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THE JOURNAL OF PORTFOLIO MANAGEMENT SUMMER 2015

The Triumph of Mediocrity: A Case Study of Naïve BetaEDWARD QIAN, NICHOLAS ALONSO, AND MARK BARNES

EDWARD QIAN

is chief investment officer in the multi asset division of PanAgora Asset Man-agement in Boston, [email protected]

NICHOLAS ALONSO

is portfolio manager in the multi asset division of PanAgora Asset Man-agement in Boston, [email protected]

MARK BARNES

is director in the multi asset division of PanAgora Asset Management in Boston, [email protected]

Smart beta, like many names for invest-ment products,1 is surely a misnomer. There are two different kinds of smart betas: one based on factors and

the other on diversification. Neither is nec-essarily “smart,” although they may qualify as beta and may even outperform traditional capitalization-weighted indices.

The factor-based smart beta approach builds portfolios with exposure to factors that are expected to deliver excess returns. These factors include the overall market, value, size, quality, momentum, and so on. They probably were “smart” in the 1960s when the capital asset pricing model (CAPM) originated (Markowitz [1952]; Lintner [1965]; Sharpe [1964]) and shortly after, when f inancial economists rediscovered the value and size factors2 and used them to explain empirical failures of a single-factor CAPM (Black et al. [1972]; Fama and MacBeth [1973]; Fama and French [1992]). The momentum factor came a little later ( Jagadeesh and Titman [1993]).

They cannot be that smart now: quan-titative and fundamental equity managers have been using these factors for decades. In addition, the portfolio construction process of factor-based smart beta often reverts to simple weighting schemes without any con-sideration of risk. To many who are familiar with quantitative equity strategies, factor-based smart beta looks like a medieval ver-sion of its old self. It is repackaged to meet

the demand for transparency and low cost by investors who have grown disillusioned with active managers. That is smart.

The second type of smart beta is diver-sification beta. Faced with uncertainty and without any theoretical inclination, investors who do not hold any prior belief regarding the future performance of different invest-ment choices would naively assume that things are going to be equal and diversify their portfolios accordingly. There are mul-tiple dimensions along which different invest-ments could be equal. They include portfolio weight, expected return, risk-adjusted return, and risk contribution. A naïve assumption of equality along these four dimensions has led to four different betas, respectively: equally weighted, minimum variance (Haugen and Heins [1972]; Clarke et al. [2006]), max-imum diversification (Choueifaty and Coi-gnard [2008]), and risk parity (Qian [2005], [2006]). These diversification betas are the focus of this article. We call them naïve beta collectively, because of their equality assump-tions. They are beta in a broad sense, in that they capture equity risk premium with a sys-tematic process.

Could naïve beta be smart after all? A simple answer to this question would be yes, provided they could outperform a capitaliza-tion-weighted index: the old beta. We present an example in this article. The true answer is more subtle and perhaps even philosophical.

JPM-QIAN_Color.indd 19JPM-QIAN_Color.indd 19 7/21/15 10:22:34 AM7/21/15 10:22:34 AM

THE TRIUMPH OF MEDIOCRITY: A CASE STUDY OF NAÏVE BETA SUMMER 2015

In investing, there is often a blurred line between being smart and being naïve. Investing with bold forecasts might appear smart, but it could turn out to be fool-hardy. In contrast, investors who appear ambiguous and invest accordingly might prove correct in the end. In this regard, the capitalization-weighted indices represent the former, as their weights are embedded with bold forecasts, and diversification betas represent the latter. The outperformance of naïve beta marks the triumph of mediocrity in investing.

However, not all naïve betas are created equal. For example, the equally weighted portfolio does not consider risk, while the other three do ( Jurczenkoa et al. [2013]). Among the three that are risk based, minimum-variance and maximum-diversification portfolios opti-mize on risk inputs, while the risk parity portfolio uses risk budgeting without optimization. It is well known that optimized portfolios are highly sensitive to both return and risk estimates; as a result, the optimal port-folio weights tend to be highly concentrated and unin-tuitive with high turnover (Best and Grauer [1991]). Even though minimum-variance and maximum-di-versif ication portfolios partly reduce the severity of this issue with their equality assumptions, the problem largely persists. In contrast, risk parity portfolios tend to be well diversified, because the risk budgeting process is robust to risk inputs.

In this article, we first introduce the four naïve beta portfolios and define their portfolio weights. We then provide an empirical example based on sector portfolios constructed from sector returns of the S&P 500 Index. In the in-sample case, optimized betas such as minimum-variance and maximum-diversification portfolios have lower risk and higher returns, with highly concentrated allocations to defensive sectors at the expense of the growth sectors. To understand this portfolio behavior, we next provide an analytic framework that models sec-tors into two groups of assets, which are homogeneous in correlations within the groups but different across the groups. Analytic solutions of the portfolio weights for these two groups provide a reasonable explanation for the sector concentration of the two optimized beta portfolios. In contrast, the risk parity portfolio is always diversified.

We follow this analysis with a more realistic out-of-sample backtest, which shows that the two optimized beta portfolios suffer from concentration as well as high turnover. Their net returns are much lower than those

of the in-sample portfolios and lower than the return of risk-parity portfolio, which has less turnover and stable sector allocation. The fact that the naïve betas mostly outperformed the S&P 500 in risk-adjusted terms is partly attributable to the sector bias in the index. We next show that over time the S&P 500 has added weights to the two sectors with the highest volatility and poorest risk-adjusted returns.

FOUR NAÏVE PORTFOLIOS

We have N investable assets for a long-only, fully invested portfolio. The commonality among the four naïve betas is that there are no explicit benchmark, no return forecasts, and no explicit factor positioning. Rather, the assumption is that the investments are con-sidered equal in some way. The first and obvious option is an equally weighted (EQ) portfolio. The other three options take risk into account, with risk prescribed by a covariance matrix ∑. The second naïve portfolio assumes that the expected returns are equal. Under this assump-tion, all portfolios, regardless of their asset weights, have the same expected return. All that remains is to minimize portfolio risk, leading to a minimum vari-ance (MV) portfolio. The MV portfolio asset weights are determined by optimization

min ,ww www www subject to 11 2w w1 wN′ Σ +,wwww subject to w1 + +� = (1)

The objective function is the portfolio variance, and the budget constraint is that the sum of weights equals 100%. The constraint can be written as: w′ ⋅ i = 1. The weight vector is wwww ( ), ,1w w, , N

′ and the vector i is a vector of ones: ii ( )

(1, , 1…= ′. We have omitted the

long-only constraint for now, because there is no ana-lytic solution for the optimization with those constraints. The solution of Equation (1) is

ww iww ii1 1

MVMV

1

( )ii ii1λΣ =iiii1 Σ− (2)

The solution equals the inverse of the covariance matrix ∑−1, times the return vector i. The budget con-straint is satisfied with a scale factor λ

MV = i′ ∑−1i.

The third naïve portfolio assumes an equal Sharpe ratio for all investments and therefore their expected excess returns are proportional to their volatilities:

, 1, ,k i Ni ikμ =i ,,i … . The parameter k is the constant



risk-adjusted return. In a vector form, we have μ = kσ. The maximum diversification (MD) portfolio is a port-folio that maximizes the Sharpe ratio

wwww

ww www www ww www wwwmax

μμ σwwwwμμ σσσμσ

= ′ ⋅′ Σ

= ′′ Σ

k (3)

The weight of the MD portfolio can be solved ana-lytically, when free of long-only constraints. We have

wwww1 1

MDMD

1

( )ii 1=λ

Σ σ1 = 1 Σ σ1 (4)

Note the similarity between Equations (2) and (4). Because the covariance matrix ∑ also contains σ, as we shall see shortly, Equation (4) is less sensitive to σ than is Equation (2). The fourth naïve beta is based on the equality of risk contribution. The financial interpreta-tion of risk contribution is the expected return contribu-tion from individual assets (Qian [2005, 2006]). Hence, the risk parity portfolio (RP) has the naïve assumption of equal return contribution.3 Given covariance matrix ∑ and a set of portfolio weights w, the marginal risk contribution is given by ∑w. Then individual asset’s risk contributions to portfolio variance is weight times the marginal contribution, that is,

RRCRRCCC ( )wwww= ⊗wwww (5)

The symbol ⊗ denotes element-by-element multi-plication. The weights of a RP portfolio are determined by a system of equations that equate Equation (5) to a linear multiple of vector i,

wwww iiiiRP RP( )wwwwRP⊗ ( = λ (6)

The scalar λRP

is used to satisfy the budget con-straint w′ ⋅ i = 1.

RISK-MODIFIED WEIGHTS AND CORRELATION MATRIX

The MV and MD weights in Equations (2) and (4) are derived analytically but are not easily interpreted. The RP weights given in Equation (6) in general are not analytically solvable. To gain insight about port-folio weights, especially about their dependence on risk estimates, we will use both empirical and theoretical

examples of special cases. Before we do so, we f irst decompose the covariance matrix into volatilities and correlation matrix

CCCCdiag diag( ) ( )Σ = ⋅) ⋅diag ( (7)

In Equation (7), C is the correlation matrix and diag(σ) is the diagonal matrix, with volatilities as the diagonal elements. The inverse of covariance matrix is

CCCCdiag g1 di 1 di( )1 ( )1Σ =1 ⋅)1 ⋅diagdi (di1 −1 di (8)

The matrix C−1 is the inverse of C and diag(σ−1) is the diagonal matrix with reciprocals of volatilities as the diagonal elements. Using this expression in the weights of MV, MD, and RP portfolios, we arrive at simplified relationships between the products of weights and volatilities and the inverse of correlation matrix. We thus define risk-modified weight as the product of volatility and weight

WW wWW www, 1, , , WWWW σ ⊗W w= σ N1, ,…1i iσWW σ i (9)

Then the MV portfolio weights given in Equation (2) can be rewritten in terms of risk-modified weights

WW CWW CCC1

MVWWWMV

1 1

λσ−1 (10)

The MD portfolio weights given in Equation (4) can be rewritten as

WW CWW C iCC iii1

MDWWWMD

1

λ (11)

And Equation (6) for RP portfolio weights can be rewritten as

WWWW iiiiRPWWWW RP( )RP⊗ =( )CCWCCWWWRPWWWW λ (12)

Without explicitly finding the solutions for the risk-modified weights, we can glean from Equations (10), (11), and (12) how these three portfolios differ in their dependence on volatilities.

Equation (10) shows the risk-modified weight WMV

has an inverse dependence on volatility. In addition, because risk-modif ied weights depend on volatili-ties as well, the MV portfolio weights have an inverse



dependence of volatility of second degree, or volatilities squared. On the other hand, both Equations (11) and (12) no longer contain additional volatility terms. This implies that both the MD and RP portfolio weights have an inverse dependence on volatility of first degree. As a result, MV portfolio weights are more sensitive to volatilities and changes in volatilities. MD and RP port-folio weights behave similarly in terms of dependence on volatilities. The major difference between MD and RP portfolios is the sensitivity to correlations.

EMPIRICAL EXAMPLE: IN-SAMPLE ANALYSIS

We now present an empirical example to study the four naïve beta portfolios. We consider sector allocation portfolios based on the 10 S&P 500 index sectors, for which we have monthly returns from January 1990 to November 2014. We choose this example for several rea-sons. First, sector allocation is a practical issue in equity investing. Second, the number of investment choices is still small enough for us to derive intuition about the portfolio weights. Third, it has not been an easy task for active managers to outperform the S&P 500 over this period, so it would be interesting to see how naïve betas did.

For the time being, we will just derive in-sample portfolio weights using an estimated covariance matrix, based on the monthly returns of the entire period. Exhibit 1 lists the return statistics of the 10 sectors over this period: the returns are cumulative annual-ized returns, the volatilities are the standard deviations of monthly returns multiplied by square root of 12, and

the Sharpe ratios are the cumulative excess returns over cash (return of three-month Treasury bills), divided by volatility. Six sectors (consumer staples, consumer dis-cretionary, energy, health care, industrial, technology) had returns of between 10% and 12%. Another three sectors (financials, materials, and utilities) delivered an annualized return of around 8%. The telecom sector had the lowest annualized return, at less than 6%. The return volatility is divided across defensive and cyclical sectors. Three defensive sectors (consumer staples, health care, and utilities) had the lowest volatilities and two cyclical sectors (f inancials and technology) had the highest volatilities. The Sharpe ratio follows a similar pattern, with defensive sectors having relatively higher Sharpe ratios (except telecom) and cyclical sec-tors having lower Sharpe ratios. Based on these results, we expect all three risk-based naïve betas to have higher weights in the consumer staples, health care, and utili-ties sectors and lower weights in the financials and tech-nology sectors. These tilts are more pronounced for the MV portfolio because of its heightened dependence on volatility.

Exhibit 2 displays the 10 sectors’ correlation matrix, together with the average correlation for each sector (including correlation one). The average correla-tion measures the aggregate diversification opportunity of each sector versus all the other sectors. In this regard, the utility, telecom, energy, and technology sectors have more diversification potential than do the indus-trial, consumer discretionary, financials, and materials sectors.

E X H I B I T 1Annualized Return, Volatility, and Sharpe Ratio of the 10 Sectors Based on Monthly Returns from 01/1990 to 11/2014



With the risk inputs of volatilities and correla-tions, we can find the MV and MD portfolio weights by solving optimizations 1 and 3 numerically under the budget constraint, as well as long-only constraints. To obtain the RP portfolio weights, we use a numerical algorithm that solves a system of equations. With RP portfolios, the weights are generally positive without additional constraints.4

Exhibit 3 shows the sector weights of the MV, MD, and RP portfolios. The MV portfolio is highly concentrated in the two sectors with the lowest vola-tility: consumer staples and utilities. It has small alloca-tions to the energy, health care, telecom, and technology sectors but is virtually unexposed to the remaining four sectors.

Compared with the MV portfolio, the MD port-folio is less concentrated, but only among the six sec-tors that have non-zero weights in the MV portfolio.

The utility sector still has a significant weight in the MD portfolio, and the same four sectors that are largely unrepresented in the MV portfolio are similarly under-represented in the MD portfolio.

The RP portfolio is much more diversified across sectors than are the two optimized portfolios. All sector weights range from 7% to 14%, with the highest weights in low-volatility sectors, such as consumer staples and utility, and the lowest weights in high-volatility sec-tors, such as financials and technology. Even though all three naïve beta portfolios are based on assumptions of equality, the MV and MD portfolios are concentrated across sectors, due to the optimization procedure. In contrast, the RP portfolio is more diversified across sec-tors, due to its risk budgeting process. In some sense, the RP portfolio improves on the EQ portfolio by incor-porating risk inputs. Its solution is also less sensitive to small differences in these inputs.

E X H I B I T 2Pairwise Correlation Matrix of the 10 Sectors and Average Correlation of Each Sector

E X H I B I T 3In-Sample Portfolio Weights of MV, MD, and RP Portfolios



Because it is in-sample, the MV portfolio should have the lowest risk, followed (in sequential order) by the MD portfolio, the RP portfolio, and the EQ portfolio. Because the lower-volatility sectors in general have had higher returns than have the higher-volatility sectors during the sample period,5 we expect the realized port-folio returns in the opposite order of realized portfolio risk. Exhibit 4 displays the four portfolios’ volatilities and total returns, together with that of the S&P 500. The portfolios are rebalanced monthly to their respective fixed weights, resulting in an average turnover of about 30% to 35% (two-way) per year. We have subtracted transaction costs from the returns, which are assumed to be 1% for 100% turnover. The four naïve beta portfolios’ net annualized returns are very close to one another. In contrast, the S&P 500 delivered a materially lower return and higher risk over the same period.

Mainly due to differences in portfolio volatility, the Sharpe ratios are arranged in the inverse order of their respective risk. At 0.57, the MV portfolio has the highest Sharpe ratio, followed by the MD portfolio at 0.53, the RP portfolio at 0.50, and the EQ portfolio at 0.47. The S&P 500 has the lowest Sharpe ratio at 0.41. Thus the in-sample performances of the MV, MD, and RP portfolios are much superior to those of the S&P 500.

SENSITIVITY OF PORTFOLIO WEIGHTS TO CORRELATIONS

This section is devoted to understanding portfolio weights’ dependence on asset correlations. Because the inverse of the correlation matrix is generally not ana-lytically tractable, we rely on a special case in which the assets are sorted into two groups, with identical correlation within each group and different correla-tions across the groups. This structure can be used as a simple model for sectors, in terms of their defensive and cyclical styles. The analytic results that follow this simple model show that the MV and MD portfolios are highly sensitive to the difference between the group correlations, while the RP portfolio is not, thus illus-trating both previous results and out-of-sample results that appear later in the article. Because the analysis is more technical, readers who care most about empirical research could be forgiven for skipping ahead to the next section.

We start with an even simpler case in which all correlations are identical, which has been analyzed many times in previous research (Maillard et al. [2010]). We denote the special correlation matrix as follows:

E X H I B I T 4The In-Sample Risk/Return Scatter Plot of the Four Naïve Beta Portfolios together with the S&P 500 Index



CCCC

1

1

1

=

ρ ρρ ρ1

ρ ρ

⎛

⎝

⎜⎛⎛

⎜⎜⎜

⎜⎜⎜

⎜⎜⎜⎜⎝⎝⎜⎜

⎞

⎠

⎟⎞⎞

⎟⎟⎟

⎟⎟⎟

⎟⎟⎟⎟⎠⎠⎟⎟� � � �

�

(13)

The inverse of C is given by

CCCC1

1

1

1

1

� � � ��

[ ]1( )1- 1N

( )2N

( )2N

( )2N

= [1) 1

×

ρ −ρ −… ρ

−ρ ρ −� ρ

−ρ −ρ ρ

⎛

⎝

⎜⎛⎛

⎜⎜⎜

⎜⎜⎜

⎜⎜⎜

⎜⎝⎝⎜⎜

⎞

⎠

⎟⎞⎞

⎟⎟⎟

⎟⎟⎟

⎟⎟⎟

⎟⎠⎠⎟⎟

−

(14)

Substituting Equation (14) into Equation (10), we have

WWWW

1

1MVWWW

1 2 3

1 2 1

M

M M1 2

�

�

�

( )2N

( )2N

∝

ρσ

− ρσ

− ρσ

− −�ρ

σ

ρσ

− ρσ

− ρσ

− ρσ

⎛

⎝

⎜⎛⎛

⎜⎜⎜

⎜⎜⎜

⎜⎜⎜

⎜⎜⎜

⎜⎝⎝⎜⎜

⎞

⎠

⎟⎞⎞

⎟⎟⎟

⎟⎟⎟

⎟⎟⎟

⎟⎟⎟

⎟⎠⎠⎟⎟

−

(15)

In Equation (15), we have absorbed the constant scalar, because we are interested in the relative magnitude of the portfolio weights. The final weights will be scaled to sum up to 1. When asset volatilities are different, the terms in solution (15) would be different. The smaller the volatility, the larger the relative magnitude of the asset weight. Substituting Equation (14) into Equation (11) leads to a simple solution for the MD portfolio

WW iWW iiiMDWWW ( )1 1 1�=iiii ′ (16)

In other words, the relative magnitude of terms in W

MD is the same for all assets. This is not entirely sur-

prising, as the correlation matrix is completely symmetric for all assets. Because W

MD is the product of volatility

and portfolio weight, we conclude that the MD portfolio is inversely proportional to volatility. The same is true for the RP portfolio weights, because of the symmetry

embedded in the correlation matrix. Therefore, in this special case, the MD and RP portfolios are identical.

We now consider a correlation matrix that is par-titioned into two groups. The correlations within each group are identical, and the correlations across the groups are different. We denote this correlation matrix by

CCCCCC CCC CCC

CC CCC CCC11 12

21 22

=⎛

⎝⎜⎛⎛

⎜⎝⎝⎜⎜

⎞

⎠⎟⎞⎞

⎟⎠⎠⎟⎟ (17)

and

CCCC

CCCC

CC CCC CCC

1

1

1

1

1

1

1 1 11 1 1

1 1 1

11

1 1

1 1

1 1

22

2 2

2 2

2 2

12 21 12

1 1

2 2

1 2

N N1

N N2

N N1

� � � ��

� � � ��

��

� � � ��

=

ρ ρ1 �

ρ ρ11 1 �

ρ ρ1

⎛

⎝

⎜⎛⎛

⎜⎜⎜

⎜⎜⎜

⎜⎜⎜

⎜⎝⎝⎜⎜

⎞

⎠

⎟⎞⎞

⎟⎟⎟

⎟⎟⎟

⎟⎟⎟

⎟⎠⎠⎟⎟

=

ρ ρ2 �

ρ ρ12 1 �

ρ ρ2

⎛

⎝

⎜⎛⎛

⎜⎜⎜

⎜⎜⎜

⎜⎜⎜

⎜⎝⎝⎜⎜

⎞

⎠

⎟⎞⎞

⎟⎟⎟

⎟⎟⎟

⎟⎟⎟

⎟⎠⎠⎟⎟

′ = ρ

⎛

⎝

⎜⎛⎛

⎜⎜⎜

⎜⎜⎜

⎜⎝⎝⎜⎜

⎞

⎠

⎟⎞⎞

⎟⎟⎟

⎟⎟⎟

⎟⎠⎠⎟⎟

(18)

Correlation matrices (17) and (18) describe a case with two groups of assets. Each group is homogeneous with the same correlation, but the correlations could be different for the different group when ρ

1 ≠ ρ

2. In addi-

tion, the correlations between assets from the different group are also constant, denoted by ρ

12, which could

be different from both ρ1 and ρ

2. However, when all

three correlations are the same, i.e., ρ1 = ρ

2 = ρ

12, the

two groups merge into one, which is described by cor-relation matrix 13.

This correlation structure represents a major improvement over matrix (13). In addition to the two-group structure, the number of assets in each group is also f lexible. We denote the number of assets in the first group as N

1 and the number of assets in the second group

as N2. This structure is particularly suitable for the sector



portfolios we introduced earlier. Fundamentally, the sec-tors can be broadly grouped into cyclical and defensive sectors. The cyclical sectors tend to have higher cor-relations among themselves, due to common exposure to economic growth. On the other hand, the defensive sectors tend to exhibit lower correlations between its members. The dispersion of correlations within each group is often smaller than the dispersion across the two groups.

on asset volatilities. However, this situation resolves itself when asset volatilities within each group are also equal. With this assumption, the risk-modified weights for the MV portfolio are also of the form of Equation (19), and studying the same ratio of W

1 to W

2 suffices.

The detailed derivation of this ratio as a function of correlations and the numbers of assets for the two groups are given in an appendix. (See http://www.iijpm.com.) We summarize the results in Equation (20).

Our objective is to see how the weights of the MV, MD, and RP portfolios react to this correlation struc-ture. A related question is how portfolio turnover would be affected by this correlation structure. The benefits of studying portfolio weights under correlation matrix (17) is that solutions for all portfolio weights are still analyti-cally tractable when there is no long-only constraint.

A crucial observation is that assets within indi-vidual groups are identical in terms of their own pair-wise correlation and correlations with the other group. Hence, we expect the solution of risk-modified weights W to have identical elements within each individual group for at least the MD and RP portfolios. In other words, the risk-modified weights can also be partitioned into two groups

WWWWWWWW

WWWWWWWW ,, WWW

1WWWW

2WWWW 1WWWWWW1

1

2

2

2

W1

W1

W2

W2

=⎛

⎝⎜⎛⎛

⎜⎝⎝⎜⎜

⎞

⎠⎟⎞⎞

⎟⎠⎠⎟⎟

⎛

⎝

⎜⎛⎛

⎜⎜⎝⎝⎜⎜

⎞

⎠

⎟⎞⎞

⎟⎟⎠⎠⎟⎟

⎛

⎝

⎜⎛⎛

⎜⎜⎝⎝⎜⎜

⎞

⎠

⎟⎞⎞

⎟⎟⎟

⎟⎠⎠⎟⎟�WWWW, 2WWWW�

⎟ = ⎜⎜⎜⎜

(19)

The vector W1 is of length N

1 and the vector W

2 is

of length N2. When ρ

1 = ρ

2 = ρ

12, the ratio of W

1 to W

2 is

1. When the correlations are different, we are interested in the ratio of W

1 to W

2. Equation (19) is not true for the MV

portfolio, because the risk-modified weights still depend

1

1

1

1

4 1 1

2 1

1

2 MV

2 2 12

1 1 12

1

2 MD

2 2 12

1 1 12

1

2 RP

12 12

2

1 2

1

W1

W2

N

N1

W1

W2

N

N

W1

W2

11

( )12N 12 1 ( )/1 2/ 2

( )/1 2 ( )11N 11 1

( )12N 12 1

( )11N 11 1

( )1 2N N1 ( )1 2N N1 ( )11N 11 ( )12 1N

( )11 1N

⎛⎝⎜⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

= 22 1 //1 / ρ/1 / + (N1⎡⎣⎡⎡ ⎤⎦⎤⎤ − ρ1N

⎛⎝⎜⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

= 22 ρ

11 ρ

⎛⎝⎜⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

=ρ)2N (N1+ ρ⎡⎣⎡⎡ ⎤⎦⎤⎤ +1+ 4 11⎡⎡⎣⎡⎡ ⎤⎦⎤⎤ 2N ρ⎡⎣⎡⎡ ⎤⎦⎤⎤

+ ( 1N ρ⎡⎣⎡⎡ ⎤⎦⎤⎤ (20)

Several special cases can help to gain some under-standing of the relationships. First, when the volatilities are the same for the two groups, that is, σ

1/σ

2 = 1, the W

ratios of MV and MD portfolios are the same. Second, when all three correlations are identical, the ratio is one for both the MD and RP portfolios. For the MV port-folio, the ratio still depends on σ

1/σ

2, as discussed earlier.

Third, the ratio of W1 to W

2 is in general positively

correlated with ρ2 and negatively correlated with ρ

1.

It makes intuitive sense that high correlation within a group would lower its risk-modified weights.

The major differences between the three solutions of Equation (20) lie in their functional forms of the algebraic expressions, which determine the sensitivity of the W ratio to various risk inputs. First, the MV and MD ratios are quotients of linear functions of correla-tions and number of assets. In contrast, the RP ratio is the square root of quotients of similar linear functions.6 Over a wide range, a square root function dampens the sensitivity of the W ratio.

Second, for the RP solution, its numerator and denominator are never zero.7 This means the risk-mod-ified weights of the RP portfolio are always positive, a desirable feature for a long-only diversified portfolio. This is not true for either the MV or the MD portfolios. In fact, it is easily violated. For instance, in the MD portfolio, we have (W

1)

MD = 0 when



112 2

2

2Nρ =12 ρ +2

− ρ (21)

When this condition occurs, the MD portfolio would have zero group 1 weight. This is true even when ρ

12,

the correlation across the two groups is only slightly larger than ρ

2, the cor-

relation within group 2. Possibly for the first time, these solutions mathematically demonstrate the sensitivity of weights for both the MD and MV portfolios to the differences in correlations, which lead to concentration in one group and zero weight in the other group.

The analytic solution of Equation (20) provides fertile ground to explore the theoretical and empirical relationships between multiple risk inputs and port-folio weights. An exhaustive examination is beyond the scope of this article. We present a close examination of the solu-tion for cases that to some degree resemble the earlier sector example. For group 1, we select sectors that have zero weights in the MD and MV portfolios (Exhibit 3). There are four (N

1 = 4) sectors in this group: consumer discretionary,

financials, industrials, and materials. The remaining six sectors (N

2 = 6): consumer staples, energy, health care,

technology, telecom, and utilities make up group 2. From Exhibit 2, we calculate that the average correlation within group 1 is 0.78 (ρ

1 = 0.78), the average correlation within

group 2 is 0.40 (ρ2 = 0.40), and the average correlation

across the two groups is 0.51 (ρ12

= 0.51). In addition, the volatility of the sectors in group 1 is also higher than the volatility of the sectors in group 2. On average, we have σ

1 = 19.3% and σ

2 = 17.8%. In group 2, consumer staples

and utilities have the lowest volatilities.When we substitute these parameters into Equation

(20), we have W1/W

2 = 0.81 for the RP portfolio. How-

ever, the ratio is negative for the MV and MD portfolios, because W

1 has turned negative while W

2 remains posi-

tive. In other words, if the MV and MD portfolios are unconstrained, the weights for the group 1 sectors—the high volatility and highly correlated group—would have been negative and both portfolios would be long group 2 and short group 1. The long-only constraints make

the weights of group 1 sectors roughly all zero in the MV and MD portfolios. Thus, the solution given by Equation (20) seems consistent with the fact that the MV and MD portfolios are concentrated in defensive sectors while the RP portfolio is diversified across both groups.

We can also use Equation (20) to study the change of W ratios as we vary the correlations. As an illustration, we choose the cross correlation ρ

12 as the variable while

fixing other parameters. We have, as in our example, N

1 = 4, N

2 = 6, ρ

1 = 0.75, ρ

2 = 0.4, and σ

1/σ

2 = 1.08.

Exhibit 5 displays the ratio W1/W

2 for the three port-

folios as the cross correlation ρ12

varies from 0.4 to 0.5. The dramatic difference in the three curves’ heights and slopes highlights the differences in three portfolios.

We make several remarks about the graph. First, the ratios are all less than one, mainly due to the fact that group 1 has a higher correlation than group 2. Second, the RP portfolio is not sensitive to the cross-group cor-relation. The ratio starts at around 0.85 and decreases slightly as ρ

12 increases. In contrast, the ratios of the MV

and MD portfolios behave rather differently. First, both ratios are much lower. The ratio of the MV portfolio is

E X H I B I T 5The Ratio W1/W2 of the MV, MD, and RP Portfolios as a Function of Cross Correlation ρ12 between the Two Groups



the lowest, as the volatility ratio also disfavors group 1. Second, as ρ

12 increases, the weight

of group 1 in both the MV and MD portfolios quickly drops to zero.

EMPIRICAL EXAMPLE—OUT-OF-SAMPLE ANALYSIS

We now examine the performance of the four naïve beta portfolios out of sample. With the insights gained from the previous analysis, we focus on the portfolio turnover and concentration of the three risk-based approaches. It is obvious that the equally weighted (EQ) portfolio will not change. To obtain the out-of-sample perfor-mance of the MV, MD, and RP portfolios, for which an ex ante covariance matrix is required, we carry out a backtest where the covariance matrix is estimated using past monthly returns with exponentially decayed weights based on a five-year half-life. The initial window consists of 24 months from 1990 to 1991. The live period is from January 1992 to November 2014. For each month in the live period, we re-estimate the covariance matrix using available returns, derive the new weights of the MV, MD, and RP portfolios, and rebalance the three portfolios to the new targeted weights. For transaction costs, we calculate turnover based on the difference of the new weights and return-adjusted old weights.

We first examine the sector weights of the three portfolios out of sample. Exhibit 6 plots the history of MV sector weights from 1992 to 2014. Several obser-vations are noteworthy. First, similar to the in-sample results, the weights are always concentrated in one or two sectors. Initially, the portfolio is dominated by the utility sector alone (over 50%). In the last few years, it is concentrated in the utility and consumer staple sec-tors (over 50%). Only during the period from 2003 to 2008 is the portfolio less concentrated, with some moderate weights in the health care, energy, and con-sumer discretionary sectors. Second, the financial and industrial sectors have almost no weight in the portfolio throughout the whole period. Third, the weights show large variations, indicating high turnover. The annual two-way turnover is about 77%.

Exhibit 7 plots the history of sector weights for the MD portfolio. They are less concentrated than the MV

weights. Instead of being dominated by a few low-vol-atility sectors, the MD portfolio tends to be dominated by the five or six sectors with low correlations. This is similar to the in-sample results. The difference is that, in the out-of-sample period, those dominating sectors show some rotation. For example, the consumer staples sector only shows up after 2000 and the materials sector disappears after 2009. Five sectors are always present in the MD portfolio. They are the utility, telecom, tech-nology, health care, and energy sectors. The changes in portfolio weight are slightly smoother than in the MV portfolio, but the turnover remains high, at 64%.

Exhibit 8 plots the history of sector weights for the RP portfolio. We make several remarks. First, all the sectors have non-zero weights. Second, it is generally the case that the cyclical sectors have lower weight than do the defensive sectors. Third, the weights are much smoother than the weights of the MV and MD portfolios, but they have been adaptive over time. For example, the weights of the consumer staples and health care sectors have been increasing, while the energy and technology sector weights have been declining. The turnover of the RP portfolio has been quite low, at 32%.

E X H I B I T 6Time Series of Sector Weights of MV Portfolios



With portfolio weights and subsequent realized return, we calculate out-of-sample port-folio returns for the MV, MD, and RP portfolios, net of transaction costs. We selected two cost levels, one at 1% and the other at 0.5% for 100% turnover to show the effect of portfolio turnover. Exhibit 9 displays the risk/return plots of the backtest. There are several striking similarities and differences between the out-of-sample and in-sample results shown in Exhibit 4. In terms of portfolio risk, the relative order of the out-of-sample volatilities is the same as that of the in-sample volatilities, with the MV portfolio having the lowest volatility and the S&P 500 having the highest volatility. The MD portfolio’s volatility is the second lowest, followed by the volatility of the RP portfolio and then the volatility of the EQ portfolio. Since these four naïve betas use risk inputs in varying degrees and the outcome of volatilities follows the same order we expected, it shows that the out-of-sample risk estimate is reasonably accurate.

However, as far as the four naïve betas are concerned, the order of out-of-sample portfolio returns is completely opposite that of the in-sample returns. Exhibit 9 shows that under both cost assumptions, the EQ portfolio has the highest return, followed by the RP portfolio, the MD portfolio, and the MV portfolio. Compared with the S&P 500 index, the EQ and RP portfolios have higher returns under both cost assumptions. The MV portfolio always underperforms the index. For the MD portfolio, it outperforms the index if the transaction cost is 0.5% and under-performs it if the cost is 1%. The effect of transac-tion cost assumptions shows that the MV and MD portfolios are subject to higher transaction costs because of their higher turnover.

The Sharpe ratios of the out-of-sample port-folio returns are lower than the in-sample ratios, with RP portfolio having the highest Sharpe ratio and the S&P 500 having the lowest. When the transaction cost is 0.5%, the descending order of Sharpe ratio is RP (0.53), MD (0.52), EQ (0.51), MV (0.47), and the S&P 500 (0.43). When the transaction cost is 1%, the effect of turnover fur-ther reduces the net returns of MV and MD port-folios. The Sharpe ratio in descending order is

E X H I B I T 8Time Series of Sector Weights of RP Portfolios

E X H I B I T 7Time Series of Sector Weights of MD Portfolios



RP (0.52), EQ (0.50), MD (0.49), MV (0.43), and the S&P 500 (0.43).

The comparison between the in-sample and out-of-sample results indicates that the in-sample analysis tends to favor optimized portfolios (MV and MD), as they make the full use of information in hindsight. But when there is much more uncertainty, the out-of-sample optimization process generates high turnover, due to its sensitivity to changes in the risk inputs. As we have seen in this example, the higher turnover is not necessarily rewarded with higher returns. In contrast, the risk-budgeting process in the RP portfolio is much more robust to changes in risk inputs and its portfolio weights maintain true diversif ication throughout the period, delivering high return on both an absolute as well as a risk-adjusted basis.

WHAT IS WRONG WITH THE S&P 500 INDEX?

If the naïve betas can outperform the S&P 500 either with risk-adjusted or absolute returns, is there anything wrong with the index itself ? This is often a controversial question. We do not intend to provide a

complete answer here. We shall focus our analysis on the index’s sector allocation and compare it with the sector alloca-tions of the naïve beta portfolios. This comparison partly explains why the index has higher risk and a lower risk-adjusted return.

Exhibit 10 shows the history of the S&P 500 index’s sector weights. As the colored bands expand and contract, the sector weights change due to price f luc-tuations of sectors and changes in index constituents. Change due to price f luc-tuation might be thought of as passive, as capitalization-weighted indices do not rebalance. Changes in constituents affect sector allocation, and they should be con-sidered active decisions.

We aim to separate these two effects. But we first note a few stylized facts evi-dent in Exhibit 10. First, comparing the beginning and ending weights, we note the sector allocation of the index was more balanced in 1990 than it was in 2014, where the utility, telecom, and materials sectors have negligible weights. Over the

entire period, the two sectors that have gained the most weight are technology and financials: 7.7% for finan-cials and 13.7% for technology. Health care is the only other sector with a weight increase, one of 5.8%. The remaining seven sectors declined in weight.

Second, two sectors were severely affected by investment bubbles and the financial crisis. The first is the technology sector, which experienced a dramatic expansion in the late 1990s, followed by the subsequent contraction during the tech bubble burst. The other is the financial sector, which experienced a gradual expan-sion before the credit crisis, followed by a collapse during the crisis. However, both two sectors maintained their substantial weights after those events.

From these observations, it is easy to see how the weight in the financial and technology sectors played a role in the volatility of the S&P 500. As seen in Exhibit 1, these two sectors have the highest volatilities among all 10 sectors. The MV portfolio has little to no weight in either sector, the MD portfolio has no weight in the financial sector, and the RP portfolio has lower weights in both sectors, relative to the other sectors. The EQ

E X H I B I T 9The Out-of-Sample Risk/Return Scatter Plot of the Four Naïve Beta Together with the S&P 500 Index, for Two Levels of Transaction Cost (1% and 0.5% for 100% Turnover)



portfolio, of course, has 10% weight in each sector, which is still less than the index weights of those sectors. Because of its concentration in volatile sectors, the S&P 500 had the highest risk of these five portfolios. If we are to believe CAPM’s assumption that high risks would lead to high returns, then the index’s concentration in the two volatile sectors might be justified. However, the actual returns and risks of the 10 sectors have refused to obey the relationship proposed in CAPM. Often the opposite is true.

There are only two possible reasons for the changes in the index’s sector weights. One possibility is differ-ences in sector returns that cause drifts in portfolio weights, and the other is changes in index constituents over time. To isolate the latter, we calculate the monthly changes in sector weights, net of price drift, by sub-tracting the return-adjusted weight derived from the prior month weight and prior month return from the current sector weight. Without changes in the index, the difference between the two sets of weights would be zero.

Exhibit 11 shows the cumulative changes in the net sector weights from February 1990 to November 2014.

Only the financials and technology sectors show net gains, each of close to 12% over the entire period. The health care sector had a small gain of about 2% until 2009, but it has since lost all the gain. Furthermore, the graph reveals when the gains in the two sectors occurred. For the tech-nology sector, the gain occurred throughout the 1990s and accelerated from 1998 to 2000, after which the increase has been gradual. For the financial sector, the gain was from two periods, one from 1990 to 1998 and the other from 2008 to 2014. The tech bubble burst effectively put a stop to the technology sector’s rapid gain, but sur-prisingly, the global financial crisis reignited the gain in the financial sector. Among the eight sec-tors with declining net weights, the losses in the health care, utility, telecom, and materials sectors have been rather modest. However, four sectors—consumer discretionary, consumer staples, energy, and industrials—have each lost roughly 5%.8 We have also examined changes in the number of stocks in each sector over time and found that the financial and technology sectors have gained the most, often at the wrong times, while two other sectors, health care and energy, also saw increases in their counts.

It is natural to study the effect of changes in index constituents on portfolio risk and return. It is not a simple task to disentangle the stock- and sector-specific impacts. We use a sector-constrained approach to cal-culate the effect of changes in sector weights. At the beginning of each calendar year, we use the S&P 500 index’s sector weights as a starting portfolio. We then follow a buy-and-hold sector portfolio based on the S&P index’s subsequent sector returns. In other words, when the index has a change in specific stocks, we make the change, but we also rebalance our portfolio to the targeted sector weights determined by the buy-and-hold approach from the beginning of the period. The return difference between this hypothetical portfolio and the S&P 500 can be attributed entirely to sector allocation.

Exhibit 12 displays the annualized return dif-ference between the S&P 500 and the buy-and-hold portfolios starting in a given calendar year and ending in 2014. In all cases, the S&P 500 trails the buy-and-hold portfolios that started out with identical sector weights to the index itself. We conclude that all the

E X H I B I T 1 0Time Series of Sector Weights of the S&P 500 Index



sector shifts induced by changes in index con-stituents have been detrimental to the index’s return. The damages caused by changes before 2000, which can be characterized as significant increases in the financials and technology sec-tors, were especially severe, detracting between 40 and 50 basis points per year. Changes since 2001, which are less dramatic, still resulted in negative alpha on the index’s return, though of smaller magnitude.

Investors can be easily led to believe that capitalization-weighted indices like the S&P 500 are passive investments. This is only partially true when one considers the narrow definition of the weighting scheme for already selected index members. As we have illustrated, the selection of index members is a highly active and inf luential process, even if the selection cri-terion are systematic. By any standard, raising the sector weights of the financials and tech-nology sectors by more than 10% each while cutting other sectors’ weights is an extremely active decision. Nevertheless, investors who use the S&P 500 as either a passive investment or benchmark are willing participants in this active process.

Of course, one might object to this line of reasoning by claiming that one of the index’s objectives, or the definition of passivity, is to ref lect the state of the U.S. economy, and thus the equity market. From this perspective, it may seem sensible that an equity index must adapt to the growing importance of the f inancial and technology sectors, as other sectors have become relatively less important due to either outsourcing or industries’ natural declines. The counter arguments to this fuzzy notion are two-fold. First, the timing of these changes is still highly subjective. Second and perhaps more importantly, it is unclear why investors should allocate their investments according to a non-diversified economy, which might lead to low risk-adjusted returns.

Contrary to popular perception, the theo-retical arguments, such as the capital asset pricing model, the efficient market hypothesis, and the fundamental consideration that one’s investment portfolio should ref lect the broad market and

E X H I B I T 1 1Cumulative Increments of Sector Weights due to Changes in Index Constituents

E X H I B I T 1 2The Return Difference between the Annualized Returns of the S&P 500 Index and the Annualized Return of a Buy-and-Hold Portfolio, Based on the Initial S&P 500 Sector Weights



economy, actually make the S&P 500 a smart index rather than a passive one. Specifically, the significant shifts in its sector allocation are akin to making a smart bet on different sectors’ relative future performance. As we have shown in this article, those bets have been det-rimental to both the index’s return and risk. Once again, being smart might not be a winning strategy.

CONCLUSION

In this article, we study diversification-based sector portfolios within the S&P 500 universe. We have found that the naïve betas—the EQ, MV, MD, and RP port-folios with an equality assumption in weights, expected returns, expected Sharpe ratios, and risk contribution, respectively—all have outperformed the traditional beta of the S&P 500 index, in terms of risk-adjusted returns. These results still largely hold after transaction costs in out-of-sample tests.

The case of the S&P 500 is not an isolated instance. The naïve beta portfolios also tend to outperform their capitalization-weighted counterparts in the non-U.S., global, and emerging-market equity universes. The fundamental reason is that the capitalization-weighted indices have strong biases in their sector, country, and stock allocations, due to two interconnected reasons. One is the short-term market momentum and the other is the change in index constituents, which is a highly active process on the part of index providers. Our study of the S&P 500 shows the index has added additional weights in excess of 10% to both the technology and financial sectors over the period from 1990 to 2014. In addition, these shifts often occurred at inopportune times. From this perspective, one cannot help but arrive at the conclusion that the capitalization-weighted indices are only passive from a mechanical perspective and are quite active from an investment perspective. In contrast, naïve beta based on an assumption of equality for some aspect of all investments is a truly passive investment. Its performance advantage, in a sense, is a triumph of mediocrity.

Our analysis also reveals the differences between three risk-based naïve beta portfolios. The MV and MD portfolios with optimization are highly sensitive to risk inputs such as volatility and correlation, resulting in con-centrated portfolio holdings and high portfolio turnover. In addition, we have also found that these portfolios’

backtest performance changes signif icantly when we use a different risk model. On the other hand, the RP portfolio incorporating a risk-budgeting approach is robust to the choice of risk model while consistently delivering a diversif ied portfolio with low portfolio turnover.

ENDNOTES

The authors thank Bryan Belton and Bryan Hoffman for helpful comments and suggestions.

1Other names are securities for stocks, high-yield bonds for junk bonds, private equity for leveraged buyouts, balanced funds for 60/40 portfolios, and hedge funds for not-hedged investments and high fees.

2Value investing existed long before financial econo-mists got interested in the topic. For example, Graham and Dodd published Security Analysis in 1934.

3In a more general sense, RP portfolios can have equal risk contribution along multiple dimensions.

4This is true when pairwise correlations are positive.5This is part of low-volatility/low-beta anomaly.6This becomes apparent after a simple algebraic

manipulation.7This is true as long as the correlation matrix is posi-

tive definite.8We have also used price return instead of total return

to adjust the weights. The results are similar.

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To order reprints of this article, please contact Dewey Palmieri at [email protected] or 212-224-3675.


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Boston, MA 02210

The opinions expressed in this article represent the current, good faith views of the author(s) at the time of publication, are provided for limited purposes, are not definitive investment advice, and should not be relied on as such. The information presented in this article has been developed internally and/or obtained from sources believed to be reliable; however, PanAgora does not guarantee the accuracy, adequacy or completeness of such information. Predictions, opinions, and other information contained in this article are subject to change continually and without notice of any kind and may no longer be true after the date indicated. The views expressed represent the current, good faith views of the author(s) at the time of publication. Any forward-looking statements speak only as of the date they are made, and PanAgora assumes no duty to and does not undertake to update forward-looking statements. Forward-looking statements are subject to numerous assumptions, risks and uncertainties, which change over time. Actual results could differ materially from those anticipated in forward-looking statements. HYPOTHETICAL PERFORMANCE RESULTS HAVE MANY INHERENT LIMITATIONS, SOME OF WHICH ARE DESCRIBED BELOW. NO REPRESENTATION IS BEING MADE THAT ANY ACCOUNT WILL OR IS LIKELY TO ACHIEVE PROFITS OR LOSSES SIMILAR TO THOSE SHOWN. IN FACT, THERE ARE FREQUENTLY SHARP DIFFERENCES BETWEEN HYPOTHETICAL PERFORMANCE RESULTS AND THE ACTUAL RESULTS SUBSEQUENTLY ACHIEVED BY ANY PARTICULAR INVESTMENT PROGRAM. ONE OF THE LIMITATIONS OF HYPOTHETICAL PERFORMANCE RESULTS IS THAT THEY ARE GENERALLY PREPARED WITH THE BENEFIT OF HINDSIGHT. IN ADDITION, HYPOTHETICAL TRADING DOES NOT INVOLVE FINANCIAL RISK, AND NO HYPOTHETICAL TRADING RECORD CAN COMPLETELY ACCOUNT FOR THE IMPACT OF FINANCIAL RISK IN ACTUAL TRADING. FOR EXAMPLE, THE ABILITY TO WITHSTAND LOSSES OR TO ADHERE TO A PARTICULAR INVESTMENT PROGRAM IN SPITE OF TRADING LOSSES ARE MATERIAL POINTS WHICH CAN ALSO ADVERSELY AFFECT ACTUAL TRADING RESULTS. THERE ARE NUMEROUS OTHER FACTORS RELATED TO THE MARKETS IN GENERAL OR TO THE IMPLEMENTATION OF ANY SPECIFIC INVESTMENT PROGRAM WHICH CANNOT BE FULLY ACCOUNTED FOR IN THE PREPARATION OF HYPOTHETICAL PERFORMANCE RESULTS AND ALL OF WHICH CAN ADVERSELY AFFECT ACTUAL TRADING RESULTS.

The S&P 500 Index is an unmanaged list of common stocks that is frequently used as a general measure of U.S. stock market performance.

RISK CONSIDERATIONSInternational investing involves certain risks, such as currency fluctuations, economic instability, and political developments. Additional risks may be associated with emerging market securities,including illiquidity and volatility. Active currency management, like any other investment strategy, involves risk, including market risk and event risk, and the risk of loss of principal amount invested.

Derivative instruments may at times be illiquid, subject to wide swings in prices, difficult to value accurately and subject to default by the issuer. Strategies that use leverage extensively to gain exposure to various markets may not be suitable for all investors. Any use of leverage exposes the strategy to risk of loss. In some cases the risk may be substantial.

This material is directed exclusively at investment professionals. Any investments to which this material relates are available only to or will be engaged in only with investment professionals. Past performance is no guarantee of future results. PanAgora is exempt from the requirement to hold an Australian financial services license under the Corporations Act 2001 in respect of the financial services. PanAgora is regulated by the SEC under US laws, which differ from Australian laws.

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