Morningstar - Q4 2020 - 67

u"(1 + E ) =

(1 + E )
m

∑

U

max

m

xi i ,

i =1

x1 , x2 , ... , xm

q

1+ O
O

q

j =1

n

m

∑∑

(1 ) (1 )

∑

xi xj

c
q

P ( y; c, T ) =
i j

ij

∑

because liabilities include housing costs,
˜
there
)' , ( T between
)
max could
+correlation
+ B ) ' (RTL and
+ the
U ( T be
A
B
A
A
B
,
returns
on
real
estate
investment
trusts.
A B
s.t. A' = , B' = 1 - , A ≥ 0, B ≥ 0

(

)

The expected return and standard deviation of
return of net worth can be written as functions of
the asset-allocation vector of financial assets. I
denote these functions EW ( ∙ ) and VW ( ∙ ),
L to
˜ = F R˜ +ThisHallows
R˜ - us
R˜ write the MVO
Rrespectively.
W
W F W H W L
problem for the net worth joint asset-allocation
and -location problem as follows:

(

)

Asset Class
U.S. Large-Cap Stocks

Effective
Tax Rate (%)

( )

( )
( )

)

E X H I B I T 4 shows the efficient frontier in terms
of the expected return and standard deviation of

f- j

( ) (1 + )
c
q

j -f
q

y
q

(

y

+ T 1+ q

Taxable Account

24.50

U.S. Small-Cap Stocks

25.93

Global DM Ex-U.S. Stocks

19.34

Emerging-Markets Stocks

20.50

Global REITs

26.39

U.S. Bonds

40.00

c
q

Global Bonds Ex-U.S.

40.00

Cash

40.00

y
q

return of net worth. It also shows the optimal
portfolio and the indifference curve that is tangent
to the efficient frontier at the optimal portfolio.
shows the optimal portfolio with
the allocations in the two accounts broken out.
As expected, municipal bonds are held in the
taxable account, and taxable bonds are held in
the tax-advantaged account. While non-U.S.
stocks are held, U.S. stocks are not. This could be
due to the U.S.-stock exposure of human
capital. Finally, REITs are held as a counter to
the REIT exposure of liabilities.
E X H I BIT 5

The risk and expected return of a portfolio depend
on the context in which we look at it. In terms
of net worth, the optimal portfolio in EXHIB IT S 4
and 5 has an expected return of 5.82% and
a standard deviation of 8.80%. But if we look at
this portfolio in the context of the financial
assets alone (as is usually done), both its expected
return and standard deviation of return are
much lower, 4.23% and 6.70%, respectively. I have
plotted this point in EXHIB IT 2 , where it is just
below the efficient frontier and has lower risk and
expected return than the optimal portfolio from
a financial assets-only perspective. This shows
how a net worth perspective can lead to a solution

10

) (1 + )

y f
q

f- j

(1 + )
P ( y; c, ) (1 + )
y
q

c
q

y f
q

n
q

-j

(1 + )
P ( y; c, )
y
q

n
q

11.51 D ( y ) =
c
15.65

0%

=

P ( y; c, T )

13.70
Pc ( y ) =

0.00

f- j

(1 + )

15.85
= qc

8.95

n
q

Tax-Advantaged Account

22.79

U.S. Mid-Cap Stocks

Municipal Bonds

(

1
y

(1 + )

1
y

(1 + ) -

y f
q

34.35

y
q

f
q

Dm (y; c, T ) =

P ( y; c, T )
20P ( y; c, T )

Dm (y; c, T ) =

D ( y; c, T )
y
1+ q

30

40

that is very different from looking at financial
∆P
≈ - Dm ∆y
assets in isolation.
P
Conclusion
D
DV 01 =portfolios
P
= individual
P Dm investors
y for
Constructing
1+ q
is complicated by the fact that investors pay taxes
on returns in some accounts and can postpone
or avoid taxes on returns in other accounts. This is
P ( y; c, T )
(y; c, T ) = by the fact that investors have
furtherCcomplicated
P ( y; c, T )
income from sources outside of their portfolios
and that they have unavoidable expenses. In this
issue of Quant U, I have presented some
C ( y; c, T ) =
extensions to the standard mean-variance model
to incorporate
these
complications
f -in
j - 2a single
c
y
f-j
f- j-1
n
1 + q thus
q ∑ j =1 q
integrated
optimizationq framework,
making it applicable
P ( y; c, to
T ) the problems faced by
individual investors. K - qT - 2
y
1
T T+ q 1+ q
Paul D. +
Kaplan, Ph.D., CFA, is director of research with
P ( y; He
c, Tis) a member of the editorial board
Morningstar Canada.
*

( )(

)(

)(

)

(

of Morningstar
∆ P magazine.

)

2

≈ - Dm ∆y + 12 C ( ∆y )
P thanks Thomas
The author
Idzorek for his helpful edits.
Wt = Ht + Ft
Wt = (1 + r) (Wt -1 - ct-1 )
T

-t

∑ (1 + r) ct-1 = W0

t=0

morningstar.com/lp/magazine
-1

- qT

)

P ( y; c, T )

P ( y; c, T ) = P y; c,

≥0

This means that subject to the non-negativity
and budget constraints, the optimizer
finds the allocations to the different asset classes
1- 1
O
in each account type, while
accounting
,O≠1
u(c) = { ln(c), O = 1 c -1
1
for the investor's human
and liabilities,
1-capital
O
that maximize expected utility from a net
1
worth perspective.
c~t+1 - O ~t
1
= Qt+1
ct
1+
To illustrate asset allocation and location in the
net
that an investor
c~t+1 worth framework,
O
~t I-assume
1
= ,000 inO financial
Qt+1
wealth,
$125,000 in
has
ct $125(1
+ )
human capital, and liabilities with a present value
of $150,000. I ~model
- O the return on human
t
Qt+1
~
capital
of
a
portfolio
consisting of 20% U.S. stocks
ROt+1 =
~t 1 - O - 1
and 80% EUt.S. Qbonds.
When coupled with the
t+1
expected returns of these two asset classes, they
1-O
~t discount
form
a weighted
rate that is applied
Et Qt+1
c~t+1
~
+ Rcash
1
to the=future human
capital
flows to arrive
Ot+1
O
ct
+
(1 )
at the $125,000. Similarly, I model the return
on the value of liabilities as a portfolio consisting
of 10% REITs and 90% U.S. bonds and use
the expected returns on these two asset classes
to form a weighted discount rate that is applied
to the future liability cash flows to arrive at
the $150,000.

(

- qT

y
q

Optimal Asset Allocation and Location for Financial Wealth as Part
of Net Worth

Source: Morningstar.

B

∑ j =1

D ( y; c, T ) =

EXHIBIT 5

i =1

However, there could be some equitylike
( x human
) s.t. xthe
max
' , x ' xcapital,
' =investor's
1, x ≥ 0 wages
risks.UFor
could be subject to the risk of economic
downturns that could be reflected in the stock
= ( 1 -  i ) i
Ai
market,
leading to correlation between R˜H
=
(
1
- 
) i on equity asset classes. Similarly,
and
the returns
i
Ai

( )

y
q

n

xi = 1, xi ≥ 0

max U EW ( T A + B ) , VW ( T A + B )
,
A B
s.t. A' = , B' = 1 - , A ≥ 0,

f- j

∑ (1 + ) + (1 + )
j =1

i =1 j=1

m

s.t.

q

67
http://www.morningstar.com/lp/magazine
Morningstar - Q4 2020

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