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Financial Planning Research Journal
A
Keywords:
ABSTRACT
79
10.2478/fprj-2022-0004
Financial
Planning
Research Journal
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Introduction
Research
has
shown
that investment
success
is
largely driven
by
asset
allocation
(Brinson
et.
al
(1991
)), Although many
investors
choose exchange traded funds
(ETFs)
and mutual funds
over
individual
stocks,
most must
make
asset
allocations because they hold multiple
positions
in
their
portfolios.
One
rule
of thumb that
is
commonly
used
is
to allocate
60
percent of the investable
asset
in
stocks
and
40
percent
in
bonds (Chaves et
al.
(2011),
Brinson
et
al.
(1991);
and Ziemba
(2013)),
As
ETFs
have expanded beyond
simply
indexing equities
or
bonds to include indices based
on currencies and commodities,
it
has
become
simple
and affordable
for
investors
to employ non-
traditional
asset
classes
to gain further diversification.
In
the aftermath of the
2008
global financial
crisis,
investors
are
looking to non-traditional investment alternatives to further
diversify
their
portfolios.
Simple
allocation
strategies
such "60/40'
are
no longer valid to the extent they overlook the potential
for
further diversification.
With
a
universe
of
some
2,792
ETFs
available
in
2021
(Bloomberg
LP,
2021),
investors
have awide variety of to choose
from
in
gaining representation
for
many
asset
classes
within
their
portfolios.
The
willingness
of
investors
to
use
ETFs
is
reflected
in
the fact that
ETF
assets
grew
from
less
than one hundred
billion
in
2000
to
seven
trillion
dollars
in
2021
(Board
of Governors of the
Federal
Reserve
System
US).
While
many
studies
have
shown
that
asset
allocation
is
of more strategic importance than active
security selection,
few
has
examined efficient
asset
allocation
within
aportfolio made up
exclusively
of
ETFs,
This
paper
aims
to create efficient portfolios
using
only
ETFs
and to compare the performance
of these portfolios
with
naive allocation
schemes.
Producing efficient portfolios
from
EFTs
alone
is
feasible given the acceptance of
ETFs
by the
investor
community and the proliferation of
ETFs
that
make
it
possible
to
find
an
ETF
for
virtually any objective,
focus,
or
asset
class.
The
motivation
for
this
research
study
is
to make
useful
and
simple
investment recommendations
for
individual
investors,
Although
ETFs
have received considerable attention among
researchers
and practitioners, there
is
little
literature
addressing
asset
allocation
using
ETFs.
Only a
few
studies
have
used
ETFs
exclusively
to provide allocation recommendations
for
individual
investors.
Our
study attempts to
fill
this
void
in
the
literature
by
producing
asset
allocations among
34
ETFs
using
Markowitz's
mean-variance optimization technique along
with
Ledoit
and
Wolfs
(2004)
modification
for
structured and unstructured covariance matrices,
This
study
is
ameaningful contribution to
avery
few
research
that
seeks
optimal
asset
allocations
using
ETFs
exclusively.
Our objective
is
to provide
useful
information
for
individual
investors,
particularly
in
the
form
of
simple
allocation
recommendations, Individual
investors
often make arbitrary allocations among
asset
classes
that
result
in
poor diversification and higher management
costs,
At
the same
time,
wealth management
companies charge substantial
fees
for
recommending efficient allocations,
Efficient
portfolios made
up
entirely
of
ETFs
can provide
exposure
to broad indices
for
equities,
bonds,
as
well
as
to non-
traditional
asset
classes
at a
relatively
low
cost.
Our expectation
is
portfolios that
use
an optimized
allocation among
several
ETFs
will
provide individual
investors
with
more
effective diversification and
potentially higher
returns.
Based
on
ex-post
analysis
from
2012
to
2017,
the top performing optimized portfolio produced an
average annualized
return
comparable to that of the top performing
ETF.
Similarly,
another optimized
portfolio produced ahigher Sharpe
ratio
than
85%
of the
ETFs
in
our
study.
Nevertheless,
when the
same optimization methods
is
implemented
using
ex-ante
returns,
the optimized portfolios do not
provide superior future performance
relative
to the
ex-post
analysis
and
also
underperforms the best
80
Financial
Planning
Research Journal
performing
ETFs
during
the
same time period. Further, these
results
are sensitive to
the
length
of
the
historical time period used to determine
the
optimized allocation weights or
the
frequency of portfolio
rebalancing.
Using
the
modified shrinkage variance-covariance matrix
to
solve
the
issue
of
assigning
excessive weight to certain
ETFs
(a
general occurrence with the Markowitz optimization
approach)
improves the portfolio performance,
Section 2
of
this
paper
provides
an
overview of
the
literature
on
ETFs
and
portfolio optimization,
In
section
3,
the
ETFs,
the
data
sources.
and
the
fund
selection criteria are described with summary
statistics.
In
section
4,
the empirical model used
to
determine the optimal weights for
the
portfolio
based
on
Markowitz Optimization
is
described.
In
section
5,
the empirical
results
of
optimized portfolio
performance
in
comparison with
the
individual
ETFs
are presented.
In
addition,
the
performance of
the
optimized portfolio using the modified variance covariance matrix with shrinkage
technique
and
average optimal weight are also evaluated, Concluding remarks are provided
in
section
6,
Literature Review
Asset allocation has often
been
cited
as
amore important factor
than
active security selection
in
contributing
to
investment success, Brinson et. al
(1991)
found
that
91
%
of
the
variance
in
pension
fund returns
is
explained
by
asset allocation.
This
result
is
also supported
by
Ibbotson
and
Kaplan
(2000),
who
report
that
asset allocation
alone
explains
87.6%
of
mutual fund returns
and
90.7%
of
pension fund returns,
The
literature addressing efficient allocation
began
with the seminal
paper
of
Markowitz
(1952),
who
demonstrated
that
allocating assets
among
indices provides superior return
ex
post, Over
the
years,
many
studies applied these principles
to
find
an
ideal portfolio allocation
but
with mixed
results.
An issue with
the
Markowitz model stems from
its
use of ahistorical variance
covariance matrix
to
determine
the
efficient allocation.
As
a
result,
the
Markowitz process often
leads to
so
called 'error maximization" which
is
reftected
in
the
assignment
of
unusually high or low
allocation weights driven by outliers
in
the
data,
Consequently,
the
weights
produced
are not stable
and
are very sensitive to
the
data,
Optimization process does
not
perform well
out
of sample, Therefore, alternatives
and
solutions are
used
in
the
literature to overcome these
issues.
One
common
alternative
is
to
use no'lve or simplistic
approaches as
an
alternative
to
portfolio optimization. Some
of
these
approaches
are 60/40 equity/
bond
portfolio mentioned earlier
and
equal
weighting.
For
example, Jacobs, et.al
(2014)
shows
that
simple heuristic allocations offer substantial benefits
and
often
produce
better
results
than
classical
Markowitz's optimization. There are
many
more studies show
that
the no'ive portfolio allocation
techniques outperform
the
mean-variance optimization
(See:
Chaves
et
al.
(2011)
and
DeMiguel
et
ai,
(2009)).
The
second branch
of
approaches
attempt
to
fix
issues
with relying
on
historical variance-covariance
matrix
that
leads to unstable estimates. Bayesian methods
and
factor models are used
to
overcome
these problems. Recently, robust optimization
technique
is
also used to solve
the
same problems.
See
Kolm
et
al.
(2014)
for amore thorough review
of
the
literature. Bayesian methods allows aresearcher
or investor
to
input his/her view into parameter estimation, With this input,
the
estimation becomes
more robust.
Many
different versions of Bayesian models are used
in
the
literature,
All
these models
essentially
combine
astructured covariance matrix with asample covariance matrix. Putting all
the
weight
in
the sample covariance matrix would lead
to
an
unbiased estimation with potentially
large estimation
error,
On
the
other hand, putting
all
the
weight
in
astructured matrix would
81
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Planning
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minimize
the
estimation error
but
increase
the
estimation bias potentially
due
to
misspeci"fication
of
the
structured matrix, Single index models, for example, capital asset pricing model (CAPM)
is
an
example of amodel with ahighly structured covoriance,
The
other extreme would
be
regulor
Markowitz optimization with
an
unstructured covariance,
Also,
sample covariance model
could
be
considered amulti-factor model considering
each
asset
in
the sample as a
unique
factor, A
signi"ficant
part
of
the
literature searches for the sweet spot between these two extremes (Marakbi
2016), Examples
of
such models are Treynor
and
Black (1973)
and
Black
and
Litterman (1992)
models,
All
Bayesian or shrinkage models differ
because
of
the
different weights chosen between
the
structured
and
unstructured covariance matrixes (also called shrinkage constant)
and
the
composition
of
the
structured covariance matrix,
For
example,
Frost
and
Savarino (1986)
obtained
a
better
out
of sample performance using Bayesian
approach
to
optimization, Ledoit
and
Wolf (2003
and
2004)
offers
awidely
accepted
shrinkage methodology, Our
paper
also
utilizes
this approach,
The
weight
put
into structured matrix
is
called
the
shrinkage constant,
This
approach
helps
to
overcome
the
error maximization
and
the
invertibility
of
variance covariance matrix,
In
other words,
the
number
of assets
could
be
more
than
the
number
of
periods,
Out
of
the
sample performance
with
the
shrinkage
approach
is
better
than
the
performance
obtained
from using only sample
covariance
in
optimization process,
Minimum-Variance Portfolio (MVP) strategies are studied
by
many
papers,
It
may
not
be
desirable
by
many
investors
to
have a
low-risk
portfolio (since usually
the
return
will
be
lower also) initially, However,
estimation problems
we
mentioned
above
is
a
lesser
problem with aglobal minimum portfolio,
In
other words, theoretically, minimum variance portfolio
may
have alow return expectation,
but
in
practice
its
return
could
be
better
than
even a
tangency
portfolio, (Clarke
et
ai, (2011), Frahm
(201
0),
and
Richardson (1989)),
There are some papers which study
the
portfolio allocation using exclusively mutual funds
like
our
approach
in
this
paper
limiting
the
allocation
to
ETFs,
Given the longer history of mutual funds, these
results
would provide a
good
perspective
of
the
potential
of
portfolios constructed exclusively with
ETFs,
Pastor
and
Stambaugh (2002) used equity mutual funds to create better portfolios,
The
study
"finds
that
investing
in
active mutual funds
along
with passive indexes
could
increase
the
Sharpe ratio
of
the
portfolio, Louton
and
Saraoglu (2008) also used exclusively mutual funds to
"find
out
how
many
mutual funds
is
necessary for a
well-diversi"fied
fund, Their
"findings
suggest
that
10
to
12
mutual
funds are required
to
get
reduction of most of
the
risk
and
drawn
down
in
the
portfolio, They did not
offer
an
allocation strategy, Moreno
and
Rodriguez (2013)
"find
that
most of
the
actively
managed
mutual funds are
not
well
diversi"fied,
The study used several optimization techniques to minimize
the
idiosyncratic
risk,
The
optimal portfolio
that
minimizes the idiosyncratic
risk
had
a
good
out
of
sample performance
and
this allocation provided
the
best
alpha
for
the
overall portfolio, Saraoglu
and
Detzler (2002) recommends using
the
analytic hierarchy process
(AHP)
to
make asset allocation
decisions using mutual funds,
The
study does
not
offer
an
out
of
sample performance
result,
Using
36
Swiss
ETFs,
Milonas
and
Rompotis (2006)
"find
that
ETFs
underperform their underlying indices
in
terms
of
both
risk
and
return, Miffre (2007) compares
the
performance of country-speci"fic
ETFs
with
that
of
open
or closed-end country funds
and
"finds
that
ETFs
are superior
due
to
lower costs, lower
tracking
error,
and
being more tax efficient,
82
Financial
Planning
Research Journal
More
closely
related to our paper,
Ma,
Maclean,
Xu,
and Zhao
(2011)
employ aregime-switching
risk
factor
in
determining that sector
ETFs
allocations perform better than
no'lve
allocation
strategies.
Furthermore,
DiLellioa
and
Stanley
(2011)
state a
similar
conclusion after comparing
several
ETF
strategies
with
the Standard and
Poor's
500
index
as
well
as
other benchmarks and finding that the
ETF
strategies
outperform the benchmarks. Agrrawal
(2013)
demonstrates that
multi
asset
class
ETF
portfolios dominate
stock
only
investment options.
Utilizing
neural networking
models,
Zhao,
Stasinakis,
Sermpinis,
&
Shi
(2018)
are
able to improve the portfolio efficiency of three
ETFs
compared
to traditional mean-variance optimization,
Hlawitschka
and
Tucker
(2008)
used
exclusively
ETFs
to
test
whether
stock
selection
has
any merit
in
addition to optimal
asset
allocation.
The
study
finds
ajustification
for
investors
to choose active
stock
selection
over
asset
allocation
strategies.
Data
This
study
uses
monthly price data obtained
from
Bloomberg
for
the period beginning August
2007
and ending December
2017.
Because
portfolio optimization
is
typically based on
60
monthly
observations,
it
was
necessary to
limit
the number of
ETFs
in
this
study.
There
are
too many
ETFs
with
similar
objectives.
In
fact, some of the
ETFs
dominates
others
if it
compared
in
most
important
categories
(see
Brown
et.al.
(2021
)),
The
optimization
process
computes the
ETF
weights based on
the sample covariance.
For
the
solution,
the
inverse
of the sample covariance
matrix
is
necessary.
The
sample covariance
matrix
could be singular
if
the number of observations
is
less
than the
number of variables.
In
other
words,
there
will
not be a
solution
and the weights can't be computed,
Therefore,
the number of
ETFs
used
needs to be under
60
which
is
the number of observations
used
in
the estimation,
However,
by modifying the covariance
matrix,
this
issue
can be
resolved,
For
example,
Moore-Penrose
inverse
can be
used
to
solve
this
problem
(Pappas,
Kiriakopoulos,
and
Kaimakamis
(201
0)).
In
choosing asubset
for
our study
from
the
2,792
or
so
U.S.
ETFs,
we
applied
three screening criteria,
First,
the largest
ETFs
by market
cap
were
selected because they
are
popular
and meaningful
from
an
investor's
perspective. Second,
this
study includes the
ETFs
with
the longest
possible
history.
The
final
criteria implemented
is
to include
ETFs
from
as
many
asset
categories
as
possible
to improve the diversification potential of our
portfolios,
Where
possible
at
least
one
ETF
is
included
from
each category within an
asset
class.
For
example,
if
two
ETFs
tracked the performance
of the
S&P
500
index,
the larger of the two
is
included
in
the study and the other
is
eliminated.
By
applying the three criteria, we selected
34
ETFs
for
our
study,
Table
1
provides
information on
key
characteristics of the
34
ETFs
included
in
the
study,
Among
34
ETFs
under investigation,
SPDR
S&P
500
ETF
Trust
(SPY)
is
the largest
ETF
in
the
US
with
more than
420
billion
dollars
in
assets
(in
December
2021)
and
has
been
in
the market
for
the longest period of
time,
for
about
27
years.
The
ETF
with
the
smallest
market capitalization
is
the
United
States
Oil
Fund
(USO)
which
tracks
U,S.
crude
oil
prices,
Currently,
the
usa
has
atotal
asset
value of about 2
billion
dollars.
The
fund
with
the
shortest
history
in
our sample, Vanguard
FTSE
Developed
Markets
ETF
(YEA)
has
been around
for
10
years,
Despite
its
short
life,
VEA
is
among the top four
ETFs
in
our study
in
terms
of market capitalization,
VEA
targets the
performance of the
FTSE
Developed
All
Cap
ex
US
Index
and
tracks
stocks
in
developed market other
than the
U.S.
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Financial
Planning
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Table
1:
Lists
and
Characteristics
of
Exchange
Traded
Funds
(ElFs) (June 2007
to
December
2017)
Total_II
Fund
Fund
Fund
Fund
Marlc8tcap Hillary
llcker
Name
(mllllanDailars)
°blocllV8
Geagraphlcal Focu.
_Clas.
StraI8gy Facu.
Longth
(Days)
I
SPY
SPDR
S&P
5aJ Elf musr
272,676
Large-cap
United
states
Equity
Bend Large-cap
9107
2
VTI
VANGUARD
TOTAL
STOCK
MKf
Elf
93,194
Broad
Market
United
states
Equity
Bend
Broad
Market
6063
3
EFA
ISHAI<1'S
MSCI
EAFE
Elf
85,325
Internatiorol
International
Equity
Bend
Large-cop
5985
4
VEA
VANGUARD
FTSE
DEVELOPED
ETF
69,281
Infernatiorol
International
Equity
Bend
Large-cap
3816
5
VIVO
VANGUARD
FTSE
EMERGING
MARKE
68.452
Emerging
rv1arkets
International
Equity
Bend
Broad
Market
4684
6QQQ
POWERSHAK£S
QQQ
TRUST
SERIES
57,747
Large-cap
UnITed
States
Equity
G'OWIh
Large-cap
6876
7
AGG
ISHAI<1'S
CORE
U,S,
AGGREGATE
53,629
Aggregate
Bond
UnITed
states
Fixed
Income
Aggregate
NA
5215
8
UH
ISHAK£S
CORE
S&P
MIDCAF
Elf
44,841
Mid-cap
United
states
Equity
Bend Mid-cap
6433
9
IWM
ISHAK£S
RUSSELL
2CIXJ
Elf
42,166
SmalkxJp
United
states
Equity
Bend
Sman-cap
6433
10
IWD
ISHARESRUSSELL
ICXXJVALUE
E
41,631
Large-cap
United
states
Equity
Value
Large-cap
6433
II
IWF
ISHARES
RUSSELL
I
CXXJ
GRCWTH
40,961
Large-cap
United
states
Equity
GR:>wIh
Large-cop
6433
12
LQD
ISHAK£S
IBOXX
UWESTMENT
GRA
38.431
Corporate
United
states
Fixed
Income
Corporate
NA
5642
13
GUO
SPDR
GOLD
SHARES
35,343
Precious
tv1etals
Global Commodity
Precious
N1etals
NA
4796
14
VNQ
VANGUARD
REIT
Elf
34,626
Real
Estate
UnITed
states
Equity
Bend
Brood
Market
4846
15
TIP
ISHAI<1'S
TIPS
BOND
ETF
24,339
Inflation
Protected
UnITed
States
Fixed
Income Inlotion
PR:>tected
NA
5145
16
BSV
VANGUARD
SHORHERM
BOND
ETF
23,884
Aggregate
Bond
Urited
stotes
Fixed
Income
Aggregate
NA
3923
17
VEU
VANGUARD
FTSE
ALL-WORlD
EX-U
23.484
International International
Equity
Bend Largecap
3956
18
VGK
VANGUARD
FTSE
EURCPE
Elf
18,598
European
Region
European
Region
Equity
Bend Largecap
4684
19
HYG
ISHARES
IBOXX
USD
HIGH
YIELD
17,946
Corporate
United
States
Fixedhcome
COrpoiOte
NA
3922
20
FfF
ISHARES
US
PREFERRED
STOCK
E
17,653
Preferred
United
states
Fixed
Income
Preferred
NA
3934
21
Br.!
VANGUARD
INTERMEDAlE-TERM
B
15,303
Aggregate
Bond
Urited
stotes
Fixed
Income Aggregate
NA
3923
22
VBR
VANGUARD
SMALL-CAP
VALUE
ETF
12,763
Smallmp
United
states
Equity
Value
Smol-cap
5089
23
MBB
ISHARES
MBS
ETF
11,859
Martgoge-&lcked
United
states
Fixed
Income Mortgoge-llocked
NA
3948
24
SHY
ISHARES
I-J
YEAR
TREASLRI
BO
11,261
Government
LJrited
stotes
Fixed
Income Government
NA
5642
25
IWS
~HARES
RUSSELL
MIJ.CAPVALU
11,125
Mid-cap
United
states
Equity
Value
Mid-cap
IiI1I
26
IWO
~HARES
RUSSELL
2CIXJ
GROWTH
9,188
Smallmp Urited
States
Equity
G'OWIh
Smol-cap
6370
27
IWP
ISHARES
RUSSELL
MIIJ.CAP
GROW
8,615
Mid-cap Urited
stotes
Equity
Growth
Mid-eap
6001
28
SHV
ISHARES
SHORT
TREASURY
BOND
8,057
Government
United
states
Fixed
Income Govemment
NA
4012
29
TLT
ISHAIi{S
20+
YEAR
TREASURY
BOND
7,185
Government
United
States
Fixed
Income Govemment
NA
5642
30
SLV
~HARES
SILVER
TRUST
5,454
Precious
Metals
Global Commodity
Precious
N1etals
NA
4270
31
RWX
SPDR
OJ
NlffiNATIONAL
REAL
E
3,765
Real
Estate
International
Equity
Bend
Broad
Market
4039
32
EFG
SHARES
MSa
EAFE
GRCWTH
Elf
3,600
International International
Equity
GR:>wIh
Lorge-cap
4536
33
DBC
POWERSHARES
DB
CCMMODflY
NO
2,292
Broad
Based
Global Commodity
Broad
Based
NA
4354
34
USC
UNITED
STATES
OL
FUND
LP
2,077
Energy
United
states
Commodity
Energy
NA
4288
Notes:
The
trading price
observations
of
each
ETF
is
obtained
from
Bloomberg
from
June
2007
to December
2017.
Out
of
approximately2200
funds.
the
34
ETFs
funds
are
used
based
on
three screening criteria including
funds
with
the largest market
capitalization longest
history,
and
within
each category
asset
class,
The
ETFs
are ranked
from
the largest
size
to
smallest
size
in
terms
of
total
assets
value
expressed
in
billion
dollars,
Beginning
in
June
2007
and ending
in
December
2017,
atatal af
126
manthly
prices
were
obtained
for
each of the
study's
34
ETFs,
Using
the monthly price observations
(P
t)of each
ETF,
the monthly
returns
for
agiven
ETF
ion day t
are
calculated,
ri't
=In(Pu/Pt-lJ
As
shown
in
Table
2,
the best
performing
ETF
during the study period
with
an average annualized
return
of
11
,4
percent
is
QQQ
(Powershares
Trust
Series),
QQQ
is
a
U,S.
large
cap
growth
ETF
tracking the Nasdaq
100
index
with
total
assets
of approximately
58
billion
dollars,
Of the
34
ETFs
under investigation, the
worst
performing
was
usa
(United
States
Oil
Fund)
with
-15.2
percent annualized
return,
USa.
an
oil
commodity fund,
is
the
smallest
ETF
in
the
study,
Among the study
ETFs,
SHY
(iShares
ShortTreasury
Bond)
has
the least
risk
with
astandard deviation
(0,33
percent) and the
lowest
positive
average annual
return
of
(0,06
percent),
SHY
invests
in
1-3
year
U,S,
Treasury
securities
has
assets
of approximately
11
billion
dollars,
SLY
and
usa
were
the
riskiest
ETFs
in
the study
with
standard deviations of approximately
34,00
percent,
SLY
(iShares
Silver
Trust)
tracks
the performance of
silver
as
aprecious metal
investment,
Note that
usa
has
one of the highest standard deviations
(33,77
percent)
yet
produced the
lowest
annualized average
return
(-18,5
percent) among the study
ETFs,
84
Financial
Planning
Research Journal
Table
2:
Hislorical
Performances
01
Exchanges
Traded
Funds
(Augus12007
10
December
2017)
TIcker
Name
Mean
Standard
Deviation
1QQQ
POWERS
HARES
QQQ
TRUST
SERIES
11,4000 18,0800
2
IWF
ISHARES
RUSSELL
1
000
GROWTH
8,0300
15,3500
3
IWO
ISHARES
RUSSELL
2000
GROWTH
7,9000
20,1800
4
IJH
ISHARES
CORE
S&P
MIDCAP
ETF
7.7200
17.7100
5
IWP
ISHARES
RUSSELL
MID-CAP
GROW
7.4300
18,0600
6
IWM
ISHARES
RUSSELL
2000
ETF
6,5400
19,6100
7
VBR
VANGUARD
SMALL-CAP
VALUE
ETF
6,2700
19,6000
8
vn
VANGUARD
TOTAL
STOCK
MKT
ETF
6,1900
15,5900
9
GLD
SPDR
GOLD
SHARES
6,0600
19,0600
10
SPY
SPDR
S&P
500
ETF
TRUST
5,8100
15,0600
11
IWS
ISHARES
RUSSELL
MID-CAPVALU
5,6600
17,9600
12
IWD
ISHARES
RUSSELL
1000
VALUE
E
3,9000
15.7400
13
TLT
ISHARES
20+
YEAR
TREASURY
BOND
3,5500
13,9300
14
VNQ
VANGUARD
REIT
ETF
2,3300
25,5100
15
SLV
ISHARES
SILVER
TRUST
2,1300
34,1100
16
LQD
ISHARES
IBOXX
INVESTMENT
GRA
1,5700
7.7400
17
TIP
ISHARES
TIPS
BOND
ETF
1,1900
6,3400
18
BIV
VANGUARD
INTERMEDIATE-TERM
B
1,1100
5,5500
19
AGG
ISHARES
CORE
U,S,
AGGREGATE
0,9500
3.7800
20
EFG
ISHARES
MSCI
EAFE
GROWTH
ETF
0,7300
18,6200
21
MBB
ISHARES
MBS
ETF
0,6900
2,9700
22
BSV
VANGUARD
SHORT-TERM
BOND
ETF
0,4600
2,4300
23
SHY
ISHARES
1-3
YEAR
TREASURY
BO
0,3800
1,2800
24
SHV
ISHARES
SHORT
TREASURY
BOND
0,0600 0,3300
25
VWO
VANGUARD
FTSE
EMERGING
MARKE
-D,28oo
23,7500
26
VEU
VANGUARD
FTSE
ALL-WORLD
EX-U
-D,28oo
19,9500
27
VEA
VANGUARD
FTSE
DEVELOPED
ETF
-D,63oo
19,0900
28
HYG
ISHARES
IBOXX
USD
HIGH
YIELD
-1,0400
11
,5100
29
EFA
ISHARES
MSCI
EAFE
ETF
-1,1100
19,2800
30
VGK
VANGUARD
FTSE
EUROPE
ETF
-2,1700
20,9500
31
PFF
ISHARES
US
PREFERRED
STOCK
E
-2,2500
19,1300
32
RWX
SPDR
DJ
INTERNATIONAL
REAL
E
-3,9500
21,1800
33
DBC
POWERSHARES
DB
COMMODITY
IND
-4,3700
20,6900
34
USO
UNITED
STATES
OIL
FUND
LP
-15,2000
33.7700
Notes:
ETFs
funds
are
ordered
from
the
highest
to
lowest
average
monthly
returns
from
August
2007
to
December
207
7,
The
monthly
return
is
the
percentage change
in
price
ofeach
ETF.
Out
of
approximately
2,200
funds,
the
34
ETFs
funds
are
used
based
on
three
screening
criteria
including
funds
with
the
largest
market
capitalization
longest
history.
and
limiting
ETFs
from
each category of
asset
classes.
The
ETFs
are
ranked
from
the
highest
to
lowest
returns.
The
average
monthly
return
and
standard
deviation
are
expressed
in
percent.
Methodology
To
investigate whether optimized
ETF
portfolios
yield
superior
returns
as
compared to individual
ETFs,
we
employed classical
Markowitz
optimization
(1952)
which
identifies
the minimum variance portfolio
for
agiven
return.
First,
the
return
data
from
the
first
60
months
was
used
to identify the optimal
ETF
allocations
for
10
optimized
portfolios,
These
10
portfolios
represent
the entire efficient frontier
for
our
34
ETFs
from
the minimum
risk-return
portfolio
(Portfolio
1)
to the maximum
risk-return
portfolio
(Portfolio
10),
Second month
61
returns
are
calculated
for
each of our ten optimized portfolios
and the portfolios
are
ranked based on average annualized
return
from
the maximum.
Third,
atthe
85
Financial
Planning
Research Journal
..
..
\l
~
end of each month beginning
with
month
61,
the
ETF
composition of each optimized portfolio
is
re-
estimated
using
returns
from
a60-month
rolling
period (the oldest price observation
is
removed and
the
most
recent
(from
the
prior
month)
is
added).
The
returns
on the ten
newly
optimized portfolios
(based on the
re-estimated
ETF
allocations) are then calculated and
ranked
for
the subsequent
month,
The
re-estimation
process
for
the
ETF
allocations of each portfolio and the calculation of
returns
based on those re-estimated
ETF
allocations
are
repeated
for
each month until December
2017,
atotal of
65
times,
Next,
amodel
ETF
allocation
is
determined
for
each of our ten portfolios
by averaging the
65
monthly weights assigned to each
ETF
in
each of the ten
portfolios,
Finally,
the
annualized average
returns
of the ten model portfolios
over
the 65-month period are compared
with
the average
returns
of individual
ETFs
during the same time period. Note that the replication of
the efficient
asset
allocation
for
65
times
also
allows
us
to examine
if
there
are
any
persistent
weights
assigned to acertain
asset
class,
See
the Appendix to understand the methodology
in
more
detail.
Based
on the classical
Markowitz
optimization, the efficient allocation
(Markowitz
1952)
can be
determined
as
follows:
The
objective function
is
to
minimize
the
risk
with
certain
constraints.
Objective =Min(v'};v)
Subject to
VOi!:O
v'r=1l
v'I=1
where
v=the vector of weights put
in
each
ETFs
in
the portfolio.
r=the vector of
ETF
returns
11=
portfolio target
return
};=
covariance
matrix
of
ETF
returns
" " "
..
(1)
Using
the Matlab optimization package,
10
portfolios are constructed on the efficient frontier formed
by the
34
ETFs
in
the
study,
Because
these
10
portfolios
represent
the whole efficient
frontier,
investors
can choose aparticular portfolio along the efficient frontier that
suits
their personal
risk
tolerance,
The
returns
of the
10
optimized portfolios
are
also
compared
with
no'lve
portfolio allocation
strategies
as
well
as
the individual
ETFs,
The
naive portfolio can be constructed
in
such
away that
all
34
ETFs
are
equally weighted.
Empirical Results
Optimized Portfolio
based
on Historical
Variance-Covariance
Matrix
Table
3
presents
the comparison of the performance of the ten optimized portfolios
with
the
performance of the individual
ETFs
during the period of August
2012
to December
2017,
The
optimized portfolios performed quite
well
producing Sharpe
ratios
higher than
most
of the individual
ETFs,
As
shown
in
Panel
Aof
Table
3,
the Sharpe
ratios
of
Portfolio
7
(1.01)
and 8
(1.00),
which
are
on
the far-right corner of efficient
frontier,
are
larger than
most
ETFs
except 5top
ETFs.
Portfolio
10
has
higher
return
than any
ETF
except QQQ.
1
Only
5
out
of
34
ETFs
provide
returns
thaI
are
larger
than
12.80%.
These
ETFs
are
QQQ,
IWF,
IWO,
IWP,
and
IJH.
86
Financial
Planning
Research Journal
Table
3:
Comparison
of
Performances
of
Optimized
Portfolio
and
Individual
Exchange
Traded
Funds
Based
on
Actual
Return
(August
2012
to
December
2017)
Portfolio
Return
Risk
Sharpe
Ratio
Panel
A:
Performance
of
Optimized
Portfolio
Lowest
RisklReturn
Portfolio
1
0,0004 0.0012
-18,3066
Portfolio
2
0,0177
0.Dl34
-D,3074
Portfolio
3
0,0350 0,0268 0,4910
Portfolio
4
0,0523
0,0404 0,7544
Portfolio
5
0,0695 0,0540 0,8840
Portfolio
6
0,0868
0,0677 0,9608
Portfolio
7
0,1041
0,0818
1.0061
Portfolio
8
0,1214 0,0994
1,0018
Portfolio
9
0,1387
0,1246
0,9387
Highest
RisklReturn
Portfolio
10
0,1560 0,1804
0.7441
Panel
B:
Performance
of
Individual
ETFs
QQQ
POWERSHARES
QQQ
TRUST
SERIES
16,2000 12,1000
1,1600
IWF
ISHARES
RUSSELL
1
000
GROWTH
13,8000
9,8000
1,1900
IWO
ISHARES
RUSSELL
2000
GROWTH
13,5000 14,1000
0,8000
IWP
ISHARES
RUSSELL
MID-CAP
GROW
13,2000 10,6000
1,0400
IJH
ISHARES
CORE
S&P
MIDCAP
ETF
13,0000 10.9000
0,9900
VBR
VANGUARD
SMALL
-CAP
VALUE
ETF
12,4000 12,0000
0,8500
VTI
VANGUARD
TOTAL
STOCK
MKl
ETF
12,3000
9,6000
l.D6oo
IWM
ISHARES
RUSSELL
2000
ETF
12,3000 13,4000
0,7500
SPY
SPDR
S&P
500
ETF
TRUST
12,2000
9,3000
1,0800
IWS
ISHARES
RUSSELL
MID-CAPVALU
12,0000 10,0000
0,9900
IWD
ISHARES
RUSSELL
1000
VALUE
E
10,9000
9,5000
0,9200
EFG
ISHARES
MSCI
EAFE
GROWTH
ETF
7,5000
11,1000
0,4800
VEA
VANGUARD
FTSE
DEVELOPED
ETF
6,4000
11,3000
0,3800
EFA
ISHARES
MSCI
EAFE
ETF
6,3000
11,6000
0,3500
VGK
VANGUARD
FTSE
EUROPE
ETF
5,9000
12,8000
0,2900
VEU
VANGUARD
FTSE
ALL-WORLD
EX-U
5,2000
11,4000
0,2700
VNQ
VANGUARD
REIT
ETF
4.0000
13,2000
0,1400
VWO
VANGUARD
FTSE
EMERGING
MARKE
2,5000
14,5000
0,0200
RWX
SPDR
DJ
INTERNATIONAL
REAL
E
1,2000
12.7000
-D,08oo
LQD
ISHARES
IBOXX
INVESTMENT
GRA
0,0000
4,9000
-D,4400
SHV
ISHARES
SHORTTREASURY
BOND
0.0000 0,1000
-21,3300
SHY
ISHARES
1-3
YEAR
TREASURY
BO
-D,lOoo
0.7000
-3,2100
TLT
ISHARES
20+
YEAR
TREASURY
BOND
-D,40oo
11,0000
-D,2400
MBB
ISHARES
MBS
ETF
-D,4oo0
2,4000
-1,0600
BSV
VANGUARD
SHORT-TERM
BOND
ETF
-D,50oo
1,3000
-2,0100
AGG
ISHARES
CORE
U,S,
AGGREGATE
-D,50oo
3.0000
-D,9200
PFF
ISHARES
US
PREFERRED
STOCK
E
-D,60oo
4,5000
-D,63oo
HYG
ISHARES
IBOXX
USD
HIGH
YIELD
-D,9OO0
5,1000
-D,61
00
TIP
ISHARES
TIPS
BOND
ETF
-1.2000
4,5000
-D.74oo
BIV
VANGUARD
INTERMEDIATE-TERM
B
-1,3000
4,3000
-D,81
00
GLD
SPDR
GOLD
SHARES
-4,3000
15,8000
-D,4100
DBC
POWERSHARES
DB
COMMODITY
IND
-9,1000
14.7000
-D,7700
SLV
ISHARES
SILVER
TRJST
-9,8000
25,2000
-D,47oo
USO
JNITED
STATES
OIL
FUND
LP
-18,5000
29,0000
-D,71
00
Notes:
The
optimizedportfolio return
is
calculatedbased on optimal allocation
in
each
asset
class
using
7-month rolling period return
and
historical variance-covariance
matrix.
These
performance
was
achieved
using
the last 60 months
returns
to find the optimal
weights
and
applying
these
weights to the current month's
returns
and
repeating the
process
for
the following month
by
adding the
latestmonth
and
dropping the
first
month,
and
keeping the sample
at
60
months.
This
process
was
repeated
65
times.
65 optimal
returns
are averaged for each portfolio
and
standard deviations are calculated
and
both annualized. Optimized portfolio
returns
are
ordered
from
the lowest risk/lowest return
(Portfolio
1)
to the highest risk/highest
returns
(Portfolio
70).
Returns
on each Individual
ETFs
are
the average monthly
returns
of
the same time period
as
optimizedportfolio Individual
ETFs
Performance are ordered
from
the highest
to lowest
returns.
Sharpe
ratio
is
the
return
on
portfolio or
ETFs,
minus
risk-free
rate divided
by
their standard deviation where
ID-year
Treasury
rate
of
2,
18
%
is
used
as
the
risk-free
rate, All
returns
and
standard deviations are
expressed
in
percent,
87
Financial
Planning
Research Journal
..
..
\l
~
To
better understand the
ETF
allocations of each of these
10
efficient
portfolios,
we calculate the
average
ETF
weights
over
the 65-month period.
Table
4
presents
the average weights of each
ETF
in
the
10
efficient
portfolios.
These
average weights could provide guidance to
investors
in
forming a
better performing portfolio which
prOVides
superior
returns,
For
example, the minimum
risk
portfolio
(Portfolio
1)
is
dominated by
SHY
(Short-Term
Treasury
Bond) and only four
ETFs
entered
in
the
highest
return
portfolio
(Portfolio
10).
These
ETFs
ore
QQQ
(Lorge
Cap Growth),
SLY
(Silver),
GLD
(Gold), and
IWO
(Small
Cap Growth).
During
the study period
from
August
2007
to December
2017,
it
is
interesnng
to note that
SHY
accounts
for
99.70%
of
lowest
return
portfolio
(Portfolio
1)
as
SHY
generated an average annual
return
close to
0%.
Similarly,
QQQ
constitutes
78,50%
of
Portfolio
10,
This
is
due to the fact that
QQQ
with
an average annual
return
of
16.2%
is
the top performer of the
34
study
ETFs,
As
is
evident
in
Table
4,
investors
who want to achieve higher
returns
along the efficient
frontier should increase their allocation to
QQQ
while
decreasing their allocation to
SHY
In
moving
from
portfolio 1to portfolio
10
the average weights of
QQQ
increase
from
0,0001
to
0.7895
while
the
weights of
SHY
decrease
from
0,9970
to
0.0020,
Table
4:
Optimal
Allocation
based
on
Average
Weights
in
each
ETFs
for
Efficient
Corner
Portfolios
(August
2012
to
December
2017)
Ticker
Name
Portlollo
1
Portlol102
Portlollo3
Portlollo
4
PortroIIo5
Porllollo6
Portlollo7
Portlollo8
Portlollo9
PortroIIo
10
Low
Risk/Relurn
High
Risk/Relurn
1llT
~HARES
20+
YEAR
TREASURY
BOND
0.1100 0.1690 0.2290
0.2890
0.3200
0.2630 0.1210
2
SHV
~HARES
1-3
YEAR
TREASURY
BO
0.9970
0.7210 0.5020 0.3590 0.2370
0.1200
0.0290
0.0020
3
PNS
~HARES
RUSSELL
MIOCAPVALU
0.0160
0.0320
0.0480 0.0640
0.0800
0.1000
0.1000
0.0930
4
SHY
ISHARES
I-J
YEAR
TREASURY
BO
0.1180
0.1740
0.1520
0.1060 0.0600
0.0210
0.0020
5
PND
~HARES
RUSSELL
1000
VALUE
E
0.0100 0.0200 0.0300 0.0390 0.0490
0.0460
0.0040
6
rNP
~HARES
RUSSELL
MIOCAP
GROW
0.0040 0.0080 0.0120 0.0160
0.0190 0.0100
0.0030
0.0020
7
rNF
ISHARES
RUSSELL
1000
GROWTH
0.0020 0,0030 0,0040 0,0050 0,0050
0,0060 0.0020
8
MBB
~HARES
MBS
ETF
0.0020 0.0030 0.0050 0.0060
0.0030
9
UH
ISHARES
CORE
S&P
MIDCAP
ETF
0.0010 0.0020 0.0030
0.0030
0.0040
0.0040 0.0050
10
LQD
ISHARES
1l0XX
INVESTMENT
GRA
0.0010 0.0020 0.0020
0.0030
0.0020
11
TlP
ISHARES
TPS
BOND
ElF
0.0010 0.0020
0.0020
0.0010
12
QQQ
PCWERSHARES
QQQ
TRUST
SERIES
0.0010
0.0640 0.1240 0.1840
0.2450 0.3050
0.3830
0.5120 0.6390
0.7850
13
SLV
ISHARES
SILVER
TRUST
0.1080
14
GUO
SPDR
GOLD
SHARES
0.0090
0.0210
0.0330 0.0460
0.0E90
0.0770 0.1030
0.1370
0.0770
15
PNO
ISHARES
RUSSELL
2000
GROWTH
0.0010
0.0010
0.0020
0.0310
Notes:
Portfolio
optimization conducted
for
65
times.
Each
of
these
optimization
processes
computes the optimal
weights
in
each
of
these
34
ETFs
for
10
model
portfolios.
In
other
words,
each
process
produces aweight table
with
34
rows
and
10
columns.
In
most
cases,
the
weights
are
zero.
To
summarize,
we
averaged
weights
by
columns
and
rows.
For
example, the average weight
of
SHV
(I-shore's
7-3-
year
Treasury
bonds
ETF)
in
portfolio 7
is
0.997.
This
is
the average weight
of
SHV
in
portfolio 1
for
65
estimations.
It dominates portfolio 7
since
this
is
aportfolio
with
minimum
risk.
It
is
natural to
see
that
low-risk
Treasury
bonds
makes
up
the bulk
of
this
minimum
risk
portfolio.
Some
ETFs
have
zero
average weight
across
the spectrum
of
portfolios.
Therefore,
we
did notinclude
these
ETFs
in
the table
since
all
the
values
are
zero.
Having
observed that the optimized
ETF
portfolios produced better performance compared
with
individual
ETFs
on
an
ex-post
basis,
we now investigate whether the optimization
process
can
produce better portfolio performance
in
the
future,
Using
an ex-onte approach,
we
calculated the
ETF
allocations
for
ten efficient portfolios
using
four different approaches.
The
first
three approaches
rebalance ten optimized portfolios monthly to allocations determined by
using
1)
returns
from
the
prior
6o-months,
2)
returns
from
the prior
4o-months,
and the
3)
the monthly cumulative
returns.
The
fourth approach rebalances each of the
10
portfolios annually
using
the
returns
from
the
prior
60
months.
88
Financial
Planning
Research Journal
Panel
Aof
Table
5
presents
the
results
of the ex-onte performances
from
August
2012
to December
2017
of
10
efficient portfolios based on past
60-months
return,
Portfolio
optimization
using
the
historical variance and covariance
matrix
for
out of sample forecasting does not appear to produce
useful
results,
The
best Sharpe
ratio
produced
using
this
strategy
is
0,572
from
optimized
Portfolio
8,
This
is
significantly
lower
than the Sharpe
ratio
of 1
,01
for
Portfolio
7based on applying optimal
allocation on
ex
post
returns
and
is
also
lower
than the Sharpe
ratio
of 1
,16
for
the
QQQ
ETF.
Further,
the highest
return
forecasted
from
efficient portfolio
is
only
8.00
%,
which
is
almost the half of the top
performing portfolio
using
the
ex
post approach
or
the
QQQ
ETF
alone.
While
prior
6D-month
return
data
is
commonly
used
in
reallocating portfolios and measuring their
performance, one particular
interest
of
this
study
is
to
see
whether the performance of optimized
portfolio can be improved by
relying
only
on
more recent
return
data,
Thus,
we
next
utilized
return
data
from
the
prior
40
months to recalculate the weights
for
our optimized
portfolios.
The
use
of
40
month
return
data
is
acceptable given that the study
is
limited to portfolios formed
from
only
34
ETFs
inasmuch
as
the number of observations that can produce an efficient optimization procedure
has
to be greater than
34.
Panel
Bof
Table
5
reports
the performance
results
for
the
10
portfolios
optimized
using
return
data
from
the
prior
40
months,
Using
the
4D-month
data, the highest Sharpe
ratio
produced
by
an optimized portfolio improved to
0,601.
The
highest
return
produced by an
optimized portfolio
using
40-month data
is
9.70%,
which
is
significantly
lower
than
15,6
%
return
produced by an optimized portfolio
using
ex
post data,
To
examine whether historical data
from
longer periods increase the
effectiveness
of our optimized
portfolio
strategy,
acumulative
return
method
is
used
where each month's
return
data added
without eliminating the earliest observation,
For
example,
while
the optimized
ETF
weights
for
month
61
were
based on 60-month
return
data, the optimized weights
for
month
62
portfolios
are
based
on
return
data
from
the prior
61
months and
so
on
until
month
125
when the optimized
ETF
weights
were
based on
return
data
from
the
prior
124
months.
Results
reported
in
Panel
Cof
Table
5
show
a
surprising
finding
where
longer
history
does,
in
fact, decreases the
effectiveness
of
this
strategy.
When
cumulative
history
of the
returns
up to the optimization month
is
used
instead of just past
60
months,
the Sharpe
ratios
of each of the
10
optimized portfolios
are
lower,
The
best performing portfolio
(Portfolio
7)
has
the Sharpe
ratio
of only
0,089
with
a
return
of only
2,9%,
Monthly portfolio rebalancing
is
costly
for
investors
due to the transactions
costs
incurred and
the higher
tax
rate
imposed on short term capital gains
as
comparted to long
term
gains,
In
reallocating portfolio
weights
monthly,
the
investor
is
forced to
taxes
currently on any gains and
thereby
loses
the ability to defer those potential
tax
liabilities.
To
address
this
issue,
we investigate how
reallocating portfolios annually rather than monthly might affect the performance of the
10
optimized
portfolios.
The
results
of portfolio performance based on annual optimization
are
reported
in
Panel
Dof
Table
5.
The
portfolio performance based on annual reallocation of the optimized
portfolios,
as
reported
in
Table
5,
is
worse
than when rebalancing occurred
monthly.
The
best performing portfolio
in
terms
of Sharpe performance
is
Portfolio
8
with
aSharpe
ratio
of
0,31,
declining
from
0,57
and
0,60
when monthly optimization based on
60-months
and
4D-months
returns
are implemented,
When
looking at the performance
from
return
perspective, the best performing portfolio generates only
5.40%,
which
is
significantly
lower
than highest
returns
from
efficient portfolio based on monthly
asset
allocation obtained
from
6D-month
(7.00%)
and
4D-months
(8,90%)
optimization
process.
89
Financial
Planning
Research Journal
..
..
\l
~
Table
5:
Out
ot
Sample
Forecast
of
Optimized
Portfolios
Performance
Using
Historical
Variance-Covariance
Matrix
(August
2012
to
December
2017)
Return
[
Panel
A:
Performance
of
Optimized
Portfolio
based
on
6O-Months
Optimization
Lowest
Risk/Return
Portfolio 1
0.0000
Portfolio 2 1
.0000
Portfolio 3
1,9000
Portfolio 4
2,7000
Portfolio 5
3,6000
Portfolio 6
4,4000
Portfolio 7
5,5000
Portfolio 8
7,0000
Portfolio 9
8,0000
Highest
Risk/Return
Portfolio
10
5,5000
Panel
B:
Performance
of
Optimized
Portfolio
based
on
40-Months
Optimization
Lowest
Risk/Return
Portfolio 1
0,0000
Portfolio 2 1
.0000
Portfolio 3
2.0000
Portfolio 4
3,1000
Portfolio 5
4,2000
Portfolio 6
5,1000
Portfolio 7
6.0000
Portfolio 8
7.7000
Portfolio 9
8,9000
Highest
Risk/Return
Portfolio
10
9,7000
Panel
C:
Performance
of
Optimized
Portfolio
based
on
Cumulative
Optimization
Lowest
Risk/Return
Portfolio 1
0,1000
Portfolio 2
0,5000
Portfolio 3 1
.0000
Portfolio 4 1
,SOOO
Portfolio 5
2.0000
Portfolio 6
2,5000
Portfolio 7
2,9000
Portfolio 8
2,4000
Portfolio 9
2,3000
Highest
Risk/Return
Portfolio
10
0,3000
Panel
A:
Performance
of
Optimized
Portfolio
based
on
Annual
Optimization
Portfolio 1
0,0000
Portfolio 2
0,8000
Portfolio 3
1,5000
Portfolio 4
2,1000
Portfolio 5
2,7000
Portfolio 6
3,3000
Portfolio 7
4,3000
Portfolio 8
5,4000
Portfolio 9
5,8000
Portfolio
10
-1
.0000
Notes: All the performances
are
calculated
based
on
out-of-sample forecast
and
historical variance covariance matrix,
PanelA
uses
historical 60 months obseNations on a7-month rolling basis to determine the
optimal
weight
for
each
month, while Panel B
uses
past
4O-months observations. Panel C
uses
acumulative sampling.
The
initial sample starts with
6O-months observations
and
the currentmonth's observation
is
added
each
month without removing the oldestobservation.
Unlike
performances reported
In
Panels A
Rand
Cwhere portfolio composition
is
changing
once
amonth, the portfolio
performance reported
in
Panel Dusing aproccess
that
reshuffles the portfolio annually.
90
Financial
Planning
Research Journal
Overall findings suggest
that
emphasis on more recent return observations
and
increased frequency
in
rebalancing portfolio
will
improve efficiency of portfolio optimization.
This
is
the
case for
the
ex-ante
approach.
However,
the
best performing portfolio based
on
out-of-sample forecast
cannot
beat
the
performance based
on
ex
post
approach.
In
addition, portfolio performance using optimization
methodology
did
poorly
compared
to
the
top
performing
ETFs,
These disappointing
results
could
be
due
to
aknown fiaw
in
Markowitz's portfolio allocation
method
that
relies
on
the
historical variance
covariance matrix,
To
address this concern,
we
construct optimized portfolios
and
gauge
their
performance using amodified sample covariance matrix,
Optimized
Portfolio
Performance
based
on
Modified
Sample
Covariance
Matrix
The
results
presented
in
Table 5indicate
that
the
ex-onte,
out
of
sample performance of
the
optimized
portfolios
not
as
good
as
the
ex-post performance
of
an
optimal portfolio.
One
well-known problem
with
the
Markowitz optimization
method
is
its
assigning
the
excessive weights
to
spedfic asset classes.
In
fact, this
can
be
observed from
the
weights assigned
to
specific
ETFs
in
several
of
the
optimized
portfolios.
As
shown earlier
in
Table 4
of
average weight
of
each
ETF
in
each
of
10
efficient portfolios,
only
15
out
of
34
ETFs
had
anon-zero weight
in
any
of
the
optimized portfolios. Ledoit
and
Wolf
(2003)
suggests
the
use
of
ashrinkage
technique
to
modify
the
covariance matrix,
The
shrinkage combines
sample covariance with ahighly structured covariance matrix. Ledoit
and
Wolf used constant
correlation matrix
as
aproxy for ahighly structured covariance matrix. Ashrinkage constant
is
used
to
combine
these two covariance matrixes.
:E
Shrunk
=
2(X)
+
(1
-
A)Y
(2)
Where 2
is
the
shrinkage constant. X
is
the
structured covariance matrix,
and
Y
is
the
sample
covariance
matriX2
.
Table
6:
The
Performance
of
Optimized
Portfolios
Using
The
Modified
Variance
Covariance
Matrix
with
the
Shrinkage
Technique
(August
2012-
Decemer
2017)
Lowest
Risk/Return
Highest
Risk/Return
Portfolio
Return
Risk
Sharpe
Ratio
Portfolio
1
0,0000 0,1000
-20.7570
Portfolio
2
1,0000 1,2000
-D,9540
Portfolio
3
2,0000 2,5000
-D,0880
Portfolio
4
2,9000
3.7000
0,1830
Portfolio
5
3.7000
5.0000
0,3130
Portfolio
6
4,6000
6,3000
0,3940
Portfolio
7
5,5000
7,5000
0,4500
Portfolio
8
6,9000
8.5000
0,5520
Portfolio
9
8,2000
10,4000
0,5760
Portfolio
10
5,5000
16.5000
0,2040
Notes:
This
table
presents the performance
of
optimized portfolios
in
the 65-month
period
starting from August31,2072
to
December
29,
2077.
At
the beginning
of
each
month,
the
optimization process configured the most efficientallocation.
The
money
is
invested
based
on
the
best
allocation
and
the performance
is
measured
at
the
end
of
the month
on
the ex
ante
basis.
The
procedure
is
repeated
for
the
65 months.
In
this
optimization the sample variance
and
covariance matrix
is
modified with ashrinkage technique.
Risk
is
measured as standard deviation. Return
and
risk
are expressed in percent.
2Please
see
Ladoit
and
Wolf
for
more
details
on
shrinkage
covorionce
matrix.
91
Financial
Planning
Research Journal
..
..
\l
~
We
estimated shrinkage constant
and
constant correlation matrix.
The
shrinkage constant estimate
is
0,1514,
Using
Ledoit
and
Wolf technique, ashrunken covariance matrix was created
and
used
in
repeating
the
optimization process,
The
optimization
ETF
allocations are
calculated
based
on
6o-months histarical
data
and
out
of sample forecast far portfolio returns. New performance metrics
for
10
optimized portfolios
obtained
using
the
shrunken covariance matrix
technique
are reported
in
Table
6,
Eight
out
of
ten optimized portfolios show amarginally improved performance
in
terms of
returns
and
Sharpe ratio when
compared
with the performance of optimized portfolios determined
using
the
historic variance-covariance matrix
and
reported
in
Panel AofTable
5.
Only portfolios 7
and
8show poorer performance when the shrunken covariance matrix
is
used.
The
return
on
the
best
performing portfolio (portfolio 9) has improved from yielding 8.00 percent
to
8,20 percent return
and
the
highest Sharpe ratio increased from 0.572 to
0,576,
Optimized
Portfolio
based
on
Average
Weight
Based
on
the
findings, rebalancing
the
optimized portfolio by using
the
actual
optimized weights
for
each
month
for
each
ETFs
does
not
appear
to
create better portfolio performance,
The
out
of
sample forecast performance does
not
yield abetter return
than
ex
post performance even when
using amodified covariance matrix. Investigating
the
asset allocation weight closely, we further
examine whether some informed strategies
that
provide optimized return
to
investors
can
be
achieved
using different weight measure. Alternatively,
the
average weights for
each
composition
of
ETF
in
each
portfolio
is
utilized instead of
the
actual
weights optimized
each
month.
The
portfolio weights
will
first
be
created by the optimization process
at
the
beginning
of
each
month
using 60-month historical
data.
Portfolio return for
that
month
is
determined based
on
ex
ante
approach, Optimized weight for
next
month
is
re-estimated on a1-month rolling basis by the most recent
month
and
the
dropping
the
first
month
in
the
data
to
keep
the
number
of
past observations
to
be
constant
at
sixty
observations,
Then,
the
optimized weight for
each
ETF
composition
in
each
portfolio for
the
second
month
is
the
average of
the
first
two months
and
investment allocation
is
constructed accordingly,
The
process
is
repeated
till
the
end
of
study period.
In
particular,
the
optimized weight for 65-month periods
will
be
the
average
of
65
optimized weights
obtained
for
each
month.
The
performance
of
10
efficient
performance
is
simply
the
average of monthly optimized portfolio performance.
The
average weights over
the
last
65
months of
each
ETF
in
these
10
efficient portfolios was presented
in
Table
4,
As
pointed
out
earlier, fewer
than
50%
of
ETFs
under
investigation are given consideration
based
on
the
optimized weights.
QQQ
is
the
dominant
ETF
as
average weight increases with
the
portfolio return. On
the
other hand, average allocation
in
SHY
decreases when investors prefer
to
invest
in
riskier
portfolio with higher returns.
This
is
understandable since
QQQ
is
an
ETF
investing
in
large-eap growth stocks
in
the
U.S,
and
provides
the
highest return
of
16.20%,
while
SHY
focusing
on
short-term
T-Bond
returns nothing
to
investors.
Utilizing
the average weights
methodology
outlined above,
the
performances
of
each
of these
10
portfolios based
on
6o-month historical
data
and
ex
ante
approach
reported
in
Table 7are lower
than
ex
post optimized portfolio performance. However,
the
performance
is
much
better
than
ex
ante
optimized portfolio
results
reported
in
Panel AofTable
5,
The
Sharpe ratio of
top
performing
portfolio (Portfolio 9) analyzed from applying average weight
on
one-month
ahead
return
is
1.008
is
practically
the
same from applying
actual
weight
on
end
of
the
month
return as reported
in
Table
4.
In
comparison with ex
ante
optimized portfolio performance analyzed from
actual
weight.
the
top
performing portfolio shows Sharpe ratio of only 0.5729, which
is
less
than
half
of
top
performing
92
Financial
Planning
Research Journal
performance from using average weight,
In
addition,
the
performance
of
best portfolio when
optimized weights are averaged remains superior
to
any
of
top
performing portfolios regardless of
the
length
of
historical
data
used
and
the frequency
of
portfolio rebalance
conducted,
As
reported
in
Table
5,
using shorter period
of
4o-month historical
data,
the Sharpe ratio
is
0,601
0
and
it
is
even
worse with cumulative
data
which yields 0,089 Sharpe ratio, Rebalancing portfolio
once
ayear
to
avoid transaction cost
and
tax
issues
does
not
lead
to
better Sharpe performance which remains
at
0.3110,
When
comparing
with individual
ETF
performance,
the
average weight
method
provides
the
portfolio return
that
is
higher
than
the
returns of
70%
(24/34) of individual
ETF
and
the Sharpe
ratio
that
is
higher
than
85%
(29/34)
of
ETF
under
investigation, Overall,
the
main information to
be
gleaned
from Table 7
is
averaging
the
weights
is
better
than
using the
actual
optimized weights for
the
month,
Table
7:
Out
of
Sample
Forecast
Performance
of
Optimized
Portfolios
Using
The
Average
Optimal
Portfolio
Weights
(August
2012-
December
2017)
Lowest
Risk/Return
Highest
Risk/Return
Portfolio
Return
Risk
Sharpe
Ralio
Portfolio
1
0.0000 0,1000
-20,8870
Portfolio
2
1,3000 1,2000
-0,7410
Portfolio
3
2,6000
2,3000
0,1840
Portfolio
4
3,9000 3,4000
0,4950
Portfolio
5
5,2000
4,6000
0,6510
Portfolio
6
6.4000
5.7000
0.7460
Portfolio
7
7.7000
6.7000
0,8220
Portfolio
8
9,2000
7,6000
0,9180
Portfolio
9
10.9000
8.7000
1,0080
Portfolio
10
11.7000
10,6000
0,8990
Notes:
This
table
presents
the
performance
of
optimizedportfolios
In
the
65-month
period
starting from August31,2072
to
December
29,2077,
This
performance
Is
obtained
with
average
optimal
weights, Using
the
average
of
the
optimal
weights, 65
month
returns
are
calculated,
The
average
of
these 65 returns
and
standard
deviations are
reported
In
this table.
Conclusions
In
this article, efficient asset allocation
among
ETFs
are investigated based
on
the
Markowitz's portfolio
optimization
approach,
Using
the
data
from
2007
to
2017,
we empirically examine whether investors
can
achieve abetter portfolio performance with various
risk
and
return levels
by
simply allocating
investment
among
different
ETFs,
The optimized portfolios are
compared
with best performing
individual
ETFs
during
the
same time period,
To
determine optimal
weight
for
each
ETF
composition
in
each
of
10
efficient portfolios, we utilize 1-month rolling returns of past
60
months, past
40
months,
and
60 months cumulative returns,
Markowitz's portfolio optimization
approach
is
known for putting
too
much
weight
on
acertain asset
class,
We
remedy this issue by implementing
the
modified variance-covariance matrix
in
optimization
process,
The
true test of aportfolio allocation strategy
is
the
out
of
sample ex-ante portfolio selection,
We
compared
the
performance
of
out
of
sample mean-variance efficient portfolios with the
actual
ETFs
performance,
93
Financial
Planning
Research Journal
..
..
\l
~
Overall
findings suggest that optimization procedure
using
historical
60-months
data
provides
an
efficient portfolio performance that
is
comporable
with
individual top performing
ETFs,
On
the
ex-post
basis,
the top performing portfolio
yields
the highest average monthly
returns
of
15,6%
with
highest
Sharpe
ratio
of 1
.01
,
similar
to
investing
100%
in
QQQ
ETF
that
emphasizes
on the large-eap high
growth
stocks
in
the
U.S.
However,
this
efficient allocation
using
the
Markowitz's
portfolio optimization
approach, does not predict asuperior future performance,
In
2012-201
7period,
this
approach
underperformed many
ETFs,
The
Sharpe
ratio
of the top performing optimized portfolio of
0.5720
is
less
than half of the Sharpe
rano
of top performing QQQ,
The
out of sample forecast of optimized
portfolio
is
a
little
better when only more recent historical data
is
used
in
determining optimal weight
because of quicker feedback,
The
performance
is
worse
when portfolio
is
rebalanced annually
instead of
monthly,
When
amodified shrinkage covariance
matrix
was
employed to reduce
the problem of
excessive
weights assigned to
some
ETFs,
the performance of optimized portfolio
improved
slightly
by
0.2%
in
terms
of
returns
and
0,004
in
Sharpe
ratio.
Lastly,
constructing efficient
portfolios based on the average of optimized weights
improves
the
returns
of top performing
portfolio by
370
basis
points
and
increases
Sharpe
ratio
significantly
(+0.5).
Efficient
portfolios
using
the average of optimized weight
performs
better than
85%
of the individual
ETFs
in
terms
of Sharpe
ratio,
In
conclusion, the average weight approach to portfolio optimization
represents
the best
overall
strategy.
This
is
the only out of sample portfolio optimization approach that
was
comparable
to the
best
ETFs
in
our
list
in
terms
of mean-variance
efficiency,
Investors
should consider
using
this
technique
as
an investment
strategy.
While
this
study
is
limited to the extent to which
Markowitz
portfolio theory
holds
for
ETFs,
it
has
clear practical applications
for
financial
professionals
as
a
possible
alternative to other portfolio allocation approaches. Additionally, the
US-based
research
provides
a
useful
platform
for
further
research
in
the Australian
market.
94
Financial
Planning
Research Journal
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96
Financial Planning Research Journal
APPENDIX
This
appendix
explains
the methodology
in
more
detail.
1.
Using
the equation
1,
the optimization
will
be obtained
using
the
returns
of
ETFs
for
the
last
60
months,
10
optimal
portfolios
will
be constructed to
represent
the whole spectrum of the efficient
frontier,
The
output of
this
process
will
create aweight
matrix
with
34
rows
and
10
columns,
Each
of
34
ETFs
weight
will
be calculated
for
each of the
10
corner
portfolios,
2.
Optimal
returns
will
be calculated
using
these
weights
and the
returns
for
that month
(Table
3)
fJ.t=WtY
t
where,
P,
is
a
return
vector of
10
portfolios
for
the month
t.
W,
is
the transpose
of
the weight
matrix
from
step
1,
and
Y,
is
the
return
vector of
34
ETFs
for
the month
t.
For
out
of
sample portfolio performances, the following month's
returns
are
used
rather
than the
current month's
returns
(Table
5and
6).
3.
The
following month, the same
process
(step
1and
2)
will
be repeated
with
removing
the
returns
of
ETF
of the
first
month and adding the
return
of the
61
st
month,
Therefore,
the sample
remains
at
60
monthly
returns
This
process
was
repeated
65
times,
65
W=
LW
t
t=l
where,
W
is
the
item-by-item
sum
of optimized
weights
--~W
/1
-
65
fl
is
the
34X1
0
matrix
of average
weights
of each of
34
ETFs
in
each of
10
portfolios
(Table
4),
4.
The
performance of each of ten
portfolios
are
calculated
[
/11,1
M-
:
/110,1
R=
~Ml
=
[?]
65 T
10
97
Financial
Planning
Research Journal
Where
M
is
a
lOX65
return
matrix
of optimal
returns
calculated
for
each of
10
portfolios
for
65
months
(For
out of sample, there
will
be only
64
returns),
1
is
a
65Xl
vector of Is,
R
is
the vector of 65-month average
returns
for
each portfolio,
65
2_ 1
"'\'(
)2
(Ji -
64
L
Jli,t
-Ti
t=l
1/J
is
the vector of standard deviations of each portfolio
if>.
=
T--,i_-_
O
_,O_2_1_8
!(Ji
if>
is
the Sharpe
ratio.
Risk
free
rate
is
the average 1
o-year
constant maturity
Treasury
bond
yields
for
the corresponding 65-month period.3
3
Boord
of
Governors
of
the
Federal
Reserve
System
(US),
Market
Yield
on
u.s.
Treasury
securities
at
1
0-
Year
Constant
Moturily
[DGS10],
retrieved
from
FRED,
Federal
Reserve
Bonk
of
St.
Louis;
https:/lfred.stlouisfed.orglseries/DGS10,
December
3,2021.
98
