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Essays in Equity Portfolio Management

DISSERTATION

of the University of St. Gallen,

School of Management,

Economics, Law, Social Sciences

and International Affairs

to obtain the title of

Doctor of Philosophy in Economics and Finance

submitted by

Ulrich Carl

from

Germany

Approved on the application of

Prof. Paul Soderlind, PhD.

and

Prof. Dr. Axel Kind

Dissertation no. 4572

Print Difo-Druck GmbH, Bamberg, 2016

The University of St. Gallen, School of Management, Economics, Law, Social Sciences and

International Affairs hereby consents to the printing of the present dissertation, without

hereby expressing any opinion on the views herein expressed.

St. Gallen, May 19, 2016

The President:

Prof. Dr. Thomas Bieger

III

Contents

Acknowledgments VII

Introduction IX

Abstract XIII

Zusammenfassung XV

Equity Factor Predictability 1

Understanding Rebalancing and Portfolio Reconstitution 49

The Power of Equity Factor Diversification 97

V

Acknowledgments

First of all, I would like to express my gratitude to my supervisor Prof. Paul Soderlind,

PhD, whose guidance and support was invaluable for the completion of this thesis. His

concise feedback was very helpful to guide my research and to learn what matters for a

good academic paper. He encouraged me to pursue the topics that I am interested in and

that are applicable in the asset management industry. Moreover, I would like to thank

Prof. Dr. Axel Kind for his efforts and support as the co-supervisor of my thesis.

The stimulating and supportive environment at Finreon Ltd. was also invaluable for

the success of this thesis. Foremost, my gratitude goes to Dr. Ralf Seiz for creating this

environment, giving me the flexibility to pursue this PhD and for many fruitful discussions.

Moreover, I am grateful to my colleagues, particularly Dr. Julius Agnesens and Lukas

Plachel, whose input was decisive to shape my research.

Additionally, I am indebted in particular to Christian Finke, Marcial Messmer and

Felix Dietrich for the discussions and feedback to my draft papers. A big thank-you goes

also to all the innumerable friends that supported me along the way.

A very special thank-you goes to Feifei, who stood by my side during this whole

endeavour. Finally, my deepest gratitude goes to my parents for their loving support

throughout all these years.

St.Gallen, January 28th, 2016

Ulrich Carl

VII

Introduction

The unifying element of this thesis is the topic of factors driving equity returns. ”What

drives equity returns?” is one of the most fundamental questions in financial economics

and asset pricing. The starting point for this question is the capital asset pricing model,

in which the general equity market is the only factor driving equity returns. However, the

assumptions of this model are too strict such that they poorly reflect the real world. Soon

inter-temporal expansions and multi-factor expansions followed. Empirical factors with

limited theoretical background have become most common. Today, research in equity

factors is exploding, particularly when it comes to discovering new potential factors.

However, even the oldest empirical factors are not fully understood yet. Even though

there is no shortage of research, academia is still struggling to join all the pieces of the

puzzle, and many pieces of the puzzle are likely still missing. The goal of this thesis is

to undercover some missing pieces and to contribute to the understanding of some of the

oldest and most recognized factor premia and their interaction. The main focus is on the

market excess return, the size factor, the value factor and the momentum factor. The first

and third chapter also include the low beta factor and the quality factor, while the second

chapter appends the short-term reversal factor and the long-term reversal factor. This

thesis illuminates three perspectives on these factors. Chapter 1 covers the perspective

of predictability, chapter 2 discusses the links between factors and portfolio adjustments,

while chapter 3 analyses the diversification properties of these equity factors.

The first chapter focuses on the prediction of the returns to equity factors. Return

predictability in the sense of predicting market excess returns is a large and controversial

field of research within financial economics. Weak signals hidden underneath a large

amount of noise make this field highly vulnerable to estimation errors and data mining.

Nevertheless, there have been considerable advances over the last decade, particularly in

terms of methods. Transferring these insights in order to predict the returns of equity

premia besides the classical market excess return is the major contribution of this chapter.

I find predictability for the low beta factor and moderate predictability for the size factor,

while the results for other factors are mixed. Moreover, predicted returns for the market

excess return, the size factor, the value factor and the momentum factor are to a large

extent driven by a common component. This common component is partly related to

the business cycle: the market excess return, the size factor and the value factor are

anti-cyclical, while the momentum factor is pro-cyclical. However, the state of the macro-

economy can only explain a small part of this common component.

The cyclical nature of equity factors also plays a crucial role in the second chapter. This

chapter discusses rebalancing and portfolio reconstitution and how they are related

to equity factors. Rebalancing and portfolio reconstitution are both fundamental activities

IX

for any security portfolio. Obtaining a better understanding of the characteristics of

these activities is therefore highly relevant. Rebalancing is the process of adjusting the

portfolio weights back to the target weights. It is a systematically anti-cyclical process

of selling stocks that performed well, while buying stocks that performed poorly since

the last rebalancing. Portfolio reconstitution is the process of determining, which stocks

are included in or excluded from the portfolio. For a portfolio that selects the eligible

stocks based on the market capitalization rank, this process is inherently pro-cyclical at

the inclusion threshold. Thus, both equity factors as well as rebalancing and portfolio

reconstitution show a distinct cyclicality. This cyclicality shows up as distinct patterns

in relative factor exposures when varying the rebalancing and the portfolio reconstitution

frequency. These patterns are symmetric for rebalancing and portfolio reconstitution.

Short term reversal drives the returns at high frequencies, momentum at intermediate

frequencies, while value and long term reversal stand out at low frequencies. The variation

in returns at different frequencies can be linked to macro-economic variables, in particular

the cross-sectional volatility.

The last chapter looks at equity factors from a diversification perspective. In

highly correlated risk-on risk-off markets, investors are desperately looking for investment

opportunities that show low correlations to traditional assets classes. One potential solu-

tion is factor based investing, which is currently very popular as an investment strategy.

To get a better understanding of the potential benefits, this chapter analyses the diversifi-

cation properties of country equity factors across six equity factors and twenty developed

markets. I find substantial diversification benefits along the country dimension as well

as the factor dimension. The cross-country correlations within each of the factors are

moderate with the exception of the market excess return and they are slightly elevated

for the momentum factor. However, the gains through international diversification of

the single factors are diminishing over time for the majority of factors. The cross-factor

correlations are very low. Moreover, there is no indication of increasing correlations be-

tween the six factors from 1991 to 2015 even though factor based strategies massively

increased in popularity over time. The portfolio construction exercise demonstrates the

diversification gains in a portfolio context. International diversification reduces portfolio

volatilities and increases Sharpe ratios for each factor compared to single country invest-

ing. The same holds for local factor diversification, which reduces portfolio volatilities

and increases Sharpe ratios for each country compared to single factor investing.

X

Abstract

All chapters of this thesis cover different aspects of equity factors. The overall goal is

to contribute to the understanding of several of the oldest and most recognized equity

factors. The first chapter focuses on the predictability of equity factors. I find some

predictability, particularly in the low beta factor and this predictability is driven by a

common component across factors. This common component is partly related to the

business cycle. The second chapter discusses portfolio adjustments and how they are re-

lated to equity factors. Varying the rebalancing and the portfolio reconstitution frequency

leads to distinct patterns in factor exposures. The patterns are symmetric for rebalancing

and portfolio reconstitution and they are due to the cyclical nature of these portfolio ad-

justments. Macro-economic variables contribute to explaining the return variation of the

portfolio adjustments. Finally, the third chapter analyses the diversification properties of

country equity factors across six factors and twenty countries. There is strong evidence

for substantial diversification benefits along the country as well as the factor dimension.

XIII

Zusammenfassung

Alle Kapitel dieser Doktorarbeit beleuchten unterschiedliche Aspekte des Themas Aktien-

faktoren. Das zentrale Ziel ist es, zum Verstandnis von einigen der altesten und anerkan-

ntesten Aktienfaktoren beizutragen. Das erste Kapitel geht uber die Vorhersagbarkeit

von Aktienfaktorrenditen. Aktienfaktorrenditen lassen sich teilweise schatzen, insbeson-

dere der Low Beta Faktor. Diese Vorhersagbarkeit wird uber Faktoren hinweg durch eine

gemeinsame Komponente getrieben. Diese gemeinsame Komponente steht teilweise mit

dem Konjunkturzyklus in Beziehung. Das zweite Kapitel diskutiert Portfolioanpassun-

gen und inwiefern diese zu Aktienfaktoren in Beziehung stehen. Die Veranderung der

Rebalancing- und der Portfoliorekonstitutionsfrequenz fuhrt zu ausgepragten Mustern

in den Faktorladungen. Diese Muster sind zwischen Rebalancing und Portfoliorekon-

stitution symmetrisch und bedingt durch die zyklische Natur der Portfolioanpassungen.

Makrookonomische Variablen konnen dazu beitragen, die Renditevariabilitat der Portfo-

lioanpassungen zu erklaren. Das dritte Kapitel analysiert die Diversifikationseigenschaften

von Landeraktienfaktoren uber sechs Faktoren und zwanzig Lander hinweg. Sowohl uber

die Lander- als auch die Faktordimension lassen sich ausgepragte Diversifikationsvorteile

erkennen.

XV

Equity Factor Predictability

Ulrich Carl∗

Draft: January 28th, 2016

Abstract

This article comprehensively reviews the predictability of six equity factors. These fac-

tors are the market excess return, size, value, momentum, low beta and quality. I find

predictability for the low beta factor and moderate predictability for the size factor. The

results for other factors are mixed. Moreover, predicted returns for the market, size, value

and momentum factors are to a large extent driven by a common component. This com-

mon component is partly related to the business cycle: the market, size and value factors

are anti-cyclical, while the momentum factor is pro-cyclical. However, business cycles can

only explain a small part of this common component.

JEL CODES: C53 G11 G12 G17

Key words: return predictability, forecasting, model uncertainty, factor model, forecast

combination, principal components

∗Finreon Ltd., Oberer Graben 3, 9000 St.Gallen, Switzerland and University of St.Gallen, School ofEconomics and Political Science, Bodanstrasse 8, 9000 St.Gallen, Switzerland. The views expressed inthis paper are my own and do not necessarily reflect those of Finreon Ltd. and of the University ofSt.Gallen. I would like to thank Paul Soderlind, Francesco Audrino, Michael Lechner, Ralf Seiz, JuliusAgnesens, Lukas Plachel, Sebastian Buchler, Marcial Messmer and Christian Finke and the seminarparticipants at the University of St.Gallen for helpful comments.contact: [email protected], +41 76 210 03 12

Ulrich Carl Equity Factor Predictability

1 Introduction

Risk-related and behavioural factors drive equity returns. Understanding these drivers

and their interaction is at the core of modern finance. While the market return is the main

driver, the empirical literature uncovered a wide range of further equity factors. These

advances in financial research have spun over to financial industry practice in the recent

years making factor based investing one of the most prominent topics in quantitative asset

management. While predicting the market return receives wide attention, other factors

driving equity returns are mostly neglected.

The goal of this paper is to close this gap and systematically analyse predictability of

the market, size, value, momentum, low beta and quality factors in a unified framework.

This approach allows uncovering the relations between the predictability of those factors

and seeing how they relate to the state of the economy. Using a broad range of economic

and financial data sets as well as a wide range of methods ensures the robustness of the

presented results and limits the risk of data mining.

I find strong and consistent levels of predictability for the low beta factor. This

predictability is mostly due to lagged returns. Moreover, there is some predictability for

the size factor, which is related to the business cycle. The results for the market, value

and momentum factors are mixed, while there is no predictability for the quality factor.

The predicted returns for the market, size, value and momentum factors interact

closely and one common component is able to capture a large part of the variation in

the predicted returns of these four factors. In a visual analysis, the market, size and

value factors show a distinct anti-cyclical behaviour, while the momentum factor shows a

distinct pro-cyclical behaviour. Using regression analysis, I can attribute a moderate, but

statistically significant, part of the predicted returns to business cycles. The factor struc-

ture of the predicted returns is stronger for the financial data set than for the economic

data set.

In most cases, simpler methods such as forecast combination (Rapach & Strauss, 2010)

or the most basic specification of the principal component regression perform best. The set

of explanatory variables is more likely to have a factor structure than a sparse structure.

The structure of the remainder of this paper is as follows. Section 2 gives a short

overview of the literature of equity market and equity factor predictability. Section 3

first discusses the econometric forecasting methods used. These are forecast combination,

principal component regressions, targeting procedures, partial least squares and the three-

pass regression filter as well as the lasso method. Then, I review forecast evaluation

methods. Section 4 presents the data used for the empirical analysis. Section 5 exhibits

the empirical results. Finally, section 6 concludes.

2


2 Literature

Equity return predictability has been controversial for decades as model uncertainty and

instability limit the scope of predictability. Moreover, the predictive ability is strongly

dependent on the phase of the business cycle (Henkel, Martin, & Nardari, 2011) such

that the time periods chosen for evaluation can have a strong impact on the results.

Traditionally, the research on equity return predictability focused on in-sample analysis

and a limited amount of explanatory variables. However, as the comprehensive study

of Welch and Goyal (2007) points out, linear regressions based on individual predictors

perform poorly out-of-sample, mostly underperforming a historical average estimate.

To reduce estimation errors, Campbell and Thompson (2008) build upon the findings

of Welch and Goyal (2007) and show that results improve once weak restrictions on

the signs of the coefficients and the signs of the return forecasts are imposed. Out-of-

sample predictive power, however, remains small. Ferreira and Santa-Clara (2011) pursue

a similar idea by forecasting stock market returns by a sum-of-the-parts-strategy, which

estimates the dividend-price ratio, the earnings growth and the price-earnings ratio growth

separately. This method also manages to significantly outperform the historical mean or

other predictive regressions mainly due to a large reduction in the estimation error.

Recently, two shrinkage type approaches gained a lot of attention in academia: fore-

cast combination and factor based approaches.1 Rapach, Strauss, and Zhou (2010) show

that forecast combination incorporates information from many economic variables and

reduces the forecast volatility substantially. Their combination forecasts manage to sig-

nificantly and consistently outperform the historical average estimate. From a theoretical

perspective, Huang and Lee (2010) show that combining forecasts can be superior to com-

bining information in case of high parameter uncertainty and explanatory variables with

similarly low predictive content.

While standard regression analysis focuses on a relatively small number of often pres-

elected predictors, factor based approaches, popularized by Stock and Watson (2002), for

forecasting macroeconomic time series, use a large cross-section of data and statistically

derive the main components using principal component analysis. Using these components

in linear regressions, the curse of dimensionality can be overcome. Ludvigson and Ng

(2007), Ludvigson and Ng (2009), Bai (2010) and Neely, Rapach, Tu, and Zhou (2014)

apply these approaches to predict equity and bond risk premia and find that these fore-

casts can beat historical average or AR(1) forecasts respectively.

However, in principal component analysis, factors are selected in order to explain the

most important common components between the predictors. These are not necessar-

1Rapach and Zhou (2013) give a very good overview of current forecasting techniques.

3


ily the components relevant for prediction. Bai and Ng (2008) and Cakmakli and van

Dijk (2010) use targeting procedures for economic time series and for equity returns and

volatility as predicted variables respectively. The predictors are pre-selected based on t-

statistics in single linear regressions (hard thresholding) or the order of selection by lasso

or lars (soft thresholding).

Another approach constructs the factors by directly incorporating information of the

predicted variables. One such method is partial least squares by Wold (1975). Kelly and

Pruitt (2014) propose a very similar approach, the three-pass regression filter. They claim

that this approach is well suited to predict the equity risk premium.

While the major focus of return predictability is on the market risk premium, fore-

casting other equity factors such as the size, value and momentum factors is the domain

of practitioner oriented research. The focus of forecasting other equity factors is on rather

short time intervals and economically significant in-sample returns. Studies on return pre-

dictability, in contrast, mostly focus on statistically significant out-of-sample R2. Levis

and Liodakis (1999) find that there is much scope for size timing, and timing strate-

gies manage to outperform a buy-and-hold strategy of small capitalization stocks after

transaction costs. For value timing, however, the prospects are limited and the timing

strategies fail to add value after taking transaction costs into account. Copeland and

Copeland (1999) find that increases in the volatility index (VIX) lead to outperformance

of large capitalization and value stocks. Asness, Friedman, Krail, and Liew (2000) show

that using value spreads and growth spreads as predictors, the value factor can be timed

successfully. More in the lines of classical return predictability, Kong, Rapach, Strauss,

and Zhou (2011) predict the returns to Fama French size-value portfolios. These portfolios

have a strong exposure to the size and the value factors. However, they are not market

neutral and the market risk premium is still the dominating driver of these returns.

Another strand of literature analyses regimes in equity factors. Especially momentum

is known for its infrequent but strong drawdowns (”momentum crashes”). Daniel and

Moskowitz (2013) find that those crashes are forecastable as they occur when the market

rebounds after a market crash. Daniel, Jagannathan, and Kim (2012) show that a hidden

Markov model can capture those drawdowns and can significantly improve the Sharpe ra-

tio of momentum returns. For the value factor, Guirguis, Theodore, and Suen (2012) show

that the earnings yield dispersion performs well in predicting the value-growth spread,

but its sign depends on the market regime.

Hence, while some papers deliver results on individual equity factors using various

methods, no paper has yet evaluated all the popular equity factors using a unified method-

ological framework.

4


3 Methods

For this paper, I use a large set of forecasting approaches such as combination methods,

factor based methods and methods focusing on variable selection. This section gives

a short overview over these methods. To evaluate the forecasting methods, I contrast

statistical methods based on the mean squared prediction error with methods investors

use to evaluate portfolio performance such as mean variance utility and information ratios.

3.1 Forecasting

3.1.1 Forecast combination

The first and most basic method is forecast combination (Rapach et al., 2010). It

consists of a very simple two step procedure.

(I) Running separate single linear regressions for each of the N predictors xi on the target

variable r and obtaining the fitted values (predictions) ri

ri,t+1 = αi + βixi,t (1)

(II) Combining these fitted values ri to obtain a final prediction rc

rc,t+1 =N∑

i=1

ωi,t+1ri,t+1 (2)

There are different specifications for the weights ωi of the individual fitted values. A

simple and effective method is to use the mean of the fitted values, i.e. ωi = 1/N ∀i =

1, ..., N . Other specifications include the median or the trimmed mean of the fitted values.

There also exist more complicated specifications based on past predictive performance,

which, however often perform worse than the simple mean (Rapach et al., 2010). We

can interpret this approach as a shrinkage method, that constrains the multiple linear

regression coefficients to 1/Nβi and performs especially well compared to other methods

if all predictors are similarly weak (Huang & Lee, 2010). The baseline case presented

herein uses mean and median combination forecasts.

3.1.2 Factor based methods

Another recently very popular forecasting method in economics (Stock & Watson, 2002)

as well as return predictability (Ludvigson & Ng, 2007) is principal component regres-

sion. It assumes that the predictors xi follow a latent factor structure f and the number

of relevant factors p is significantly smaller than the number of predictors N . Directly

using estimated factors f instead of the original predictors in the predictive regression

5


massively reduces the dimensionality and thus avoids model overfitting. The estimates f

of the true factors f are usually obtained by principal component analysis.

xi,t = λift + ǫi,t (3)

rt+1 = αPCR + βPCRft (4)

The decisive aspect about this methodology is to appropriately select the factors f

used in the regression. However, the specifications have varied widely in existing papers

about factor based regressions. In principle, the standard approaches to model selection

apply, so that we can select the models using information criteria such as the Akaike

information criterion (AIC) and the Bayesian information criterion (BIC), cross validation

or bootstrap techniques as well as validation samples.

However, several layers of complexity and potential data mining arise. First, only

the first few components explain the large majority of the variance of the predictors x,

such that a cut-off point for the relevant components p needs to be established. While

most papers choose this cut-off based on ad hoc measures, Bai and Ng (2002) propose a

selection mechanism for this problem.

Next, the question arises, which subsets of the factors ft we allow to be selected in the

model selection step. A very restrictive approach is to use model selection to determine

number of the q largest components. A more flexible approach allows for a separate

selection of all components below the cut-off level p, e.g. the model can select the 2nd

and the 7th component. Finally, to make the model selection step even more flexible,

functions of the factors ft such as quadratic factors e.g. f 21 or interactions of factors e.g.

f2 × f5 could be allowed in the selection step. Obviously, the more flexible the approach,

the more likely it is that the true model is included. However, the likelihood increases to

simply maximize the in-sample fit and it is difficult to find an economic interpretation of

the functions of factors.

We also need to pay attention to the fact that when changing time windows, the

factors, their loadings and thus their interpretation may change and may significantly

differ from the factors identified over the whole sample. The baseline analysis relies

on three different approaches: (I) using only the first principal component (first), (II)

selecting the largest q components via BIC allowing for a maximum of p = 10 factors

(ascend) and (III) selecting the relevant components separately without transformations

via BIC from the p = 10 largest factors (step). My results indicate that the larger and the

more flexible the model the worse its performance. This makes intuitive sense as leaving

too many degrees of freedom in estimation leads to overfitting the model.

At the same time, the problem arises, that the factors f constructed to explain the

6


cross-section of the predictors x are not necessarily relevant for explaining the target

variable r. Thus, we can apply targeting techniques to the predictors x and construct

the factors f only from those predictors that help explaining the target. To do so, two

targeting procedures have been proposed in the literature so far.

Hard thresholding (Bai & Ng, 2008; Cakmakli & van Dijk, 2010) first runs single

linear regressions for each single predictor on the target. Then it selects only those

predictors, which have t-statistics larger in absolute value than a certain threshold level

e.g. the 15% one-sided confidence interval. Hard thresholding suffers from considering

only the univariate relationship between xi and ri while neglecting the interaction between

xi and xj. Thus, it is likely to select highly collinear predictors.

The goal of soft thresholding is to alleviate this concern. It ranks the variables in

order of their importance understood as the position of inclusion in a lasso, elastic net or

LARS model (Bai & Ng, 2008; Cakmakli & van Dijk, 2010). All these three methods are

intimately related and in the following, I use the LARS approach. The baseline tests set

the threshold t-statistic for hard thresholding to 1.04 following Cakmakli and van Dijk

(2010) and set the percentage of predictors included in the soft thresholding version to

30%. Both thresholding versions fix the minimum number of predictors to 20.

While only factor based regression approaches use these targeting approaches in the

literature, a similar problem of including irrelevant predictors is present in the forecast

combination setting. Thus, I also apply the respective pre-selection approaches to forecast

combination with the intention to reduce noise in the estimation. I find no benefits to

using targeting procedures compared to the untargeted estimates.

Instead of using targeting procedures to select the relevant predictors for constructing

the factors f , we can explicitly use the target variable ri in factor construction. A common

method to do so is partial least squares (PLS) by Wold (1975). While PLS has not

been proposed to equity risk premium prediction yet, it is a natural next step from

principal component regressions.

It is also a special case of the three-pass regression filter proposed by Kelly and

Pruitt (2014) for return predictability. The three-pass regression filter assumes that the

factors relevant for ri are a strict subset of the factors relevant for explaining the predictors.

Based on this assumption, this filter determines the relevant factors while abandoning the

irrelevant factors. We can represent this approach as a set of three regressions:

(I) N different time-series regressions of the predictors x on the proxies Z. (II) T

separate cross-sectional regressions of the predictors xt on the first-stage coefficients φi

and finally (III) a single time series forecasting regression of the target r on the factors

f . The critical part of this approach is the selection of the number and type of proxies Z

in the first regression stage. The standard specification uses one proxy and sets it equal

7


to the target, Z = r.

xi,t = φiZt + ǫi,t (5)

xi,t = Ftφi + ζi,t (6)

rt+1 = β0 + Ftβ + ηi,t (7)

3.1.3 Least absolute shrinkage and selection operator (LASSO)

Medeiros and Mendes (2016) proposed the lasso for equity premium forecasting in a draft

version of their paper. Moreover, Rapach, Strauss, Tu, and Zhou (2015) use the adaptive

lasso for predicting industry returns. The lasso is a shrinkage estimator that penalizes

the absolute size of the coefficients of the estimated parameter vector θ compared to a

simple least squares problem as in

θ = argminθ

||Y −Xθ||2 + λN∑

i=1

ωi|θi| (8)

The classical lasso (setting ωi = 1 ∀i = 1, ..., N) is successful in shrinking the irrelevant

parameters to zero and can handle more variables than observations under some condi-

tions. It, however, requires the ”irrepresentable condition” and does not have the oracle

property. 2 By weighting the coefficients in the penalty term by ωi, the adaptive lasso

(Zou, 2006) overcomes these problems. The estimation strongly depends on the parame-

ter λ. We can estimate this parameter by means of classical model selection techniques.

This paper uses the BIC criterion.

This method differs from the approaches presented before as it assumes that a limited

number of explanatory variables are relevant for the forecast, so called sparsity. Forecast

combination and factor based methods, in contrast, assume that all variables are impor-

tant for the prediction. Thus, contrasting the lasso approach with the other methods

has implications for the underlying data structure of the problem at hand. Overall, the

results hint to a non-sparse data structure.

2An estimator that has the oracle property selects the correct subset of variables with non-zero coef-ficients. Moreover, it has asymptotically normally distributed non-zero coefficient estimates.

8


3.2 Forecast Evaluation

The most common statistic in evaluating out-of-sample performance is the out-of-sample

R2 (Campbell & Thompson, 2008) which is closely related to the well-known in-sample

R2 statistic.

R2OOS = 1−

∑Tt=1(rt − rt)

2

∑Tt=1(rt − rt)2

= 1−ˆMSFE¯MSFE

(9)

where rt is the realized return, rt is the forecasted return and rt is the historical average

return. A positive out-of-sample R2 suggests an improvement in the predictive perfor-

mance compared to the historical average estimate, while a negative out-of-sample R2

indicates a deterioration. The historical mean is a hard to beat benchmark in forecasting

the market risk premium. While the out-of-sample R2 just averages over the whole time

period, Henkel et al. (2011) show that the forecasting performance varies widely between

the different stages of the business cycle. Thus, I also calculate the out-of-sample R2

using only the data points at recession time periods as defined by the National Bureau

of Economic Research (NBER), R2Rec, and only the data points at NBER expansion time

periods, R2Exp. The out-of-sample R2 itself does not give us any information about the

significance of the forecast improvement. By now, the mean squared forecast error

(MSFE) adjusted statistic by Clark and West (2007) has become the standard for

evaluating the significance of nested forecast models. It adjusts the mean squared predic-

tion errors to account for the noise of the forecasting model. After this adjustment the

statistic is approximately normally distributed, but it is rather conservative such that the

nominal 0.10 (0.05) tests have an actual size of 0.05-0.10 (0.01-0.05). Moreover, this ad-

justment leads to differences compared to the out-of-sample R2 such that we can observe

negative R2, but a statistically significant forecast improvement, particularly for noisy

estimates. The statistic defines

di,t = u20,t − [u2

i,t − (rt − ri,t)2] (10)

where u0,t = rt − rt are the residuals from the forecast model and ui,t = rt − ri,t are the

residuals from the benchmark model. We then test if di,t is different from zero assuming

normality. For the overall significance tests, I use Newey and West (1987) standard errors,

while I use classical standard errors for expansion and recession periods, as they are not

necessarily adjacent in time.

For the out-of-sample R2, I also calculate adjusted p-values based on the Holm-

Bonferroni method to account for the familywise error rate across the different esti-

mation methods. This approach sorts the p-values p of the hypotheses in ascending order

from 1 to m. Then the adjusted p-value p is given by

9


pi = maxj≤i

(min ((m− j + 1)pj) , 1) . (11)

While the mean squared prediction error (MSPE) is a statistical concept, investors

care about utility gains due to predictability, see e.g. Dichtl, Drobetz, and Kryzanowski

(2014). As widely discussed in the literature, e.g. Rapach and Zhou (2013), there is

only a moderate correlation between utility gains and out-of-sample R2. In the following,

the investor has mean-variance preferences with risk aversion γ = 2 and can switch

between the risk free rate rft and the investment in a particular factor rt. The yearly

performance gain of the investor U is the difference in the annualized mean of the single

period utilities for the estimation method U(t) and the benchmark historical mean esti-

mate U(t). The volatility estimate σ is the historical volatility across the training period

for all specifications.

wt =1

γ

rtσ2t

(12)

U(t) = wtrt + rft − 0.5γw2t σ

2t (13)

U =12

T

(

T∑

t=1

U(t)−T∑

t=1

U(t)

)

(14)

Finally, the information ratio is a common measure to evaluate the relative perfor-

mance of a given strategy by calculating the excess return over the relative risk. I present

the annualized differences in the information ratios of a mean-variance investor that either

uses the respective predictive method or the historical mean estimate.

4 Data Sets

4.1 Explained data: six equity factors

This analysis focuses on the prediction of six equity factors: (1) the market excess return

traditionally used in return predictability corresponds to the return of an investment in

the broad market capitalization weighted index minus the risk-free rate. (2) The size

factor (Banz, 1981; Fama & French, 1992, 1993) holds a long position in small capital-

ization stocks and a short position in large capitalization stocks. (3) The value factor

(Basu, 1983; Fama & French, 1992, 1993) invests in stocks with high book-value-to-

market-capitalization and shorts low book-value-to-market-capitalization stocks. (4) The

momentum factor (Jegadeesh & Titman, 1993; Carhart, 1997) holds a long position in

stocks with the highest returns in the last year (excluding the latest month), while shorting

10


those stocks with the lowest returns in the same time period. (5) The betting-against-beta

factor (Frazzini & Pedersen, 2014) is a low beta factor, which takes a market beta neutral

investment in low beta stocks while selling high beta stocks. (6) Finally, the quality factor

(Asness, Frazzini, & Pedersen, 2013) defines a quality company as a profitable, stable,

growing and dividend paying company and forms long-short portfolios analogously to the

other equity factors.

I obtain the factor data from Andrea Frazzini’s web page3. The value factor is an

adjusted HML factor (Asness & Frazzini, 2013), which uses current market capitalization

to calculate the book-to-market ratio and is a more realistic proxy for value factors used

in practice.

4.2 Explanatory data sets

The explanatory data consists of four different data sets: (1) A small financial data set,

(2) a broad financial data set, (3) an economic data set and (4) the combination of the

broad financial data set and the economic data set.

The small financial data set corresponds to the data of Welch and Goyal (2007)

and is available in an updated version on Amit Goyal’s web page4. This data set is

common for studies of return predictability such as in Rapach et al. (2010) and Ferreira

and Santa-Clara (2011). The data set contains monthly data from 1927 to 2013 for

15 economic variables: dividend-price ratio, dividend yield, earnings-price ratio, 10-year

average earnings-price ratio, dividend-payout ratio, stock variance, book-to-market ratio,

net equity expansion, Treasury bill rate, long-term yield, long-term return, term spread,

default yield spread, default return spread and inflation.

The large financial data set is a combination of several existing data sets. First,

it contains the stock price and interest rate data of the collection of macroeconomic data

series provided by the Federal Reserve Bank of St. Louis (FRED). Additionally, it contains

the data of Amit Goyal used in the small financial data set. Moreover, it contains lagged

Fama French factors (Liew & Vassalou, 2000), lagged returns on 49 industries (Hong,

Torous, & Valkanov, 2007; Rapach et al., 2015) and the returns on 25 size-value-portfolios

(Wahal & Yavuz, 2013) obtained from Kenneth French’s web page5.

The economic data set starts with a collection of 130 macroeconomic data series

provided by the Federal Reserve Bank of St. Louis (FRED). This collection is based on

Stock and Watson (2005) and contains income and production time series, employment

data, housing market data, orders and inventories, monetary series, stock prices, interest

3http://www.econ.yale.edu/~af227/data library.htm4http://www.hec.unil.ch/agoyal/5http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html

11


rates, exchange rates and inflation data. The economic data set excludes stock prices

and interest rates as they are used in the large financial data set. It also excludes incom-

plete data series. Transforming the data using year-over-year absolute or relative changes

ensures stationarity.

The combination data set takes Ludvigson and Ng (2007) as a guideline, which

construct similar macroeconomic and financial data sets. It simply combines the large

financial and the economics data sets described above.

In most of the paper, I use one month ahead forecasts and a one month gap for all

data, e.g. the forecast for January 1975 is based on data up to November 1974. While

this is standard for using macroeconomic data in predictability, there is often no gap for

financial data, i.e. observation and implementation take place at the same time. Correct

lag specification is especially relevant if illiquid assets have a large impact on factor returns

such as for the size factor. Most of the economic data is only available since the end of

1961 and there is some reasonable minimum amount of training data required. Thus, the

out-of-sample evaluation period starts in January 1975 up until November 2013 for the

baseline specification. For each data set, the training sample starts as early as possible

given that enough data is available at the starting point. Hence, the two financial data sets

start the training period in December 1927 (except for the low beta factor in December

1931 and the quality factor in December 1957), while the economic and combined data

sets start in December 1961. The baseline estimates use expanding windows. To be able

to process the data with all different methods, I standardize the data before estimation

and exclude data series with missing values. For each non-binary series, I truncate outliers

at four times the interquartile range.

5 Empirical Results

5.1 Predictive Performance

5.1.1 Low Beta Factor

Out of the six equity factors considered in this paper, the low beta factor shows the most

prominent forecasting ability. This predictability, however, varies substantially between

the four data sets.

For the large financial data set in panel B of table 1 we see that the out-of-sample

R2 (3) are highly significant for each method used. The significance also holds up to

familywise testing (4). All successful principal component regression models select the

first principal component which loads relatively equally on lagged industry returns and

the lagged size-value-portfolios. Similar loadings hold for the PLS method. The lasso

12


mostly loads on two industry portfolios (Agriculture, Consumer Goods) as well as the

long-term yield.

For the small financial data set in panel A as a subset of the large data set in panel B,

in contrast, there is no predictability. Most financial variables not included in the small

financial data set contain different types of lagged returns. This also confirms the findings

that lagged returns are dominant drivers of predictability for the financial data.

Moreover, there is some significant predictability for the economic data set in panel

C. The significance is, however, concentrated in the forecast combination specifications

and does not hold up to familywise testing.

The combination data of financial and economic data in panel D overall shows slightly

improved predictive performance compared to the large financial data set in panel B for

most estimation methods. In particular, the results are stronger for forecast combinations

and restricted principal component regression models in terms of all performance metrics.

For more complex principal component regression models, the results, however, become

weaker. In the principal component regression case, the first component gets enhanced

with information about the current state of the economy. This additional information is a

proxy for the trough of the business cycle as it loads negatively on industrial production,

inflation and employment, and loads positively on unemployment and surprisingly the

real estate market (building permits as well as housing starts).

Utility gains (7) and information ratios (8) mirror the findings on the out-of-sample

R2, while they are less powerful and significance is mostly limited to the simple models

such as forecast combination and restricted principal component regression.

While the absolute sizes of the recession R2 (5) are higher than the expansion R2

(6) for most significant specifications, the significance levels in expansions are higher due

to the reduced estimation variance as there are much more expansionary months than

recessionary months in the sample.

Table 2 shows that using all the data available for the financial data sets by starting

the analysis 1950 has very limited impact on the results. For the small financial data set

in panel A, the results are unchanged. For the large financial data set in panel B, the

significant predictive ability gets slightly attenuated, but remains statistically significant.

13

Ulrich

Carl

Equity

Facto

rPredicta

bility

Panel A: Small Financial Dataset (Start 1932, Eval 1975-2013)

Method Details R2 HB R2

RecR2

ExpU(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 0.04 1.000 -0.75 0.30* 0.30 -0.02

FC median/none -0.02 1.000 -0.28 0.07 -0.10 -0.01PCR first/none -0.34 1.000 -0.91 -0.15 -1.54 -0.07PCR ascend/none -0.58 1.000 -0.91 -0.47 -2.76 -0.12PCR step/none -0.46 1.000 -0.89 -0.32 -2.12 -0.11

LASSO -0.35 1.000 -0.45 -0.32 -1.78 -0.07LASSO adaptive -0.87 1.000 -0.95 -0.85 -4.38 -0.153 PRF -1.82 1.000 -5.76 -0.52* -8.64 -0.16PLS -0.99 1.000 -0.40 -1.18 -4.23 -0.13

Panel B: Large Financial Dataset (Start 1932, Eval 1975-2013)


RecR2

ExpU(MV) IR

FC mean/none 1.51*** 0.003*** 2.05* 1.33*** 6.45*** 0.14*

FC mean/hard 1.77*** 0.002*** 2.11* 1.66*** 7.65*** 0.16FC mean/soft 1.25*** 0.002*** 1.51 1.16*** 5.42*** 0.11**

FC median/none 1.52*** 0.004*** 2.12* 1.32*** 6.44*** 0.14*

PCR first/none 2.42*** 0.004*** 3.74** 1.99*** 10.24*** 0.22PCR ascend/none 1.87*** 0.018** 3.74** 1.26** 7.30** 0.12PCR step/none 2.13*** 0.018** 2.55 1.99*** 8.95 0.07PCR step/hard 3.44*** 0.015** 6.43* 2.46*** 15.07* 0.15PCR step/soft 1.59*** 0.018** -3.11 3.14*** 8.69 0.08LASSO 1.54** 0.018** -2.04 2.72*** 8.54* 0.08LASSO adaptive 1.55*** 0.018** -2.26 2.81*** 8.74 0.063 PRF 3.44*** 0.004*** 0.65 4.36*** 14.95* 0.15PLS 2.67*** 0.004*** 4.50** 2.06*** 11.22*** 0.21

Panel C: Economics Dataset (Start 1962, Eval 1975-2013)


RecR2

ExpU(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 0.68** 0.414 0.08 0.88*** 4.62** 0.07**

FC mean/hard 0.92** 0.371 0.22 1.16*** 6.25** 0.09*

FC mean/soft 1.00** 0.170 0.57 1.14*** 6.22** 0.09**

FC median/none 0.45** 0.414 0.18 0.54*** 3.22** 0.05***

PCR first/none -0.30 1.000 -1.53 0.11 -0.39 -0.02PCR ascend/none -0.24 1.000 -1.71 0.25 2.50 -0.01PCR step/none -4.13 1.000 -7.45 -3.02 -14.77 -0.21PCR step/hard -0.19* 0.534 0.95 -0.56** -0.02 -0.01PCR step/soft -0.83* 0.534 -0.51 -0.94** -2.85 -0.04LASSO -0.21 1.000 0.65 -0.49 -1.76 -0.05LASSO adaptive -0.93 1.000 1.21 -1.64 -6.90 -0.153 PRF 1.39** 0.294 0.23 1.77*** 11.07 0.12PLS -0.36* 0.534 4.74** -2.06 1.11 -0.01

Panel D: Combination Dataset (Start 1962, Eval 1975-2013)


RecR2

ExpU(MV) IR

FC mean/none 1.61*** 0.000*** 1.87** 1.52*** 7.94*** 0.14***

FC mean/hard 2.25*** 0.000*** 2.48** 2.17*** 11.68*** 0.19***

FC mean/soft 1.70*** 0.000*** 2.45** 1.45*** 7.95*** 0.14***

FC median/none 0.81*** 0.005*** 1.05** 0.74*** 3.68*** 0.07***

PCR first/none 4.14*** 0.003*** 6.41** 3.38*** 17.89*** 0.29PCR ascend/none 3.53*** 0.008*** 6.56** 2.53*** 15.24* 0.18PCR step/none 0.40** 0.081* 5.74 -1.37** -1.91 -0.04PCR step/hard 2.31*** 0.037** 3.64 1.87*** 14.23 0.16PCR step/soft -1.01* 0.101 1.94 -1.99* -2.41 -0.05LASSO 0.83* 0.101 -0.13 1.15** 5.46 0.07LASSO adaptive 1.80*** 0.037** 1.19 2.01*** 9.76 0.103 PRF 5.53*** 0.000*** 5.31** 5.60*** 30.64*** 0.37PLS 4.55*** 0.008*** 9.54*** 2.89*** 22.98* 0.27

Table 1: Forecasting performance of the low beta factorThe low beta factor is forecasted for four different data sets (panel A-D) and the methods (1) FC (forecast combination), PCR (principle component regression), LASSO, 3 PRF (Three-passregression filter) and PLS (partial least squares). The details (2) further specify the method. If applicable, the targeting procedure is denoted as ’none’ (no targeting), ’soft’ (targeting basedin LARS), ’hard’ (targeting based on OLS t-stat). For PCR, ’first’ uses only the first component, ’ascend’ selects the maximum number of adjacent factors, ’step’ allows for non-adjacentfactors. The evaluation measures R2 (3), R2

Rec (5), R2

Exp (6) are the out-of-sample R2 for the whole, the recession and the expansion period in percent. Significance is calculated using MSFEadjusted statistics for the respective time periods. (4) are the Holm Bonferroni adjusted p-values of the R2. U(MV) (7) are the annualized certainty equivalent utility gains in percent for amean variance investor. IR (8) are the annual information ratio differences. *, **, and *** represent one-sided statistical significance at the 10, 5 and 1% level.

14



Method Details R2 HB R

2Rec

R2Exp

U(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 0.09 0.452 -0.31 0.23* 0.32 -0.02FC median/none -0.02 0.330 -0.13 0.02 -0.09 -0.01PCR first/none -0.43 0.452 -0.91 -0.26 -1.51 -0.07PCR ascend/none -0.64 0.452 0.41 -1.01 -2.28 -0.10PCR step/none -0.21 0.678 0.69 -0.54 -0.77 -0.07LASSO -0.50 1.000 0.03 -0.69 -1.73 -0.07LASSO adaptive -1.24 1.000 -0.27 -1.58 -4.14 -0.143 PRF -2.58 0.424 -3.41 -2.28* -9.02 -0.13PLS -2.58 1.000 -0.70 -3.24 -8.70 -0.20



2Rec

R2Exp

U(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 1.23***0.006*** 1.19 1.25*** 3.87*** 0.09FC mean/hard 1.43***0.005*** 1.07 1.56*** 4.53*** 0.10FC mean/soft 1.03***0.004*** 0.89 1.08*** 3.26*** 0.07FC median/none 1.24***0.009*** 1.22 1.24*** 3.86*** 0.10PCR first/none 1.96***0.009*** 2.18* 1.88*** 6.08*** 0.15PCR ascend/none 1.45***0.036** 2.18* 1.20** 4.13* 0.07PCR step/none -0.18** 0.053* -1.16 0.17*** -1.57 -0.05PCR step/hard 1.64***0.036** 4.07* 0.79*** 4.88 0.05PCR step/soft -2.85** 0.053* -5.89 -1.78*** -8.53 -0.11LASSO 1.17** 0.053* -2.00 2.30*** 4.85* 0.06LASSO adaptive 0.82** 0.046** -2.89 2.14*** 3.76 0.023 PRF 1.80***0.009*** -1.91 3.12*** 5.54 0.06PLS 1.68***0.009*** 4.00** 0.85*** 5.15** 0.08

Table 2: Forecasting performance of the low beta factor - long periodThe low beta factor is forecasted for two different data sets (panel A and B) and the methods (1) FC (forecast combination),PCR (principle component regression), LASSO, 3 PRF (Three-pass regression filter) and PLS (partial least squares). Thedetails (2) further specify the method. If applicable, the targeting procedure is denoted as ’none’ (no targeting), ’soft’(targeting based in LARS), ’hard’ (targeting based on OLS t-stat). For PCR, ’first’ uses only the first component, ’ascend’selects the maximum number of adjacent factors, ’step’ allows for non-adjacent factors. The evaluation measures R2 (3),R2

Rec (5), R2

Exp (6) are the out-of-sample R2 for the whole, the recession and the expansion period in percent. Significanceis calculated using MSFE adjusted statistics for the respective time periods. (4) are the Holm Bonferroni adjusted p-valuesof the R2. U(MV) (7) are the annualized certainty equivalent utility gains in percent for a mean variance investor. IR (8)are the annual information ratio differences. *, **, and *** represent one-sided statistical significance at the 10, 5 and 1%level.

15


5.1.2 Size Factor

While being much less pronounced than the predictability for the low beta factor, there

is some significant predictability in the size factor in table 3.

Especially economic variables in panel C are helpful in predicting the size factor and

the majority of methods find statistically significant predictability. More complicated

principal component regression models and partial least squares, however, cannot beat

the historical average estimate. Some simple models such as forecast combination and

restricted principal component regression models also remain significant after controlling

for the family-wise error rate across methods (4). The predictability in the size factor is

related to economic growth. The two successful principal component regression models

both use the first factor, which has strongly positive loadings on industrial production

and negative loadings on unemployment.

The results of the small financial data set in panel A are mixed with some moderately

significant findings particularly for forecast combination and lasso. These, however, do not

withstand family-wise testing. The default yield spread, known to be strongly correlated

with growth, plays an important role in the most successful models. The large financial

data set in panel B cannot make use of the additional data and shows less predictability

than the small financial data set.

The combination data set in panel D can retain most of the predictability compared

to the economics data set in panel C, but overall, adding the financial data deteriorates

the performance. This is in line with the weak findings for the financial data sets.

Neglecting transaction costs, investors can achieve significant utility gains (7) using

simple methods on the economics data set in panel C and the combination data set in

panel D. The gains in information ratios (8) remain insignificant.

Overall, predictability is particularly strong in expansions (6), not only because of

reduced estimation error but also because of differences in the out-of-sample R2. These

results are, however, not stable in an in-sample setting or when using a longer time

interval.

Using a longer time interval for the financial data sets in table 4, overall significance

is increased. The gains are particularly strong for the small financial data set in panel A.

It now shows significant out-of-sample R2 (3) across all estimation methods, which even

holds up to family-wise tests (4). For the large financial data set in panel B, the results

improve as well, particularly during expansions (6). Compared to the small financial data

set, the gains in predictability remain moderate. Controlling for the family-wise error rate

(4), the results remain mostly insignificant. Also here, having a larger range of financial

data does not improve the forecasting results.

16

Ulrich

Carl

Equity

Facto

rPredicta

bility



RecR2

ExpU(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 0.77* 0.309 0.28 0.89* 1.61 0.15

FC median/none 0.46** 0.284 0.13 0.55* 0.96 0.11PCR first/none -3.69 0.322 -2.34 -4.01 -7.60 -0.34PCR ascend/none -2.44* 0.309 -1.84 -2.59* -5.02 -0.21PCR step/none 0.06** 0.165 0.42 -0.02** -0.15 -0.07

LASSO 1.05** 0.222 0.27 1.24** 2.09 0.01LASSO adaptive 0.60** 0.165 0.97 0.51** 1.17 -0.043 PRF -1.94 0.322 -0.53 -2.28 -3.61 -0.21PLS -2.97 0.322 -6.27 -2.18* -6.25 -0.25



RecR2

ExpU(MV) IR

FC mean/none -0.15 1.000 -2.38 0.38 -0.29 -0.04FC mean/hard -0.22 1.000 -2.99 0.45* -0.42 -0.06FC mean/soft 0.27 1.000 -1.65 0.73** 0.54 0.01FC median/none -0.22 1.000 -2.39 0.31 -0.43 -0.05PCR first/none -0.94 1.000 -4.61 -0.06 -1.89 -0.16PCR ascend/none -5.99 1.000 -12.00 -4.53 -12.06 -0.40PCR step/none -9.06 1.000 -10.37 -8.74 -17.54 -0.51PCR step/hard -6.39 1.000 -12.32 -4.96* -12.75 -0.31PCR step/soft -8.59 1.000 -12.86 -7.56 -17.73 -0.31LASSO 0.97** 0.256 -1.13 1.48** 1.94 0.00LASSO adaptive -1.33** 0.256 -3.60 -0.78** -2.84 -0.113 PRF -9.52 1.000 -14.06 -8.42 -18.61 -0.38PLS -6.41** 0.525 -10.66 -5.38** -13.21 -0.25



RecR2

ExpU(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 0.78*** 0.097* 0.03 0.96*** 2.14** 0.09FC mean/hard 0.81** 0.168 -0.85 1.22*** 2.28* 0.05FC mean/soft 0.63** 0.170 -0.12 0.81** 1.73* 0.07FC median/none 0.67*** 0.044** 0.16 0.79*** 1.80*** 0.10PCR first/none 1.83*** 0.031** -0.19 2.32*** 5.06** 0.12PCR ascend/none 1.83*** 0.031** -0.19 2.32*** 5.06** 0.12PCR step/none -0.15* 0.270 -1.23 0.12** 0.05 -0.07PCR step/hard -1.59 0.346 1.89 -2.44 -4.53 -0.17PCR step/soft -4.56 0.638 -0.59 -5.53 -11.76 -0.41LASSO 0.66* 0.270 0.74 0.64** 1.78 0.03LASSO adaptive 0.74** 0.270 0.58 0.78** 1.99 -0.013 PRF 0.62** 0.168 -5.88 2.21*** 2.17 0.00PLS -0.77 0.346 -0.48 -0.84 -1.80 -0.14



RecR2

ExpU(MV) IR

FC mean/none 0.42** 0.216 0.10 0.50** 1.16** 0.06FC mean/hard 0.73** 0.216 -0.75 1.09*** 2.03* 0.05FC mean/soft 0.22 0.614 0.14 0.25 0.61 0.02FC median/none 0.17*** 0.089* 0.38* 0.12** 0.47** 0.03***

PCR first/none -0.33 1.000 -0.53 -0.28 -0.82 -0.08PCR ascend/none 0.59 0.614 1.98 0.25 1.77 0.01PCR step/none -1.57 1.000 -2.02 -1.47 -3.81 -0.21PCR step/hard -1.38*** 0.059* -7.86 0.20*** -3.23 -0.03PCR step/soft -6.09 1.000 -1.08 -7.31 -16.33 -0.40LASSO 0.79** 0.274 0.74 0.81** 2.19* 0.05LASSO adaptive 0.68** 0.274 0.58 0.71** 1.83 -0.013 PRF 0.34*** 0.110 -6.58 2.03*** 1.37 0.00PLS -1.48 1.000 0.18 -1.89 -3.72 -0.25

Table 3: Forecasting performance of the size factorThe size factor is forecasted for four different data sets (panel A-D) and the methods (1) FC (forecast combination), PCR (principle component regression), LASSO, 3 PRF (Three-passregression filter) and PLS (partial least squares). The details (2) further specify the method. If applicable, the targeting procedure is denoted as ’none’ (no targeting), ’soft’ (targeting basedin LARS), ’hard’ (targeting based on OLS t-stat). For PCR, ’first’ uses only the first component, ’ascend’ selects the maximum number of adjacent factors, ’step’ allows for non-adjacentfactors. The evaluation measures R2 (3), R2

Rec (5), R2


17




2Rec

R2Exp

U(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 1.06***0.041** 1.97** 0.77** 1.73 0.19FC median/none 0.68** 0.041** 1.46** 0.43** 1.10 0.13PCR first/none -3.82** 0.041** 3.04** -6.00* -6.91 -0.22PCR ascend/none -2.95***0.041** 3.18** -4.90** -5.26 -0.13PCR step/none -0.77*** 0.01** 2.56** -1.84*** -1.46 -0.02LASSO 0.96***0.024** 1.44* 0.80*** 1.61 0.07LASSO adaptive -0.37*** 0.03** 1.11* -0.84*** -0.55 -0.013 PRF -2.22***0.041** 4.15** -4.26** -3.83 -0.12PLS -2.65** 0.041** 0.48* -3.65** -4.74 -0.13



2Rec

R2Exp

U(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 0.01 0.946 -1.86 0.60** -0.06 -0.01FC mean/hard -0.04 0.946 -2.26 0.66** -0.16 -0.02FC mean/soft 0.39* 0.429 -1.46 0.98*** 0.58 0.04FC median/none -0.14 0.946 -2.20 0.51** -0.30 -0.02PCR first/none -0.89 0.946 -4.23 0.18* -1.60 -0.09PCR ascend/none -6.29** 0.389 -5.11 -6.66** -11.16 -0.26PCR step/none -9.49 0.666 -6.67 -10.39 -16.60 -0.34PCR step/hard -6.47** 0.260 -8.59 -5.80** -11.41 -0.19PCR step/soft -11.12* 0.389 -15.95 -9.57** -19.33 -0.21LASSO 0.10** 0.122 -0.97 0.44*** 0.22 0.00LASSO adaptive -3.52***0.074* -3.87 -3.41*** -5.87 -0.073 PRF -10.41** 0.389 -9.78 -10.61* -18.00 -0.24PLS -8.90** 0.122 -5.95 -9.84** -15.57 -0.19

Table 4: Forecasting performance of the size factor - long periodThe size factor is forecasted for two different data sets (panel A and B) and the methods (1) FC (forecast combination),PCR (principle component regression), LASSO, 3 PRF (Three-pass regression filter) and PLS (partial least squares). Thedetails (2) further specify the method. If applicable, the targeting procedure is denoted as ’none’ (no targeting), ’soft’(targeting based in LARS), ’hard’ (targeting based on OLS t-stat). For PCR, ’first’ uses only the first component, ’ascend’selects the maximum number of adjacent factors, ’step’ allows for non-adjacent factors. The evaluation measures R2 (3),R2

Rec (5), R2


18

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Carl

Equity

Facto

rPredicta

bility



RecR2

ExpU(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 0.20 1.000 1.21** -0.21 0.34 0.09

FC median/none 0.39* 0.651 0.34 0.42** 0.73 0.06PCR first/none -1.34 0.990 6.44*** -4.49 -2.88 -0.12PCR ascend/none -2.20 1.000 5.05** -5.13 -4.43 -0.23PCR step/none -4.78 1.000 -2.61 -5.66 -9.47 -0.40

LASSO -0.42 1.000 -0.01 -0.58 -0.97 -0.08LASSO adaptive -2.54 1.000 -1.80 -2.84 -5.30 -0.243 PRF -3.29 1.000 2.61* -5.68 -6.91 -0.33PLS -3.51 1.000 -0.79 -4.62 -7.46 -0.32



RecR2

ExpU(MV) IR

FC mean/none 0.04 1.000 0.21 -0.03 0.07 0.01*

FC mean/hard 0.04 1.000 0.87** -0.30 0.05 0.04FC mean/soft -0.03 1.000 0.08 -0.08 -0.09 0.01FC median/none -0.01 1.000 0.04 -0.02 -0.01 0.00PCR first/none -0.02 1.000 0.02 -0.04 -0.03 0.00PCR ascend/none -2.76 1.000 3.92** -5.47 -5.49 -0.29PCR step/none -2.94 1.000 5.13** -6.21 -5.70 -0.29PCR step/hard -3.31 1.000 2.82* -5.79 -7.00 -0.33PCR step/soft -5.62 1.000 -9.43 -4.07 -11.35 -0.38LASSO -0.66 1.000 -0.80 -0.61 -1.43 -0.10LASSO adaptive -3.19 1.000 -2.19 -3.60 -6.52 -0.273 PRF -3.83 1.000 4.31** -7.12 -8.03 -0.33PLS -0.41 1.000 0.17 -0.64 -0.88 -0.05



RecR2

ExpU(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 0.10 1.000 -1.03 0.55* 0.26 -0.07FC mean/hard 0.07 1.000 -1.53 0.71* 0.11 -0.12FC mean/soft 0.15 1.000 -0.63 0.45* 0.42 -0.05FC median/none 0.05 1.000 -0.57 0.30 0.13 -0.05PCR first/none -0.58 1.000 -2.76 0.29 -1.99 -0.21PCR ascend/none -1.32 1.000 -4.41 -0.09 -4.38 -0.34PCR step/none -2.09 1.000 -0.86 -2.58 -6.55 -0.44PCR step/hard -2.56 1.000 -5.03 -1.58 -8.07 -0.43PCR step/soft -4.97 1.000 -11.67 -2.31 -15.60 -0.54LASSO -0.39 1.000 0.16 -0.61 -1.25 -0.17LASSO adaptive -2.09 1.000 -0.78 -2.60 -6.61 -0.383 PRF -1.98 1.000 -3.45 -1.40 -6.50 -0.35PLS -1.52 1.000 -5.34 0.00* -4.80 -0.30



RecR2

ExpU(MV) IR

FC mean/none 0.16 1.000 -0.45 0.40* 0.46 -0.02FC mean/hard 0.11 1.000 -1.20 0.63* 0.26 -0.08FC mean/soft 0.09 1.000 -0.56 0.34* 0.24 -0.03FC median/none 0.05 1.000 0.01 0.07 0.16 -0.01PCR first/none -0.21 1.000 0.07 -0.32 -0.67 -0.07PCR ascend/none -0.80 1.000 0.07 -1.14 -2.44 -0.21PCR step/none -5.03 1.000 -4.11 -5.40 -15.81 -0.60PCR step/hard -5.55 1.000 -10.74 -3.49 -17.74 -0.58PCR step/soft -6.47 1.000 -9.88 -5.12 -19.95 -0.55LASSO -0.84 1.000 -0.27 -1.07 -2.66 -0.24LASSO adaptive -2.36 1.000 -1.52 -2.69 -7.41 -0.453 PRF -3.27 1.000 -4.82 -2.66 -10.55 -0.40PLS -2.13 1.000 -2.05 -2.16 -6.68 -0.29

Table 5: Forecasting performance of the market excess return The market excess return is forecasted for four different data sets (panel A-D) and the methods (1) FC(forecast combination), PCR (principle component regression), LASSO, 3 PRF (Three-pass regression filter) and PLS (partial least squares). The details (2) further specify the method. Ifapplicable, the targeting procedure is denoted as ’none’ (no targeting), ’soft’ (targeting based in LARS), ’hard’ (targeting based on OLS t-stat). For PCR, ’first’ uses only the first component,’ascend’ selects the maximum number of adjacent factors, ’step’ allows for non-adjacent factors. The evaluation measures R2 (3), R2

Rec (5), R2

Exp (6) are the out-of-sample R2 for the whole,the recession and the expansion period in percent. Significance is calculated using MSFE adjusted statistics for the respective time periods. (4) are the Holm Bonferroni adjusted p-values ofthe R2. U(MV) (7) are the annualized certainty equivalent utility gains in percent for a mean variance investor. IR (8) are the annual information ratio differences. *, **, and *** representone-sided statistical significance at the 10, 5 and 1% level.

19




2Rec

R2Exp

U(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 0.72** 0.081* 2.06*** 0.10 1.04* 0.13FC median/none 0.89***0.019** 1.51*** 0.60** 1.35** 0.11PCR first/none -0.71** 0.081* 7.85*** -4.67 -1.13 0.01PCR ascend/none -1.41** 0.081* 7.55*** -5.55 -2.25 -0.11PCR step/none -7.29 1.000 -5.63 -8.05 -10.82 -0.35LASSO -0.62 1.000 -1.43 -0.24 -1.17 -0.07LASSO adaptive -2.48 1.000 -2.61 -2.41 -4.35 -0.193 PRF -3.11** 0.081* 5.26*** -6.99 -5.21 -0.23PLS -5.27 1.000 -0.32 -7.56 -8.20 -0.30



2Rec

R2Exp

U(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 0.14* 0.582 0.35** 0.04 0.20 0.02***

FC mean/hard 0.44** 0.385 1.56*** -0.08 0.63 0.08FC mean/soft 0.09 1.000 0.27* 0.01 0.11 0.02FC median/none -0.02 1.000 0.04 -0.05 -0.04 0.00PCR first/none -0.06 1.000 -0.05 -0.06 -0.08 0.00PCR ascend/none -1.84** 0.385 6.84*** -5.86 -3.00 -0.15PCR step/none -3.11 1.000 3.98*** -6.38 -4.97 -0.23PCR step/hard -3.07 0.835 4.70*** -6.67 -5.24 -0.22PCR step/soft -4.13 1.000 -5.60 -3.44 -7.07 -0.25LASSO -0.63 1.000 -1.57 -0.19 -1.20 -0.07LASSO adaptive -2.87 1.000 -3.00 -2.81 -5.02 -0.213 PRF -5.30** 0.155 5.44***-10.26 -8.38 -0.28PLS 0.06* 0.630 0.10 0.04* 0.09 0.00

Table 6: Forecasting performance of the market excess return - long periodThe market excess return is forecasted for two different data sets (panel A and B) and the methods (1) FC (forecastcombination), PCR (principle component regression), LASSO, 3 PRF (Three-pass regression filter) and PLS (partial leastsquares). The details (2) further specify the method. If applicable, the targeting procedure is denoted as ’none’ (notargeting), ’soft’ (targeting based in LARS), ’hard’ (targeting based on OLS t-stat). For PCR, ’first’ uses only the firstcomponent, ’ascend’ selects the maximum number of adjacent factors, ’step’ allows for non-adjacent factors. The evaluationmeasures R2 (3), R2

Rec (5), R2

Exp (6) are the out-of-sample R2 for the whole, the recession and the expansion periodin percent. Significance is calculated using MSFE adjusted statistics for the respective time periods. (4) are the HolmBonferroni adjusted p-values of the R2. U(MV) (7) are the annualized certainty equivalent utility gains in percent for amean variance investor. IR (8) are the annual information ratio differences. *, **, and *** represent one-sided statisticalsignificance at the 10, 5 and 1% level.

20


5.1.3 Market excess return

Table 5 shows that there is no significant predictability for the market excess return

between 1975 and 2013, which is consistent with findings of no (Welch & Goyal, 2007) or

very limited predictability (Rapach et al., 2010) after the mid of the 1970s. Besides low

out-of-sample R2 (3), the utility gains (7) for investors and their information ratios (8)

are insignificant.

Henkel et al. (2011) discuss the differences in predictability between states of the

economy and claim that there is more predictability in recessions. In the financial data

sets (panel A and panel B), we find some significant results for recessionary phases (5). For

the economics data set (panel C) and the combination data set (panel D) the opposite

holds, but to a lesser extent. Overall, there is no consistent and significant difference

between predictability in expansionary and recessionary phases for the market excess

return.

The longer time period from 1950 onwards can explain the results of other authors

finding strong predictability. Table 6 shows that starting the evaluation period in 1950

instead of 1975, the forecasting performance improves significantly, especially for the

small financial data set in panel A and during recessions (5). Simple methods like fore-

cast combination and restricted principal component regressions perform best and remain

moderately significant even after controlling for the family-wise error rate (4). The sig-

nificant utility gains (7) support these findings. The most successful methods heavily

load on classical measures of the valuation level of the stock market such as the dividend-

price-ratio, the dividend-yield, the (10 year) earnings-price-ratio and the book-to-market

ratio. The findings of Henkel et al. (2011) that predictability is driven by recessions are

supported by the financial data sets starting 1950.

21

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Carl

Equity

Facto

rPredicta

bility



RecR2

ExpU(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 0.29 1.000 0.52 0.17 0.70 0.07

FC median/none -0.05 1.000 0.12 -0.14 -0.08 0.01PCR first/none -2.66 1.000 -1.10 -3.49 -4.84 -0.26PCR ascend/none -4.84 1.000 -0.34 -7.25 -8.67 -0.22PCR step/none -5.25 1.000 2.07 -9.15 -9.44 -0.24

LASSO -3.61 1.000 -0.13 -5.47 -6.53 -0.22LASSO adaptive -8.87 1.000 -2.49 -12.27 -16.53 -0.313 PRF -4.05 1.000 0.38 -6.42 -7.47 -0.28PLS -12.60 1.000 -2.74 -17.86 -22.77 -0.52



RecR2

ExpU(MV) IR

FC mean/none 0.09 1.000 0.14 0.07 0.21 0.02FC mean/hard 0.21 1.000 0.36 0.13 0.52 0.05FC mean/soft 0.13 1.000 0.17 0.11 0.31 0.03FC median/none 0.00 1.000 0.04 -0.02 0.01 0.00***

PCR first/none -0.04 1.000 0.09 -0.11 -0.07 0.00PCR ascend/none -5.26 1.000 -0.39 -7.85 -9.55 -0.25PCR step/none -7.03 1.000 -4.66 -8.29 -13.05 -0.26PCR step/hard -7.70 1.000 -6.74 -8.20 -13.98 -0.25PCR step/soft -6.58** 0.323 0.91** -10.58 -11.15 -0.11LASSO -3.96 1.000 -0.97 -5.56 -7.12 -0.22LASSO adaptive -11.31* 1.000 -4.69 -14.84 -20.97 -0.293 PRF -2.96 1.000 0.28 -4.68 -5.10 -0.19PLS -1.99 1.000 -5.70 -0.01 -3.95 -0.21



RecR2

ExpU(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none -0.25 1.000 -0.77 0.03 -0.91 -0.03FC mean/hard -0.18 1.000 -0.85 0.17 -0.55 -0.01FC mean/soft -0.02 1.000 -0.56 0.27* 0.03 0.00FC median/none -0.28 1.000 -0.70 -0.06 -1.01 -0.04PCR first/none -0.78 1.000 -1.58 -0.36 -3.31 -0.10PCR ascend/none -0.78 1.000 -1.58 -0.36 -3.31 -0.10PCR step/none -0.63 1.000 -1.54 -0.14 -1.88 -0.07PCR step/hard -2.00 1.000 -3.68 -1.11 -7.09 -0.21PCR step/soft -1.52 1.000 -3.38 -0.53* -4.70 -0.12LASSO -0.28 1.000 -0.61 -0.10 -1.16 -0.04LASSO adaptive -0.93 1.000 -1.38 -0.69 -3.85 -0.133 PRF -2.84 1.000 -3.82 -2.31 -11.22 -0.15PLS -1.50 1.000 -0.29 -2.14 -5.96 -0.19



RecR2

ExpU(MV) IR

FC mean/none -0.20 1.000 -0.35 -0.13 -0.74 -0.03FC mean/hard -0.31 1.000 -0.64 -0.13 -1.05 -0.03FC mean/soft -0.27 1.000 -0.63 -0.08 -1.06 -0.04FC median/none -0.13 1.000 -0.15 -0.11 -0.42 -0.02PCR first/none -0.29 1.000 0.49 -0.70 -0.89 -0.03PCR ascend/none -0.29 1.000 0.49 -0.70 -0.89 -0.03PCR step/none -1.09 1.000 -1.44 -0.90 -3.54 -0.12PCR step/hard -3.29 1.000 -5.45 -2.14 -12.31 -0.32PCR step/soft -2.00 1.000 -8.98 1.72*** -6.54 -0.15LASSO -0.07 1.000 0.00 -0.10 -0.28 -0.01LASSO adaptive -0.45 1.000 0.00 -0.69 -1.87 -0.063 PRF -1.89 1.000 -2.23 -1.70 -7.44 -0.13PLS 0.36 1.000 0.00 0.55 1.60 0.06

Table 7: Forecasting performance of the value factorThe value factor is forecasted for four different data sets (panel A-D) and the methods (1) FC (forecast combination), PCR (principle component regression), LASSO, 3 PRF (Three-passregression filter) and PLS (partial least squares). The details (2) further specify the method. If applicable, the targeting procedure is denoted as ’none’ (no targeting), ’soft’ (targeting basedin LARS), ’hard’ (targeting based on OLS t-stat). For PCR, ’first’ uses only the first component, ’ascend’ selects the maximum number of adjacent factors, ’step’ allows for non-adjacentfactors. The evaluation measures R2 (3), R2

Rec (5), R2


22




2Rec

R2Exp

U(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 0.50** 0.294 0.19 0.67** 0.63 0.08FC median/none 0.12 0.294 -0.17 0.28** 0.02 0.04PCR first/none -4.98* 0.294 -3.29 -5.88** -6.68 -0.30PCR ascend/none -8.00** 0.294 -6.66 -8.71** -10.85 -0.25PCR step/none -6.49* 0.294 -0.45* -9.69 -8.16 -0.18LASSO -2.76** 0.278 -0.31 -4.05** -4.09 -0.15LASSO adaptive -7.36** 0.294 -2.54 -9.91** -10.90 -0.233 PRF -6.71* 0.294 -3.16 -8.59** -9.11 -0.29PLS -11.66 0.297 -2.93 -16.28 -16.45 -0.37



2Rec

R2Exp

U(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 0.17* 0.420 0.04 0.25* 0.23 0.03FC mean/hard 0.41* 0.402 0.16 0.54* 0.56 0.06FC mean/soft 0.22 0.420 0.18 0.24 0.33 0.03FC median/none -0.01 1.000 -0.10 0.04 -0.01 0.00PCR first/none -0.08 1.000 -0.10 -0.07 -0.11 -0.01PCR ascend/none -6.81** 0.267 -2.84 -8.91* -9.42 -0.22PCR step/none -6.00***0.113 -1.09** -8.60 -8.76 -0.16PCR step/hard -7.33** 0.135 -4.98 -8.58** -10.28 -0.16PCR step/soft -8.87***0.036** -1.14** -12.97** -10.85 -0.11LASSO -2.43** 0.139 -0.06 -3.68** -3.75 -0.13LASSO adaptive -9.35***0.113 -3.67 -12.36** -13.77 -0.203 PRF -5.21** 0.305 -1.28 -7.29* -6.41 -0.21PLS -1.97 1.000 -2.38 -1.75 -2.72 -0.16

Table 8: Forecasting performance of the value factor - long periodThe value factor is forecasted for two different data sets (panel A and B) and the methods (1) FC (forecast combination),PCR (principle component regression), LASSO, 3 PRF (Three-pass regression filter) and PLS (partial least squares). Thedetails (2) further specify the method. If applicable, the targeting procedure is denoted as ’none’ (no targeting), ’soft’(targeting based in LARS), ’hard’ (targeting based on OLS t-stat). For PCR, ’first’ uses only the first component, ’ascend’selects the maximum number of adjacent factors, ’step’ allows for non-adjacent factors. The evaluation measures R2 (3),R2

Rec (5), R2


23


5.1.4 Value Factor

Table 7 shows no evidence of predictability in the value factor for any specification between

1975 and 2013. Also in terms of utility gains (7), there is no indication that investors

could benefit from timing the value factor.

Using longer financial data sets in table 8, I find moderate predictability for most

methods, which is particularly strong in expansionary phases (6). This time the simple

methods such as forecast combination and restricted principal component regressions

perform worse than the other methods in terms of MSFE adjusted statistics. However,

the opposite holds for the level of the out-of-sample R2 (3), the utility gains (7) and the

information ratio gains (8).

Though finding significant forecast improvements, the signs of the out-of-sample R2

are strongly negative for most methods, which indicates large estimation noise in the

models. As the MSFE adjusted statistic corrects for this effect, the out-of-sample R2 and

the MSFE adjusted statistic can have different signs. Gains in utility and information

ratios remain insignificant. Testing the methods family-wise (4) results in insignificant

results as well.

Overall, there is no predictability of the value factor in recent times and mixed results

for the sample starting 1950.

5.1.5 Momentum Factor

There are some significant results for predictability of the momentum factor in table 9

mostly in recessionary phases (5) and for the simple methods such as forecast combination

and restricted principal component regressions in the large financial data set in panel B

and the combination data set in panel D. Some significant utility gains (7) for the large

financial data set support these findings. However, there is little consistency, only low

significance levels and the results are not robust to family-wise testing (4).

Nevertheless, the out-of-sample R2 (3) and utility gains (7) have economically mean-

ingful sizes. The lack of statistical significance is then an indication that the predictive

gains are not persistently accumulated, but occur during few and short time periods. Ob-

serving the cumulative differences in the mean squared errors, there are distinct jumps in

predictive outperformance particularly after the financial crisis in 2008. These findings are

well in line with the finding of momentum crashes as described in Daniel and Moskowitz

(2013). Usually returns to the momentum factor are accumulated quite consistently, but

momentum returns take a blow, when markets rebound after a large market correction.

This is largely due to momentum having dynamic factor exposures.

24

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Carl

Equity

Facto

rPredicta

bility



RecR2

ExpU(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 0.29 1.000 1.31** -0.36 0.84 0.01

FC median/none 0.26 1.000 0.93** -0.16 0.69 0.01PCR first/none -2.17 1.000 1.28 -4.34 -5.80 -0.04PCR ascend/none -3.01 1.000 1.34 -5.75 -7.86 -0.09PCR step/none -3.57 1.000 2.22* -7.22 -9.32 -0.07

LASSO -3.67 1.000 -0.49 -5.68 -9.49 -0.21LASSO adaptive -7.70 1.000 -1.32 -11.73 -20.13 -0.253 PRF 0.09 1.000 6.50** -3.97 0.24 0.05PLS 0.23* 0.498 2.55* -1.23 0.54 0.01



RecR2

ExpU(MV) IR

FC mean/none 0.71** 0.592 1.53* 0.20 2.04* 0.06FC mean/hard 0.79* 0.592 1.83* 0.14 2.29* 0.07FC mean/soft 0.59* 0.794 1.61** -0.05 1.68 0.05FC median/none 0.62* 0.592 1.33* 0.17 1.76* 0.06PCR first/none 1.09** 0.592 2.31* 0.32 3.12* 0.10PCR ascend/none -3.03 1.000 2.97* -6.81 -7.84 -0.08PCR step/none -4.45 1.000 1.05 -7.92 -11.97 -0.16PCR step/hard -3.12 1.000 2.72* -6.80 -8.30 -0.10PCR step/soft -7.65 1.000 -2.08 -11.17 -20.37 -0.33LASSO -1.66 1.000 -0.28 -2.53 -4.33 -0.12LASSO adaptive -5.65 1.000 -1.35 -8.37 -14.83 -0.203 PRF 0.79** 0.592 5.59** -2.24 2.66 0.08PLS 0.28 0.794 0.82 -0.07 0.90 0.02



RecR2

ExpU(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 0.14 1.000 0.34 0.01 0.10 0.00FC mean/hard 0.06 1.000 0.60 -0.28 -0.62 -0.02FC mean/soft 0.09 1.000 0.41 -0.12 -0.05 0.00FC median/none 0.15 1.000 0.18 0.13 0.37 0.01*

PCR first/none -0.02 1.000 0.43 -0.31 -0.94 -0.02PCR ascend/none -0.02 1.000 0.43 -0.31 -0.94 -0.02PCR step/none -0.09 1.000 0.00* -0.15 -0.46 -0.01PCR step/hard -0.36 1.000 0.56** -0.94 -1.75 -0.04PCR step/soft -0.56 1.000 2.49 -2.48 -2.85 -0.07LASSO 0.77 1.000 2.05 -0.03 2.48 0.06LASSO adaptive -0.75 1.000 3.56 -3.47 -2.43 -0.043 PRF -0.49 1.000 2.48 -2.37 -4.28 -0.11PLS -0.34 1.000 0.00* -0.56 -1.20 -0.03



RecR2

ExpU(MV) IR

FC mean/none 0.50* 0.837 1.14* 0.10 1.75 0.05FC mean/hard 0.90* 0.837 2.35** -0.02 3.04 0.08FC mean/soft 0.44* 0.837 0.96* 0.10 1.55 0.04FC median/none 0.19 0.837 0.50* 0.00 0.72 0.02PCR first/none 1.21* 0.804 3.29* -0.10 5.20* 0.12PCR ascend/none 1.22* 0.693 3.29* -0.08 5.22* 0.12PCR step/none 0.20 0.837 1.24 -0.46 1.10 0.03PCR step/hard 1.31* 0.837 3.68 -0.18 5.18 0.12PCR step/soft 0.34* 0.804 0.48 0.25** 1.08 0.03LASSO -0.47 1.000 -0.01 -0.75 -1.67 -0.04LASSO adaptive -2.61 1.000 -2.57 -2.64 -8.90 -0.203 PRF -0.07 0.837 3.55** -2.36 -1.86 -0.04PLS 0.24 0.837 0.32 0.19 0.87 0.02

Table 9: Forecasting performance of the momentum factorThe momentum factor is forecasted for four different data sets (panel A-D) and the methods (1) FC (forecast combination), PCR (principle component regression), LASSO, 3 PRF (Three-passregression filter) and PLS (partial least squares). The details (2) further specify the method. If applicable, the targeting procedure is denoted as ’none’ (no targeting), ’soft’ (targeting basedin LARS), ’hard’ (targeting based on OLS t-stat). For PCR, ’first’ uses only the first component, ’ascend’ selects the maximum number of adjacent factors, ’step’ allows for non-adjacentfactors. The evaluation measures R2 (3), R2

Rec (5), R2


25




2Rec

R2Exp

U(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 0.26* 0.452 1.29** -0.36 0.56 -0.01FC median/none 0.30** 0.330 0.98** -0.10 0.57 0.00PCR first/none -3.99* 0.452 1.07* -7.04 -7.24 -0.06PCR ascend/none -4.91* 0.452 0.41* -8.12 -8.99 -0.09PCR step/none -4.27 0.678 -1.18 -6.14 -8.30 -0.13LASSO -3.35 1.000 -1.02 -4.75 -6.55 -0.17LASSO adaptive -6.96 1.000 -2.00 -9.96 -13.78 -0.213 PRF -2.61* 0.424 4.80** -7.08 -4.21 -0.02PLS -3.76 1.000 -0.14 -5.95 -6.96 -0.15



2Rec

R2Exp

U(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none 0.52* 0.657 1.21* 0.11 1.22* 0.04FC mean/hard 0.56* 0.657 1.44* 0.03 1.36* 0.03FC mean/soft 0.40 0.925 1.16* -0.07 0.94 0.03FC median/none 0.42* 0.731 1.05* 0.05 1.02* 0.03PCR first/none 0.68* 0.729 1.73* 0.04 1.70 0.06PCR ascend/none -5.52 0.925 -0.05 -8.83 -9.97 -0.11PCR step/none -5.08 1.000 -1.52 -7.23 -10.06 -0.18PCR step/hard -4.90 1.000 -1.27 -7.09 -9.30 -0.14PCR step/soft -9.65 1.000 -5.50 -12.17 -18.10 -0.32LASSO -1.44 1.000 -0.48 -2.02 -2.89 -0.11LASSO adaptive -5.64 1.000 -2.01 -7.84 -11.07 -0.193 PRF -1.07** 0.462 3.78** -4.01 -0.85 -0.01PLS 0.56 0.925 2.38* -0.54 1.50 0.04

Table 10: Forecasting performance of the momentum factor - long periodThe momentum factor is forecasted for two different data sets (panel A and B) and the methods (1) FC (forecast combina-tion), PCR (principle component regression), LASSO, 3 PRF (Three-pass regression filter) and PLS (partial least squares).The details (2) further specify the method. If applicable, the targeting procedure is denoted as ’none’ (no targeting), ’soft’(targeting based in LARS), ’hard’ (targeting based on OLS t-stat). For PCR, ’first’ uses only the first component, ’ascend’selects the maximum number of adjacent factors, ’step’ allows for non-adjacent factors. The evaluation measures R2 (3),R2

Rec (5), R2


26

Ulrich

Carl

Equity

Facto

rPredicta

bility



RecR2

ExpU(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none -0.16 1.000 0.34 -0.33 -0.79 -0.06

FC median/none -0.15 1.000 -0.02 -0.19 -0.70 -0.03PCR first/none -0.64 1.000 0.08 -0.87 -3.25 -0.15PCR ascend/none -0.67 1.000 -0.06 -0.87 -3.39 -0.16PCR step/none -2.17 1.000 -4.40 -1.45 -10.43 -0.39

LASSO 0.00 1.000 0.00 0.00 0.00 0.00LASSO adaptive 0.00 1.000 0.00 0.00 0.00 0.003 PRF -2.59 1.000 1.11 -3.79 -13.09 -0.28PLS -0.83 1.000 -1.64 -0.57 -5.41 -0.23



RecR2

ExpU(MV) IR

FC mean/none -0.17 1.000 -0.14 -0.18 -0.71 -0.03FC mean/hard -0.32 1.000 -0.20 -0.36 -1.37 -0.07FC mean/soft -0.24 1.000 -0.23 -0.24 -1.06 -0.05FC median/none -0.09 1.000 -0.13 -0.08 -0.35 -0.02PCR first/none -0.27 1.000 -0.48 -0.20 -1.08 -0.05PCR ascend/none -0.27 1.000 -0.48 -0.20 -1.08 -0.05PCR step/none -1.36 1.000 -1.35 -1.36 -6.56 -0.24PCR step/hard -3.80 1.000 -1.95 -4.40 -20.77 -0.47PCR step/soft -6.08 1.000 -6.77 -5.86 -29.42 -0.60LASSO 0.00 1.000 0.00 0.00 0.00 0.00LASSO adaptive 0.00 1.000 0.00 0.00 0.00 0.003 PRF -5.87 1.000 -3.45 -6.65 -27.38 -0.48PLS -0.13 1.000 -0.52 -0.01 -0.73 -0.03



RecR2

ExpU(MV) IR

(1) (2) (3) (4) (5) (6) (7) (8)

FC mean/none -0.30 1.000 -0.49 -0.23 -1.40 -0.07FC mean/hard -0.69 1.000 -0.98 -0.60 -3.31 -0.17FC mean/soft -0.26 1.000 -0.30 -0.25 -1.12 -0.07FC median/none -0.16 1.000 -0.29 -0.11 -0.72 -0.04PCR first/none -0.65 1.000 -1.84 -0.26 -3.08 -0.14PCR ascend/none -0.65 1.000 -1.84 -0.26 -3.08 -0.14PCR step/none -0.08 1.000 0.00 -0.10 -0.37 -0.06PCR step/hard -1.65 1.000 -1.82 -1.60 -9.27 -0.30PCR step/soft -1.79 1.000 1.29 -2.78 -8.55 -0.31LASSO -0.45 1.000 0.00 -0.60 -1.60 -0.07LASSO adaptive -1.01 1.000 0.00 -1.33 -3.62 -0.183 PRF -3.12 1.000 -1.70 -3.57 -15.91 -0.43PLS -0.74 1.000 -3.03 0.00 -2.68 -0.13



RecR2

ExpU(MV) IR

FC mean/none -0.25 1.000 -0.32 -0.22 -1.12 -0.06FC mean/hard -0.68 1.000 -0.85 -0.62 -3.14 -0.15FC mean/soft -0.23 1.000 -0.11 -0.27 -1.05 -0.06FC median/none -0.11 1.000 -0.19 -0.09 -0.48 -0.03PCR first/none -0.35 1.000 -0.64 -0.26 -1.49 -0.07PCR ascend/none -0.35 1.000 -0.64 -0.26 -1.49 -0.07PCR step/none 0.16* 0.685 0.61 0.02 0.87 0.02PCR step/hard -3.01 1.000 0.34 -4.08 -16.20 -0.46PCR step/soft -3.26 1.000 -3.28 -3.25 -16.37 -0.56LASSO -0.30 1.000 0.00 -0.39 -1.02 -0.04LASSO adaptive -0.88 1.000 0.00 -1.17 -3.01 -0.143 PRF -5.75 1.000 -5.74 -5.76 -26.70 -0.55PLS 0.00 1.000 0.00 0.00 0.00 0.00

Table 11: Forecasting performance of the quality factorThe quality factor is forecasted for four different data sets (panel A-D) and the methods (1) FC (forecast combination), PCR (principle component regression), LASSO, 3 PRF (Three-passregression filter) and PLS (partial least squares). The details (2) further specify the method. If applicable, the targeting procedure is denoted as ’none’ (no targeting), ’soft’ (targeting basedin LARS), ’hard’ (targeting based on OLS t-stat). For PCR, ’first’ uses only the first component, ’ascend’ selects the maximum number of adjacent factors, ’step’ allows for non-adjacentfactors. The evaluation measures R2 (3), R2

Rec (5), R2


27


Especially after market downturns the momentum factor invests in low beta stocks,

while shorting high beta stocks (Blitz, Huij, & Martens, 2011). Thus, as the markets pick

up after a large drawdown, momentum has a significantly negative loading on the market

beta and thus suffers commensurately. In general, for momentum factor prediction, it

is important to detect the times of large drawdowns in the momentum factor, which

are relatively rare. In this paper, the focus is on linear models, which do not allow for

time-varying regression coefficients. As the momentum factor behaves very differently

over time, regime switching models are probably more appropriate for momentum factor

predictability as done in Daniel et al. (2012). Using longer financial data sets in table 10

does not substantially alter the previously discussed results. For the small financial data

set in panel A, the levels of the statistics do not change. Due to the longer horizon,

the significance increases such that there are moderately significant out-of-sample R2

for forecast combination and restricted principal component regressions. For the large

financial data set in panel B, in contrast, the results moderately deteriorate.

5.1.6 Quality Factor

The attempts to predict the quality factor in table 11 fail completely, out-of-sample R2

(3) are consistently negative across all methods, data sets and business cycle phases.

Investors that pursue a quality factor timing strategy also suffer from consistent utility

loss (7) in all specifications. Using a longer financial data set as for the other factors is

infeasible, as the time series of the quality factor is only available since mid of 1957.

5.2 Commonality in predicted returns

5.2.1 Visual analysis

Having a closer look at the forecasted returns for the different factors, there is a distinct

cyclicality, which is closely related to business cycles. Rapach and Strauss (2010) describe

this pattern for the market excess return, but it is also present in other factors. For the

size factor in figure 2, the market excess return in figure 3 and the value factor in figure 4,

there is a decay in forecasted returns during expansions. During recessions, in contrast, the

forecasted returns increase quickly. Some methods that critically depend on selection such

as lasso, partial least squares and principal component regression produce noisier estimates

in several cases. In in-sample estimates this cyclicality is even more pronounced as long as

the respective method does not collapse to the historical mean estimate. This anti-cyclical

behaviour of the market, size and value factors with high predicted returns in recessions

and low predicted returns in expansions supports the prediction of the consumption-based

asset pricing model (Breeden, 1979) that risk premia need to be high when consumption

28


Ret

urn

in %

FC (mean)

1960 1980 2000

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Ret

urn

in %

FC (median)

1960 1980 2000

0.4

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Ret

urn

in %

PCR

1960 1980 2000

0.5

1.0

1.5

Ret

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in %

LASSO

1960 1980 2000

0.0

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1.0

Ret

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in %

3PRF

1960 1980 2000

−0.5

0.0

0.5

1.0

1.5

2.0

Ret

urn

in %

PLS

1960 1980 2000

−1.0

0.0

1.0

2.0

3.0

Figure 1: Estimated returns to the low beta factorThese graphs show the out-of-sample return forecasts for the small financial data set for six different estimation methodsfrom 1950 to 2013. The method in the top figures is forecast combination (FC) averaging across all forecasts (mean) or usingthe median forecast only (median). The figures in the middle show principal component regression (PCR) with adjacentfactors (ascent) and the least absolute shrinkage and selection estimator (lasso). The bottom figures display the three passregression filter (3PRF) and partial least squares (PLS). The grey coloured time periods correspond to NBER recessions.

Ret

urn

in %

FC (mean)

1960 1980 2000

−0.2

0.0

0.2

0.4

0.6

Ret

urn

in %

FC (median)

1960 1980 2000

−0.1

0.0

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0.5

0.6

Ret

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in %

PCR

1960 1980 2000

−1.0

0.0

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Ret

urn

in %

LASSO

1960 1980 2000

−0.5

0.0

0.5

1.0

1.5

Ret

urn

in %

3PRF

1960 1980 2000

−1.0

0.0

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Ret

urn

in %

PLS

1960 1980 2000−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

Figure 2: Estimated returns to the size factorThese graphs show the out-of-sample return forecasts for the small financial data set for six different estimation methodsfrom 1950 to 2013. The method in the top figures is forecast combination (FC) averaging across all forecasts (mean) or usingthe median forecast only (median). The figures in the middle show principal component regression (PCR) with adjacentfactors (ascent) and the least absolute shrinkage and selection estimator (lasso). The bottom figures display the three passregression filter (3PRF) and partial least squares (PLS). The grey coloured time periods correspond to NBER recessions.

29


Ret

urn

in %

FC (mean)

1960 1980 2000

0.2

0.4

0.6

0.8

1.0

Ret

urn

in %

FC (median)

1960 1980 2000

0.2

0.4

0.6

0.8

1.0

Ret

urn

in %

PCR

1960 1980 2000

−1.0

−0.5

0.0

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1.5

2.0

Ret

urn

in %

LASSO

1960 1980 2000

0.0

0.5

1.0

1.5

2.0

Ret

urn

in %

3PRF

1960 1980 2000−2.0

−1.0

0.0

1.0

2.0

3.0

Ret

urn

in %

PLS

1960 1980 2000

−2.0

0.0

2.0

4.0

6.0

Figure 3: Estimated returns to the market excess returnThese graphs show the out-of-sample return forecasts for the small financial data set for six different estimation methodsfrom 1950 to 2013. The method in the top figures is forecast combination (FC) averaging across all forecasts (mean) or usingthe median forecast only (median). The figures in the middle show principal component regression (PCR) with adjacentfactors (ascent) and the least absolute shrinkage and selection estimator (lasso). The bottom figures display the three passregression filter (3PRF) and partial least squares (PLS). The grey coloured time periods correspond to NBER recessions.

Ret

urn

in %

FC (mean)

1960 1980 2000

−0.2

0.0

0.2

0.4

0.6

0.8

Ret

urn

in %

FC (median)

1960 1980 2000

−0.2

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Ret

urn

in %

PCR

1960 1980 2000−2.0

−1.0

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5.0

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in %

LASSO

1960 1980 2000−1.0

0.0

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in %

3PRF

1960 1980 2000

−2.0

−1.0

0.0

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2.0

Ret

urn

in %

PLS

1960 1980 2000−5.0

0.0

5.0

Figure 4: Estimated returns to the value factorThese graphs show the out-of-sample return forecasts for the small financial data set for six different estimation methodsfrom 1950 to 2013. The method in the top figures is forecast combination (FC) averaging across all forecasts (mean) or usingthe median forecast only (median). The figures in the middle show principal component regression (PCR) with adjacentfactors (ascent) and the least absolute shrinkage and selection estimator (lasso). The bottom figures display the three passregression filter (3PRF) and partial least squares (PLS). The grey coloured time periods correspond to NBER recessions.

30


Ret

urn

in %

FC (mean)

1960 1980 2000

0.4

0.6

0.8

1.0

1.2

Ret

urn

in %

FC (median)

1960 1980 2000

0.4

0.6

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1.0

1.2

Ret

urn

in %

PCR

1960 1980 2000

−2.0

−1.0

0.0

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in %

LASSO

1960 1980 2000

−1.5

−1.0

−0.5

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Ret

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in %

3PRF

1960 1980 2000−2.0

−1.0

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Ret

urn

in %

PLS

1960 1980 2000

−1.0

0.0

1.0

2.0

3.0

4.0

Figure 5: Estimated returns to the momentum factorThese graphs show the out-of-sample return forecasts for the small financial data set for six different estimation methodsfrom 1950 to 2013. The method in the top figures is forecast combination (FC) averaging across all forecasts (mean) or usingthe median forecast only (median). The figures in the middle show principal component regression (PCR) with adjacentfactors (ascent) and the least absolute shrinkage and selection estimator (lasso). The bottom figures display the three passregression filter (3PRF) and partial least squares (PLS). The grey coloured time periods correspond to NBER recessions.

is low.

In contrast to these three anti-cyclical factors, the momentum factor in figure 5 is

clearly pro-cyclical with the forecasted returns rising in expansions and decreasing in

recessions. The forecasted returns to the momentum factor are the mirror image of the

forecasted returns to the market, size and value factors. This also supports the claim that

the value factor and the momentum factor are well suited complements in investing such

as Asness (1997) and Asness, Moskowitz, and Pedersen (2013). There is also a wide range

of literature that links the size factors (Vassalou & Xing, 2004; Petkova, 2006) and value

factors (Fama & French, 1995; Zhang, 2005) to systematic risk, while for the momentum

factor, behavioural based arguments dominate (Shefrin & Statman, 1985; Hong & Stein,

1999; Grinblatt & Han, 2005).

The cyclicality of the predicted returns to the low beta factor in figure 1 is not as

closely related to the business cycle as the four factors previously discussed. However, the

expected returns to the low beta factor mostly decrease before and at the beginning of a

recession, while they increase towards the end and shortly after recessions. This finding

could be due to the exposure of the low beta factor to funding liquidity risk as discussed

in Frazzini and Pedersen (2014). In general, the estimates are noisier than for the other

equity factors considered.

31


Finally, the predicted returns to the quality factor do not show a distinct cyclical

behaviour. Instead they are closely linked to the historical average estimates for most

methods and data sets with only limited and seemingly random deviations. Thus, the

quality factor is excluded in the following analyses and not displayed here.

5.2.2 Correlations and factor structure

Analysing the correlations between the deviations of the predicted returns from the histor-

ical mean estimate for the five different factors, we can observe distinct patterns. There

are two distinct correlation groups that have positive correlations within and negative

correlations across groups. The first group consists of the market excess return, the size

factor and the value factor, while the second group consists of the momentum factor and

to a lesser extent the low beta factor. However, the low beta factor has the weakest cor-

relations in absolute size and is in many settings close to uncorrelated to the other four

factors. Table 12 shows the aggregated pattern across all methods and all data sets.

The intensity of the correlation structure is strongly dependent on the forecasting

methods and data sets as is detailed in figure 6 in the appendix. This intensity is strongest

for the simple methods such as forecast combination and restricted principal component

regressions, which in tendency also show the best forecast performance. The three pass

regression filter also captures the correlation structure fairly well. Moreover, the pattern

is most visible in the small financial data set followed by the large financial data set. For

the economics and combination data sets it becomes more and more blurry. It is virtually

non-existent for complex principal component regressions, lasso approaches and partial

least squares.

Using principal component analysis to understand the structure of the deviations of

the predicted returns from the historical mean estimate for the five different factors, we

see in table 13 that depending on the data set, on average 55-80% of the variance of

Mkt SMB HML UMD BAB

Mkt 1.00 0.42 0.38 -0.48 -0.02SMB 0.42 1.00 0.33 -0.36 0.17HML 0.38 0.33 1.00 -0.57 -0.14UMD -0.48 -0.36 -0.57 1.00 0.21BAB -0.02 0.17 -0.14 0.21 1.00

Table 12: Correlations between predicted return deviationsThis table shows the correlations between the predicted return deviations from the historical mean estimate for each of thefive factors averaged across all data sets and methods. The five factors are the market excess return (Mkt), size (SMB), value(HML), momentum (UMD) and low beta (BAB). The average is formed across four data sets and 13 different predictionmethods.

32


Component 1 2 3 4 5

All 66.9 17.7 8.4 4.5 2.5GW 79.9 9.7 5.4 3.2 1.9

Financial 71.9 16.2 6.0 3.9 2.1Economics 60.9 21.0 10.1 5.1 2.9

Combination 54.9 23.8 12.3 5.8 3.2

Table 13: Factor structure of predicted return deviationsThis table shows the percentage of the variance of the predicted return deviations from the historical mean estimateexplained by principal components. Five equity factors, the market excess return, size, value, momentum and low betaform the base assets. I present the factor structure for four different data sets as well as the mean across the data sets (All).The four data sets are (1) the small financial data set (GW), (2) the large financial data set (Financial), (3)the economicsdata set (Economics) and (4) the combination data set (Combination). For all data sets, the factor structure is averagedacross all 13 methods used in this paper.

the predicted return deviations can be explained by the first principal component. The

differences in the explained variance between data sets follow the pattern already found

in the correlation analysis. Figure 7 in the appendix gives more details on the factor

structure for each method and data set separately.

The first principal component often loads similarly on the first four factors (market

excess return, size, value, inverted momentum). Figure 8 in the appendix details the

principal component regression loadings. These results hint towards a common component

driving the predicted returns of the different factors particularly the market, size, value

and momentum factor. This effect is pronounced for simple forecasting techniques and

for the small financial data set followed by the large financial data set. This reflects the

findings of the correlation analysis.

5.2.3 Explanatory regressions

According to the consumption based asset pricing model (Breeden, 1979), the predictabil-

ity in the factor returns is linked to economic conditions. Thus, in a first step I regress

the predicted return deviations on a dummy representing the National Bureau of Eco-

nomic Research (NBER) recessions and expansions. Table 14 shows the coefficients of the

recession dummy and the significance level for each of the five equity factors across four

data sets and 13 estimation methods.

For most specifications, recessions have a statistically significant impact on the pre-

dicted return deviations. On average, the predicted returns rise during recessions by 1.73%

for the market excess return, by 2.97% for the size factor and by 2.83% for the value fac-

tor. For the momentum factor (-2.16%) and the low beta factor (-1.82%), in contrast, the

predicted returns fall during recessions. The significance is particularly pronounced for

the financial data sets, while it is substantially weaker for the economic data sets. This is

interesting, as we would intuitively expect higher significance for the economics data set

33


if macroeconomic conditions were a strong driver of the predicted return deviations.

Even though the results are mostly significant, the R2 remains low. The state of the

business cycle explains only between 0.3% and 8.9% of the variation in the predicted

return deviations. On average only about 3% of the variation can be explained. One

exception is the size factor with an R2 of around 7%, which shows a stronger link to the

business cycle than the other four factors.

To sum up, while the state of the business cycle plays a statistically significant role

in explaining the predicted returns, it is far from being the dominant driver. Using more

sophisticated measures of the business cycle such as OECD Composite Indicators does

not change the basic results.

Finally, to see if there is one common component in the predicted return deviations, I

form a synthetic factor consisting of a simple average of the predicted return deviations for

the market excess return, the size factor, the value factor and the (inverse) momentum

factor. Table 15 shows the regression R2 obtained from a regression of the predicted

returns on the synthetic factor. Across 13 methods and four data sets, this synthetic

factor can on average explain more than 50% of the variance in the predicted return

deviations of these four factors. The low beta factor, in contrast, shows only a very

moderate relation to the synthetic factor.

However, the results again depend heavily on the respective data sets and methods

for most factors. The synthetic factor is by far strongest for the small financial data set,

while it is weakest for the combination data set. As in the previous findings, the simple

methods such as forecast combination and restricted principal component regression are

best explained by the synthetic factor except for the large financial data set.

Overall, the predicted return deviations from the historical mean estimate for the

market, size, value and momentum factors are strongly related. One single synthetic

factor is driving most of the predicted return deviations.

34

Ulrich

Carl

Equity

Facto

rPredicta

bility

Mkt SMB HML UMD BAB

Panel A: Small Financial Dataset

FC mean/none 1.47*** 1.75*** 2.17*** -1.89*** -0.49***

FC mean/hard 1.47*** 1.75*** 2.17*** -1.89*** -0.49***

FC mean/soft 1.47*** 1.75*** 2.17*** -1.89*** -0.49***

FC median/none 1.34*** 0.97*** 1.22*** -1.15*** -0.20***

PCR first/none 3.39*** 2.77*** 3.20*** -2.99*** -0.54***

PCR ascend/none 3.57*** 6.02*** 4.42*** -1.93 -1.83***

PCR step/none 3.42*** 3.68*** 4.29*** -1.41 -3.39***

PCR step/hard 3.42*** 3.68*** 4.29*** -1.41 -3.39***

PCR step/soft 3.42*** 3.68*** 4.29*** -1.41 -3.39***

LASSO 3.80*** 4.76*** 7.14*** -5.20*** -0.51**

LASSO adaptive 4.38*** 7.08*** 9.85*** -7.13*** -1.13**

3 PRF 5.17*** 5.42*** 7.92*** -6.24*** -4.01***

PLS 4.53*** 6.72*** 2.36 -2.87*** -6.19***

Panel B: Large Financial Dataset

FC mean/none 0.35*** -0.20 0.50*** -1.02*** -0.84***

FC mean/hard 1.07*** -0.24 1.62*** -1.63*** -1.09***

FC mean/soft 0.68*** 0.49** 0.86*** -1.45*** -0.92***

FC median/none 0.17*** -0.41** 0.02 -0.58*** -0.69***

PCR first/none 0.27*** -0.98*** 0.01 -1.01*** -1.18***

PCR ascend/none 2.21** 1.80* 10.46*** -9.40*** -1.51***

PCR step/none -0.32 6.49*** 11.15*** -8.11*** -5.23***

PCR step/hard 2.61** 9.63*** 12.63*** -7.00*** -4.65***

PCR step/soft 9.82*** 8.93*** 15.68*** -13.70*** -5.56***

LASSO 3.86*** 3.82*** 7.99*** -5.05*** -2.08***

LASSO adaptive 4.57*** 6.13*** 9.73*** -9.43*** -3.44***

3 PRF 7.26*** 2.20 9.61*** -8.04*** -7.25***

PLS -0.23 9.05*** -0.87 -0.93** -1.57***

Mean

Mkt SMB HML UMD BAB

Panel C: Economics Dataset

FC mean/none 1.00*** 1.55*** 0.18 -0.19 0.53***

FC mean/hard 2.26*** 2.48*** 0.20 -0.14 0.75***

FC mean/soft 0.57* 1.34*** 0.11 -0.17 0.02

FC median/none 0.62*** 1.10*** 0.07 -0.16 0.45***

PCR first/none 3.39*** 4.69*** 0.93*** -1.10** 2.85***

PCR ascend/none 3.27*** 4.69*** 0.93*** -1.10** 1.70**

PCR step/none -0.31 1.17 0.47 0.10 -3.64***

PCR step/hard -1.57* 2.77*** -0.90* 0.82 -2.40**

PCR step/soft -2.77** 0.93 1.90** 0.30 -2.12*

LASSO -0.64 1.06*** 0.33* -0.07 -0.85***

LASSO adaptive -0.91 2.41*** 0.73* 0.17 -1.95***

3 PRF 3.14*** 5.39*** 2.54*** -0.37 2.11***

PLS -0.43 3.19*** 0.33 0.38 -1.55

Panel D: Financial & Economics Dataset

FC mean/none 0.80*** 0.88*** 0.18** -0.43*** -0.15

FC mean/hard 1.56*** 2.07*** 0.39* -1.15*** -0.09

FC mean/soft 0.42** 0.67*** 0.10 -0.35** -0.63***

FC median/none 0.62*** 0.31*** 0.09* -0.16** 0.05

PCR first/none 0.29 -0.28*** 0.27 -0.96** -1.79***

PCR ascend/none 0.22 2.16*** 0.27 -0.65 -3.45***

PCR step/none -1.44 0.87 0.31 0.64 -7.22***

PCR step/hard 3.57** 4.66*** -0.36 -1.12 -4.99***

PCR step/soft -2.55 1.80* -0.76 -0.60 -5.79***

LASSO -0.18 1.20*** 0.15 0.19 -1.23**

LASSO adaptive -0.09 2.93*** 0.34 0.89 -1.79

3 PRF 6.75*** 6.98*** 3.23*** -2.47** 1.53

PLS -1.03 0.60 0.08 0.26 -2.68**

1.73 2.97 2.83 -2.16 -1.82

Table 14: Recessions as drivers of predicted factor returnsThis table shows the coefficients of a regression of the predicted factor returns on a dummy representing the National Bureau of Economic Research (NBER) recessions and expansions. Thereare predicted returns for five factors, which are the market excess return (Mkt), the size factor (SMB), the value factor (HML), the momentum factor (UMD) and the low beta factor (BAB).There is one panel for each of the four data sets. The four different data sets are (1) the small financial data set, (2) the large financial data set, (3) the economics data set and (4) thecombination data set. The 13 rows of plots represent the 13 methods used in this paper: four forecast combination versions, five principal component regression versions, two lasso versions, thethree pass regression filter and partial least squares. The methods in column 1 are FC (forecast combination), PCR (principle component regression), LASSO, 3 PRF (Three-pass regressionfilter) and PLS (partial least squares). The details in column 2 further specify the method. If applicable, the targeting procedure is denoted as ’none’ (no targeting), ’soft’ (targeting basedin LARS), ’hard’ (targeting based on OLS t-stat). For PCR, ’first’ uses only the first component, ’ascend’ selects the maximum number of adjacent factors, ’step’ allows for non-adjacentfactors. *, **, and *** represent two-sided statistical significance at the 10, 5 and 1% level.

35

Ulrich

Carl

Equity

Facto

rPredicta

bility

Mkt SMB HML UMD BAB

Panel A: Small Financial Dataset

FC mean/none 0.93 0.87 0.98 0.96 0.08FC mean/hard 0.93 0.87 0.98 0.96 0.08FC mean/soft 0.93 0.87 0.98 0.96 0.08FC median/none 0.77 0.84 0.91 0.80 0.12PCR first/none 0.98 0.99 0.99 1.00 0.21PCR ascend/none 0.80 0.56 0.88 0.80 0.03PCR step/none 0.60 0.60 0.79 0.56 0.06PCR step/hard 0.60 0.60 0.79 0.56 0.06PCR step/soft 0.60 0.60 0.79 0.56 0.06LASSO 0.82 0.60 0.97 0.94 0.06LASSO adaptive 0.73 0.59 0.94 0.92 0.043 PRF 0.89 0.76 0.95 0.91 0.20PLS 0.59 0.46 0.39 0.39 0.01

Panel B: Large Financial Dataset


Mean

Mkt SMB HML UMD BAB

Panel C: Economics Dataset


Panel D: Financial & Economics Dataset


0.60 0.50 0.54 0.59 0.11

Table 15: Explanatory power of a synthetic factorThis table shows the coefficients of determination (R2) of a regression of the predicted factor returns on a synthetic factor. There are predicted returns for five factors, which are the marketexcess return (Mkt), the size factor (SMB), the value factor (HML), the momentum factor (UMD) and the low beta factor (BAB). The synthetic factor is an equal weighted combination ofthe market excess return, size, value and inverse momentum. There is one panel for each of the four data sets. The four different data sets are (1) the small financial data set, (2) the largefinancial data set, (3) the economics data set and (4) the combination data set. The 13 rows of plots represent the 13 methods used in this paper: four forecast combination versions, fiveprincipal component regression versions, two lasso versions, the three pass regression filter and partial least squares. The methods in column 1 are FC (forecast combination), PCR (principlecomponent regression), LASSO, 3 PRF (Three-pass regression filter) and PLS (partial least squares). The details in column 2 further specify the method. If applicable, the targeting procedureis denoted as ’none’ (no targeting), ’soft’ (targeting based in LARS), ’hard’ (targeting based on OLS t-stat). For PCR, ’first’ uses only the first component, ’ascend’ selects the maximumnumber of adjacent factors, ’step’ allows for non-adjacent factors. *, **, and *** represent two-sided statistical significance at the 10, 5 and 1% level.

36


6 Conclusions

Predictability remains limited for most equity factors. Results are often heavily affected

by the data set, the time period and the methods applied. Nevertheless, there is distinct

predictability for some factors, particularly low beta and size, which is much stronger

than for the market excess return used in classical return predictability literature.

I find the most pronounced predictability for the low beta factor, mostly due to lagged

return series like industry returns and size-value-portfolio returns. There is also some

significant predictability for the size factor, which is most closely related to macroeconomic

growth as evidenced by positive loadings on industrial production, and negative loadings

on unemployment as well as default yield spreads. While the focus of return predictability

is on the market excess return, there is virtually no predictability after 1975 with stronger

results in longer samples starting in 1950. For the value factor, there is no evidence

of predictability after 1975 with mixed results in the longer samples. The predictive

ability for the momentum factor is limited, mostly due to the fact that the predictive

gains are not accumulated consistently, but usually during short periods of ”momentum

crashes”. Thus, classification or regime-switching approaches may be better suited for the

momentum factor. Finally, there is no indication of any predictive ability for the quality

factor.

The predicted factor returns are related to the business cycle with the exception of

the quality factor. The predicted returns to the market excess return, the size factor and

the value factor are pro-cyclical with respect to consumption, increasing in recessions and

decreasing in expansions. This also relates predictability of these factors to time-varying

risk premia in the consumption based asset pricing model (Breeden, 1979). The predicted

returns to the momentum factor, in contrast, are anti-cyclical, decreasing in recessions

and increasing in expansions.

The predicted returns of these four factors are tightly interrelated. A synthetic factor

constructed by averaging across the predicted factor return deviations for the market

excess return, the size factor, the value factor and the momentum factor can explain more

than 50% of the variance in most cases. For the returns to the low beta factor, the links

to the business cycle are not as pronounced as for the other factors and could be related

to funding liquidity.

37


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41


A Robustness Checks: Predictive Performance

In general, the results are robust to rolling versus expanding window specifications and a

larger forecast horizon. Lag specification can play a large role. In the baseline settings,

the specification is conservative and ensures that all information is available at the time

of the forecast and there is enough time to implement the respective trading strategy.

A.1 Low Beta Factor

Rolling windows increase out-of-sample R2 as well as utility gains, but do not impact the

statistical significance of the results.

The results depend crucially on the lag specification. Shortening the lag specification

to adjacent periods massively increases predictability for all financial data sets. Especially

the large financial data set and the combined data set achieve very high levels of out-of-

sample R2, statistical significance as well as utility gains. The impact on the economics

data set, however, is negligible. Increasing the lag by another period massively decreases

the out-of-sample R2, significance levels as well as utility gains for the large financial data

set and the combined data set. However, the results remain borderline significant for the

overall out-of-sample R2 and highly significant for the expansion out-of-sample R2. The

economic data set, again, is not impacted.

Increasing the forecast horizon to 3 months does not have a material impact on the

previously reported results.

A.2 Size Factor

Switching to rolling windows does not have an impact on the results.

Especially for the size factor, it is important to correctly specify the forecasting lags.

Having no implementation lag for the financial data set for the one-period ahead forecasts

as often done in the literature, leads to massively overblown predictability results as small

capitalization stock returns show auto-correlation due to bid-ask-bounces and illiquidity.

Thus, shortening the lag compared to the baseline specification massively increases pre-

dictability and utility gains for the datasets containing lagged returns. Increasing the time

lag compared to the baseline specification by another month, however, does not alter the

results. This clearly supports the conservative strategy of a gap of one month for strategy

implementation applied in this paper.

Increasing the forecast horizon supports the previous findings and even increases the

statistical significance of the predictability, particularly for the large financial data set,

which mostly showed insignificant results before.

42


A.3 Market excess return

Using rolling instead of expanding windows produces similar results.

Reducing the time lag to directly adjacent periods slightly but consistently improves

forecasting performance though not in a statistically significant way. Increasing the gap to

two periods deteriorates the performance. Thus, there is some information that gets lost,

if longer lags are considered. Generally, shortening the time lag increases the patterns

found in the baseline scenario, while increasing the lag weakens those patterns.

Increasing the forecast horizon to three months does not have a large impact.

A.4 Value Factor

A rolling window approach leads to increased estimation noise and rather deteriorates the

forecasting performance.

Shortening the lag to directly adjacent periods improves the predictability of the large

financial data set massively, leading to significant results, particularly during expansionary

phases. This increased predictability is again related to lagged returns and very similar

to the effect observed for the size factor. Increasing the lag compared to the baseline

specification does not have an impact on the results. This again supports the conservative

strategy of a gap of one month for strategy implementation applied in this paper.

Increasing the forecast horizon leads to a little bit more predictability (barely signifi-

cant) in the expansionary state in the financial data sets, while reducing predictability in

the recessionary state.

A.5 Momentum Factor

Using rolling instead of expanding windows as well as altering the lag specification by

shortening or increasing the time lag by one period does not have an impact on the

results.

Increasing the forecast horizon to three months consistently and in parts significantly

reduces the predictive performance in the recessionary state of the economy, while the

impact on predictability in expansionary states is mixed. One explanation could be that

momentum crashes happen relatively quickly and recessions are usually short. Reducing

the data frequency leads to longer reaction times, thus lowering predictive performance.

43


A.6 Quality Factor

Rolling windows do not have an impact on the results.

Shortening the time lag to directly adjacent periods massively improves the forecast

performance for the large financial data set to a level approaching borderline statistical

significance and economically significant predictability. As for the size and the low beta

factor, this effect is again due to lagged returns. For the other data sets except the com-

bination data set or in the case of increasing the lag, predictive results are not impacted

by lag specifications.

Increasing the forecast horizon increases the predictive performance in the recessionary

state, while overall the results remain more or less unchanged.

44


B Commonality in predicted returns

B.1 Correlations between predicted return deviations

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 6: Correlations between predicted return deviationsThis graph shows the 5x5 correlation matrices of the predicted return deviations to the historical average estimate for fiveequity factors for four data sets and for 13 different methods. The five equity factors are the market excess return, size,value, momentum and low beta. Averaging across all 52 plots results in table 12. The four columns represent the fourdifferent data sets: (1) the small financial data set, (2) the large financial data set, (3) the economics data set and (4) thecombination data set. The 13 rows represent the 13 methods used in this paper: four forecast combination versions, fiveprincipal component regression versions, two lasso versions, the three pass regression filter and partial least squares.

45


B.2 Factor structure of predicted return deviations

B.2.1 Variance explained by principal components

0 10 20 30 40 50 60 70 80 90 100PLS

3PRFLasso(2)Lasso(1)PCR(5)PCR(4)PCR(3)PCR(2)PCR(1)

FC(4)FC(3)FC(2)FC(1)

Percentage of Variance Explained

Small Financial Dataset

0 10 20 30 40 50 60 70 80 90 100PLS




Large Financial Dataset

0 10 20 30 40 50 60 70 80 90 100PLS




Economics Dataset

0 10 20 30 40 50 60 70 80 90 100PLS




Combination Dataset

Figure 7: Variance explained by principal componentsThis graph shows the percentage of the variance of the predicted return deviations from the historical mean estimateexplained by principal components. Five equity factors, the market excess return, size, value, momentum and low betaform the base assets. The factor structure is presented for four different data sets: (1) the small financial data set, (2)the large financial data set, (3) the economics data set and (4) the combination data set. For each data set, there are the13 methods used in this paper, four forecast combination versions, five principal component regression versions, two lassoversions, the three pass regression filter and partial least squares.

46


B.2.2 Loadings of the principal components

−0.4 −0.2 0 0.2 0.4 0.6

Figure 8: Loadings of the principal componentsIn this graph, each sub-plot shows the principal component loadings of the predicted return deviations from the historicalmean estimate. Five equity factors, the market excess return, size, value, momentum (inverted) and low beta form the baseassets in the rows. The five columns show the loadings of the first until the fifth component.The four columns of plots represent the four different data sets: (1) the small financial data set, (2) the large financial dataset, (3) the economics data set and (4) the combination data set. The 13 rows of plots represent the 13 methods used inthis paper: four forecast combination versions, five principal component regression versions, two lasso versions, the threepass regression filter and partial least squares.

47

Understanding Rebalancing and Portfolio

Reconstitution

Ulrich Carl∗

Draft: December 23rd, 2015

Abstract

This paper analyses the impact of rebalancing and portfolio reconstitution on portfolio

returns and factor exposures. Varying the rebalancing frequency and the portfolio recon-

stitution frequency leads to distinct patterns in relative factor exposures. These patterns

are symmetric for rebalancing and portfolio reconstitution. Short term reversal drives

the returns at high frequencies, momentum at intermediate frequencies, while value and

long term reversal stand out at low frequencies. The variation in returns at different

frequencies can be linked to macro-economic variables, in particular the cross-sectional

volatility.

JEL CODES: G11, G12

Key words: rebalancing, portfolio reconstitution, portfolio additions, portfolio deletions,

momentum, trending, reversal

∗Finreon Ltd., Oberer Graben 3, 9000 St.Gallen, Switzerland and University of St.Gallen, School ofEconomics and Political Science, Bodanstrasse 8, 9000 St.Gallen, Switzerland. The views expressed inthis paper are my own and do not necessarily reflect those of Finreon Ltd. and of the University ofSt.Gallen. I would like to thank Paul Soderlind, Francesco Audrino, Ralf Seiz, Julius Agnesens, LukasPlachel, Christian Finke and the seminar participants at the University of St.Gallen for helpful comments.contact: [email protected], +41 76 210 03 12

Ulrich Carl Understanding Rebalancing and Portfolio Reconstitution

1 Introduction

Rebalancing and portfolio reconstitution are fundamental activities for each security port-

folio. Rebalancing is crucial to keep a portfolio diversified and to avoid concentration risk

in assets that outperformed in the past. On the asset class level, in particular, rebal-

ancing is a standard procedure for most institutional investors in order to keep their

portfolios aligned to their strategical asset allocation. Portfolio reconstitution, primarily

using market capitalization as a measure of relevance, is central for the portfolio to reflect

all important securities in a market. Thus, understanding these two types of portfolio

adjustments is essential for portfolio management but also asset pricing more generally.

So what is the definition of these key concepts in this paper?

(1) Rebalancing is the process of adjusting the portfolio weights back to the target

weights e.g. 1/N for an equally weighted portfolio. Rebalancing takes place either at

regular time intervals or when the weight deviations exceed a predefined threshold. It is

a systematically anti-cyclical process of selling stocks that performed well, while buying

stocks that performed poorly since the last portfolio rebalancing.

(2) Portfolio reconstitution is the process of determining, which stocks are included

in or excluded from the portfolio. For a portfolio that selects the eligible stocks based

on the market capitalization rank, this process is inherently pro-cyclical at the inclusion

threshold. On one hand, stocks below the threshold that perform well will be included

at the next portfolio reconstitution. On the other hand, stocks above the threshold that

perform poorly will be excluded. Using market capitalization rank as criterion for portfolio

reconstitution is essential for the pro-cyclical portfolio reconstitution effect described in

this paper. This effect is independent of the so called indexing effect (index reconstitution

effect). This indexing effect is the return impact of announcing and performing additions

and deletions to well-known indices like the S&P 500.

The cyclicality in rebalancing and portfolio reconstitution is key in this paper. Due

to this cyclicality, the return impact of rebalancing and portfolio reconstitution depends

heavily on the relative trending or reversal behaviour in the cross-section of equity re-

turns. Rebalancing profits from relative mean-reversion, while portfolio reconstitution

profits from relative trends. Using non-market-capitalization-based weighting schemes

(smart beta) amplifies the anti-cyclical respectively pro-cyclical effects of rebalancing and

portfolio reconstitution.

This paper is the first to link the cyclical nature of rebalancing and portfolio recon-

stitution to trending and reversal effects observed in the cross-section of equity returns.

Equity factors, pre-dominantly short-term reversal, momentum, value and long-term re-

versal are the focus of this analysis. Further insights can be gained by linking the returns

to rebalancing and portfolio reconstitution to macro-economic variables.

50


I find that the rebalancing frequency and the portfolio reconstitution frequency have

a major impact on the portfolio returns. Lowering the rebalancing frequency decreases

returns, except during time frequencies where momentum dominates. Lowering the recon-

stitution frequency, in contrast, increases returns. Again the exceptions are the frequencies

where momentum is the main driver. Moreover, changes in rebalancing and portfolio re-

constitution frequency show up as distinct patterns in the relative factor loadings. These

patterns are symmetric for rebalancing and portfolio reconstitution. The pattern size for

portfolio reconstitution is chiefly driven by the weighting scheme and to a lesser extent

the exact policy specification and the number of stocks in the portfolio. The overall shape

of the patterns is very robust to changes in specifications. Short term reversal drives the

returns at high frequencies, momentum at intermediate frequencies, while value and long

term reversal stand out at low frequencies. Finally, business cycles, time series volatility

and particularly cross-sectional volatility drive the previously mentioned returns at high

rebalancing frequencies and portfolio reconstitution frequencies. For returns at interme-

diate and low rebalancing frequencies and portfolio reconstitution frequencies, only the

cross-sectional volatility remains robust.

After this introduction, section 2 gives an overview of the literature on rebalancing and

portfolio reconstitution. Section 3 describes the empirical approach. Section 4 presents

the data. Section 5 exhibits the empirical results, while section 6 concludes.

2 Literature

The role of rebalancing as a generator of excess returns receives wide attention among

academics as well as practitioners alike.

The focus in this field of research is predominantly on the effect of ”excess growth” of

diversified and rebalanced portfolios. Particularly in mathematical finance, there are many

authors that take a technical approach such as the stochastic portfolio theory of Fernholz

and Shay (1982) or ”growth optimal portfolios” in Cover (1991), Dempster, Evstigneev,

and Schenk-Hoppe (2007) and Dempster, Evstigneev, and Schenk-Hoppe (2009). A more

intuitive description is presented in Booth and Fama (1992), Stein, Nemtchinov, and

Pittman (2009) and Bouchey, Nemtchinov, Paulsen, and Stein (2012). However, as shown

by Cuthbertson, Hayley, Motson, and Nitzsche (2015) and Chambers and Zdanowicz

(2014), this effect is purely based on a change in the distribution of terminal payoffs.

The expected terminal payoff itself does not change, as long as the underlying return

process is independent and identically distributed. More precisely, the effect of ”excess

growth” occurs when using performance measures that are concave transformations of the

expected terminal wealth - such as growth rates or geometric returns. As rebalancing leads

51


to a more diversified and as such less risky portfolio, the expected growth rates increase

due to Jensen’s inequality1. Expressed differently, it is lower dispersion of the terminal

wealth that leads to an increase in the growth rates. The expected terminal wealth,

however, remains the same as long as there exist no momentum or reversal effects. Thus,

rebalancing in a setting of independently and identically distributed returns does not

boost the return in a conventional sense, but reduces portfolio risk.

Focusing on the effect of ”excess growth” led to a lot of confusion and distracted from

another influencing factor, which directly impacts the expected payoffs of a rebalancing

strategy: The role of (relative) mean-reversion (Perold & Sharpe, 1995; Nardon &

Kiskiras, 2013; Granger, Greenig, Harvey, Rattray, & Zou, 2014). Rebalancing entails

systematic purchases of assets that have dropped in value, while systematic sales of assets

that have increased in value. This is a classical buy-low sell-high (also known as contrar-

ian) investment strategy. These strategies evidently profit if asset prices revert to some

long-term trend. This reversion does not have to take place for each individual asset, but

a weak relative reversion condition like a market diversity condition (Fernholz, 1999) is

sufficient. In contrast, rebalancing strategies suffer from relative trending in the markets.

Using history-based simulations, Dichtl, Drobetz, and Wambach (2014) illustrate that

rebalancing strategies outperform non-rebalanced strategies.

Having discussed the cyclical nature of rebalancing, what is the trending and re-

versal behaviour in financial markets?

In the financial markets, there are two types of trending and reversal we can distin-

guish: (a) time-series (absolute) trending and reversal for each individual security and (b)

cross-sectional (relative) trending and reversal that takes place relative to other securities

in the cross-section. While both types of trending and reversal impact the returns to re-

balancing and reconstitution, time-series trending and reversal is the stronger assumption.

For this paper, the weaker assumption of cross-sectional trending and reversal suffices.

The traditional view in academia is the random walk hypothesis: There is no auto-

correlation in equity returns, at least for large capitalization stocks. However, among

practitioners so called commodity trading advisors (CTAs) have been surprisingly suc-

cessful using trend-following strategies. Moskowitz, Ooi, and Pedersen (2012) challenge

the random walk hypothesis by showing that there exists distinct time series momentum,

which is consistent across asset classes and across markets.

In the cross-section of equity returns, relative trending and reversal is a widely accepted

finding. Jegadeesh (1990), Jegadeesh and Titman (1995) and Conrad, Gultekin, and

Kaul (1997) observe that in the short term i.e. up to one month, recent underperformers

1Jensen’s inequality states that a concave function evaluated at the mean is larger or equal than themean of the concave function e.g. log 1

N

∑

xi ≥1

N

∑

log(xi)

52


outperform recent outperformers. This is the so called short term reversal effect. For

intermediate periods, in contrast, stocks that outperformed during the last 3 to 12 months

continue to outperform. This is the famous momentum effect (Jegadeesh & Titman,

1993; Carhart, 1997; Asness et al., 2013). For long investment periods, we observe a

long term reversal effect (De Bondt & Thaler, 1985; Poterba & Summers, 1988; Chopra,

Lakonishok, & Ritter, 1992): stocks that underperformed in the last three to five years,

tend to outperform. Thus, the autocorrelation patterns in equity returns depend heavily

on the time horizon considered.

These effects are empirically stable and persistent. In terms of potential explana-

tions, behavioural explanations such as investor under-reaction and overreaction are most

prevalent (Shefrin & Statman, 1985; Lo & MacKinlay, 1990; Chopra et al., 1992; Bar-

beris, Shleifer, & Vishny, 1998; Daniel, Hirshleifer, & Subrahmanyam, 1998; Hong &

Stein, 1999; Grinblatt & Han, 2005). From the efficient markets and risk-based perspec-

tive, Berk, Green, and Naik (1999) use a dynamic model to show that momentum and

reversal are the consequence of a firm’s optimal investment choices. In the Johnson (2002)

model, time-varying dividend growth rates cause momentum effects. Sagi and Seasholes

(2007) provide a model of firms with revenues, costs, growth options and shutdown op-

tions that can explain the historical size, value and momentum premia. Vayanos and

Woolley (2013) propose a theory based on flows between investment funds. Moreover,

there is a wide range of literature on risk-based explanations of the value effect (Fama

& French, 1995; Griffin, Ji, & Martin, 2003; Zhang, 2005; Choi, 2013), which is closely

linked to long term reversal. Another potential explanation for size, value and reversal

effects are pricing errors that correct over time (Arnott, 2005; Treynor, 2005; Arnott &

Hsu, 2008; Arnott, Hsu, Liu, & Markowitz, 2011).

Due to their pro-cyclical respectively anti-cyclical nature, rebalancing as well as port-

folio reconstitution are closely linked to these cross-sectional effects in equity returns.

While rebalancing is a popular topic, the concept of portfolio reconstitution and its

relationship to reversal and momentum is new in the academic literature. Only Banner,

Papathanakos, and Whitman (2012) use the notion of a (portfolio) reconstitution drag in

an opinion piece. Their definition of (portfolio) reconstitution drag, however, depends on

forward looking data.

The academic literature discusses reconstitution only from an indexing standpoint.

The indexing effect, also known as the index reconstitution effect, represents the mar-

ket impact of the announcement that a stock enters or exits a renowned stock index.

Stocks entering indices such as the S&P 500 temporarily increase in price due to price

pressure (Harris & Gurel, 1986; Beneish & Whaley, 1996; Lynch & Mendenhall, 1997),

downward-sloping long-run demand curves (Lynch & Mendenhall, 1997; Kaul, Mehrotra,

53


& Morck, 2000; Chakrabarti, Huang, Jayaraman, & Lee, 2005), increased investor aware-

ness (Chen, Noronha, & Singal, 2004) and decreased information cost or liquidity effects

(Beneish & Gardner, 1995). The reverse holds for index exclusions. There are distinct

trading patterns around index adjustments (Kappou, Brooks, & Ward, 2010). Moreover,

index adjustments are not information-free in the sense that newly included companies ex-

perience significant increases in earnings-per-share forecasts and realized earnings (Denis,

McConnell, Ovtchinnikov, & Yu, 2003).

The indexing effect does not impact the portfolio reconstitution effect discussed in

this paper. The portfolio reconstitution effect is due to momentum and reversal in the

cross-section. It is independent of front-running behaviour, market impact or information

contained in index changes by major index providers. Thus, this paper gives a completely

new perspective on the economically and statistically significant systematic effects of

rebalancing and portfolio reconstitution. The portfolio reconstitution effect occurs in

every portfolio that selects its constituents based on market capitalization rank.

3 Approach

The main goal of this paper is to link rebalancing and portfolio reconstitution to trending

and reversal effects in the cross-section of equity returns. As these trending and reversal

effects occur at certain time horizons, the focus is on differences between in portfolios that

are rebalanced or reconstituted at different frequencies. The appendix contains details on

further portfolio reconstitution specifications.

There are two sets of portfolios of primary interest: (1) The discussion of rebalancing

uses equally weighted portfolios that are rebalanced at frequencies varying from one day

up to five years. The portfolio reconstitution frequency remains constant at one year such

that we can separate the rebalancing and portfolio reconstitution effects. (2) The dis-

cussion of portfolio reconstitution uses equally weighted portfolios that are reconstituted

at frequencies varying from one day up to five years. Analogously to the first case, the

rebalancing frequency remains constant at one day in order to separate the rebalancing

and portfolio reconstitution effects.

Changing the rebalancing or reconstitution frequencies leads to varying time gaps be-

tween portfolio adjustments. This on one hand determines, how far the portfolio can

depart from a purely theoretical portfolio of rebalancing and portfolio reconstitution in

continuous time. On the other hand, it enables trending and reversal effects in the portfo-

lio constituents to take place before the portfolio is adjusted back to its theoretical weights

and constituents.

An additional focus is on the number of constituents in the portfolio. It has a sub-

54


stantial impact on the relative importance of the effects discussed in this paper, particu-

larly for portfolio reconstitution. The relative performance depends on the percentage of

the portfolios invested in stocks near the inclusion threshold. I perform all analyses on

portfolios ranging from 50 to 500 constituents, which de-facto limits the results to large

capitalization stocks.

Performance

Figure 1 demonstrates the annualized geometric returns of rebalancing and reconstitution

at different frequencies. Already on the return level, there is an intimate connection

between returns and trending and reversal effects observed in the cross-section. The

appendix shows that these findings hold when considering Sharpe ratios in figure 10 and

information ratios in figure 11. The information ratios use the market capitalization

weighted benchmark as described in section 4 as benchmark.

There are three distinct sub-sections in the frequency domain: (1) The horizon at

which we can observe short term reversal in the cross-section of equity returns (one day

up to three months), (2) the horizon at which there is momentum in the cross-section of

equity returns (three months up to 18 months), and (3) the horizon at which long term

reversal is present in the cross-section of equity returns (18 months up to five years).

Last, I discuss the robustness of the return differences at the three different frequency

sub-sections across time.

Factor Exposures

The returns can already give us a crude first picture, but we can gain much more insight,

when having a closer look at the relative factor exposures. As for the returns, the factor

loadings are obtained from equal weighted portfolios that are rebalanced respectively

reconstituted at different frequencies. However, the factor loadings are relative to those

of daily rebalanced respectively reconstituted portfolios. The factor model used for this

purpose is based on the Carhart model, which includes the market excess return (MKT),

the size factor (SMB), the value factor (HML) and the momentum factor (UMD). I

enhance this model by the two reversal factors by Fama and French, which are short

term reversal (STR) and long term reversal (LTR). Thus, the regressions are of this type:

Rwl −Rw

s = α + β1RMKT + β2RSMB + β3RHML + β4RUMD + β5RSTR + β6RLTR + ǫ (1)

Herein, Rwl is the return to the long portfolio with weighting scheme w, Rw

S is the return to

the short portfolio (the portfolio with daily rebalancing or daily portfolio reconstitution)

55


with weighting scheme w. βx are the factor loadings presented in figure 3. Tables present

the regression coefficients as well as Newey-West standard errors for rebalancing and

portfolio reconstitution portfolios for 50 to 500 stocks each.

Macroeconomic Drivers

After discussing the factor exposures, the focus is on the macroeconomic drivers of the

return differences at the three afore mentioned time horizons. These are the one day

to three months horizon (short term reversal), the three months to 18 months horizon

(momentum) and the 18 months to five years horizon (long term reversal). The regressions

are the following:

Rwl − Rw

s = β0 + β1NBER + β2CSV + β3TSV + ǫ (2)

The differences Rwl − Rw

s are the return differences for the three sub-sections in the fre-

quency domain described before: the short term reversal horizon (1 day to 3 months),

the momentum horizon (3 months to 18 months) and the long term reversal horizon (18

months to 5 years). The direction of the differences is selected such that differences con-

forming to the theoretical expectations have a positive sign. βx are the regression loadings

to the National Bureau of Economic Research recession dummy NBER, the normalized

cross-sectional volatility CSV , and the GARCH(1,1) time-series volatility TSV . The

tables present the regression coefficients as well as Newey-West standard errors for rebal-

ancing and portfolio reconstitution portfolios for 50 to 500 stocks each.

4 Data

The underlying data is the cross-section of daily U.S. equity returns from January 1926

to December 2014 obtained from the Center for Research in Security Prices (CRSP). The

sample only uses common stocks (CRSP share codes 10 and 11), thus it excludes certifi-

cates, American depository receipts (ADRs), shares of beneficial interest (SBIs), units,

foreign companies, closed-end funds as well as real estate investment trusts (REITs).

Non-traded prices are treated like traded prices. The weights in the market capitalization

weighted benchmark are based on the number of shares outstanding times the reported

prices and are updated on each trading day. The benchmark includes all stocks with valid

prices and a valid number of shares outstanding at the respective date. The analysis is

limited to data from January 1927 to December 2014 for return calculations and from

January 1931 to December 2014 for the factor analyses as well as for portfolio reconstitu-

tion specifications using lagged market capitalization data. This gap is necessary as the

56


long term reversal factor is only available from March 1930 and the lagged specifications

need a lag of up to five years.

The factor analysis of the portfolios mainly uses daily factor data provided by Kenneth

French2. The foundation is the Carhart (1997) model. It uses the market excess return,

the SMB factor (small-minus-big stocks), the HML factor (high-minus-low book value to

market capitalization) and the UMD factor (past winners minus past losers). The market

excess returns are the daily returns of the market capitalization weighted benchmark (as

calculated before) minus the risk free rate provided by Kenneth French. I enhance this

model by adding the factors on short-term reversal (STR) and long-term reversal (LTR)

to the factor model. Both additional factors play a crucial role in explaining the returns

to rebalancing and portfolio reconstitution.

Business cycle information from the National Bureau of Economic Research (NBER)3

serves as proxy for the state of the economy. I create recession dummies and convert them

to the daily frequency. The cross-sectional volatility is the standard deviation of all eligi-

ble stock returns on a particular day. To avoid the impact of outliers, only returns within

five interquartile ranges around the median are eligible. This makes the results slightly

more robust by reducing noise, but does not change them substantially. The time series

volatility is based on a Garch(1,1) model of the market capitalization weighted bench-

mark. For ease of interpretation, the time-series volatility and cross-sectional volatility

are normalized in the regressions.

5 Empirical Results

The first subsection discusses the effects of different rebalancing and portfolio reconsti-

tution frequencies on portfolio returns and the stability of the resulting return patterns

over time. The next section links the return variation observed at different rebalancing

and portfolio reconstitution frequencies to the equity factor exposures. Finally, the paper

identifies the macroeconomic drivers of these return differences.

5.1 The impact on returns

As described in the introduction, portfolio reconstitution is pro-cyclical and thus profits

from trending. On the opposite, rebalancing is anti-cyclical and therefore profits from

reversal. At the same time, there is time-horizon dependent trending and reversal in

the cross-section of equity returns. When looking at the returns to rebalancing and

portfolio reconstitution at different frequencies in figure 1, the link between rebalancing,

2http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html3http://www.nber.org/cycles/recessions.html

57


Ret

urn

in %

p.a

.

Rebalancing Frequency

Rebalancing (Equal Weights)

1D 2D 1W 2W 1M 2M 3M 4M 6M 9M 1Y 18M 2Y 3Y 4Y 5Y9

9.5

10

10.5

11

11.5

12

12.5

13

13.5

50100150200250300350400450500

Ret

urn

in %

p.a

.

Reconstitution Frequency

Reconstitution (Equal Weights)

1D 2D 1W 2W 1M 2M 3M 4M 6M 9M 1Y 18M 2Y 3Y 4Y 5Y

9

10

11

12

13

14

50100150200250300350400450500

Figure 1: Returns of equally weighted portfolios at different rebalancing and portfolioreconstitution frequenciesThe two figures represent the annualized geometric returns of equally weighted portfolios containing the 50 to 500 largeststocks of the U.S. equity market from January 1927 to December 2014. (1) The left figure shows the return impact ofvarying the rebalancing frequency, while keeping the reconstitution frequency constant at one year. (2) The right figureshows the return impact of varying the reconstitution frequency, while keeping the rebalancing frequency constant at oneday. The frequencies vary between one day and five years in both cases. The grey area marks frequencies at which thereis momentum in the cross-section, while there is cross-sectional reversal at the uncoloured frequencies. All portfolios arereconstituted based on market capitalization ranks to reflect the largest stocks in the universe. There are no transactioncosts.

portfolio reconstitution and the characteristics of the cross-section of equity returns shows

up prominently. First, the returns to rebalancing and reconstitution are mirror images

of each other. Second, the sign of the changes in returns to rebalancing and portfolio

reconstitution switches between horizons where momentum is observed in the cross-section

(grey shaded area) and the horizons where reversal is present. For rebalancing, increasing

the rebalancing frequency at the reversal horizons increases returns. The reversal horizons

are for frequencies below 3 months for short term reversal and for frequencies above

18 months for long term reversal. As rebalancing profits from reversal, the more you

rebalance, the higher are the returns. At the momentum horizon (3 months to 18 months),

in contrast, the trending in the cross-section harms rebalancing. Thus, at this horizon,

decreasing the rebalancing frequency increases the returns. The pro-cyclical portfolio

reconstitution is the mirror image of the anti-cyclical rebalancing. Therefore, the reverse

argument holds for portfolio reconstitution. More frequent reconstitution is bad in times

of reversal and good in times of trending. These patterns in the returns to rebalancing and

portfolio reconstitution are not due to risk. Adjusting for absolute risk (Sharpe ratios)

and relative risk (Information ratios) does not change the results, as presented in the

appendix. The different lines represent portfolios containing a different number of stocks.

They show that the return patterns are robust in the number of stocks in the portfolio.

The continuous increase in the returns with a rising number of stocks in the portfolio is

due to the well-known size effect. Note again that the discussion is about relative reversal

and relative trending compared to the other stocks in the cross-section of equity returns.

58


Panel A: Annualized Return Differences (short term rev. horizon: 1 day to 3 months)

50 100 150 200 250 300 350 400 450 500

Rebalancing 0.69*** 1.06*** 1.29*** 1.44*** 1.58*** 1.73*** 1.87*** 2.10*** 2.29*** 2.59***

(0.08) (0.10) (0.11) (0.12) (0.13) (0.14) (0.15) (0.17) (0.18) (0.20)

Reconstitution 1.29*** 1.29*** 1.43*** 1.31*** 1.41*** 1.70*** 1.81*** 1.73*** 1.80*** 1.99***

(0.15) (0.13) (0.14) (0.12) (0.12) (0.13) (0.15) (0.14) (0.14) (0.16)

Panel B: Annualized Return Differences (momentum horizon: 3 to 18 months)

50 100 150 200 250 300 350 400 450 500

Rebalancing 0.03 0.09 0.08 0.10 0.10 0.13 0.15 0.18 0.20 0.20(0.10) (0.11) (0.13) (0.13) (0.14) (0.14) (0.14) (0.15) (0.15) (0.16)

Reconstitution 0.24 0.06 0.09 0.20 0.18 0.19 0.13 0.15 0.10 0.10(0.17) (0.19) (0.18) (0.18) (0.18) (0.17) (0.17) (0.17) (0.16) (0.16)

Panel C: Annualized Return Differences (long term rev. horizon: 1.5 to 5 years)

50 100 150 200 250 300 350 400 450 500

Rebalancing 0.05 0.06 0.11 0.13 0.16 0.17 0.14 0.15 0.12 0.10(0.09) (0.10) (0.11) (0.11) (0.12) (0.13) (0.15) (0.15) (0.16) (0.16)

Reconstitution 0.47** 0.52** 0.58** 0.73*** 0.70*** 0.66*** 0.66*** 0.77*** 0.89*** 0.93***

(0.22) (0.21) (0.23) (0.23) (0.22) (0.20) (0.20) (0.20) (0.20) (0.19)

Table 1: Annualized return differences at different rebalancing and portfolio reconstitu-tion frequenciesThis table shows the annualized arithmetic return differences when varying the rebalancing frequency respectively recon-stitution frequency of a portfolio. Panel A presents the differences at the short term reversal horizon (1 day to 3 months),Panel B presents the differences at the momentum horizon (3 months to 18 months) and Panel C presents the differences atthe long term reversal horizon (1.5 to 5 years). The direction of the differences is selected such that according to theory allreturns should be positive. For rebalancing that means one day minus 3 months, 18 months minus 3 months and 18 months- 5 years. For portfolio reconstitution the direction is reversed. There are portfolios containing the 50 to 500 largest stocksof the U.S. equity market from January 1927 to December 2014. The Newey West standard errors are given in brackets. *stands for two sided significance at the 10% level, ** for significance at the 5% level and *** for significance at the 1% level.

It is not about trending and reversal in the aggregate equity market.

The next aspect is the robustness and statistical significance of the previous

return patterns in table 1. For the interpretation of the return differences, the direction

of the difference is key e.g. 3 months minus one day or one day minus three months. Here, I

define the direction such that a positive return difference means that the results correspond

to the theoretical predictions. This means that rebalancing profits from reversal and

suffers from trending, while portfolio reconstitution profits from momentum and suffers

from reversal.

The return changes when varying the frequency within the short term reversal horizon

(1 day to 3 months) are economically substantial and statistically highly significant for

both rebalancing and portfolio reconstitution for all portfolios containing 50 to 500 stocks.

High frequency rebalancing and low frequency portfolio reconstitution pay off during the

short term reversal horizon. Rebalancing a portfolio of the 500 largest stocks at daily

frequency yields an annualized excess return of 2.59% over rebalancing at the quarterly

frequency. In contrast, reducing the portfolio reconstitution frequency in a portfolio of the

500 largest stocks from daily to quarterly increases the return by 1.99%. The Newey-West

standard errors at the optimal lag length are given in brackets.

59


Looking at the intermediate momentum horizon (3 months to 18 months), the signs

are positive in each specification and thus the effects have the right sign. However, the

effects remain statistically insignificant. They are also much weaker in economic terms

with return gains of around 10 to 20 basis points annually.

For the long term reversal horizon (1.5 years to 5 years), the results are mixed. While

increasing the rebalancing frequency slightly increases the returns in each specification

as expected, the results are statistically insignificant and economically weak. Decreasing

the portfolio reconstitution at this horizon, in contrast, leads to statistically as well as

economically significant return gains. These gains are up to 0.93% p.a. for a portfolio

of the 500 largest stocks that is reconstituted every five years compared to reconstitution

every 18 months.

The role of the number of stocks in the portfolio also shows distinctly in the return

differences. For the equal weighted portfolio increasing the number of stocks in the port-

folio increases the return differences for rebalancing as well as portfolio reconstitution.

Rebalancing is driven by all stocks in the portfolio according to their portfolio weight.

Thus, when increasing the number of stocks in the portfolio, there is a significant portfolio

overlap. As despite the overlap, the return increase is still large, the rebalancing effect

must be present much more in relatively smaller capitalization stocks. Note that the port-

folios do not contain small capitalization stocks in the conventional sense as only the 50

to 500 largest stocks are selected. The largest gains when increasing the number of stocks

are at the short term reversal horizon. Ball, Kothari, and Wasley (1995) and Conrad et

al. (1997) find that the profitability of short-term reversal strategies is biased upwards

because of the bid-ask-bounce, which matters particularly for small, illiquid stocks with

large spreads. In contrast, De Groot, Huij, and Zhou (2012) find no indication that the

profits to short term reversal increase between portfolios of the largest 100, 500 and 1500

U.S. stocks. For the momentum horizon, there are small, but insignificant increases in

the return differences for a larger number of stocks in the portfolio. This observation is

in line with the findings of Israel and Moskowitz (2013) and Asness, Frazzini, Israel, and

Moskowitz (2014) that there is no reliable relation between momentum returns and size.

Finally, for the long term reversal horizon, there is no relationship between the number of

stocks in the portfolio and the return differences. Israel and Moskowitz (2013) and Asness

et al. (2014), however, find that the value factor is much stronger for small capitaliza-

tion stocks and weak for large capitalization stocks. Chopra et al. (1992) find that the

overreaction effect is stronger for smaller firms. Zarowin (1990) also links the returns to

long-term reversal to the size effect as the losers are on average smaller than the winners.

While all stocks in the portfolio affect rebalancing, portfolio reconstitution is only

driven by the stocks near the inclusion threshold. Only stocks near the threshold are

60


likely to be included respectively excluded depending on their past returns. Thus, there is

no or barely any overlap between the affected stocks. In total, there are three components

that drive the percentage of the portfolio impacted by portfolio reconstitution: (a) The

relative weight of the stocks that are far from the inclusion threshold, (b) the relative

spacing in the market capitalizations of the stocks (i.e. flatness of the capital distribution

curve) near the inclusion threshold and (c) the variability and sensitivity to trending and

reversal of the stocks at the inclusion threshold.

For an equal weighted portfolio, each stock in the portfolio receives the same weight.

Consequently, the relative weight of the stocks far from the inclusion threshold is relatively

small compared to market capitalization weighted approaches. Increasing the number of

stocks increases the number of stocks far from the inclusion threshold. Thus, the first

component diminishes the impact of portfolio reconstitution when increasing the number

of stocks in the portfolio.

However, if the relative spacing in the market capitalizations of the stocks becomes

smaller and smaller at the inclusion threshold, there will be more and more stocks in the

vicinity of the inclusion threshold. This can overcompensate the effect of the increase in

the number of relatively large stocks. Stock market diversity and the capital distribution

curve is discussed in Fernholz (2005).

Finally, as for rebalancing, the return characteristics of the stocks near the inclusion

threshold play a role. The volatility of stocks tends to increase as the market capitaliza-

tion decreases, smaller stocks are more ”junky” (Asness, Frazzini, Israel, Moskowitz, &

Pedersen, 2015). At the same time, short term reversal and long term reversal are more

prominent for smaller stocks as discussed for the case of rebalancing.

Which effects dominate, depends primarily on the weighting scheme of the portfolio,

but also the structure of the particular market. For the U.S. data considered here, effects

(b) and (c) dominate for equally weighted portfolios, while effect (a) dominates for market

capitalization weighted portfolios. For further details, I present the role of reconstitution

for market capitalization weighted portfolios in a separate section in the appendix. The

portfolio reconstitution effect impacts alternative weighting schemes (smart beta) that

have a sizeable tilt towards smaller capitalization stocks and invest in broad portfolios

much more than classical market capitalization weighting. Thus, for smart beta ap-

proaches taking this effect into account is vital.

While rebalancing is driven by all stocks in the portfolio according to their portfolio

weight, portfolio reconstitution is only driven by the stocks near the inclusion threshold.

Therefore, the relevant stocks for portfolio reconstitution are on average smaller. However,

even the smallest stocks considered here are large capitalization stocks or even mega

capitalization stocks in the conventional sense, depending on the number of stocks in the

61


1940 1960 1980 2000

2

4

6

8

10

Rel

. Out

perfo

rman

ce

RebalancingShort Term Reversal Period

50100150200250300350400450500

1940 1960 1980 20001

2

3

4

5

6

Rel

. Out

perfo

rman

ce

ReconstitutionShort Term Reversal Period

50100150200250300350400450500

1940 1960 1980 2000

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

Rel

. Out

perfo

rman

ce

RebalancingIntermediate Momentum Period

50100150200250300350400450500

1940 1960 1980 2000

1

1.1

1.2

1.3

1.4

Rel

. Out

perfo

rman

ce

ReconstitutionIntermediate Momentum Period

50100150200250300350400450500

1940 1960 1980 2000

0.9

0.95

1

1.05

1.1

1.15

Rel

. Out

perfo

rman

ce

RebalancingLong Term Reversal Period

50100150200250300350400450500

1940 1960 1980 2000

1

1.5

2

2.5

Rel

. Out

perfo

rman

ce

ReconstitutionLong Term Reversal Period

50100150200250300350400450500

Figure 2: Performance stability at different rebalancing and portfolio reconstitution fre-quenciesThis figure represents the cumulative return differences between different rebalancing frequencies (on the left side) andportfolio reconstitution frequencies (on the right side). There are three time horizons considered: (1) The top-most figuresshow the cumulative difference between highest frequency (one day) and the three months frequency, which is the short-termreversal horizon. (2) The two figures in the middle present the cumulative difference between the three months frequencyand the 18 months frequency, which is the momentum horizon. (3) The bottom two figures display the cumulative differencebetween the 18 months and the five year frequency, which is the long term reversal horizon. The direction of the differencesin each figure is chosen such that there is a positive cumulative difference for improved visibility. In each figure, there areportfolios containing the 50 to 500 largest stocks of the U.S. equity market from January 1927 to December 2014.

portfolio. Therefore, the effects to rebalancing and portfolio reconstitution are not due

to small and illiquid stocks or trading frictions. This common criticism of many equity

factors does not apply. In contrast, the effects are present in the largest and most liquid

stocks in the U.S. equity universe.

Besides the average return differences for the whole period from 1927 to 2014, the

inter-temporal stability is key. In how far are these rebalancing and portfolio recon-

stitution effects stable across time or are they driven by extraordinary events? Mirroring

62


the results of statistical significance, we find that in figure 2 for rebalancing and portfolio

reconstitution during the short term reversal period the excess returns are accumulated

very smoothly. The excess returns are smoother and in particular higher the larger the

number of stocks in the portfolio. However, the financial crisis of 1929, the tech bust of

2001 and the recent financial crisis of 2008 have a visible impact on the relative returns

with jumps in the relative returns after crisis periods.

For the intermediate momentum period, the insignificant excess returns are primarily

driven by few events such as the aftermaths of financial crisis of 1929, the tech bust of

2001 and the financial crisis of 2008. This is consistent with Daniel and Moskowitz (2013),

which find pronounced momentum crashes after all of these three events. Reducing the

rebalancing frequency leads to a smoother excess return trajectory with a high number of

stocks in the portfolio as we would assume. For portfolio reconstitution, in contrast, there

is no link between excess return smoothness and the number of stocks in the portfolio.

Finally, for the long term reversal period, the results are mixed. The statistically

significant excess returns from portfolio reconstitution are accumulated in a relatively

smooth manner, though the three previously mentioned events play a decisive role. The

excess returns and the smoothness are clearly increasing in the number of stocks. For the

statistically insignificant excess returns from rebalancing, the three events stand out and

their impact closely mirrors their effects at the intermediate momentum period. There is

no distinct link to the number of stocks in the portfolio.

5.2 The impact on relative factor exposures

The return patterns become more prominent, when we consider them in terms of the

relative factor exposures. In figure 3, there is a distinct branch-like structure of factor

exposures. In particular the four equity factors linked to trending and reversal in the

cross-section dominate. These are short-term reversal, momentum, value and long-term

reversal. As for the returns, the factor loadings between rebalancing and portfolio re-

constitution are mirror images for the four equity factors that are linked to trending and

reversal in the cross-section of equity returns.

The factor exposures are relative to a portfolio with the highest rebalancing frequency

respectively reconstitution frequency. The factor model uses the Carhart (1997) model as

foundation and further adds a short term reversal factor and a long term reversal factor.

The exact numerical values as well as the Newey-West standard errors at the optimal lag

length are presented in tables 2 and 3. The standard errors are very low as there are more

than 80 years of daily data and long-short portfolios show only very moderate variation.

Thus, the results are highly statistically significant. All changes in the factor loadings are

for a portfolio of 250 stocks for illustrative purposes, but the results are robust to changing

63


Fact

or E

xpos

ure

in %


Rebalancing (Equal Weights)

1D 2D 1W 2W 1M 2M 3M 4M 6M 9M 1Y18M2Y 3Y 4Y 5Y

−4

−2

0

2

4

6

8

AlphaMktSMBHMLUMDSTRLTR Fa

ctor

Exp

osur

e in

%


Reconstitution (Equal Weights)

1D 2D 1W 2W 1M 2M 3M 4M 6M 9M 1Y18M2Y 3Y 4Y 5Y−12

−10

−8

−6

−4

−2

0

2

4

6

8

AlphaMktSMBHMLUMDSTRLTR

Figure 3: Changes in factor loadings of portfolios rebalanced and reconstituted at differentfrequenciesThis figure represents the relative factor loadings of equally weighted portfolios containing the 250 largest stocks of the U.S.equity market from January 1931 to December 2014. (1) The left figure shows the relative factor loadings when varying therebalancing frequency, while keeping the portfolio reconstitution frequency constant at one year. The factor loadings arerelative to the factor loadings at the daily rebalancing frequency. (2) The right figure shows the relative factor exposuresvarying the reconstitution frequency, while keeping the rebalancing frequency constant at one day. The factor loadings arerelative to the factor loadings at the daily portfolio reconstitution frequency. The frequencies vary between one day andfive years in both cases. All portfolios are reconstituted based on market capitalization ranks to reflect the largest stocksin the universe. There are no transaction costs. The factor model is a six factor model that enhances the Carhart model(market excess returns [MKT], size [SMB], value [HML], momentum [UMD]) by short term reversal (STR) and long termreversal (LTR). The alpha is annualized for ease of interpretation.

the number of stocks in the portfolio. The appendix presents the graphical results for

portfolios containing 100 and 500 stocks.

As expected, at the short term reversal horizon (up to 3 months), rebalancing

profits from short term reversal. At each rebalancing, we sell stocks that have gained in

value since the last rebalancing, while buying stocks that have lost in value. This is a bet

on the reversion of the relative asset prices. As this effect is very short term, the higher

the rebalancing frequency, the more profitable the strategy. Fernholz and Maguire (2007)

demonstrate that without transaction costs such a high frequency ”statistical arbitrage”

strategy can be highly profitable. Decreasing the rebalancing frequency from one day

to three months decreases the short term reversal loading by -4.23%. Changes to other

factors during the short term reversal horizon are very small. Thus, most of the return

benefits of high frequency rebalancing can be attributed to the short term reversal factor.

At frequencies lower than three months, the exposure to short term reversal remains

relatively constant. Other factors capture longer term trending and reversal.

When considering the intermediate momentum horizon from 3 to 18 months,

reducing the rebalancing frequency leads to an extraordinary increase in the momentum

exposure. At lower rebalancing frequencies, momentum stocks are allowed to build up in

the portfolio before the next rebalancing. Rebalancing at higher frequencies, in contrast,

directly sells all stocks that performed well in the past and thus does not profit from the

increased trending in these stocks. Reducing the rebalancing frequency from 3 months

64


Alpha MKT SMB HML UMD STR LTR

2D-1D -0.06*** 0.04* 0.13*** -0.05 -0.06** -0.44*** 0.03(0.02) (0.02) (0.05) (0.04) (0.03) (0.04) (0.04)

1W-1D -0.06 0.01 0.22** -0.09 -0.13** -1.17*** -0.01(0.05) (0.04) (0.09) (0.09) (0.06) (0.09) (0.10)

2W-1D 0.07 -0.06 0.26** 0.00 -0.15* -1.82*** -0.05(0.06) (0.06) (0.10) (0.12) (0.09) (0.11) (0.12)

1M-1D 0.54*** -0.15** 0.29** 0.11 0.00 -3.00*** -0.02(0.08) (0.07) (0.12) (0.15) (0.11) (0.14) (0.16)

2M-1D 1.00*** -0.29*** 0.26* 0.30 0.54*** -3.96*** 0.02(0.10) (0.10) (0.15) (0.22) (0.17) (0.17) (0.18)

3M-1D 1.01*** -0.57*** 0.29* 0.45* 1.34*** -4.23*** 0.12(0.11) (0.12) (0.16) (0.24) (0.18) (0.18) (0.20)

4M-1D 1.12*** -0.67*** 0.16 0.72** 1.85*** -4.49*** 0.30(0.13) (0.17) (0.24) (0.30) (0.22) (0.22) (0.32)

6M-1D 0.96*** -1.09*** 0.10 0.71** 3.30*** -4.43*** 0.59*

(0.15) (0.18) (0.26) (0.36) (0.26) (0.24) (0.32)

9M-1D 0.90*** -1.53*** -0.09 0.86** 4.79*** -4.47*** 0.09(0.16) (0.17) (0.24) (0.39) (0.28) (0.23) (0.32)

1Y-1D 0.89*** -1.79*** 0.11 0.76* 6.26*** -4.52*** 0.48*

(0.17) (0.18) (0.27) (0.43) (0.33) (0.25) (0.29)

18M-1D 1.01*** -2.05*** 0.07 0.52 7.44*** -4.57*** 0.02(0.21) (0.20) (0.28) (0.48) (0.36) (0.28) (0.33)

2Y-1D 0.77*** -2.21*** -0.02 0.21 8.26*** -4.34*** -0.45(0.23) (0.22) (0.30) (0.47) (0.38) (0.30) (0.36)

3Y-1D 0.75*** -2.27*** 0.00 -0.88** 8.36*** -4.21*** -2.01***

(0.22) (0.20) (0.28) (0.40) (0.33) (0.28) (0.39)

4Y-1D 0.76*** -1.80*** -0.21 -1.09*** 8.69*** -4.18*** -2.33***

(0.22) (0.20) (0.29) (0.38) (0.34) (0.28) (0.34)

5Y-1D 0.83*** -2.81*** -0.17 -2.64*** 8.59*** -4.24*** -4.45***

(0.25) (0.22) (0.30) (0.39) (0.35) (0.30) (0.51)

Table 2: Changes in factor loadings at different rebalancing frequenciesThis table represents the factor loadings of a strategy that invests in an equally weighted portfolio that is rebalanced atdifferent frequencies while shorting an equally weighted portfolio rebalanced daily. Thus, the table represents the relativefactor loadings compared to daily rebalancing. The first column specifies the rebalancing frequencies of the long (firsttwo characters) and short sides (last two characters) of the strategy, where D is days, W is weeks, M is months and Y isyears. The portfolios consist of the 250 largest stocks of the U.S. equity market from January 1931 to December 2014.The portfolios are reconstituted annually based on market capitalization ranks to reflect the largest stocks in the universe.The factor model is a six factor model that enhances the Carhart model (market excess returns [MKT], size [SMB], value[HML], momentum [UMD]) by short term reversal (STR) and long term reversal (LTR). The alpha is annualized for easeof interpretation, all values are in percent. The Newey West standard errors are given in brackets. * stands for two sidedsignificance at the 10% level, ** for significance at the 5% level and *** for significance at the 1% level.

to 18 months increases the momentum factor loadings from 1.34% to 7.44%. After the

momentum horizon, the momentum loading remains relatively stable as is the case for the

short term reversal loading at frequencies lower than three months. Classical momentum

strategies usually use portfolio formation periods of 3 to 12 months, while the holding

period is shorter. The most common portfolio formation period is 12 months, where the

last month is discarded to avoid a negative impact of short term reversal and the holding

period is one month, e.g. Carhart (1997). When considering rebalancing, however, the

portfolio formation period and the holding period are taken together and the most recent

data is not discarded, such that rebalancing does not optimally exploit the momentum

effect. However, as Jegadeesh and Titman (1993) demonstrate, momentum is fairly robust

to variations in the formation period, holding period and discarding the most recent data.

65



2D-1D 0.12*** -0.00 -0.12** -0.01 0.04 0.26*** -0.03(0.03) (0.02) (0.05) (0.04) (0.03) (0.04) (0.05)

1W-1D 0.21*** 0.03 -0.21** -0.08 0.17** 0.79*** 0.09(0.05) (0.04) (0.09) (0.08) (0.07) (0.07) (0.09)

2W-1D 0.17*** 0.07 -0.26** -0.14 0.16* 1.25*** 0.11(0.06) (0.05) (0.11) (0.11) (0.09) (0.09) (0.11)

1M-1D -0.15** 0.12* -0.32*** -0.26* 0.04 2.14*** 0.05(0.07) (0.06) (0.12) (0.15) (0.11) (0.11) (0.14)

2M-1D -0.51*** 0.10 -0.33** -0.33 -0.50*** 2.94*** -0.12(0.09) (0.10) (0.15) (0.24) (0.17) (0.15) (0.17)

3M-1D -0.54*** 0.26** -0.34** -0.47* -1.21*** 3.21*** -0.25(0.10) (0.11) (0.16) (0.27) (0.18) (0.15) (0.19)

4M-1D -0.62*** 0.20 -0.35* -0.52 -1.82*** 3.45*** -0.43*

(0.13) (0.15) (0.21) (0.33) (0.20) (0.18) (0.26)

6M-1D -0.50*** 0.30* -0.53** -0.26 -3.36*** 3.41*** -0.80***

(0.16) (0.17) (0.26) (0.41) (0.28) (0.21) (0.28)

9M-1D -0.62*** 0.30 -0.76* 0.04 -5.19*** 3.65*** -0.62(0.23) (0.22) (0.39) (0.57) (0.39) (0.27) (0.38)

1Y-1D -0.43 0.26 -1.29** 1.04 -7.68*** 3.50*** -1.59***

(0.34) (0.32) (0.57) (0.87) (0.63) (0.40) (0.54)

18M-1D -0.53* 0.14 -1.34*** 1.37** -8.89*** 3.58*** -0.94**

(0.28) (0.26) (0.49) (0.68) (0.54) (0.34) (0.43)

2Y-1D 0.15 -0.01 -1.53*** 2.83*** -9.85*** 2.89*** -0.36(0.32) (0.26) (0.50) (0.64) (0.50) (0.35) (0.44)

3Y-1D 0.54 -0.72** -2.66*** 6.12*** -9.99*** 2.47*** 1.29**

(0.38) (0.33) (0.59) (0.82) (0.54) (0.42) (0.58)

4Y-1D 0.66 -1.20*** -2.66*** 6.37*** -10.78*** 2.40*** 1.94***

(0.47) (0.40) (0.74) (1.02) (0.69) (0.52) (0.67)

5Y-1D 0.71 -0.45 -2.57*** 8.11*** -11.27*** 2.47*** 4.38***

(0.49) (0.42) (0.75) (1.06) (0.71) (0.54) (0.75)

Table 3: Changes in factor loadings at different portfolio reconstitution frequenciesThis table represents the factor loadings of a strategy that invests in an equal weighted portfolio that is reconstituted atdifferent frequencies while shorting an equal weighted portfolio reconstituted daily. Thus, this table represents the relativefactor loadings compared to daily portfolio reconstitution. The first column specifies the portfolio reconstitution frequenciesof the long (first two characters) and short sides (last two characters) of the strategy, where D is days, W is weeks, M ismonths and Y is years. The portfolio reconstitution based on market capitalization ranks ensures that the portfolio containsthe largest stocks in the universe. The portfolios consist of the 250 largest stocks of the U.S. equity market from January1931 to December 2014. The portfolio is rebalanced daily. The factor model is a six factor model that enhances the Carhartmodel (market excess returns [MKT], size [SMB], value [HML], momentum [UMD]) by short term reversal (STR) and longterm reversal (LTR). The alpha is annualized for ease of interpretation, all values are in percent. The Newey West standarderrors are given in brackets. * stands for two sided significance at the 10% level, ** for significance at the 5% level and ***for significance at the 1% level.

For the long term reversal horizon from 18 to 60 months, there are two factors -

value and long term reversal - that capture the bulk of changes in the factor exposures.

Value and long term reversal are highly correlated and capture a very similar effect or even

the same effect (Fama & French, 1996; Hong & Stein, 1999; Asness, Frazzini, Israel, &

Moskowitz, 2015). For asset classes where value cannot be measured such as commodities

and currencies, long term reversal often serves as proxy e.g. Asness et al. (2013). Both

factors are mostly price-driven and invest in stocks that underperformed in the past. Thus,

these stocks are ”cheap” compared to their peers. At the same time the factors are selling

”expensive” stocks. This is exactly what rebalancing does by buying more of the past

losers and selling the past winners. When rebalancing gets less frequent than two years,

the weights of the past losers are too low and the weights of the past winners too high,

66


when the prices start to revert around two years (De Bondt & Thaler, 1985). In contrast,

if rebalancing takes place shortly before reversal sets in, the portfolio weights in past

losers are substantially higher and the portfolio weights in past winners are substantially

lower. Therefore, the portfolio profits much more from return reversal. Less rebalancing

in turn implies a reduced value and long term reversal loading. Reducing the rebalancing

frequency from 18 months to 60 months, the relative value loading decreases from 0.52%

to -2.64% and the relative long term reversal loading decreases from 0.02% to -4.45% .

The relative factor loadings for the portfolio reconstitution and their economic motiva-

tion are the exact mirror image of rebalancing for short term reversal, momentum, value

and long term reversal. Therefore, I forgo a detailed discussion of the factor loadings for

portfolio reconstitution. One conceptual difference is that rebalancing only moderately

adjusts the weights of each stock in the portfolio, while portfolio reconstitution is a binary

in-out-decision only at the inclusion threshold. This, however, has no distinct impact in

terms of factor exposures.

It is important to note that varying the rebalancing and reconstitution frequency is

not an optimal strategy to profit from the implicit factor exposures to rebalancing and

portfolio reconstitution. The portfolio formation period and the holding period are jointly

determined by the rebalancing or the reconstitution frequency respectively and cannot be

separated. When only varying the rebalancing frequency, the frequency for momentum

stocks should cover both the portfolio formation period and the holding period. Like

this, the weights of the momentum stocks are high, when the momentum returns are

high. For reversal stocks, in contrast, the rebalancing frequency should be equal to the

portfolio formation period. This again ensures that the weights of the reversal stocks are

high, when the reversal returns are high. For the reconstitution frequency, the mirror

image holds. A momentum stock should be reconstituted after the formation period to

be included in the portfolio when momentum returns are high. A reversal stock should

only be reconstituted after both the portfolio formation period and the holding period to

not suffer from excluding stocks that will outperform. Optimizing the portfolio formation

period and the holding period separately can further improve the returns to rebalancing

and reconstitution. In this context neutralizing unwanted factor exposures e.g. by par-

tially synthetic time series at reversal horizons for rebalancing or momentum horizons

for reconstitution could be an interesting approach. Another approach for portfolio re-

balancing could be a momentum overlay as proposed by Granger et al. (2014). Portfolio

reconstitution using lagged market capitalization data as described in the appendix can

also be used to separate portfolio formation periods and holding periods.

Besides the factors linked to trending and reversal in the cross-section of equity re-

turns, we can also observe distinct changes in other factors. One remarkable effect is

67


the reduction of the size loading by up to -2.57% as when curtailing the reconstitution

frequency. As stocks that grow in value do not enter the portfolio, while stocks that

decline in value remain in the portfolio, the portfolio becomes more concentrated in large

capitalization stocks. For rebalancing, there are similar, but less pronounced and statis-

tically insignificant concentration effects. The size exposure moderately, but consistently

decreases with a reduction in the rebalancing frequency. These findings reflect the nat-

ural concentration tendencies in a buy-and-hold portfolio in the best performing assets,

when no rebalancing and reconstitution takes place (Stein et al., 2009; Cuthbertson et

al., 2015). The impact of reconstitution on the size factor dominates compared to the

insignificant effect of rebalancing.

Decreasing the rebalancing frequency also significantly decreases market risk by up

to -2.81%. This indicates that frequent rebalancing increases the risk (in the sense of

market beta) of the portfolio. A potential explanation is that with less rebalancing,

the portfolio becomes more concentrated in defensive large capitalization stocks. This

could be due a combination of two effects. First, The low volatility anomaly of defensive

stocks performing better (Blitz & van Vliet, 2007; Baker & Haugen, 2012) and thus

receiving a higher weight over time. Second, the natural concentration tendencies of a

buy-and-hold portfolio (Stein et al., 2009; Cuthbertson et al., 2015) that is also responsible

for the reduced size exposure. This result is also in line with the findings of Granger

et al. (2014), which see rebalancing as a combination of a buy and hold portfolio and

a straddle, which induces negative convexity and increases draw-downs. Reducing the

portfolio reconstitution frequency for an equal weighted portfolio also reduces market

risk, but in an inconsistent and statistically mostly insignificant way. Using lagged data

for portfolio reconstitution or market capitalization as weighting scheme as demonstrated

in the appendix, the reduction of market risk becomes highly statistically significant. One

potential explanation is that defensive and stable stocks have a higher survival probability

and thus concentrate in the portfolio.

Moreover, daily rebalancing has a significantly lower annualized alpha than rebal-

ancing only monthly or less frequently. This is to some extent at odds with the notion

of rebalancing itself being a source of alpha, e.g. Nardon and Kiskiras (2013). One po-

tential explanation lies in the construction of the short term reversal factor used in the

factor model. The returns to short term reversal are potentially overstated as they are

constructed using illiquid small capitalization stocks. These stocks profit from market

frictions such as bid-ask-bounces (Ball et al., 1995; Conrad et al., 1997). The short term

reversal factor has a substantially larger loading at high rebalancing frequencies. There-

fore, returns at high rebalancing frequencies can be incorrectly attributed to the short

term reversal factor instead of the alpha and thus lowering the alpha.

68


Looking at absolute factor loadings - in contrast to the relative discussion here - a

daily rebalanced equally weighted portfolio still has a significant alpha that cannot be

explained by the six factors in the model. Thus, even though the analysis of relative

factor loadings delivers valuable insights into the role of factor loadings in rebalancing,

there still remains a residual that cannot be explained.

Reducing the reconstitution frequency decreases the alpha at intermediate frequencies,

while at low frequencies it substantially increases the alpha. For an equal weighted port-

folio, only the decrease at intermediate frequencies is significant, while it is insignificant

at the low frequencies. For alternative specifications in the appendix, the results are also

highly significant at the low frequency end. Short term reversal overstatement can again

be one potential explanation for the negative alphas. Alpha decreases when the short

term reversal loadings increase at high to intermediate frequencies. Alpha increases as

the short term reversal loadings decrease at low frequencies. However, there is an addition

drift in the alphas that short term reversal overstatement cannot explain. The results for

the alphas indicate that the effect of reducing the reconstitution frequency cannot fully be

explained by this factor model. There are additional gains to reducing the reconstitution

frequency besides those reaped by taking the compensated factor exposures considered in

this factor model.

To sum up, varying rebalancing and portfolio reconstitution frequencies leads to very

distinct patterns in the relative factor loadings. We observe distinctly negative short term

reversal, value and long term reversal loadings when reducing the rebalancing frequency,

while at the same time there are distinct positive momentum loadings. For portfolio

reconstitution, the pattern is reversed. The results are economically as well as statistically

highly significant and robust to diverse specifications that are further discussed in the

appendix.

5.3 Macroeconomic Drivers

Besides the factor exposure perspective, the literature often links the returns to rebal-

ancing to measures of cross-sectional volatility or market volatility and the state of the

economy.

Particularly in the literature on the ”excess growth” of rebalanced portfolios, one

of the major drivers that are discussed is the volatility e.g. Willenbrock (2011) and

Bouchey et al. (2012). Moreover, there is a wide literature relating the relative trending

and reversal in the equity markets to measures of time series as well as cross-sectional

volatility. Stivers and Sun (2010) find that the cross-sectional dispersion in stock returns is

positively related to the subsequent value return and negatively related to the subsequent

momentum return. Wang and Xu (2015) discuss the relation between excess volatility and

69


Panel A: Drivers of Rebalancing (short term rev. horizon: 1 day to 3 months)

50 100 150 200 250 300 350 400 450 500

Const 0.66*** 0.96*** 1.17*** 1.30*** 1.42*** 1.54*** 1.66*** 1.84*** 1.99*** 2.20***

(0.08) (0.09) (0.09) (0.10) (0.11) (0.11) (0.12) (0.13) (0.14) (0.16)

NBER 0.15 0.53* 0.62** 0.71** 0.82** 0.93*** 1.04*** 1.28*** 1.43*** 1.90***

(0.28) (0.28) (0.31) (0.30) (0.32) (0.34) (0.36) (0.41) (0.44) (0.51)

CSV 1.59*** 2.21*** 2.49*** 2.77*** 2.98*** 3.19*** 3.34*** 3.61*** 3.79*** 4.21***

(0.29) (0.36) (0.41) (0.52) (0.53) (0.55) (0.57) (0.60) (0.61) (0.67)

TSV 0.12 0.42 0.64** 0.72** 0.88*** 1.02*** 1.16*** 1.34*** 1.51*** 1.77***

(0.26) (0.26) (0.29) (0.28) (0.31) (0.33) (0.35) (0.37) (0.39) (0.46)R2 1.41 2.99 3.85 4.70 5.58 6.47 7.14 7.89 8.88 10.49

Panel B: Drivers of Reconstitution (short term rev. horizon: 1 day to 3 months)

50 100 150 200 250 300 350 400 450 500

Const 1.04*** 1.21*** 1.23*** 1.13*** 1.22*** 1.48*** 1.49*** 1.49*** 1.57*** 1.70***

(0.14) (0.13) (0.13) (0.12) (0.11) (0.11) (0.12) (0.12) (0.11) (0.12)

NBER 1.23*** 0.42 0.96*** 0.86*** 0.94*** 1.06*** 1.57*** 1.16*** 1.10*** 1.45***

(0.47) (0.36) (0.35) (0.31) (0.31) (0.31) (0.36) (0.33) (0.35) (0.44)

CSV 2.10*** 2.22*** 2.44*** 2.01*** 2.25*** 2.49*** 2.99*** 2.89*** 2.74*** 2.70***

(0.49) (0.40) (0.46) (0.32) (0.45) (0.41) (0.48) (0.44) (0.43) (0.41)

TSV 0.60 0.58 0.98*** 0.57** 0.49* 1.12*** 1.07*** 1.06*** 1.23*** 1.55***

(0.40) (0.36) (0.33) (0.24) (0.28) (0.27) (0.32) (0.29) (0.28) (0.34)R2 1.31 2.09 3.41 2.74 3.31 5.69 7.60 7.43 7.77 8.25

Table 4: Drivers of rebalancing and portfolio reconstitution at the short term reversalhorizonThis table shows the regression loadings of potential drivers of the rebalancing and portfolio reconstitution effects. They arecalculated for portfolios containing the 50 to 500 largest stocks of the U.S. equity market from January 1927 to December2014. The explained variables are the annualized arithmetic return differences when varying the rebalancing frequency(Panel A) respectively reconstitution frequency (Panel B) of a portfolio. The differences are at the short term reversalhorizon (1 day to 3 months). The direction of the differences is selected such that according to theory all returns should bepositive. For rebalancing that means one day minus 3 months. For portfolio reconstitution the direction is reversed. Theregressors are a business cycle dummy (NBER), the normalized cross-sectional volatility (CSV) and the normalized timeseries volatility (TSV). The Newey West standard errors are given in brackets. * stands for two sided significance at the10% level, ** for significance at the 5% level and *** for significance at the 1% level.

momentum. Du Plessis (2013) analyses the relationship of momentum with cross-sectional

dispersion and volatility.

Moreover, the consumption based asset pricing model of Breeden (1979) states that risk

premia should vary with consumption. Risk premia should be high, when consumption is

low. To be able to give return patterns of rebalancing and portfolio reconstitution a risk

premium interpretation, they should depend on the state of the economy.

Considering the rebalancing and portfolio reconstitution effects at the short term

reversal horizon, we observe in table 4 that all three potential drivers (recessions, cross

sectional volatility, time series volatility) have a highly significant impact on the relative

returns. A dummy indicates recessions, while the cross-sectional volatility and the time

series volatility are normalized and, thus, the coefficients represent the changes in the

returns due to a one standard deviation change in these variables. For ease of interpre-

tation, the returns as explained variable are annualized arithmetic returns. The Newey

West standard errors are given in brackets.

For rebalancing (in panel A), the effects of these drivers rise in size, significance and

70


Panel A: Drivers of Rebalancing (momentum horizon: 3 to 18 months)

50 100 150 200 250 300 350 400 450 500

Const -0.01 0.07 0.03 0.05 0.04 0.04 0.06 0.11 0.14 0.12(0.09) (0.11) (0.12) (0.13) (0.14) (0.14) (0.14) (0.15) (0.15) (0.16)

NBER 0.22 0.10 0.24 0.24 0.32 0.45 0.46 0.34 0.30 0.38(0.32) (0.36) (0.42) (0.44) (0.45) (0.45) (0.46) (0.48) (0.49) (0.51)

CSV -0.98*** -0.87*** -1.02*** -1.14*** -1.28*** -1.46*** -1.54*** -1.62*** -1.63*** -1.71***

(0.27) (0.32) (0.35) (0.38) (0.40) (0.42) (0.41) (0.44) (0.44) (0.45)TSV 0.31 0.13 0.03 0.12 0.17 0.31 0.33 0.40 0.41 0.33

(0.25) (0.30) (0.32) (0.34) (0.36) (0.38) (0.37) (0.39) (0.38) (0.40)R2 0.30 0.23 0.32 0.35 0.42 0.50 0.54 0.55 0.55 0.59

Panel B: Drivers of Reconstitution (momentum horizon: 3 to 18 months)

50 100 150 200 250 300 350 400 450 500

Const 0.31* -0.00 -0.05 0.08 0.08 0.15 0.09 0.16 0.10 0.06(0.18) (0.19) (0.19) (0.19) (0.20) (0.19) (0.19) (0.19) (0.19) (0.19)

NBER -0.31 0.30 0.66 0.58 0.52 0.22 0.21 -0.07 0.02 0.23(0.59) (0.60) (0.58) (0.59) (0.56) (0.53) (0.52) (0.51) (0.49) (0.49)

CSV -1.05** -1.11** -1.10** -0.96* -0.80* -0.81* -0.56 -0.43 -0.51 -0.58(0.50) (0.50) (0.49) (0.50) (0.47) (0.45) (0.44) (0.43) (0.42) (0.41)

TSV 0.31 -0.17 -0.09 0.08 -0.14 -0.07 -0.44 -0.41 -0.33 -0.25(0.45) (0.47) (0.43) (0.36) (0.36) (0.34) (0.29) (0.28) (0.29) (0.28)

R2 0.10 0.21 0.20 0.13 0.13 0.14 0.18 0.13 0.14 0.14

Table 5: Drivers of rebalancing and portfolio reconstitution at the momentum horizonThis table shows the regression loadings of potential drivers of the rebalancing and portfolio reconstitution effects. They arecalculated for portfolios containing the 50 to 500 largest stocks of the U.S. equity market from January 1927 to December2014. The explained variables are the annualized arithmetic return differences when varying the rebalancing frequency(Panel A) respectively reconstitution frequency (Panel B) of a portfolio. The differences are at the momentum horizon(3 months to 18 months). The direction of the differences is selected such that according to theory all returns should bepositive. For rebalancing that means 18 months minus 3 months. For portfolio reconstitution the direction is reversed. Theregressors are a business cycle dummy (NBER), the normalized cross-sectional volatility (CSV) and the normalized timeseries volatility (TSV). The Newey West standard errors are given in brackets. * stands for two sided significance at the10% level, ** for significance at the 5% level and *** for significance at the 1% level.

the coefficient of determination when the number of stocks in the portfolio increases. In

the case of 500 stocks, being in a recession raises the returns of the strategy of being long

in the daily rebalanced portfolio and short in the portfolio rebalanced at the three month

frequency by 1.90% annually. Increasing the cross-sectional volatility by one standard

deviation increases the annualized returns by 4.21%, while increasing the time series

volatility by one standard deviation adds 1.77% annually to portfolio returns. Thus,

in particular the cross-sectional volatility is an exceptionally strong driver of returns.

The coefficient of determination of 10.5% is relatively high. In short, a high rebalancing

frequency pays off in highly volatile markets during recessions.

Interpreting the higher realized returns to rebalancing during recessions as conditional

expected returns, the finding is in line with risk based asset pricing. It claims that higher

systematic risk should be compensated with higher expected returns. Higher returns

to rebalancing when volatility is high are also in line with the literature on the ”excess

growth” of rebalanced portfolios, even though I explicitly use arithmetic returns and not

geometric returns in this analysis. In contrast the ”excess growth” is often expressed as

a difference between arithmetic and geometric returns e.g. Bouchey et al. (2012).

71


Panel A: Drivers of Rebalancing (long term rev. horizon: 1.5 to 5 years)

50 100 150 200 250 300 350 400 450 500

Const 0.07 0.12 0.13 0.16 0.21 0.27* 0.25 0.26 0.22 0.22(0.11) (0.11) (0.13) (0.13) (0.14) (0.15) (0.16) (0.17) (0.18) (0.19)

NBER -0.10 -0.27 -0.08 -0.18 -0.26 -0.49 -0.53 -0.55 -0.49 -0.59(0.26) (0.29) (0.34) (0.34) (0.38) (0.42) (0.46) (0.49) (0.50) (0.53)

CSV 0.69*** 0.68*** 0.67** 0.76** 0.88** 1.00** 1.16*** 1.38*** 1.41*** 1.57***

(0.23) (0.26) (0.32) (0.32) (0.35) (0.39) (0.42) (0.46) (0.49) (0.53)

TSV -0.31* -0.13 -0.05 -0.17 -0.16 -0.20 -0.30 -0.46 -0.48 -0.58(0.18) (0.24) (0.27) (0.28) (0.32) (0.36) (0.40) (0.43) (0.46) (0.49)

R2 0.16 0.16 0.15 0.16 0.21 0.25 0.28 0.34 0.34 0.40

Panel B: Drivers of Reconstitution (long term rev. horizon: 1.5 to 5 years)

50 100 150 200 250 300 350 400 450 500

Const 0.42* 0.45* 0.60** 0.71*** 0.66*** 0.58** 0.53** 0.60*** 0.67*** 0.70***

(0.25) (0.24) (0.26) (0.27) (0.25) (0.23) (0.23) (0.23) (0.22) (0.21)

NBER 0.21 0.38 -0.09 0.09 0.19 0.40 0.64 0.82 1.09** 1.10**

(0.67) (0.63) (0.67) (0.70) (0.65) (0.59) (0.57) (0.56) (0.55) (0.52)

CSV 0.98* 1.49*** 1.74*** 2.16*** 2.08*** 1.85*** 1.92*** 2.09*** 2.27*** 2.12***

(0.57) (0.51) (0.57) (0.64) (0.59) (0.52) (0.52) (0.53) (0.53) (0.49)TSV -0.00 -0.37 -0.59 -0.65 -0.58 -0.44 -0.54 -0.51 -0.53 -0.42

(0.50) (0.44) (0.46) (0.50) (0.44) (0.37) (0.35) (0.35) (0.35) (0.32)R2 0.09 0.18 0.20 0.32 0.36 0.34 0.38 0.49 0.62 0.62

Table 6: Drivers of rebalancing and portfolio reconstitution at the long term reversalhorizonThis table shows the regression loadings of potential drivers of the rebalancing and portfolio reconstitution effects. They arecalculated for portfolios containing the 50 to 500 largest stocks of the U.S. equity market from January 1927 to December2014. The explained variables are the annualized arithmetic return differences when varying the rebalancing frequency(Panel A) respectively reconstitution frequency (Panel B) of a portfolio. The differences are at the long term reversalhorizon (18 months to 5 years). The direction of the differences is selected such that according to theory all returns shouldbe positive. For rebalancing that means 18 months minus 5 years. For portfolio reconstitution the direction is reversed.The regressors are a business cycle dummy (NBER), the normalized cross-sectional volatility (CSV) and the normalizedtime series volatility (TSV). The Newey West standard errors are given in brackets. * stands for two sided significance atthe 10% level, ** for significance at the 5% level and *** for significance at the 1% level.

For portfolio reconstitution (in panel B), I observe a very similar behaviour. Note that

the direction of the explained difference is reversed. Thus, for 500 stocks decreasing the

reconstitution frequency from daily to quarterly increases the return by 1.45% in reces-

sions and by 2.70% (1.55%) for an one-standard-deviation change in the cross-sectional

(time-series) volatility. What both, increasing the rebalancing frequency and reducing

the reconstitution frequency have in common at the short term reversal horizon is their

increased short term reversal loading.

The findings are also consistent with figure 2 that demonstrates that the financial

crises of 1929 and 2008, the tech bust 2001 and their aftermath show distinctly in the

cumulative returns to these rebalancing and reconstitution strategies.

At the intermediate momentum horizon, table 5 shows that only the cross-sectional

volatility is a significant driver of returns. Moreover, this only holds consistently for re-

balancing in panel A, while the results for reconstitution in panel B disappear for a large

number of stocks in the portfolio. The sign of the coefficient flips. Thus, reducing rebal-

ancing frequency (less rebalancing) increases returns in line with the increasing momentum

72


exposure, but these returns accumulate much more in times of low cross-sectional volatil-

ity. Reducing the rebalancing frequency from 3 months to 18 months, reduces returns

by -1.71% for 500 stocks, when the volatility increases by one standard deviation. For

reconstitution, as before, the direction of the explained difference is switched. Increasing

the portfolio reconstitution frequency from 18 months to 3 months increases returns, par-

ticularly when the cross-sectional volatility is low. A one standard-deviation increase in

the cross-sectional volatility decreases this return difference for 50 stocks by -1.05%.

What both, reducing the rebalancing frequency and increasing the reconstitution fre-

quency have in common at the momentum horizon is their increased momentum loading.

The finding that momentum pays off particularly when (cross-sectional) volatility is low is

consistent with the findings of Stivers and Sun (2010) and Wang and Xu (2015). The in-

significantly positive relation between momentum and recessions in the data could be due

to momentum crashes (Daniel & Moskowitz, 2013) that happen during times of market re-

covery after crises. Again, the finding that less rebalancing pays off when (cross-sectional)

volatility is low is consistent with the literature on excess growth. An open question re-

mains, why the effect of the cross-sectional volatility on portfolio reconstitution decreases

in the number of stocks in the portfolio.

Finally, at the long term reversal horizon, table 6 shows that the cross-sectional

volatility is a very consistent and significant driver of returns. The difference of the returns

to rebalancing at the 18 months and 5 year frequency increases by 1.57% for 500 stocks

as the cross-sectional volatility increases by one standard deviation. For portfolio recon-

stitution, the difference of the returns at the 5 year and 18 months frequency increases by

2.12% for 500 stocks as the cross-sectional volatility increases by one standard deviation.

Time series volatility and the business cycle are insignificant in most specifications.

Increasing the rebalancing frequency or reducing the reconstitution frequency leads to

an increased value and long term reversal exposure. The positive relationship between

the value factor and volatility is discussed among others in Li, Brooks, and Miffre (2009),

Stivers and Sun (2010), Arisoy (2010) and Simlai (2014). The insignificantly negative

loadings on recessions for rebalancing are counter-intuitive, while the partially significant

positive loadings on recessions for the portfolio reconstitution are in line with the counter-

cyclical nature of value e.g. Zhang (2005) and Gulen, Xing, and Zhang (2011).

To conclude, the the cross-sectional volatility stands out as a driver to rebalancing

and reconstitution, while the time series volatility and the business cycle dummy are only

consistently significant at the short term reversal horizon. The empirical findings are

consistent with the literature, particularly on the pro-cyclical nature of the momentum

factor and the anti-cylical nature of the value factor.

73


6 Conclusions

Variations in the rebalancing and portfolio reconstitution frequencies have a substantial

impact on the returns and show up as distinct patterns in the relative factor loadings.

Momentum, short term reversal, long term reversal and value are the predominant fac-

tors, while other factors only play a moderate role. At high rebalancing frequencies and

portfolio reconstitution frequencies, the effect of short term reversal dominates. At inter-

mediate frequencies, the effect of momentum prevails, while at low frequencies, value and

long term reversal are the main drivers. Thus, rebalancing and portfolio reconstitution

are intimately connected to trending and reversal patterns observed in the cross-section of

equity returns. The impact of rebalancing and portfolio reconstitution on relative factor

loadings is close to symmetric. This is due to the anti-cyclical nature of rebalancing that

mirrors the pro-cyclical nature of portfolio reconstitution.

Overall, factor loadings describe the majority of return differences between different re-

balancing and reconstitution frequencies. The remaining alpha is moderate and probably

to quite some extent driven by overstating the returns to short term reversal. Increasing

the rebalancing frequency does not result in higher alphas, which raises questions towards

supposed independent alpha generation capability through rebalancing.

From a macroeconomic perspective, the cross-sectional volatility stands out as a driver

of rebalancing and portfolio reconstitution returns. Time series volatility and recessions

are strong drivers at the short term reversal horizon (1 day to 3 months), while they

are mostly insignificant at lower frequencies. The direction and the significance of the

coefficients can be linked to the literature via the dominating factor exposures at the

respective horizons. The value premium is counter-cyclical and has a high risk premium

in volatile, recessionary markets. The momentum premium is pro-cyclical and has a low

risk premium in volatile, recessionary markets.

Last, rebalancing and portfolio reconstitution impact an equally weighted portfolio

much more than a market capitalization weighted portfolio. This shows up in the rela-

tive returns, in changes of factor exposures as well as in the importance of macroeconomic

drivers to explain these effects. Particularly for non-market-capitalization weighted strate-

gies, taking factor exposures of rebalancing and portfolio reconstitution into account is of

paramount practical importance.

74


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A Robustness Checks

A.1 Reconstitution using lagged market capitalization data

Ret

urn

in %

p.a

.


Reconstitution Frequencies (Equal Weights)


9

10

11

12

13

14

50100150200250300350400450500

Ret

urn

in %

p.a

.Reconstitution Lag

Reconstitution Lags (Equal Weights)

0D 1D 2D 1W2W 1M 2M 3M 4M 6M 9M 1Y18M2Y 3Y 4Y 5Y9

10

11

12

13

14

15

16

17

50100150200250300350400450500

Figure 4: Returns of equal weighted portfolios using different reconstitution specificationsThis figure represents the annualized geometric returns of equal weighted portfolios containing the 50 to 500 largest stocksof the U.S. equity market from January 1927 (left side) respectively 1931 (right side) to December 2014. The portfoliosare reconstituted using different reconstitution specifications. (1) On the left hand side, the portfolio is reconstituted atvarying frequencies between one day and five years along the x-axis based on market capitalization ranks. (2) On the righthand side, the portfolios are reconstituted daily based on market capitalization ranks. However, the market capitalizationused during portfolio reconstitution is lagged by different time periods between zero days (no lag) up to five years along thex-axis. All portfolios are rebalanced daily. There are no transaction costs.

Fact

or E

xpos

ure

in %


Reconstitution Frequencies (Equal Weights)


−10

−8

−6

−4

−2

0

2

4

6

8


ctor

Exp

osur

e in

%

Reconstitution Lag


0D 1D 2D 1W2W 1M 2M 3M 4M 6M 9M 1Y18M2Y 3Y 4Y 5Y

−10

−5

0

5

10


Figure 5: Changes in factor loadings for equally weighted portfolios using different port-folio reconstitution policiesThis figure represents the relative factor loadings of equal weighted portfolios containing the 250 largest stocks of the U.S.equity market from January 1931 to December 2014. The portfolios are reconstituted using different reconstitution specifi-cations. (1) On the left hand side, the portfolio is reconstituted at varying frequencies between one day and five years alongthe x-axis based on market capitalization ranks. The factor loadings are relative to the factor loadings at the daily portfolioreconstitution frequency. (2) On the right hand side, the portfolios are reconstituted daily based on market capitalizationranks. However, the market capitalization used during portfolio reconstitution is lagged by different time periods betweenzero days (no lag) up to five years along the x-axis. The factor loadings are relative to the factor loadings at the dailyportfolio reconstitution frequency using current data. All portfolios are rebalanced daily. There are no transaction costs.The factor model is a six factor model that enhances the Carhart model (market excess returns [MKT], size [SMB], value[HML], momentum [UMD]) by short term reversal (STR) and long term reversal (LTR). The alpha is annualized for easeof interpretation.

80



1D-0D 0.23*** -0.00 -0.23** -0.01 0.07 0.52*** -0.06(0.05) (0.04) (0.10) (0.08) (0.06) (0.08) (0.11)

2D-0D 0.28*** 0.02 -0.24* -0.13 0.14 0.84*** -0.01(0.07) (0.06) (0.14) (0.12) (0.10) (0.10) (0.13)

1W-0D 0.23*** 0.05 -0.35** -0.18 0.20* 1.48*** 0.11(0.08) (0.07) (0.14) (0.14) (0.11) (0.11) (0.17)

2W-0D -0.06 0.18** -0.30** -0.27 0.10 2.21*** 0.23(0.09) (0.08) (0.13) (0.18) (0.12) (0.12) (0.14)

1M-0D -0.70*** 0.17* -0.28 -0.50** -0.27 3.39*** 0.10(0.12) (0.10) (0.18) (0.26) (0.17) (0.16) (0.19)

2M-0D -0.65*** 0.23 -0.44 -0.74* -1.96*** 3.88*** -0.39(0.15) (0.17) (0.27) (0.43) (0.29) (0.22) (0.30)

3M-0D -0.58*** 0.20 -0.51 -0.33 -3.58*** 3.87*** -1.09***

(0.20) (0.23) (0.36) (0.52) (0.33) (0.27) (0.41)

4M-0D -0.58** 0.14 -0.69 -0.30 -5.06*** 3.78*** -1.42***

(0.26) (0.27) (0.48) (0.65) (0.45) (0.32) (0.49)

6M-0D -0.35 -0.10 -0.94 0.54 -7.75*** 3.50*** -1.81***

(0.39) (0.36) (0.65) (0.95) (0.70) (0.46) (0.65)

9M-0D -0.35 0.07 -1.19 1.60 -11.14*** 3.69*** -2.63***

(0.43) (0.42) (0.75) (1.15) (0.81) (0.53) (0.69)

1Y-0D -0.18 -0.39 -1.59** 2.97*** -13.22*** 3.44*** -2.77***

(0.45) (0.42) (0.76) (1.12) (0.86) (0.53) (0.70)

18M-0D 0.68 -1.22*** -2.53*** 6.23*** -13.30*** 2.62*** -0.65(0.55) (0.47) (0.86) (1.17) (0.82) (0.64) (0.77)

2Y-0D 1.07** -1.56*** -3.04*** 8.60*** -12.75*** 2.49*** 1.48*

(0.54) (0.47) (0.80) (1.20) (0.82) (0.60) (0.76)

3Y-0D 1.78*** -2.38*** -2.79*** 11.30*** -10.37*** 1.66** 5.27***

(0.58) (0.51) (0.89) (1.28) (0.86) (0.67) (0.84)

4Y-0D 1.25** -2.83*** -2.39*** 11.94*** -9.30*** 2.30*** 9.31***

(0.56) (0.52) (0.87) (1.32) (0.81) (0.63) (0.89)

5Y-0D 1.58*** -3.58*** -2.35*** 13.03*** -9.95*** 2.16*** 10.41***

(0.53) (0.50) (0.82) (1.29) (0.80) (0.59) (0.86)

Table 7: Changes in factor loadings when using lagged data for portfolio reconstitutionThis table represents the factor loadings of a strategy that invests in an equal weighted portfolio that is reconstitutedusing lagged market capitalization ranks while shorting an equal weighted portfolio reconstituted using current marketcapitalization ranks. Thus, the table represents the relative factor loadings compared to portfolio reconstitution withcurrent market capitalization ranks. The first column specifies the reconstitution lags of the long (first two characters) andshort sides (last two characters) of the strategy, where D is days, W is weeks, M is months and Y is years. The portfolioreconstitution based on market capitalization ranks ensures that the portfolio contains the largest stocks in the universe.The portfolio is reconstituted daily. The portfolios consist of the 250 largest stocks of the U.S. equity market from January1931 to December 2014. The portfolio is rebalanced daily. The factor model is a six factor model that enhances the Carhartmodel (market excess returns [MKT], size [SMB], value [HML], momentum [UMD]) by short term reversal (STR) and longterm reversal (LTR). The alpha is annualized for ease of interpretation, all values are in percent. The Newey West standarderrors are given in brackets. * stands for two sided significance at the 10% level, ** for significance at the 5% level and ***for significance at the 1% level.

Instead of changing the portfolio reconstitution frequency, there are also other poten-

tial portfolio reconstitution policies that make use of the pro-cyclical effect at the inclusion

threshold. One example is to use lagged market capitalization data to determine the rank

of stocks at portfolio reconstitution. Another option is to use average market capital-

ization data for ranking e.g. over 36 months. Here, I will focus on reconstitution using

lagged market capitalization data, shortly referred to as ”reconstitution lags”.

By setting reconstitution lags accordingly, trending and in reversal effects at different

horizons can be captured similarly to varying the reconstitution frequency. However, as

we can separate reconstitution lags and reconstitution frequency, there is much better

81


control over the portfolio formation period and the holding period. Using lagged data is

not a scheme to reduce portfolio turnover, as there is no averaging across time to smooth

out short term fluctuations. Instead market values at one particular date in the past

determine the ranking. The focus here is on market capitalization data that is lagged

between no lag (current data) and a lag of five years, while there is daily reconstitution.

The similarity of the return patterns in figure 4 and the factor exposure patterns in

figure 5 confirms that there is an intimate connection between the impact of reconstitution

frequencies and reconstitution lags. Also in terms of statistical significance, the changes

in the factor loadings in table 7 are similar.

Overall, reconstitution lags show a stronger effect in terms of return differences and

relative factor exposures. This is due to the separation of portfolio formation period and

holding period, where the holding period is very short. The short term reversal loading is

up to 3.88% at a three months lag compared to 3.21% at a three months reconstitution

frequency. The momentum loading is down to -13.22% (vs. -7.68%) at the 12 months

horizon. The value loading reaches 13.03% (vs. 8.11%) and the the long term reversal

loading reaches 10.41% (vs. 4.38%) at the five year horizon. Due to this separation,

the momentum horizon is only until 12 months (only portfolio formation period), while

it is until 18 months (portfolio formation period plus holding period) for reconstitution

frequencies. This separation is also likely to cause the reversal of the momentum loading

at long lags.

Also in terms of macro-economic drivers, the results in table 7 are comparable in terms

of sign and significance and overall larger in magnitude for reconstitution lags. For the

short term reversal interval and a portfolio of 500 stocks for example the coefficients for

the business cycle dummy, the cross-sectional volatility and the time series volatility are

1.63, 2.95 and 1.71 respectively. This is moderately more than in the case of changing

the reconstitution frequency with 1.45, 2.70 and 1.55 respectively.

82


Panel A: Drivers of Reconstitution (short term rev. horizon: 1 day to 3 months)

50 100 150 200 250 300 350 400 450 500

Const 1.15*** 1.22*** 1.40*** 1.22*** 1.35*** 1.55*** 1.49*** 1.60*** 1.60*** 1.81***

(0.20) (0.20) (0.20) (0.19) (0.19) (0.17) (0.18) (0.17) (0.17) (0.18)

NBER 1.40** 0.77 0.83 0.97* 1.05* 1.21** 2.19*** 1.42*** 1.43*** 1.63***

(0.67) (0.62) (0.60) (0.54) (0.55) (0.52) (0.55) (0.51) (0.49) (0.55)

CSV 2.26*** 2.56*** 2.71*** 2.03*** 2.22*** 2.59*** 2.94*** 2.79*** 2.70*** 2.95***

(0.58) (0.57) (0.57) (0.46) (0.49) (0.45) (0.49) (0.48) (0.46) (0.48)

TSV 0.58 0.75 1.08** 0.71* 0.81** 1.30*** 1.41*** 1.50*** 1.62*** 1.71***

(0.51) (0.49) (0.45) (0.37) (0.38) (0.38) (0.41) (0.40) (0.37) (0.42)R2 0.80 1.27 1.80 1.25 1.60 2.77 3.84 3.85 4.16 4.87


50 100 150 200 250 300 350 400 450 500

Const 0.56** -0.05 0.23 0.16 0.28 0.43 0.20 0.31 0.23 0.20(0.27) (0.27) (0.27) (0.28) (0.28) (0.27) (0.26) (0.27) (0.26) (0.25)

NBER -0.04 1.04 0.86 0.72 0.53 0.11 0.33 0.05 0.19 0.16(0.95) (0.97) (0.97) (0.89) (0.90) (0.87) (0.83) (0.82) (0.79) (0.78)

CSV -0.39 -0.90 -0.76 -1.09* -0.71 -0.34 -0.65 -0.37 -0.38 -0.20(0.58) (0.59) (0.62) (0.66) (0.59) (0.58) (0.56) (0.58) (0.58) (0.53)

TSV 0.04 -0.23 -0.07 0.12 -0.28 -0.14 -0.40 -0.32 -0.24 -0.52(0.50) (0.57) (0.52) (0.47) (0.44) (0.43) (0.39) (0.38) (0.38) (0.37)

R2 0.01 0.08 0.05 0.08 0.07 0.02 0.09 0.04 0.04 0.05

Panel C: Drivers of Reconstitution (long term rev. horizon: 1 to 5 years)

50 100 150 200 250 300 350 400 450 500

Const 0.95* 0.85* 1.43*** 1.47*** 1.43*** 1.45*** 1.26*** 1.44*** 1.72*** 1.74***

(0.48) (0.44) (0.43) (0.42) (0.38) (0.36) (0.34) (0.35) (0.34) (0.33)

NBER 1.54 3.22** 1.26 1.35 1.19 1.51 2.07 3.06** 3.34*** 3.88***

(1.55) (1.48) (1.51) (1.52) (1.46) (1.38) (1.28) (1.29) (1.27) (1.23)

CSV 3.35*** 2.66*** 3.21*** 3.36*** 3.09*** 3.14*** 2.84*** 4.11*** 4.40*** 4.33***

(1.17) (0.96) (1.03) (1.03) (0.97) (0.92) (0.96) (1.02) (0.99) (0.91)TSV -0.60 -0.80 -0.90 -0.82 -1.12 -0.89 -0.73 -0.94 -0.70 -0.70

(1.02) (0.87) (0.87) (0.82) (0.72) (0.67) (0.68) (0.69) (0.67) (0.67)R2 0.22 0.20 0.23 0.30 0.24 0.31 0.29 0.65 0.85 0.89

Table 8: Drivers of portfolio reconstitution using lagged market cap ranks for equal weightsThis table shows the regression loadings of potential drivers of the portfolio reconstitution effects. They are calculatedfor equally weighted portfolios containing the 50 to 500 largest stocks of the U.S. equity market from January 1931 toDecember 2014. The explained variables are the annualized arithmetic return differences when using variably laggedmarket capitalization ranks for portfolio reconstitution between no lag and a lag of five years. The differences are at theshort term reversal horizon (current to 3 months) in Panel A, the momentum horizon (3 months to 12 months) in Panel Band the long term reversal horizon (1 to 5 years) in Panel C. The direction of the differences is selected such that accordingto theory all returns should be positive. This means 3 months minus current, 3 months minus 12 months and 5 yearsminus 1 year. The regressors are a business cycle dummy (NBER), the normalized cross-sectional volatility (CSV) andthe normalized time series volatility (TSV). The Newey West standard errors are given in brackets. * stands for two sidedsignificance at the 10% level, ** for significance at the 5% level and *** for significance at the 1% level.

83


A.2 Reconstitution for market capitalization weighted portfo-

lios

Ret

urn

in %

p.a

.


Reconstitution Frequencies (Market Cap)


8.9

9

9.1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

50100150200250300350400450500

Ret

urn

in %

p.a

.

Reconstitution Lag

Reconstitution Lags (Market Cap)


9.2

9.4

9.6

9.8

10

10.2

50100150200250300350400450500

Figure 6: Returns of market weighted portfolios using different reconstitution specifica-tionsThis figure represents the annualized geometric returns of market capitalization weighted portfolios containing the 50 to 500largest stocks of the U.S. equity market from January 1927 (left side) respectively January 1931 (right side) to December2014. The portfolios are reconstituted using different reconstitution specifications. (1) On the left hand side, the portfolio isreconstituted at varying frequencies between one day and five years along the x-axis based on market capitalization ranks.(2) On the right hand side, the portfolios are reconstituted daily based on market capitalization ranks. However, the marketcapitalization used during portfolio reconstitution is lagged by different time periods between zero days (no lag) up to fiveyears along the x-axis. All portfolios are rebalanced daily. There are no transaction costs.

Fact

or E

xpos

ure

in %


Reconstitution Frequencies (Market Cap)


−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5


ctor

Exp

osur

e in

%

Reconstitution Lag



−4

−3

−2

−1

0

1

2

3

4


Figure 7: Changes in factor loadings for market capitalization weighted portfolios usingdifferent portfolio reconstitution policiesThis figure represents the relative factor loadings of market capitalization weighted portfolios containing the 250 largeststocks of the U.S. equity market from January 1931 to December 2014. The portfolios are reconstituted using differentreconstitution specifications. (1) On the left hand side, the portfolio is reconstituted at varying frequencies between one dayand five years along the x-axis based on market capitalization ranks. The factor loadings are relative to the factor loadingsat the daily portfolio reconstitution frequency. (2) On the right hand side, the portfolios are reconstituted daily based onmarket capitalization ranks. However, the market capitalization used during portfolio reconstitution is lagged by differenttime periods between zero days (no lag) up to five years along the x-axis. The factor loadings are relative to the factorloadings at the daily portfolio reconstitution frequency using current data. All portfolios are rebalanced daily. There are notransaction costs. The factor model is a six factor model that enhances the Carhart model (market excess returns [MKT],size [SMB], value [HML], momentum [UMD]) by short term reversal (STR) and long term reversal (LTR). The alpha isannualized for ease of interpretation.

84



2D-1D 0.01** 0.01* -0.03** -0.01 0.01* 0.04*** -0.00(0.01) (0.00) (0.01) (0.01) (0.01) (0.01) (0.01)

1W-1D 0.04*** 0.02*** -0.05** -0.03** 0.03*** 0.12*** 0.01(0.01) (0.01) (0.03) (0.01) (0.01) (0.01) (0.01)

2W-1D 0.04*** 0.03*** -0.07** -0.05*** 0.03** 0.20*** 0.01(0.01) (0.01) (0.03) (0.02) (0.02) (0.02) (0.02)

1M-1D -0.00 0.04*** -0.09*** -0.09*** 0.01 0.36*** -0.00(0.02) (0.01) (0.03) (0.03) (0.02) (0.02) (0.02)

2M-1D -0.07*** 0.03 -0.14*** -0.09* -0.11*** 0.51*** -0.05(0.02) (0.02) (0.04) (0.05) (0.03) (0.03) (0.04)

3M-1D -0.07*** 0.05* -0.18*** -0.12** -0.25*** 0.55*** -0.07*

(0.02) (0.02) (0.04) (0.06) (0.04) (0.03) (0.04)

4M-1D -0.08*** 0.02 -0.22*** -0.09 -0.38*** 0.59*** -0.11**

(0.03) (0.03) (0.05) (0.07) (0.04) (0.04) (0.05)

6M-1D -0.05 -0.00 -0.34*** -0.01 -0.70*** 0.58*** -0.20***

(0.04) (0.04) (0.07) (0.09) (0.06) (0.05) (0.06)

9M-1D -0.06 -0.09 -0.53*** 0.16 -1.10*** 0.61*** -0.25**

(0.07) (0.06) (0.14) (0.16) (0.11) (0.08) (0.11)

1Y-1D -0.00 -0.19** -0.76*** 0.45* -1.61*** 0.56*** -0.48***

(0.10) (0.10) (0.20) (0.27) (0.18) (0.12) (0.18)

18M-1D -0.00 -0.33*** -0.95*** 0.58*** -1.86*** 0.55*** -0.34**

(0.09) (0.08) (0.18) (0.21) (0.15) (0.10) (0.15)

2Y-1D 0.17* -0.46*** -1.13*** 1.00*** -2.09*** 0.36*** -0.22(0.10) (0.09) (0.19) (0.24) (0.15) (0.11) (0.16)

3Y-1D 0.26* -0.84*** -1.74*** 1.86*** -2.17*** 0.27* 0.09(0.14) (0.13) (0.23) (0.34) (0.20) (0.14) (0.21)

4Y-1D 0.32* -1.05*** -2.16*** 2.08*** -2.41*** 0.21 0.18(0.17) (0.16) (0.29) (0.42) (0.25) (0.18) (0.27)

5Y-1D 0.35** -1.04*** -2.47*** 2.39*** -2.54*** 0.18 0.59**

(0.17) (0.16) (0.29) (0.44) (0.26) (0.18) (0.28)

Table 9: Changes in factor loadings for market capitalization weighted portfolios recon-stituted at different frequenciesThis table represents the factor loadings of a strategy that invests in a market capitalization weighted portfolio that isreconstituted at different frequencies while shorting a market capitalization weighted portfolio reconstituted daily. Thus,this table represents the relative factor loadings compared to daily portfolio reconstitution. The first column specifies theportfolio reconstitution frequency of the long (first two characters) and short sides (last two characters) of the strategy,where D is days, W is weeks, M is months and Y is years. The portfolio reconstitution based on market capitalization ranksensures that the portfolio contains the largest stocks in the universe. The portfolios consist of the 250 largest stocks of theU.S. equity market from January 1931 to December 2014. The portfolio is rebalanced daily. The factor model is a six factormodel that enhances the Carhart model (market excess returns [MKT], size [SMB], value [HML], momentum [UMD]) byshort term reversal (STR) and long term reversal (LTR). The alpha is annualized for ease of interpretation, all values arein percent. The Newey West standard errors are given in brackets. * stands for two sided significance at the 10% level, **for significance at the 5% level and *** for significance at the 1% level.

As already mentioned, the weighting scheme plays a crucial role. Market-capitalization

weighting is not affected by rebalancing as long as there are no corporate actions or divi-

dends to be reinvested. Moreover for market-capitalization weighting, the relative weight

of large stocks dominates. Therefore, the relative importance of the inclusion threshold is

relatively low. Each stock near the inclusion threshold only obtains a negligible weight.

Increasing the number of stocks in the portfolios just increases the dominance of the large

stocks in the portfolio. The average market capitalization and consequently the portfolio

weight of each stock in the vicinity of the inclusion threshold decreases. Thus, increas-

ing the number of stocks in the portfolio decreases the impact of portfolio reconstitution

for market capitalization weighted portfolios. As portfolios containing a large number

85



1D-0D 0.02** 0.02* -0.05** -0.01 0.02* 0.07*** -0.01(0.01) (0.01) (0.03) (0.01) (0.01) (0.01) (0.02)

2D-0D 0.05*** 0.03** -0.06* -0.04** 0.04** 0.12*** -0.01(0.01) (0.01) (0.04) (0.02) (0.02) (0.02) (0.02)

1W-0D 0.05*** 0.03** -0.09** -0.06** 0.04** 0.24*** 0.01(0.02) (0.01) (0.04) (0.03) (0.02) (0.02) (0.02)

2W-0D 0.02 0.05*** -0.09*** -0.10*** 0.02 0.37*** 0.02(0.02) (0.02) (0.03) (0.04) (0.02) (0.03) (0.03)

1M-0D -0.10*** 0.04** -0.12*** -0.13** -0.05 0.60*** -0.00(0.03) (0.02) (0.04) (0.05) (0.04) (0.04) (0.04)

2M-0D -0.08** 0.04 -0.26*** -0.14 -0.38*** 0.66*** -0.12**

(0.03) (0.04) (0.07) (0.09) (0.06) (0.05) (0.06)

3M-0D -0.08 -0.01 -0.36*** -0.01 -0.73*** 0.66*** -0.25***

(0.05) (0.05) (0.09) (0.11) (0.07) (0.06) (0.08)

4M-0D -0.08 -0.06 -0.50*** 0.08 -1.07*** 0.66*** -0.33***

(0.07) (0.07) (0.14) (0.17) (0.11) (0.08) (0.12)

6M-0D -0.02 -0.20** -0.70*** 0.35 -1.66*** 0.57*** -0.47**

(0.11) (0.10) (0.21) (0.28) (0.18) (0.13) (0.20)

9M-0D -0.00 -0.33** -1.03*** 0.78** -2.38*** 0.56*** -0.68***

(0.14) (0.13) (0.27) (0.36) (0.23) (0.16) (0.24)

1Y-0D 0.05 -0.54*** -1.30*** 1.21*** -2.81*** 0.48*** -0.71***

(0.15) (0.14) (0.28) (0.39) (0.25) (0.16) (0.26)

18M-0D 0.29 -0.96*** -1.83*** 2.11*** -2.92*** 0.23 -0.22(0.18) (0.17) (0.32) (0.46) (0.27) (0.20) (0.29)

2Y-0D 0.37** -1.16*** -2.31*** 2.75*** -2.75*** 0.15 0.13(0.18) (0.18) (0.30) (0.47) (0.26) (0.20) (0.29)

3Y-0D 0.52*** -1.47*** -2.89*** 3.24*** -2.46*** -0.01 0.98***

(0.19) (0.19) (0.33) (0.50) (0.29) (0.21) (0.31)

4Y-0D 0.51** -1.66*** -3.41*** 3.50*** -2.36*** 0.03 1.87***

(0.21) (0.21) (0.33) (0.54) (0.30) (0.23) (0.35)

5Y-0D 0.62*** -2.08*** -3.97*** 4.07*** -2.35*** -0.07 2.15***

(0.23) (0.23) (0.36) (0.60) (0.34) (0.26) (0.40)

Table 10: Changes in factor loadings of market capitalization weighted portfolios whenusing lagged data for portfolio reconstitutionThis table represents the factor loadings of a strategy that invests in a market capitalization weighted portfolio that is re-constituted using lagged market capitalization ranks while shorting a market capitalization weighted portfolio reconstitutedusing current market capitalization ranks. Thus, the table represents the relative factor loadings compared to portfolioreconstitution with current market capitalization ranks. The first column specifies the portfolio reconstitution lags of thelong (first two characters) and short sides (last two characters) of the strategy, where D is days, W is weeks, M is monthsand Y is years. The portfolio reconstitution based on market capitalization ranks ensures that the portfolio contains thelargest stocks in the universe. The portfolio is reconstituted daily. The portfolios consist of the 250 largest stocks of theU.S. equity market from January 1931 to December 2014. The portfolio is rebalanced daily. The factor model is a six factormodel that enhances the Carhart model (market excess returns [MKT], size [SMB], value [HML], momentum [UMD]) byshort term reversal (STR) and long term reversal (LTR). The alpha is annualized for ease of interpretation, all values arein percent. The Newey West standard errors are given in brackets. * stands for two sided significance at the 10% level, **for significance at the 5% level and *** for significance at the 1% level.

of constituents are common for systematic equity strategies and indices, the portfolio

reconstitution effect often becomes close to negligible for market capitalization weighted

portfolios.

Equal-weighting, in contrast, requires regular rebalancing and displays a relatively

larger part of the portfolio close to the inclusion threshold. Thus the impact of trending

and reversal on portfolio returns are much stronger for the equal weighted portfolio.

I present the impact of market capitalization weighting only for portfolio reconstitu-

tion, as rebalancing does not make sense for a market capitalization weighted portfolio.

86



50 100 150 200 250 300 350 400 450 500

Const 0.39*** 0.36*** 0.30*** 0.26*** 0.24*** 0.24*** 0.20*** 0.18*** 0.17*** 0.16***

(0.05) (0.04) (0.03) (0.03) (0.03) (0.02) (0.02) (0.02) (0.02) (0.02)

NBER 0.41** 0.08 0.18** 0.11 0.12* 0.11** 0.17*** 0.12*** 0.07* 0.07*

(0.16) (0.11) (0.08) (0.07) (0.06) (0.05) (0.05) (0.04) (0.04) (0.04)

CSV 0.68*** 0.58*** 0.52*** 0.36*** 0.30*** 0.28*** 0.29*** 0.24*** 0.21*** 0.17***

(0.16) (0.11) (0.10) (0.07) (0.06) (0.05) (0.04) (0.04) (0.04) (0.03)

TSV 0.22* 0.11 0.19** 0.08 0.03 0.11*** 0.08** 0.06* 0.07** 0.07*

(0.13) (0.10) (0.08) (0.05) (0.05) (0.04) (0.04) (0.03) (0.03) (0.03)R2 1.16 1.51 2.29 1.49 1.19 1.83 2.28 1.83 1.66 1.31


50 100 150 200 250 300 350 400 450 500

Const 0.14* 0.02 -0.00 0.03 0.04 0.05 0.04 0.05 0.04 0.04(0.08) (0.07) (0.06) (0.06) (0.06) (0.06) (0.06) (0.06) (0.06) (0.06)

NBER -0.34* -0.10 0.02 -0.01 -0.02 -0.05 -0.04 -0.09 -0.09 -0.07(0.21) (0.18) (0.15) (0.13) (0.13) (0.12) (0.11) (0.11) (0.10) (0.10)

CSV -0.20 -0.22 -0.17 -0.07 -0.05 -0.04 0.02 0.04 0.03 0.03(0.18) (0.15) (0.14) (0.13) (0.12) (0.12) (0.12) (0.12) (0.12) (0.11)

TSV 0.04 -0.09 -0.07 -0.03 -0.04 -0.03 -0.09 -0.10 -0.08 -0.08(0.16) (0.14) (0.10) (0.08) (0.08) (0.07) (0.07) (0.07) (0.07) (0.06)

R2 0.05 0.13 0.08 0.02 0.02 0.01 0.02 0.03 0.03 0.02

Panel C: Drivers of Reconstitution (long term rev. horizon: 1.5 to 5 years)

50 100 150 200 250 300 350 400 450 500

Const 0.06 0.05 0.07 0.08 0.07 0.06 0.05 0.06 0.06 0.06(0.11) (0.09) (0.09) (0.09) (0.08) (0.08) (0.07) (0.07) (0.07) (0.07)

NBER 0.26 0.34 0.20 0.18 0.16 0.17 0.19 0.18 0.19 0.19(0.26) (0.21) (0.19) (0.18) (0.17) (0.16) (0.16) (0.15) (0.15) (0.14)

CSV 0.06 0.11 0.12 0.13 0.13 0.10 0.10 0.10 0.09 0.07(0.25) (0.19) (0.18) (0.17) (0.17) (0.16) (0.16) (0.16) (0.16) (0.15)

TSV 0.16 0.02 -0.02 -0.04 -0.02 -0.01 -0.01 -0.02 -0.03 -0.02(0.19) (0.14) (0.12) (0.12) (0.11) (0.11) (0.10) (0.10) (0.10) (0.09)

R2 0.03 0.03 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

Table 11: Drivers of portfolio reconstitution at different frequencies for market cap weightsThis table shows the regression loadings of potential drivers of the portfolio reconstitution effects. They are calculated formarket capitalization weighted portfolios containing the 50 to 500 largest stocks of the U.S. equity market from January1927 to December 2014. The explained variables are the annualized arithmetic return differences when varying the portfolioreconstitution frequency between one day and five years. The differences are at the short term reversal horizon (1 day to 3months) in Panel A, the momentum horizon (3 months to 18 months) in Panel B and the long term reversal horizon (1.5 to5 years) in Panel C. The direction of the differences is selected such that according to theory all returns should be positive.This means 3 months minus one day, 3 months minus 18 months and 5 years minus 1.5 years. The regressors are a businesscycle dummy (NBER), the normalized cross-sectional volatility (CSV) and the normalized time series volatility (TSV). TheNewey West standard errors are given in brackets. * stands for two sided significance at the 10% level, ** for significanceat the 5% level and *** for significance at the 1% level.

As in the previous section of the appendix, there are two different portfolio reconstitu-

tion specifications: (a) reconstitution frequencies between one day and five years and (b)

reconstitution using lagged market capitalization ranks with the lags ranging from no lag

to a lag of five years.

In terms of returns in graph 7, the patterns discussed in the main section remain.

The return differences, however are much smaller and decrease with the number of stocks

in the portfolios. Again using reconstitution lags leads to stronger effects than varying

the frequency. However, the effect of the weighting scheme dominates. The graph for

87



50 100 150 200 250 300 350 400 450 500

Const 0.43*** 0.36*** 0.33*** 0.28*** 0.25*** 0.24*** 0.20*** 0.19*** 0.17*** 0.17***

(0.08) (0.06) (0.06) (0.05) (0.05) (0.04) (0.04) (0.03) (0.03) (0.03)

NBER 0.49** 0.13 0.08 0.08 0.08 0.08 0.20** 0.10 0.07 0.04(0.25) (0.18) (0.14) (0.13) (0.11) (0.09) (0.09) (0.07) (0.07) (0.07)

CSV 0.73*** 0.67*** 0.55*** 0.33*** 0.29*** 0.30*** 0.29*** 0.22*** 0.20*** 0.19***

(0.21) (0.16) (0.14) (0.11) (0.09) (0.08) (0.08) (0.07) (0.06) (0.06)

TSV 0.18 0.13 0.20* 0.12* 0.08 0.12** 0.11** 0.10** 0.10** 0.09*

(0.17) (0.13) (0.11) (0.08) (0.06) (0.06) (0.06) (0.05) (0.05) (0.04)R2 0.61 0.85 1.04 0.56 0.51 0.81 1.00 0.75 0.71 0.67


50 100 150 200 250 300 350 400 450 500

Const 0.27** 0.02 0.06 0.06 0.10 0.12 0.07 0.08 0.08 0.07(0.12) (0.09) (0.09) (0.09) (0.09) (0.09) (0.09) (0.08) (0.08) (0.08)

NBER -0.25 0.18 0.12 0.14 0.09 0.02 0.09 0.02 0.01 -0.01(0.32) (0.26) (0.24) (0.21) (0.19) (0.18) (0.17) (0.16) (0.15) (0.14)

CSV 0.11 -0.20 -0.20 -0.17 -0.08 0.02 0.00 -0.01 0.03 0.01(0.22) (0.17) (0.18) (0.17) (0.16) (0.17) (0.17) (0.16) (0.16) (0.15)

TSV -0.08 -0.09 0.01 0.08 0.01 0.00 -0.05 -0.02 -0.03 0.00(0.19) (0.15) (0.14) (0.11) (0.10) (0.09) (0.08) (0.08) (0.08) (0.07)

R2 0.01 0.05 0.03 0.02 0.01 0.00 0.00 0.00 0.00 0.00

Panel C: Drivers of Reconstitution (long term rev. horizon: 1 to 5 years)

50 100 150 200 250 300 350 400 450 500

Const 0.11 0.12 0.15 0.17 0.17 0.18 0.10 0.11 0.12 0.10(0.24) (0.18) (0.16) (0.15) (0.14) (0.13) (0.12) (0.12) (0.12) (0.11)

NBER 0.67 1.16** 0.70 0.63 0.63 0.56 0.62* 0.60* 0.52 0.53*

(0.65) (0.53) (0.48) (0.44) (0.42) (0.39) (0.37) (0.34) (0.32) (0.31)CSV 0.38 0.08 -0.03 0.05 0.11 0.17 0.05 0.09 0.05 -0.03

(0.57) (0.39) (0.35) (0.31) (0.30) (0.29) (0.26) (0.25) (0.25) (0.25)TSV 0.01 -0.08 -0.01 0.04 -0.03 -0.03 -0.00 -0.01 0.01 0.09

(0.41) (0.30) (0.29) (0.23) (0.23) (0.21) (0.20) (0.18) (0.18) (0.18)R2 0.02 0.04 0.02 0.02 0.02 0.03 0.03 0.03 0.02 0.03

Table 12: Drivers of portfolio reconstitution using lagged market cap ranks for marketcap weightsThis table shows the regression loadings of potential drivers of the portfolio reconstitution effects. They are calculated formarket capitalization weighted portfolios containing the 50 to 500 largest stocks of the U.S. equity market from January1931 to December 2014. The explained variables are the annualized arithmetic return differences when using variably laggedmarket capitalization ranks for portfolio reconstitution between no lag and a lag of five years. The differences are at theshort term reversal horizon (current to 3 months) in Panel A, the momentum horizon (3 months to 12 months) in Panel Band the long term reversal horizon (1 to 5 years) in Panel C. The direction of the differences is selected such that accordingto theory all returns should be positive. This means 3 months minus current, 3 months minus 12 months and 5 yearsminus 1 year. The regressors are a business cycle dummy (NBER), the normalized cross-sectional volatility (CSV) andthe normalized time series volatility (TSV). The Newey West standard errors are given in brackets. * stands for two sidedsignificance at the 10% level, ** for significance at the 5% level and *** for significance at the 1% level.

the reconstitution lags looks substantially noisier, particularly if there is a small number

of stocks in the portfolio. This is due to the lack of smoothing for reconstitution lags

discussed in the methodological remark at the end of the appendix.

In terms of relative factor exposures in figure 6 as well as in tables 9 and 10, the overall

patterns remain very similar. However, the size of the factor loadings gets substantially

reduced. The short term reversal loading is up to 0.55% at the three months frequency

for the market capitalization compared to 3.21% for the equal weighted portfolio. The

88


momentum loading is down to -1.61% (vs. -7.68%) at the 12 months frequency. The value

loading reaches 2.39% (vs. 8.11%) and the the long term reversal loading reaches 0.59%

(vs. 4.38%) at the five year horizon.

Reconstitution at low frequencies or using long lags shows a very distinct reduction

in the loadings to the market risk premium (-1.04% and -2.08%) and to the size factor

(-2.47% and -3.97%). This effect is much stronger relatively to the other factor exposures

than for the equally weighted portfolio, where it is -0.45% and -3.58% for the market risk

premia and -2.57% and -2.35% for size. As discussed before, the reduced size loading is

due to the fact that stocks growing in size do not enter the portfolio, while stocks declin-

ing in size remain in the portfolio. Therefore, the portfolio becomes more concentrated

in large capitalization stocks. In terms of market beta, with less reconstitutions the port-

folio becomes more concentrated in defensive and large capitalization stocks with a lower

market beta. This could be due a combination of two effects. First, The low volatility

anomaly of defensive stocks performing better (Blitz & van Vliet, 2007; Baker & Haugen,

2012) and thus receiving a higher weight over time. Second, the natural concentration

tendencies of a buy-and-hold portfolio (Stein et al., 2009; Cuthbertson et al., 2015) that

is also responsible for the reduced size exposure.

Finally for the macroeconomic drivers, market capitalization weights substantially

weaken the results. In tables 9 and 10, we can observe that only the cross-sectional

volatility remains highly significant at the short term reversal horizon with return impacts

of 0.17 respectively 0.19 for 500 stocks. For equal weighted portfolios in contrast, the

return impacts were 2.70 and 2.95. The significance becomes much weaker for business

cycles and the time series volatility. For the momentum horizon as well as the long term

reversal horizon, more or less all the results are insignificant and there is virtually no

explanatory power left.

89


A.3 Relative factor exposures to number of stocks

Relative factor exposures for portfolios of 100 largest stocks

Fact

or E

xpos

ure

in %


Rebalancing (Equal Weights 0BP)


−2

0

2

4

6


Fact

or E

xpos

ure

in %


Reconstitution Frequency (Market Cap)


−3

−2

−1

0

1

2

3


ctor

Exp

osur

e in

%


Reconstitution Frequency (Equal Weights)


−10

−5

0

5


Fact

or E

xpos

ure

in %

Reconstitution Lag


0D 1D 2D 1W2W 1M 2M 3M 4M 6M 9M 1Y18M2Y 3Y 4Y 5Y−4

−3

−2

−1

0

1

2

3

4

5


ctor

Exp

osur

e in

%

Reconstitution Lag



−10

−5

0

5

10


Figure 8: Changes in portfolios containing 100 stocks using different portfolio reconstitu-tion policiesThis figure represents the relative factor exposures of different rebalancing and portfolio reconstitution policies for marketcapitalization weighted portfolios and equal weighted portfolios containing the 100 largest stocks of the U.S. equity marketfrom January 1931 to December 2014. The top figure presents the relative factor exposures for equal weighted portfoliosusing different rebalancing frequencies. The factor loadings are relative to the factor loadings at the daily rebalancingfrequency. The middle figures display the relative factor exposures when varying the reconstitution frequencies for marketcapitalization weighted portfolios (middle left figure) and equal weighted portfolios (middle right figure). The factor load-ings are relative to the factor loadings at the daily portfolio reconstitution frequency. Finally, the lower figures visualize therelative factor exposures when using lagged market capitalization ranks to determine the largest stocks in the portfolio atportfolio reconstitution. The lags vary between no lag and a lag of five years. The factor loadings are relative to the factorloadings at the daily portfolio reconstitution frequency using current data. The portfolio weighting schemes are marketcapitalization weighting (bottom left figure) and equal weighting (bottom right figure).

The main section demonstrates the relative factor exposures for 250 stocks in the

portfolio. Repeating the exercise for portfolios of 100 and 500 stocks for rebalancing as

well as four versions of portfolio reconstitution, similar patterns emerge. In all cases,

90


Relative factor exposures for portfolios of 500 largest stocks

Fact

or E

xpos

ure

in %




−4

−2

0

2

4

6

8

10


Fact

or E

xpos

ure

in %



1D 2D 1W 2W 1M 2M 3M 4M 6M 9M 1Y18M2Y 3Y 4Y 5Y−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


ctor

Exp

osur

e in

%




−8

−6

−4

−2

0

2

4

6

8


Fact

or E

xpos

ure

in %

Reconstitution Lag



−4

−3

−2

−1

0

1

2

3


ctor

Exp

osur

e in

%

Reconstitution Lag



−10

−5

0

5

10


Figure 9: Changes in portfolios containing 500 stocks using different portfolio reconstitu-tion policiesThis figure represents the relative factor exposures of different rebalancing and portfolio reconstitution policies for marketcapitalization weighted portfolios and equal weighted portfolios containing the 500 largest stocks of the U.S. equity marketfrom January 1931 to December 2014. The top figure presents the relative factor exposures for equal weighted portfoliosusing different rebalancing frequencies. The factor loadings are relative to the factor loadings at the daily rebalancingfrequency. The middle figures display the relative factor exposures when varying the reconstitution frequencies for marketcapitalization weighted portfolios (middle left figure) and equal weighted portfolios (middle right figure). The factor load-ings are relative to the factor loadings at the daily portfolio reconstitution frequency. Finally, the lower figures visualize therelative factor exposures when using lagged market capitalization ranks to determine the largest stocks in the portfolio atportfolio reconstitution. The lags vary between no lag and a lag of five years. The factor loadings are relative to the factorloadings at the daily portfolio reconstitution frequency using current data. The portfolio weighting schemes are marketcapitalization weighting (bottom left figure) and equal weighting (bottom right figure).

the four factors capturing the trending and reversal effects in the cross-section of equity

returns show the same behaviour. The magnitude of the coefficients and the relative

importance, however, varies moderately.

91


A.4 Adjusting for risk: Sharpe ratios

Adjusting the portfolios discussed so far for absolute risk does not change the results. All

portfolios show relatively similar portfolio volatility such that the patterns remain the

same.

Shar

pe R

atio




0.35

0.4

0.45

0.5

0.55

50100150200250300350400450500

Shar

pe R

atio



1D 2D 1W 2W 1M 2M 3M 4M 6M 9M 1Y 18M 2Y 3Y 4Y 5Y0.2

0.25

0.3

0.35

0.4 50100150200250300350400450500

Shar

pe R

atio



1D 2D 1W 2W 1M 2M 3M 4M 6M 9M 1Y 18M 2Y 3Y 4Y 5Y0.32

0.33

0.34

0.35

0.36

0.37

0.38

50100150200250300350400450500

Shar

pe R

atio




0.35

0.4

0.45

0.5

0.55

0.6 50100150200250300350400450500

Shar

pe R

atio

Reconstitution Lag



0.34

0.35

0.36

0.37

0.38

0.39

0.4

0.41

50100150200250300350400450500

Shar

pe R

atio

Reconstitution Lag



0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

50100150200250300350400450500

Figure 10: Sharpe ratios of rebalancing and portfolio reconstitution policiesThis figure represents the annualized Sharpe ratios of different rebalancing and portfolio reconstitution policies for marketcapitalization weighted portfolios and equal weighted portfolios containing the 50 to 500 largest stocks of the U.S. equitymarket from January 1927 (frequencies) respectively January 1931 (lags) to December 2014. The top figures present theSharpe ratios to equal weighted portfolios using different rebalancing frequencies assuming no transaction costs (top leftfigure) and one-way transaction costs of 50 basis points (top right figure). These portfolios are reconstituted annuallybased on market capitalization ranks to reflect the largest stocks in the universe. The middle figures display the Sharperatio impact of varying the reconstitution frequencies for market capitalization weighted portfolios (middle left figure) andequal weighted portfolios (middle right figure). These portfolios are rebalanced daily. Finally, the lower figures visualizethe Sharpe ratio impact when using lagged market capitalization ranks to determine the largest stocks in the portfolio atportfolio reconstitution. The lags vary between no lag and a lag of five years. Meanwhile the portfolios are daily rebalancedand reconstituted. The portfolio weighting schemes are market capitalization weighting (bottom left figure) and equalweighting (bottom right figure).

92


A.5 Adjusting for relative risk: Information ratios

Adjusting the portfolios discussed so far for relative risk does not change the results.

All portfolios show a relatively similar portfolio tracking error versus the broad market

capitalization weighted benchmark containing all stocks in the universe. Therefore, the

patterns remain the same.

Info

rmat

ion

Rat

io




−0.2

0

0.2

0.4

0.6

0.8

50100150200250300350400450500

Info

rmat

ion

Rat

io



1D 2D 1W 2W 1M 2M 3M 4M 6M 9M 1Y 18M 2Y 3Y 4Y 5Y−1

−0.8

−0.6

−0.4

−0.2

0

0.2

50100150200250300350400450500

Info

rmat

ion

Rat

io




−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

50100150200250300350400450500

Info

rmat

ion

Rat

io




−0.2

0

0.2

0.4

0.6

0.8

1

50100150200250300350400450500

Info

rmat

ion

Rat

io

Reconstitution Lag



−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

50100150200250300350400450500

Info

rmat

ion

Rat

io

Reconstitution Lag


0D 1D 2D 1W2W 1M 2M 3M 4M 6M 9M 1Y18M2Y 3Y 4Y 5Y−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

50100150200250300350400450500

Figure 11: Information ratios of rebalancing and portfolio reconstitution policiesThis figure represents the annualized information ratios of different rebalancing and portfolio reconstitution policies formarket capitalization weighted portfolios and equal weighted portfolios containing the 50 to 500 largest stocks of the U.S.equity market from January 1927 (frequencies) respectively January 1931 (lags) to December 2014. The top figures presentthe information ratios to equal weighted portfolios using different rebalancing frequencies assuming no transaction costs(top left figure) and one-way transaction costs of 50 basis points (top right figure). These portfolios are reconstitutedannually based on market capitalization ranks to reflect the largest stocks in the universe. The middle figures display theinformation ratio impact of varying the reconstitution frequencies for market capitalization weighted portfolios (middle leftfigure) and equal weighted portfolios (middle right figure). These portfolios are rebalanced daily. Finally, the lower figuresvisualize the information ratio impact when using lagged market capitalization ranks to determine the largest stocks in theportfolio at portfolio reconstitution. The lags vary between no lag and a lag of five years. Meanwhile the portfolios are dailyrebalanced and reconstituted. The portfolio weighting schemes are market capitalization weighting (bottom left figure) andequal weighting (bottom right figure).

93


A.6 The impact of transaction costs on rebalancing and portfo-

lio reconstitution

Ret

urn

in %

p.a

.



7

7.5

8

8.5

9

9.5

10

10.5

50100150200250300350400450500

Figure 12: Returns of equally weighted portfolios at different rebalancing frequenciesassuming 50 basis points of one-way transaction costsThis figure represents the annualized geometric returns of equally weighted portfolios containing the 50 to 500 largest stocksof the U.S. equity market from January 1927 to December 2014. The portfolios are reconstituted annually based on marketcapitalization ranks to reflect the largest stocks in the universe. The rebalancing frequency of the portfolios varies betweenone day and five years along the x-axis. There are one-way transaction costs of 50 basis points.

For rebalancing in a portfolio context, transaction costs play a crucial role. As figure 12

shows for 50 basis points of one-way transaction costs, the picture changes dramatically

compared to figure 1 without transaction costs. The theoretical return gains get eaten

away by transaction costs fairly quickly, particularly at very high frequencies. Especially

for a low number of stocks in a portfolio, 20 to 30 basis points in one-way transaction

costs are enough to counteract the return gains of rebalancing at high frequencies. At 40

basis points one-way transaction costs, high frequency rebalancing is unattractive to all

analysed portfolios. In the case of 50 basis points in figure 12, high frequency rebalancing

is already a massive return detractor. Looking at more recent time periods such as 2000

to 2014, the basic shape does not change, but the gains from high frequency rebalancing

become weaker, such that already at costs of 20 basis points, a high frequency strategy

becomes unattractive. Thus, high frequency rebalancing is only suited in case of low

transaction costs as for example for market makers. For most investors, low to very low

rebalancing frequencies are likely to be the best option. The goal here, however, is to

discuss the general properties and characteristics of rebalancing and not find an optimal

investment strategy.

For portfolio reconstitution, in contrast, the returns increase with a decreasing re-

94


constitution frequency. This leads to less portfolio turnover and, thus, lower transaction

costs. Therefore, portfolio reconstitution becomes even more attractive after costs at

lower frequencies. Smoothing market capitalization data for reconstitution also reduces

turnover, but less than reducing reconstitution frequency. Finally, reconstituting using

lagged market capitalization data does not reduce portfolio turnover, such that it cannot

profit from reduced transaction costs.

B Methodological remark: smoothing potential sea-

sonality effects

To obtain robust estimates of the sets of portfolios, we have to take into account that

setting a particular first rebalancing or portfolio reconstitution date becomes random as

the frequency becomes lower and lower. It is an arbitrary decision to perform a yearly

portfolio reconstitution every last trading day of December or on the last trading day in

March for example. Just fixing one date could, however, capture unwanted systematic

seasonality effects or just create unnecessary noise. Thus, at frequencies lower than 10

trading days, the portfolio engine randomly draws 10 potential first rebalancing or portfo-

lio reconstitution dates and averages the returns across these 10 dates to ensure that the

choice of a particular portfolio adjustment day does not impact the results. At frequencies

higher than ten trading days, it calculates all potential shifts and averages across them.

This smooths out potential seasonality effects, reduces noise and at the same time effi-

ciently uses limited calculation resources. Without computational constraints, smoothing

across all first rebalancing or portfolio reconstitution dates while keeping the desired fre-

quency would obviously be preferred. In the case of reconstitution lags, smoothing the

time points is not feasible, which shows in less robust patterns.

95

The Power of Equity Factor Diversification

Ulrich Carl∗

Draft: January 26th, 2016

Abstract

This paper analyses the diversification properties of country equity factors across six

equity factors and twenty developed markets from 1991 to 2015. The factors considered are

the market excess return, size, value, momentum, low beta and quality. I find substantial

diversification benefits along the country dimension as well as the factor dimension. In

a portfolio setting, country diversification significantly reduces the volatility compared to

single country investing for each of the six equity factors. Factor diversification works in

each of the twenty markets by means of reducing the portfolio volatility.

JEL CODES: G11 G12 G15 C38

Key words: equity factors, factor diversification, international diversification, factor

investing

∗Finreon Ltd., Oberer Graben 3, 9000 St.Gallen, Switzerland and University of St.Gallen, School ofEconomics and Political Science, Bodanstrasse 8, 9000 St.Gallen, Switzerland. The views expressed inthis paper are my own and do not necessarily reflect those of Finreon Ltd. and of the University ofSt.Gallen. I would like to thank Paul Soderlind, Francesco Audrino, Lukas Plachel, Marcial Messmer,Christian Finke, Julius Agnesens and Ralf Seiz and the seminar participants at the University of St.Gallenfor helpful comments.contact: [email protected], +41 76 210 03 12

Ulrich Carl The Power of Equity Factor Diversification

1 Introduction

Traditionally, institutional investors manage their strategic asset allocation from an asset

class perspective by setting quotas for stocks, bonds and other asset classes. However,

during the last years, a new approach - considering a portfolio increasingly from the

perspective of risk factors - has become popular. Some of the largest institutional investors

worldwide such as the Norwegian sovereign wealth fund have led the trend. Particularly

in the equity domain, multi-factor products have seen substantial growth in assets under

management.

From an academic perspective, the knowledge of diversification properties of the com-

monly used factors is of crucial importance for the understanding of these factors, their

economic rationale and their interdependencies. On the one hand, we need to understand

if factor risk is idiosyncratic or if it is systematic. Empirical work shows that the risk

of individual factors can be to a large extent diversified away. This is at odds with the

traditional view of rational and efficient capital markets, where only systematic risk is

compensated. On the other hand, to be able to specify factor models correctly, we need

to know in how far these factors are global or if the factors are segmented across markets.

This paper contributes to the research on factor investing by analysing the diversifica-

tion properties of country equity factors. The focus is on six equity factors - market excess

return, size, value, momentum, low beta and quality - across twenty developed markets

from January 1991 to April 2015. These factors are arguably the most recognized factors

in empirical asset pricing. The analysis looks at diversification in three dimensions: (a)

across twenty countries for each factor (international diversification), (b) across six factors

for each country (local factor diversification) and (c) across twenty countries and across

six factors jointly (international factor diversification). Finally, a portfolio construction

exercise demonstrates the diversification benefits in a portfolio context.

I find that all six factors except the size factor achieve robust positive returns across

countries. All equity country factors offer substantial diversification benefits across the

country dimension as well as factor dimension. International investing substantially re-

duces volatility compared to single country investing. Moreover, forming local portfolios

of factors (local factor diversification) works across all twenty countries in the sample by

reducing the portfolio volatility compared to single factor investing.

The structure of the remainder of this paper is as follows. Section 2 gives a short

overview of the literature on factor diversification. Section 3 describes the empirical

approach and the methods used. Section 4 presents the data. Section 5 is the main

section and covers the empirical results. Last, section 6 concludes.

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2 Literature

In the capital asset pricing model (Treynor, 1961; Sharpe, 1964; Lintner, 1965; Mossin,

1966), there is only one single factor driving equity returns: the equity market risk pre-

mium. However, this simple model cannot adequately capture the prices observed in the

capital markets. The intertemporal capital asset pricing model of Merton (1973) hints

towards factors that capture changes in the investment opportunity set. The arbitrage

pricing theory of Ross (1976) lays the theoretical groundwork for multi-factor models to

describe asset returns. However, it gives no indication about the relevant factors. Breeden

(1979) uses aggregate consumption as the only state variable. The first multi-factor mod-

els focused on macro-economic variables such as in Chen, Roll, and Ross (1986). Data

limitations are the main drawback of these approaches. There is a multitude of economic

variables, which are measured at low frequencies and with considerable measurement er-

ror and which are subject to frequent data revisions. Therefore, empirical asset pricing

models using investable portfolios to create factors are more popular.

The first popular empirical factor model is the Fama and French (1992, 1993) model.

It includes a size factor (SMB) and a value factor (HML) besides the market excess return

from the capital asset pricing model.

The size factor (Banz, 1981) capitalizes on the higher returns of small-capitalization

stocks compared to large-capitalization stocks. The explanations for the existence of the

size factor mostly focus on risk. Small companies tend to be unproductive and highly

leveraged (Chan, Hamao, & Lakonishok, 1991). The size factor is often related to de-

fault risk (Vassalou & Xing, 2004), to innovations in investment opportunities (Petkova,

2006) or credit risk (Hwang, Min, McDonald, Kim, & Kim, 2010). The size factor has

been widely criticised for its widespread recent poor returns across international markets.

However, Asness, Frazzini, and Pedersen (2013) and Asness, Frazzini, Israel, Moskowitz,

and Pedersen (2015) show how to resurrect the size effect when correcting for negative

quality exposure1.

The value factor (Basu, 1977, 1983) profits from higher returns of high book-value-to-

market-capitalization stocks compared to low book-value-to-market-capitalization stocks.

It has been the most recognized factor throughout the last 30 years. There is a wide range

of literature on risk-based explanations of the value effect such as distress risk (Fama

& French, 1995), cost reversibility (Zhang, 2005) or the interaction of asset risk and

financial leverage (Choi, 2013). Behavioural explanations focus on investor overreaction,

e.g. Barberis et al. (1998), Daniel et al. (1998) and Hong and Stein (1999)2.

Another common factor in the literature is the momentum factor (UMD) of Jegadeesh

1For a detailed review of the size factor see van Dijk (2011)2For a detailed review of the value factor see Asness, Frazzini, Israel, and Moskowitz (2015)

99


and Titman (1993), which capitalizes on the return differential between past winners and

past losers over the 6-12 months period. The Carhart (1997) model, including momentum

along with the market excess return, size and value is still one of the work horses for

performance analysis nowadays. Behavioural explanations for momentum such as investor

underreaction and overreaction are most prevalent (Shefrin & Statman, 1985; Barberis

et al., 1998; Daniel et al., 1998; Hong & Stein, 1999; Grinblatt & Han, 2005). From the

risk-based perspective, momentum is linked to a firm’s optimal investment choices (Berk

et al., 1999), time-varying dividend growth rates (Johnson, 2002), revenues, costs, growth

options and shut-down options (Sagi & Seasholes, 2007) or investment flows (Vayanos &

Woolley, 2013)3.

Particularly in the last years, the number of potential factors has grown massively

such that Cochrane (2011) describes it as a ”zoo of new factors”. Harvey, Liu, and Zhu

(2016) catalogue 314 different factors as of 2012 and discuss tests of statistical significance

in light of data mining for factors.

Moreover, I add two more factors to the analysis that prove very robust across time

and markets and receive wide attention from academics and practitioners alike.

The low risk factor, which is based on the finding that low risk stocks outperform

high risk stocks in terms of risk-adjusted returns (low beta) or even absolute returns

(low volatility) dates back to Black (1972) and Haugen and Heins (1975). While being

neglected for a long time, this anomaly achieved high visibility with the papers of Ang,

Hodrick, Xing, and Zhang (2006) and Blitz and van Vliet (2007) as well as with the

increasing risk aversion of investors in the wake of the 2008 financial crisis. The betting-

against-beta factor (BAB) of Frazzini and Pedersen (2014) is the most recognized factor

to capture the low beta anomaly. Most explanations focus on market imperfections and

behavioural biases such as benchmarking of institutional investors (M. Baker, Bradley, &

Wurgler, 2011), leverage and margin constraints (Black, 1972; Frazzini & Pedersen, 2014),

over-optimism of sell-side analysts (Hsu, Kudoh, & Yamada, 2013), option-like manager

compensation and agency issues (N. Baker & Haugen, 2012). It is hard to reconcile the

empirical findings with explanations based on systematic risk.

Finally, quality investing in several guises has always been a part of investment prac-

tice. However, with some exceptions such as Piotroski (2000), the academic commu-

nity picked these ideas up only recently with the quality-minus-junk factor (QMJ) of

Asness, Frazzini, and Pedersen (2013) and the gross-profitability factor of Novy-Marx

(2013, 2014). Several recognized newer factor models use profitability as pricing factor

e.g. Fama and French (2015) and Hou, Yue, and Zhang (2015). The QMJ factor defines

3For a detailed review of the momentum factor see Jegadeesh and Titman (2011) or Asness et al.(2014)

100


quality as a combination of profitability, growth, safety and payout. There has not been

much of a discussion on explanations of the returns to the quality factor yet. The dis-

position effect (Shefrin & Statman, 1985) or arguments similar to those used to explain

the low beta anomaly are obvious candidates. The defensive characteristics and a flight

to quality during crashes (Asness, Frazzini, & Pedersen, 2013) contradict a risk-based

explanation.

While most of the factor research focuses on the U.S. equity market, the factors in

this paper (except for SMB) are stable internationally. Fama and French (2012) test

the size, value and momentum factors in four regions (North America, Europe, Japan,

and Asia Pacific) and find that value and momentum are strong in all regions except in

Japan. They find that local factor models are better at describing the cross-section of

asset returns, which hints to potential gains from international factor diversification. The

findings of Griffin (2002) and Hou, Karolyi, and Kho (2011) support the outperformance of

local factor models, while Asness, Moskowitz, and Pedersen (2013) find a strong common

factor structure of value and momentum across markets and asset classes.

Even though the previously discussed factors achieve significantly positive returns in

the long term, each single factor shows pronounced cyclicality and draw-downs. Thus,

from an investor’s perspective, the tracking error is often too large and thus the informa-

tion ratio is too low to warrant an investment. The intuitive solution is the combination

of moderately correlated factors in order to reduce the tracking error. In academia,

particularly the combination and interaction of value and momentum and their link to

macroeconomic risk, funding liquidity risk and stock market liquidity risk receives wide

attention (Asness, 1997; Asness, Moskowitz, & Pedersen, 2013; Cakici & Tan, 2013).

In the investment management community, factor investing gained traction with the

report of Ang, Goetzmann, and Schaefer (2009) to Norway’s Government Pension Fund

Global, one of the largest institutional investors worldwide. In their report, they show

that factors drive two thirds of the return differential of active management and they

propose to harvest risk premia in a systematic way4. Bender, Briand, Nielsen, and Ste-

fek (2010) and Carhart, Cheak, De Santis, Farrell, and Robert (2014) demonstrate the

performance and diversification benefits of investing in risk premia across asset classes in

a long-short setting. Ghayur, Heaney, and Platt (2013) approach factor based investing

in equities in a long-only context. Idzorek and Kowara (2013) contrast a factor-based

asset allocation with an asset-class-based asset allocation. Cazalet and Roncalli (2014)

give a detailed overview about factor investing and the challenges when transferring the

4For further information: Chambers, Dimson, and Ilmanen (2012) (details about the Norwegianmodel), Ang (2014) (case study about the Norwegian sovereign wealth fund in the chapter on factorinvesting), Ang, Brandt, and Denison (2014) (follow-up report to the Norwegian sovereign wealth fund)and Bambaci et al. (2013) (MSCI report to the Norwegian sovereign wealth fund).

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theoretical findings to an implementable portfolio.

To my knowledge, Eun, Lai, de Roon, and Zhang (2010) are the only ones to discuss

international diversification using equity factors as assets in an investment portfolio. Their

focus is on mean-variance-optimization using the broad equity market, SMB, HML and

UMD for ten developed markets. I put more emphasis on the diversification properties

of the equity factors and discuss international diversification for each factor separately.

Moreover, I expand the range of factors as well as countries and use portfolio construction

techniques that are more common in practice.

3 Approach and Methods

After a factor performance overview, this paper gives a detailed analysis of the correlation

structure of country equity factors across factors as well as across countries. In terms of

correlations, I focus on (a) the cross-country correlations for each of the six equity

factors, (b) the cross-factor correlations for each of the twenty countries, (c) the cross-

country cross-factor correlations, (d) the dynamics in the cross-country correlations for

each of the six factors, (e) the dynamics in the cross-factor correlations for each of the

twenty countries and (f) the conditional cross-country correlations for each of the six

equity factors for market tail events. The analysis of the dynamics uses rolling windows

with a length of 36 months. I define market tails in terms of the tails of the monthly

market excess return distribution, where the left tail corresponds to the 30% lowest market

excess returns and the right tail corresponds to the 30% highest market excess returns.

This is a compromise between capturing the extreme values in the tails and the availability

of enough data. The appendix demonstrates the tail correlation analysis for each factor

in terms of tails in the respective factor instead of the tails in the market excess returns.

In the next step, I use principal component analysis to evaluate in how far the

components explain (a) the cross-country variation for each of the six factors, (b) the

cross-factor variation for each of the twenty countries as well as (c) the total cross-country

cross-factor variation in the data set of the 120 country equity factors.

To decompose the data into principal components, there are two equivalent ways: an

eigenvalue decomposition of the covariance matrix or a singular value decomposition of

the data, which I detail here. The singular value decomposition decomposes the data X

in X = UΣV ′, where U are the left singular vectors, the diagonal elements of Σ are the

singular values and V are the right singular vectors. Important elements for this analysis

are (a) the score matrix UΣ, which is the representation of the data X in the principal

component space, (b) the right singular values V that can be interpreted as coefficients to

map the data X to the principal component space UΣ and (c) the squares of the singular

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values Σ, which correspond to the eigenvalues of the covariance matrix of X. The Σ2 are

used to calculate the percentage of the total variance explained by each of the principal

components (columns in the score matrix UΣ).

The goal in the portfolio construction exercise is to demonstrate the risk reduction

through international diversification and local factor diversification. For both dimensions,

I contrast a risk parity weighted portfolios of country factors constructed along the country

dimension (20 country factors form a global factor) and the factor dimension (6 local

factors form a local factor portfolio) with synthetic factors. These synthetic factors reflect

the average returns and the average volatilities of the portfolio constituents, but neglecting

the diversification effect.

The goal of the risk parity approach (Maillard, Roncalli, & Teıletche, 2010) is to ensure

that all portfolio components (i.e. countries or factors in this case) contribute equally to

the portfolio volatility. The following objective function minimizes the differences between

the risk contributions, where x is the vector of weights, N is the number of assets in the

portfolio and Σ is the covariance matrix:

x⋆ = argminx

N∑

i=1

N∑

j=1

(xi(Σx)i − xj(Σx)j)2 (1)

u.c. 1′x = 1 and 0 ≤ x ≤ 1

One crucial input for the estimation of the risk parity portfolio is the covariance matrix.

Simply taking the sample covariance matrix, particularly when using monthly data, leads

to an unstable estimator of the true covariance matrix. There are several approaches

to stabilize the estimates while moderately increasing the bias (classical variance-bias

trade-off): (1) Dimensionality reduction of the problem by means of a factor model for

estimating the covariance matrix, e.g. the CAPM or statistical factors. (2) Shrinkage

approaches, where we shrink the sample covariance matrix towards a factor-based co-

variance matrix (Ledoit & Wolf, 2003), the constant correlation matrix (Ledoit & Wolf,

2004a) or the identity matrix (Ledoit & Wolf, 2004b). (3) Matrix cleansing through the

use of random matrix theory (Laloux, Cizeau, Potters, & Bouchaud, 2000).

As recommended by Coqueret and Milhau (2014), I use an eigenvalue clipping ap-

proach based on random matrix theory. Random matrix theory is concerned with the

distributions of random matrices and helps to discriminate between eigenvectors contain-

ing signals and eigenvectors containing noise. Eigenvalues of random matrices (Wishart

matrices5) are Marcenko-Pastur distributed and thus bounded. A common approach is

5A Wishart matrix W is a special case of a random matrix. It is a symmetric N × N matrix of the

form W = ATA

Twhere A is a T ×N matrix of iid standard normally distributed random variables with

T ≥ N . Therefore, if the returns are iid standard normal, the correlation matrix is a Wishart matrix.

103


to use the upper bound of this distribution as a cut-off level to discard eigenvalues that

are small and thus likely caused by noise. This approach retains only those eigenvalues

that are too large to be a result of noise.

The tests in the paper mostly rely on block bootstrapping methods to obtain boot-

strap means and bootstrap standard errors and calculate the t-statistics, where the block

length is 6 months (T 1/3) and there are 1000 bootstrap iterations. This allows testing

correlations as well as returns, volatilities and Sharpe ratios with one consistent and in-

tuitive approach. The first exception are the annualized factor returns in figure 1, which

use Newey-West Standard errors. The second exception is the estimation of linear time

trends for the dynamics in the cross-country correlations. Here, I use the White estimator

i.e. a HAC estimator with a truncated kernel and a 36 month window length to capture

the distinct form of auto-correlation in the 36 months rolling correlations. Beyond 36

months, the estimated standard error is insensitive to the window length used.

4 Data

The country factor data set used in this paper was created by Asness, Frazzini and Peder-

sen. Nowadays, the asset manager AQR6 maintains this data set in its data library. The

data set contains monthly data for six major equity factors for 24 developed markets. The

equity factors are the market excess return (MKT), size (SMB), value (HML), momentum

(UMD), low beta (BAB) and quality (QMJ).

The papers of Fama and French (1992, 1993, 1996), Asness, Frazzini, and Pedersen

(2013) and Frazzini and Pedersen (2014) form the basis for the factor construction in

this data set. This data set is based on the following sources: The U.S. price data from

1926 to 2015 is a combination of the Center of Research in Security Prices database and

the Compustat / XpressFeed Global database. The accounting data is from Moody and

the Compustat / XpressFeed Global database. The international data for 23 developed

markets from 1981 to 2015 originates from the Compustat / XpressFeed Global database.

All returns are in U.S. dollars. Therefore, it is the perspective of a USD investor with

unhedged equity exposure. For unleveraged long-short factors, i.e. all factors except for

the market excess return and the low beta factor, the net currency exposure is zero. For

the low beta factor, it is hard to quantify, but the overall currency impact should rather

be small.

The market excess return (MKT) is the market capitalization weighted return on all

available stocks in the respective country market minus the one-month Treasury bill rate.

The size factor (SMB) and the value factor (HML) result from double sorting on market

6https://www.aqr.com/library/data-sets

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capitalization and the ratio of book-value to market capitalization. The momentum factor

(UMD) uses a conditional sort on market capitalization and 12 months minus 1 month

total returns. The low beta factor (BAB) capitalizes on the low beta anomaly as a self-

financing market-neutral long-short portfolio buying low beta stocks and shorting high

beta stocks. For the low beta factor, estimated quantities are used for sorting. This

is an errors-in-variables problem if the true betas are known, but imprecisely measured.

However, the true beta as the true book value of a stock is not known and all investors

work with the same limited type of information to obtain characteristics based portfolios.

Moreover, using Bayesian methods to reduce the estimation error as often done to estimate

forward looking betas does not change the ranking of the betas. This ranking is the only

relevant information for factor based portfolio selection. Therefore, the use of estimated

quantities is not a problem. The quality-minus-junk factor (QMJ) uses a conditional sort

on market capitalization and quality. The quality score herein is based on profitability,

growth, safety and payout. All factors are market capitalization weighted. Except for the

market excess return all portfolio are rebalanced monthly.

This paper uses monthly return data of twenty developed markets (excluding Greece,

Israel, New Zealand and Portugal) starting in January 1991 due to incomplete data before.

The portfolio construction uses 36 months of data for covariance matrix estimation such

that the performance calculations of the constructed portfolios start in January 1994. The

data set uses data up to April 2015.

5 Empirical Results

This section gives a short overview of the returns to the six factors in twenty countries

before I continue to analyse the diversification properties by means of correlations and

principal components. Finally, a portfolio construction exercise demonstrates the im-

proved risk-return-trade-off by diversifying across countries and across factors.

5.1 Factor Performance

Research has shown that the six equity factors discussed here generate significant excess

returns in the long run for the U.S. equity market. For the period of January 1991 to

April 2015, all factors except for SMB achieve substantially positive log returns in most of

the 20 countries considered. These findings are summarized in graphical form in figure 1

and in tabular form in table 7 in the appendix.

The market excess return (MKT) has a median return of 6.1% p.a. across countries.

With Japan, there is only one country which has a negative equity risk premium of -1.0%

p.a. in the period considered. The realized equity premium is highest in Hong Kong

105


AUS AUT BEL CAN CHE DEU DNK ESP FIN FRA GBR HKG IRL ITA JPN NLD NOR SGP SWE USA

0

5

10Market Excess Return (MKT)

Ret

urn

in %


−5

0

Size (SMB)

Ret

urn

in %


0

5

10

Value (HML)

Ret

urn

in %

AUS AUT BEL CAN CHE DEU DNK ESP FIN FRA GBR HKG IRL ITA JPN NLD NOR SGP SWE USA0

5

10

15

Momentum (UMD)

Ret

urn

in %


101520

Low Beta (BAB)

Ret

urn

in %


0

5

10

Quality (QMJ)

Ret

urn

in %

Figure 1: Annualized factor returns in twenty developed countriesThis figure represents the annualized log returns in U.S. dollars of the six equity factors (MKT, SMB, HML, UMD, BABand QMJ) for 20 different developed markets from January 1991 to April 2015. The twenty country markets are Australia(AUS), Austria (AUT), Belgium (BEL), Canada (CAN), Switzerland (CHE), Germany (DEU), Denmark (DNK), Spain(ESP), Finland (FIN), France (FRA), Great-Britain (GBR), Hong Kong (HKG), Ireland (IRL), Italy (ITA), Netherlands(NLD), Norway (NOR), Singapore (SGP), Sweden (SWE) and the United States (USA). The bar colours represent the two-sided significance using Newey-West standard errors, where black represents significance at the 5% level and grey representssignificance at the 10% level.

106


(9.0% p.a.). Due to the short time period, the market excess return is only significant in

Switzerland and the U.S., but 13 out of 20 markets have t-statistics larger than one.

In contrast to the long-term returns of the SMB factor in the U.S. equity market, SMB

performs mostly negative across countries with a median return of -1.5% p.a. There are

only 5 out of 20 countries with a positive size premium. SMB performs worst in Germany

with a return of -8.3% p.a. and best in the U.S. with a return of 2.3% p.a. Thus, the

U.S. size factor is not representative of the country size factors during the observation

period. Except for the significantly negative factor returns in Germany, the size factor is

insignificant in all countries.

With a median return of 4.4% p.a. and negative returns only in Denmark with -2.9%

p.a., the HML factor generates substantially positive returns. The value premium is

particularly strong in Austria (10.9% p.a.), Australia (8.6% p.a.) and Germany (8.1%

p.a.). Overall, the value factor is significant in 5 out of 20 cases.

The momentum factor (UMD) performs even better with a median return of 6.7% p.a.

and positive returns across all countries. The returns on the momentum factor are widely

dispersed, ranging from 17.6% p.a. in Canada and 17.4% p.a. in Australia down to 0.9%

p.a. in Japan. The poor momentum performance in Japan is consistent with the findings

of Asness (2011), Asness, Moskowitz, and Pedersen (2013) and Chui, Titman, and Wei

(2010). Asness, Moskowitz, and Pedersen (2013) argue that this could be due to strong

value performance as well as highly negative correlation between value and momentum in

Japan. Chui et al. (2010) find that individualism is positively associated with momentum

profits, where Japan is one of the least individualistic countries. The momentum factor

achieves significantly positive returns in 10 out of 20 markets.

The best performing factor between January 1991 and April 2015 is the BAB factor

with a median return of 9.1% p.a. As for the UMD factor, the BAB factor has a positive

return in each of the twenty countries and has a wide range of returns ranging from 22.3%

p.a. in Australia, 20.9% p.a. in Canada and 20.1% p.a. in Hong Kong down to 1.5% p.a.

in Japan. It is significant in 14 out of 20 markets.

Finally, the QMJ factor achieves a fairly consistent positive return across countries

with a median return of 4.2% p.a. The worst performer is the Netherlands with a return

of -2.2% p.a., while QMJ performs particularly well in Canada with a return of 10.6%

p.a. The return to quality is significant in 7 out of 20 markets.

Though the significance in some cases remains limited due to the short observation

period from 1991 to 2015, positive factor returns are very consistent across countries

except for the size factor.

107


5.2 Correlation Analysis

Starting with simple correlations in figure 2, I observe that the cross-country cor-

relations vary distinctly between factors. The average pairwise correlations between

countries for each country and factor are given in table 1. Note that the standard errors

and significance levels are for the deviations to the country average in the last line.

As to be expected due to the tight integration of global equity markets, the market risk

factor (MKT) is strongly correlated across countries with an cross-country correlation

of 0.646. Panel B of table 2 shows that this cross-country correlation is statistically

significantly higher than the cross-country correlation for all other equity factors. The

cross-country correlation is lowest for Japan (0.423), followed by Hong Kong (0.535) and

Singapore (0.579), while the cross-country correlations are highest for major European

markets such as Netherlands (0.747), France (0.736) and Great-Britain (0.727). All these

correlations differ significantly from the country average of 0.646.

The SMB factor shows the lowest cross-country correlation (0.160). The SMB column

in table 1 shows that Great-Britain (0.280) and France (0.260) have the highest cross-

country correlations for the size factor and are statistically significantly higher than the

country average of 0.160, but remain moderate in economic terms.

For the HML factor, the cross-country correlation of 0.177 is statistically indistin-

guishable to the cross-country correlation of the SMB factor as presented in panel B

of table 2. Again, France (0.285) and Great-Britain (0.282) stand out with the high-

est cross-country correlation. The U.S. (0.280), Germany (0.276), Sweden (0.268) and

Canada (0.251) complete a cluster of significantly increased correlations.

Besides the MKT factor, UMD is the most tightly linked factor across countries with

an cross-country correlation of 0.401. In table 1, the highest correlated markets are again,

Great-Britain (0.542), France (0.531) and the U.S. (0.517), while Ireland (0.259) and the

countries in Asia-Pacific i.e. Japan (0.274), Singapore (0.264) and Hong Kong (0.324)

have correlations that are significantly below the country average.

With 0.214, the cross-country correlations of the BAB factor remain moderate, but

statistically distinguishably higher than for the size factor as presented in panel B of

table 2. In particular Ireland (0.098) as well as the countries in the Asia-Pacific region i.e.

Australia (0.067), Japan (0.093), Singapore (0.095) and Hong Kong (0.119) make good

diversifiers with significantly reduced correlations compared to the country average.

In contrast to SMB, HML and BAB the average correlations for QMJ are slightly

elevated with 0.263 but still significantly below UMD and MKT. While Finland (0.076)

is a good diversifier, the U.S. (0.367), Great-Britain (0.359), France (0.372), Switzer-

land (0.379), the Netherlands (0.359) and Belgium (0.331) form a cluster of significantly

increased cross-country correlations.

108


MKTAU

SAU

TBE

LC

ANC

HE

DEU

DN

KES

PFI

NFR

AG

BRH

KG IRL

ITA

JPN

NLD

NO

RSG

PSW

EU

SA

AUSAUTBELCANCHEDEUDNKESPFIN

FRAGBRHKG

IRLITA

JPNNLDNORSGPSWEUSA

SMB

AUS

AUT

BEL

CAN

CH

ED

EUD

NK

ESP

FIN

FRA

GBR

HKG IR

LIT

AJP

NN

LDN

OR

SGP

SWE

USA


FRAGBRHKG

IRLITA

JPNNLDNORSGPSWEUSA

HML

AUS

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BEL

CAN

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FRA

GBR

HKG IR

LIT

AJP

NN

LDN

OR

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SWE

USA


FRAGBRHKG

IRLITA

JPNNLDNORSGPSWEUSA

UMD

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AUT

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ED

EUD

NK

ESP

FIN

FRA

GBR

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FRAGBRHKG

IRLITA

JPNNLDNORSGPSWEUSA

BAB

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AUT

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CAN

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ED

EUD

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ESP

FIN

FRA

GBR

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FRAGBRHKG

IRLITA

JPNNLDNORSGPSWEUSA

QMJ

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AUT

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EUD

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FRAGBRHKG

IRLITA

JPNNLDNORSGPSWEUSA

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 2: Cross-country correlations for each of the six equity factorsThis figure represents the cross-country correlations calculated from monthly U.S. dollar returns across twenty developedmarkets for each of the six different equity factors (MKT, SMB, HML, UMD, BAB and QMJ) from January 1991 to April2015. The twenty country markets are Australia (AUS), Austria (AUT), Belgium (BEL), Canada (CAN), Switzerland(CHE), Germany (DEU), Denmark (DNK), Spain (ESP), Finland (FIN), France (FRA), Great-Britain (GBR), Hong Kong(HKG), Ireland (IRL), Italy (ITA), Netherlands (NLD), Norway (NOR), Singapore (SGP), Sweden (SWE) and the UnitedStates (USA).

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Mean MKT SMB HML UMD BAB QMJ

AUS 0.284 0.652 0.192 0.172 0.377 0.067*** 0.241(0.015) (0.023) (0.020) (0.019) (0.026) (0.021)

AUT 0.250 0.641 0.134 0.015*** 0.302* 0.141* 0.266(0.023) (0.024) (0.053) (0.054) (0.039) (0.025)

BEL 0.314 0.672** 0.126 0.087*** 0.431 0.236 0.331**

(0.013) (0.030) (0.034) (0.031) (0.022) (0.027)

CAN 0.337 0.664* 0.227*** 0.251** 0.447** 0.201 0.232(0.011) (0.014) (0.033) (0.019) (0.024) (0.033)

CHE 0.343 0.659 0.163 0.128 0.476*** 0.253 0.379***

(0.015) (0.024) (0.038) (0.014) (0.030) (0.017)

DEU 0.376 0.715*** 0.176 0.276*** 0.478*** 0.319*** 0.294(0.013) (0.022) (0.026) (0.022) (0.019) (0.024)

DNK 0.286 0.679*** 0.091** 0.115** 0.365 0.271*** 0.197*

(0.011) (0.031) (0.025) (0.028) (0.016) (0.036)

ESP 0.326 0.667 0.193 0.140 0.408 0.268*** 0.279(0.014) (0.022) (0.035) (0.024) (0.020) (0.024)

FIN 0.256 0.586*** 0.113 0.205 0.325*** 0.234 0.076***

(0.022) (0.034) (0.022) (0.028) (0.020) (0.037)

FRA 0.422 0.736*** 0.260*** 0.285*** 0.531*** 0.350*** 0.372***

(0.010) (0.018) (0.025) (0.015) (0.014) (0.015)

GBR 0.416 0.727*** 0.280*** 0.282*** 0.542*** 0.306*** 0.359***

(0.011) (0.014) (0.023) (0.011) (0.016) (0.019)

HKG 0.246 0.535*** 0.167 0.149 0.324*** 0.119*** 0.180***

(0.023) (0.017) (0.036) (0.029) (0.031) (0.025)

IRL 0.212 0.581** 0.106** 0.081*** 0.259*** 0.098** 0.150***

(0.028) (0.024) (0.033) (0.035) (0.051) (0.042)

ITA 0.308 0.624 0.151 0.127 0.445** 0.236 0.262(0.022) (0.021) (0.036) (0.020) (0.027) (0.022)

JPN 0.215 0.423*** 0.126* 0.191 0.274*** 0.093*** 0.185**

(0.028) (0.019) (0.026) (0.035) (0.034) (0.035)

NLD 0.370 0.747*** 0.195* 0.146 0.482*** 0.289*** 0.359***

(0.008) (0.020) (0.024) (0.019) (0.020) (0.016)

NOR 0.288 0.671* 0.105** 0.189 0.337 0.215 0.211**

(0.015) (0.024) (0.021) (0.046) (0.020) (0.025)

SGP 0.230 0.579*** 0.079** 0.144 0.264*** 0.095*** 0.220*

(0.023) (0.032) (0.025) (0.036) (0.025) (0.022)

SWE 0.350 0.682*** 0.170 0.268*** 0.444* 0.241 0.295(0.012) (0.024) (0.023) (0.026) (0.023) (0.023)

USA 0.375 0.678*** 0.153 0.280*** 0.517*** 0.254* 0.367***

(0.010) (0.018) (0.021) (0.013) (0.021) (0.021)

Mean 0.310 0.646 0.160 0.177 0.401 0.214 0.263

Table 1: Average cross-country correlations for each country and factorFor each country and each factor, this table presents average pairwise correlations of a country equity factor with the sameequity factor in different countries. Column 1 shows the factor averages for each country, while the last row displays thecountry averages for each factor. The standard errors in parentheses are block bootstrapped standard errors of the deviationsfrom the country averages in the last row. Significance also refers to the deviations from the country averages on the lastrow. * stands for two sided significance at the 10% level, ** for significance at the 5% level and *** for significance at the1% level. There are six different equity factors (MKT, SMB, HML, UMD, BAB and QMJ) and twenty different developedmarkets. The country markets are Australia (AUS), Austria (AUT), Belgium (BEL), Canada (CAN), Switzerland (CHE),Germany (DEU), Denmark (DNK), Spain (ESP), Finland (FIN), France (FRA), Great-Britain (GBR), Hong Kong (HKG),Ireland (IRL), Italy (ITA), Netherlands (NLD), Norway (NOR), Singapore (SGP), Sweden (SWE) and the United States(USA). The correlations are based on monthly U.S. dollar returns from January 1991 to April 2015.

From a country perspective, France and Great-Britain followed by Germany and

the U.S. show high cross-country correlations, whereas Ireland as well as the countries

in the Asia-Pacific region (Australia, Japan, Singapore and Hong Kong) offer attractive

diversification benefits. Even though the data is at the monthly frequency, there could

potentially still be a residual effect of non-synchronous trading hours. This would lead to

110


increased correlations between countries with similar trading hours, i.e. clustering among

American, European and Asia-Pacific countries. Figure 12 in the appendix visualizes the

cross-country correlations for local factor portfolios.

Analysing the correlation structure across all 120 country equity factors (cross- coun-

try cross-factor dimension) in figure 3, we can observe a distinct factor grouping rec-

ognizable as 6x6 checker pattern of 20x20 tiles. The high cross-country correlations within

the MKT factor and less so the UMD factor significantly stand out on the high correlation

end.

In contrast, UMD and in particular QMJ offer very attractive diversification benefits

to the MKT factor. They are significantly better diversifiers to the market excess returns

than the other equity factors as the t-statistics in Panel C of table 2 show. The good

diversification properties of QMJ are likely due to the negative market beta of the quality

factor, which comes from the safety component (Asness, Frazzini, & Pedersen, 2013).

For the momentum factor, these good diversification properties are not that self-evident

as there is a lot of time variation in the market beta loadings of the momentum factor.

When past market returns are high, momentum loads on high beta stocks, while it loads

on low beta stocks when past market returns are low (Blitz et al., 2011). Potentially, the

low correlation is due to the particular time period that contains two pronounced market

crashes and recoveries and particularly the pronounced momentum crash in 2009.

Additionally, the cross-correlations of UMD, BAB and QMJ form a significant corre-

lation cluster with a 0.185 increased correlation compared to the remaining off-diagonal

elements of the correlation matrix as tested in panel D of table 2. Kolanovic and Wei

(2014) discuss a similar finding that there are two sets of factors, which they term value

and generalized momentum. The later consists of momentum, low beta and quality.

BAB is the only factor, which shows increased cross-factor correlations to all other

equity factors: on average a significant 0.114 higher correlation compared to the other

five factors.

Even though the diversification between value and momentum receives most attention

in academia (Asness, 1997; Asness, Moskowitz, & Pedersen, 2013; Cakici & Tan, 2013),

it is not that prominent in the data. The construction of the value factor is key for this

effect. In this paper, the classical Fama-French approach using lagged market prices mixes

value and momentum. The approach by Asness and Frazzini (2013) using current market

prices, however, gives a much purer value factor. This approach shows a more negative

correlation to momentum.

Using only cross-factor correlations within the same country and therefore neglecting

cross-country effects in table 8 in the appendix, increases the range of the cross-factor

correlations compared to table 2. The effects of clustering between UMD, BAB and QMJ

111


MKT SMB HML UMD BAB QMJ

MKT

SMB

HML

UMD

BAB

QMJ

−0.6

−0.4

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0.2

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Figure 3: Cross-country cross-factor correlationsThis figure represents the cross-country cross-factor correlations between the six different equity factors (MKT, SMB, HML,UMD, BAB and QMJ) and 20 different developed markets. This 120 * 120 correlation matrix shows a distinct pattern of 20* 20 tiles, which show the cross-country correlations between the same factors. The twenty country markets are Australia(AUS), Austria (AUT), Belgium (BEL), Canada (CAN), Switzerland (CHE), Germany (DEU), Denmark (DNK), Spain(ESP), Finland (FIN), France (FRA), Great-Britain (GBR), Hong Kong (HKG), Ireland (IRL), Italy (ITA), Netherlands(NLD), Norway (NOR), Singapore (SGP), Sweden (SWE) and the United States (USA). The correlations are based onmonthly U.S. dollar returns from January 1991 to April 2015.

and the increased correlation of the BAB factor increase when neglecting cross-country

effects.

Finally, the positive as well as negative correlation effects between factors are stronger

for the same countries, which becomes visible as more or less pronounced diagonal lines

within the 6 x 6 smaller squares in figure 3.

Looking at the dynamic evolution of the correlations over time, I find that the cross-

country correlations for the six factors in figure 4 mostly show a distinct cyclicality and

have risen over the last twenty years.

In particular for the market excess return (MKT), the correlations are highest and

show a distinct upward trend across time, with only moderate cyclicality. In table 3,

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Panel A: Average cross-factor correlations


MKT 0.65 0.04 -0.01 -0.20 -0.01 -0.34SMB 0.04 0.16 0.02 -0.09 0.07 -0.09HML -0.01 0.02 0.18 -0.01 0.07 -0.02UMD -0.20 -0.09 -0.01 0.40 0.12 0.22BAB -0.01 0.07 0.07 0.12 0.21 0.07QMJ -0.34 -0.09 -0.02 0.22 0.07 0.26

Panel B: t-Statistics of correlation differences of diagonal elements

MKT*MKT SMB*SMB HML*HML UMD*UMD BAB*BAB QMJ*QMJ

MKT*MKT 0.00 13.65 9.39 4.81 13.02 15.08SMB*SMB -13.65 0.00 -0.48 -4.77 -2.03 -3.40HML*HML -9.39 0.48 0.00 -3.84 -0.98 -1.83UMD*UMD -4.81 4.77 3.84 0.00 3.73 3.15BAB*BAB -13.02 2.03 0.98 -3.73 0.00 -1.78QMJ*QMJ -15.08 3.40 1.83 -3.15 1.78 0.00

Panel C: t-Statistics of correlation differences to the market excess return

MKT*MKT MKT*SMB MKT*HML MKT*UMD MKT*BAB MKT*QMJ

MKT*MKT 0.00 20.61 12.86 9.45 14.86 14.00MKT*SMB -20.61 0.00 1.02 2.69 1.13 5.89MKT*HML -12.86 -1.02 0.00 2.64 -0.01 6.35MKT*UMD -9.45 -2.69 -2.64 0.00 -2.83 3.23MKT*BAB -14.86 -1.13 0.01 2.83 0.00 5.63MKT*QMJ -14.00 -5.89 -6.35 -3.23 -5.63 0.00

Panel D: Other tests

UMD, BAB and QMJ vs Rest 0.185***

(0.042)

BAB vs Rest 0.114***

(0.011)

Table 2: Average cross-factor correlationsThis table displays the cross-factor correlation structure and tests for differences in the correlations. Panel A representsthe cross-factor correlation structure after averaging over all countries. This means that each 20x20 cross-country tile ofthe 120x120 cross-country cross-factor correlation matrix in figure 3 is condensed to a single number, resulting in a 6x6cross-factor correlation matrix. Panel B shows the t-statistics of the differences between the diagonal elements of PanelA. It therefore tests the differences between the cross-country correlation structure of one factor versus the cross-countrycorrelation structure of another factor. Panel C gives the t-statistics between the elements in the first column of Panel A(MKT). This tests the diversification of each factor against the market. Panel D tests the difference in the average correlationbetween UMD, BAB and QMJ and the other factors and the differences in the average correlation between BAB and theother factors. For Panel D, block bootstrapped standard errors are in brackets. * stands for two sided significance at the10% level, ** for significance at the 5% level and *** for significance at the 1% level. There are six different equity factors(MKT, SMB, HML, UMD, BAB and QMJ) and 20 different developed markets. The twenty country markets are Australia(AUS), Austria (AUT), Belgium (BEL), Canada (CAN), Switzerland (CHE), Germany (DEU), Denmark (DNK), Spain(ESP), Finland (FIN), France (FRA), Great-Britain (GBR), Hong Kong (HKG), Ireland (IRL), Italy (ITA), Netherlands(NLD), Norway (NOR), Singapore (SGP), Sweden (SWE) and the United States (USA). The correlations are based onmonthly U.S. dollar returns from January 1991 to April 2015.

a simple linear trend regression shows a significant annual increase in the cross-country

correlation of 2.2%. The regression uses a HAC estimator with a truncated kernel with

a window length of 36 months to account for the strong autocorrelation. This represents

the increased globalization and financial market integration, which is in line with the

findings of Longin and Solnik (1995) for 1960 to 1990. Bekaert, Hodrick, and Zhang

(2009) only find increasing correlations for European stock markets. These, however,

dominate the sample in this paper with 14 out of 20 countries being European. There

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Constant 0.404*** 0.090*** 0.090 0.104 0.151*** 0.047(0.022) (0.018) (0.062) (0.075) (0.058) (0.040)

Time 0.022*** 0.007*** 0.003 0.022*** 0.005 0.018***

(0.003) (0.001) (0.003) (0.005) (0.005) (0.003)R2 0.84 0.65 0.04 0.49 0.11 0.70

Table 3: Time trends in cross-country correlations for each factorThis table represents a simple linear time trend in the rolling 36 months cross-country correlations across twenty developedmarkets for each of the six different equity factors (MKT, SMB, HML, UMD, BAB and QMJ). The Time variable isannualized for ease of interpretation. The White standard errors with a window length of 36 months are given in brackets.* stands for two sided significance at the 10% level, ** for significance at the 5% level and *** for significance at the 1%level. The correlations are based on monthly U.S. dollar returns from January 1991 to April 2015.

is also an intimate connection to the finding of mean-variance convergence (Eun & Lee,

2010), gaining importance of industry factors vs. country factors (Cavaglia, Brightman, &

Aked, 2000) and increasing volatility spillover effects (Karolyi & Stulz, 2003; Baele, 2005).

However, in the years since the financial crisis, the correlations decreased, indicating more

divergence in cross-country market returns.

For the size factor (SMB), I find slightly, but significantly increasing cross-country cor-

relations over time of about 0.7% p.a. with some moderate cyclicality. The cross-country

correlation of the value factor (HML) is dominated by a strongly cyclical component

peaking during the burst of the technology bubble (2001/2002) and a second, less pro-

nounced peak around the Euro crisis (2012). There is no distinct trend in the HML factor

(0.3% p.a.).

As for the value factor (HML), the momentum factor (UMD) shows two distinct

peaks in the cross-country correlation. The peaks around the burst of the technology

bubble (2001/2002) and the recovery of the financial crisis (2009/2010) match the HML

factor fairly well. For momentum, the second peak, however, is more pronounced. This

peak is likely due to the phenomenon of a momentum crash (Daniel & Moskowitz, 2013)

during early crisis recovery. Overall, there is a significant time trend of about 2.2% p.a.

in the cross-country correlations of the momentum factor.

The low beta factor (BAB) correlations also exhibit a distinct peak around the burst

of the technology bubble (2001/2002) with mostly decreasing correlations since then.

There is no significant time trend in the cross-country correlations between country BAB

factors (0.5% p.a.). Finally, the quality factor (QMJ) is another example of a cyclical

factor with two correlation peaks around 2005 and 2012. Those peaks are also similar to

the HML and UMD peaks. I also find a distinct time trend of 1.8% p.a. in the QMJ

cross-country correlations.

In general, the technology bubble and its burst (2000-2005) and less so the financial

crisis and the recovery (2009-2012) lead to distinct cycles in the cross-country correlations

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1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 20140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Rol

ling

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MKTSMBHMLUMDBABQMJ

Figure 4: Time trends in cross-country correlations for each factorThis figure represents the rolling 36 months cross-country correlations based on twenty developed markets for each of thesix different equity factors (MKT, SMB, HML, UMD, BAB and QMJ). The correlations are based on monthly U.S. dollarreturns from January 1991 to April 2015.

1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014

−0.15

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AUSAUTBELCANCHEDEUDNKESPFINFRAGBRHKGIRLITAJPNNLDNORSGPSWEUSA

Figure 5: Time trends in cross-factor correlations for each countryThis figure represents the rolling 36 months cross-factor correlations based on six equity factors (MKT, SMB, HML, UMD,BAB and QMJ) for each of the 20 developed markets. The bold black line represents the average of the 20 cross-factorcorrelations. The twenty country markets are Australia (AUS), Austria (AUT), Belgium (BEL), Canada (CAN), Switzerland(CHE), Germany (DEU), Denmark (DNK), Spain (ESP), Finland (FIN), France (FRA), Great-Britain (GBR), Hong Kong(HKG), Ireland (IRL), Italy (ITA), Netherlands (NLD), Norway (NOR), Singapore (SGP), Sweden (SWE) and the UnitedStates (USA). The correlations are based on monthly U.S. dollar returns from January 1991 to April 2015.

115


of the factors. While the cycles between the cross-country correlations of the factors

indicate dependence, their peaks are not fully synchronous. Overall, the data window of

around 20 years is relatively short, as the cycle length is long. With more data available

in the future, cycles in equity factors - be it for factor returns or factor correlations - will

be an interesting avenue for more research.

Repeating the analysis of dynamic correlations, but this time across factors for each

country separately in figure 5, I find very low average correlations without distinct time

trends and barely any cyclicality. Country factor portfolios are well diversified for all

countries across time with correlations between -0.15 and 0.2, thus practically uncorre-

lated. The bold line represents the mean across rolling cross-factor correlations. The

country factor portfolios become slightly more correlated around 2005 and 2008, while

they are least correlated around 2000. However, these differences are economically barely

meaningful with an average peak-to-through correlation difference of less than 0.10. There

is no indication of a time trend in cross-factor correlations.

Also when forming global factors, the average correlation among the factors does not

show a time trend. Thus, even though the benefits of international diversification within

each factor are decreasing, this does not impact the benefits of factor diversification.

Currently, research mostly prefers local factor models (Griffin, 2002; Hou et al., 2011), but

when the trend to decreasing international diversification for the single factors continues,

global factor models could be better suited in the future.

Further insights can be gained by using more advanced methods from the literature on

international diversification. Another direction is analysing the time trends in the returns

to these factor strategies or the time variation in the explanatory power of the factors

in an asset pricing model. In this context, Chordia, Subrahmanyam, and Tong (2014)

find that the returns of a portfolio strategy based on prominent anomalies shows a strong

downward trend over time.

If there is a common cause to the factors and there is crowded directional investment

activity in these factor premia, an increase of the cross-factor correlations would likely be

observed. The findings here give no indication for such a scenario. However, this simple

correlation analysis cannot give a definitive account on how markets will react to the

increased popularity of factor based investing.

Besides the time trends, the conditional cross-country correlations are of interest.

Therefore, table 4 presents the correlations during market tails. The graphical repre-

sentation is given in the appendix. The left (right) tail is defined as the months with the

30% lowest (highest) global market excess returns. The cross-country correlations of the

factors besides the market (MKT) and quality (QMJ) are only moderately influenced by

market tails.

116


Left Tail Right Tail Difference t-statistic

MKT 0.54 0.34 0.20 2.09SMB 0.19 0.17 0.02 0.37HML 0.22 0.18 0.04 0.48UMD 0.39 0.45 -0.06 -0.62BAB 0.25 0.23 0.02 0.44QMJ 0.24 0.15 0.08 1.70

Table 4: Differences in cross-country correlations for each factor during market tailsFor each of the six factors in column 1, this table presents the cross-country correlation across 20 developed countries for the30% worst global market months (left tail) and for the 30% best global market months (right tail) and tests the differencefor significance. The t-statistic is obtained via block-bootstrapping. The correlations are based on monthly U.S. dollarreturns from January 1991 to April 2015.

As expected, the cross-country correlations for the market excess return significantly

rise from 0.34 to 0.54, when going from the top 30% returns to the bottom 30% returns.

Thus, markets show higher correlation in times of crisis. This is in line with findings

of increased correlations during crises (Longin & Solnik, 1995; Asness, Israelov, & Liew,

2011) and of financial market interdependence (Forbes & Rigobon, 2002).

The cross-country correlations for the quality factor (QMJ) increase as well from 0.15

to 0.24, which is significant at the 10% level. One explanation for this observation could

be the substantially negative loading on market beta for the quality factor. As a mirror

image to the widely negative market excess return during crises across countries, the

quality factor widely outperforms during crises across countries (flight to quality).

For size (SMB), the increase is negligible from 0.17 to 0.19. The same holds for value

(HML) with 0.18 to 0.22 and low beta (BAB) from 0.23 to 0.25. For the factor with the

highest cross-country correlation besides the market - momentum (UMD) - however, the

cross-country correlation substantially decreases in times of market stress, reducing from

0.45 to 0.39. Due to large standard errors, however, this decrease is insignificant. The

reduction in correlation could be due to synchronous momentum crashes that usually

happen when markets rebound after a crash, i.e. in times of strongly positive market

returns.

In general, the high standard errors are due to conditioning on the 30% most extreme

values, which substantially reduces the dispersion in the sub-sample and does not fully

incorporate the data available.

5.3 Principal Component Analysis

Besides correlations, principal component analysis gives us valuable insight into the factor

structure of the country equity factors. Figure 6 shows that the first principal components

explain relatively little of the variance of the cross-section of 120 country equity factors

with the first factors explaining 22.6%, 8.8% and 5.8% of the variance respectively.

117


20 40 60 80 100 1200

5

10

15

20

25Pe

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tage

of V

aria

nce

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aine

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Number of Components

20 40 60 80 100 1200

20

40

60

80

Cum

ulat

ive

Perc

enta

geof

Var

ianc

e Ex

plai

ned

Number of Components

Figure 6: Percentage of the variance explained by principal components of the cross-section of 120 country equity factorsThe top figure shows the percentage of the total variance of the cross-section of 120 country equity factors (six factors for20 countries each) explained by the ordered principal components. The bottom figure displays the cumulative percentageof the total variance of the cross-section of country equity factors (six factors for 20 countries each) explained by theordered principal components. The six equity factors are the market excess return (MKT), size (SMB), value (HML),momentum (UMD), low beta (BAB) and quality (QMJ). The twenty country markets are Australia (AUS), Austria (AUT),Belgium (BEL), Canada (CAN), Switzerland (CHE), Germany (DEU), Denmark (DNK), Spain (ESP), Finland (FIN),France (FRA), Great-Britain (GBR), Hong Kong (HKG), Ireland (IRL), Italy (ITA), Netherlands (NLD), Norway (NOR),Singapore (SGP), Sweden (SWE) and the United States (USA). The calculations are based on monthly U.S. dollar returnsfrom January 1991 to April 2015.

Figure 7 exposes that the first component loads strongly positively on all country

market risk factors (MKT), while loading consistently negatively on the country momen-

tum (UMD) and quality (QMJ) factors. The second component loads positively on the

market (MKT), momentum (UMD) and low beta (BAB) country factors, while the third

components loads mostly, but less consistently positively on size (SMB), value (HML) and

low beta (BAB) country factors while shorting momentum (UMD) and market (MKT)

country factors. Components are less clustered along the equity factor dimension after

the third component.

In order to explain more than 80% of the cross-section of 120 country equity factors, it

takes the first 33 principal components. The six first principal components alone explain

a mere 47.8% of the variance. This indicates that not only the six equity factors drive

118


Percentage of the Variance Explained by Component

22.6 8.8 5.8 4.2 3.3 3.1 2.5 2.3 2.1 1.9

MKT

SMB

HML

UMD

BAB

QMJ

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Figure 7: Coefficients of the principal components of the cross-section of 120 countryequity factorsThis figure displays the coefficients (right singular values V ) of the first ten principal components of the cross-section of 120country equity factors (6 factors, 20 countries). The vertical axis displays the 120 country equity factors sorted according tothe six equity factors and twenty developed markets, while the horizontal axis represents the ten first principal componentswith the percentage of the variance explained by the respective component. The six equity factors are the market excessreturn (MKT), size (SMB), value (HML), momentum (UMD), low beta (BAB) and quality (QMJ). The twenty countrymarkets are Australia (AUS), Austria (AUT), Belgium (BEL), Canada (CAN), Switzerland (CHE), Germany (DEU),Denmark (DNK), Spain (ESP), Finland (FIN), France (FRA), Great-Britain (GBR), Hong Kong (HKG), Ireland (IRL),Italy (ITA), Netherlands (NLD), Norway (NOR), Singapore (SGP), Sweden (SWE) and the United States (USA). Thecalculations are based on monthly U.S. dollar returns from January 1991 to April 2015.

the cross-section of country equity factors, but also the country versions of these factors

offer a large diversification potential.

Focusing on the country dimension for each factor separately in figure 8, the insights

of the correlation analysis are confirmed. The market risk factor (MKT) can be captured

with only a few components, where the first component already explains 65% of the port-

folio variation and four components suffice to explain 80% of the cross-sectional variance.

The second most concentrated factor is momentum with 42% of the variation explained

by the first component. It takes already eight factors to explain 80% of the cross-section.

For the other equity factors, the first component only explains between 20% (SMB) and

28% (QMJ) of the variation and it requires about half of all components to explain 80%

of the cross-sectional variance. This again confirms the diversification potential in the

country dimension of equity factors.

Figure 15 in the appendix illustrates the loadings of these components. For all six

factors, the first principal component loads relatively equally on all 20 countries. The

second component is an Ireland component for each factor except for the market excess

119


Factors

Perc

enta

ge o

f Var

ianc

e Ex

plai

ned

MKT SMB HML UMD BAB QMJ0

10

20

30

40

50

60

70

80

90

100

Figure 8: Percentage of the cross-country variation explained by principal componentsFor each of the six equity factors (MKT, SMB, HML, UMD, BAB and QMJ), this graph displays the percentage of the cross-country variance across twenty developed markets explained by the principal components. The twenty country marketsare Australia (AUS), Austria (AUT), Belgium (BEL), Canada (CAN), Switzerland (CHE), Germany (DEU), Denmark(DNK), Spain (ESP), Finland (FIN), France (FRA), Great-Britain (GBR), Hong Kong (HKG), Ireland (IRL), Italy (ITA),Netherlands (NLD), Norway (NOR), Singapore (SGP), Sweden (SWE) and the United States (USA). The calculations arebased on monthly U.S. dollar returns from January 1991 to April 2015.

Perc

enta

ge o

f Var

ianc

e Ex

plai

ned


10

20

30

40

50

60

70

80

90

100

Figure 9: Percentage of the cross-factor variation explained by principal componentsFor each of the twenty developed markets, this graph displays the percentage of the cross-factor variance explained by theprincipal components. The six equity factors are the market excess return (MKT), size (SMB), value (HML), momentum(UMD), low beta (BAB) and quality (QMJ). The twenty country markets are Australia (AUS), Austria (AUT), Belgium(BEL), Canada (CAN), Switzerland (CHE), Germany (DEU), Denmark (DNK), Spain (ESP), Finland (FIN), France (FRA),Great-Britain (GBR), Hong Kong (HKG), Ireland (IRL), Italy (ITA), Netherlands (NLD), Norway (NOR), Singapore (SGP),Sweden (SWE) and the United States (USA). The calculations are based on monthly U.S. dollar returns from January 1991to April 2015.

120


return. The third component is dominated by Finland for the market excess return,

size and value, while Hong Kong and Singapore dominate for momentum, low beta and

quality. This third component also reflects the grouping of the factors discussed during

the correlation analysis.

Moreover, even though there is a distinct link between the components and factors

in figure 9, their contribution to the variance explained varies widely so that the first

component explains about 40% of the cross-sectional variance, while it is around 4%

for the last component. While Denmark and Norway are particularly well diversified

in terms of component contributions with inverse Herfindahl indices of 4.75 and 4.67

respectively, Singapore (3.22), Japan (3.50) and the U.S. (3.54) are least diversified in

their component contributions. Thus, just taking the U.S. as reference for the potential

of factor diversification likely underestimates the diversification gains within each single

country.

Figure 16 in the appendix illustrates the loadings of the components. Analysing the

cross-factor variation, I find that for many countries, one equity factor clearly dominates

each component. However, the components are far from forming a one-to-one relationship

to the factors. The market excess return (MKT) in one direction and some combination

of momentum (UMD), low beta (BAB) and quality (QMJ) in the other direction form the

first principal component. Again, the first component reflects the grouping of the factors

discussed during the correlation analysis.

5.4 Portfolio Construction

After laying out the theoretical diversification benefits of country equity factors across

the country dimension as well as the factor dimension, I form portfolios to demonstrate

how these diversification gains translate into the reduction of risk and the improvement

of the risk-return trade-off.

5.4.1 International diversification

Compared to international diversification for each factor, investing in a single country

factor does not profit from diversification. In this exercise, I represent single country

investing by synthetic factors that have the average returns and the average volatilities

across the twenty countries. I contrast these synthetic factors with the widely used risk

parity approach to ensure that each local factor contributes equally in terms of risk to the

global factor. In the appendix, the results are also given for an equal weighted portfolio

of local factors in tabular form.

In figure 10 and table 5, the theoretical benefits of international diversification for each

121


MKT SMB HML UMD BAB QMJ Multi0

5

10

15

20

25

Vola

tility

in %

p.a

.

Volatilities

Country Risk ParityCountry Equal WeightsSynthetic Factor

MKT SMB HML UMD BAB QMJ Multi0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Shar

pe R

atio

Sharpe Ratios

Country Risk ParityCountry Equal WeightsSynthetic Factor

Figure 10: Risk and Sharpe ratio impact of international diversification for each factorThe two figures display volatilities (left) and Sharpe ratios (right) of internationally diversified factors versus syntheticundiversified factors. There are six individual factors (MKT, SMB, HML, UMD, BAB, QMJ). Twenty country factors formthe base assets for each internationally diversified factor, in which each country factor contributes equally to the factorrisk (country risk parity) or to portfolio weights (country equal weights). ’Multi’ is the equal weighted combination ofthe six factor portfolios. The synthetic factors are factors constructed from the average returns and average volatilitiesof the single countries without benefiting from diversification. The twenty country markets are Australia (AUS), Austria(AUT), Belgium (BEL), Canada (CAN), Switzerland (CHE), Germany (DEU), Denmark (DNK), Spain (ESP), Finland(FIN), France (FRA), Great-Britain (GBR), Hong Kong (HKG), Ireland (IRL), Italy (ITA), Netherlands (NLD), Norway(NOR), Singapore (SGP), Sweden (SWE) and the United States (USA). The calculations are based on monthly U.S. dollarlog returns from January 1994 to April 2015.

of the six factors translate into a substantial and statistically highly significant reduction

in the portfolio volatilities for each factor.

For the market excess return, increasing international diversification significantly de-

creases the volatility from 21.64% to 17.21%. However, this risk reduction of about 20%

is substantially lower than for other equity factors. This is consistent with the previous

findings that the equity markets are highly correlated such that the improved diversifica-

tion can only moderately reduce portfolio volatility. Even more than in this short term

correlation analysis, international diversification works in the long term, when economic

growth matters more than synchronous short-lived panics (Asness et al., 2011).

For the size factor, the value factor, the quality factor and the low beta factor, the

volatility reduction is around between 50-60%, while it is slightly lower for the momentum

factor (-39%). After a reduction of the volatility from 12.38% to 5.37% the global size

factor has the lowest volatility. The global value factor is only slightly more risky with

6.00% (down from 14.19%). The global momentum factor has the highest volatility of the

alternative equity factors with 12.16% (down from 20.05%). The low beta factor (18.20%

to 8.11%) and the quality factor (14.68% to 7.04%) form a middle ground. This is in line

with the findings of slightly elevated cross-country correlations for the momentum factor

compared to the other alternative equity factors.

For an equal weighted portfolio of the six factor premia (global factor diversification),

there is also a significant risk reduction when improving the international diversification.

The volatility is reduced by -46% from 6.45% to 3.48%.

122


Volatility Sharpe Ratio

RP SF Delta SE RP SF Delta SE

MKT 17.21 21.64 -4.43*** 0.27 38.65 29.65 9.00 5.85

SMB 5.37 12.38 -7.01*** 0.25

HML 6.00 14.19 -8.19*** 0.37 77.10 35.57 41.52** 18.16

UMD 12.16 20.05 -7.89*** 0.60 90.15 52.50 37.65** 18.59

BAB 8.11 18.20 -10.09*** 0.45 143.23 66.97 76.26*** 17.15

QMJ 7.04 14.68 -7.63*** 0.46 77.27 36.93 40.34*** 14.89

Multi 3.48 6.45 -2.97*** 0.17 190.60 105.85 84.75*** 16.80

Table 5: Risk and Sharpe Ratio impact of international diversification for each factorThis table shows the annualized volatilities (in percent) and annualized Sharpe ratios (multiplied by 100) between globalportfolios for the equity factor specified in the first column. For the global portfolios, twenty countries are weighted accordingto risk parity (RP) and contrasted to synthetic factors (SF) with average returns and volatilities of the countries. ∆ isthe difference between these two portfolios and SE is the block-bootstrapped standard error of the difference. * stands fortwo sided significance at the 10% level, ** for significance at the 5% level and *** for significance at the 1% level. Thefour statistics are presented for volatilities as well as Sharpe ratios. The Sharpe ratio for the size factor is omitted as it isnegative. The performance calculations are based on monthly U.S. dollar log returns from January 1994 to April 2015.

Figure 17 in the appendix demonstrates the volatility reduction over time by means

of rolling 36 months strategy volatilities. Besides the persistent reduction of the volatili-

ties, the cyclicality of the volatility is moderately diminished compared to single country

investing. This is particularly the case for the factors that show low cross-country cor-

relations such as the size factor. This is also consistent with the finding of moderate or

even slightly reduced cross-country correlations during times of poor performance in the

respective global factor.

This volatility reduction translates into extensive Sharpe ratio improvements as ev-

idenced on the right side of figure 10 and columns 5 to 8 in table 5. On average, the

Sharpe ratios can be close to doubled. The gains are particularly strong for value, low

beta and quality, reaching more than 100%. The Sharpe ratios for the size factor are

omitted due to their negative values. The significance of the Sharpe ratio gains is slightly

reduced compared to the volatility reductions as there is the additional estimation error

of the return estimates.

Overall, we can see that international diversification of equity factors helps to substan-

tially reduce the volatilities. The volatility reductions are particularly strong for those

factors that show only a moderate cross-country correlation.

5.4.2 Local factor diversification

Besides the international dimension of diversification, there are large diversification ben-

efits of factor diversification within each country (local factor diversification). For this

purpose I again contrast factor risk parity portfolios in each country with synthetic factors.

To construct this factor, the returns and the volatilities of the six factors are averaged,

while neglecting the impact of diversification. The appendix presents the same exercise

123


0 5 10 15 20 25 30 35

USA

SWE

SGP

NOR

NLD

JPN

ITA

IRL

HKG

GBR

FRA

FIN

ESP

DNK

DEU

CHE

CAN

BEL

AUT

AUS

Volatility in % p.a.

Volatilities

Factor Risk ParityFactor Equal WeightsSynthetic Factor

0 0.5 1 1.5 2 2.5

USA

SWE

SGP

NOR

NLD

JPN

ITA

IRL

HKG

GBR

FRA

FIN

ESP

DNK

DEU

CHE

CAN

BEL

AUT

AUS

Sharpe Ratios

Sharpe Ratios

Factor Risk ParityFactor Equal WeightsSynthetic Factor

Figure 11: Risk and Sharpe ratio impact of local factor diversification for each countryThe two figures display volatilities (left) and Sharpe ratios (right) of local factor portfolios versus synthetic local factorinvestments. Six local factors (MKT, SMB, HML, UMD, BAB, QMJ) form the base assets for each local factor portfolio, inwhich each local factor contributes equally to the risk (factor risk parity) or to portfolio weights (factor equal weights). Thesynthetic factors are local factors constructed from the average returns and average volatilities of the single factors withoutbenefiting from diversification. The twenty country markets are Australia (AUS), Austria (AUT), Belgium (BEL), Canada(CAN), Switzerland (CHE), Germany (DEU), Denmark (DNK), Spain (ESP), Finland (FIN), France (FRA), Great-Britain(GBR), Hong Kong (HKG), Ireland (IRL), Italy (ITA), Netherlands (NLD), Norway (NOR), Singapore (SGP), Sweden(SWE) and the United States (USA). The calculations are based on monthly U.S. dollar log returns from January 1994 toApril 2015.

using equal weighted factor portfolios and contrasting it to the synthetic factors.

In figure 11 and table 6, it is obvious that the combination of the six nearly uncorrelated

factors substantially reduces the volatility in each of the twenty countries. The factor risk

parity portfolios have on average less than 40% of the volatility of the synthetic factors

(6.0% vs. 16.5%) and the difference is highly significant in each country. The volatility

reduction is similar for most countries ranging from -72% in the UK to -54% in the U.S.

Using only U.S. data therefore potentially underestimates the gains through local factor

diversification.

Figure 18 in the appendix demonstrates the volatility reduction over time by means of

rolling 36 months strategy volatilities. Besides the persistent reduction of the volatilities,

the cyclicality in the volatility is substantially diminished compared to single factor in-

vesting. This reduced cyclicality is more pronounced for local factor diversification than

for international diversification of the six factors. The local factor portfolios are highly

effective in terms of diversification and show limited increases in volatility during times

of market stress.

Even though the returns are on average -1.2% lower for the factor risk parity port-

folio, the massive risk reduction translates into significant and substantial Sharpe ratio

124



RP SF ∆ SE RP SF ∆ SE

AUS 4.88 13.82 -8.94*** 0.66 187.67 70.64 117.03*** 17.03

AUT 6.48 18.60 -12.12*** 1.21 99.45 38.41 61.04*** 15.79

BEL 5.05 15.38 -10.33*** 0.90 104.38 38.65 65.73*** 15.74

CAN 6.53 16.00 -9.47*** 0.69 133.56 72.96 60.61*** 22.92

CHE 5.92 14.60 -8.68*** 0.77 84.57 44.11 40.46** 20.26

DEU 6.93 15.87 -8.94*** 1.06 78.41 47.49 30.92** 13.21

DNK 5.40 15.71 -10.31*** 0.73 111.47 42.70 68.77*** 15.43

ESP 5.44 16.26 -10.82*** 0.84 59.91 27.31 32.59* 16.97

FIN 7.27 21.06 -13.80*** 1.12 87.20 35.75 51.44*** 14.01

FRA 5.24 14.92 -9.68*** 0.67 106.68 46.99 59.69*** 21.43

GBR 3.93 13.93 -10.00*** 1.02 144.81 41.28 103.54*** 20.09

HKG 6.97 20.36 -13.39*** 1.40 101.94 44.05 57.89*** 22.15

IRL 9.55 29.10 -19.55*** 2.54 42.10 17.84 24.26 14.94

ITA 5.28 15.44 -10.16*** 0.64 114.63 38.34 76.29*** 16.73

JPN 4.73 12.99 -8.26*** 0.50 54.05 19.43 34.62 21.43

NLD 5.34 16.18 -10.83*** 0.81 88.60 35.32 53.28*** 18.41

NOR 6.39 18.70 -12.31*** 0.84 104.42 43.78 60.64*** 14.33

SGP 5.91 17.31 -11.40*** 1.71 65.92 29.59 36.33* 21.67

SWE 7.15 17.99 -10.85*** 0.89 86.52 42.84 43.68*** 15.82

USA 5.91 12.91 -6.99*** 1.11 55.76 40.44 15.32 19.00

Table 6: Risk and Sharpe Ratio impact of local factor diversification for each countryThis table shows the annualized volatilities (in percent) and annualized Sharpe ratios (multiplied by 100) between localfactor portfolios for the country specified in the first column. In the local portfolios six factors are weighted according to riskparity (RP) and contrasted to synthetic factors with average returns and average volatilities. ∆ is the difference betweenthese two portfolios and SE is the block-bootstrapped standard error of the difference. * stands for two sided significance atthe 10% level, ** for significance at the 5% level and *** for significance at the 1% level. The four statistics are presentedfor volatilities as well as Sharpe ratios. The performance calculations are based on monthly U.S. dollar log returns fromJanuary 1994 to April 2015.

improvements. These improvements are on average +137% (from 0.41 to 0.96), while

there is a huge variability ranging from +38% in the U.S. to +251% in the UK. Also in

terms of Sharpe ratios the U.S. shows the lowest gains and does not constitute the best

reference point to gauge the gains to local factor diversification. The Sharpe ratio gains

are significant at the 10% level in 17 out of 20 countries and significant at the 1% level

in 13 out of 20 countries. These findings are consistent with the substantial Sharpe ratio

improvements in Eun et al. (2010), when enhancing the local market portfolios with local

size, value and momentum factors.

The significant return reduction through factor risk parity compared to the synthetic

factors in 9 out of 20 countries is somewhat surprising. For each factor, this would imply

that the higher the factor volatility, the higher the factor returns. Traditional asset pricing

models assume a positive relationship between return and systematic risk, but there is no

indication that the factor volatility is systematic risk in the classical sense. As there is

no theory behind this finding and the scope of this paper is to show the benefits of risk

reduction, I refrain from a more detailed discussion.

Overall, we can see that local factor diversification can massively reduce risk of factor

based strategies for each of the twenty developed markets discussed in this paper.

125


6 Conclusions

In this paper, I discuss the diversification properties of six widely recognized equity factors

across twenty countries from 1991 to 2015. These equity factors are the market excess

return, size, value, momentum, low beta and quality. In the time period considered five

of these six equity factors yield consistently positive returns across countries. The size

factor is the exception with mostly negative returns.

The cross-country correlations within each of the factors are moderate with the ex-

ception of the market factor and they are slightly elevated for the momentum factor. In

line with the findings on decreasing benefits of international diversification for the mar-

ket excess return, the cross-country correlations have mostly been rising over the last

twenty years, even though they are highly cyclical. There is a significant increase in

cross-country correlations for the market excess return, size, momentum and quality. The

gains through international diversification of the single factors are thus diminishing, but

remain substantial.

The cross-factor correlations are very low. Especially momentum and quality are good

diversifiers to the market. There is a slightly increased correlation between momentum,

low beta and quality. Moreover, there is no indication of increasing correlations between

the six factors from 1991 to 2015.

Using principal component analysis confirms the findings that there are large diver-

sification benefits across factors as well as across countries. The diversity in the factor

dimension is larger than in the country dimension and the first principal components re-

flect the factor dimension, when analysing the factor and country dimension jointly using

the 120 time series. However, the first six components only explain 47.8% of the total

variance, such that country effects still play a distinct role.

The portfolio construction exercise demonstrates the diversification gains in a portfolio

context. International diversification reduces portfolio volatility for each factor. This risk

reduction is particularly strong for the factors that show low cross-country correlations

such as size, value, low beta and quality (-50 to -60%), while it is less pronounced for

the market excess return (-20%) and momentum (-39%). The volatility reduction through

improved international diversification leads to significant Sharpe ratio increases compared

to single country investing.

Factor diversification significantly reduces volatilities and thus increases Sharpe ratios

compared to single factor investing. The gains from factor diversification are significant

for each of the twenty developed markets, with volatility reductions ranging from -72% in

the UK to -53% in the U.S. These findings also demonstrate why factor investing currently

receives wide attention from academics and practitioners alike.

126


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132


A Factor Performance


AUS 7.46 -3.09 8.56*** 17.35*** 22.30*** 5.89**

(4.99) (3.02) (2.44) (4.43) (5.23) (2.78)

AUT 1.26 1.78 10.93*** 5.03 7.12 0.23(5.89) (2.74) (3.63) (5.30) (5.11) (3.00)

BEL 6.61 -1.45 4.90* 9.18** 6.55** 1.84(4.94) (2.19) (2.98) (4.46) (3.12) (3.13)

CAN 6.23 -1.26 5.16 17.61*** 20.92*** 10.63***

(4.59) (2.36) (3.53) (5.47) (5.46) (3.71)

CHE 7.80** -1.88 3.62 9.41** 7.23** 4.82*

(3.83) (2.43) (2.51) (4.06) (3.51) (2.61)

DEU 4.13 -8.33*** 8.11** 12.47*** 9.00** 5.95**

(4.39) (2.67) (3.35) (4.56) (4.18) (2.42)

DNK 7.90* -4.29 -2.86 13.34*** 7.64* 7.64**

(4.71) (2.79) (3.37) (4.12) (4.30) (3.29)ESP 5.64 -3.44 2.63 4.49 5.07 1.90

(4.83) (2.96) (3.34) (3.80) (3.81) (2.90)

FIN 6.16 -1.92 3.99 11.58** 10.59** -0.16(6.75) (3.21) (5.32) (4.69) (5.03) (4.21)

FRA 4.92 -0.63 4.37 7.40** 12.73*** 4.95**

(4.30) (1.85) (2.90) (3.78) (3.95) (2.48)

GBR 4.53 -0.80 4.89 11.61** 6.38 2.81(4.00) (2.80) (3.03) (4.64) (4.59) (2.32)

HKG 9.04 0.42 4.43 6.01 20.08*** 6.76*

(5.94) (3.45) (3.59) (4.70) (7.16) (3.54)IRL 5.63 -1.58 1.77 5.65 4.12 0.32

(5.95) (4.00) (4.48) (7.39) (7.78) (6.43)

ITA 1.80 -1.89 2.97 6.06 6.45** 5.93**

(5.02) (2.24) (3.01) (3.72) (3.28) (2.91)

JPN -1.00 -0.50 5.73** 0.86 1.50 1.76(4.65) (2.02) (2.31) (3.73) (3.22) (2.30)

NLD 6.10 1.48 5.58* 2.46 10.26*** -2.24(4.67) (2.14) (3.39) (4.38) (3.84) (3.04)

NOR 5.93 0.58 1.51 11.51** 13.22*** 4.47(6.11) (2.62) (3.72) (4.86) (4.82) (4.30)

SGP 6.76 -5.33 6.26** 1.84 14.21*** 1.34(5.68) (3.31) (3.02) (5.27) (4.28) (3.36)

SWE 7.98 -2.57 2.40 5.15 12.95*** 7.44**

(5.75) (2.32) (5.17) (4.90) (4.72) (3.38)

USA 7.16** 2.34 2.01 5.19 9.23** 3.83(3.55) (1.96) (2.36) (3.98) (3.71) (2.45)

Table 7: Annualized factor returns across countriesThis table represents the annualized log returns in U.S. dollars of the six equity factors (MKT, SMB, HML, UMD, BABand QMJ) for 20 different developed markets from January 1991 to April 2015. The twenty country markets are Australia(AUS), Austria (AUT), Belgium (BEL), Canada (CAN), Switzerland (CHE), Germany (DEU), Denmark (DNK), Spain(ESP), Finland (FIN), France (FRA), Great-Britain (GBR), Hong Kong (HKG), Ireland (IRL), Italy (ITA), Netherlands(NLD), Norway (NOR), Singapore (SGP), Sweden (SWE) and the United States (USA). The Newey West standard errorsare given in parentheses. * stands for two sided significance at the 10% level, ** for significance at the 5% level and *** forsignificance at the 1% level.

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B Correlation Analysis

B.1 Average cross-factor correlations ex cross-country effects

Panel A: Average cross-factor correlations

MKT SMB HML UMD BAB QMJMKT 1.00 -0.05 -0.01 -0.25 -0.01 -0.48SMB -0.05 1.00 -0.08 -0.16 0.16 -0.20HML -0.01 -0.08 1.00 -0.08 0.05 -0.18UMD -0.25 -0.16 -0.08 1.00 0.21 0.36BAB -0.01 0.16 0.05 0.21 1.00 0.21QMJ -0.48 -0.20 -0.18 0.36 0.21 1.00

Panel B: Other tests

UMD, BAB and QMJ vs Rest 0.383***

(0.0500)

BAB vs Rest 0.240***

(0.0103)

Table 8: Average cross-factor correlations ex cross-country effectsThis table displays the cross-factor correlation structure and tests for differences in the correlations. Panel A representsthe cross-factor correlation structure after averaging over all countries. This repeats table 2, while neglecting cross-countryeffects in the sense that e.g. I only consider the correlation between MKTUSA and SMBUSA, but not between MKTUSA

and SMBJPN . Panel B tests the difference in the average correlation between UMD, BAB and QMJ and the other factorsand the differences in the average correlation between BAB and the other factors. For Panel B, block bootstrapped standarderrors are in brackets. * stands for two sided significance at the 10% level, ** for significance at the 5% level and *** forsignificance at the 1% level. There are six different equity factors (MKT, SMB, HML, UMD, BAB and QMJ) and 20different developed markets. The twenty country markets are Australia (AUS), Austria (AUT), Belgium (BEL), Canada(CAN), Switzerland (CHE), Germany (DEU), Denmark (DNK), Spain (ESP), Finland (FIN), France (FRA), Great-Britain(GBR), Hong Kong (HKG), Ireland (IRL), Italy (ITA), Netherlands (NLD), Norway (NOR), Singapore (SGP), Sweden(SWE) and the United States (USA). The correlations are based on monthly U.S. dollar returns from January 1991 to April2015.

Table 8 eliminates cross-country effects by only comparing cross-factor correlations

for the same country. This way, I only consider the correlation between MKTUSA and

SMBUSA, but not between MKTUSA and SMBJPN . This leads to more variation in the

cross-factor correlation structure in panel A and strengthens the findings of clusters in

panel B. Testing for the cross-country correlation differences between factors as in panel

B and C of table 2 are not possible when eliminating the cross-country effects.

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B.2 Cross-country correlation for local factor portfolios

AUS

AUT

BEL

CAN

CH

E

DEU

DN

K

ESP

FIN

FRA

GBR

HKG IR

L

ITA

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NLD

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R

SGP

SWE

USA

AUS

AUT

BEL

CAN

CHE

DEU

DNK

ESP

FIN

FRA

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HKG

IRL

ITA

JPN

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USA0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 12: Cross-country correlation for local factor portfoliosThis figure represents the cross-country correlations between local factor portfolios consisting of six different equity factors(MKT, SMB, HML, UMD, BAB and QMJ). There are the following twenty country markets in the sample: Australia(AUS), Austria (AUT), Belgium (BEL), Canada (CAN), Switzerland (CHE), Germany (DEU), Denmark (DNK), Spain(ESP), Finland (FIN), France (FRA), Great-Britain (GBR), Hong Kong (HKG), Ireland (IRL), Italy (ITA), Netherlands(NLD), Norway (NOR), Singapore (SGP), Sweden (SWE) and the United States (USA). The correlations are based onmonthly U.S. dollar returns from January 1991 to April 2015.

The local factor portfolios show a distinct pattern of cross-country correlations. Par-

ticularly Ireland the countries in the Asia Pacific region (Japan, Hong Kong, Singapore,

Australia) show a very low correlation with local factor portfolios in other countries. The

local factor portfolios of European and American countries in contrast show a substan-

tially higher correlation. This is particularly the case for France and Great-Britain.

135


B.3 Tail Correlations: Tails in the global market excess returns

MKT: Left TailAU

SAU

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HML: Left Tail

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HML: Right Tail

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FRAGBRHKG

IRLITA

JPNNLDNORSGPSWEUSA

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Figure 13a: Cross-country correlations in the tails of the market excess returns for eachequity factorThis figure represents the cross-country correlations across twenty developed markets for each of the six different equityfactors (MKT, SMB, HML, UMD, BAB and QMJ) for the 30% worst global market months (left tail) and for the 30%best global market months (right tail). The twenty country markets are Australia (AUS), Austria (AUT), Belgium (BEL),Canada (CAN), Switzerland (CHE), Germany (DEU), Denmark (DNK), Spain (ESP), Finland (FIN), France (FRA),Great-Britain (GBR), Hong Kong (HKG), Ireland (IRL), Italy (ITA), Netherlands (NLD), Norway (NOR), Singapore(SGP), Sweden (SWE) and the United States (USA). The correlations are based on monthly U.S. dollar returns fromJanuary 1991 to April 2015.

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UMD: Left TailAU

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UMD: Right Tail

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BAB: Left Tail

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QMJ: Left Tail

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QMJ: Right Tail

AUS

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FRAGBRHKG

IRLITA

JPNNLDNORSGPSWEUSA

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Figure 13b: Cross-country correlations in the tails of the market excess returns for eachequity factor - continuedThis figure represents the cross-country correlations across twenty developed markets for each of the six different equityfactors (MKT, SMB, HML, UMD, BAB and QMJ) for the 30% worst global market months (left tail) and for the 30%best global market months (right tail). The twenty country markets are Australia (AUS), Austria (AUT), Belgium (BEL),Canada (CAN), Switzerland (CHE), Germany (DEU), Denmark (DNK), Spain (ESP), Finland (FIN), France (FRA),Great-Britain (GBR), Hong Kong (HKG), Ireland (IRL), Italy (ITA), Netherlands (NLD), Norway (NOR), Singapore(SGP), Sweden (SWE) and the United States (USA). The correlations are based on monthly U.S. dollar returns fromJanuary 1991 to April 2015.

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B.4 Tail Correlations: Tails in the respective factor returns

Left Tail Right Tail Difference t-statistic

MKT 0.54 0.34 0.20 2.09SMB 0.08 0.14 -0.06 -1.90HML 0.09 0.16 -0.07 -1.02UMD 0.35 0.22 0.13 1.38BAB 0.14 0.15 -0.01 -0.22QMJ 0.12 0.17 -0.05 -1.32

Table 9: Differences in the cross-country correlations in the tails of each equity factorFor each of the six factors in column 1, this table presents the average cross-country correlation for the 30% worst monthsfor the respective global factor (left tail) and for the 30% best months (right tail) and tests the difference for significance.The t-statistic is obtained via block-bootstrapping. The correlations are based on monthly U.S. dollar returns from January1991 to April 2015.

While I discuss the cross-country correlations during tails of the global market excess

returns in table 4, it is also interesting to see, how cross-country correlations behave in

the tails of the respective global factor. In table 9 and figures 14a and 14b, I can observe

that except for the market excess return (MKT) and the momentum factor (UMD), the

cross-country correlations are even moderately higher during times of high global factor

returns compared to low global factor returns. Thus, there are no increasing correlations

across markets during times of bad factor returns for the size (SMB), value (HML), low

beta (BAB) and quality factors (QMJ).

For the market factor (MKT), the results correspond to the cross-country correlations

during market stress in table 4, which increase from 0.34 to 0.54. A similar increase

in the left factor tail can be observed for momentum (UMD), where the average cross-

country correlation rises from 0.22 to 0.35. This finding is consistent with momentum

crashes (Daniel & Moskowitz, 2013) that are global and happen concurrently in many

countries due to dynamics in the loadings of the market beta. The increase is economically

significant but statistically insignificant.

For the size factor (SMB), the cross-country correlation decreases from 0.14 to 0.08

during periods of stress in the size factor. This indicates a decoupling of size factors

in times of poor global size factor returns. Similarly, the correlations drop from 0.16

to 0.09 for value (HML), from 0.15 to 0.14 for low beta (BAB) and from 0.17 to 0.12

for quality (QMJ). As for the market tail events, the high standard errors are due to

conditioning on the 30% most extreme values, which substantially reduces the dispersion

in the sub-sample and does not fully incorporate the data available. To conclude, except

for the market factor and insignificantly for the momentum factor, there is no indication

of factor contagion or increased interdependence across countries.

138


MKT: Left TailAU

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MKT: Right Tail

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Figure 14a: Cross-country correlations in the tails of each equity factorThis figure represents the cross-country correlations for each of the six different equity factors (MKT, SMB, HML, UMD,BAB and QMJ) for the 30% worst months of the respective global factor (left tail) and for the 30% best months of therespective global factor (right tail). The twenty country markets are Australia (AUS), Austria (AUT), Belgium (BEL),Canada (CAN), Switzerland (CHE), Germany (DEU), Denmark (DNK), Spain (ESP), Finland (FIN), France (FRA),Great-Britain (GBR), Hong Kong (HKG), Ireland (IRL), Italy (ITA), Netherlands (NLD), Norway (NOR), Singapore(SGP), Sweden (SWE) and the United States (USA). The correlations are based on monthly U.S. dollar returns fromJanuary 1991 to April 2015.

139


UMD: Left TailAU

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JPNNLDNORSGPSWEUSA

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Figure 14b: Cross-country correlations in the tails of each equity factor - continuedThis figure represents the cross-country correlations for each of the six different equity factors (MKT, SMB, HML, UMD,BAB and QMJ) for the 30% worst months of the respective global factor (left tail) and for the 30% best months of therespective global factor (right tail). The twenty country markets are Australia (AUS), Austria (AUT), Belgium (BEL),Canada (CAN), Switzerland (CHE), Germany (DEU), Denmark (DNK), Spain (ESP), Finland (FIN), France (FRA),Great-Britain (GBR), Hong Kong (HKG), Ireland (IRL), Italy (ITA), Netherlands (NLD), Norway (NOR), Singapore(SGP), Sweden (SWE) and the United States (USA). The correlations are based on monthly U.S. dollar returns fromJanuary 1991 to April 2015.

140


C Principal component analysis

C.1 Component loadings of the cross-country variation

MKT

Percentage of the Variance Explained by Component65.2 6.9 5 3.6 3.1 2.7 2.1 1.9 1.4 1.3


FRAGBRHKG

IRLITA

JPNNLDNORSGPSWEUSA

SMB

Percentage of the Variance Explained by Component20.6 9.8 8.1 7 6.5 5.6 4.8 4.5 4.2 4.1


FRAGBRHKG

IRLITA

JPNNLDNORSGPSWEUSA

HML

Percentage of the Variance Explained by Component25.4 9.9 8.7 6.8 6.2 5.2 4.8 4.5 3.9 3.7


FRAGBRHKG

IRLITA

JPNNLDNORSGPSWEUSA

UMD

Percentage of the Variance Explained by Component42.3 11.6 7.3 5.1 4.8 4.1 3.4 2.9 2.6 2.2


FRAGBRHKG

IRLITA

JPNNLDNORSGPSWEUSA

BAB

Percentage of the Variance Explained by Component25.5 16.5 9 6.5 5.8 5.2 4.2 3.7 3.1 2.7


FRAGBRHKG

IRLITA

JPNNLDNORSGPSWEUSA

QMJ

Percentage of the Variance Explained by Component

28.4 16.1 8 7.5 5.5 4.7 4 3.6 3.1 2.9


FRAGBRHKG

IRLITA

JPNNLDNORSGPSWEUSA

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

Figure 15: Coefficients of the principal components of the cross-country variation for eachfactorThis figure displays the coefficients (right singular values V ) of the first ten principal components of the cross-countryvariation for each of the six equity factors. The vertical axis displays the twenty developed markets, while the horizontalaxis represents the ten first principal components with the percentage of the variance explained by the respective component.The six equity factors are the market excess return (MKT), size (SMB), value (HML), momentum (UMD), low beta (BAB)and quality (QMJ). The twenty country markets are Australia (AUS), Austria (AUT), Belgium (BEL), Canada (CAN),Switzerland (CHE), Germany (DEU), Denmark (DNK), Spain (ESP), Finland (FIN), France (FRA), Great-Britain (GBR),Hong Kong (HKG), Ireland (IRL), Italy (ITA), Netherlands (NLD), Norway (NOR), Singapore (SGP), Sweden (SWE) andthe United States (USA). The calculations are based on monthly U.S. dollar returns from January 1991 to April 2015.

For all six factors, the first principal component loads relatively equally on all 20

countries. The second component is an Ireland component for each factor except for

the market excess return. The third component is dominated by Finland for the market

excess return, size and value, while Hong Kong and Singapore dominate for momentum,

low beta and quality. This third component also reflects the grouping of the factors

discussed during the correlation analysis.

141


C.2 Component loadings of the cross-factor variation

AUS

43.4 19.3 16.7 10.2 6.7 3.7

MKTSMBHMLUMDBABQMJ

AUT

34.9 24.1 15.2 12.2 8.1 5.4

MKTSMBHMLUMDBABQMJ

BEL

41.7 20 14.4 11.9 7.7 4.4

MKTSMBHMLUMDBABQMJ

CAN

36.6 28.2 17.5 7.8 6.3 3.6

MKTSMBHMLUMDBABQMJ

CHE

40 20.2 16.7 11.4 8.3 3.4

MKTSMBHMLUMDBABQMJ

DEU

44.2 21 16.7 9.8 5.4 2.9

MKTSMBHMLUMDBABQMJ

DNK

29.2 24.2 19 14.7 8 4.9

MKTSMBHMLUMDBABQMJ

ESP

42.7 20.1 13.2 11.8 7.9 4.3

MKTSMBHMLUMDBABQMJ

FIN

42.2 18.4 14.9 13.4 6.6 4.5

MKTSMBHMLUMDBABQMJ

FRA

43.1 21.5 16.2 10.1 6.1 3

MKTSMBHMLUMDBABQMJ

GBR

36.9 27 16.3 11.9 6.1 1.7

MKTSMBHMLUMDBABQMJ

HKG

37.2 27.9 14.9 10.3 6.9 2.7

MKTSMBHMLUMDBABQMJ

IRL

36.5 26.7 13.9 8.7 7.6 6.5

MKTSMBHMLUMDBABQMJ

ITA

45.6 18.7 13.7 11.7 5.8 4.4

MKTSMBHMLUMDBABQMJ

JPN

43.7 23.7 16.7 7.8 6 2

MKTSMBHMLUMDBABQMJ

NLD

44.2 18.3 14.5 11.3 7.3 4.4

MKTSMBHMLUMDBABQMJ

NOR

31.6 22.4 18.6 13.7 9.1 4.5

MKTSMBHMLUMDBABQMJ

SGP

49.7 18.5 12.4 8.7 7.6 3.1

MKTSMBHMLUMDBABQMJ

SWE

42 20.3 17.6 9.3 7.2 3.6

MKTSMBHMLUMDBABQMJ

USA

44.1 22.2 16.2 9.3 5.6 2.6

MKTSMBHMLUMDBABQMJ

−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Figure 16: Coefficients of the principal components of the cross-factor variation for eachcountryThis figure displays the coefficients (right singular values V ) of the six principal components of the cross-factor variation foreach of the twenty developed markets. The vertical axis displays the six equity factors, while the horizontal axis representsthe six principal components with the percentage of the variance explained by the respective component. The six equityfactors are the market excess return (MKT), size (SMB), value (HML), momentum (UMD), low beta (BAB) and quality(QMJ). The twenty country markets are Australia (AUS), Austria (AUT), Belgium (BEL), Canada (CAN), Switzerland(CHE), Germany (DEU), Denmark (DNK), Spain (ESP), Finland (FIN), France (FRA), Great-Britain (GBR), Hong Kong(HKG), Ireland (IRL), Italy (ITA), Netherlands (NLD), Norway (NOR), Singapore (SGP), Sweden (SWE) and the UnitedStates (USA). The calculations are based on monthly U.S. dollar returns from January 1991 to April 2015.

Analysing the cross-factor variation, I find that for many countries, one equity factor

clearly dominates each component. However, the components are far from forming a

one-to-one relationship to the factors. The market excess return (MKT) in one direction

and some combination of momentum (UMD), low beta (BAB) and quality (QMJ) in the

other direction form the first principal component. Again, the first component reflects

the grouping of the factors discussed during the correlation analysis.

142


D Portfolio construction

D.1 Time trends in risk reduction

International diversification

2000 2005 2010 20150

20

40

Vola

tility

in %

MKT

2000 2005 2010 20150

10

20

Vola

tility

in %

SMB

2000 2005 2010 20150

10

20

30

Vola

tility

in %

HML

2000 2005 2010 20150

10

20

30

Vola

tility

in %

UMD

2000 2005 2010 20150

10

20

30

Vola

tility

in %

BAB

2000 2005 2010 20150

10

20

Vola

tility

in %

QMJ

Figure 17: Risk reduction through international diversification over timeThis figure displays the 36 months rolling volatilities for the country risk parity weighted portfolios (black line) and syntheticfactors representing single country investments (grey line). There is one graph for each of the six individual factors (MKT,SMB, HML, UMD, BAB, QMJ). Each global factor is constructed from twenty country factors. The twenty country marketsare Australia (AUS), Austria (AUT), Belgium (BEL), Canada (CAN), Switzerland (CHE), Germany (DEU), Denmark(DNK), Spain (ESP), Finland (FIN), France (FRA), Great-Britain (GBR), Hong Kong (HKG), Ireland (IRL), Italy (ITA),Netherlands (NLD), Norway (NOR), Singapore (SGP), Sweden (SWE) and the United States (USA). The calculations arebased on monthly U.S. dollar log returns from January 1994 to April 2015.

Figure 17 in the appendix demonstrates the volatility reduction over time by means

of rolling 36 months strategy volatilities. Besides the persistent reduction of the volatili-

ties, the cyclicality of the volatility is moderately diminished compared to single country

investing. This is particularly the case for the factors that show low cross-country cor-

relations such as the size factor. This is also consistent with the finding of moderate or

even slightly reduced cross-country correlations during times of poor performance in the

respective global factor.

143


Local factor diversification

2000 2005 2010 20150

5

10

15

20AUS

2000 2005 2010 20150

10

20

30

40AUT

2000 2005 2010 20150

10

20

30BEL

2000 2005 2010 20150

10

20

30CAN

2000 2005 2010 20150

10

20

30CHE

2000 2005 2010 20150

10

20

30DEU

2000 2005 2010 20150

10

20

30DNK

2000 2005 2010 20150

10

20

30ESP

2000 2005 2010 20150

10

20

30

40FIN

2000 2005 2010 20150

10

20

30FRA

2000 2005 2010 20150

10

20

30GBR

2000 2005 2010 20150

10

20

30

40HKG

2000 2005 2010 20150

20

40

60IRL

2000 2005 2010 20150

5

10

15

20ITA

2000 2005 2010 20150

5

10

15

20JPN

2000 2005 2010 20150

10

20

30NLD

2000 2005 2010 20150

10

20

30NOR

2000 2005 2010 20150

10

20

30

40SGP

2000 2005 2010 20150

10

20

30

40SWE

2000 2005 2010 20150

10

20

30USA

Figure 18: Risk reduction through local factor diversification over timeThis figure displays the 36 months rolling volatilities for the factor risk parity weighted portfolios (black line) and syntheticfactors representing single factor investments (grey line). There is one graph for each of the twenty countries. The twentycountry markets are Australia (AUS), Austria (AUT), Belgium (BEL), Canada (CAN), Switzerland (CHE), Germany(DEU), Denmark (DNK), Spain (ESP), Finland (FIN), France (FRA), Great-Britain (GBR), Hong Kong (HKG), Ireland(IRL), Italy (ITA), Netherlands (NLD), Norway (NOR), Singapore (SGP), Sweden (SWE) and the United States (USA).The base assets for each factor portfolio are six individual factors (MKT, SMB, HML, UMD, BAB, QMJ). The calculationsare based on monthly U.S. dollar log returns from January 1994 to April 2015.

Figure 18 demonstrates the volatility reduction over time by means of rolling 36 months

strategy volatilities. Besides the persistent reduction of the volatilities, the cyclicality

in the volatility is substantially diminished compared to single factor investing. This

reduced cyclicality is more pronounced for local factor diversification than for international

diversification of the six factors. The local factor portfolios are highly effective in terms

of diversification and show limited increases in volatility during times of market stress.

144


D.2 Equal weighted portfolios

International diversification


EW SF Delta SE EW SF Delta SE

MKT 17.98 21.64 -3.66*** 0.30 36.18 29.65 6.53 5.37

SMB 5.69 12.38 -6.69*** 0.27

HML 6.84 14.19 -7.34*** 0.36 73.97 35.57 38.40** 16.77

UMD 13.21 20.05 -6.84*** 0.50 81.74 52.50 29.24* 14.95

BAB 9.29 18.20 -8.91*** 0.48 131.34 66.97 64.38*** 14.61

QMJ 7.68 14.68 -7.00*** 0.42 70.74 36.93 33.82** 13.13

Multi 3.93 6.45 -2.52*** 0.14 174.24 105.85 68.40*** 13.82

Table 10: Risk and Sharpe Ratio impact of international diversification for each factorThis table shows the annualized volatilities (in percent) and annualized Sharpe ratios (multiplied by 100) between globalportfolios for the equity factor specified in the first column. For the global portfolios, twenty countries are weighted accordingto equal weights (EW) and contrasted to synthetic factors (SF) with average returns and volatilities of the countries. ∆is the difference between these two portfolios and SE is the block-bootstrapped standard error of the difference. * standsfor two sided significance at the 10% level, ** for significance at the 5% level and *** for significance at the 1% level. Thefour statistics are presented for volatilities as well as Sharpe ratios. The Sharpe ratio for the size factor is omitted as it isnegative. The performance calculations are based on monthly U.S. dollar log returns from January 1994 to April 2015.

Using country equal weights instead of country risk parity to improve international

diversification of each of the six factors yields similar results. However, equal weighting

is a rather naive diversification scheme and does not make use of the information in the

covariance matrix. Thus, the risk reduction is less pronounced. The average volatility

is 10.1% compared to 9.3% in the case of country risk parity. This is also reflected in

the reduced significance of the Sharpe ratio improvement. Nevertheless, the bulk of the

volatility reduction compared to single country investing with an average volatility of

16.9% can also be captured by means of country equal weights.

145


Local factor diversification


EW SF Delta SE EW SF Delta SE

AUS 5.19 13.82 -8.62*** 0.57 187.72 70.64 117.08*** 18.87

AUT 7.54 18.60 -11.07*** 1.02 94.31 38.41 55.90*** 16.61

BEL 5.45 15.38 -9.93*** 0.84 108.92 38.65 70.27*** 15.25

CAN 7.10 16.00 -8.90*** 0.72 164.59 72.96 91.63*** 19.83

CHE 6.19 14.60 -8.41*** 0.67 104.20 44.11 60.09*** 13.74

DEU 6.34 15.87 -9.53*** 0.84 118.64 47.49 71.15*** 13.90

DNK 6.09 15.71 -9.61*** 0.68 109.51 42.70 66.82*** 15.49

ESP 5.89 16.26 -10.37*** 0.80 75.41 27.31 48.10*** 16.77

FIN 7.45 21.06 -13.62*** 0.98 100.96 35.75 65.21*** 13.23

FRA 6.25 14.92 -8.67*** 0.61 112.71 46.99 65.72*** 16.02

GBR 5.28 13.93 -8.65*** 0.92 108.84 41.28 67.56*** 21.41

HKG 7.24 20.36 -13.12*** 1.21 123.78 44.05 79.73*** 22.72

IRL 10.90 29.10 -18.20*** 2.07 47.42 17.84 29.58** 14.75

ITA 5.78 15.44 -9.66*** 0.57 102.65 38.34 64.31*** 15.80

JPN 4.92 12.99 -8.07*** 0.65 50.76 19.43 31.33* 16.12

NLD 5.96 16.18 -10.22*** 0.65 96.05 35.32 60.73*** 17.70

NOR 7.36 18.70 -11.34*** 0.81 111.19 43.78 67.42*** 15.91

SGP 5.84 17.31 -11.47*** 1.79 85.71 29.59 56.12*** 20.31

SWE 7.72 17.99 -10.28*** 0.82 100.07 42.84 57.22*** 14.21

USA 4.55 12.91 -8.36*** 0.82 114.84 40.44 74.40*** 19.34

Table 11: Risk and Sharpe Ratio impact of local factor diversification for each countryThis table shows the annualized volatilities (in percent) and annualized Sharpe ratios (multiplied by 100) between localfactor portfolios for the country specified in the first column. In the local portfolios six factors are equal weighted (EW) andcontrasted to synthetic factors with average returns and average volatilities. ∆ is the difference between these two portfoliosand SE is the block-bootstrapped standard error of the difference. * stands for two sided significance at the 10% level, **for significance at the 5% level and *** for significance at the 1% level. The four statistics are presented for volatilities aswell as Sharpe ratios. The performance calculations are based on monthly U.S. dollar log returns from January 1994 toApril 2015.

Using factor equal weights for local factor diversification instead of factor risk parity

also yields distinct diversification benefits compared to single factor investing. As before,

equal weights as a naive weighting scheme are not as efficient in reducing the volatility

in the local factor portfolio. The average volatility is 6.5% compared to 6.0% in the case

of factor risk parity. As for international diversification, the bulk of the risk reduction

compared to single factor investing with 16.9% can also be achieved when using factor

equal weights.

146

Curriculum Vitae

Ulrich Joachim Carl, born on December 3rd, 1985, in Kempten (Germany)

Education

09/2012-05/2016 Ph.D. in Economics and Finance, University of St. Gallen (Switzer-

land)

09/2010-05/2012 Master of Arts in Quantitative Economics and Finance,

University of St. Gallen (Switzerland)

08/2011-12/2011 Exchange Semester, University of Southern California, Los Angeles

(USA)

10/2006-02/2010 Bachelor of Arts in Economics, University of St. Gallen (Switzerland)

10/2006-02/2010 Bachelor of Arts in Business, University of St. Gallen (Switzerland)

08/2008-12/2008 Exchange Semester, Singapore Management University (Singapore)

09/1996-05/2005 Abitur, Allgau Gymasium, Kempten (Germany)

Professional Experience

since 05/2012 Portfolio Manager / Quantitative Strategist,

Finreon AG, St. Gallen (Switzerland)

01/2012-04/2012 Quantitative Analyst, Finreon AG, St. Gallen (Switzerland)

01/2011-07/2011 Student Worker, Finreon AG, St. Gallen (Switzerland)

03/2010-08/2010 Intern Market Risk Data, Commerzbank AG, Frankfurt (Germany)

149

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