Morningstar - Q1 2024 - 42

Strategies
Solving the Retirement Equations, Part IV
From annuity payouts to consumption
levels, Milevsky has the formulas.
QUANT U
Paul D. Kaplan
This is the fourth and final installment of Quant U
in which I discuss the equations presented
by Moshe Milevsky of York University's Schulich
School of Business in Toronto in two of
his books:
1 The 7 Most Important Equations for Your
Retirement (Wiley, 2012)
2 Life Annuities: An Optimal Product for
Retirement Income (CFA Institute Research
Foundation, 2013)
The equations in these two books answer
specific questions in retirement-income planning
and the pricing of insurance instruments.
In this installment, I discuss four equations, each
of which answers a different question.
Payout of an Immediate Variable Annuity
Life Annuities, Equations 11a, 11b
I discussed immediate variable annuities, or IVAs,
in the Q3 2022 issue of Quant U, so this section
is somewhat of a review. An IVA is an annuity
that continues to make payouts until the investor
dies. But unlike a fixed annuity, payments
vary over time depending on the performance of
an underlying portfolio of risky assets. The
payouts of an IVA also depend on its assumed
interest rate.
The equations that Milevsky presents focus on
the calculation of the payouts of an IVA in
real time. (As usual, my notation is a bit different
than Milevsky's but is consistent with that of
previous issues of this series.) The first equation
gives the initial payment:
42
Morningstar Q1 2024
P0
W0
=
A(a, 0, h)
where:
P0
=
h
1
amax
A(a, 0, h) =
W0
Pt
1 + Rt
1+h
q =
a + t
LI1 B =
a
t
a
t
t
-a
∑
t=1
qa
t
(1 ) t
+h
= the initial payout (time 0)
= the amount of wealth invested in
the IVA
Pt-1
= the assumed interest rate
A(a,0,h) = the price of a fixed annuity that
pays $1 for life for a person of age a,
with no guaranteed period, with
discount rate h. This is the price of
the IVA.
qa
qa
t +1
1 q -
1 + r
1
amax
-a
a + t
B
From the equation for the price of an immediate
fixed annuity that I present in Part II of this series
(in the Q3 2023 issue), we have:
P0 = ∑ (1 ) v - t
LI B = W0
A(a, 0, h)
v= t
p =
a + t
v - t
LI B =
Pt
a
where:
amax
t
=
ln(ct +1
qa
1
t
=
LI1 B =
a
t
a
t
q =
a + t
qa + t
A(a, 0, h) =
1 + Rt
1+h
∑
v= t+1
) - ln(ct
t +1
qa
qa
t
1
Using the truncated Gompertz model that I describe
in Part I of this series (Q2 2023) for a 25-year-old
woman and an assumed interest rate of 4%,
I calculate the price of the IVA, A(25, 0, 0.04), to be
$22.37. Assuming that the investment in the IVA
(W0
μ- rf
2 1
1 q -
1 + r
amax
-a
v= t
a + t ( F + H )
B
P0 = ∑ (1 ) v - t
LI B = W0
p =
a + t
The second equation gives subsequent payouts
as follows:
v - t
) is $100,000, the initial payout (P0
qa + t
A(a, 0, h)
.
v - t - 1
A(a, 0, h) =
LI B =
Pt
a
t
=
ln(ct +1
1
1 + Rt
1+h
∑
v= t+1
q =
a + t
) - ln(ct
t +1
qa
qa
t
1
1
μ- rf
a + t ( F + H )
ln(ct +1
) - ln(ct
) =
amax -a - 1
∑
amax -a- qa + t
v - ttqa
t=1
(1 ) t
pa + t
+h
v - t
(1 ) v - t
Pt-1
+r
) =
ln(q1
a + t) + -
B
p =
a + t
v - t
LI B =
t
In year t, the probability of the person dying in year
t+1 is 1- q1
qa + t
amax
∑
v - t - 1 - qa + t
v - t
in year t is: -a - 1
a
v= t+1
a+t. The premium for term life insurance
pa + t
+r
v - t
(1 ) v - t
ln(q1
B
o
a + t) + -
D(
qa + t
+r
v - t
LI1 B
a + t
v - t
) is $4,469.
q =
a + t
1
LI1 BWhere, as before, qa
a
t
=
LI B =
a
t
1 q -
1 + r
1
-a
∑
v= t
(1 ) v - t
qa + t
+r
v - t
qa
qa
t
a + t
age a surviving t or more years. Hence, q1
amax
LI1 B
a + t
v - t
tB is the probability of someone
a+t
means someone of age a+t surviving at least one
more year. Note that I define a to be the person's
age at time 0, so their age is a+t in year t.
Dm
P(
f =
n
Dm
c
q
∆
P
P
P(
=
DV
Pc
t +1
(
Dc
v - t - 1
amax -a - 1
∑
qa + t
+r
v - t
amax -a- qa + t
v - ttqa
t=1
(1 ) t
pa + t
+h
v - t
(1 ) v - t
Pt-1
+r
= the highest possible age (set to 115)
) =
= the probability of someone age a
surviving t or more years
ln(q1
a + t) + -
B
LI1 B
a + t
v - t
I am going to express the Huebner equation
in a bit of a different way than Milevsky by tying it
directly to term life insurance. With term life
insurance, each year you pay a term life premium
that keeps the insurance in place for the following
year. You must pay the premium each year so
long as you want to keep the policy in effect.
However, the premium rises each year because the
probability of dying the following year also rises.
P0
W0
=
A(a, 0, h)
amax
A(a, 0, h) =
Pt
=
1 + Rt
1+h
-a
∑
To calculate what a fair premium would be, the
first step is to calculate the probability of
dying in one year, given that you are alive in year t.
This is given by:
t=1
(1 ) t
+h
Pt-1
o
qa
t
Value of a Financial Legacy (Solomon S. Huebner,
1882-1964)
The 7 Most Important Equations, Equation 6
Solomon S. Huebner, whom Milevsky attributes
the equation I discuss in this section, was a
major figure in the life insurance industry. Milevsky
writes that in the early 20th
century, the life
insurance industry had a terrible reputation. In the
1920s, '30s, and '40s, Huebner was the person
most responsible for rehabilitating that reputation.
He was also a great advocate for life insurance.
o
D(
P(
f =
n
P
where
t is the realized return in year t on the
underlying portfolio of risky assets.
To show how an IVA works, I took the 500
Monte Carlo trials on the 50/50 stock/bond mix
that I use in Part III of this series (in the
Q4 2023 issue) to illustrate self-annuitization to
represent here the underlying risky portfolio
of an IVA. In this illustration, I go out 90 years.
EXHIBIT 1 shows the evolution of various
percentiles of the payout.
P(
f =
n
c
q
P
=
P(
Pc
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