Morningstar - Q1 2021 - 48

Strategies

Solving the Asset-Location Problem, Part III
Practitioners can take multiple approaches
to finding the right spot.

QUANT U

Paul D. Kaplan
Individual investors typically have at least two types
of accounts: taxable and tax-advantaged.1
This leads to complexities not faced by tax-exempt
investors, such as foundations and endowments.
Individual investors with different tax-account
types not only have to decide on an overall asset
allocation but also how to invest each account.
This second decision is commonly known as the
asset-location problem.
To solve the asset-location problem, we start
by estimating the effective tax rate for each asset
class. In the first installment of this series
on asset location (which appeared in the Q3 2020
issue), I presented a model of aftertax returns
that takes into account the various components
of total return and how each account is taxed.
These components include qualified income,
nonqualified income, and long-term and short-term
capital gains. The model also considered the
impact of turnover and of liquidation on taxes. For
each asset class, I derived an effective tax
rate from the ratio of the aftertax expected return
to pretax expected return.
In the second installment (in the Q4 2020 issue),
I showed how to use these effective tax rates
to extend mean-variance optimization, or MVO, to
jointly solve the asset-allocation and asset-location
problems. In the extended version of MVO,
there were two accounts, the taxable account and
the qualified account. There was an asset mix

for each account. For the taxable account, the
effective tax rates affected the expected
returns and the standard deviations. The expected
return and standard deviation of an overall
allocation that encompassed both accounts were
calculated by the effective tax rates to the
taxable account but not to the qualified account.
The optimizer selected the asset mix of both
accounts simultaneously.
In this installment of the series, I compare the
results from the extended MVO model to the
results from three other methods for asset allocation and location. I then compare the measure
of success (the utility) of each of the three
approaches to that of the extended MVO model.
Risk Aversion and the Utility Function
In this issue of Quant U, I use a fairly standard
form of the utility function ( U ) in the expected
return of the portfolio in question ( E ) and in
the variance (standard deviation squared) of the
return of the portfolio in question ( V ):

U ( E,V ) = E -

2

where is the coefficient
of risk aversion, which
RjP =
(1 - i ) xiT + xiQ RijA
reflects
the investor's
attitude toward risk. That
i =1
is, the utility of the portfolio is equal to expected
return minus the investor's coefficient of risk
times[Rthe
Investors should seek
Prob
≤ -variance
VaR(p)]of=return.
p
a portfolio that maximizes this function.

∑(

)

CVaR(p)
-E [ R | R ≤ -ofVaR(p)]
In the first= installment
this series, I derived
pretax expected returns on the asset classes
m

m

Methods for Asset Allocation and Location
The four methods of asset allocation and location
that I examine are:
1 Joint The joint method solves for the ideal asset

allocation for each account type using a
single optimization. This is the most sophisticated
approach. It is a version of the extended MVO
method I used in the second installment of
this series. It uses the utility function I present
here with the value of consistent with the
reference portfolio being optimal.

i

optimizations using the same value of .
I maximize the utility function using the effective
tax rates on the taxable account and separately
maximize the utility function on the
tax-advantaged account without tax rates.

∑ x = 1,
i

f
P

3 Duplicate In this method, I maximize the utility

function on the tax-advantaged account in
isolation and then simply duplicate the resulting
asset mix for the taxable account. This is the
most basic approach.

i j i j

P

D

ij

=1
i =1
max
1 Tax-advantaged accounts can be further classified as tax-deferred and
tax-exempt iaccounts,
soj=1
there can be up to three different types of accounts. For simplicity, in this series of Quant U, I
x1 , x2 , ... , xm
m
treat all tax-advantaged accounts as being tax-exempt, so that there
only two account
types.

s.t.

n

m

∑ x ,∑ ∑ x x
i

In this installment, I continue to assume the
reference portfolio is on the efficient frontier. I also
calculate the value of (that is, the risk-aversion
coefficient) that makes the reference portfolio
the utility maximizing portfolio in the absence of
taxes. One might think of as identifying the
rate at which an investor is willing to take on more
risk for more expected return. I use this value
of to illustrate all four methods of asset
allocation and location, which I describe below.

2 Separate In this method, I perform two separate

V

M

U

using a method called reverse optimization.
In reverse optimization, we assume that for a
given asset mix, the reference portfolio is
on the tax-free mean-variance efficient frontier.
From the reference portfolio, the covariance
matrix, the expected return on the reference
portfolio, and the expected return on cash,
reverse optimization derives the set of expected
returns that places the reference portfolio
on the efficient frontier.

xi ≥ 0

P

i =1

48

Morningstar Q1 2021

max U ( x ' , x ' x ) s.t. x ' = 1, x ≥ 0

c
q
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