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THE JOURNAL OF PORTFOLIO MANAGEMENT 57FALL 2014

The Drivers of Predicted BetaJACKSON WANG AND JOSE MENCHERO

JACKSON WANG

is vice president of MSCI in San Francisco, [email protected]

JOSE MENCHERO

is managing director of MSCI in Berkeley, [email protected]

Beta measures the sensitivity of an asset’s return to the broad market return. Assets with beta greater than one tend to outperform in

up markets, while underperforming in down markets. Assets with beta between zero and one are also positively correlated with the overall market, but tend to outperform in down markets, while lagging in up markets. Some assets have negative beta, meaning that they tend to move in the opposite direction of the broad market.

An asset’s beta is certainly one of its most important attributes. According to the Capital Asset Pricing Model (CAPM) described by Sharpe [1964], the expected return of any asset is completely determined by the product of two terms: a) the asset’s beta, relative to the broad market, and b) the expected return of the market itself. The former represents the asset’s market exposure, and the latter represents the market portfolio’s return premium.

The CAPM rests on the assump-tion that the market portfolio is eff icient, meaning that no other portfolio has higher expected risk-adjusted performance. Notions of market efficiency, of course, have not dis-suaded active managers from their quest to beat the market. Even for active managers, however, beta remains a critical attribute, as it represents a major driver of risk and return. Many active managers explicitly construct

portfolios to have beta close to one, so that their active return is not directly linked to the ups and downs of the overall market. Other managers, by contrast, intentionally tilt their portfolios toward either low beta or high beta, in order to express a tactical view of the market’s direction.

One common approach to estimating beta is known as historical beta, which is based on a time-series regression of asset returns against market returns. In theory, the market portfolio consists of all investable assets. In practice, a benchmark—typically a broad equity index—serves as a market proxy.

Historical beta suffers from several known shortcomings. For instance, historical betas tend to be quite noisy. This may lead not only to large estimation errors, but also to undesirable biases. Vasicek [1973] described a Bayesian technique designed to mitigate such biases. Another challenge in estimating his-torical betas pertains to asynchronous stock returns, which may cause the estimated betas to be spuriously low. Scholes and Williams [1977] developed an econometric technique for estimating betas from asynchronous returns.

Rosenberg [1985] pioneered the use of multi-factor risk models to estimate stocks’ predicted betas. Predicted betas have two main advantages over historical betas. First, the factor structure helps to f ilter noise in

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58 THE DRIVERS OF PREDICTED BETA FALL 2014

the beta forecasts. Second, as we show in this article, predicted betas can be attributed to their underlying factors in the model. Historical betas, by contrast, simply represent the output of a time-series regression, and as such cannot be further decomposed.

This article examines and analyzes the drivers of predicted beta in two distinct ways. The first concerns how to attribute the predicted beta of an individual stock or portfolio to the underlying factors. The second is to investigate which factors drive the observed differ-ences in predicted betas across a universe of stocks. In this article, we consider both aspects, showing that the concept of factor beta forms the common link between the two.

We begin this article with a brief overview of factor models and a discussion of factor portfolios. Next, we introduce the notion of factor beta and show how it can be used to attribute the predicted beta of a stock or portfolio. We also examine the time evolution of factor betas for a select group of style and industry fac-tors. Finally, we study the cross-sectional dispersion of predicted betas, showing how it can be decomposed into contributions from individual factors. This lets us identify which factors drive the differences in the pre-dicted betas of stocks and how these vary across time. We provide mathematical details in the Appendix.

FACTOR MODELS

Factor models decompose stock returns into a factor contribution and a remainder term, known as the idiosyncratic, or stock-specific, return. As described in Appendix, the return contribution from an individual factor is the product of the stock exposure to the factor and the factor return. The idiosyncratic return represents the component of stock return that the factors cannot collectively explain.

In this article, we use the Barra U.S. Equity Model (USE4) to conduct our analysis. USE4 contains 73 factors in total: one country factor, 60 industry fac-tors, and 12 style factors. Every stock has unit exposure to the country factor, which represents the intercept term in the cross-sectional regression. Stocks have unit exposure to their industry of membership, and zero exposure to other industry factors.1 The style factor exposures are expressed as standardized z-scores, with mean zero and standard deviation equal to one. Factor returns are estimated by cross-sectional regression and

represent the returns of pure factor portfolios. Such portfolios are powerful constructs, as they capture the unique effect of the individual factor, net of all other factors.

As discussed by Menchero [2010], the country factor portfolio is 100% net long and essentially repre-sents the cap-weighted U.S. market portfolio. Industry factor portfolios are dollar neutral, going 100% long the industry in question, 100% short the U.S. market, and neutral to all style factors. In other words, industry factor portfolios capture the industry’s performance, net of the market and net of all style factors. Style factor portfolios have unit exposure to the factor in question, and zero exposure to all other factors. This implies that they are dollar neutral within each industry.

FACTOR BETA

Factor beta is the predicted beta of a pure factor portfolio, relative to a selected benchmark. Equivalently, as shown in the Appendix, factor beta is the predicted correlation of the pure factor portfolio with the bench-mark, multiplied by the volatility ratio between the pure factor portfolio and the benchmark. Consequently, fac-tors with high volatility and strong correlation with the benchmark will have large factor betas.

As shown in the Appendix, stock betas can be decomposed into a factor contribution and a stock-spe-cif ic component, termed the idiosyncratic beta. The factor contribution is simply the product of the stock exposure and the factor beta, summed over all factors. The idiosyncratic beta represents the beta of the stock-specif ic return, relative to the benchmark. For most stocks, the idiosyncratic beta is extremely small and may often be neglected. However, stocks with high specif ic risk and very large weight in the index may have a significant idiosyncratic beta, as shown in the following example.

Example 1

Apple has experienced a long period of spectac-ular performance since 2003. In September 2012, the stock price peaked at more than $700 per share, a nearly 100-fold increase since January 2003, making Apple the largest publicly traded company in the world. Following the peak, however, its price fell dramatically, dipping below $400 per share by April 2013.



We compute Apple’s predicted beta on May 31, 2013, using USE4. For the benchmark, we select the MSCI USA Investable Market Index (IMI), a broad index representing the U.S. equity market. The pre-dicted beta was 1.34, well above the cap-weighted mean beta of 1.

In Exhibit 1, we attribute Apple’s predicted beta to the USE4 factors. The country factor had a factor beta of 0.99. Because the country factor portfolio essen-tially represents the U.S. market, we expect a factor beta very close to 1. Furthermore, because each stock has unit exposure to the country factor, it follows that the country factor contributes 0.99 to the predicted beta of every stock.

The USE4 beta factor is constructed from the historical betas of stocks, which are then transformed into standardized z-scores. Interestingly, Apple had a negative exposure to this factor on the analysis date, indicating that its historical beta was below average. The beta factor portfolio goes long high-beta stocks and takes short positions in low-beta stocks. This factor portfolio is typically quite volatile and has strong posi-tive correlation with the U.S. market. As a result, it tends to have a large factor beta (0.33 on the analysis date). The negative exposure to this factor reduced Apple’s predicted beta by 0.05.

Apple’s relatively high predicted beta can be under-stood by its exposures to other factors. In particular,

Apple had a large positive exposure (1.38) to the residual volatility factor. This factor portfolio, which goes long high-volatility stocks and shorts low-volatility stocks, tends to be quite volatile and significantly correlated with the U.S. market. As a result, residual volatility typi-cally has a fairly large factor beta (0.10 in this case). The large positive exposure to residual volatility, coupled with the large factor beta, contributed 0.14 to Apple’s predicted beta.

Apple also had a large negative exposure to the momentum factor, ref lecting its poor performance fol-lowing the peak in September 2012. Since the financial crisis of 2008, the momentum factor has been negatively correlated with the U.S. market, resulting in a negative factor beta. The large negative exposure to momentum, combined with the negative factor beta, contributed another 0.07 to Apple’s predicted beta.

Another signif icant contributor to Apple’s pre-dicted beta was the computer electronics industry, which had a factor beta of 0.08. This implies that computer electronics stocks, net of all style factors, had above-average predicted betas. Since Apple had unit exposure to computer electronics, this factor contributed an addi-tional 0.08 to Apple’s predicted beta.

Finally, specific risk contributed 0.10 to Apple’s predicted beta. This is not representative of typical stocks, which usually have much smaller idiosyncratic betas. Apple’s unusually large idiosyncratic beta is due to two effects: a) Apple’s extremely large weight in the benchmark, and b) Apple’s high specific risk.

TIME EVOLUTION OF FACTOR BETAS

We have seen that factor beta depends on the pre-dicted correlation of the factor portfolio relative to the benchmark, and the volatility ratio of the factor portfolio versus the benchmark. Consequently, changes in pre-dicted volatilities and/or correlations cause factor betas to vary across time.

In Exhibit 2, we plot factor beta versus time for three important USE4 style factors: residual volatility, momentum, and beta. The analysis period is from June 1995 to June 2013. The factor beta of the residual vola-tility factor was remarkably stable during this period, generally falling within the narrow range of 0.08 to 0.12. The volatility of the residual volatility factor was typically 20% of the benchmark volatility; equivalently, the volatility ratio was 0.20. The predicted correlation

E X H I B I T 1Attribution of Predicted Beta for Apple, Relative to the MSCI USA IMI. Analysis Date Is May 31, 2013



with the U.S. market was roughly 0.50. The product of these two, therefore, led to the observed average factor beta of approximately 0.10.

Unlike the residual volatility factor, the momentum factor beta has changed significantly over time. In the late 1990s, it was strongly positive, but it became highly negative following the 2008 financial crisis. This sign change was caused by time-varying correlations with the U.S. market. These correlations averaged about 0.25 in the late 1990s, but hit a low of −0.33 in 2009. The vola-tility ratio for momentum f luctuated in the range of 0.20 to 0.40, with an average value of approximately 0.30.

Of the three factors in Exhibit 2, the beta factor consistently had the highest factor beta. This result is intuitive, as beta is typically the most volatile style factor, and also tends to have the highest correlation with the U.S. market. The most prominent feature in this plot is the sharp rise in factor beta immediately following the collapse of the Internet bubble. In early 2000, the factor beta stood below 0.20, but more than doubled roughly one year later.

In order to gain further insight into the time evo-lution of factor betas, it is useful to separately plot the predicted correlation and the volatility ratio. In Exhibit 3, we plot these two quantities for the beta factor. Over the period from 1995 to 1998, the correlation ranged from 0.5 to 0.6, then increased steadily to a level approaching 0.90 in recent years. Following the col-lapse of the Internet bubble, the volatility ratio increased

from 0.25 to 0.64. The volatility ratio has since declined somewhat, although this effect was largely offset by the rising correlation, leading to a fairly stable factor beta since 2004. It is clear from Exhibit 3 that the sharp rise in factor beta over the period from 2000 to 2001 was due to an increase in the volatility ratio, not a sudden change in predicted correlation.

In Exhibit 4, we plot the time-series evolution of factor betas for a select group of USE4 industry fac-tors. Semiconductors, an industry in the information technology sector, exhibited a persistently positive factor beta. The factor beta was particularly large in the 1990s, but has declined since then to more modest levels.

The factor beta of the airlines industry has changed across time, sometimes abruptly so. For example, in Sep-tember 2001, the factor beta rose sharply, from zero to approximately 0.30. The events of that month caused air-line stocks to crash at precisely the same time that the broad market plummeted. In October 2008, it was the opposite story; the sharp drop in the price of fuel caused airline stocks to outperform, just as the market was crashing.

The electric utilities factor has displayed a consis-tently negative factor beta. This behavior is typical of industries in defensive sectors. As of May 31, 2013, the factor beta of electric utilities was approximately −0.14.

E X H I B I T 3Correlation and Volatility Ratio for the USE4 Beta Factor

The correlation represents the predicted correlation between the beta factor portfolio and the benchmark (MSCI USA IMI). The volatility ratio rep-resents the ratio of the beta factor volatility versus the benchmark volatility. The product of the two curves is the factor beta.

E X H I B I T 2Time Evolution of Factor Betas for Select USE4 Style Factors



This implies that a fully invested, style-neutral portfolio of electric utility stocks would have a predicted beta of approximately 0.86. In reality, electric utility stocks tend to have low historical beta, leading to negative expo-sure to the beta factor. This typically also makes a large negative contribution to the predicted beta of a utility portfolio, as shown in the following example.

Example 2

In Exhibit 5, we attribute the predicted beta of the MSCI USA IMI Utilities index to the USE4 factors on May 31, 2013. The predicted beta was 0.57, well below the average beta of one for the U.S. market. The country factor contributed 0.99 to the portfolio’s predicted beta. The three utility industries collectively contributed −0.12 to the predicted beta. The largest negative con-tribution (−0.33) came from the beta factor, owing to the large negative exposure to this factor. This example shows that the utilities portfolio’s low predicted beta was explained primarily by the fact that utility stocks tend to have low historical beta, and less by any intrinsic characteristic of a utility stock.

CROSS-SECTIONAL DISPERSION OF PREDICTED BETA

It is instructive to examine the distribution of pre-dicted betas over time. Since the cap-weighted mean of the distribution is always exactly one, the most salient feature of the distribution is its dispersion, defined as the cross-sectional standard deviation of the predicted betas. The average dispersion over the sample period is approximately 0.4. Assuming that the predicted betas are normally distributed, this implies that roughly two-thirds of the predicted betas typically fall within the range of 0.6 to 1.4.

The dispersion of predicted betas, however, may vary considerably over time. In Exhibit 6, we plot the cap-weighted dispersion versus time. The most striking feature is the Internet bubble and its aftermath. In late 1998, the dispersion was less than 0.3, which rose to nearly 0.7 in early 2001.

The dispersion of predicted betas can be decom-posed into contributions from individual factors. To decompose the dispersion of predicted betas, we apply the same technique that Menchero and Morozov [2011] used to decompose the cross-sectional volatility of stock returns. As shown in the Appendix, the dispersion in predicted betas can be decomposed into contributions from individual factors, as well as a stock-specific con-tribution. The contribution from an individual factor is given by the product of three terms: a) the factor beta, b) the dispersion of factor exposures, and c) the cross-sectional correlation between the predicted betas and the factor exposures.

E X H I B I T 4Time Evolution of Factor Betas for Select USE4 Industries

E X H I B I T 5Attribution of the Predicted Beta of MSCI USA IMI Utilities, Relative to the MSCI USA IMI on Analysis Date May 31, 2013



USE4 style factors are standardized to have an equally weighted, cross-sectional standard deviation of exactly one. The cap-weighted dispersion may deviate somewhat from this value. For industry factors, by con-trast, Menchero and Morozov [2011] showed that the dispersion is approximately equal to the square root of the weight in the industry. Since industry weights are typically quite low, industry dispersion tends to be much smaller than the corresponding dispersion for style fac-tors. As a result, industry factors may explain a relatively small fraction of the observed cross-sectional variation in predicted betas, despite having relatively large factor betas.

Example 3

In Exhibit 7, we report the top 10 contributors to dispersion of predicted betas on the analysis date: May 31, 2013. The total dispersion was 0.3588. The largest contribution came from the beta factor (0.2716). This result can be understood as the combined effect of large factor beta, high dispersion, and strong cross-sectional correlation with predicted betas. The next six top con-tributors are all style factors. The momentum factor had a negative cross-sectional correlation, indicating that high-momentum stocks tended to have low predicted beta. Pharmaceuticals were the largest industry contributor.

In Exhibit 8, we consider which factors have been the most important drivers of cross-sectional variation in predicted beta over time. We plot the contribution to dispersion in predicted beta from four sources: a) the beta factor, b) all other style factors, excluding beta, c) all industry factors, and d) stock-specific contributions. From 1995 to 2001, we see that the contribution from the beta factor was comparable to the contributions from other styles and other industries. Beginning in 2001, however, we see that the beta factor explained between 70% and 80% percent of the total variation in predicted betas. Over the last 10 years, industries and other styles have become comparatively less important in explaining predicted beta than they were in the 1990s. Note that

E X H I B I T 7Attribution of Dispersion in Predicted Betas for MSCI USA IMI on Analysis Date May 31, 2013

E X H I B I T 6Cross-Sectional Dispersion of MSCI USA IMI Predicted Betas vs. Time

E X H I B I T 8Percent Contribution to Dispersion of Predicted Beta for Select USE4 Factor Groupings



the stock-specific contribution to dispersion was virtu-ally negligible.

SUMMARY

A stock’s predicted beta is one of its most impor-tant attributes. In this article, we developed the notion of factor beta to study which factors represent the key drivers of predicted beta. We showed how to attribute the predicted beta of a stock or portfolio to individual factors. We also analyzed which factors made the largest contribution to the observed cross-sectional variation in predicted beta.

A P P E N D I X

Factor Betas. The return of stock n can be written in terms of a factor model

r X f un nk kff

kn∑X fk ff (A-1)

where Xnk is the exposure of stock n to factor k, f

k is the return

of the factor, and un is the specific return. The specific returns

are assumed to be mutually uncorrelated and also uncorre-lated with the factors.

We use the benchmark as an operational definition of the market. The beta of stock n relative to the benchmark is given by the usual definition,

cov2nB

( )r R,n B,R,β =n σ

(A-2)

where RB is benchmark return and 2

Bσ is the variance of the benchmark. Substituting Equation (A-1) into Equation (A-2), we find

12 X cov covnB

nkk

∑ ( )f Rk B,ff R ( ),u Rn B, Rβ =n σ⎡⎣⎢⎡⎡⎣⎣

⎤⎦⎥⎤⎤⎦⎦

(A-3)

The benchmark return can be written via the factor model as

R w uB

lmB

mm∑ ∑X fl

BlffX flX B ff (A-4)

where XlB is the benchmark exposure to factor l, and wm

B is the benchmark weight in stock m. Now, substituting Equa-tion (A-4) into Equation (A-3), and using the fact that the factor returns are uncorrelated with the specific returns, we find

1cov2 wn

B lkmB

m∑ ∑X X Fnk l

BklFF∑ ( ),u u,n m,β =n σ

⎡⎣⎢⎡⎡⎣⎣

⎤⎦⎥⎤⎤⎦⎦

(A-5)

where Fkl is the covariance between factor k and factor l, taken

from the factor covariance matrix. Since the specific returns are mutually uncorrelated, the last summation is immedi-ately determined as wn

BnΔ , where Δ

n is the specific variance

of stock n.We define the factor beta as the beta of the factor with

respect to the benchmark,

cov2kB

� ( )f R,k Bf Rf ,β =k σ

(A-6)

Substituting Equation (A-4) into Equation (A-6), and using the result that stock-specific returns are uncorrelated with factors, we find

∑β =

σ⎛⎝⎛⎛⎝⎝

⎞⎠⎟⎞⎞⎠⎠

1cov

⎛⎝⎜⎛⎛⎝⎝2 f X∑, fk

Bk l∑f Xf ∑, B

lffl

� (A-7)

Equation (A-7) can be rewritten as

∑β =

σ1

2 X FkB

lB

klFFl

� (A-8)

It is also useful to rewrite Equation (A-6) as

k

k

B

� ( )f R,kff Bβ =k ρσσ

(A-9)

This expresses factor beta as a product of the predicted correlation and the volatility ratio. We further define the idiosyncratic beta of a stock as the beta of the stock-specific return with respect to the benchmark,

2

wn

nB

n

B

φ =n

Δσ

(A-10)

Substituting Equations (A-8) and (A-10) into Equation (A-5), the stock beta can now be expressed as

Xn nk k

kn

�∑β =n β +k φ (A-11)

The portfolio beta is given by the weighted average of stock betas,

wP nwP

nn

∑β =P β (A-12)



Substituting Equation (A-11) into Equation (A-12), we find

wP

knP

nn

�∑ ∑XkX Pk

�β =P k φ (A-13)

The first term is the factor contribution to beta; the second term represents the specific contribution. Equation (A-13) represents an attribution of portfolio beta.

Decomposing Dispersion of Predicted Betas

Menchero and Morozov [2011] showed how to decom-pose the cross-sectional volatility of stock returns into con-tributions from individual factors. Starting with Equation (A-1), they show

r fkff

k∑ ( )Xk ( ),X rk ( ),u r,(( ) ( )uσ (r σ ρ( )XkX + σ ρ (A-14)

We apply their methodology to decompose the cross-sectional dispersion of predicted betas. Using the benchmark as the analysis universe and the observation that the cap-weighted beta of the benchmark is exactly one, the cross-sectional variance of predicted betas is

2 2

nB

n∑ ( )1n( )σ2 ( wn

B= ∑ ( (A-15)

where wnB is the benchmark weight in stock n. Comparing

Equation (A-1) to Equation (A-11), we make the identifica-tion that stock returns are replaced by predicted betas, factor returns are replaced by factor betas, and idiosyncratic returns are replaced by idiosyncratic betas. This identification allows us to immediately translate Equation (A-14) into the cor-responding equation for the cross-sectional decomposition of predicted betas,

k

k

�∑ ( )k( )X ( ),Xk( ) ( ),( )σ ( = β∑ ρσ ( )XkX +) σ ( ρ( (A-16)

where σ(Xk) is the cross-sectional dispersion of factor

exposures,

2wnwB

n∑ ( )X Xnk k( )k( )Xσ (Xk (A-17)

and ρ(Xk, β) is the cross-sectional correlation between the

factor exposures and the predicted betas,

wnB

n∑ ( )Xnk k( )k

( )1n

( )k( )X ( )ρ( ,Xk =)Xk (

σ σ( )Xk

(A-18)

The cross-sectional variance of the idiosyncratic beta is

2 2

nB

n∑ ( )n( )σ2 ( wn

B=∑ ( (A-19)

and ρ(φ, β) is the cross-sectional correlation between the idiosyncratic betas and the predicted betas,

∑ ( )( )( ) ( )( )ρ( =) )(σ ( σ (

wnB )()(

n (A-20)

Because the idiosyncratic betas are themselves clustered near zero, the dispersion σ(φ) is typically very small.

ENDNOTE

1Some stocks in USE4 have exposure to multiple indus-tries, although these exposures sum to one by construction.

REFERENCES

Menchero, J. “The Characteristics of Factor Portfolios.” Journal of Performance Measurement, Vol. 15, No. 1 (Fall 2010), pp. 52-62.

Menchero, J., and A. Morozov. “Decomposing Global Equity Cross-Sectional Volatility.” Financial Analysts Journal, Sep-tember/October 2011, pp. 58-68.

Rosenberg, B. “Prediction of Common Stock Betas.” The Journal of Portfolio Management, Vol. 11, No. 2 (Winter 1985), pp. 5-14.

Scholes, M., and J. Williams. “Estimating Betas from Non-Synchronous Data.” Journal of Financial Economics, Vol. 5, No. 3 (December 1977), pp. 309-327.

Sharpe, W. “Capital Asset Prices: A Theory of Market Equi-librium under Conditions of Risk.” Journal of Finance, Vol. 19, No. 3 (1964), pp. 425-442.

Vasicek, O. “A Note on Using Cross-Sectional Information in Bayesian Estimation of Security Betas.” Journal of Finance, Vol. 28, No. 5 (December 1973), pp. 1233-1239.

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


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