Essays on Financial Markets

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Essays on Financial

Markets

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Abstract

My dissertation consists of three main chapters: one on portfolio optimiza-tion, and two on empirical asset pricing.

The first chapter employs monthly and daily returns of US stocks to eval-uate the out-of-sample performance of investment rules stemming from the mean-variance, Kelly and universal portfolio literature. We find that none of the strategies considered is significantly better or worse than all the others. Moreover, we show that the theoretical goal of the different strategies, be it either the maximization of the risk-adjusted portfolio return or the final wealth, is not related to their out-of-sample performance relative to the dif-ferent measures adopted. Conversely, agents should take into account the properties (return, risk and correlation) of the set of stocks selected for in-vestment when they are choosing the portfolio model to follow. Specifically, on the one hand if stocks are highly heterogeneous in terms of return and they have a low risk profile, the Kelly investor will get richer. On the other hand, if stocks have similar returns the minimum-variance and the univer-sal portfolio will increase performance. However, relative performance of the latter remains poor. Finally, although the performance of the mean-variance rule is not significantly influenced by portfolio characteristics, it performs no worse than the other strategies when stocks have heterogeneous returns. The second chapter adopts the latent variables approach by Hwang & Salmon (90) to analyse style herding in the value-growth and size dimensions of US domestic equity mutual funds. We document that mutual fund herding in styles is significant and persistent. Furthermore, the results show that mutual fund herding tends to increase after periods of high cumulative re-turns and market volatility. A higher sentiment is followed by an increase

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in mutual fund herding towards small stocks. Instead, mutual fund herd-ing in value stocks significantly decreases after an improvement in economic conditions. Mutual fund herding in styles causes overpricing in the market portfolio, and SMB and HML factors; this effect is stronger when the av-erage fund flows are higher. We also observe that mutual fund herding in styles in some cases affects the autocorrelation structure of factor returns. Finally, we find that mutual fund herding in styles impacts the average fund flows while it has no effect on the average performance of the industry. The third chapter incorporates a speculative bubble subject to a surviving and a collapsing regime into the present-value model by Binsbergen et al. (15), who pioneer the latent variables approach to estimate expected returns and expected dividend growth rates. To estimate this new high-dimensional model, we develop an efficient Markov chain Monte Carlo sampler to simu-late from the joint posterior distribution. We apply our present-value model to artificial as well as real-world datasets. Our setup is able to correctly identify 92.27% of all the bubble collapsing dates in the artificial datasets. And it never signals a bubble when there is none in the data generating process. We then show the existence of significant Markov-switching struc-tures in real-world stock price bubbles. The results indicate that dividend growth rates are highly predictable. Further, we find that in the surviving bubble regime, bubble variation accounts for most of the variation in the price-dividend ratio in the US, UK, Malaysia and Japan, and more than 35% of the price-dividend variation in Brazil. Moreover, bubble variation explains a large share of unexpected return variation in the surviving bubble regime.

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Acknowledgements

The completion of my PhD has been a long journey, for this I have to acknowledge all the people I met on the way, who helped me grow personally and professionally. I could not have succeeded without their support. First and foremost I would like to thank my PhD supervisor, Giulio Bottazzi, for his invaluable support and enthusiasm. His constant encouragement and constructive suggestions were determinant for the accomplishment of the work presented in this thesis.

I also would like to thank my mentor during my visit at the Vrije Univer-siteit in Amsterdam, Remco Zwinkels, for insightful discussions, constant guidance and his helpfulness.

Special thanks go to Joshua Chan, for sponsoring my visit at the University of Technology Sydney which has been a wonderful research and personal experience, and for his academic support and personal cheering.

I also would like to thank Fulvio Corsi, Alessio Moneta, Edoardo Otranto, and Mathijs van Dijk for taking the time of reading my thesis and for their invaluable comments and suggestions.

Another special thank goes to my dearest friends, Annarita and Angelo for their support and help during this journey. Likewise, I want express my grat-itude to all my current and former colleagues and friends: Giovanna Cap-poni, Andrea Giovannetti, Wenqian Huang, Li Le, Stefania Manetti, Lorenzo Napolitano, Matteo Pittiglio, Lilit Popoyan, Emanuele Russo, Pietro San-toleri, Fernando Sossordof, Jacopo Staccioli, Jonathan Taglialatela, Shihao Yu, Dieter Wang, and all the others.

This thesis is dedicated to my parents, Lorella and Maurizio, and to Francesco and Chiara for their unconditional love and support.

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List of Figures

2.1 Scatter plot of range and volatility - best strategy (Sharpe ratio) . . . . 42

2.2 Scatter plot of range and volatility - worst strategy (Sharpe ratio) . . . . 42

2.3 Scatter plot of range and volatility - best strategy (Final wealth) . . . . 43

2.4 Scatter plot of range and volatility - worst strategy (Final wealth) . . . 43

3.1 Herding towards Large Value stocks . . . 61

3.2 Herding towards Small Value stocks . . . 61

3.3 Herding towards Large Growth stocks . . . 62

3.4 Herding towards Small Growth stocks . . . 62

3.5 Predicted market returns . . . 67

3.6 Predicted SMB returns . . . 68

3.7 Predicted HML returns . . . 69

4.1 Artificial dataset 1 . . . 94

4.2 Price-dividend ratio by country . . . 95

4.3 Smoothed Surviving-probabilities and Log-price-dividend - Artificial datasets 1 to 3 . . . 99

4.4 Smoothed Surviving-probabilities and Log-price-dividend - Artificial datasets 4 to 5 . . . 100

4.5 Smoothed Surviving-probabilities and Log-price-dividend - United States 101 4.6 Smoothed Surviving-probabilities and Log-price-dividend - United King-dom . . . 101

4.7 Smoothed Surviving-probabilities and Log-price-dividend - Malaysia . . 102

4.8 Smoothed Surviving-probabilities and Log-price-dividend - Japan . . . . 102

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LIST OF FIGURES

4.10 Realized and Expected series - Artificial datasets 1 to 2. . . 105

4.11 Realized and Expected series - Artificial datasets 3 to 5. . . 106

4.12 Realized and Expected series - United States . . . 107

4.13 Realized and Expected series - United Kingdom . . . 108

4.14 Realized and Expected series - Malaysia . . . 108

4.15 Realized and Expected series - Japan . . . 108

4.16 Realized and Expected series - Brazil . . . 109

A.1 Excess Sharpe ratio and Range of Stock returns . . . 114

A.2 Excess Sharpe ratio and Portfolio Volatility . . . 114

A.3 Excess Final Wealth and Range of Stock returns . . . 115

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List of Tables

2.1 List of the Portfolio Strategies . . . 15

2.2 List of Datasets . . . 21

2.3 Sharpe Ratios - Monthly Returns . . . 26

2.4 Sortino Ratios - Monthly Returns . . . 27

2.5 Rachev Ratios - Monthly Returns . . . 28

2.6 Final Wealth - Monthly Returns . . . 29

2.7 Spearman Rank Correlation Matrix - Monthly Returns . . . 30

2.8 Sharpe Ratios - Daily Returns . . . 32

2.9 Sortino Ratios - Daily Returns . . . 33

2.10 Rachev Ratios - Daily Returns . . . 34

2.11 Final Wealth - Daily Returns . . . 35

2.12 Spearman Rank Correlation Matrix - Daily Returns . . . 35

2.13 Wilcoxon test for the difference in Sharpe ratios . . . 38

2.14 Wilcoxon test for the difference in Final Wealth . . . 39

2.15 Performance and portfolio characteristics . . . 44

3.1 Fund Characteristics . . . 54

3.2 Benchmark Portfolios . . . 54

3.3 Log cross-sectional standard deviation of the betas . . . 58

3.4 Maximum likelihood estimates . . . 59

3.5 Correlation of herding towards different styles . . . 63

3.6 Mutual fund herding and market conditions . . . 64

3.7 Mutual fund herding and fund flows . . . 71

3.8 Mutual fund herding and fund performance . . . 72

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LIST OF TABLES

3.10 Maximum likelihood estimates (Value-weighted CSSD) . . . 75

4.1 Priors and Starting values . . . 87

4.2 Parameter specification for artificial datasets . . . 93

4.3 Descriptive Statistics of the Log-Price-Dividend ratio . . . 95

4.4 Parameter estimates - Artificial datasets . . . 96

4.5 95% credible intervals - Artificial datasets . . . 96

4.6 Parameter estimates - Real-world datasets . . . 97

4.7 95% credible intervals - Real-world datasets . . . 97

4.8 Regression results - Artificial datasets . . . 104

4.9 Regression results - Real-world datasets . . . 105

4.10 Variance decomposition - Real-world datasets . . . 109

A.1 Performance and portfolio characteristics - OLS . . . 113

A.2 Average Weights 5 Industry Indices (monthly data) . . . 116

A.3 Average Weights 10 Industry Indices (monthly data) . . . 117

A.4 Average Weights Size & BTM Portfolio (monthly data) . . . 118

A.5 Average Weights Size & Momentum Portfolio (monthly data) . . . 119

A.6 Average Weights 5 Industry Indices (daily data) . . . 120

A.7 Average Weights 10 Industry Indices (daily data) . . . 121

A.8 Average Weights NASDAQ Mega Cap (daily data) . . . 122

A.9 Average Weights NYSE Mega Cap (daily data) . . . 123

A.10 Average Weights NASDAQ Large Cap (daily data) . . . 124

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1 Introduction

“Although academic models often assume that all investors are rational, this assumption is clearly an expository device not to be taken seriously.” Rubinstein (133).

There is now a vast literature that proposes evolutionary and behavioural approaches for the modeling of financial markets which aims at creating a plausible alternative to the conventional Walrasian equilibrium theory based on the hypothesis of full rationality of market participants.

Once we abandon the hypothesis of full rationality, market participants may have a whole range of patterns of behaviour depending on their individual psychology. In-vestors’ strategies may involve, for example, mimicking and rules of thumb based on experience. In this situation, also social interaction starts to play a major role in in-vestment decisions. Market participants might take into consideration the behaviour of other participants or compare the performance of their strategies to those adopted by others. They might also have objectives of a different nature: survival (especially in crisis environments), domination in a market segment, or fastest capital growth.

This work wants to contribute to the empirical investigation of these issues. My PhD thesis consists of three main chapters: one on portfolio selection, specifically on the performance evaluation of mean-variance versus growth optimal portfolio strategies, and two on asset pricing. In particular, we study two well-known phenomena which may induce prices to deviate from fundamentals, namely herding behaviour and periodically collapsing bubbles.

The first chapter, Relative Performance of Mean-Variance, Kelly and Universal Portfolio in the Equity Market, contributes to the literature on the evaluation of

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port-folio allocation rules. In particular, we provide evidence on the relative performance of strategies stemming from the mean-variance, Kelly and universal portfolio literature across ten different datasets. We evaluate the Markowitz model and some of its exten-sions developed to reduce the estimation error. We focus on the Bayesian estimator of the mean and variance-covariance matrix (94,95,97) of asset returns, and on methods that ignore the estimation of averages (91,109). However, given that the mean-variance model performs poorly out-of-sample (53, 58, 143), we study also the Kelly criterion which aims at maximizing the growth rate of wealth and it has been proved to be the global attractor of the market selection mechanism (18). Additionally, we analyse mod-els that do not rely on the distributional properties of asset returns. Among them, we focus on the Cover’s universal portfolio algorithm and the Exponential Gradient-update algorithm proposed by Helmbold et al. (84).

To implement the analysis we employ data for the US equity market, in particular we consider ten datasets differing in the frequency of stock returns which is either monthly or daily, the number of assets N = {4, 5, 6, 9, 10, 15}, and the composition of the datasets which can either be portfolios of stocks or individual stocks. Moreover, the datasets employed have been extensively used in the literature which make our results comparable with previous findings (see i.e.53,58,143, and others).

In the vast literature on portfolio optimization, our paper is the first to compare the performance of strategies stemming from the mean-variance, Kelly and universal portfo-lios literature. From a theoretical perspective the merits of the different approaches are clear: mean-variance portfolio should have the highest risk-adjusted return, the Kelly portfolio should maximize final wealth, and the universal portfolio should perform as good as the best constant rebalanced portfolio in hindsight. However, when these in-vestment rules are brought to real stock data, the theoretical prescriptions may not always be considered a useful guidance.

The main findings of the paper are the following. First, we show that there is not a strategy that always outperforms or underperforms the others. This evidence is con-firmed using four performance measures, two estimation methods of portfolio weights, and across several datasets. In particular, the mean-variance and the Kelly portfo-lios achieve almost the same performance. Furthermore, the minimum-variance strat-egy generally achieves a high performance, while the UP and EG algorithms perform poorly. We also find that it may be possible to significantly rank strategies within a

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certain dataset, which suggests the importance of portfolio characteristics for strategies performance. Second, we find that strategies rankings in terms of Sharpe ratio, Sortino ratio, Rachev ratio and Final Wealth is highly correlated. Hence, investors’ goals should not be a major concern in the choice of the portfolio rule. Indeed, the theoretical goal of the different strategies, be it either the maximization of the risk-adjusted portfolio return or the final wealth, is not related to their out-of-sample performance relative to the different measures adopted. Finally, the results indicate that the choice of the asset-allocation model should depend on the properties (return, risk and correlation) of the selected set of stocks. In particular, if stocks are highly heterogeneous in terms of return and they have a low risk profile, the Kelly criterion gains a higher final wealth and it performs well relative to the other strategies. In the case of stocks with similar returns, the minimum-variance rule performs better according to both the risk-adjusted portfolio return and final wealth. Similarly, the UP algorithm has a higher Sharpe ratio in portfolios composed by stocks with lower volatility, however its relative performance remains poor. Although the performance of the mean-variance rule is not significantly influenced by portfolio characteristics, it performs no worse than the other strategies when stocks have heterogeneous returns.

In short, this evidence brings two important implications: first, there is a need of further research on the construction of an optimal portfolio strategy since the issue of how investors should optimally split their wealth among different stocks is still open. Second, more effort should be devoted on the study of the relation between strategies performance and stock characteristics. In particular, the properties of the stocks should be exploited in the construction of the portfolio. Only some recent contributions have proposed to take into account the size, value and momentum anomalies for building the optimal portfolio (23,85). However, these papers focus only on the mean-variance rule and some of its extensions.

The second chapter, Exploring Style Herding by Mutual Funds, has been developed during my visiting at the Vrije Universiteit Amsterdam where I worked with Prof. Remco C.J. Zwinkels.

The chapter analyses style herding in the value-growth and size dimensions of US domestic equity mutual funds. Specifically, the added value of the paper lies in the focus on the imitation behaviour of mutual funds when facing the choice of the investment

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style. Indeed, style investing is particularly appealing for fund managers since the iden-tification of a fund within a certain asset class simplify problems of choice of individual investors, and it allows to easily evaluate the performance of the fund through compari-son with a benchmark specific to the style. Both individual investors and mutual funds allocate more at the style level than at the stock level (see e.g.,70,71), moreover funds tend to increase their exposure to styles which are expected to outperform. The paper explores herding in styles in the value-growth and size dimensions, while the literature has focused on industry styles (see e.g.29,39). The second contribution concerns the methodology used for detecting herding. We apply the approach proposed by Hwang & Salmon (90) which, to the best of our knowledge, has not been applied to style herding by mutual funds. The authors treat herding as an unobservable variable. Specifically, they argue that in the presence of herding, the factor loadings for individual funds will be biased and their cross-sectional dispersion smaller. Unlike the LSV and Sias herding measures, the Hwang & Salmon (90) approach allows to capture adjustments in the mutual fund investment in styles due to herding rather than fundamentals. An other relevant advantage of the Hwang & Salmon (90) approach is that it only requires re-turns and not holding data. Although rere-turns data may appear to be less informative than portfolio holdings, they are collected with a higher frequency hence allow to detect herding even if it occurs within shorter periods. Indeed, while previous papers estimate institutional herding quarterly (39,106,145, and others), we estimate it monthly. We believe that this is an element of strenght of the current work since the empirical in-vestigation of institutional herding has been severely affected by the impossibility to detect herding at a higher frequency (see e.g. 105). Furthermore, portfolio holdings information may be problematic when fund managers are following window dressing practices to disguise the actual portfolios held. Third, we focus on herding of open-end mutual fund, while the great majority of research in the field has analysed the broader category of institutional investors including banks, insurance companies, mutual funds (investment companies), independent investment advisors, and other institutions (see e.g.39,106,139). However, the heterogeneity of institutional investors may make more difficult the identification of herding. Furthermore, herding of mutual funds is particu-larly interesting since they are considered more prone to herd than other institutional investors due to reputational risk.

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We observe several interesting results. First, we find that mutual fund herding in styles is significant and persistent, independent from fundamentals (66). We also observe that mutual fund herding in small-cap stocks tends to be more persistent and less volatile than herding in large-cap stocks. This can be explained by the fact that information asymmetry is smaller for large firms than for small firms, causing investors in small firms to rely more on other sources of information. Second, we show that in general mutual fund herding tends to increase after periods of high market returns. Furthermore, uncertainty positively and significantly impacts mutual fund herding. In line with Celiker et al. (29), we find that months of positive sentiment tend to be followed by an increase in mutual fund herding towards small stocks. Only mutual fund herding in value stocks significantly decreases after an improvement of economic conditions, while herding towards the growth stocks is not affected. Similarly to Hwang & Salmon (90), who show that crises are turning points in herding behaviour of investors, we observe that ceteris paribus the level of mutual fund herding was lower during the burst of the dotcom bubble and the recent global financial crisis. Third, we report that mutual fund herding in styles destabilizes the return of the Fama-French market portfolio, Small Minus Big (SMB) and High Minus Low (HML) factor. This effect is stronger when the average flows to the mutual fund industry are higher. Furthermore, mutual fund herding in styles in some cases impact the autocorrelation structure of factor returns. Thus, the presence or absence of mutual fund herding in styles is a valuable information for investors since it can predict factor returns. Finally, mutual fund herding in styles generally affects average fund flows. Specifically, flows to the mutual fund industry tend to increase when there is herding in large growth stocks and it decreases when there is adverse herding in these styles. Differently, both mutual fund herding and adverse herding in large value and small growth stocks significantly decreases fund flows, while herding towards small value stocks does not have any effect. We also document that mutual fund herding in all the styles considered does not affect the average outperformance (underperformance) of the mutual fund industry as measured by the average alpha of the Carhart (28) four-factor model computed with daily data over monthly intervals for each fund in the sample.

These results have important implications for individual investors since the infor-mation concerning the herding behaviour of mutual fund managers may help to predict factor returns. Furthermore, we have shown that herding is more likely to occur after

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particular states of the market, hence the tendency of mutual fund to herd in styles is predictable.

The third chapter, Speculative Bubbles in Present-Value Models: A Bayesian Markov-Switching State Space Approach, has been developed during my visiting at the University of Technology Sydney where I worked with Prof. Joshua C.C. Chan.

The chapter contributes to the literature on the identification of speculative bubbles in stock prices. Specifically, we extend the state space specification of the present-value model in Binsbergen et al. (15) by allowing prices to deviate from fundamentals because of a latent rational bubble component subject to a surviving and a collapsing regime as in Al-Anaswah & Wilfling (1) and Lammerding et al. (107). Our framework allows to estimate expected returns and expected dividend growth rates, as well as to identify bubble’s collapse dates. To estimate this new high-dimensional model, we develop an efficient Markov Chain Monte Carlo (MCMC) sampler to simulate from the joint posterior distribution. Indeed, when the number of model parameters is quite large, standard maximum likelihood estimation tends to be numerically unstable and may result in estimates that are locally, but not globally, maximal. Differently, MCMC methods are numerically more robust. The key novel feature of our approach is that it builds upon the band and sparse matrix algorithms for state space models developed in Chan & Jeliazkov (32), Chan (31) which are shown to be more efficient than the conventional Kalman filter-based algorithms.

In line with previous research on the identification of speculative bubbles, we employ artificial as well as real-world datasets. The artificial bubble processes are defined in the sense of Evans (62), whereas the real-world datasets are drawn from Datastream. We consider 20 years of monthly data (November 1997 - October 2017) for the dividend yield and the price index. We use data for five countries: United States, United Kingdom, Malaysia, Japan and Brazil. We have decided to conduct the analysis on this set of countries because of their economic relevance and the severe bubble episodes experienced in the past (104). The advantage of the artificial datasets with respect to real-world data is that the bubbles’ collapse dates are known, hence they allow us to assess the accuracy of our bubble-detection method. Instead, for the real-world datasets we rely on what economic historians have classified as bubble periods (104).

The major findings of the chapter are the following: first, we show that our approach is able to correctly identify 92.27% of all the bubble collapsing dates in the artificial

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datasets, moreover it never signals a bubble which has no counterpart in the price process. These results represent an improvement with respect to the methodology discussed in Al-Anaswah & Wilfling (1). Also, we find that our framework is able to identify most of the bubble periods classified as such by Kindleberger & Aliber (104). Second, consistent with Al-Anaswah & Wilfling (1) and Lammerding et al. (107), we document the existence of statistically significant Markov-switching in the data-generating process of real-world stock price bubbles. Third, our setup is also able to predict dividend growth rates as well as returns with R2 values ranging from 74.07% to 78.89% for dividend growth rates and 4.04% and 20.71% for returns in the artificial datasets. In the real-world datasets, the R2 values for dividend growth rates are quite high, the highest value is recorded for the US where it is equal to 70.49% while the lowest value is registered for Brazil where it is equal to 49.10%. However, the R2 values for returns are less than 1% with the exception of Brazil where it is above 3%. Finally, we document that in the surviving bubble regime, most of the variation in price-dividend ratio is related to the bubble variation. Specifically, bubble variation accounts for more than 50% of the price-dividend variation in all the countries under study with the exception of Brazil where it accounts for about 36%. Further, bubble variation explains also a large share of unexpected return variation in the surviving bubble regime.

In sum, our setup allows to model jointly expected dividend growth rates, expected returns and the bubble component of stock prices. As such it may improve conventional methods for the detection of real-time stock-price bubbles allowing an early detection of future bubbles. Moreover, this methodology allows for hypothesis testing of some features of expected dividend growth rates and expected returns such as their average and/or persistence.

Concluding, the thesis brings evidence on the role played by investors’ psychology and social interaction on what we observe in financial markets. We show that portfolio strategies do not perform as they should according to their theoretical prescriptions. Mutual fund managers may get involved in herding behaviour when there is uncertainty about the future prospects of the market or because of optimism towards a particular style which lead them to undervalue its risk. Further, stock prices may incorporate a speculative bubble which is generated by factors not related to fundamentals and it is driven by self-fulfilling expectations. Thus, future research should advocate more efforts

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on the evolutionary and behavioral approaches to finance as we need to improve our understanding of financial markets that, so far, has been built on totally unrealistic assumptions about the behavior of people acting in them.

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2 Relative Performance of

Mean-Variance, Kelly and Universal

Portfolio in the Equity Market

2.1 Introduction

Whenever an individual wants to invest his money in the equity market he has to decide how optimally allocate his wealth among the available stocks. Although contributions in the portfolio selection literature attempted to solve the investor’s allocation problem, there is still no agreement on the optimal portfolio rule. Given this context, the article aims at providing evidence on the relative performance of investment rules stemming from the mean-variance, Kelly and universal portfolio literature. Moreover, we attempt to deliver some criteria that may guide agents in their investment choice.

The seminal work by Markowitz (115), focusing on investors who care only about returns and risk, suggests to construct portfolios that either minimize risk for a given expected return, or maximize expected return for a given risk level; in other words portfolios that maximize the Sharpe ratio of the investment. However, some recent studies have found that the Markowitz model performs poorly out-of-sample, moreover it does not consistently outperform the naive (1/N) or equal-weights rule (53,58,143). In particular, Chopra & Ziemba (41), Kallberg & Ziemba (100) demonstrate that the mean-variance allocations are sensitive even to small variations in means and covariances of asset returns. In fact, over the last few years, many researchers proposed alternative

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2.1 Introduction

methods able to reduce the severity of the estimation error. These studies mainly focus on portfolio rules that constrain short selling (41, 73), on the Bayesian approach to estimation error which consists in shrinking the asset return toward a common target (94,95,97), and finally on methods that ignore the estimation of average asset returns (91,109).

Although the Markowitz model is appealing for its simplicity and it is also commonly used by financial practitioners, its generalization to multiple periods is still debatable (24,114,119). Moreover, investors generally find risk-adjusted returns difficult to digest. Indeed, in their periodic financial statements investors tend to look at whether or not their invested capital grows rather than the Sharpe ratio of their investments. Similarly, fund management companies tend to summarize performance with the mean compound return of the portfolio. Thus, maximization of capital growth seems a plausible goal, especially for long horizon investments. Breiman (25) has demonstrated that investing proportionally to the probability of the states maximizes the growth rate of wealth. This rule is known as the Kelly criterion (102, 108), and it is equivalent to maximizing the expected logarithm of relative returns. The Kelly criterion has the appealing property of exhibiting the highest growth rate of wealth in any population of portfolio rules (18), moreover under general conditions it maximizes the median of terminal wealth (61). Furthermore, Bottazzi & Dindo (22) have recently shown that when the Kelly rule is present in the market, prices asymptotically reflect the best available information on dividends. Otherwise, when prices are endogenous, persistent asset mispricing may be observed and prices do not convey anymore all the available information.

Notwithstanding this, the Kelly criterion is built on the assumption of known proba-bility distribution of stock returns, therefore its implementation yields significant issues. In order to overcome these problems, Cover (46) proposes an algorithm for portfolio se-lection, known as universal portfolio, able to perform as good as the best constant rebalanced portfolio based on perfect knowledge of the future without making any as-sumption on the assets return dynamics1. In particular, given the observed past returns, the universal portfolio strategy dictates to invest a share of wealth in asset n equal to the weighted mean of the share invested by all possible constant rebalanced portfolios, where the weights are determined by portfolio past performance. However, the Cover’s

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See Cover & Thomas (48), Chapter 16 for a formal discussion of the application of information theory to portfolio theory.

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2.1 Introduction

algorithm is exponential in the number of assets, thus its application on large portfolios requires big computational efforts. Kalai & Vempala (99) suggest an efficient implemen-tation of the Cover’s algorithm based on non-uniform random walks that are rapidly mixing. Alternatively, Helmbold et al. (84) discuss an on-line investment algorithm which has properties similar to those of the Cover’s universal portfolio, and a required computational effort linear in the number of stocks.

Our work contributes to the existing empirical literature on optimal portfolio strate-gies in several ways: first, previous research mainly focus on either extensions of the mean-variance approach in order to reduce the estimation error, on the comparison of the mean-variance and Kelly criterion, or on the development of efficient implementa-tion methods of the Cover’s algorithm. To the best of our knowledge, there are not works that compare the performance of strategies stemming from the mean-variance, Kelly and universal portfolios literature. Although from a theoretical perspective the mean-variance portfolio should have the highest risk-adjusted return, the Kelly portfolio should maximize final wealth, and the universal portfolio should perform as good as the best constant rebalanced portfolio in hindsight; when we apply these investment rules to real stock data, the theoretical prescriptions may not be verified. In light of this, we aim at filling this gap in the literature by providing evidence on the relative out-of-sample performance of the aforementioned strategies. Second, since the portfolio rules analysed have different objectives, we assess whether the investors’ goal should be taken into account in the choice of the portfolio-allocation model. In other words, we study how strategies ranking changes according to different performance measures which rep-resent different objectives, such as maximization of the risk-adjusted portfolio return or final wealth. Third, we investigate whether the characteristics of the stocks included into portfolio (return, risk and correlation) influences strategies performance, indeed the optimal strategy may vary according to the set of stocks chosen for investment.

To implement our work we employ data for the US equity markets, in particular we consider ten datasets differing in the frequency of stock returns which is either monthly or daily, the number of assets N = {4, 5, 6, 9, 10, 15}, and the composition of the datasets which can either be portfolios of stocks or individual stocks. Moreover, the datasets employed have been extensively used in the literature which make our results comparable with previous findings (see i.e. 53,58,143, and others). Given the recent interest on the performance of trivial investment strategies relative to portfolio

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2.1 Introduction

optimization (53,58,143), we compare all the portfolio models with the naive or equal-weights strategy. Indeed, this rule does not require any estimation of the moments of asset returns, moreover agents tend to adopt simple heuristics such as fixed portfolio rules in case of uncertainty - see for example Benartzi & Thaler (12) and Gigerenzer & Brighton (75). Also, we compute the in-sample mean-variance strategy and the best constant rebalanced portfolio in order to evaluate the performance of these strategies in case of perfect knowledge of the future. In order to keep the analysis simple we do not take into account transaction costs, nevertheless their presence in the market may affect the relative performance of portfolio strategies (51,113).

The paper shows that there is not a strategy that always outperforms or under-performs the others. This evidence is confirmed using four performance measures, two estimation methods of portfolio weights, and across several datasets. In particular, the mean-variance and the Kelly portfolios achieve almost the same performance. Further-more, the minimum-variance strategy generally achieves a high performance, while the universal portfolio and exponential gradient algorithms perform poorly. We also find that it may be possible to significantly rank strategies within a certain dataset, which suggests the importance of portfolio characteristics for strategies performance.

Furthermore, we find that strategies rankings in terms of Sharpe ratio, Sortino ratio, Rachev ratio and Final Wealth is highly correlated. Hence, investors’ goals should not influence the choice of the optimal portfolio rule. Indeed, the theoretical goal of the different strategies, be it either the maximization of the risk-adjusted portfolio return or the final wealth, is not related to their out-of-sample performance relative to the different measures adopted.

Finally, the results indicate that the choice of the asset-allocation model should depend on the properties (return, risk and correlation) of the selected set of stocks. In particular, if stocks are highly heterogeneous in terms of return and they have a low risk profile, the Kelly criterion gains a higher final wealth and it performs well relative to the other strategies. In the case of stocks with similar returns, the minimum-variance rule performs better according to both the risk-adjusted portfolio return and final wealth. Similarly, the universal portfolio algorithm has a higher Sharpe ratio in portfolios composed by stocks with lower volatility, however its relative performance remains poor. Although the performance of the mean-variance rule is not significantly

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2.2 Related Literature

influenced by portfolio characteristics, it performs no worse than the other strategies when stocks have heterogeneous returns.

The paper is structured as follows: next section reviews some empirical contributions in the field, Section 2.3 introduces the portfolio models under study. In Section 2.4, we describe the datasets, then Section 2.5treats the methodology adopted to compute the optimal portfolio weights, and the performance measures. Section2.6discusses the results on the out-of-sample performance of the strategies, and Section 2.6.4 presents the relation between stock characteristics and strategies performance. Finally, Section 2.7concludes.

2.2 Related Literature

Theoretical financial papers assume that investors know the true parameters of the model. However, economic agents generally do not dispose of information on the model parameters, for this reason the performance of an investment rule depends on how good the estimates of the unknown parameters are.

A number of studies, mainly focusing on the US, argue that the naive (1/N) in-vestment rule is no worst than the out-of-sample mean-variance model (53, 58, 143). Indeed, the estimation error of asset return moments is so severe to undermine the performance of optimal portfolio strategies. In light of this, a large literature explicitly deals with methods able to reduce the estimation error. In a recent paper Pantaleo et al. (123), concentrating on a number of estimators of the second moment of asset returns, find that the choice of the covariance matrix estimators should depend on the ratio between estimation period T and number of stocks N into the portfolio, on the presence or absence of short selling, and on the performance metric considered.

Concerning the Kelly rule, studies have mainly tackled the comparison with the mean-variance strategy both in terms of performance and portfolio composition1. Hunt (88, 89), using Australian and US equity data, shows that Kelly portfolios produce growth rates that are up to twice those of the naive (1/N), minimum-variance and 15% drift portfolios, however also volatility is almost double. More recently, Estrada (60) points out that the mean-variance strategy may be more plausible than the Kelly cri-terion for investors relatively more risk-averse, with a shorter investment horizon, and

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2.3 Description of Portfolio Strategies

uncertain about the length of their holding period. Concerning the differences in terms of composition of the mean-variance and Kelly portfolios the evidence is mixed: Grauer (78), using a sample of 20 Dow stocks and 20 NYSE portfolios, finds that the mean-variance and the Kelly criterion rarely select the same asset mix; in contrast Pulley (128) with 10 samples of US stocks shows that the optimal portfolios are remarkably similar. Györfi et al. (80) introduces a small computational complexity alternative to the Kelly rule, called semi-log-optimal strategy. In a later paper Ottucsák & Vajda (122) describe the relationship between the mean-variance and the semi-log-optimal portfolio strategies, specifically when investors have a risk aversion parameter of one the two strategies coincide. The empirical results in Györfi et al. (80) show that the proposed semi-log-optimal and the Kelly strategies have essentially the same performance mea-sured on past NYSE daily data, confirming previous evidence that the mean-variance and Kelly rule perform essentially the same.

Regarding algorithms that do not rely on the distributional properties of asset re-turns, Cover (46) studies the performance of the universal portfolio on daily returns of four portfolios composed of two NYSE stocks during 22 years. He documents that the universal portfolio outperforms the best stock in terms of wealth. Helmbold et al. (84) test the EG-update algorithm on NYSE stocks and find that it outperforms the best single stock as well as the universal portfolio. Moreover, it earns about 90% of the wealth achieved by the best constant rebalanced portfolio in hindsight.

2.3 Description of Portfolio Strategies

This section briefly introduce the portfolio strategies we compare in this paper (see the list in Table 2.1). It must be stressed that we only take into consideration strategies that constrain short selling. Indeed, DeMiguel et al. (53) show that the short selling constraint on the mean-variance portfolio weights is a form of shrinkage on the expected returns, while in the case of the minimum-variance portfolio it is equivalent to shrinking the elements of the variance-covariance matrix (91). Thus, constraining short selling should reduce estimation error and improve performance as highlighted in DeMiguel et al. (53). Moreover, the UP and EG algorithms have been developed in a framework which does not allow for short selling. Hence, motivated by previous results and for

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Table 2.1: List of the Portfolio Strategies

Sect. Model Abbreviation

2.3.1 1/N Rebalancing rebalancing

2.3.1 1/N Buy & Hold B&H

2.3.2 Mean-Variance (in-sample) mv-in

2.3.6 Best Constant Rebalanced Portfolio BCRP

2.3.2 Mean-Variance with Sample Mean and Covariance Matrix

mv

2.3.2 Mean-Variance with Bayes-Stein Estimates mv-bs

2.3.3 Minimum-Variance min

2.3.4 Generalized-Minimum-Variance g-min

2.3.5 Kelly Criterion Kelly

2.3.6 Universal Portfolio UP

2.3.7 Exponential Gradient EG

This table lists the portfolio strategies considered. The first column reports the Section in which the strategy is described, the second one the name of the portfolio model, and the last column reports the abbreviation used to refer to the strategies.

the sake of comparability we restrain from considering the strategies performance in the presence of short selling.

Each portfolio strategy determines the N -vector of portfolio weights w using the observation of the past M realized returns of the N securities. Let rt and Rt with

t = 1, . . . , M denote, respectively, the N -vector of net and gross returns observed at time t, while rf_t and Rf_t stand for the contemporaneous net and gross return of the risk-free asset. 1_N stands for a N -vector of ones.

2.3.1 Naive Portfolios

Gigerenzer & Brighton (75) argue that in case of uncertainty, agents tend to adopt simple heuristics such as fixed portfolio rules. In particular, the naive strategy consists in holding an equal fraction of wealth in each of the stocks into the portfolio. Thus, this rule does not involve any optimization or estimation. We consider both a rebalancing 1/N rule in which investors at each time step trade the assets in order to maintain constant the portfolio weights, and a buy and hold 1/N rule where investors initially split equally their wealth among the assets and then stop trading until the end of the investment is reached. Indeed, Barber & Odean (8) claim that, on average, the most active traders earn lower returns than investors who trade the least.

2.3.2 Mean-Variance Portfolio

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choices either minimizing risk for a given expected return, or maximizing expected return for a given risk level.

The mean-variance strategy takes into account traders whose preferences are fully described by the mean and variance of a given portfolio. Specifically, at each time instant, investors choose the portfolio weights w that maximize their mean-variance utility max w w 0_{µ −}γ 2 w0Σw , s.t. w ≥ 0, 1N0w = 1 , (2.1)

where µ is the N -vector of expected returns, Σ is the N × N covariance matrix of returns and γ is the risk aversion parameter which balances the relative importance of reduced variance and increased returns. Notice that the choice of adopting the variance as a measure of risk is rooted in the assumption of normality of asset returns. To implement this model, we simply maximize Eq. (2.1) in which we replace the mean µ and covariance matrix Σ of asset returns with either their sample counterparts or their Bayes-Stein estimates. In our empirical investigations we set the risk aversion parameter γ = 11.

2.3.2.1 Sample Mean and Covariance Matrix

The sample mean and covariance matrix of asset returns are computed according to the usual formulas ˆ µ = 1 M M X t=1 rt and Σ =ˆ 1 M − 1 M X t=1 (rt− ˆµ)(rt− ˆµ)0 .

The reason to consider the sample covariance matrix relies in the fact that that when short selling is forbidden and the number of time observations is greater than the num-ber of assets included into portfolio (M > N ), alternative estimators are unable to outperform the sample covariance matrix in terms of realized risk (see, 123).

1

See Ottucsák & Vajda (122) for a description of the relationship between mean-variance and semi-log optimal portfolio introduced in Györfi et al. (80).

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2.3.2.2 Bayes-Stein Shrinkage Estimator

A number of studies, including Dickinson (57), Frankfurter et al. (69), Jobson & Korkie (95), has shown that parameter uncertainty may lead to suboptimal portfolio choices. In particular, in the context of multivariate problems (N > 2), such as problems of optimal portfolio selection, the sample mean is not the best estimator of the population mean (141).

In order to take into account the estimation error, Jorion (97) develops a Bayes-Stein estimator of the population mean,

ˆ

µ_BS = (1 − λ) ˆµ + λ ˆµmin1N, (2.2)

with a shrinkage target equal to the return of the (unconstrained) minimum variance portfolio computed with the rescaled covariance matrix ˜Σ = (T − 1) ˆΣ/(T − N − 2)

ˆ µmin =

1NΣ˜−1µˆ0

10_NΣ˜−1₁ N

and where λ represents the weights given to the shrinkage target

λ = N + 2

N + 2 + M ( ˆµ − ˆµmin1N)0Σ˜−1( ˆµ − ˆµmin1N)

.

The associated Bayes-Stein estimator of the population covariance matrix reads ˆ ΣBS = 1 + 1 M + z ˆ Σ + z M (M + 1 + z) 1N10N 10_NΣ˜−11N , (2.3)

in which z = M λ/(1 − λ). Consider that, differently from Jorion (98), we use the Bayes-Stein estimator also for the covariance matrix and we apply both estimators in a constrained optimization problem that forbids short positions.

2.3.3 Minimum-Variance Portfolio

Expected stock returns can be hardly estimated accurately (120). Specifically, estima-tion errors lead to a suboptimal choice of portfolio weights (41,98). Thus, Jagannathan & Ma (91), Ledoit & Wolf (109) propose to ignore average asset returns and to minimize

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the variance of returns.

min

w w

0_Σw,

s.t. w ≥ 0, 10_Nw = 1.

(2.4)

This strategy is optimal in mean-variance sense under the assumption that the average return of all considered assets is the same. The minimum-variance model is implemented using the sample estimates of the covariance matrix of returns.

2.3.4 Generalized-Minimum-Variance Portfolio

Given that expected returns are difficult to estimate, we follow DeMiguel et al. (53) and we consider also a generalization of the minimum-variance portfolio which imposes an additional constraint to problem (2.4), namely w ≥ p1_n, with p ∈ [0, 1/N ]. This strategy is again a sort of shrinkage procedure similar in spirit to the one adopted in the Bayes-Stein estimator and it can be seen as a combination of the minimum-variance and the naive strategy: setting p = 0 we obtain the minimum-variance portfolio, whereas for p = 1/N we get the naive portfolio. Following DeMiguel et al. (53), in the imple-mentation of this strategy we set p = 1/(2N ).

2.3.5 Kelly Criterion

The seminal papers by Kelly Jr (102) and Latané (108) started the literature on growth optimal portfolio also known as the Kelly portfolio1.

In gambling the Kelly criterion is a formula used to determine the optimal wealth allocation in a sequence of bets. In particular, Kelly Jr (102) shows that investing a fraction of wealth proportional to the probability of winning of each bet maximizes the expected growth rate of the logarithm of wealth. In our framework one needs to solve the following maximization problem

max w E ln w0R , s.t. wt≥ 0, 1N0wt= 1 , (2.5)

where R is the N -dimensional vector of returns. The unique requirements for the validity of this criterion are the reinvestment of winnings and the ability to change the amount

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of money bet in different gambles. However, the main drawback of this strategy is that it is built on the assumption of known probability distribution of stock returns, whereas it is generally unknown. We replace it with the empirical joint distribution of realized returns. Specifically, the most recent M periods of realized returns are recorded, then it is assumed that the M joint realizations have 1/M probability to occur in the next period. Thus, we find the portfolio weights w by maximizing the corresponding sample analog1 max w 1 M M X t=1 ln w0Rt , s.t. w ≥ 0, 10_Nw = 1 . (2.6)

In this way estimates are obtained on a moving basis and there is no loss of information.

2.3.6 Universal Portfolio

Cover (46) introduces an algorithm for portfolio selection able to asymptotically outper-form the best stock in the market. In particular, Cover’s universal portfolio algorithm aims at building a portfolio which performs as good as the best constant rebalanced portfolio (BCRP), that is the constant rebalanced portfolio that maximizes the final wealth. Of course the BCRP is only known in hindsight, that is once the performance of all the assets have been observed. Hence, in practice, the Universal Portfolio rep-resents the best feasible portfolio. Notice that the good asymptotic performance on the Universal Portfolio are not based on any particular assumption about the data generating process. In particular, if stock returns are i.i.d. according to the unknown distribution F , the universal portfolio achieves the same asymptotic wealth growth rate of the growth optimal portfolio computed using F .

Given the observed past returns, the universal portfolio strategy dictates to invest a share of wealth in asset n equal to the weighted mean of the share invested by all possible constant rebalanced portfolios, where the weights are determined by portfolio past performance. Formally,

w = R ∆xW(x)dx R ∆W(x)dx (2.7)

1_{The paper uses the sequential quadratic problem method to solve the optimization problem in}

(2.6). The method produces a higher sum of the optimal objective function value as compared to the BFGS, simplex algorithms and the method of moving asymptotes.

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where ∆ is the N -simplex and W(x) is the wealth achieved by portfolio x

W(x) =

M

Y

t=1

x0Rt. (2.8)

The number of assets considered in our portfolios is small enough that we can effectively compute the integrals in (2.7) using adaptive Monte Carlo techniques.

2.3.7 Exponential Gradient Algorithm

Helmbold et al. (84) discuss an investment algorithm which gains almost the same wealth as the BCRP. Similarly to Cover (46), this algorithm does not require any assumption on the distribution properties of stock returns. Moreover, the computational effort required by the algorithm is linear in the number of stocks. According to the proposed rule, at the beginning of each period t, new portfolio weights wt+1 are obtained as a

function of past weights w_t and lastly observed returns R_tas

wt+1= argmax w η ln(w0Rt) − N X n=1 wnln wn wn,t , s.t. w ≥ 0, 1N0w = 1. (2.9)

The first term of the objective function is proportional to the logarithmic relative wealth achieved, under the assumption that the last realized returns R_t will be observed also in the next period. The second term represents a penalty which is proportional the the information divergence of the new weights with respect to the old ones. The parameter η controls the relative importance of the two terms and capture the speed at which new information is incorporated into portfolio weights. Since the objective function is difficult to maximize, as it depends non-linearly on wt+1, Helmbold et al. propose

to substitute the first term with its first-order Taylor approximation around w = wt.

Doing so, they finally obtain a portfolio updating rule which they call Exponential Gradient wn,t= wn,texp ηRn,t−1 w0_tRt−1 PN j=1wj,t−1exp ηRj,t−1 w0_tRt−1 , 1 ≤ n ≤ N . (2.10)

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2.4 Data Description

The final weights for t = M are those adopted by the strategy, w = wM.

In the empirical analysis the choice of the learning rate parameter η is crucial for assessing the performance of this strategy. We explore three values for the learning rate, specifically 0.1, 1 and 31.

2.4 Data Description

Table 2.2: List of Datasets

Dataset Source N Time Period Frequency

1 Five Industry Indices Ken French’s website 5 01/Jan/1953-31/Dec/2015 monthly 2 Ten Industry Indices Ken French’s website 10 01/Jan/1953-31/Dec/2015 monthly 3 Size & BTM Indices Ken French’s website 6 01/Jan/1953-31/Dec/2015 monthly 4 Size & Momentum Indices Ken French’s website 6 01/Jan/1953-31/Dec/2015 monthly 5 Five Industry Indices Ken French’s website 5 20/Oct/2004-31/Dec/2015 daily 6 Ten Industry Indices Ken French’s website 10 20/Oct/2004-31/Dec/2015 daily 7 NASDAQ Mega Cap Yahoo Finance website 4 20/Oct/2004-31/Dec/2015 daily 8 NYSE Mega Cap Yahoo Finance website 9 20/Oct/2004-31/Dec/2015 daily 9 NASDAQ Large Cap Yahoo Finance website 15 20/Oct/2004-31/Dec/2015 daily 10 NYSE Large Cap Yahoo Finance website 15 20/Oct/2004-31/Dec/2015 daily This table lists the datasets considered in our analysis, the data sources, the number of assets N in each dataset, the time span and the returns frequency. Notice that in the case of daily data only the last 756 observations are used to compute the performance measures, previous observations are employed in the regression analysis.

We apply the investment strategies described in the previous section to a selection of datasets built to differ in the return frequency, size and composition (see the list in Table 2.2).

The first group of six datasets are retrieved from the Kenneth French’s Web site2. The Industry Indices (datasets number 1, 2, 5, and 6) consist of either monthly or daily returns on North American industry portfolios. Portfolios are constructed by assigning each NYSE, AMEX, and NASDAQ stock to a specific industry at the end of June of year t, based on its four-digit SIC code of principal activity at that time3. The Size & Book-to-Market Indices, and Size & Momentum Indices (datasets number 3 and 4) consider all NYSE, AMEX, and NASDAQ stocks. At the end of each June the portfolios are

1

Since the results for the learning rate equal to 0.1 are almost identical to the ones of the naive strategies as the portfolio weights do not move away from the initial portfolio, we do not report them. However, the results can be made available upon request.

2

For a more detailed description of the procedure used to construct the portfolios seehttp://mba. tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

3

The 5 Industry Indices includes the Consumer, Manufacturing, High Tech, Health and Other in-dustry portfolios, while the 10 Inin-dustry Indices contains Consumer Non-Durables, Consumer Durables, Manufacturing, Energy, High Tech, Telecommunication, Shops, Health, Utilities and Other industry portfolios.

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2.4 Data Description

built as the intersections of two portfolios formed on market equity and three portfolios formed on the ratio of book equity to market equity for the case of the Size & BTM Indices, and three portfolios formed on the monthly prior (2-12) return for the Size & Momentum Indices. The size breakpoint for year t is the median NYSE market equity at the end of June of year t. The breakpoints for the ratio of book equity to market equity and the monthly prior (2-12) return are the 30th and 70th NYSE percentiles.

The components of the above datasets are portfolios of stocks; because of lower levels of idiosyncratic volatility, we expect that the naive 1/N rule will perform better in these datasets than in portfolios of individual stocks.

The other four datasets are composed of equity prices obtained from the Yahoo Fi-nance website1. Stocks are assigned to these portfolios according to the market in which they are quoted (NASDAQ or NYSE), and their market capitalization (Mega Cap or Large Cap). Stocks are classified as Mega Cap if they have a market capitalization greater than $ 200 billion, and as Large Cap in case of a market capitalization greater than $ 10 billion. We select all Mega Cap stocks, and the first 15 Large cap stocks with data available for the time period under study2. Returns at time t are computed as the ratio of the increment/decrement of the adjusted closing price between time t and time t − 1, and the adjusted closing price at time t − 1. The Yahoo Finance adjusted closing price is the closing price after adjustments for all applicable splits and dividend distri-butions. Data is adjusted using appropriate split and dividend multipliers, according to Center for Research in Security Prices (CRSP) standards. We have 756 monthly and

1

Concerning the reliability of Yahoo financial data, Flanegin et al. (68) show that the adjusted returns retrieved from Yahoo Finance are not significantly different from that of CRSP.

2

The NASDAQ Mega Cap portfolio includes: Apple Inc. (AAPL), Alphabet Inc. (GOOGL), Microsoft Corporation (MSFT), and Amazon.com Inc. (AMZN).

The NYSE Mega Cap portfolio includes: Exxon Mobil Corporation (XOM), General Electric Com-pany (GE), Johnson & Johnson (JNJ), Wells Fargo & ComCom-pany (WFC), China Mobile (Hong Kong) Ltd. (CHL), AT& T Inc. (T), Procter & Gamble Company (PG), J P Morgan Chase & Co (JPM), and Wal-Mart Stores, Inc. (WMT).

The NASDAQ Large Cap portfolio includes: Intel Corporation (INTC), Comcast Corporation (CM-CSA), Gilead Sciences Inc. (GILD), Cisco Systems Inc. (CSCO), Amgen Inc. (AMGN), Starbucks Corporation (SBUX), Walgreens Boots Alliance Inc. (WBA), Celgene Corporation (CELG), Mondelez International Inc. (MDLZ), Costco Wholesale Corporation (COST), QUALCOMM Inc. (QCOM), Biogen Inc. (BIIB), The Priceline Group Inc. (PCLN), Texas Instruments Inc. (TXN), Twenty-First Century Fox Inc. (FOX).

The NYSE Large Cap portfolio includes: Verizon Communications Inc. (VZ), Toyota Motor Corp Ltd Ord (TM), Pfizer Inc. (PFE), Coca-Cola Company (KO), Novartis AG (NVS), Chevron Corporation (CVX), Home Depot Inc. (HD), Walt Disney Company (DIS), Oracle Corporation (ORCL), Novo Nordisk A/S (NVO), Pepsico Inc. (PEP), Bank of America Corporation (BAC), Merck & Company Inc. (MRK), HSBC Holdings plc (HSBC), and Medtronic plc (MDT).

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2.5 Methodology

2820 daily observations of stock returns. In addition, we use the 1-month Treasury Bill return drawn from Kenneth French’s Web site as the risk-free asset return.

2.5 Methodology

Given that investors do not dispose of a perfect knowledge of the future, the actual potential gains of a strategy can be assessed only by means of its out-of-sample perfor-mance. We explore two different approaches to obtain out-of-sample statistics for the considered strategy. In the “rolling-window” approach we consider a fixed time window of K time steps to estimate the parameters necessary to implement the strategies. At each time t we use the previous K realized returns, that is the returns in period (t−K, t). The estimated parameters are then employed to compute the portfolio weights, and the weights are used to compute the ex-post portfolio return in t + 1. Instead, in the “increasing-window” approach, at each time t the parameters of the strategy, and the portfolio weights, are computed using all the previously observed returns starting from an initial period t₀, that is the returns in period (t₀, t). Thus, in the rolling-window case the sample length M of the strategies described in Section 2.3 is set equal to K, while in the increasing-window case it is initially t₀ and then it increases as more recent time periods are considered. The two approaches differ with respect to their underlying assumptions. If the data generating process is stationary, the increasing-window ap-proach should lead to better results as the sample on which the strategies are estimated becomes progressively larger. On the other hand, if the data generating process changes in time as some structural breaks induce a different behaviour in the time series, the rolling-window approach should be preferred.

In the rolling-window approach we consider two time window lengths, K = 240 and K = 120. To maintain comparability of the results between the two choices and with the increasing-window we always start to collect statistics from t = 2401. For the same reason, when using daily data, we restrict the analysis to the last 756 available returns. Thus, in all three cases considered, the two rolling-window with different lengths and the increasing-window, we end up with a series of T = 516 out-of-sample returns generated by each of the portfolio strategies considered. Then, we use this series of out-of-sample portfolio returns rp_i for the different portfolios i to compute four

1

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2.5 Methodology

measures of performance: the Sharpe ratio (SR), Sortino ratio (S), Rachev ratio (R) and Final Wealth (W).

The SR of strategy i is calculated as the ratio of the mean and the standard deviation of excess portfolio returns over the risk-free asset

SRi =

E[r_ip− rf_]

pV[rp i − rf]

. (2.11)

The SR (137) measures the risk-adjusted portfolio return, higher is the ratio and higher is the performance. However, this ratio does not distinguish between upside and down-side volatility. Hence, in case of non-Gaussian distribution of returns the SR is a poor performance metric. In fact, it overestimates risk for positively skewed return distri-butions, conversely it understates risk for negatively skewed return distributions. To address this issue, Sortino & Hopelain (140) propose to use downside deviation rather than standard deviation as the measure of risk in the SR. In other words, they suggest to consider only those returns below a certain benchmark as risky. In formula the S ratio is defined as Si= E[r_ip− rf_] s 1 T PT t=1 min{0, r_i,tp − rf_t} 2 , (2.12)

where the numerator is the portfolio average excess return over the risk-free asset, and the denominator is the downside deviation computed taking the riskless-return as the minimum acceptable return at each time period.

In the case of heavy-tail distributed returns, Biglova et al. (13) suggest an alternative reward-to-risk measure known as the Rachev ratio (R) defined as

Ri =

ETLα(rf − r_ip)

ETL_β(r_ip− rf₎, (2.13)

where ETLxstands for Expected Tail Loss above x, that is the expected value computed

on the upper x-quantile of the loss distribution, that is the lower x-quantile of the distribution of portfolio returns. In words, the R ratio measures the magnitude of the expected tail return relative to the expected tail loss1_.

Since some investors may be interested in maximizing their wealth rather than the risk-adjusted portfolio return, and the Kelly portfolio is specifically constructed to this

1

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2.6 Results

end; we calculate also the final wealth generated by each portfolio strategy given an initial unitary investment

Wi = T Y t=1 (1 + rp_i,t). (2.14)

2.6 Results

This section presents the results obtained implementing the asset-allocation models described in section 2.3 across all the datasets listed in Table 2.2. Tables report the performance of the best strategy in bold characters, while italic characters denote strate-gies with the worst performance. Last column shows the average rank of each strategy. In Panel A, we report the results achieved by the naive strategies based on the sym-metric Buy & Hold and rebalanced portfolio. They can be used as a first benchmark to which compare the performance of the other, more sophisticated, strategies. Panel B contains a second benchmark, that is the performance of the in-sample mean vari-ance and best constant rebalvari-anced portfolios (BCRP). Consider that these strategies are estimated ex-post, after all the returns realization are known. Thus they cannot be actually implemented, as they would require a perfect knowledge of the future. In Panel C and D, we present the out-of-sample results for the rolling-window cases, K = 240 and K = 120, respectively. Panel E reports the out-of-sample performance computed with the increasing-window. The average portfolio weights of the different strategies in the different cases are reported in AppendixA.2.

2.6.1 Strategies Performance with monthly data

Let us start analyzing the performance of the strategies when monthly data are consid-ered. Being indeces computed over several assets, these are time series with moderate volatility and also moderate dispersion of growth rates. Table2.3reports the results for Sharpe ratio, Table 2.4for Sortino ratio, Table 2.5for Rachev ratio and Table 2.6 the final wealth achieved by the different strategies over the period of observation. Looking at Panel B of Tables 2.3,2.4,2.5, and2.6, it is interesting to note that with respect to all the measures considered, the performance of the two in-sample strategies is basically identical. This consideration reinforces the idea that the relative efficiency of the differ-ent strategies can be reliably assessed only by means of their out-of-sample performance.

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2.6 Results

Table 2.3: Sharpe Ratios - Monthly Returns

Industry Industry Size & Size & Average Indices Indices BTM Momentum Rank N=5 N=10 N=6 N=6

Panel A: Naive Strategies

rebalancing 0.1281 0.1338 0.1373 0.1184 -B&H 0.1244 0.1308 0.1504 0.1556 -Panel B: In-Sample Strategies

mv-in 0.1267 0.1267 0.1765 0.1879 -BCRP 0.1267 0.1279 0.1765 0.1879 -Panel C: Rolling Window K=240

mv 0.0840 0.0771 0.1754 0.1879 4.50 mv-bs 0.1014 0.0989 0.1754 0.1877 3.75 min 0.1497 0.1682 0.1399 0.1188 3.75 g-min 0.1411 0.1589 0.1390 0.1153 5.00 Kelly 0.0836 0.0770 0.1754 0.1879 5.00 UP 0.1268 0.1330 0.1396 0.1265 4.75 EG(1) 0.1197 0.1221 0.1532 0.1629 4.50 EG(3) 0.1028 0.0956 0.1678 0.1827 4.75 Panel D: Rolling Window K=120

mv 0.0777 0.0781 0.1712 0.1859 4.75 mv-bs 0.0973 0.0988 0.1618 0.1891 4.00 min 0.1527 0.1607 0.1432 0.1222 3.75 g-min 0.1438 0.1514 0.1434 0.1190 4.50 Kelly 0.0779 0.0780 0.1723 0.1853 4.75 UP 0.1269 0.1330 0.1381 0.1221 5.25 EG(1) 0.1212 0.1226 0.1445 0.1423 4.50 EG(3) 0.1077 0.0989 0.1566 0.1647 4.50 Panel E: Increasing Window

mv 0.1265 0.1044 0.1756 0.1879 4.00 mv-bs 0.1127 0.1131 0.1748 0.1879 4.50 min 0.1400 0.1560 0.1302 0.1081 4.50 g-min 0.1331 0.1476 0.1384 0.1143 4.50 Kelly 0.1266 0.1047 0.1755 0.1879 4.25 UP 0.1283 0.1334 0.1407 0.1338 4.50 EG(1) 0.1279 0.1270 0.1577 0.1746 4.50 EG(3) 0.1216 0.1088 0.1713 0.1860 5.25 Num. obs. 756

Sharpe ratio of rebalancing and buy & hold 1/N strategies, of mean-variance strategy and best constant rebal-anced portfolio (in-sample), and of strategies described in Section2.3(out-of-sample). Bold characters denote strategies with the highest performance, while italic characters denote strategies with the worst performance.

The second consideration is that, even if the mean-variance and BCRP portfolios are built with the idea to be optimal with respect to different objective functions, specifi-cally a weighted combination of mean and variance for the first and the final wealth for the second, their ultimate performance is basically the same. On theoretical basis, one would expect to find that the BCRP portfolio has better performance in terms of final wealth, while the mean-variance should be superior with respect to Sharpe or similar ratios. In practice, this is not the case and we will see that a similar consideration can also be extended to the out-of-sample performance of the strategies in Section2.3, reported in Panel C, D and E of Tables 2.3,2.4,2.5,and 2.6.