Morningstar - Q2 2022 - 52

85.7
Strategies
25
Age
EXHIBIT 3
Evolution of Human Capital and the Value of Liabilities
Age
25
35
45
25
Age
35
45
55
65
35
45
55
65
28.2
35.7
41.8
42.3
53.9
28.2
35.7
59.8
58.1
28.2
35.7
41.8
42.3
53.9
55
65
30
30
45
30
45
60
75
15
15
Value
$2.5M
15
2.0
LIABILITIES
25
Age
Source: Morningstar.
25
Age
EXHIBIT 4
Fraction of Net Worth Spent on Discretionary Consumption
65
LIABILITIES
25
Age
45
85
LIABILITIES
45
65
85
105
120
73
73
45
65
85
105
120
1.5
1.0
0.5
Value
$2.5M
2.0
Value
$2.5M
1.5
1.0
0.5
2.0
1.5
1.0
0.5
Fraction of Net Worth
80%
105 Fraction of Net Worth
80%
120
60
40
20
73
25
Age
25
Age
Source: Morningstar.
25
Age
EXHIBIT 5
Financial Wealth and Term Life Insurance Over the Lifecycle
Term Life Insurance
Annuity Wealth
Term Life Insurance
25
Age
25Term Life Insurance 45
Age
25
Age
Source: Morningstar.
45
65
Regular Financial Wealth
85
105
115
45
65
Annuity Wealth
Regular Financial Wealth
0.70
0.35
1.05
Wealth
$1.40M
Regular Financial Wealth
85
105
115
0.70
0.35
1.05
0.70
0.35
65
85
105
115
45
Annuity Wealth
65
85
105
115
45
65
85
105
115
45
65
85
105
115
60
40
20
60
Fraction of Net Worth
80%
40
20
Wealth
$1.40M
Wealth
$1.40M
1.05
pv
∑ ∑=∑qv
T qv
T1+r )Tv= t
(
v= t
T
T
L =It
Ut
v= t( cv
( cv
∑
vT= t
∑ (1+ )
(1+r )
v= tqv
qv
T
v= t
The Intertemporal Budget Constraint and
the Solution
The investor is subject to an intertemporal budget
constraint to make sure that all consumption
and life insurance can be funded. Letting Ft
denote financial wealth in year t, The intemporal
budget constraint is:
t
v= tg ∑ - ) = (1+g ) ( cv-1
Lt
==qv
(1+r )
t
g =tpv
H =t
v= tv- t xv
1+r
1+
qv T
( ∑ - qv t
= qv -1
L =It
T
t =1+r ) qv
v= t
v=Tt
T
∑
H =tt = v= tT qv (1+r ) - t
ct = + (1+rt )qv
∑
T
Lt =∑ (1+r )
v= t
∑
v= t
v= t
Lt IG=
Ut
pvIG
Ut
t
v= ttc t= +( 1 - )
pv
= qv -1 - qv
v=tt
L =It
LT =It
t = qv -1
t
Dt = ∑ (
t
g = qv
v= t
=∑
∑
∑ ∑Dt
∑
T
v= t
g IG=
Ut
∑
v= t
v= t
T
Tt =(1+r )qv
t
t
( cv - ) = (1+g ) ( cv-1
v= t
v= tqv
(1+r )
ct = +
g =
=
t =
ct = +∑ (t Ft + Ht
v= t
So, growth is positive if the market rate of
return is greater than the subject discount rate,
Dt = qv 1+ ) - Lt - LIt
v= t
t =
∑
∑
3 It shrinks if g is negative.
52
Morningstar Q2 2022
T qv
1+rt- v
t
ct = +
Ut
Ut
Ft + Ht
- Lt - LIt
t
B
IG = ( 1 - ) u (ct - ) +
IG = ( 1 - ) u (ct - ) +
T
D ∑
t v
u
u
B
Dt
B
Dt
v= t
T
t
1+r
1+g
v- t
B
v- t
The growth rate g is given by:
( cv - ) = (1+g ) ( cv-1
t
1+r
1+
qv
Ft + Ht
- 1
T( 1 - )
t
1+r
1+
1+gu (ct - ) +
v- t
- 1
u
- )
( cv
= qv (1+1+g
T
t
v- t
v= tqv (1+r ) t- v
( )cv
1+r
1+
t
v- tFt + Ht( cv
t
qv
(1+tr )u
T ( 1 - ) u (ct - ) +
- 1 )
t
tpv
pv
(1+ r )
t
t- vv- t
v- t
v- t(ct - ) +
t
- Lt - LIt FT
B
The optimal path of lifetime consumption is the
path that maximizes lifetime utility subject
to the intertemporal budget constraint. One of the
properties of the optimal consumption path
is that discretionary consumption grows at a
constant rate, g, so that:3
1+r
( cv - ) = (1+g ) ( cv-1
T - qv
- )
u
B
Dt
- yvv- t ) = - LIT
FT
- yv = - LIT
- Lt - LIt
) - )FT
B
B
Dt
B
B
- yv ) = - LIT
u
B
Dt
B
∑ ∑=(1+r )
T qv
t Ft + Htv yv
1+r
qv
T v- tt xv1+g
t
1+r
1+
qv
(1+r )
ptv
qv
- 1
(1+r )
v- t
t - 1
- t
v- t
1+rv yv
t
1+gv- t - Lt - LIt
v- t
v- t
B
- )
Ut =∑ xv
v= t
qv
(
v- t
Ut
Ut
∑ (1+ )
T
v= t
∑ ( v= t
T
v= t
∑ ( t
u ( =x)qv
1+r )
t
T
denote the probability of the investor
dying in year v (given that the investor is alive
in year t), we have:
T
H =t
Lt
Ut =∑ ((1+r ) v- t
=
∑
T
T
T
T
∑ ∑pv
Lt
v= t
u ( =x)
L =tIt
t
u ( =x)= qv -1 (1+r )
(1+ )
t = t qv
v= t
The lump-sum price for guaranteeing a bequest
of $1 is:
= qvtT-1 - qv t
qv
∑
T
t
Hence, the lump-sum price for guaranteeing a
bequest of size B is LIt
t
(1+ ( )1+r )
t
t
t
(v- t ) u (ct
( cv
v yv) = ∑ (1+ )
- t
∑ ∑H =tT - ) = (1+g ) ( cv-1
v= t
t 1+ )qv
t
qv
(v- t ) u (ct) B.= ∑ (1+ )
v- t
- yv ) = - LIT
T
- )v= t
FT
v= t
Bt
(v- t )
u (ct
)
qv
(v- t )
t
- 1 (x
t
- qv
(v- t )
t u (ct
v- t
(1+r )- 1 (xv- t
t
pv
(1+r( )(1+r )( cvv- t u ()cv
v- t
t 1+ t )pv
qv
ln(x),
-1
- 1 ), (if ≠ 1
- 1 ),
∑
v= t
- )
- )FT
B
t
u (ct
)
if
v- t u ( cv
v- t - yv = - LIT
T
if ≠ 1
qv
qv
qv
v= t
v= t
v= t
(1+r )
t
Aget
t
v yv
1+ ) v- t
- t u ( cv
ln(x),
) =-1
- )
T
if qv= 1t
1+ )
= 1
(v- t )
u (ct
)
qv
qv
qv
T =∑ ( (v- t )
t
1+ )
t
v= t qv1+ )
(1+r )
25
Age
v- t xv
25
Age
1+ )u (ct) = ∑ (1+ )
65
t Working
v- t u ( cv
45
45
- )
v= t
T
T
v= t
65
qv
of size B, no matter when they die. (Below, I
discuss how this could be achieved with
either term or whole life insurance.) To price the
lump-sum policy, we first need to know the
probability of dying in each year. We can calculate
this from the survival probabilities. Letting pt
qvt
(v- t ) u (ctWorking ∑ (1+ )
) =
qv
H =t( ∑ (1+r )- 1 (x- t
v- t xv
t
qv
v= t
25
45
65
85
v
115
ln(x),
-1
v yv
(v- t )Retired (ct
u
85
- 1 ),
if SOCIAL SECURITY
if ≠ 1
= 1
SOCIAL SECURITY
Retired
t
(v- t )
SOCIAL SECURITY 115
t
85
)
115
25
u (ct
)
50
75
25
25
50
1 (
v- t u ( cv
) )
HUMAN CAPITAL
HUMAN CAPITAL
HUMAN CAPITAL
W
A
G
S
E
Morningstar - Q2 2022

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