Morningstar - Q1 2021 - 51

EXHIBIT 4

Similar Numbers for the Four AssetAllocation and -Location Methods

Method

Expected
Standard
Return (%) Deviation (%)

Utility
(%)

Sharpe
Ratio

Joint

5.79

10.33

3.80

0.367

Separate

5.78

10.41

3.76

0.363

Duplicate

5.23

9.12

3.68

0.354

Two-Step

5.37

9.73

3.60

0.346

Source: Morningstar.

be solved jointly in an optimization that accounts
for differences in the tax treatments of each
investment. I have shown how do that with an
extension of Harry Markowitz's mean-variance
optimization framework. However, this is
not typically done in practice. In practice, either
each account is handled separately or the
allocations to each account are made after making
the overall investment allocations. In this
issue of Quant U, I have shown that the various
methods can lead to very different asset
allocations and locations across account types.

However, there are only modest differences in the
expected returns and standard deviations
of the allocations, leading to very small differences
in utility. Thus, there is a great deal of latitude
in how to approach the asset-allocation and
asset-location problems. K
Paul D. Kaplan, Ph.D., CFA, is director of research with
Morningstar Canada. He is a member of the editorial board
of Morningstar magazine.
The author thanks Thomas Idzorek, CFA, and David Blanchett,
Ph.D., CFA, CFP, for their helpful edits and comments.

Asset Location With Markowitz 2.0
U ( E,V ) = E -
In previous issues of Quant U, I have discussed
a generalization of Harry Markowitz's meanvariance model of portfolio construction, called
Markowitz 2.0. Here, I briefly review Markowitz 2.0
and how to use it to jointly solve the assetallocation and asset-location problems.
Scenario Approach
Rather than using a set of parameters (such as
expected returns, standard deviations, and
correlations) to describe the joint distribution of
asset-class returns, the model builder in
Markowitz 2.0 creates a list of all possible
scenarios and the return on each asset class under
each scenario. This allows for the possibility
of fat tails. Also, it places no restrictions on the
relationships between asset-class returns, so
nonlinear relationships are possible. The scenarios
can be based on historical data (adjusted
for forward-looking expected returns) or Monte
Carlo simulation based on statistical models
of returns that can include fat tails.
The scenario approach can be extended to include
taxable and tax-advantaged accounts. Let
M	
=	 the number of asset classes
	
=	 the fraction of assets in the taxable
		account
=	 the allocation of asset class i in
xiT	
		 the taxable account

=	 the allocation of asset class i in the
xiQ	
		 tax-advantaged (qualified) account
	
=	 the effective tax rate for asset class i
i
		 (See Quant U in the Q3 2020 issue.)
=	 the return on asset class i in scenario j
RijA	
P
=	 return on the portfolio in scenario j
Rj 	
U ( E,V ) = E - V
2
We have:
M

RjP =

∑ ((1 -
i =1

i

)

) xiT + xiQ RijA

2

∑∑

m

∑

) s.t.
max
, x ' x at
U ( x ' Value
Conditional
Riskx ' = 1, x ≥ 0
Because to investors, risk is a matter of possible
losses, it has been argued that any meaningful risk
= ( 1 -  i ) i
Ai
measure
should take into account only possible
= ( 1 - 
) ai risk measure is called a " downside
losses.
Such
Ai
i
risk measure. " One downside risk measure is value
atmax
risk,Uor (VTaR.+VaR)is
) ' respect
( T A + toB )a
' defined
, ( T A + with
A
B
B
,
A B
s.t. A' = , B' = 1 - , A ≥ 0, B ≥ 0

(

given percentile, which we denote as p. VaR(p) of
M
a Preturn distribution
is the number such that
Rj =
(1 - i ) xiT + xiQ RijA
therei =1
is a p probability
of losing VaR(p)-or more.
In the notation of probability theory:

∑(

)

Prob [R ≤ - VaR(p)] = p
~

where R denotes the single-period return on a
[ R | R ≤ - VaR(p)]
CVaR(p)
portfolio =as-E
a random
variable. The most common
value for p is 5%.
m m
U ( E,V ) = E - m V
VaR is notUvery2useful
xi i , itself because
xi xj i j ijit only tells
the
location ofi =1the lefti =1tailj=1of the distribution.
max
x , x , ... ,Mx
m
It Pdoes not measure
the severity of the losses of
Rj =
(1s.t.
) x T 1,xiQx ≥RijA0
-
i xi i =+This
i measured by
the left-tail
returns.
is
i =1
i =1
conditional value at risk, or CVaR, also called
max U ( x 'shortfall. "
, x ' x ) s.t.
x ≥ 0the average
CVaxR'( p)=is1,minus
" expected
V
a
R
(
p).
In
the
notation
of
returns
less
than
-
Prob [R ≤ - VaR(p)] = p
of probability theory:
= ( 1 -  i ) i
Ai
1

Geometric Expected Return
CVaR(p) = -E [ R | R ≤ - VaR(p)]
In the mean-variance model, reward is expected
return, which is the arithmetic mean of
possible futurem returns,m one
m period out. However,
over the long
be more
U run,xi ani , investor
xi xwould
j i j ij
i =1
i =1mean.
j=1
max
2.0
interested
in
geometric
Markowitz
x , x , ... , x
m
provides the option of using expected geometric
s.t. xi = 1, xi ≥ 0
mean as reward.
i =1
1

V

∑

The allocations are subject to the constraints
M
M
Prob
that ∑[Ri = ≤1 x-iT =VaR(p)]
and =∑ i p= 1 xiQ = 1 -

∑

2

)

2

m

∑( ∑

∑∑

)

CVaR(p)
= ( 1 - =i )-Ei [ R | R ≤ - VaR(p)]
Ai
Markowitz( 2.0 supports
several downside risk
max U T A m+ B )' ,m( TmA + B ) ' ( T A + B )
CV
a
R. A Markowitz 2.0
measures,
including
,
A B
U
xi i ,
xx
=
1 - i ,j iA j≥ ij0,between
'
≥ 0 risk
s.t.
=
,
'
efficient
frontier
shows
the
B i =1 j=1 trade-off
B
A i =1
max
x , x , ... , x
and reward using
whatever measures the
m
0 as CVaR for risk
1, xi ≥such
xi =selects,
user of thes.t.
model
i =1
and geometric
mean for reward. Using the
method
) s.t.first
max
x ' , x ' inx the
x ' equation,
= 1, x ≥it0 can be
U ( shown
L
˜ +
˜R asset-allocation
˜R = F Rto
extended
the Hjoint
and
-
R˜
W
W F problem.
W H W L
asset-location
= ( 1 -  i ) i
Ai

(

1

2

∑

)

∑∑

m

∑

Ai

= ( 1 -  i )

(

i

max U EW ( T A +

) , VW ( T A +
B

)

)
B

51
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