# 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
∆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

Contents
Morningstar - Q2 2023 - Cover1
Morningstar - Q2 2023 - Cover2
Morningstar - Q2 2023 - Contents
Morningstar - Q2 2023 - 2
Morningstar - Q2 2023 - 3
Morningstar - Q2 2023 - 4
Morningstar - Q2 2023 - 5
Morningstar - Q2 2023 - 6
Morningstar - Q2 2023 - 7
Morningstar - Q2 2023 - 8
Morningstar - Q2 2023 - 9
Morningstar - Q2 2023 - 10
Morningstar - Q2 2023 - 11
Morningstar - Q2 2023 - 12
Morningstar - Q2 2023 - 13
Morningstar - Q2 2023 - 14
Morningstar - Q2 2023 - 15
Morningstar - Q2 2023 - 16
Morningstar - Q2 2023 - 17
Morningstar - Q2 2023 - 18
Morningstar - Q2 2023 - 19
Morningstar - Q2 2023 - 20
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Morningstar - Q2 2023 - 43
Morningstar - Q2 2023 - 44
Morningstar - Q2 2023 - 45
Morningstar - Q2 2023 - 46
Morningstar - Q2 2023 - 47
Morningstar - Q2 2023 - 48
Morningstar - Q2 2023 - 49
Morningstar - Q2 2023 - 50
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Morningstar - Q2 2023 - 55
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Morningstar - Q2 2023 - 64
Morningstar - Q2 2023 - 65
Morningstar - Q2 2023 - 66
Morningstar - Q2 2023 - 67
Morningstar - Q2 2023 - 68
Morningstar - Q2 2023 - 69
Morningstar - Q2 2023 - 70
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Morningstar - Q2 2023 - 76
Morningstar - Q2 2023 - 77
Morningstar - Q2 2023 - 78
Morningstar - Q2 2023 - 79
Morningstar - Q2 2023 - 80
Morningstar - Q2 2023 - Cover3
Morningstar - Q2 2023 - Cover4
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2024q3
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2024q2
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2024q1
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2023q4
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2023q3
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2023q2
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2023q1
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2022q4
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2022q3
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2022q2
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2022q1
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2021q4
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2021q3
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2021q2
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2021q1
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2020q4
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2020q3
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2020q2
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2020q1
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2019winter
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2019fall
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2019summer
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2019spring
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20191201
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20181011
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20180809
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20180607
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20180405
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20180203
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20181201
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20171011
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20170809
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20170607
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20170405
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20170203
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20171201
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20161011
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20160809
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20160607
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20160405
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20160203
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20161201
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20151011
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20150809
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20150607
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20150405
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20150203
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20151201
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20141011
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20140809
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20140607
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20140405
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20140203