Morningstar - Q2 2023 - 52

r� = ln(1 + r)Strategies
t =
t =
1
r�
ln
c
c - Wr
q(t; a, m, b) = exp 1 - exp
to limit the range of t. To do this, I select a
very small number,
q(t; a, m, b) = exp 1 - exp
following equation for :
q( ; a, m, b) =
I then round to the nearest integer, which I call T.
For example, for a generic 25-year-old female and
setting = 10-9
q(t; a, m, b) =
q(t; a, m, b) - q( ; a, m, b)
1 - q( ; a, m, b)
, I get T = 93.
p(t; a, m, b) = q(t-1; a, m, b) -
T
LE (a, m, b) = a + t
t =1
∑
EXHIBIT 2
LE (a, m + m, b) =
Time to Exhaust a Nest Egg by Rate of Return and Withdrawal Rate
Rate of Return
50
40
50
10
20
30
40
10
20
30
Source: Author's calculations.
4%
Withdrawal Rate
EXHIBIT 3
Death Age Probability Distributions for 25-Year-Old Females
5%
3
4
3
4
2
1
25
Age
Source: Author's calculations.
Wt = + Ft
Ht
52
Morningstar Q2 2023
Wt = (1+r) (Wt-1
- ct-1
)
55
85
115
25
Age
55
85
115
Life Expectancy
2
1
5%
Isabela
Generic
25-Year-Old
Female
Isabela
Generic
25-Year-Old
Female
Life Expectancy
7
10
13
15
r = 2.5%
7.4
r = 3.0%
7.5
r = 2.0%
7.2
r = 2.5%
7.4
4%
Withdrawal Rate
7
10
13
15
r = 2.0%
7.2
Rate of Return
r = 3.0%
7.5
I limit t to be between 0 and T. To make the
probability of survival for T years into the future
p(t; a, m, b)
q(t; a, m, b)
(epsilon), and solving the
t
b
exp
a - m
b
q( ; a, m, b) =
q(t; a, m, b) =
t
b
exp
o
a - m
b
zero, I form a truncated version of the Gompertz
function, which I define as:
q(t; a, m, b) - q( ; a, m, b)
1 - q( ; a, m, b)
Life Expectancy
A person's life expectancy is the average age
at which the person dies, according to the
probability distribution of their age of death. To
compute it, we first need their death age
p(t; a, m, b) = q(t-1; a, m, b) -
T
LE (a, m, b) = a + t
t =1
∑
LE (a, m + m, b) =
p(t; a, m, b)
q(t; a, m, b)
1
r�
ln
c
c - Wr
t =
tr� = ln(1 + r)c - Wr
=
1
r�
1
f = n- qT
n = qT
ln
ln
q(t; a, m, b) = exp 1 - exp+ 1+ t )
j=1
r�P =
c
q
c - Wr
∑
probability distribution. This can be calculated
from their survival probabilities given by
the truncated Gompertz formula. Specifically, the
probability of dying t years from now (where
t is between 1 and T) is:
n
q(t; a, m, b) = = ∑(
q( ; a, m, b) =
P(y; c, T)exp 1 - exp1+
j=1
c
q
q(t; a, m, b) =
q( ; a, m, b) =
p(t; a, m, b) =
LE (a, m, b) = a += t
1
r�
LE (a, m, b) = a + t
f-jt =1
LE (a, m + m, b)
Life expectancy is given by:
p(t; a, m, b) = q(t-1; a, m, b) -
T
c
q
t =
=
(
P(y; c, T)
1+
y
q
)
ln
distribution and life expectancy for a generic
25-year-old female (LE = 86.4) and a hypothetical
25-year-old female who has a life expectancy
of 94, whom I named Isabela and discuss below.
LE (a, m + m, b)
EXHIBIT 3 shows the death age probability
c
q
(
r� = ln(1 + r)
c (y)
t =
1
r�
Adjusting the m-Parameter to Get to a Target
Life Expectancy
Isabela is 25, and both of her parents have families
with a history of long lifespans. Based on
this, it makes sense to adjust the m-parameter
for her to reflect the longevity in her family.
Based on her family history, her financial planner
estimates her life expectancy to be 94.
P =
ln
D =
c (y)
P(y; c,
(c
(
1
y
1
y
1+ )- q
y
q
P(y; c, T)
P (y; c, T)
Dm (y; c, T) =
q(t; a, m, b) =
Let
∆P
P
denote the target life expectancy
≈ -Dm∆y
(94 for Isabela) and Δm denote the change to the
m-parameter that makes the life expectancy
. To personalize the Gompertz formula
LE (a, m, b) = a + t
1+ y = Dm
P
*
for a given value of we solve the following
equation for Δm:
∑
LE (a, m + m, b) =
C (y; c, T) =
C(y; c, T) =
In the next issue of Quant U, I will continue
discussing the important equations for retirement
and annuities as presented by Milevsky. K
c
q
∑ n
j=1
( )(
f -j
q
Paul D. Kaplan, Ph.D., CFA, is director of research at
Morningstar Canada. He is a member of the editorial board
of Morningstar magazine.
+
The author thanks Thomas Idzorek for his helpful edits
and comments.
∆P
P
P(y; c, T)
1+ y
1
q
q
≈ -Dm∆y + C(∆y)
1
2
2
W
W
t =
T
∑
f -j- 1
q
)(1+
)
y
q
)
P(y; c, T)
T T+ )(
(
- qT-2
∆
P
T q
t =1
P(y; c, T)
P (y; c, T)
For Isabela, the solution is Δm = 7.6, so for her,
the modal death age is 91 + 7.6 = 98.6.
What's Next
f-j-2
p(t; a, m, b)
q(t; a, m, b)
p(t; a, m, b) = q(t-D1; a, m, b) -
DV01 P=
C
C(
c
q
D(y; c, T)
1+ y
q
q(t; a, m, b) - q( ; a, m, b)
1 - q( ; a, m, b)
q(t; a, m, b) = exp 1 - exp
Dm (y; c, T) =
q( ; a, m, b) =
∆
P
DV
t
b
exp
a - m
b
Dm
y
1+ q
)
c - Wr�
=
n
q
)
1+ y
q
c - Wr
f
Dm
)
f
c
∑
∑
=
t =1
-j
Pc
Dc
c
q
D(y; c, T) =
∑ n
j=1
j -f
q
y
q
q(t; a, m, b)
)
q(t; a, m, b) = q(t-1; a, m, b)
P(y; c, T) = P(y; c,
T
n
q
f-j
t
1 - q( ; a, m, b)
p(t; a, m, b)
q(t; a, m, b)
p(t; a, m, b)
q(t; a, m, b)
)(1+
(
y
q
)
f
1+
P(y; c,
n
q
y
q
)
f-j
)(1+
y
q
)
f
=
( ) (- q( ; a, m, b)+ 1+T
P(y; c, T)
c
q
b ( exp )
f-j
+ 1+
1+
y
q
)
q(t; a, m, b) - q( ; a, m, b)
1 - q( ; a-, m, b)
c
q
y
q
(
- qT
a - m
b
y
q
)
- qT
-j
(
y
q
b
n
-n
exp(1+ a - m )b
o
y
q
f
P(
D(
c
c
f =
P
P(
+
Probability of Dying
Probability of Dying
Years to Exhaustion
Years to Exhaustion
https://www.morningstar.com/authors/528/paul-kaplan
Morningstar - Q2 2023

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