Morningstar - Q3 2020 - 49

Strategies

Solving the Asset-Location Problem, Part I
The first step is to form capital
market assumptions for both beforeand aftertax returns.

QUAN T U

Paul D. Kaplan

As Benjamin Franklin famously wrote, "In this
world nothing can be said to be certain,
except death and taxes."1 Yet, the standard
asset-allocation paradigm that is so well
established in the investment literature and
applied with the mean-variance optimization
model of Harry Markowitz (1952, 1959) does
not consider taxes.
Mean-variance optimization, or MVO, is usually
applied to a single account or the aggregation
of multiple accounts. However, for many individual
investors, different accounts have different
tax consequences. Therefore, an individual investor
not only faces the problem of asset allocation
but also that of asset location-that is, which
asset class to locate in which account.
This issue of Quant U is the first of a three-part
series on an extension of the MVO model
that simultaneously solves the asset-allocation
and the asset-location problems. I developed
the extension with Thomas Idzorek of Morningstar
Investment Management. In Part I, I discuss
how to form capital market assumptions for
the tax-aware model. In the next issue, I'll
discuss how to use the model to simultaneously
perform asset allocation and location. In
the third installment, I will compare the results

of performing asset allocation and location
simultaneously with the results of performing
them sequentially, as is common practice.
Pretax Reverse Optimization
In the standard MVO model, there are sets of
inputs that I denote as follows:

	
=	
	i
=	
		
	
=	
ij
		
i

the expected returns of asset class i
the standard deviation of returns
of asset class i
the correlation between the returns
of asset classes i and j

In principle, all of these parameters should be
forward-looking. In practice, the standard
deviations and correlations are often estimated
from long-term historical return data under
the assumption that these parameters are
stable over time. However, given how poor past
performance is as a predictor of future
performance, it is not good practice to estimate
expected returns from past returns. We need
an alternative forecasting method.
One method of forming expected returns is reverse
optimization. First proposed by William Sharpe,
reverse optimization takes standard deviations and
correlations as given and assumes that a particular
portfolio or asset allocation is mean-variance
efficient.2 From these assumptions, it infers the set
of expected returns that would in fact make the
portfolio efficient.

I call the asset mix assumed to be efficient the
reference portfolio. Typically, the reference
portfolio is based on the market values of the asset
classes. In reverse optimization, the covariance
matrix is used to calculate the sensitivity of
each asset class to the reference portfolio. The
sensitivity to each asset class to the reference
portfolio is called its beta. It is similar to the
beta in the capital asset pricing model, or CAPM.
If the reference portfolio is the market portfolio,
the betas of reverse optimization are the same
as those of the CAPM.
In addition to the beta of each asset class, reverse
optimization requires an assumption regarding
the expected return of two asset classes or asset
mixes. The two assumed expected returns are
typically for cash and for the reference portfolio
itself. If the reference portfolio is the market
portfolio, the difference between the expected
return on the market portfolio is the market
premium of the CAPM.
EXHIB IT 1 presents an example of reverse
optimization. To create this example, I estimated
a covariance matrix from historical returns on
indexes that represent the 10 asset classes.
EXHIB IT 1 shows the standard deviations on the 10
asset classes that come from the covariance
matrix. It also shows the reference portfolio, betas,
and expected returns for the 10 asset classes.

The reference portfolio is based on the market
values of the asset classes, so I am assuming,
as in the CAPM, that the market portfolio is
an optimal portfolio. Note that while this model
includes municipal bonds and cash, their
allocations in the reference portfolio are both zero.
I did this for municipal bonds because while
munis would not be held in a nontaxable
account, I included them as an asset class in
tax-advantaged accounts. I didn't include
the results for municipal bonds where they are
not relevant.
I included cash with a zero allocation because
while cash is not part of the reference portfolio, it

1 This saying did not originate with Franklin. See https://en.wikipedia.org/wiki/Death_and_taxes_(idiom).
2 Sharpe, William F. 1974. "Imputing Expected Security Returns From Portfolio Composition." The Journal of Financial and Quantitative Analysis, Vol. 9, No 3. (June), PP. 463-472.

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