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Morningstar Indexes 2014 15
Back to Markowitz
Paul D. Kaplan, Ph.D., CFA, Director of Research, Morningstar Research, Inc.
Volatility-based weighting is taking us back to what Harry Markowitz told us to do in the first
place, inspiring us to create non-market-cap weighted indexes based on defined risk and return
assumptions and not empirical anomalies alone.
The principles of Markowitz’s portfolio construction model,
which requires explicit risk and expected return assumptions,
are widely accepted. In practice, however, the most widely
used portfolio construction techniques—market-cap weighting
and its main rival, fundamental weighting—make no explicit
assumptions about these very same risk and return parameters.
The most recent departures from market-cap weighted
indexing are a variety of weighting schemes based on market
anomalies rather than standard investment theory. These
schemes include minimizing the variance of the overall portfolio
and maximizing measures of portfolio diversification.
1. Although the Italian actuary
Bruno de Finetti presented
what was essentially the
same as Markowitz’s model
12 years earlier in 1940,
his paper had no impact
on investment theory since
it dealt with the actuarial
problem of optimizing a
reinsurance portfolio
rather than a portfolio of
investible securities. It was
in Italian and therefore
not on the radar screen of
English-speaking investment
professionals and academics.
2. The CAPM relaxes the
assumption of Markowitz’s
original model that investors
can only take long positions.
It is striking how similar these techniques are to the original
portfolio construction technique that Harry Markowitz first
described in 1952. To use Markowitz’s mean-variance optimization model, the investor first needs to develop estimates
for the expected return on each stock, the standard deviation of
return on each stock, and the correlation of returns for each
pair of stocks. This is essentially the same procedure used by
many of today’s volatility-based index providers. The only
material difference is that rather than explicitly estimating the
expected return of each stock, these procedures make implicit
assumptions about the expected returns. In some cases, the
assumption is that expected return is a function of risk. In this
article I argue that we should return to Markowitz’s complete
model of portfolio construction, which requires explicit risk and
expected return assumptions, so that non-market-cap weighted
indexes are clearly based on investment theories and do not rely
on empirical anomalies alone.
A Circular History of Portfolio Theory
Figure 1 presents a circular view of the history of portfolio
construction techniques, highlighting the original departure from
Markowitz’s theory in the 1960s and ultimately the return
to it in the 2000s and 2010s.
The Beginning: The Markowitz Mean-Variance Model
It is no understatement to say that Harry Markowitz revolutionized investment theory and practice with a 12-page paper
simply titled “Portfolio Selection” in the Journal of Finance in
1952. Before the introduction of his mean-variance model,
for all practical purposes, there simply was no theory of portfolio construction (at least not one that explicitly takes risk
into account).1 What Markowitz achieved was the creation of
a mathematical model that takes both expected return and
risk into account in portfolio construction. A key element to his
approach is that the risk part of the model addresses
diversification through a matrix of correlation coefficients.
First Step Away from Markowitz:
The Capital Asset Pricing Model
In the 1960s William F. Sharpe and others asked what would be
the implications if every investor used Markowitz’s approach
to portfolio construction,2 every investor used the same set of
2
Back to Markowitz
Figure 1. The Circular Evolution of Equity Indexes
Markowitz Mean-Variance Model
gExplicit risk and expected return assumptions required
CAPM Weighting Model
gTheoretical basis of market-cap weighted indexes
gNo inputs other than market cap
gNo optimization
1952
1960s
Back to Markowitz: Where We Started
gExplicit expected return assumptions
are reintroduced
gA return to Markowitz’s original model
2014
2005
2009
Volatility-Based Weighting Models
gRisk modeling reintroduced into
portfolio construction
gSilence on expected returns, leaving them implicit
gOptimization sometimes used
Fundamental-Weighting Model
gSecurities weighted in proportion to
“fundamental”measures of size such as
earnings, revenue, dividends, book value
gRequires fundamental data
gNo optimization
gValue-tilted strategy
Source: Morningstar
assumptions about the expected returns, risks, and return
correlations of every security; and every investor could borrow
or lend at the same risk-free rate of return. The startling
conclusion is that investors would not need to perform meanvariance in the first place! Rather, all investors would hold
the market portfolio in combination with cash, with the exact
combination depending on the investor’s risk tolerance.
In other words, the conservative investors would lend to the
aggressive investors so that the aggressive investors could
take levered positions in the market portfolio. These assumptions and resulting conclusions constitute the Capital
Asset Pricing Model, or CAPM. The CAPM is the intellectual
foundation of market-cap weighted indexing.
3. In the CAPM, market value
is inversely related to risk
and plays an implicit role in
portfolio construction.
The CAPM has another important implication: namely that the
expected return on each security in excess of the risk-free
rate is proportional to its beta, or systematic risk, and unrelated
unsystematic risk. The CAPM implies that only systematic
risk is priced (i.e., is rewarded with positive expected return),
and that unsystematic risk is not rewarded and therefore
should be avoided.
Second Step Away from Markowitz: Fundamental Weighting
In 2005, Rob Arnott, Jason Hsu, and Philip Moore published
an article in the Financial Analysts Journal titled “Fundamental
Indexation” that challenges the wisdom of market-cap weighting by presenting an alternative: fundamental weighting. In a
fundamental weighting scheme, market values are ignored and
the security weights are set using a group of “fundamental”
measures of firm size such as revenues, earnings, dividends, and
book value. As Gideon Magnus and I discuss in “Investing At
Full Tilt” (pages 22–26 in this publication), doing so introduces a
value tilt to a portfolio. This value tilt explains much of the
long-term outperformance of a fundamentally weighted portfolio
relative to its market-cap weighted counterpart. What is striking
about this fundamental weighting approach is that it is
devoid of any measure of risk, and thus ignores a core pillar of
Markowitz’s portfolio construction principles.3
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Morningstar Indexes 2014 15
First Step Back to Markowitz: Volatility-Based Weighting
4. For a comprehensive analysis
and rationale of the lowvolatility effect, see, Nardin
L. Baker and Robert A.
Haugen, “Low Risk Stocks
Outperform within All
Observable Markets of the
World,” www.lowvolatilitystocks.com, April 2012.
5. See Clarke, R., H. de Silva
and S. Thorley, “MinimumVariance Portfolio
Composition,” Journal of
Portfolio Management,
Winter 2011.
Low Volatility
As Gideon Magnus and I discuss in “Investing At Full Tilt,”
academic research since the late 1970s shows that the CAPM’s
predication that long-term stock returns are related solely
to market beta does not hold empirically. Rather they are related
to several factors such as size (as measured by market
capitalization) and value/growth orientation (as measured by
price/earnings and price/book ratios). More recent research
shows that long-term stock returns are inversely related
to historical volatility (standard deviation), although whether or
not this low-volatility effect is truly distinct from the size and
value effects remains an open question.4
Just as the value effect can be acted upon through tilting a portfolio by screening stocks to include those with low price
multiples or by fundamental weighting, the low-volatility effect
can be acted upon in similar ways. To form a low-volatility
portfolio, either screen for stocks that have low historical standard deviation, or weight each stock in inverse proportion to
its historical volatility. The latter approach is known as volatility
weighting or risk weighting. Figure 2 illustrates this method.
Some low-volatility indexes combine these methods by first
screening for low-volatility stocks and then weighting them in
inverse proportion to standard deviation.
Figure 2. Example of Volatility Weighting
Standard
Deviation
(%)
Reciprocal of
Standard
Deviation
Weighting
(%)
9.75
0.1025
13.33
B
5.15
0.1940
25.23
C
10.74
0.0931
12.11
D
6.42
0.1557
20.25
E
4.47
0.2236
29.08
—
0.7690
100.00
Stock
A
Total
Source: Morningstar
Minimum Volatility
Volatility weighting takes the first step back to Markowitz and
pure investment theory by explicitly bringing risk back into the
portfolio construction equation. However, it still does not
incorporate the correlations of returns. This is done in another
volatility-based strategy known as minimum volatility, or
min.-vol. for short. As with low-volatility strategies, min.-vol.
strategies are motivated by research that shows they have
both lower risk and higher long-term returns than their marketcap weighted counterparts, contrary to the predictions of the
CAPM.5 To create a min.-vol. portfolio, an index provider needs
a complete risk model that provides the standard deviation of
return of each stock and the correlations of returns of all stocks.
This is typically achieved using a factor model as illustrated
in Figure 3. In a factor model, the correlation of the returns on
different stocks is explained by a set of market-wide factors,
while the total risk of each stock is explained by exposures to
these factors and stock-specific risk.
Figure 3. Factor Model Structure
Market-Wide Factors
Stock-Specific Factors
Stock-Specific Factors
Stock Standard
Deviations
Stock
Correlations
Source: Morningstar
In some factor models, the factors are explicitly stated to be
specific macroeconomic variables such as gross domestic
product growth, inflation, etc., and each stock’s exposures to
the factors are estimated using statistical analysis. In other
models, factor exposures are based on observable characteristics of the stocks, such as market capitalization, price/earnings
ratio, and industry sector.
After choosing and estimating a factor model, an index provider
uses a Markowitz-style optimization algorithm to minimize the
standard deviation of the overall portfolio. Typically this includes
constraints to ensure that the portfolio takes only long positions,
has sufficient sector diversification, and has enough stocks to
represent the target universe. The only difference between this
procedure and the one used by many quantitative active managers is the absence of any explicit assumptions about expected
returns. This is because a min.-vol. strategy implicitly assumes
that the expected returns on all stocks are equal. Figure 4 shows
where the min.-vol. portfolio falls on a Markowitz efficient frontier that a quantitative active manager who has explicit expected return assumptions might use when selecting a portfolio.
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Back to Markowitz
Figure 4. Minimum-Volatility and Active Portfolios on the Markowitz Efficient Frontier
Minimum-Volatility Portfolio
Stocks
0.8
0.7
0.6
Active Portfolios
0.5
0.4
0.2
0.1
0
Standard Deviation
2
4
6
8
10
0
12
Expected Return
0.3
Source: Morningstar
6. Risk parity is best known
as an asset-allocation
strategy, but is also being
used as an equity
index weighting scheme.
Equal-Risk Contribution
In an equal-risk contribution weighting scheme, also known
as risk parity,6 the weight of each stock is inversely proportional
to its exposure to the variance of the overall portfolio. In this
scheme, stock-specific contributions to the volatility of the
overall portfolio (the stock-by-stock products of weights and
exposures to overall portfolio risk) are equal for all stocks.
Figure 5 illustrates how this works. Because the stock-specific
exposures and the weights depend on all of the weights
simultaneously, finding the weights requires solving a system
of nonlinear equations. Like optimization, this can be a
computationally intensive process. Therefore, providers
of min.-vol. indexes and providers of equal-risk contribution
indexes both perform tasks similar to those of quantitative
active managers. Figure 6 illustrates this by comparing the
three volatility-based weighting schemes we have reviewed
with each other and with quantitative active management.
Figure 6.Comparison of Volatility-Based Weighting Schemes and
Quantitative Active Management
Weighting Scheme Motivation
Weighting
(%)
Contribution to
Portfolio Risk
(%)
A
1.46
13.68
20.00
B
0.75
26.76
20.00
C
1.81
11.06
20.00
D
1.01
19.85
20.00
E
0.70
28.65
20.00
—
100.00
100.00
Stock
Total
Source: Morningstar
Weighting
Method
Expected
Returns
Volatility
Weighting
Empirical evidence
that low-vol.
stocks outperform
high-vol. stocks
No
Inverse proportion to standard
deviation
Implicit
Minimum
Volatility
Empirical evidence
that low-risk
portfolios outperform the market
Yes
Markowitz
optimization
Implicitly
all equal
Equal-Risk
Contribution
Diversify by equalizing contributions
of all stocks
to portfolio risk
Yes
Solve nonlinear
system
of equations
Implicit
Quantitative
Active
Management
Exploit insights of
active managers
Yes
Markowitz
optimization
Explicitly
based on
manager
insights
Figure 5. Example of Equal-Risk Contribution
Exposure to
Portfolio Risk
Factor
Model
Needed?
Source: Morningstar
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Morningstar Indexes 2014 15
Implied Expected Returns
Recall that in Markowitz’s original model, the expected returns
and the complete risk model are inputs, and the portfolio
weights are the output. But in volatility-based schemes (such
as equal-risk contribution and min.-vol.), the portfolio weights
are calculated solely from the risk model with no explicit
expected return parameters. However, given the explicit risk
model that these schemes deploy and given each scheme’s
resulting portfolio, there is an implied set of expected returns
for each scheme that make the portfolio weights optimal in
Markowitz’s model. Figure 7 presents this idea with a diagram
I call the Markowitz Triangle. The corners of the triangle are
the expected returns, the risk model, and the portfolio weights.
By knowing any two of the three corners, with a few additional
assumptions, you can derive the third. The CAPM provides
a good example of this. Recall that the CAPM implies that the
market portfolio is optimal, so that by combining it with a risk
model, we obtain the market beta of each stock. By assuming
a market risk premium and a risk-free rate of return, we can
derive the expected return of each stock from its market beta.
The same principle can be applied to the equal-risk contribution
model. From this risk model, we can derive the portfolio weights
and the corresponding portfolio covariances. Like the betas
in the CAPM, the portfolio covariances are linearly related to any
set of expected returns that would make the portfolio weights
optimal. Because the equal-risk contribution model’s portfolio
weights are inversely proportional to the portfolio covariances,
they are also inversely related to the implied expected
returns. Figure 8 illustrates how this works under a singlefactor risk model.
Getting Back to Markowitz: Explicit Expected Returns
Volatility-based weighting schemes such as min.-vol. and equalrisk contribution require a complete, explicit risk model and
use of complicated mathematical algorithms to form portfolios.
This raises the question: Why stop there? If we’re explicit about
risk, why not also create explicit models of expected return
and use the Markowitz portfolio construction to form weights—
fully returning to investment theory? I believe this is the next
logical step in the evolution of indexing, which is why I show it
as the final stage on the circular evolution of equity indexes
diagram (Figure 1). It is a complete return to where portfolio construction started by being explicit about both risk and expected
return to form portfolios with clear links to investment theory,
that do not rely solely on empirical anomalies. The difference
between now and 1952 is that since then, there has been extensive research to identify variables that are predictive of equity
returns, allowing us to make more-informed models.
Through our extensive equity research, Morningstar has
capsulized this research into a few stock-specific measures:
Figure 7. The Markowitz Triangle
g Quantitative Valuation—This is the ratio of a stock’s
Quantitative Fair Value Estimate to its most recent close price.
It is similar to the analyst-driven Fair Value Estimate to last
market close price ratio.
Portfolio Weights
Expected Returns
Source: Morningstar
g Quantitative Uncertainty—Any equity valuation involves some
degree of uncertainty. Quantitative Uncertainty describes our
level of uncertainty about the accuracy of our Quantitative Fair
Value Estimate. In this way, it is analogous to the Morningstar
Uncertainty Rating. The lower the Quantitative Uncertainty,
the narrower the potential range of outcomes for that particular company.
Risk Model
g Quantitative Financial Health—This rating reflects the probability that a firm will face financial distress in the near future.
Our calculation uses a predictive model designed to anticipate
when a company may default on its financial obligations.
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Back to Markowitz
Figure 8. Weights and Implied Expected Returns in the Equal-Risk Contribution Model with a Single Risk Factor
300
250
Higher
Weight
200
150
Higher
Expected
Return
50
0
0.2
Systematic Risk
0.4
0.6
0.8
Source: Morningstar
g Quantitative Economic Moat—This rating is analogous to the
Morningstar® Economic Moat™ Rating in that both are meant to
describe the strength of a firm’s competitive position. It is
calculated using an algorithm designed to predict the Economic
Moat Rating a Morningstar analyst would assign to the stock.
Our research shows that these measures are useful as predictors of returns and therefore can be valuable when used to
form models of expected returns. Combining these quantitative
return measures with a chosen risk model in the Markowitz
portfolio construction model could well be the basis for the next
generation of “strategic beta” indexes. K
1.0
1.2
1.4
1.6
0
Unsystematic Risk
100