Morningstar - Q3 2020 - 52

Strategies

EXHIBIT 4

Postliquidation Capital Market Assumptions
The aftertax expected returns in E X H I BI T 2 are
preliquidation returns with unrealized capital gains T
that=are
taxed. However, it is likely that ∆ =
IRi not
+ CG
Bi
t
i
at some point, the assets will be liquidated and v = t
T
the=capital
gains will be realized and taxed. ∆ =
IRi + CG
Bi
t
i
Let=T qbe the+number
years from now until the v = t
(1 - qi) of
i LTCG
Ii
OI
assets are liquidated. Let:
T
+ CG
∆t =
+ (1
== qIR
i
i - qi) OI
i LTCG
IiBi
(1 - value
LTi ) OIof assets in asset class i v =T t
	 = LT
=	i LTCG
the +initial
VCGi
0i
value
of
assets in asset class i ∆t =
VTiBi	 = IR=	i + the
CGi
v= t
+
(1
-
q
)
=
q
		
at time
+ (1T-i just
LT )before liquidation
LTCG
IiCGi = i LT
) IRi + CGOIi -i TOOIi (1 + CGi - COSTi ) CGI
= (1 i- LTCG
Ii
Ai

∑
∑
∑
∑
T

∑

We
+ (1
- qi) OI
=have:
q i + CG
∆ =
IiBi = IR
i
- Ii ) IR
- TO (1 + CGi - COSTi ) CGIt v = t
= i(1LTCG
+i +
(1 CG
- LT
i i ) OIi
Ai = LTi LTCG
CGi
VTi = V0i (1 - Ai ) T
) OI
+ (1+ -(1q-i) LT
==q LTi LTCG
IiCGi = i(1LTCG
OI-i TO
- Ii ) IRi +T CG
(1 + CGi - COSTi ) CGI
i
i
Ai = CG
(1
-
)
VThe
V
fraction
of
assets
that
are
taxed at liquidation is:
Ti
= 0i i (1 - AiTOi )
i
Bi
= (1 - Ii ) IRi + CGi - TOi (1 + CGi - COSTi ) CGI
Ai = CG
LTi LTCG + (1 - LTi ) OI
CGi
) Ti )
VTii == V0iBi (1(1-- AiTO
TLi = ( VTi - V0i ) i LTCG
IRi )+T CGi - TOi (1 + CGi - COSTi ) CGI
== CG
VThe
V(10i -i(1onIi-) the
Ai realized capital gains at
TiAi taxes
(1
-
=
TLi i = ( BiVTi - V0i TO
) ii ) LTCG
liquidation
come
to:
L
VTi = VCG
- TLi
Ti
i
(1 - TO )
=
VTii = V0iBi (1 -1 Ai ) Ti
VTLTiLLi == V(TiVVTiTi-L -TLTVi 0i ) i LTCG
=
-1
Ai
V0i 1
CG
i
TLi i == ( VVTiL -(1TV-0i TO
) i ) LTCG
BiTi at liquidation, the value of assets is:
LL =
-1
=
TL
VTherefore,
V
-
Ai
Ti
TiV
i
0i L
Ai
=
1
-
1
Ei
BiT
VTLTiLLi == V(TiVTi-L -TL
V ) i LTCG
L i 0i
=
-1
Ai
Ai
V
=
1
-
0i 1
Ei
V L BiT
L =postliquidation
(1 -Ti Ei ) -Bi1 aftertax expected return is:
The
=
AiAi
VTiL = VTiV0i- TL
L i
= 1 - Ai
Ei
= (1 - EiBiH)1 Bi- L
Ai
L Tt
t
L
xtL = 1V+
= Ti Ai -F1
Ai = 1 -
t
V0i
Ei
BiH - L
t
xAit = (11-+ ~Ei ) Bi t
)F ~
1 u'(ct+1
L t = Qt
E X H I B I T 3 shows
thet+1calculation of postliquidation
Ai
1 += 1 - u'(ct)
Ei = (1 - H
) L
~
Ai
EiBi t Bi-assumptions
capital
market
on the 10 asset
+ t+1 ) t ~t
xt 1= 1u'(c
1 2 , assuming a $1,000
classes
in
E X HFIB=ITQ
S t+1
1 and
1-
1+
) t
c O -1 , O ≠ 1
{ u'(c
ln(c),
u(c)
Ht t -O L=tin1 each asset
initial=investment
class and a
1- O1
+~)
xt =
= (11-holding
1
20
Ai -year u'(c
Bi period.
Ei )F
~t 1- O
1
t+1 t
ln(c), O1 ==Q1t+1c -1
,O≠1
u(c)
1 + = { u'(c
)
~ t-O
1- O1
c
~
~
1
t
t+1
-
E X H I B I Tu'(c
4 demonstrates
the logic of the calculation
H) L
t
x1t +1= 1 +ct t+1t == tQQ~t+1
t+1 1- 1 returns. It shows the
of
aftertax
F
1
1 +postliquidation
u'(c
~ t)- OOt = 1 c O -1 , O ≠ 1
u(c)
~t the preliquidation
1 = { cln(c),
t+1 between
1
relationship
aftertax
1-
= Qt+1
~
-O O
c1t+1
+
c
~
1 paid at liquidation, and
t taxes
~
t1
expected
return,
the
1-
= u'(ct+1 ) O Q~t+1
t c O -1
c1t = {(1
+ )O1 ==Q1t+1
ln(c),
,O≠1
u(c)
the
aftertax
1 expected return
~
-
1 + postliquidation
~
)
u'(c
ct+11t O ~t 1-
-O O
c
1
t
ont+1U.=
S. large-cap
A $1,000 investment in
= stocks.
Q- t+1
Ot+1
ct+ ~
1c+t
Ot
(1
)
1
1-
~
~ -QO1t+1
O
1 ,O≠1
ROt+1
1 =={ cln(c),
Ot== Q1~1-t Oc - -1
u(c)
t+1 ~
1
~
t+1 1-
-
E
c
O
Q
-
O
O
+
c
~
~
1t+1 Morningstar
tt 1
t
t Q3 2020
t+1
52
Q Qt+1
~ =
= (1 + )~Ot+1 1 - O - 1
RcOt+1
t
~
~ -~1t 1 - O

( )

( )
( )
(( ()) )

Logic of Calculation of Postliquidation Aftertax Expected Return: U.S.
Large-Cap Stocks
1
(1+ r )v- t
1
(1+ r )v- t
1
(1+ r )v- t
= IRi + CGi
Bi1
-t
( 1Bi+ =
+ CGi
r ) vIR
i
+ (1 - qi) OI
=q
1 i LTCG
( 1Ii +=r )qvi- tLTCG + (1 - qi) OI
0
5
= LTi LTCG + (1 - LTi )
CGi
Years
Source:
LTi LTCG + (1 - LTi )
= Morningstar.
CGi
Ai

2,952

n

∑
∑

∑
∑

10
OI
OI
CGI

= (1 - Ii ) IRi + CGi - TOi (1 + CGi - COSTi ) CGI
large-cap stocks, growing at its preliquidation
(1 - Ai ) T return of 6.12%, increases
= V0i expected
Vaftertax
Ti
) T as the top curve shows. At
= ,279
VtoTi $3
V0i (1in -20Aiyears,
CGi $326.83 is paid in taxes, leaving
liquidation,
(1 - TOi )
=
ai postliquidation
value of $2,952.17. The rate of
Bi
CG
i
)
(1
-
TO
=
return
for which $1
i ,000 grows to $2,952.17 is
i
Bi
5.56%, as the bottom curve shows.
TLi = ( VTi - V0i ) i LTCG
Toi calculate
standard deviation of
= ( VTi - the
V0i ) aftertax
TL
i LTCG
return on each asset class i, we first need
VTiL = VTi - TLi
to calculate the asset class' effective tax rate. The
L
is the single tax rate that,
= VTi -tax
TL1 rate
Veffective
Ti
i
VTiL toT the asset class' pretax return, yields
if Lapplied
=
-
1
Ai
VV0iL T1return. It is given by:
the aftertax
L
= Ti - 1
Ai
V0i
L
Ai

BiL
Ai

=1-
Ei
The aftertaxBi standard deviation is:
= (1 - Ei ) Bi
Ai
= (1 - Ei ) Bi
Ai
H -L
xt = 1 + t t
H F-t L
In
xt addition
= 1 + tot the tcalculation of postliquidation
aftertax return,
~ Ft EXHIB IT 3 shows the
~t
1 u'(ct+1 )
Qt+1
postliquidation =effective
tax rate and standard
1 + u'(c
~ t) )
u'(c
1
deviation
for
asset
class.
t
t+1 each ~
= Qt+1
1+
u'(ct)
1- 1
O
= { ln(c), O = 1 c -1
,O≠1
u(c)
Conclusion
1-1 O1
1-
O -1
An
individual
investor
not
only
c
, Oneeds
≠ 1 to decide
u(c) = { ln(c), O = 1
1
1-
on asset allocation
for
the
overall
portfolio
O
1
c~t+1 - O ~t
1
= Qt+1
c 1
1+
c~t+1t - O ~t
1
= Qt+1
ct
1+
c~t+1
~t - O
1
=

RN

-j

(

-j

(

y

+ 1+ q

n

∑(

3,000

-n

y f
q

-n

y f

) (1 + ) 2,500

) ( )
) ( )

+ 1 +f-qj
1 + -qqT 2,000
y
y
1+ q + 1+ q

j =1
n -j

y

-yqT f 1,500
f-yj - n
+ -f1qy +c q + 1y +1f-qy+j q
y
n 1 j+
∑j =1j =1 q q 1 + q + T 1,000
1+ q
j =1
n =
(
)
D
y;
c,
T
n = 15qT
-j
20y - n
c
y f
f- +
j
n +j -f1 +cPq( y; c,y T )1
P= q
y
n
-qqT
f- j
c∑
y
y
+
+
) = q j =1 1 +q q q 1+ 1q + q T 1 + q
j =1
c, TqT
fPD(=(y;
n-
y; c, T ) =
j =1
P ( y; c, T )
n
n
- qT
- j n y f- j y- n f
c
f
y
c
1c +qy +q+ 1y +1f-q+j+ qyT + y
q+
PP (=y; c,q T ) = Pq∑y;n c,1+j+
q-f1
1
1 q
q
q
j =1 q
j
=1
)=
D ( y; c, Tj =1
n
P ( y;y c,fyTf-)f-j j and
but alsoP (how
=f-Pj y;n c,between
y; c, Tto)y allocate
jq-f 1c + taxable
y
q y
- qT
∑
qf-1j1++ q
q
+
c
y+ T 1+ q
y
j
=1
1
q accounts.
q assettax-advantaged
This
is
the
)
=
PDc ((y;
c,
T
+
+
+
1 q
y; c, T ) = q = qc 1 q
( y;n c, Tf-)jtoy f
locationq problem.
practice
P ( y; c,y Tf-)Itj isj =1common
Pn y; Pc,
1+
y
+qqy qf f- j q
+T )q = P y; c, q and
y; 1c,asset-allocation
1 +1asset-location
addressPc (the
y
y
n c j -f c
+y Tf 1 + q
-=j
q
q 1+ q
y ∑ j =1q qwhich
n
problems
sequentially-in
two
(
)
P
y;
c,
T
+
P
+
1
(
)
D y; c, T =q
y; c, qy f 1 q
)y =f-performed:
Pj y; c, qn one
Tf-)jon aftertax
optimizations
based
=P ( y;qc c, Tare
1P+( y;
q c,
y
n
+q
+
1
1
q c,y - j
P
y;
capital market
c
+ q q = qc leading to taxable
1assumptions,
q
y f
n
Tf-) j
f- jon
asset-allocation
and
one
= qcP ( y; c,y models,
P y;
c, based
q yf 1 + q
n y f
+
+
y
1
1
n
q
q
1
)y =c,1market
P+q qy; cc,assumptions,
before-tax
leading
)c,P=T y;
1+ q
Pcc ((y;ycapital
q
-=j q
q
y f
n
P (asset-allocation
y; c,1 +T )qy
to qualified
models.
P y; c, q 1 + q
c
f
=P (qy ) = 11 1f-+j yy
f- j
y
Dcc ( y )P=+ yyy 1 +n -qqj - qf
+
1
1
y;
c,
yq
q
q
In the next
issue
U, I will present a new
c
1 +ofq Quant
= qc
y f
=q qcPwhich
n
(
)
y extension
approach,
an
y;
c,
1 Tis
P
1+ q
Dc ( y )P= 1y 1 +n yq ( f - qf y;) c,ofq Markowitz's
P
y;
c,
T
y;
c,
(
)
q
PDc y(y;=c, Tyoptimization
mean-variance
model and requires a
) 1=+ q
m
-j
( y; c, T )
y Passumptions
set of capital market
for both before1+ q y f
11
) I presented
P
y;
c,
T
y ( this
f issue,
)
=D
=
Pcc ((qcy(y;
+
and aftertax
returns.
In
1
y ) =c, Tyy ) =
1 + qq - q
m
P y; c,toqnPD
((y;
) these two sets
y; c,c, TT)at
some approaches
arriving
Dm (y; c, T ) =
y
1 I will yapply
f
+
of returns.
Next,
the
two sets of capital
Dc ( y ) = y 1 +Pq1( y;
- qq
y f c, T )
1
(
)
(
)
c, T
marketPDcassumptions
to y;
demonstrate
how the
y(y;=c, T ) 1=+ D
Dmm (y; c, Ty ) = Pq( y; c,y T )
asset-allocation
and asset-location
problems can
+q
∆P
1
≈ - Dm ∆y P ( y; c, T )
y
f
be solved
simultaneously.
DPcm ((y;
y ) =c, Ty1 ) =
D
1 + q( - K
PD (y;
y; c,c,qTT))
P
∆
=
Dm (y;≈ c,- TDm) ∆y
y
D 1 is+ director
Paul D. Kaplan,
CFA,
of research with
P 01 =Ph.D.,
q
DV
P
y = P Dm
Morningstar Canada.1He
Pisq((y;ay;member
c,TT)) of the editorial board
+
D
c,
(y; c,
c,
=
Dmm (y;
TT )) =
D
of Morningstar
magazine.
y
∆ P01 = P ∆yDP (1y;+=c,q TP) D
DV
≈ - Dm
m
y
P thanks Thomas
The author
1 + q Idzorek and David Blanchett for
their helpful edits and discussions.
PD ( y; c, T )
∆P
Cm(y;
(y;≈c,c,-TTD) m)=∆y
=D
D
y
P
c,q TP) Dm
DV 01 = P P (y1y;+=
1 +P q( y; c, T )
C (y; c, T ) = D
∆ P01 = P ∆yP (yy; =
c, TP) Dm
DV
≈ - Dm
CP( y; c, T ) =1 + q
P ( y; c, T )
C (y; c, T ) =
f- j-2
DPf -(jy;
c( n
y
f)- j
- 1 c, T )
CDV
y;
c,
T
=
1 +Dqm
q ∑01
qy = P
j =1 = qP
1 +P q( y; c, T )
C (y; c, TP )( y;= c, T )
f- j-2
c
y
f-j
- 1 c, T )
n
Pf -(jy;
∑
1+
n

c

∑ ∑ ((
∑
∑ ((

Pf (==y;n-
c,q TqT) =

= (1 - Ii ) IRi + CGi - TOi (1 + CGi - COSTi )

=1-

fn == n-qTqT
c

Ai

Ei

3,279

ETU
f == n-q qT
URN P
T
NR
RET 1
O
I
N
T
∆ItD=AT I O
ntU=I D A qTj =1n
vI-Q
L I Q U v =T t ( 1P+O1Sr T)L P = c
E
R
q
TP
∆t =W T H AT
TH A
) = qc
j =1
c,qTTqT
G R O v = t ( 1 + r ) v -fnPt (==y;n-
GROW

Ii

$3,500

Taxes Paid $327
n = qT

(
(
(
(
(

c
q

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

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

(

)

*

*

*

n

fn
- qT

)
)
)

) (
)
(
)

)

)
)

P

P

) )(

)

f
P

- qT

- qT

- qT

P

D

D
- qT

)

P

P

c
q

c
q

=

=

Pc

Pc
D

D

D

D

D

D

∆
P
∆
P
D

*

( )(

q

j =1

( )(
q

)(

q

)(

)

*

q

)

D

C

C
Morningstar - Q3 2020

Table of Contents for the Digital Edition of Morningstar - Q3 2020
