Morningstar Magazine - December 2017/January 2018 - 49

₀ ∞
1+r
q(t)
A₀■=Human
∑ Capitalt ■ Regular Account
p( t ) =t = 0q ((1
t -+1)r)- q ( t )
∞
p(t)
L₀ ==+ 1|t)
∑∞ = q(tt + 1)
q(t
+ r)q(t)
t = 0 (1q(t)
A₀ = ∑
t
t = 0 (1 + r)

1 - p(t)
q(t + 1|t)
L1₀==== ∑∑ q(t)y
H₀
L₀
(11+++r)r)rtt
t t==00 (1

■ Annuities

■ Annuity Purchases ■ Annuity Income

$1M

()

exp -1t - exp bt
(1 + )
-1

exp

( )
a-m
b

750K

exp
500K

∞L

250K

1 - exp

EXHIBIT 1

()
t
b

exp

a-m
b

Survival Probability
Survival Probability

j =1 f- j
y
q

n
nj =1c
q

c
q

EXHIBIT 2

j -f
q

y
q

c
q
f- j

f- j
f- j
y
y
q
q

- qT
y y f
qq

n
q

y
q

- qT

j =1

-j
Gompertz
Model:f-Death
y
20K Mortality
j
- qT
+ q ) n j -f c
y
y
1
(
c
∑
q ( 1y+ fq ) + T ( 1 + q )
(
)
q
Probabilities
for
a
30-Year-Old
j
=1
=( q
n
)

Gompertz Mortality Model: Survival
Probabilities for a 30-Year-Old

20++1|t)
30 40 50 60 70
0
10 20 30 40
H₀0create
+ F₀ 110
=- A₀c
L₀B
q(tq(t
+ 1)
100%
To
L1₀Year
Year
q(t
+= 1|t) a=guaranteed income for consumption of
+
1
r
q(t)
c and a guaranteed bequest of B, the investor
F₀ - L₀B
needsH₀
to+spend
A₀ c on annuities and L₀ B on
75
c =
A₀
permanent
life
insurance.
The
amount
that
can
be
L
q(t)y
q(t + 1)
H₀ =+= 1|t)
∑ these
q(t
= ist limited by the intertemporal
spent
on
50
t = 0 (1 + r)q(t)
budget constraint,
which
is
as
follows:
1+r
(Ht-1 - yt-1 )
HtLife
= Insurance Premiums
25
q(t|t - 1)
$400
A₀c + L₀B
H₀ + F₀L = q(t)y
0
H₀ == ∑
t
t = 0 (1 + r)
0
17 0 34 17 51 34 68 51 85 68
85
1 + r level of financial
300
Years0,into
the Future
Years into the Future
where
wealth
in
year
+
-
(FA
)
AP
AI
FAt =H₀F₀+isF₀the
- L₀B t-1
t-1
t-1
- 1)
c = I q(t|t
Source: Morningstar.
which
takeA₀as
given. Taking B as given, I solve for
F₀ = A₀c + L₀B
200
cH₀as+ follows:

(∑ ( )() () )
∑
( ) ( )
1 + ) ∑ ( ) ((11++ )) + T (1 + )
(40K
D ( y;P (c,y;Tc,) T=) =
P ( y; 30K
c, T ) = ∑ (1P+( y;)Pc,(+y;()c,1(1T++) ) )
c
q

( )

0

f = n- qT
n
y f
- qT
P (( y; c, T ))n = Pc y; c, qn y 1f-+j q
P y; c, T = q -1j + q +- n1 + qy
c
y
y f
j =1 + 1 + q
P =$50K
1+ q
q

)(

()

)

P (y; 10K
c, T ) == Pn y; c, q 1 + q
D
P y; c, q
P ( y; c, T )
0
Death Probability
Death Probability
50 60 70
f- j
f- j
y
y
y f
100%
+q
Pc ( y )1=+ y1q 1 + q
1
c
c
y f
Pq ( y;P (c,y;Tc,) =T ) P =y; qc, qn 1 + nq
y
P
1+ q
y;
c,
q
75
y
Dc ( y ) = y1 1 + q - qf

(

(

50
25
0

(

)

()(

(

)
))( )

4%

f

3

( )
+
1
(1 + )
(1 +( ) )=
( y; c, T )
P
P(y;(Py;c,( c,y;
T )Tc,=) ) P ( y; c, ) (1 + )
P ( y; c, T )
f-yj - j
y
q q

c c
q q

=
Dm

()

n
q

y
q

c
q

-j f
y
q

Pc ( y ) = 1 +1 +
)
D ( y; c,17T 51
0 = qc 17 0 34
34 68
(y; c,theT )Future
= n intoy the Future
Dm into
Years
P y; c,Years
q 1+ q
y
y ) = y1 1 + q - qf
Dc (Morningstar.
Source:
1
y

1
y

c

2

y f
q

n
q

(( ))
( )
( )
P∆ (Py )≈=- D(1∆y+ )
P
y
q

f- j

1
0

51 85

68

85

y f
q

m

P ( y; c, T )
)
FRt = (1 1+ +r)r(FRt-1 + yt-1 + AIt-1 - c - APt-1 - LPt-1100
Dm (y; c, T ) =
Ht = H₀ + F₀ - L₀B(Ht-1 - yt-1 )
y
1
EXHIBIT
3
P
( y ) = y 1Cash
c = q(t|t - 1)
Dc Flow
+ q( y;
-c,qfT )
Cash
Wealth
Wealth
D Flow
0
A₀
DV 01
= P Annuities
P Dand
■ Human CapitalTotal
■ Human
■ Regular
Capital
Account
■and
Regular
■ Annuities
Account
■ Annuities
■ Annuity
Purchases
■ Annuity
Annuity
Purchases
■ Annuity
Income
Wealth
Cash
Flows
in Lifecycle Model
With
Life
Insurance
mIncome
y ■=
10 20 30 40 50 60 70
0
1+ q
(
)
D
y;
c,
T
$50K
$1M
$1M
Year
Annuities
Dm (y; c, T ) = P ( y; c,y T )
+ rLife Insurance Over the Lifecycle
1and
+
)
1
r
+
-
(y;
c,
T
=
D
(FA
)
AP
AI
FA
=
+
1 q
m
t-1
t-1
t-1
- yannuities
t-1 )
ItHtist =not q(t|t
necessary(H
tot-1
buy
and
life
P ( y; c, T )
40K
q(t|t --1)1)
750K
750K
insurance in a single lump sum. The same level of
P ( y; c, T )
P c, T ) =
C∆(y;
30K
≈ - Dm ∆yPD((y;y;c,c,TT))
lifetime consumption and the same level of
P
500K
500K
D
+
1
r
m (y; c, T ) =
+
+
+
-
c
-
-
y
Human
Capital
(1
r)
(FR
)
y
AI
AP
LP
FR
=
guaranteed
bequest
achieved
in
the
lump-sum
t-1(FA t-1+ APt-1 - AI ) t-1
t-1
FAtt =
1+ q
t-1
t-1
t-1
20K
- 1)achieved using a dynamic
solutionq(t|t
can be
D
250K
250KLabor DV 01 = P
= P Dm
approach in which the investor varies the levels of
10K
C∆(Py; c, T ) =1 + qy
Income
≈ - Dm ∆y
0
0
0
annuitization and life insurance over time.
P
) 200 3010 4020 5030 6040 7050 60 70
FRt = (1 + r) (FRt-1 + yt-1 + AIt-1 - c - APt-10 - LPt-1
f - j30
- 2 6040 7050
10
20
30
10
40
20
50
60
70
0
10
0
c
y
f-j
f- j-1
n
To do this, the investor first places initial financial
1+ q
Year
Year
Yearq ∑ j =1 q Year q
Annuity Income
Consumption
The Investor
D
wealth (F₀) in a regular investment account
DV 01 = P P ( y;=c, PT ) Dm
C (y; c, TP )( y;=1c,+Tqy)
that earns an annual rate of return of r. Hence,
P ( y; c, T )
- qT - 2
Life Insurance
financial wealth can consist of two accounts: the
y
Annuity Purchases
Life Insurance
1
+
+Premiums
Company
1
Annuity Account
Regular Account T T q
q
regular account and an annuities account.
+
P ( y; c, T )
I denote the values of these in year t as FRt and
y; c,c, TT)) == P ( y; c, T )
CC ((y;
Source: Morningstar.
FAt respectively.
Life Insurance Premiums
Life Insurance Premiums
P ( y; c, T ) 2
∆P
≈ - Dm ∆y + 12 C ( ∆y ) f - j - 2
$400
$400
cP n
y
f-j
f- j-1
1+ q
q ∑ j =1 q
q
In each year t, the investor does the following as
long as he or she lives:
C ( y; c, TP)( y;
= c, T )
300
300
+
Ft account, are related through the
W
=
H
t
t
and the
regular
annuity income.
- qT - 2
y
1
f- j-2
+
c
y each year t, in
f
-
j
<
1f -+j - 1qthem.
T n T between
1 The investor receives labor income, yt. If t L, this
5 . The investor consumes c.
cash∑flows
q
200
200
1 + In
q
q
q
j =1 q
+
6 If labor income (if any) and annuity income
is y; else it is 0.
addition toP labor
yt, and consumption, c,
( y; c, Tincome,
)
r)c,(W
Wt = (1P +( y;
T t)-1 - ct-1 )cash flows, all
2 If the balance of regular account is less than B, the
(if any) less life insurance premiums
(if
there
are
three
additional
100
100
2
∆ P to insurance.
-( ∆y
qT - 2) are:
any), annuity purchases (if any), and0consumption,0
related
investor buys enough life insurance, in year t,
≈ - D ∆y + 1 CThere
PT T +T +-1qmt 1 + 2qy
∑ (1 r) ct-1 = W0
to fill the gap between B and the balance. This
invested
3010 40the
20 net
5030amount
6040 is
7050
60 in 70
0 is10 200is positive,
+t = 0
P ( y; c, T )
the
regular
investment
account.
If
this
quantity
single-year life insurance payable in year t +Year
1
Annuity
Income
Year
is negative, it is withdrawn from the regular
which costs L1t ( B - FRt ).
TheW
that the investor
has purchased pay
tP = Ht + Ft
2
∆annuities
-1 D ∆y + 1 C ( ∆y )
Wm0
cP0 = ≈
∆-0income.
3 Once the bequest is secured by reaching a balance
account to cover the difference between income
a level
of
I 2denote this AIt.
and spending.
of B, the investor stops buying life insurance
(1 + r) (Wt -1 - ct-1 )
Wt = Purchases
and starts buying annuities, making incremental
Annuity
Human Capital Human Capital
Wt the
= year
Ht + Fint which the investor meets the
E X H I BIT 3 shows how the three parts of total
purchases until retirement.
From
T
-t
4 Once the investor has annuities, he or she receives
wealth, human capital, the annuity account,
bequest
the regular accountLabor
until
∑ (1 +goal
r) cin
Labor
t-1 = W0

(

)

*

*

( )(

)(

)

*

(

)(

)

( )(

)(

)

( ( )) (( ))(

)

(

)(

)

t=0

Wt = (1 + r) (Wt -1 - ct-1 )

Income

global.morningstar.com/Morningstarmagazine
-1
Annuity Income
W
cT0Annuity
= ∆ 0 Income
-t0
∑ (1 + r) ct-1 = W0

Income

49

Consumption
http://global.morningstar.com/Morningstarmagazine
Table of Contents for the Digital Edition of Morningstar Magazine - December 2017/January 2018
