Morningstar Magazine - October/November 2017 - 55

EXHIBIT 2

future income. Under our assumptions, the value
of human capital in year 0 is:
L

H0 = y ∑ (1 + r)

The Evolution of Human Capital, Financial Wealth, and Total Wealth
Financial Wealth

Human Capital

Total Wealth

-t

L

-t -
if t ≤ L
t-1 y),as follows:
H0 =
= ycapital
∑(1 +
(1r)+ (H
r)evolves
Human
H
t
t=
0
0,
if t > L

-
+
FH0t = 0 (1 r) (Ht-1 y),
0,

1.5M
USD

n = qT

t=0

if t ≤ L
if t > L

(1 + r) (Ft-1 + y - ct-1 ), t ≤ L +1
F0t = 0 +
(1 r)year
(Ft-1L-+ c1t-1
t ≥ L +there
1
Starting with
, H),t = 0 because
is no
more income forthcoming.
(1 + r) (Ft-1 + y - ct-1 ), t ≤ L +1
Ft =
+ +Ftr) (Ft-1 - ct-1 ),
t ≥ L +1
Wt = Ht(1
Financial LWealth
-t
+
H0 = y ∑ (1 r)
Financial
is what is conventionally thought
t =wealth
0
of
as
assets.
In
this
Wtt = (1
Ht+L+r)Ft(Wt -1- t- model,
ct-1 ) it is just money
+
H
=
y
∑
(1
r)
0
-
+
if thave
≤ L assumed no
(1 r) (HBecause
earning tinterest.
t-1 y), we
Ht = = 0
0,
if thave:
>L
T
financial
assets in year 0, we
-t
+ r) +ct-1 = W-0
∑
(1
) if t ≤ L
W = (1 (1r)+ (W
t -1 -ct-1
t = 0t
r) (H
t-1 y),
H
F0t == 0 0,
if t > L

L

H0 = y ∑ (1 + r)

-t

t=0

Ht =

(1 + r) (Ht-1 - y),
0,

if t ≤ L
if t > L

F0 = 0
25
Age
F =

35
45
55
(1 + r) (Ft-1 + y - ct-1 ), t ≤ L +1
t
(1 + r) (F - ct-1 ),
t ≥ L +1
Source: Morningstar. t-1

f = n- qT
n = qT n
-j
c
y
y
P= q
1+ q + 1+ q
f = n- qT
j =1

∑(

) (

n

c

n -j
y

-n

) (

1.0

)

y f

1+ q

-n

f=
0.5

∑ ( ∑)( ( ) ) ( ( ) )
f-yj

-yqT f

y+
+c - t + y +
P= L
H0 P=( y; yc,q∑T )(1=+1r)q q 1 + 1q q + 1 +1q q
t = 0j =1

j =1
n

f- j - qT

∑ (( ) ) ( ( ) ) (

j -f y
c n
65 P ( y; c,(1
T )+=r)75(H∑ j =1- y),
+q
Ht D=( y; c,LT ) = q -t-1t 1 q
H0 = y ∑0, (1 + r) j =1
t=0

cf- j
q

y
y
y
if851+
t +≤ 1Lq + q + T 1 + q
Pif(ty;>c,LT )
f- j

( )( )
( )( )

(

( ∑)( ( ) ()( ( ) ) )(
( )(
( ∑)( ∑)( ( )( ) ( )(
( )
( )(
( )∑ (( )) ( ( )
( )
( ( ))( ) ( )
( ( ) )( )
∑(( )) ( ) (
( )( ( ) )( ( ) )
∑ ( ( ) ()(
( )
( )
∑
(
)
(
( ) ( ( ) ( ) )( ()
( )
( )∑ ( ) (
( ( ))
( ( ))( ( ) )
( )
( )
( )
( ) ( )(
( )( )

)

c

j -f

y

y

)

( )
(( ) )

(

- qT

)

)
) (

)

(

)

)

))

)
)(

( )
( ) () ) (
( ( ))
( )( )(

)

*

)

)

)

(

)

)
)( )

(

P(

D(

P(

))

*

*

P

0.0

- qT

n
+ T 1+ q
∑
q 1+ q
q
F0 nD=(=y;0c,qTT ) = j =1
(1 + r) (Ht-1 - y),n Pif(ty;y≤c,fLT )
Ht P=( y; c, T ) = P y; c,
Wt = Ht + Ft
1if+t q> L
q
f = n-0,
qT
financial
wealth.
During
From this, it follows that in year 0, the present
+ y - retirement,
+ r) (Ft-1
L +1investor
ct-1 ), t ≤ the
(1
n
=
qT
Ft = down
n financial wealth until it is fully
spends
value of consumption must equal total wealth in
+
+
-
n
-
j
-
c
(1
r)
(F
),
t
≥
t-1
f- j t-1
f- j L y1 f
y
y
y yf
n
F0 PP=(=y;at
01c,qc+death.
q c,+ 1 +
TqT)qy = 1P1+ y;
W
1 +1q +q q 1 + q
spent
year
fc = n-
q
t =0:(1 + r) (Wt -1 - ct-1 )
j =1
= qc
q
n )
y f
n- n
(
P
y;
c,
T
-j
T
T
1≤+ Lq+y1 f
yn + P y; c,y q
WtP =Pattern
Hqtc(1+ +Fof
t
y
-
c
(F
),
-t
-t
f-
j+t-1
f-
j
tr)
=
f-
j
t-1
+
+
+
The
Consumption
1
1
1
Ft P=( y; c,+T )y = c q + y q ++ y + y -qqT
∑ (1 +-1r) ct-1 = W0
∑ (1 + r) ct-1 = W0
1 (1j =1+
c=0 0= ∆ 0 Wwealth
q -t-1
t ≥qL +1 of the
q r) (F
0
j -1ct-1 q), a 1full q1
tFinancial
t = 0 Income
evolves
as
follows:
In
the
Appendix,
statement
c
Consumption
y I present
FF0 == 0 (1 + r) (Ft-1 + y - ct-1 ), t ≤ L +1
=j =1 qc
+
q
1
q
t
y f
)utility
n optimization
intertemporal
(1 + r) (Ft-1 - ct-1 ),
t ≥ L +1
60K
P yy; f-c,j qn problem
1 +f-yj q - qT
==qcP(1( y;+ c,r) T(W
- qT
c - c j -f
)
W
-
c + y+
y
tP ( y; c, T ) = tn-1 n t-1+
-1 +
-1
USD
1 q and
1q q + T The
q∑model
This
is
called
the
intertemporal
budget
constraint.
in
the
barebones
solution.
P
q q its
+
+
y;
c,
1
1
1
r
q
q
j
=1
+
cc0 =
= ∆0 W
∆
c
=
W
F
W
=
H
+
+
t
≤
L
1
y
-
c
(1
),
-
j
0
0
0
L 0+ r) (Ft-1
t
t
t
t-1
y j =1
D ( y; c,gives
T1) += the
t
Ftt = 1 + + c-t-1
It says that the present discounted value, calculated
solution
level of consumption in
T
H0 = y ∑(1 +
(1r) (F
r) - c ),
q
t ≥ L +1
P ( y; c, T f-) j
-t
c qT
- qT
S
A
V
I
N
G
S
n
=
Wt = Ht t=+0 Ft t-1 t-1
+
=
∑ (1(qyear.
r)
=
W
c
yn00,f consumption
y is proportional
j -f c
y
1In
each
year
at
the
market
rate
of
return
of
all
consumption,
t-1
-
-
+ T 1+ q
q 1+ q
t = 0Pc y ) P= y 1 +n∑q j =1 q
y;
c,
2
+
+
1 r
q year 0:
1 rthe initial value of total wealth,
totDtotal
equal
( (1c,wealth
-
+T r)) =(Win
W
ct-1
ct =
c
cmust
t =
f ==y;n-
qT t -1 ct-1 ) P ( y; c, T )
T
max=Wealth
+ financet-1and human capital. The investor
y), if t ≤-1L Budget
(1+++FTand
r) (Hthe
1
1
t-1 --Intertemporal
30
Total
Constraint
the
sum
of
t
-
t
W
H
y f
t
t
y
n
cH
(1t -1+ -)ct-1 ) if tc>t L s.t. ∑ (1 + r) ct
Pc( y;
( y-1)c,=T )1ny1 =1PW+ Iyqy;
(1,ct+
W1t ,c
D
2=, ...
- qqf A W1A+L qS- n
T Hfc,
0,T r)t∑=(W
T =
-Dj R
- 1hand in hand
(
)
f
=
y
P
∆
c
W
+
0
t = 0 with the
Thet lifecycle
approach
goes
can
schedule
consumption
over
time,
subject
1
y
y
0 c
0 c- t 0
qy
L
∑ P(1=+ r)q cyt-1 1=+ W
1+ q
q 0 + 1+ q
+ r) - t
to
this
constraint.
To
increase
future
consumption,
total
wealth
approach.
Human
capital
and
H
=
y
∑
(1
t=0
-1
-1
T0 W
=
f
j
=1
0
T
max t -= t0 T
T
T
max
y
n
f- j
nP (=y; c,qTT )y =f-Pj y;
t )
-t
-t
-t
-0)c-t-1
=,+...0(1
W
r),c+wealth,
cr)t-1(W=t -1W
- Py ( y;
f T )r,1 +, q , yand T as presented
t (1
where
function
consumption
financial
into
cF∑0 ,c
cthe
, ... ,cT ∑can
ct total
s.t.wealth,
∑ (1 + r)is the
ct
(1 +forgo
) some current
ct s.t.
∑ (1 + r) ct
y )11=0+c,+isTqy1ra) =
D
T ∑ (1 +combined
+ qn c,- c,qqof
1 ,c2investor
1+ q
1
c ((y;
t =1 0 2
D
-
-
c
c
m
1
1
t=0
t=0
t=0
t=0
- qT
c=t-1
cint the
= Appendix.
y
( y;q c,+year
)y f-0j, consumption
evolves0
and, in doing
so, earn additional market
basis for consumption, savings, and investment
-1
fP=q (=y;
n-
) =T ) qc PAfter
T+qT
T
1PT qy; c,85+n 1 1+f-+jq y f
+ r) (Ht-1 - y), if t ≤ L
(c,1y;W
P
c,
(1t 1-
∆
c
0
0
0
25
35
45
55
65
75
-
f-
j
=
W
=
W
+ r)(1=(Fwealth
q
q y
H =0 (1
t ≤ sum
L +1 of human
+t-1
return0 on savings. Conversely, the investor can
as follows: ny
decisions.
the
r)cTotal
)0 y -iscsimply
t-1 ),
W+
j =1
t-1
F∑t =(1=+(1r)+ 0,
if t > L
1 c+ q
Age
- j c, T ) 1 +- nq
Py ( y;
tc=t 0= ∆ -1 W
ct-1 ),
(10+financial
r) (Ft-1 -wealth:
t ≥ L +1
y
y f
c
c
increase current consumption by forgoing future
capital
0
0 and
-
j
)
(y;
c,
T
=
D
- qT
Pq m= q
1y-+D=q(n( y;q +c,j -fT1))+c q y 1f-+j y q f
y
-T
)
(y;
c,
T
=
D
n
+
-1
P
y;
c,
T
1
(
)
-1-
maxm Pc y;
consumption
and some market return.
1 +jc,=1r TT q ∑ j1=1+- t qPyq y;qc, 1q+ q 1 + +q TT 1 + q - t
1-
∆
=
cct1 ,c=D2(,y;
F00= (1-1
0+ r) (1 + )
q...c,
,cTT ) ∑= (1 +ct-1)
c
s.t. ∑ (1 + r) ct
= (1 + r) (1 + )
-Pf-1(j y;t c, T ) t-=qT0
c0 ==∆ 0H1W+-0rF1 -
P1 +y;t =c,0 y qn - jn
W
t
t
t
c D ( y; c, T y)
y
ct-1
ct =
E X H I BIT 2 plots
the evolution of human capital,
1y+ q + 1 + q
∆m( Py;
(y;c,c,T 1)T+=) =q q
1+
= =PDW
0c ≈ - D ∆y
-+ r)-T(Ft-1 + y - ct-1 ), t ≤ L +1
- -T
(1
m
+
q
j
=1
1
There
points -1
about this equation:
financial
wealth, and total wealth over the
∆0 =
P are several notable
F∆t 0 =
-
y f q
-1
Pc ( y ) P= y;y1 Tc,1 +qn q
T
max
t ≥ L +1
+-+r1r) (Ft-1 - ct-1 ),
1(1
y f f- j
-t n
- -t qT
Because
we
have
assumed
that
financial
wealth
lifecycle.
At
the
beginning,
total
wealth
is
made
up
c
cW
=
(
)
-
+
c
y s.t. ∑ (1 + r)
j
-f
y
+
=
P
y;
c,
T
P
t t = (1 r) (Wt -1 t-1 ct-1 )
+
c
,c
,
...
,c
(1
c
ct
∑
n )c, q
1
y;
1
2
T
q
t
+
+
+
T
+
1
-1
∑
1
1
q
-
q
q
q
1
j
=1
t
=
0
t
=
0
∆ P + no1-reference
T
max
D+ )f to retirement. By saving
is
0 in year 0,T total wealth
in year 0 is just
1 It makes
entirely of human capital. During the working
-t
-t
∆y
≈
-
D
=
(1
r)
(1
(
)
D
y;
c,
T
=
y
y
m
DV01
=
P
=
P
D
1
m
c1T ,c2 , ... ,cT ∑ (1 + )
ct s.t. ∑ (1 + r) ct
) == y1working
PDW
- qf the
1+++ qqy years,
c ((yy)the
cP
y 1
P ( y;investor
c, Tf-)j is
=
human
in year 0. Total
evolves
during
years, the investor saves to accumulate
- 1 wealth
0
-t
t=0
f-1j q
W
Hcapital
t +ctF=t 0 = W
y
y
∑ t (1=+ r)
t-1
0
-T
+
+
-1
1
1
as
follows:
financial
wealth.
Over
this
period,
human
capital
able
to
accumulate
enough
financial
wealth to
q
q
t=
= 0W
-
T
T
max
c
c
0
-t
-t
D
∆
=
=
y
1
f
0q ( )
q
c1 ,c2 , ... ,cT ∑ (1 + )
ct s.t. ∑ (1 + r) ct
declines
as
financial
wealth
increases.
At
create
a
smooth
pattern
of
consumption
y ( y;==-c,yP1T 1) +Pqy ( y;-=c,qPT ) Dm n
D
y f that
DV01
c P
-1
Pn y; c, qy f 1 + q
1-
t=0
t=0
) =(1
+
(y;
c,
T
D
1
m
q
+
+
=
(1
r)
)
(
)
-
+
= PP y;((years
P y;the
c, Tworking
retirement, total wealth is entirely made up of
spans
retirement years.
(1 r) (Wt -1 ct-1 )
Wt = -1
) 1 + the
y;c, c,q T)and
q
C (y; c, T ) = P- j y; c, T
=
c0 =W0∆ 0 W01-
y P ( y; c, T )
+
+
= (1 r) (1 )
(
)
T
-1 +-Tq P y; c, T
c
-t
f- j
∆0=D=m q(y; c, Ty ) f-=j PD((y;y; c,c,TT))
y
∑ (1 + r) ct-1 =- W0
1P+c,-y;Tq1)c,= nP ( y;c c,y T ) 1 + q
-T
(y;
D
t=0
m
+
c
1-
-
1 r
q
c, T )U,=I will
=1introduce
1 The
uncertainty about
the
ct-1) down to 0 at death is the assumption that the investor knows T, the number of years until death. In the next issueCq of(y;Quant
c∆t 0=reason
= (1 + r)that it(1is+drawn
q+ q
f
1 +- 1
C ( y;P (c,y;Tc,) T=) P ( y; c,PT )y; c, qn 1 + qy
time to death.
Dy ( y;f c, T )
-1 because we did not assume any financial wealth in year 0, this is entirely human capital in year 0.
2 Recall that
D
P∆cm(Py(y;) =c, Ty1 ) 1=+ -qj y
c0 = ∆ 0 W-0 -T
f- j-2
y f -1
∆0 =
c
y
-m
j ∆y
j -+
1 q
n≈ - fD
+
-
1
1
-1
qP∑ j =1 1q+ q
q
q
c
T
T
max
c, T ) = y
=C ( y;
-t
-t
q
c1 ,c2 , ... ,cT ∑ (1- + )
ct s.t. ∑ (1 + r) ct
D∆c P( y )P=Py;y1( y;c,1c,+n Tq) - qf
global.morningstar.com/Morningstarmagazine
55
-
+
1
1 tr= 0
t=0
≈ - Dm ∆yqD
f- j-2
ct-1
ct =
cP n = f -Pj
f - j - 1 = P Dy
DV01
m
+
y
1
- qT
1 +- 2 q
q ∑ j =1 q
= W0
1 1+ q y

)

n

)

c
q

=

Pc (
Dc

Dm

Dm
∆
P
D

C(

C(
c
q


http://global.morningstar.com/Morningstarmagazine

Table of Contents for the Digital Edition of Morningstar Magazine - October/November 2017

Contents
Morningstar Magazine - October/November 2017 - Cover1
Morningstar Magazine - October/November 2017 - Cover2
Morningstar Magazine - October/November 2017 - 1
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Morningstar Magazine - October/November 2017 - Contents
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