Morningstar - Q1 2020 - 52

t

t+1

=

t

t+1

t+1

Strategies

n = qT

~t
1
=
Et Qt+1
~t ~
~
+
1
Xt+1
Pt Xt+1 = Et RQftt+1
~
Xt+1
~
EXHIBIT
2
Rt+1~=
~1 - 1
t
Et Qt+1 P=t Xt+1Portfolio
Simulated
1 + Rft
Risk Tolerance
~
~t
~
X50%
+
1
Rt+1ft =-1175%
Qt+1
ER~t 25%
=
t+1
~
Pt Xt+1
Rft
~t ~
Rft =
Et Qt+1
+
~t
~ 1 Rft
1 + Rft = 1
Et Qt+1
~~
~~
E X~Y ~= Cov RX,Y + E
t
ft
Et Qt+1 Rft =
1 + Rft
~
~
Et Rt+1 - Rft = - 1 + Rft
~~
~~
E X Y = Cov X,Y + E

f = n- qT
n

c

Portfolio Return

100%

80%

40

~
~
X E Y
~t ~
Covt Qt+1
, Rt+1
~
~
X E Y

~t ~
EXHIBIT
~ 3Covt Qt+1 , Rt+1 ~
Et =Rt+1 - Rft = t Et RBt+1- Rft
~t Portfolio
~
Simulated
Returns
Covt Qt+1
, RBt+1

0
-40
20

~
~t ~t+1
~
Qvt = Qt+1
Qt+2 ... u(c
Qvv~-1 )
t+1
+
Ut = u(ct ) + Et
1+
~t ~
Wt = ct + Et Qt+1 ct+1 + Et
~
~t
1 u'(ct+1 )
~
=Q
t+1 )
u(c
1+
t+1
)
u'(c
+
Ut = u(ct ) +t Et
1+

u'(ct)
c~ - O

Source: Morningstar.1

1
1+

t+1

ct

$50,000

∑(

y f
q

) ( )
) ( )
( )( ) (
∑( ) ( )

∑

(

25,000

0
10

f- j
To illustrate how differences
can y
y
j -fin crisk tolerance
n
+ T 1+ q
∑returns
q 1y+ fq
j =1 nq and
lead to PDdifferences
in
spending,
((y;
)
y; c,
c, TT ) == P y; c, q 1 + q
P ( y;
T ) , each
I generated 30 random values for
thec,SDF
corresponding to af- year,
and
calculated
j
f- the
j
y
y
+
+
1
1
25%, 50%,
portfolioc return qfor risk-tolerance
levels
q
c
= q n
y f
=T ) P IT
Pq ( y;
75%, and
100%
these
+ returns.
y f
y; 2c, plots
P (c,y;T.c,)EXHIB
Pq y;1c, qnq 1 + q
Because the returns on all of the portfolios
-j
are calculated from
f-yj the values of the SDF
f- j , they are
y+
y
1
q
+ q calculated
+
1then
1
csync with
q
in perfect
each
other.
I
c
c
= q
= q
q
n
y f
n
Py; c,y;Tof
the optimalP (level
c,) spending
q
P y;(consumption)
c, q 1 + q
each year given these returns. EXHIBIT 3 shows
-j f
y
the results.
+1q + qy levels move up
= 1y1spending
Pc ( y )The
= qcin accordance with the portfolio returns,
and down
n
P y;changes
c, q y corresponding to the
with the largest
Dc ( y ) = y1 1 + q - qf
riskiest portfolio ( = 100% ).
y f
Pc ( y ) = y1 1 + q
P ( y; c, T )
Powerful
(y; c, T ) =
Dm Tool
y
P
1
fT )
The concept
of
the
pricing
especially in
Dc ( y ) = y 1 + q( y;-c,kernel,
q
conjunction with expected utility theory, is
a powerful theoreticalD tool
( y; c,inT )understanding how
Dm (y; c, T ) = P ( y; c,y T )
assets are
priced
c, T ) and
= how
Dm (y;
1 + qinvestors should
)
P ( y;and
c, Tspend
manage their portfolios
over time. Key
themes∆that
emerge
from
the
models
that
P
≈ - Dm ∆y D ( y; c, T )
I have outlined
here
are:
P
Dm (y; c, T ) =
y
1+ q
1 The market value of D
on the
DV 01 = P any asset
= P depends
Dm
1 +the
∆ P between
covariance
q future value of the asset
≈ - Dm ∆y
and the PSDF.
2 Returns on optimal portfolios are linked to the
)
SDF through
the levelDP of( y;risk
c, Ttolerance.
CDV(y;01c,=T )P=1 + y = P Dm
q(
3 Changes in optimal spending
P y; c, T ) go hand in hand
with portfolio returns.

( ( ))( ( ) )

(

)

20

()(

(

30

~t
= Qt+1

(

)
))( )

(

)
)( )

(

(

(( ))
( )
( )
( )
(

u(c~t+2)
+...
(1 + ) 2

)

*

*

( )

( )
( () )

~tt 1--OO
~
Et change
QQ~t+1
Hence,
in consumption
is directly
~
~ct+1 the
t+1
-1 +1 ROt+1
RcOt+1 ==
O -
~return
related
to the
+
(1
t ) 1 Oon the portfolio:
t
Et Qt+1

( )
(
( )

~t 1 - O
Et Qt+1
c~t+1
~
=
1 + ROt+1
ct
+ )O O = 1
(1ln(c),

( )

52

Spending

u(c~t+2)
+...
(1 + ) 2
~t ~
Qt+2 ct+2 ...

~
O
~t -the
1 condition,
Toct+1meet
investor needs to
1
= this
c~t+1+ -)OO Q~t+1
c
1
t
(1
t
construct
a portfolio
= Qt+1with return:
ct
1+
~t - O
Qt+1
~
~ -O
1
=
RcOt+1
t+1
~t Q1 -t O - 1
=
ct
(1Et+ Q) Ot+1 t+1

u(c) =

-n

) (1 + )

assumptions,
calculated the expected return
f = n- IqT
n
- qT
f- j
c
y
and standard
deviation
P ( y; c, T )n = q of-1j + qy for+values
+q
- n1
c
y
y f
of between
EXHIB
standard
+ IT1 +1 plots
P = q 0 and 1.j =1
1+ q
q
deviation versus
j =1expected return for these
f- j
y
j -f c
y
n
∑ nj =1curve
1 + q to+aT 1 + q
q similar
q is
portfolios. The resulting
- qT
f- j
D ( y; c, T ) = c
y
y
Markowitz
P ( y;efficient
c, T ) = frontier.
1 + q P (+y; c,1 T+) q
q

( () )
( )

30

for Different Levels of Risk Tolerance

Risk Tolerance
~ ~~
~ 25%~t ~50%
100%
t+1
, RQt+1v -175%
Qt ...
tQ t+1
Qvt = QCov
t+1
t+2
v
=
t
~t ~
Covt Qt+1
Bt+1
~,t R~
~t ~
Wt = ct + Et Qt+1
ct+2 ...
ct+1 + Et Qt+2

~t
= Qt+1

y

(

1

c1- O -1 , O ≠ 1
1- O1

Morningstar Q1 2020

ln(c), O = 1

- qT

)

j =1

Source: Morningstar.

u'(c~t+1 )

(

+ 1+ q

j =1

Returns for Different Levels of Risk Tolerance

~
0E R~
- Rft = t Et RBt+1
~ - Rft ~10t ~
t ~t+1
-
+
Qt+1 , Rt+1
Et Rt+1 Rft = - 1 Rft Cov
Year
t

0 1
1+
Year

-j

∑

Pn = qT
q

)

)

Different Degrees of Risk Tolerance
If there are investors with different degrees of
risk tolerance, each one will manage his or her
own portfolio such that the returns on the
portfolio track the SDF according to the equation
. Because there are no inefficiencies, there
for
should be a positive relationship between
risk and expected return across the portfolios. To
follows
demonstrate this, I assume that
a lognormal distribution. I assume that the riskfree rate is 2% so that the expected value of
the SDF is 1/1.02. I also assume that the standard
deviation of ln( ) is 0.2. Based on these

( y; c, T ) construct, not
Although
thec,c,SDF
y;
TT)) ==is aP theoretical
CC ((y;
)
P ( y;the
c, Tinsights
visible in the real world,
it provides
f- j-2
c in
y
f-j
f -In
j - 1summary,
n practice.
are useful
the
theory
1+ q
q ∑ j =1 q
q
of the pricing kernel contains a kernel of truth. K
C ( y; c, TP)( y;
= c, T )

( )(

)(

)

Paul D. Kaplan, Ph.D., CFA, is director
- qT - 2 of research with
y
1
j-2
+j -a1q membery off -the
c T Canada.
f -is
Morningstar
editorial board
n T +f -qj He1
1+ q
q
q
+q ∑ j =1magazine.
of Morningstar

( ( )) (( ))(
P ( y; c, T )
P ( y; c, T )

2
∆P
≈ - D ∆y + 1 C-( ∆y
qT - 2)
P T T + 1qm 1 + 2qy
+
P ( y; c, T )
+ Ft
W
2
∆ tP = Ht ∆y
≈ - Dm + 12 C ( ∆y )
P

(

)(

)

)

- qT

)



Morningstar - Q1 2020

Table of Contents for the Digital Edition of Morningstar - Q1 2020

Contents
Morningstar - Q1 2020 - Cover1
Morningstar - Q1 2020 - Cover2
Morningstar - Q1 2020 - 1
Morningstar - Q1 2020 - 2
Morningstar - Q1 2020 - Contents
Morningstar - Q1 2020 - 4
Morningstar - Q1 2020 - 5
Morningstar - Q1 2020 - 6
Morningstar - Q1 2020 - 7
Morningstar - Q1 2020 - 8
Morningstar - Q1 2020 - 9
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Morningstar - Q1 2020 - 11
Morningstar - Q1 2020 - 12
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Morningstar - Q1 2020 - 79
Morningstar - Q1 2020 - 80
Morningstar - Q1 2020 - Cover3
Morningstar - Q1 2020 - Cover4
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2024q1
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2023q4
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2023q3
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2023q2
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2023q1
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2022q4
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2022q3
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2022q2
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2022q1
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2021q4
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2021q3
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2021q2
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2021q1
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2020q4
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2020q3
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2020q2
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2020q1
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2019winter
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2019fall
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2019summer
https://www.nxtbook.com/nxtbooks/morningstar/magazine_2019spring
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20191201
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20181011
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20180809
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20180607
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20180405
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20180203
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20181201
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20171011
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20170809
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20170607
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20170405
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20170203
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20171201
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20161011
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20160809
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20160607
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20160405
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20160203
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20161201
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20151011
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20150809
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20150607
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20150405
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20150203
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20151201
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20141011
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20140809
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20140607
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20140405
https://www.nxtbook.com/nxtbooks/morningstar/magazine_20140203
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20141201
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20131011
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20130809
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20130607
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20130405
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20130203
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20131201
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20121011
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20120809
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20120607
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20120405
https://www.nxtbook.com/nxtbooks/morningstar/investorconference2012
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20120203
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20121201
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20111011
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20110809
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20110607
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20110405
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20110203
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20111201
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20101011
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20100809_lincoln
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20100809
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20100607_lincoln
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20100607
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20100405_lincoln
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20100405
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20100203
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20101201
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20091011
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20090809
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20090607
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20090405
https://www.nxtbook.com/nxtbooks/morningstar/advisor_20090203
https://www.nxtbook.com/nxtbooks/morningstar/advisor_2008fall
https://www.nxtbook.com/nxtbooks/morningstar/advisor_2008summer
https://www.nxtbook.com/nxtbooks/morningstar/advisor_2007spring
https://www.nxtbook.com/nxtbooks/morningstar/advisor_2007fall
https://www.nxtbook.com/nxtbooks/morningstar/advisor_2007summer
https://www.nxtbook.com/nxtbooks/morningstar/advisor_2008spring
https://www.nxtbook.com/nxtbooks/morningstar/advisor_2008catalog
https://www.nxtbook.com/nxtbooks/morningstar/advisor_2008winter
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