Morningstar - Q4 2020 - 65

EXHIBIT 1

Risk and Expected Return Without Taxes
Expected Return
12%

INDI

EFFI

CIEN

T

TI
FRON

ITH
ER W

TA X E
OUT

FFE

CE
REN

CUR

VE

9

6

Reference Portfolio

S

3
0

0%
5
Standard Deviation

10

15

20

25

the optimal portfolio. This is where expected utility
is highest among efficient portfolios. Because
I have assumed that the reference portfolio is the
optimal portfolio, the point of tangency is the
point of the reference portfolio.
1
  ( )
U E,V = u (1 + E ) + u"(1 + E )V
2
Mean-Variance Optimization
for the Joint
O - 1 Problem
Asset-Allocation and -Location
(1 + E ) O , O ≠ 1
1
Suppose
has two accounts, one
u (1 + E ) =that the-investor
(1 + E ) ,
OLet
= 1 be the
taxable and onelntax-advantaged.
fraction of assets in the taxable account, so that
-1
) =fraction in the
u"(1 is
+ Ethe
1-
1 + tax-advantaged
O
1 + E ) Obe transferred between
account. Money(cannot
m
m
accounts. Money
in them taxable
account earns
,
x
U
xi xjthe
i i money in
i j tax-advantaged
ij
aftertax returns, and
i =1
i =1 j=1
max
x , x , ... , x earns pretax returns. In the last issue,
account
m
I showed how
tax rate, i ,
≥effective
0
= 1, xan
s.t. toxiderive
i
for asset classi =1i. Effective tax rates allow us
to convert pretax expected returns and standard
max U ( x ' , x ' x ) s.t. x ' = 1, x ≥ 0
deviations into their aftertax counterparts:

∑

1

2

∑∑

m

∑

Source: Morningstar.

EXHIBIT 2

Risk and Expected Return With Taxable and Tax-Advantaged Accounts

Ai

Expected Return
12%

R
RONTIE
IENT F
E F F I C O U T TA X E S
WITH

JOIN

T

IENT
EFFIC

FRONT

9

IER

6

Optimal Joint Portfolio

RVE
INDIFFERENCE CU

Optimal Joint Net Worth Portfolio

0
0%
5
Standard Deviation

10

15

20

25

Source: Morningstar.

(

increase in expected return needed to compensate
for increased risk increases.
We could draw indifference curves that are
above the efficient frontier and that intersect the
efficient frontier. However, there is a unique
indifference curve that is tangent to the efficient
frontier, the point of tangency being that of

)

)' , (tax
max
' (T A + B )
T O+O- 1 B )into
U ( T A +effective
To
incorporate
B
(1 + EA)rates
, O ≠ 1the
, B =
-
1
A
(1 + problem
)
I introduce
uMVO
Es.t.
, A ≥ 0,
' = 1 -notation,
≥0
= , matrix
' using
B )
A
,
1 B
an m× m matrix,ln(T,1 +inEwhich
the iOth =diagonal
is 1- i and all off-diagonal
elements are 0.
-1
) = be the asset-allocation
u"
1+ O
Let(1 +A Eand
vectors of
B
(1 + E ) O
the taxable and tax-advantaged accounts,
m
m m
respectively. The MVO problem is to select A
R˜W = F RU˜F + Hxi R˜Hi , - L Rx˜iL xj i j ij
and
maximize
the expected utility of
i =1 W
j=1
maxBWso as toi =1W
x , x , ... , x
aftertax
returnm over the portfolio as a whole.
Hence, it solves
onlyxi for
≥ 0asset allocation but
= 1,
s.t. xnot
i
also for asseti =1
location. Thus, I refer to it as
the
solution
max
UU( xMVO
' A x+) Bs.t.
) ,and
E'W ,(xTproblem
VxW' (its
T= 1,
+x B≥)0 as the
maxjoint
A
optimal
joint
portfolio.
,
A B
s.t. A' = , B' = 1 - , A ≥ 0, B ≥ 0
=
(
1
-  i ) i the MVO problem in matrix notation
We
Ai can write
as =follows:
( 1 -  )

∑

2

∑∑

m

∑

(

)

i

Ai

indifference curve in E X HI BI T 1 . An indifference
curve shows all combinations of expected
return and standard deviation that result in the
same level of expected returns. Indifference curves
are upward sloping at an increasing rate,
showing that an investor is willing to take on
additional risk to get additional expected
return, but as we increase the level of risk, the

i

= ( 1 -  i ) i
Ai
1
U ( E,V ) = u (1 + E ) + u"(1 + E )V
2

1

3

= ( 1 -  i )

i

(

)

max U ( T A + B )' , ( T A + B ) ' ( T A + B )
1
,
A B
c1- O -1
= s.t.
{ ln(c),
u(c)
'A O= = ,1 B' = 11 - , ,O ≠A ≥1 0, B ≥ 0
1- O

-1
O subject
c~t+1 that
~t to the non-negativity and
This
1 means
= Qt+1
ct
1+
budget
constraints,
the optimizer finds the

allocations to the different asset classes in each
R˜~W = F R˜F + H R˜H -- L R˜L
ct+1 W type
OW expected utility.
~
W maximize
account
1 that
=
Qt
ct
(1 + ) O t+1

( )

~t - O
morningstar.com/lp/magazine

( )

Qt+1
~
=
-1
Rmax
Ot+1 U E ( T~ +1 - O ) , V ( T
+

(

65

)

)
http://www.morningstar.com/lp/magazine
Morningstar - Q4 2020

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