Morningstar - Q1 2021 - 49

EXHIBIT 1

4 Two-Step As the name indicates, this method

consists of two steps. In the first step, I maximize
the utility of the overall portfolio without
imposing taxes. In the second step, I perform a
second optimization in which I maximize the
expected return of the overall portfolio (applying
the effective tax rates to the taxable account).
For each asset class, I also constrain the allocation
in the taxable account plus the allocation in
the qualified to equal the optimal allocation from
the first step. At the same time, I constrain
the allocation to each account to equal to its initial
allocation. Note that although in the Separate
method, two optimizations are done in parallel,
each on one account, in the Two-Step method, two
optimizations are performed sequentially, both
for the overall portfolio. Furthermore, in the
second optimization, the objective is to maximize
expected return rather than utility.
To illustrate these four methods, I use the pretax
expected returns that I obtained using reverse
optimization. For example, on U.S. large-cap stocks,
the pretax expected return is 7.2%; on U.S.
bonds, it is 2.3%; and on cash, it is 2%. I also use
the effective tax rates that I obtained from
the model of aftertax returns. For example, on U.S.
large-cap stocks, the effective tax rate is
22.79%; on all taxable fixed income, it is 40%.
The Results
E X H I B I T 1 shows asset allocation and location
for the 10 asset classes that I worked with in the
two previous issues of Quant U. It shows that
there can be substantial differences between the
methods in both asset allocation and location.
For example, the Joint method allocates 14.3% of
the overall portfolio to U.S. bonds, placing the
entire allocation in the tax-advantaged account. In
contrast, the Two-Step method allocates 21.8%
of the overall allocation of the portfolio to U.S.
bonds but places all of it in the taxable account.
Also, only the Separate method makes any
allocation to municipal bonds: 11% of the overall
portfolio of the taxable account.

Asset Allocation and Locations Using Four Methods
U.S. Large-Cap Stocks

Method
Joint

3.84

shows how much difference the
choice of asset allocation and location has on the
portfolio's expected return, standard deviation,
and utility. In this exhibit, I show a section
of the efficient frontier of the Joint approach,

Method
Joint

8.49

Separate

8.44

Duplicate

7.62

7.62

2-Step

7.62

4.39
9

Duplicate

1.08
0%
Allocation

Duplicate

2.17
2

2-Step
4

Joint
Separate
2-Step
0%
Allocation

14.12
5.97

Separate

5.97

Duplicate

11.93

2-Step

8

16

10.90

Separate

10.90

10.90

Duplicate

21.80

2-Step

2-Step
15

30

Global Bonds ex-U.S.
Taxable Account
Tax-Advantaged Account
Method

Joint
Duplicate

11.86
11.86

2-Step
0%
Allocation

20

40

Taxable Account
Tax-Advantaged Account
2.17
2.37

0.71

0.71 0.71
1.41
0%
Allocation

2

4

6

10.96

0%
Allocation

6

12

Cash

21.29
1.06

9.64
19.27

Joint

14.31

0%
Allocation

9.64

9.64

Municipal Bonds
Taxable Account
Tax-Advantaged Account
Method

Taxable Account
Tax-Advantaged Account

Duplicate

5.0

32.04
19.13

0%
Allocation

Method
Joint

U.S. Bonds

Separate

4.37
2.5

Global REITs

8.04
5.97

2.18

0%
Allocation

1.08

Emerging-Markets Stocks
Taxable Account
Tax-Advantaged Account
Method

Method
Joint

2.18

Separate

2-Step

Duplicate

Duplicate

Joint

3.68
1.08

2.18

Global Developed-Market ex-U.S. Stocks
Taxable Account
Tax-Advantaged Account
Method

Taxable Account
Tax-Advantaged Account

Separate

0.05

Separate

18

U.S. Small-Cap Stocks

Method
Joint

Taxable Account
Tax-Advantaged Account

2-Step

10.85

0%
Allocation

Separate
EXHIBIT 2

U.S. Mid-Cap Stocks

Taxable Account
Tax-Advantaged Account

Method
Joint

Taxable Account
Tax-Advantaged Account

Separate

0.05

11.86

Duplicate

0.05

23.71
12

2-Step

7.62

0.05
0.09

0%
Allocation

0.05

0.10

Source: Morningstar.

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