Morningstar Magazine - December 2017/January 2018 - 50

Strategies

25

25

0

0
17

17 51

0 34

34 68

51 85

68

85

34 68

51 85

68

850

20

30

40

50

60

70

Years into the Future
Years into the Future

Year

EXHIBIT 4

EXHIBIT 5

EXHIBIT 6

Components of Total Wealth Over
the Lifecycle

Annuity Cash Flows Over
the Lifecycle

Life Insurance Premiums Over
the Lifecycle

Cash Flow
Cash Flow
■ Annuity
Purchases
■ Annuity Income
■ Annuity Purchases
■ Annuity
Income

Life Insurance Premiums

()
()
q( t ) = exp 1 - exp ( )
p( t ) = q( t - 1) - q ( t )
q( t ) = exp

1 - exp

t
b

b
t
b

p( t ) = q( t - 1) - q ( t )

( )
( )
$1M
exp ( )
exp

a-m
b

b

a-m
b

750K

$1M

$50K
-t

(1 + )

750K

500K
500K
p( t =
) =∑ q( t q(t)
- 1) - q ( t )
A₀
t
t = 0 (1 + r)
∞
q(t)
A₀ = ∞∑
250K
250K
+ r) t
t = 0 (1p(t)
L₀ == ∑∞ q(t) t
0
0
A₀ = t ∞∑
= 0 (1 + r)
t
+ r)50
(1p(t)
= 0 40
20
30 6040 7050 60 70
200L₀ ==30t10
∑
Year t = 0 (1 + r) t
∞
p(t)
Source:
L₀ == Morningstar.
∑ 1 - q(t + 1|t)
L1₀ =t = 0 (1 + r) t
1+r
1 - q(t + 1|t)
L1₀ =
1+r
retirement,
the+investor
1 - q(t
1|t) purchases annuities. I
L1₀ =
q(t + 1)
denote
the
amount
spent purchasing annuities AIt.
+
1
r
q(t
+
1|t)
=
Life Insurance Premiums
q(t)
q(t + 1)
$400
$400
q(t
1|t) = Premiums
Life +Insurance
q(t)
q(t + 1)
Until+ the
goal in the
L investor
q(t
=q(t)y reaches the bequest
300
300
H₀ == 1|t)
∑account,q(t)
t
regular
each
year,
he
or
she
buys
a
t = 0 (1 + r)
L
q(t)y
one-year
policy. As I noted earlier,
H₀ == ∑ life insurance
200
200
t
+
(1
r)
t
=
0
this L1tL( B - FRt ). I denote this LPt.
q(t)y
H₀ +==F₀∑ = A₀c +t L₀B
100
100
t = 0 (1 + r)
With these definitions, the equations for the
H₀ + F₀ = A₀c + L₀B
0
0
evolution+ofF₀the
three parts of total wealth are:
L₀B
10 4020- 50
30 6040 7050 60 70
200c =30H₀
H₀ + F₀ = A₀c
A₀ + L₀B
Year
H₀ + F₀ - L₀B
Human
c = Capital
A₀
H₀ + +F₀ - L₀B
cHt == 1 r
(Ht-1 - yt-1 )
q(t|tA₀- 1)
1+r
(Ht-1 - yt-1 )
Ht =
q(t|t - 1)
+
1
r
1 +Account
r (H - +
The
Ht t =Annuity
t-1t-1 yt-1
FA
(FA
AP) t-1 - AIt-1 )
=
q(t|t
q(t|t--1)1)
1+r
FAt =
(FAt-1 + APt-1 - AIt-1 )
q(t|t - 1)
1+r
+ AP -- cAI-t-1AP
) t-1 - LPt-1 )
FA
+ yt-1
FRtt == (1 + r) (FRt-1(FA
t-1 + AIt-1
t-1
q(t|t - 1)
The Regular
Account
FRt = (1 + r) (FRt-1 + yt-1 + AIt-1 - c - APt-1 - LPt-1 )

40K

-1

-t

30K

(1 + )

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

∞

10

17 51

Years into the Future

Wealth
t Accounta -■
■
Capital
■t Regular
Account
■ Annuities
exp m Annuities
q(Human
) = exp
1 ■- Regular
exp

10

0 34

17

0

0
10

0
0

250K

1

1

0
Year

exp
10

( ) exp ( )
( ) exp ( )
1 0- exp
20 (5030) 6040
20
30(10) 40exp
t
b

a-m
b
a-m
b
a-m
b

20K
10K
0
7050 60

Year

Source: Morningstar.

n = qT

fn$50K
== n-qTqT

$400

n

c qT
fn === n-qT
P 40K
q

-j

-n

+ (1 + ) (1 + )
300
∑
f == n- qT
+ (1 + ) (1 + )
P 30K
∑
200
(=y; c, T ) = ∑ (1 + ) + (1 + )
+
+
+
PP 20K
1
1
∑
( ) ( )
100
P (10K
y; c, T ) = ∑ (1 + ) + (1 + )
0
∑ ( ) (1 + ) + T (1 + ) 0
== ∑
30(1 +40 ) +50(1 + 60) 70
700 PD( y;10c, T ) 20
P ( y; c, T )
c
q
c
q

n
j =1

y
q

-n
y
y f
q
q
- qT
f- j
c
y
y
-n
-j
y
y f
q
q
q
q
q
n
j =1
- qT
f- j
c
y
y
q
q
f-q j
y
j -f c
nn
j =1
- qT
qf- j
q
c j =1 q y
y
q
q
f-q j
y
j -f c
n
j =1
q
q
j =1 q
-j

n

n
j =1

j =1

In other words, I assume that the investor is 30
years old, makes $50,000 a year, has $25,000
in savings, and will retire in 30 years at age 60. I
assume that the capital markets provide a
return of 4% a year. Based on these assumptions,
using the formulas that I gave above, the level of
consumption, c, turns out to be $36,991.

y
q

- qT

- qT

( ) (1 + ) + T (1 + )
D ( y; c, T ) =
∑ ( ) P((1y;+c, T)) + T (1 + )
D ( y; c, T ) =
P ( y; c, T ) = P ( y; c, ) (1P+( y; )c, T )
been
P ( y;reached.
c, T ) = POver
(1 + )annuity income
( y; c,this)period,
rises and
then
levels
off
at
+
1
( )
(its1 +retirement
) level.
P ( y; c, T ) = P (=y; c, ) (1 + )
P (1y;+ c, T )
P ( y; (c,1 of
+ )life
()1 +insurance
)
( shows
) =the pattern
premiums.
rise
as
the
investor
P (1y;+ c,AtT )first, they
y; (c,1 + ) ()1 + )
( 1 +) )=in theP(regular
builds up(wealth
account. Then, they
= P ( y; c, T )
P
y;
c,
drop off
once
the
investor
has
about
( )(1 + half) of what
P ( 1y;+c, ))
he=or she needs to fulfill his or her bequest goal.
P ( 1y;+c, ))
=P ( y ) = (1 + )
Managing Uncertainty
P ( y; c, )
TheP uncertainty
( y ) = (1 +of )the date of death creates risk for
D ( y ) = Fortunately,
(1 + ) - through the tools of
investors.
(
)
=
y
P
(1 ++ insurance,
)-
annuities
investors can manage
D ( y ) =and(1life
)
this risk. Life insurance
the problem
P ( y; c, addresses
T)
)=
D ((y;
y ) =c, Ttoo
D
+P ( )y;-c, T )
with
dying
not having enough
(1 soon-namely,
P ( y; c, T )
∑

Year

Source: Morningstar.

n
j =1

c
q

j -f
q

y
q
f
y
q

n
q

L = 30
B = $500,000
a = 30
F₀ = $25,000
r = 4%
m, b as specified earlier

y f
q

y
q

c
q

y
q
EXHIB
c IT 6
q

y
q

c
q

c
q
c
q

1
y

c
c q

1
y1
y
1
y1

c

c

c

c

m
c

c
q

f- j

n
q

c
q
f- j
-j
y
c
q
q
-j
yn
q q
-j
yn
q q y f
q
n
q y f
qy
q
y f
qy

y
q

- qT

y f
q
f- j
y
y qf
q
f- j y f
n y
q
q q

n
q

f- j

f- j

y
q

n y
q q
n
q

f- j

y f
q
y f
q

f
q

y

q

f
q

1
y

y
q

f
q

E X H I BI T 4 shows the evolution of the three
assets to leave the desired bequest. Annuities
Human Capital Human Capital Dm (y; c, T ) =
P ( y; c,
T)
components of total wealth. Note that human
address the problem
P ((y;y;of
T)) a long life-or not
D
c,c,Tliving
)
c, T = assets yto keep consumption at the
Dm (y;enough
capital follows the same pattern as in the
having
P (1y;+ c,q T )
Labor
Labor
( y; c, T that
) I discuss here
bare-bones model, declining gently
until
Income desired level. TheDmodel
Income
Dm (y; c, T ) =
y
retirement. The regular account grows until it
illustrates how to use
these
tools over the lifecycle.
+
1 q
∆P
D ( y; c, T )
∆y
≈
-
D
m
reaches the bequest value in year 23, seven years
Dm (y; c, T ) =
y
The Investor
Annuity Income Annuity Income
Consumption PConsumption 1 + The
Investor
before retirement. Over the working years
In these
ofqQuant U, I have covered
∆ P two issues
≈ - Dm ∆y
P of the basics
before reaching the bequest balance, the investor
some
D of the lifecycle model.
Life Insurance
Life
DV
= P Premiums
P Insurance
Dm
P01Insurance
∆one
y =
Annuity
Purchases
Life
Annuity
Purchases
Life Insurance
Premiums
The
aspect
that
have not
covered
yet is
saves
money
in
the
regular
account
and
buys
∆y
≈
-
D
m 1 + q I Company
Company
Annuity Account Annuity Account
Regular Account Regular Account P
D
market
uncertainty
and
its
ramifications
temporary life insurance to fill the gaps between
FRt = (1 + r) (FRt-1 + yt-1 + AIt-1 - c - APt-1 - LPt-1 )
DV 01 = P
y = P Dm
1 + q I will address this next.
for asset allocation.
savings and the bequest goal.
D
DV 01 = P
y = P Dm
c, T )
1 +P q( y;
)
C
(y;
c,
T
=
Paul
D.
Kaplan,
Ph.D.,
CFA, is director of research with
A Numerical Example
E X H I BI T 5 shows the pattern of the cash flows
P ( y; c,isTa) member of the editorial board
Morningstar Canada.
P (He
y; c, T )
To illustrate how the lifecycle model with annuities
related to annuities over the lifecycle in
C (y; c, T ) =magazine.
of Morningstar
P ( y; c, T )
my example. Annuity purchases start when the
and life insurance works, I created an example
P ( y; c, T )
)) =
CC ((y;
c,
T
bequest goal has been fulfilled in the regular
using the following value for the parameters:
y; c, T = P ( y; c, T )
account and rise until retirement. Then, they stop
f- j-2
Cc ( y;nc, T f)- j= f - j - 1
because the retirement level of annuities has
y = $50,000
y
1+ q
q ∑ j =1 q
q
f- j-2
Cc ( y;nc, TPf)-( y;
y
j= c, fT- )j - 1
1+ q
q ∑ j =1 q
q
*

*

*

50

Morningstar December/January 2018

c
q

( )( )( )
( )( )( )
∑T (T(+P ( y;) c,
( 1T+) ) (1 + )
n
j =1

f -1j
qq

f - j - 1y
q q

- qT - 2 y f - j - 2
q



Table of Contents for the Digital Edition of Morningstar Magazine - December 2017/January 2018

Contents
Morningstar Magazine - December 2017/January 2018 - Cover1
Morningstar Magazine - December 2017/January 2018 - Cover2
Morningstar Magazine - December 2017/January 2018 - 1
Morningstar Magazine - December 2017/January 2018 - 2
Morningstar Magazine - December 2017/January 2018 - Contents
Morningstar Magazine - December 2017/January 2018 - 4
Morningstar Magazine - December 2017/January 2018 - 5
Morningstar Magazine - December 2017/January 2018 - 6
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Morningstar Magazine - December 2017/January 2018 - Cover3
Morningstar Magazine - December 2017/January 2018 - Cover4
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