m, b) = exp 1 - exp
∑
LE (a, m + m, b) =
q( ; a, m, b) =
ln
c
p(t; a, m, b)
q(t; a, m, b)
b
t
exp
a - m
b
q(t; a, m, b) =
q(t; a, m, b) - q( ; a, m, b)
1 - q( ; a, m, b)
c
t
b
exp
a - m
b
gFor a given rate of return, the higher the
withdrawal rate, the shorter the period until
funds are exhausted.
o
gEach curve is convex (decreasing at a
decreasing rate), and they all converge to about
seven years at a withdrawal rate of 15%.
gThe curve for 3% is above the curve for 2.5%,
which is above the curve for 2%. This shows that
for any given withdrawal rate, the higher
the rate of return, the longer the period until
funds are exhausted.
o
=
D(
P
D(
f =
n
P(
P
P(
c
q
P
P(
t =
1
r�
ln
c
c - Wr
Observe that the only difference between this
equation and the previous one is that r replaces
in the expression in parentheses (but not in the
term outside of the parentheses).
q(t; a, m, b) = exp 1 - exp
q( ; a, m, b) =
Also, to use this equation, it must be the case
that c > Wr. This is because if c ≤ Wr, returns on
the nest egg are high enough that the
withdrawals will never exhaust it. In other words,
withdrawals can be made forever.
q(t; a, m, b) =
q(t; a, m, b) - q( ; a, m, b)
1 - q( ; a, m, b)
p(t; a, m, b) = q(t-1; a, m, b) -
T
LE (a, m, b) = a + t
t =1
∑
Finally, although c and W appear as separate
variables, it is only their ratio, c/W, that
matters. This ratio is the fraction of the initial nest
egg that is withdrawn each period and is
sometimes called the withdrawal rate. Another
way of stating the condition for the equation
to work is that the withdrawal rate must exceed
the rate of return.
p(t; a, m, b)
q(t; a, m, b)
LE (a, m + m, b) =
To illustrate how the equations work,
in EXHIBIT 2 , I vary the withdrawal rate and
the rate of return to show how they
each affect the time until the nest egg is
exhausted. Each curve shows the relationship
between the withdrawal rate and the
years to exhaustion for a different discrete
rate of return (3%, 3.5%, or 4%) for withdrawal
rates between 4% and 15%. Here are a
few observations:
t
b
exp
a - m
b
c
c - Wr�
f =
n
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