Morningstar - Q4 2020 - 64

Strategies

U ( E,V ) = u (1 + E ) +

1
u"(1 + E )V
2
O -1

(1 + E ) O , O ≠ 1
-1
u (1 + E ) =
ln(1 + E ) ,
O =1
Using the Levy-Markowitz
utility
function, we can
-1
(1 +the
u"
E ) =MVO problem1 +asO follows:
write
(1 + E ) O

Solving the Asset-Location Problem, Part II
How to expand mean-variance
optimization to build a tax-aware model.

m

max
x , x , ... , x
1

2

m

∑

U

xi i ,

i =1

m

m

∑∑ x x

i j i j

ij

i =1 j=1

m

∑ x = 1,

s.t.

i

xi ≥ 0

i =1

QUANT U

Paul D. Kaplan
In the previous issue of Quant U, I introduced the
problem of simultaneously finding an optimal
asset allocation and the optimal location of asset
classes in taxable and tax-advantaged accounts.
I presented a method for calculating pretax
and aftertax expected returns that implies a set
of effective tax rates, which applies to
both expected returns and standard deviations.
In this issue, I review the basics of mean-variance
optimization, or MVO, and show how to extend it
using effective tax rates to simultaneously solve for
optimal asset allocation and location. I then show
how to further extend the model to incorporate an
investor's assets and liabilities that should
be taken into account when deciding on the asset
allocation and location of the financial assets.
These are the very assets and liabilities I discussed
in the Q2 2020 issue of Quant U ("A Better
Retirement Spending Rule for Everyone"), although
here they apply at all stages of the lifecycle.
The Mean-Variance Criterion
In an earlier Quant U ("Clearing Up the Great
Confusion," October/November 2014), I explained
the logic that Harry Markowitz uses to justify MVO.
Given a utility function in one plus return, u(.), the
expected utility of a random return can be
approximated by the expected return ( E ) and the
variance of return (V ) as follows:

1
U ( E,VU) (=x 'u (,1x+' Ex) )+s.t. xu"' (1=+1,E )Vx ≥ 0
max
2
This can be written more succinctly using matrix
O -1
notation. Let:
(1 + E ) O , O ≠ 1
-
1
u (1=+(E1)- = )
i
Ai
i
ln(1of+ pretax
E ) , expected
O = 1returns
	 =	 the vector
=
(
1
- 
)
i
Ai
i
	 =	 covariance-matrix
of pretax asset class
1
u"(1 + E ) =
1+ O
		 returns. The
ij-element
is
(1 + E ) O
( Tvector
)' allocation
max
+ Bof
, ( T A + Bto) 'the( Tasset
+ B)
	 =	 Uthe
A
A
m
m m
,
A B
		classes
s.t.U A' = xi , i ,B' = 1 -xi x,j iA j≥ ij0, B ≥ 0
	max=	 a vector
i =1 of m ones
i =1 j=1
x , x , ... , x

Because this approximation for expected utility
was first introduced by Haim Levy and
Markowitz, I refer to it as the Levy-Markowitz
utility function.1

(

I assume the constant relative risk aversion, or
1
( E,V,) utility:
CRRA
U
= u (1 + E ) + u"(1 + E )V
2
u (1 + E ) =

(1 + E )
-1
ln(1 + E ) ,

1

2

i =1

-1
u"(1 +UE( )x =' , x ' x ) s.t.1 +x O' = 1, x ≥ 0
max
(1 + E ) O
m

∑∑

xi xj i j ij
= (that
1 - Uu"
) i + Exi) isi , negative
i (1
Ai
Note
so that variance
i =1
i =1 j=1
max
x , x , ... , x
is
"bad."
=
(
1
- 
)
m
i
Ai
i
s.t. xi = 1, xi ≥ 0
i =1 Optimization Without Taxes
Mean-Variance
max U ( T A + B )' , ( T A + B ) ' ( T A + B )
To
describe
MVO
) s.t. x 'taxes,
, U ( x ' , x ' xwithout
≥ 0the following
max
= 1, I xuse
A B
s.t. A' = , B' = 1 - , A ≥ 0, B ≥ 0
notation:
1

2

m

∑

(

m

m

i =1

	

O =1

∑

∑

2

∑

, O ≠1

∑∑

m

1

∑∑

0 written this way:
1, now
xi ≥ be
s.t. xi =can
The MVO problem

m

m

∑

O -1
O

-1
u"(1 + E ) =
1+ O
where is the level
O investor's risk tolerance,
(1 + Eof) the
1 0 and 1.
which
is
usually
between
U ( E,V ) = u (1 +m E ) + m u"m(1 + E )V
2
U
xi i ,
xx
O i- 1j i j ij
The
second derivative
i =1
i =1 the
max
O , Outility
(1of
+j=1
E ) CRRA
≠1
x , x , ... , x
= m -1
(1 + E ) is:
ufunction
s.t. lnx(i 1=+1,E ) ,xi ≥ 0 O = 1

∑

)

)

f

max
U ( x ' , x ' x ) s.t. x ' = 1, x ≥ 0
R˜W = F R˜F + H R˜H - L R˜L
W
W
W

P

= ( 1 -  ) i subject to the non-negativity and
Ai meansi that
This
budget
constraints,
the optimizer finds the
= ( 1 - 
) i
i
Ai
allocations
to
the
different
) that
T A +classes
max U EW ( T A + B ) , VW (asset
B
maximize
expected
utility.
,
A B U (T
max
s.t. A' A =+ ,B )'B' , =( T1 -A +, BA )≥' 0,( T BA ≥+ 0 B )
,
A B
In the last
, A ≥ 0,pretax
' = 1U,-I derived
≥0
s.t. issue
' = of,Quant
B
B
A
expected returns on a set of asset classes using
the reverse optimization technique. In reverse
optimization, we assume that a given portfolio-
the reference portfolio1-that
1 is often a market
c OL-1 frontier
portfolio-is
O H=the1R˜ efficient
u(c)
˜R +on
˜R == F{ ln(c),
-
R˜ , O ≠ 1 and,
W
F
H 1- 1 L
W optimal
W for
WO
therefore,
investors
who have a certain
level of risk tolerance.
To
find
what
level of risk
~ - O1
c
~
tolerance
makes
the
reference
portfolio
optimal,
1
t
t+1
= Qt+1
+
c
1
I first traced
t out the entire efficient frontier
(max
EXHIBUIT E
1W
). (Then,
T A + forB )various
, VW ( T values
+ B )of , I search
A
, efficient
c~At+1
O
~t for- the
the
optimal portfolio.
1 frontier
B
=s.t. ' =O ,Qt+1' = 1 - , ≥ 0, ≥ 0
B
ct the
+A ) of B that resultsA in an optimal
(1value
I take
portfolio closest to the reference portfolio as the
~t - O for which the reference
level
Qt+1
~ of risk tolerance
-turns
1 out to be 22.7%.
ROt+1 = is optimal.
portfolio
O
~t 1 -This
Et Qt+1

(
(

(

)

)

)

	 ==	( 1 - 
thei )pretax
expected return on
Ai
i
i
		
asset
class
i
= ( 1 -  i ) i
Ai
	i =	 the pretax standard deviation
		
+ Hclass
R˜ -i L R˜L
R˜W = FonR˜Fasset
( Tcorrelation
( TWreturns
)
max
+W B )H' , of
' ( T A + asset
+ B )between
Uthe
A
A
B
	ij =	W
,
A B
		classes
s.t. A' =i and
, Bj' = 1 - , A ≥ 0, B ≥ 0
1
xi	 =	 the allocation to asset class i
To illustrate maximizing
expected
utility in the
1
~Ot =11- O c1- O -1 , O ≠ 1
U ( E,V ) = u (1 + E ) + u"(1 + E )V
~
=
{
ln(c),
u(c)
E
Qt+1framework,1 I ~include an
ct+1
2
m	 =	 the number of asset classes
mean-variance
t
1-1 +O R
=
Ot+1
max U EW ( T A + B ) , VW ( T A + B )
ct n = (1qT+ ) O
O -1
(1 + E ) O , O ≠ 1
,
A B
1
-
1
~ -O
u (1 + E ) =
s.t. A' = , B' = 1 - , A ≥ 0, B ≥ 0
~t
qT3, PP.
1 f =c , n-
1 Levy, H., & Markowitz,
H.M.
Expected Utility by
Issue
308-17.
= Qt+1
ln(1 +
E ) ,1979. "Approximating
O =1
F R˜ of+ Mean
H R˜ and- Variance."
L R˜ The American Economic Review, Vol. 69t+1
˜R a =Function
+
c
1
W
n
t
W F W H W L
-n
-j
c
y
y f
-1
+ 1+ q
P= q
1+ q
u"(1 + E ) =
~
1+ O
ct+1
1 j =1 ~t - O
64 Morningstar (Q4
1 +2020
E) O
=
Q
ct
(1 + ) O t+1 n
- qT
f- j
m
m m

(

(

)

)

n

( )

P
n

f
D
P

PP

c

Dq

=P

Pcc

( )
( )

D

( )

=
D

(

∑

( )
c

)

(

) (

y

)

y

q

Pc

D

D

∆
P
D
Morningstar - Q4 2020

Table of Contents for the Digital Edition of Morningstar - Q4 2020
