Morningstar - Q2 2022 - 51

EXHIBIT 1
EXHIBIT 2
Men
Projected Annual Wages for Ages 25-65, by Gender and Education Level
Women
Postcollege
College
Men
Postcollege
College
High School
Generic
High School
Generic
Postcollege
College
Women
Postcollege
College
High School
Generic
High School
Generic
Annual Salary
$90K
85.7
59.8
58.1
28.2
35.7
41.8
42.3
53.9
25
Age
25
Age
Source: Morningstar.
Value
$2.5M
I assume that the investor is a 25-year-old man
with a postcollege degree currently making
$50,000 a year.
25
Age
Social Insurance
Once a person retires, they may have some
income that at least in part replaces their wages.
These sources of retirement income typically
include government-sponsored social insurance.
Many countries provide lifetime income to
their residents on the basis of salary history,
contributions, years of residence, starting
age, and other factors. In the U.S., this is Social
Security. In Canada, it is the Canada
Pension Plan.2
LIABILITIES
LIABILITIES
25
Age
45
45
65
65
85
25
Age
In the example I present here, I assume
that the investor is a U.S. resident who retires
at age 65 and immediately begins collecting
Social Security. I used the formulas that
the Social Security Administration uses to estimate
how much a retiree will receive from
Social Security. I assume no other exogenous
sources of income. EXHIBIT 2 shows how
25
Age
45
45
65
65
exogenous income evolves, with wages from
age 25 to 64 and Social Security starting
at age 65.
u ( =x)
T
- 1 (x
t
Ut =∑ (1+ )
qv
v= t
T
v. Letting r denote the market rate or return
(which I assume to be constant), the price of this
contract is:
∑u ( =x) 85
v= t
T
Ut
T
=qv
(1+r )
t
(1+ ) - 1 (x) =
qv
(v- t ) u (ct
t
v= tv- t 1+ )
t
(1+ )
v= t
T
Lt =qv
85pv
∑
H =t
t = qv 85- qv
v= t
Tt
∑
L =It
Annuity Wealth
T
∑
Annuity Wealth
T
-1
T
2 The exception is workers in Quebec, who get the Quebec Pension Plan. Canada also has Old Age Security and the Guaranteed Income Supplement, which are not tied to salary history or
contributions.
(1+r )qv
Lt =∑ (1+r )
v= t
∑
v= t
t
v- t
v= ttpv
Term Life Insurance
Term Life Insurance
= qv -1
T
L =It
∑
(1+r )- qv
t
( cv
t
Regular Financial Wealth t
pv
( cv - ) = (1+g ) ( cv-1
Regular Financial Wealth
t
- )
- yv ) = - LIT
FT
B
0.70
1.05
0.70
1.05
qv
pv
t
v- t
v- t
Wealth
$1.40M
Wealth
$1.40M
ct = +
Ut
Ft + Ht
- Lt - LIt
t
IG = ( 1 - ) u (ct - ) +
B
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u
B
Dt
51
(v- t )qv
(1+r )
t
qv
(1+r )v= t
v- t(1+r )
xv
t t
(1+r )
t
- t
qv
v yv
- t
∑ ∑H =tv= t
In the context of the model that I present here,
human capital is the annuity value of present and
future exogenous income. Letting yv
t
T qv
t u (ct) = ∑ (1+ )
qv
v= t
denote
exogenous income in year v and T denote the last
possible year that the investor could still be
alive, human capital in year t is:
v- t
40
20
105
v yv
115
105
115
40
20
(v- t )
60 u (ct
)
v- t u ( cv
Fraction of Net Worth
- )
T
73
t
Fraction of Net Worth
80%
73
60
∑ ( xv
t
qv
80%
105
ln(x),
-1
T
- 1∑)105
v= t
120
= 1t
, (if ≠ 1(v- t )
1+ )
120 u (ct
)
ln(x),
-1
if
- 1 ),
v- t u ( cv
- )
= 12.0
if ≠ 1
Human Capital and Liabilities
I assume there are complete annuity markets,
meaning that in year t, it is possible to buy
or sell a contract that pays xv
1.5
1.0
0.5
in some future year 0
qv
if
Value
$2.5M
2.0
1.5
1.0
0.5
35
35
45
45
55
55
65
65
30
45
60
75
15
Annual Salary
$90K
85.7
59.8
58.1
28.2
35.7
41.8
42.3
53.9
30
45
60
75
15
75
u ( =x)
T
v= t
Working
- 1 (x
t
Ut =∑ (1+ )
25
T
qv
Age
45
25
t
∑ (1+ )
qv
t
(1+r )
v- t xv
Liabilities is a concept similar to human capital,
but rather than measuring the value of exogenous
income, it measures the annuity value of
present and future nondiscretionary spending. The
value of liabilities in year t is:
qv
T
H =t
∑
v= t
T
v= t
t
(1+r )
t
v- t
Lt =∑ (1+r )
qv
EXHIBIT 3t
pv
value of liabilities evolve in the example I present
here, with nondiscretionary spending set at
$5,000 per year. As I discuss below, human capital
and liabilities play important roles in determining
the optimal level of consumption each year.
= qv -1 - qv
t
T
L =It
T
∑
v= t
∑
v= t
t
Life Insurance
Life insurances allows an investor to leave a
bequest of a given size should the investor
die before accumulating enough financial wealth
to fulfill their desired bequest. While life
insurance policies take a variety of different forms,
for now, I assume that the investor is willing
to pay a lump sum in year t to guarantee a bequest
(1+r )
v- t
( cv
- yv ) = - LIT
FT
- )
( cv - ) = (1+g ) ( cv-1
g =
T
t =
∑
v= t
t
qv
1+r
1+
- 1
1+g
1+r
v- t
o
B
qv
o
t
(1+r )
v- t
pv
shows how human capital and the
t
v yv
- t
qv
Source: Morningstar.
v= t
(v- t )
ln(x),
-1
Retired
if
Working
- 1 ),
v- t u ( cv
65
45
Age u (ct
T
v= t
= 1
Retired
if ≠ 1
SOCIAL SECURITY
- )
85
65
t
(v- t )
u (ct
)
) = ∑ (1+ )
qv
SOCIAL SECURITY
85
115
25
115
25
50
75
50
Consumption and Exogenous Income
Over the Lifecycle
Consumption
Consumption
Annual Income
Annual Income
$100K
$100K
HUMAN CAPITAL
HUMAN CAPITAL
W
W
A
A
G
G
S
S
E
E
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Morningstar - Q2 2022

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