Can Optimized Portfolios Beat 1/N?

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2 Can Opimized Porfolios Bea 1/N? This disseraion is presened in par fulfillmen of he requiremen for he compleion of an MSc in Economics in he Deparmen of Economics, Universiy of Konsanz, and an MSc in Economics and Economerics in he School of Economics, Universiy of Noingham. The work is he sole responsibiliy of he candidae. By: Valerius Disch Period of Compleion: Word Coun: 1s Assessor: 2nd Assessor: Konsanz, Augus 21, 2018 April 21, Augus 21, ,777 / 15,000 Professor Marcel Fischer, Universiy of Konsanz Professor Parick Marsh, Universiy of Noingham

3 Absrac ii Absrac This analysis conducs an ou-of-sample horse race beween he naive diversificaion approach and 15 opimized porfolio sraegies across four US and wo European equiy daa ses. No a single opimized sraegy achieves o consisenly ouperform he naive diversificaion benchmark in a saisically significan manner based on a oal of six performance evaluaion crieria. However, unconsrained sraegies relaed o he sample-based mean-variance and minimum-variance sraegy exhibi superior performances in wo US daa ses. Timing sraegies aain consisenly good resuls for US indusry daa ses. Performance behavior appears o be sensiive o respecive daa ses, ye independen of performance evaluaion crieria.

4 Conens iii Conens Absrac Lis of Figures Lis of Tables Lis of Abbreviaions ii v v vi 1. Inroducion 1 2. Lieraure Review 2 3. Daa 4 4. Opimized Porfolio Sraegies and 1/N Mean-Variance Framework Esimaion Risk Approaches o Figh Esimaion Risk Porfolio Sraegies Equally-Weighed Sample-Based Mean-Variance Minimum-Variance Value-Weighed MacKinlay-Pásor Bayes-Sein Porfolio Weigh Consrains Mean-Variance and Minimum-Variance Equally-Weighed and Minimum-Variance Equally-Weighed and Mean-Variance Equally-Weighed and Mean-Variance and Minimum-Variance Volailiy Timing Reward-o-Risk Timing Mehodology Esimaion Procedure Performance Evaluaion Crieria Sharpe Raio Sorino Raio Omega Raio Calmar Raio Reurn on Value-a-Risk Cerainy Equivalen Reurn Empirical Resuls and Discussion Robusness Checks Esimaion Window Risk-Aversion Parameer Tuning Parameer

5 Conens iv 8. Conclusion 48 Appendices 49 Appendix A. Deails of he Daa Ses 49 A.1. US MKT, PF6, PF25, IND10, and IND A.2. European MKT, PF6, and PF Appendix B. Mean-Variance Opimizaion in Excess Reurns 51 Appendix C. Minimum-Variance Opimizaion 52 Appendix D. Robusness Checks Tables 53 References 64

6 Lis of Figures v Lis of Figures 3.1. Boxplos for he US and European MKT Series and PF6 Porfolios Monhly Cumulaive MKT Series for he US and Europe Hisogram, Normal Densiy, and Kernel Densiy Esimae for he US and European MKT Series Quanile-Quanile Plo for he US and European MKT Series Effecs of Naive Diversificaion on he Porfolio s Sandard Deviaion.. 19 Lis of Tables 3.1. Overview Daa Ses Descripive Saisics for he US and European MKT Series and PF6 Porfolios Correlaion Table for he US and European MKT Series and PF6 Porfolios Normaliy and Saionariy Tess for he US and European MKT Series Overview Porfolio Sraegies Sharpe Raios Average Minimum and Maximum Sharpe Raio Porfolio Weighs Sorino and Omega Raios Calmar Raios and Reurns on Value-a-Risk Cerainy Equivalen Reurns Performance Evaluaion Crieria Rank Correlaions Daa Ses Rank Correlaions D.1. Sharpe Raios, Esimaion Window M = D.2. Cerainy Equivalen Reurns, Esimaion Window M = D.3. Sharpe Raios, Esimaion Window M = D.4. Cerainy Equivalen Reurns, Esimaion Window M = D.5. Sharpe Raios, Expanding Window D.6. Cerainy Equivalen Reurns, Expanding Window D.7. Sharpe Raios, Risk-Aversion Parameer γ = D.8. Cerainy Equivalen Reurns, Risk-Aversion Parameer γ = D.9. Sharpe Raios, Risk-Aversion Parameer γ = D.10.Cerainy Equivalen Reurns, Risk-Aversion Parameer γ = D.11.Sharpe Raios and Cerainy Equivalen Reurns, Tuning Parameer η = 2 and η =

7 Lis of Abbreviaions vi Lis of Abbreviaions AMEX = American Sock Exchange AR(p) = Auoregressive Model of Order p BS = Bayes-Sein BSsc = Bayes-Sein Shor Sale Consrain CAPM = Capial Asse Pricing Model CE = Cerainy Equivalen Reurn CR = Calmar Raio DD = Drawdown e al. = e alia, and ohers EU = Europe EW = Equally-Weighed EW/MinV = Equally-Weighed and Minimum-Variance EW/MV = Equally-Weighed and Mean-Variance EW/MV/MinV = Equally-Weighed and Mean-Variance and Minimum-Variance HAC = Heeroskedasiciy and Auocorrelaion i.e. = id es, ha is o say IS = In-Sample JK = Jobson and Korkie LPM = Lower Parial Momens LW = Ledoi and Wolf MAR = Minimum Accepable Reurn MaxDD = Maximum Drawdown MinV = Minimum-Variance MinVsc = Minimum-Variance Shor Sale Consrain MKT = Excess Reurn on he Marke ML = Maximum Likelihood MP = MacKinlay-Pásor MSCI = Morgan Sanley Capial Inernaional MV = Mean-Variance MVlsc = Mean-Variance Long-Shor Sale Consrain MV/MinV = Mean-Variance and Minimum-Variance MVsc = Mean-Variance Shor Sale Consrain NASDAQ = Naional Associaion of Securiies Dealers Auomaed Quoaions NYSE = New York Sock Exchange OR = Omega Raio RoVaR = Reurn on Value-a-Risk RRT = Reward-o-Risk Timing SoR = Sorino Raio SR = Sharpe Raio US = Unied Saes USA = Unied Saes of America VaR = Value-a-Risk VT = Volailiy Timing VW = Value-Weighed

8 1. Inroducion 1 1. Inroducion In 1952, Harry Markowiz published his work Porfolio Selecion and arguably se he foundaion of wha is nowadays known as Modern Porfolio Theory. Regarding a se of risky asses, Markowiz (1952) raionalizes an invesmen behavior ha favors expeced reurn and dismisses risk measured in erms of variance of reurn. Since reurns ypically end o increase wih risk, his consiues a fundamenal rade-off which, however, can be addressed by a raional and risk-averse invesor via fixing a level of variance of reurn while maximizing expeced reurn. Alhough heoreically appealing, here is no guaranee ha following his proposed invesmen behavior indeed resuls in desired oucomes when esed in a real daa environmen. A naural choice of a desired oucome is he ouperformance of an opimized porfolio sraegy relaive o he naive diversificaion or simply 1/N approach. The laer describes an invesmen schedule which invess equal shares of wealh in all available risky asses of which he number is assumed o be N. In his spiri, his analysis conducs an ou-of-sample horse race beween he naive diversificaion approach and 15 opimized porfolio sraegies based on he mean-variance framework. By doing so, several conribuions o he exising relaed porfolio opimizaion lieraure are made. Firs, similar sudies such as DeMiguel e al. (2009b) are updaed by respecing well-esablished mean-variance model exensions as well as raher newly proposed approaches resuling in a comprehensive selecion of opimized porfolio sraegies. Second, he daa ses are no exclusively resriced o he Unied Saes (US) bu also accoun for European equiy daa and hence allow for a comparison beween US and European specific resuls. Third, he scope of performance evaluaion crieria is exended and conains an addiional number of four reward-o-risk raios alongside he sandard Sharpe raio and cerainy equivalen reurn. Besides, he es saisic inroduced by Ledoi and Wolf (2008) o decide on he pairwise difference beween he Sharpe raios of wo differen sraegies is adjused so as o be also valid in case of cerainy equivalen reurns. This faciliaes he idenificaion of saisically significan resuls and explicily accouns for nonnormally disribued reurn daa. The resuls of his analysis are beneficial no only for privae invesors bu also for professional praciioners who srive for a more favorable reurn-o-risk rade-off and have an ineres in answering he quesion of wheher opimized porfolios can bea 1/N. The main finding of his analysis can be summarized in he fac ha no a single opimized porfolio sraegy achieves o consisenly ouperform he naive diversificaion benchmark in a saisically significan manner. However, performances across differen daa ses are very heerogeneous. In case of wo US equiy daa ses formed on bivariae sors of marke equiy and he book-o-marke equiy raio, unconsrained sraegies relaed o he sample-based mean-variance and minimum-variance sraegy exhibi superior performances. In conras, iming sraegies only aain consisenly good resuls in he conex of US equiy daa classified according o US indusries. The European daa ses are generally shor of any ouperforming opimized sraegy. Evenually, he performance

9 2. Lieraure Review 2 behavior appears o be very sensiive o respecive daa ses. The relaive performance rankings of he sraegies under consideraion, however, are independen of he performance evaluaion crieria, i.e. he Sharpe raio is in general an adequae choice alhough relying on he assumpion of normally disribued reurn daa. This analysis is organized as follows. Secion 2 reviews relevan lieraure and presens imporan empirical resuls. Secion 3 describes he daa. Secion 4 inroduces he opimized porfolio sraegies and Secion 5 liss he performance evaluaion crieria. Resuls are presened and discussed in Secion 6. Secion 7 conains robusness checks and Secion 8 concludes. 2. Lieraure Review The pas few years of porfolio opimizaion research can be characerized by a series of publicaions conducing horse races beween differen compeing opimized porfolio sraegies. In his conex, he work by DeMiguel e al. (2009b) is of paricular ineres and also serves as he basis on which his analysis is build. DeMiguel e al. (2009b) compare he performance of 12 differen opimized porfolio sraegies relaive o he naive diversificaion benchmark wih respec o he ou-of-sample Sharpe raio, cerainy equivalen reurn, and urnover. The underlying daa are composed of monhly excess sock reurns for mainly US equiies and cover he period from July 1963 o November By applying a rolling-window esimaion approach, DeMiguel e al. (2009b) come o he somewha surprising conclusion ha none of he sraegies under consideraion achieves o consisenly ouperform he naive diversificaion approach. This resul is valid hroughou all performance crieria. DeMiguel e al. (2009b) aribue he bad performance of he opimized sraegies o he characerisic occurrence of esimaion errors when esimaing expeced reurns and variance-covariance marices in he process of building hese opimal sraegies. Tu and Zhou (2011) subsequenly inroduce four novel opimized porfolio sraegies ha are opimized combinaions of already exising approaches. In paricular, he naive diversificaion approach is combined wih he sample-based mean-variance sraegy as well as wih he sraegies suggesed by Jorion (1986), MacKinlay and Pásor (2000), and Kan and Zhou (2007), respecively. The general seing and esimaion mehodology is very similar o DeMiguel e al. (2009b) which ensures a cerain degree of comparabiliy. Tu and Zhou (2011) show evidence of an ouperformance of he opimally combined porfolio sraegies relaive o heir uncombined counerpars in erms of Sharpe raios and cerainy equivalen reurns. Some of he newly proposed opimized porfolio sraegies even achieve o ouperform naive diversificaion. This is especially rue in case of large sample sizes. These findings are promising and raionalize he use of a subse of heir proposed models also in his analysis.

10 2. Lieraure Review 3 Pflug e al. (2012) focus on he wo levels of uncerainy prevailing in he conex of porfolio opimizaion. Firs, he acual realizaions of he asse reurns are uncerain and second, and even more criical, he daa generaing process iself is no known exacly. The higher he degree of his model uncerainy he more he heoreically opimal porfolio sraegy resembles he naive diversificaion approach. On he basis of differen risk measures, Pflug e al. (2012) subsequenly demonsrae ha naive diversificaion is even opimal in case of a risk-averse invesor who cares abou boh levels of uncerainy in combinaion wih a very limied or even nonexisen knowledge of he daa generaing process. As a resul, he empirical success of he naive diversificaion approach canno only be aribued o general esimaion inaccuracies when esimaing momens of an assumed daa generaing disribuion bu also o wrong inferences abou he daa generaing process in he firs place. Following his argumen, focus should be eiher pu on he improvemen of he esimaion precision of already exising models or he inroducion of new opimized porfolio sraegies including a jusified modeling of he daa generaing process. An ineresing remark is made by Kirby and Osdiek (2012) who argue ha he design of he comparisons beween he differen porfolio sraegies ypically favors naive diversificaion. The general assumpion of a represenaive mean-variance invesor leads o opimized porfolio sraegies ha srive for high expeced reurns. This, however, can resul in exreme porfolio weigh posiions ha ake advanage of even minor asse reurn differences. In case of an imprecise esimaion of hese reurn raes, he resuling porfolio can exhibi poor ou-of-sample performance. Hence, Kirby and Osdiek (2012) propose he idea of imposing a arge reurn on he opimized porfolio sraegies equal o he ouof-sample reurn obained by he naive diversificaion approach. This indeed improves he performance of he opimized porfolio sraegies relaive o naive diversificaion and dampens he porfolio weighs o less exreme posiions. In addiion, Kirby and Osdiek (2012) inroduce a volailiy and reward-o-risk iming sraegy which achieve superior resuls relaive o he naive diversificaion benchmark for US equiy daa covering he period from July 1963 o December The performance of hese wo newly inroduced sraegies is encouraging as i suggess ha opimized porfolio sraegies can acually ouperform naive diversificaion irrespecive of he reliance on sufficienly long sample sizes. An exension of he work by DeMiguel e al. (2009b) o a broader se of inernaional asse classes including socks, bonds, and commodiies is given by Jacobs e al. (2014). For he ime period from February 1973 o December 2008, a oal of eleven mean-variance based opimized porfolio sraegies are compared o he naive diversificaion approach. As a firs resul, Jacobs e al. (2014) confirm previous findings ha hardly any opimized porfolio sraegy consisenly ouperforms naive diversificaion wih respec o equiy daa. Second, hese resuls ransmi o he oher asse classes under consideraion for which naive diversificaion also seems o be he mos successful approach in achieving

11 3. Daa 4 high ou-of-sample Sharpe raios. In he same spiri, Bessler e al. (2017) examine he period from January 1993 o December 2011 wih respec o an inernaional porfolio consising of socks, bonds, and commodiy indices and compare he naive diversificaion approach o he opimized sample-based mean-variance, minimum-variance, and Bayes-Sein sraegy. In accordance wih DeMiguel e al. (2009b), he Sharpe raios belonging o he opimized sraegies do no significanly ouperform naive diversificaion. Addiionally, he Omega raio as well as he maximum drawdown are included as furher reward-o-risk measures and according o hese crieria he obained resuls are suggesive of a superior performance regarding he opimized porfolio sraegies. This analysis aims a coninuing his series of porfolio opimizaion research by conducing a horse beween opimized porfolio sraegies ha incorporaes he main insighs of previous empirical work wih respec o boh he selecion of opimized sraegies as well as performance evaluaion crieria. 3. Daa This analysis is based on a oal of six daa ses conaining monhly excess reurns over a risk-free rae defined as he one-monh US Treasury bill rae. There are four samples based on US daa and wo samples referring o European daa. The reurns of he US daa ses are a combinaion of New York Sock Exchange (NYSE), American Sock Exchange (AMEX), and Naional Associaion of Securiies Dealers Auomaed Quoaions (NASDAQ) socks and are value-weighed porfolios eiher formed based on bivariae sors on marke equiy and he book-o-marke equiy raio or ordered by indusry affiliaion. The US daa cover he ime period from January 1970 o December The choice of he ime period is deermined by daa availabiliy. The US indusry daa are only complee as of he year The reurn series for he oher US samples originally include more observaions bu are adjused for he same ime period as he US indusry daa o ensure comparabiliy. Independen of he acual daa availabiliy, all daa ses are balanced such ha he ime period always sars in January and ends in December. This is done o guaranee an equal disribuion of monhs o miigae calendar effecs such as he well-known January effec. As a resul, he US porfolios each conain a oal of 576 monhs of excess reurn daa represening 48 years of excess reurns, respecively. The reurns of he European daa ses are based on daa provided by Morgan Sanley Capial Inernaional (MSCI) and Bloomberg and are value-weighed porfolios formed based on bivariae sors on marke equiy and he book-o-marke equiy raio. The ime period ranges from January 1991 o December 2017 and hus includes 324 observaions alogeher corresponding o 27 years of monhly excess reurn daa. All daa belong o he single-asse class of socks and are drawn from Kenneh R. French s daa library 1. The 1 hp://mba.uck.darmouh.edu/pages/faculy/ken.french/daa_library.hml

12 3. Daa 5 Table 3.1: Overview Daa Ses. This able liss a oal of six daa ses ha are included in his analysis all of which conain monhly excess reurns over a risk-free rae defined as he one-monh US Treasury bill rae. The firs four samples refer o US daa while he remaining wo samples are based on European daa. The abbreviaions are inroduced o refer o he specific daa ses in he ex. The variable N indicaes he oal number of available risky asses included in each daa se. The ime period indicaes he duraion of he daa collecion and he las column of he able liss he underlying number of observaions included in each daa se. The daa are obained from Kenneh R. French s websie hp://mba.uck.darmouh. edu/pages/faculy/ken.french/daa_library.hml. A more deailed descripion of he daa and he process of creaing he differen porfolios can be found in Appendix A. # Daa Se Abbreviaion N Time Period Obs. Unied Saes of America (USA) 1 Six US porfolios formed on marke equiy and he book-o-marke equiy raio PF6 USA 6 01/ / Tweny-five US porfolios formed on marke PF25 USA 25 01/ / equiy and he book-o-marke equiy raio 3 Ten US indusry porfolios IND10 USA 10 01/ / Fory-eigh US indusry porfolios IND48 USA 48 01/ / Europe (EU) 5 Six European porfolios formed on marke equiy and he book-o-marke equiy raio 6 Tweny-five European porfolios formed on marke equiy and he book-o-marke equiy raio PF6 EU 6 01/ / PF25 EU 25 01/ / daa ses employed in his analysis are very similar o he samples ypically used in he relaed empirical porfolio opimizaion lieraure and herefore a high degree of comparabiliy wih respec o he obained resuls is guaraneed. Table 3.1 presens an overview of he daa considered in his analysis. The remainder of his secion is dedicaed o he descripion of PF6 USA and PF6 EU as well as heir respecive value-weighed excess marke reurn (MKT) series o obain a deeper undersanding of he inheren characerisics of he daa ses underlying his analysis. In case of he PF6 porfolios, he oal sock daa are divided ino a small and a big group wih respec o a specific level of marke equiy of which each group is again subdivided ino eiher a low, medium, or high group wih respec o a specific level of book-o-marke equiy raio. A more deailed descripion of he daa and he process of creaing he differen porfolios can be found in Appendix A. Table 3.2 exhibis descripive saisics for he US and European MKT series and PF6 porfolios, respecively. Several ineresing facs appear when examining Table 3.2. Focusing on he MKT series of he USA and Europe, a firs observaion refers o he mean of he US MKT series which is subsanially higher relaive o ha of Europe. In addiion, he European marke is characerized by a higher sandard deviaion and more exreme values as indicaed by he corresponding minimum and maximum values. These facs summarize in a European MKT Sharpe raio of approximaely 0.12 which is lower han he respecive value of 0.17 for he USA. Based on his reward-o-risk raio, he descripive saisics are suggesive of an ouperformance of he European marke by he US marke. A second observaion is

13 3. Daa 6 Table 3.2: Descripive Saisics for he US and European MKT Series and PF6 Porfolios. This able liss he MKT series and he six porfolios for he US and Europe, i.e. PF6 USA and PF6 EU, respecively. The six porfolios are consruced by means of bivariae sors of he oal sock daa on marke equiy wih wo groups, small and big, and on book-o-marke equiy raio wih hree groups, low, medium, and high. The US porfolios are runcaed so as o mach he ime period of he European porfolio. Hence, he daa span from January 1991 o December 2017 conaining 324 observaions of monhly excess reurns represening 27 years. This able displays he mean, sandard deviaion, sample-bias correced skewness, sample-bias correced kurosis, minimum value, maximum value, and Sharpe raio of each reurn series. USA Small Big Saisics MKT Low Medium High Low Medium High Mean Sandard Deviaion Skewness Kurosis Minimum Maximum Sharpe Raio Europe Small Big Saisics MKT Low Medium High Low Medium High Mean Sandard Deviaion Skewness Kurosis Minimum Maximum Sharpe Raio ha hroughou all porfolios he minimum and maximum values are relaively exreme in relaion o he corresponding means and sandard deviaions. This behavior is indicaive of ouliers. Also, all porfolios show a negaive skewness as well as a level of kurosis ha exceeds he reference level of a sandard normal disribuion of hree. These observaions cas a firs doub on he general assumpion prevailing in he empirical lieraure of having normally disribued asse reurns. A graphical represenaion of some disribuional characerisics of he US and European MKT series and PF6 porfolios can be obained by boxplos as shown in Fig The lenghs of he boxes for he European sample appear o be more homogenous relaive o he US sample suggesing ha he European porfolios share more similar disribuional characerisics. Again, he number of ouliers in all boxplos speak in favor of reurn daa which are no normally disribued. Table 3.3 yields correlaions of he US MKT series and PF6 porfolios. All correlaions are highly posiive and even he lowes correlaion coefficien beween he US B-M and S-L porfolios sill amouns o In addiion, he correlaions beween he US and European porfolios are highly posiive ranging from 0.62 for he S-H porfolio o 0.79 for he MKT series. Facing an invesmen universe ha only consiss of highly correlaed asses limis he poenial benefis of diversificaion and migh resul in lower levels of

14 3. Daa 7 Boxplos for US MKT and PF6 Boxplos for European MKT and PF6 MKT MKT MKT Series and PF6 Porfolios S-L S-M S-H B-L B-M MKT Series and PF6 Porfolios S-L S-M S-H B-L B-M B-H B-H Monhly Excess Reurns (a) Boxplos for he US MKT series and PF6 porfolios. The US porfolios are runcaed so as o mach he ime period of he European porfolios. Hence, he daa spans from January 1991 o December 2017 conaining 324 observaions of monhly excess reurns represening 27 years Monhly Excess Reurns (b) Boxplos for he European MKT series and PF6 porfolios. The daa spans from January 1991 o December 2017 conaining 324 observaions of monhly excess reurns represening 27 years. Figure 3.1: Boxplos for he US and European MKT Series and PF6 Porfolios. This figure shows boxplos for he US and European MKT series and PF6 porfolios, respecively. The verical line in each box denoes he median of he disribuion. The lef and righ bounds of he box indicae he 25h and 75h perceniles, respecively. The whiskers cover he range of he observaions ha are sill wihin he 1.5 inerquarile range of he lower and upper quarile. Ouliers are denoed by a do. reward-o-risk raios due o higher levels of risk. Fig. 3.2 displays he monhly cumulaive MKT series for he USA and Europe. The US sample sars in January 1970, whereas he European sample only sars in January 1991 due o limied daa availabiliy. Boh series are upward sloping and show a clear posiive rend. Moreover, boh reurn paerns resemble each oher, however, he US cumulaive reurns are higher relaive o he European reurns for mos of he underlying ime period. Several major drops in he observed reurn paerns can be aribued o corresponding economic crises such as he do-com bubble in he beginning of he 21s cenury as well Table 3.3: Correlaion Table for he US and European MKT Series and PF6 Porfolios. This able liss correlaions for he US and European MKT series and PF6 porfolios, respecively. The US porfolios are runcaed so as o mach he ime period of he European porfolios. Hence, he daa span from January 1991 o December 2017 conaining 324 observaions of monhly excess reurns represening 27 years. This able conains several differen correlaions. In fac, columns 2 (MKT) o 7 (B-H) span wo correlaion marices a he same ime. The lower lef riangular marix conaining plain numbers represens he correlaion marix of he US sample. The upper righ riangular marix conaining ialic numbers represens he correlaion marix of he European sample. The las column USA/EU yields he correlaions of he porfolios beween he USA and Europe. Facors MKT S-L S-M S-H B-L B-M B-H USA / EU MKT S-L S-M S-H B-L B-M B-H

15 3. Daa Cumulaive US and European MKT Series MKT USA MKT Europe Monhly Cumulaive MKT Series Dae Figure 3.2: Monhly Cumulaive MKT Series for he US and Europe. This figure shows he monhly cumulaive MKT series for he US and Europe. The US sample covers he period from January 1970 o December 2017 conaining 576 observaions of monhly excess reurns represening 48 years. The European sample covers he period from January 1991 o December 2017 conaining 324 observaions of monhly excess reurns represening 27 years. as he global financial crisis in 2007 and Since he acual disribuion of he underlying reurn daa criically influences he choice of appropriae saisical procedures, several differen ess are performed. Tess for normaliy wih respec o a specific sample disribuion can rely eiher on a graphical approach or more formally uilizing acual saisical ess. Fig. 3.3 depics he hisogram, a fied normal disribuion, and a kernel densiy esimae for he US and he European MKT series. A visual inspecion of he fied normal disribuion relaive o he kernel densiy esimae does no yield a huge discrepancy neiher for he US nor for he European sample. Only he European reurn disribuion exhibis slighly fa ails, especially on he lef-hand side of he kernel densiy esimae. These fa ails are characerisic of re Hisogram, Normal Densiy, and Kernel Densiy Esimae Hisogram Normal Densiy Kernel Densiy Esimae for he US MKT Series Hisogram, Normal Densiy, and Kernel Densiy Esimae Hisogram Normal Densiy Kernel Densiy Esimae for he European MKT Series 8 8 Densiy 6 Densiy Monhly Excess Reurns (a) Hisogram, normal densiy, and kernel densiy esimae for he US MKT series based on a oal of 576 monhly excess reurn observaions from January 1970 o December The kernel funcion is he Epanechnikov wih a bandwidh of Monhly Excess Reurns (b) Hisogram, normal densiy, and kernel densiy esimae for he European MKT series based on a oal of 324 monhly excess reurn observaions from January 1991 o December The kernel funcion is he Epanechnikov wih a bandwidh of Figure 3.3: Hisogram, Normal Densiy, and Kernel Densiy Esimae for he US and European MKT Series This figure shows he hisogram, normal densiy, and kernel densiy esimae for he US and European MKT series. The kernel funcion is he Epanechnikov wih unbounded suppor.

16 3. Daa Quanile-Quanile Plo for he US MKT Series 0.15 Quanile-Quanile Plo for he European MKT Series US MKT Quaniles European MKT Quaniles Theoreical Quaniles of he Normal Disribuion (a) Quanile-quanile plo for he US MKT series covering he period from January 1970 o December 2017 conaining 576 observaions of monhly excess reurns represening 48 years Theoreical Quaniles of he Normal Disribuion (b) Quanile-quanile plo for he European MKT series covering he period from January 1991 o December 2017 conaining 324 observaions of monhly excess reurns represening 27 years. Figure 3.4: Quanile-Quanile Plo for he US and European MKT Series. This figure shows he quanile-quanile plos for he US and European MKT series. The sraigh line represens perfecly normally disribued daa and serves as a benchmark for comparison wih he acual MKT series. urn series and he reason why reurn daa is ofen beer described by a lepokuric raher han a normal disribuion. Quanile-quanile plos such as given in Fig. 3.4 allow for a comparison of wo disribuions. To be precise, he acual reurn daa of he US and European MKT samples are compared o a corresponding normally disribued sample represened by he sraigh line, respecively. Clearly, boh ends of boh reurn series do no fall along he sraigh line, hence indicaing ha he MKT series are no normally disribued. The assumpion of normally disribued reurn daa is also examined by employing wo saisical ess. Boh he Jarque-Bera es as well as he Lilliefors es are wo-sided goodness-of-fi ess of wheher he sample daa sem from a normal disribuion wih unknown and hence esimaed mean and variance. The Jarque-Bera es compares he pre- Table 3.4: Normaliy and Saionariy Tess for he US and European MKT Series. This able liss he Jarque-Bera and he Lilliefors es for normaliy as well as he augmened Dickey-Fuller es for saionariy of a ime series. The normaliy ess challenge he null hypohesis of having a normally disribued sample o he alernaive hypohesis of he sample daa being nonnormal. The es for saionariy challenges he null hypohesis of having a ime series characerized by MKT = MKT 1 + ɛ, i.e. he ime series conains a uni roo and is nonsaionary, o he alernaive hypohesis of he ime series being saionary and following an auoregressive (AR) process of order one, AR(1), i.e. MKT = φmkt 1 + ɛ wih φ < 1 and whie noise ɛ wihou any drif or deerminisic ime rend. The ess are applied o he US and European monhly MKT series covering he period January 1970 o December 2017 and January 1991 o December 2017, respecively. This able displays he es saisics of each es and he corresponding p-values. Tess for Normaliy USA Europe Saisic p-value Saisic p-value Jarque-Bera Lilliefors Tes for Saionariy Augmened Dickey-Fuller

17 4. Opimized Porfolio Sraegies and 1/N 10 diced skewness and kurosis of he normal disribuion o he acual daa sample, whereas he Lilliefors es compares he empirical disribuion funcion of he daa sample wih he cumulaive disribuion funcion of he prediced normal disribuion. As shown in Table 3.4 and already recommended by he visual inspecion of he MKT series, boh ess rejec he null hypohesis ha he sample disribuion is normal a he 5% significance level. This is rue for he US as well as for he European MKT series. Table 3.4 also presens he es saisic and he corresponding p-value of he augmened Dickey-Fuller es for saionariy. The es rejecs he null hypohesis a he 5% significance level suggesing a saionary MKT series for boh he US and Europe. This resul also implies ha he daa generaing parameers such as he mean and variance do no change over ime. To sum up, he above findings are suggesive of an ouperformance of he European marke by he US marke based on boh a higher mean reurn and smaller sandard deviaion. In addiion, he specific process of creaing he porfolios considered in his analysis resuls in high posiive correlaions of asses wihin he differen daa ses. All reurn series exhibi he common characerisics of having a negaive skewness and high kurosis. This speaks in favor of frequen small reurn gains combined wih he possibiliy of few raher exreme reurn losses. In fac, saisical ess rejec he null hypohesis of having normally disribued reurn series, an informaion ha needs o be accouned for o ensure reliable saisical inferences. 4. Opimized Porfolio Sraegies and 1/N This secion presens a brief overview of differen approaches on how o creae opimized porfolios when facing esimaion risk as is done in he relaed empirical lieraure. Subsequenly, he opimized porfolio sraegies employed in his analysis are inroduced. However, o begin wih, some noaion and he general seing are defined Mean-Variance Framework Reurn and risk measured in erms of variance of reurn are he key characerisics when considering asses and porfolios of asses. Due o heir pivoal role wih respec o porfolio opimizaion, i is crucial o undersand he basic mechanisms a work when asses wih differen properies are combined o creae a porfolio of asses. Le r = (r 1,..., r N ) T denoe he vecor of uncerain excess reurns for an invesmen universe consising of N risky asses over he risk-free rae r f, i.e. r = R r f 1 N wih he complemenary vecor of nonexcess reurns R R N 1 and 1 N is a vecor of ones wih dimension N. The expeced excess reurns of he asses µ(r 1,..., r N ) can be saed as µ = ( E(r 1 ),..., E(r N ) )T. (4.1)

18 4. Opimized Porfolio Sraegies and 1/N 11 The symmeric N N variance-covariance marix of he reurns Σ(r 1,..., r N ) is given by Σ = E ( (r µ)(r µ) T) V(r 1 ) Cov(r 1, r 2 )... Cov(r 1, r N ) Cov(r 2, r 1 ) V(r 2 )... Cov(r 2, r N ) = Cov(r N, r 1 ) Cov(r N, r 2 )... V(r N ) σ 2 1 σ σ 1N σ 21 σ σ 2N = σ N1 σ N2... σ 2 N (4.2) wih he variance of asse i denoed by σ 2 i and σ i j = σ i σ j ρ i j is he covariance beween asse i and j wih correlaion coefficien ρ i j. A porfolio can be compleely characerized by specifying he proporion ha each available risky asse akes in he porfolio. This informaion can be sored in x = (x 1,..., x N ) T wih he individual weighing of asse i denoed by x i such ha x f = 1 1 T N x is he invesmen in he risk-free asse and he represenaive invesor is fully invesed, i.e. he weighs sum up o one. Consequenly, he reurn of a porfolio in excess of he risk-free rae r p (x 1,..., x N ) akes he form r p = N x i r i = x T r. (4.3) i=1 The expeced excess reurn of a porfolio µ p (x 1,..., x N ) consising of N differen asses can be saed as N µ p = E(r p ) = x i E(r i ) = x T µ. (4.4) The corresponding variance of a porfolio σ 2 p(x 1,..., x N ) can be specified as i=1 σ 2 p = N xi 2 σ 2 i + i=1 N i=1 N x i x j σ i σ j ρ i j = x T Σx, (4.5) j=1 j i where Σ is expeced o be posiive definie, i.e. x T Σx > 0, x 0, o guaranee ha he asses under consideraion are no redundan, i.e. are linearly independen wih respec o heir reurn paern. Alhough he exisence of such an asse would no aler he resul of he underlying opimizaion problem he inveribiliy of he variance-covariance marix needs o be secured due o compuaional issues. For he represenaive invesor o behave in accordance wih he expeced uiliy hypohesis a leas one of he wo assumpions have o be fulfilled. Eiher he invesor is subjec o

19 4. Opimized Porfolio Sraegies and 1/N 12 have quadraic uiliy, alhough oher uiliies can be evaluaed uilizing a second-order approximaion, or he disribuions of asse reurns are joinly normally disribued, i.e. can be compleely characerized by only he firs wo momens, while he individual reurns are all independen and idenically disribued, respecively. Pennacchi (2008) provides a general proof. Quadraic uiliy implies an increasing relaive and absolue risk aversion wih increasing invesor s wealh, a fac ha seems o conradic realisic behavior as argued in Cohn e al. (1975) and Morin and Suarez (1983). Normally disribued reurns refer o he class of ellipical disribuions such as he normal Gaussian disribuion. However, empirically reurns raher follow a lepokuric disribuion ha canno enirely be described only by is mean and variance in accordance o he findings of Secion 3. Also, invesors aiming a maximizing expeced uiliy no necessarily only care abou he firs wo momens of he underlying disribuion bu migh have also preferences for higher momens. Despie hese poenial limiaions and assumpions, he beauy of a model is no based on he exac replicaion of he complex world in all deail bu raher on absracion and simplificaion while sill mainaining he acual purpose of a model ha is providing meaningful insighs. In his spiri and given a ime period of T observaions, he represenaive invesor chooses a composiion x a every period, = 1,..., T, of he available N risky asses o creae a porfolio so as o maximize expeced uiliy, E(u), i.e. max x E ( u(x ) ) = x T µ γ 2 xt Σ x. (4.6) The expeced uiliy is an increasing funcion in he porfolio s expeced excess reurn and decreasing in he porfolio s variance wih he invesor s specific risk-aversion parameer γ represening he srengh of his fundamenal rade-off beween expeced reurn and risk. I is no necessary o explicily consrain he weighs o sum o one as his condiion is already implicily incorporaed in he opimizaion problem due o he fac ha excess reurns raher han nonexcess reurns are considered, see Appendix B. This opimizaion problem has a closed-form soluion ha can be obained by aking he firs derivaive of Eq. (4.6) wih respec o x and akes he form x = 1 γ Σ 1 µ. (4.7) In general, his soluion implies an allocaion o he unique risky porfolio as well as o he risk-free asse wih a larger share invesed in he risk-free asse he higher he invesor s risk-aversion parameer. Finally, he adjused weighs x ω = 1 T N x = 1 γ Σ 1 µ 1 T 1 N γ Σ 1 µ = Σ 1 µ 1 T N Σ 1 µ (4.8) yield he relaive amoun allocaed o he risky porfolio a dae. The rescaling by he

20 4. Opimized Porfolio Sraegies and 1/N 13 absolue value guaranees a proporional invesmen beween he risky porfolio and he risk-free asse across he differen sraegies such ha performance differenials do no depend on an unequal invesmen share wih respec o he risk-free asse Esimaion Risk The opimizaion problem in Eq. (4.6) crucially depends on he esimaion of expeced reurns, variances, and covariances. The number of parameers one would need o esimae o compue he opimal mean-variance porfolio when considering 100 risky asses already amouns o 5, Apar from he fac ha he esimaion of a large number of quaniies can be raher ime consuming and a compuaional challenging ask, here is also a cerain level of risk inheren in every esimaion. In fac, here is no guaranee ha he obained esimaes coincide wih he acual rue underlying values. Already Meron (1980) shows evidence ha he esimaion of he variance-covariance marix is much more precise relaive o he esimaion of he corresponding expeced reurns. Chopra and Ziemba (1993) furher specify ha he mos criical esimaions wih respec o esimaion accuracy are he expeced reurns, followed by he variance esimaes and lasly he esimaes of he covariances. The difficuly in esimaing expeced reurns has is origin in he ypically observed relaively small average reurn values in combinaion wih relaively high reurn volailiies such ha only a very long ime series of observaions can guaranee an accurae esimaion. Broadie (1993) calculaes ha 26 years of monhly reurn daa are needed o correcly disinguish wih a probabiliy of 90% beween wo asses wih differen expeced monhly reurns of 1 and 1.5%, respecively. This resul assumes normally disribued reurns, a common sandard deviaion of 7%, and a correlaion coefficien of 0.5. This example nicely illusraes how difficul i is o obain precise reurn esimaes or raher how reurn esimaes criically depend on he number of observaions available for esimaion. In conras, less han five years of monhly daa are required o disinguish wih a probabiliy of 90% beween wo asses wih differen sandard deviaions of 6 and 7%, respecively. This compuaion is based on normally disribued reurns wih a common mean of 1% and a correlaion coefficien of 0.5, see Broadie (1993). Sill, having a large daa se also inroduces new challenges o he esimaion process since he probabiliy of he ime series being nonsaionary, for example, increases wih he lengh of he daa se as argued in Jobson and Korkie (1980). This consiues a rade-off beween esimaion precision and esimaion validiy ha is omnipresen in he conex of ou-ofsample porfolio opimizaion. Over he pas decades, he relaed empirical lieraure has developed differen economeric approaches on how o miigae his esimaion risk. 2 For a se of N N risky asses, N reurns r i, N variances σ 2 i, and N(N 1)/2 corresponding covariances σ i j, i = 1,..., N, i j, need o be esimaed resuling in a oal of 2N + N(N 1)/2 parameers. Addiional parameers needed are he risk-free rae r f and he risk-aversion parameer γ.

21 4. Opimized Porfolio Sraegies and 1/N Approaches o Figh Esimaion Risk The Bayesian approach sars from he premise ha he parameers of he daa generaing process are no known o he represenaive invesor. As a resul, he invesor is faced wih he ask of incorporaing an educaed guess abou he daa generaing process ino he expeced uiliy maximizaion problem. These a priori inferences, or priors, represen he invesor s beliefs abou he parameers of he daa generaing process before being presened wih evidence. The class of priors is commonly divided ino uninformaive or diffuse priors and informaive priors. The former such as proposed in Barry (1974), Klein and Bawa (1976), and Brown (1979) do no sysemaically incorporae any specific informaion abou he parameers of he reurn disribuion. Since he Bayesian approach explicily accouns for poenial esimaion errors, he already risky asses become even more risky. The risk of he risk-free asse, however, remains zero by assumpion. In comparison o he mean-variance plug-in approach, he Bayesian soluion hence ypically invess relaively less in he risky asses and more in he risk-free asse. Neverheless, he resuls obained from hese general diffuse priors are very similar o he mean-variance approach and even coincide in case of an infiniely long ime series of observaions, see Brand (2009). In conras, informaive priors do incorporae specific informaion. The arguably mos well-known approach is based on Black and Lierman (1992). Firs inroduced by Sein (1956) and hen exended by James and Sein (1961), he shrinkage echnique also aims a reducing poenial esimaion errors. The basic idea of shrinkage, nicely illusraed in Efron and Morris (1977), is ha he qualiy of he esimae of he expeced reurns should increase if he sample mean is shrinked oward a common value or grand mean, i.e. he mean of he means across all variables. In paricular, he shrunked mean is supposed o be less likely affeced by exreme observaions relaive o he sample mean. This approach can be undersood as a special case of he Bayesian echnique in which he prior is he shrinkage arge. The resuling reducion in he esimae s variance is assumed o be more beneficial han he harm ha occurs by shrinking and hus biasing he esimae. Shrinkage can hus be seen as a rade-off beween eiher incurring a bias or having low levels of variance. In general, he shrinkage facor is an increasing funcion in he number of asses and a decreasing funcion in he number of ime periods considered. Jobson and Korkie (1980), Jorion (1985), and Jorion (1986) focus on shrinking expeced reurns, whereas Fros and Savarino (1986), Ledoi and Wolf (2003), and Ledoi and Wolf (2004) exend he radiional approach and apply he shrinking echnique on variances and covariances. Kouris e al. (2012) avoid he deour and direcly shrink he inverse of he variance-covariance marix. Moreover, shrinkage is no only limied o he momens of he asse reurn disribuion bu can also be applied o opimal porfolio weighs. Applicaions can be found in Golosnoy and Okhrin (2007) and Frahm and Memmel (2010). A furher specificaion of a prior is given by facor models. Facor models ry o faciliae he esimaion of expeced reurns by idenifying a limied number of variables

22 4. Opimized Porfolio Sraegies and 1/N 15 able o explain he observed variaion in asse reurn cross-secions or ime series. The Capial Asse Pricing Model (CAPM) as suggesed by Sharpe (1964), Linner (1965), and Mossin (1966) represens a special case of a single-facor model and links he expeced asse reurn o he expeced excess marke reurn. The number of parameers one would need o esimae o compue he opimal mean-variance porfolio when considering 100 risky asses decreases from 5,150 based on he mean-variance plug-in approach o This reducion in he number of parameers o be esimaed illusraes he advanages of facor models wih respec o he esimaion procedure. Applicaions of his approach can be found in Pásor (2000) and Pásor and Sambaugh (2000). However, he workhorse of he empirical facor model lieraure is he Fama-French hree-facor model as proposed in Fama and French (1993) including is exensions o a four-facor model as in Carhar (1997) and a five-facor model as in Fama and French (2015). Addiional srands consider differen specific porfolio resricions and hence are supposed o model real-world rading in a more realisic fashion. By doing so, shor sale consrains play a key role. In his conex, Jorion (1992) proposes an ineresing hough experimen. He considers a world wih wo asses in which asse A and B are assumed o have an esimaed average reurn of 10.1 and 9.9%, respecively, alhough boh asses have a rue reurn of 10%. A shor sale resriced invesor would choose o inves mainly in asse A based on is higher reurn. Wihou shor-sale consrains, however, he invesor will heavily buy asse A and shor asse B o ake advanage of he reurn differenial of 0.2%. Since he resuling profi of he porfolio is increasing wih he long posiion in asse A and he shor posiion in asse B, he opimizaion process can resul in very exreme invesmen posiions. In response o ha finding, consraining he weighs o be nonnegaive no only miigaes he chance of adoping exreme porfolio weighs due o esimaion error bu also leads o more realisic oucomes. DeMiguel e al. (2009a) exend his approach by focusing on he minimum-variance porfolio while consraining differen norms of he porfolio weigh vecor o be smaller han a given hreshold. Furhermore, i is also possible o resric shor sales only o specific single asses or o allow shor sales in general bu only up o a limied degree. Oher real-world examples of resricions ha invesors migh face are specific consiuional regulaions or policy consrains ha only allow he invesmen in socially responsible asses such as given by companies ha do no violae human righs, suppor corrupion, or negaively affec he environmen Porfolio Sraegies The selecion of models included in his analysis aims a fulfilling wo objecives. Firs, pas empirical lieraure is respeced by including already well-esablished opimized porfolio sraegies such as he sample-based mean-variance and he Bayes-Sein sraegy. 3 There are only 3N + 2 parameers ha need o be esimaed namely N reurns, N beas, N variances, he expeced marke reurn, and he variance of he marke reurn. Addiional parameers needed are he risk-free rae r f and he risk-aversion parameer γ.

23 4. Opimized Porfolio Sraegies and 1/N 16 Table 4.1: Overview Porfolio Sraegies. This able liss a oal of 16 porfolio sraegies ha are included in his analysis. The firs model is he naive diversificaion approach ha also serves as he benchmark model on which he performances of he remaining 15 opimized porfolio sraegies are measured agains. The abbreviaions are inroduced o refer o he specific model in he ex. The las column of he able liss he number of esimaed parameers needed o implemen a specific model under he assumpion of an invesmen universe consising of N differen risky asses. This number excludes he esimaion of he risk-free rae r f and he risk aversion parameer γ. # Porfolio Sraegy Abbreviaion Number of Esimaions Naive Diversificaion 1 Equally-weighed EW 0 Sample-Based Mean Variance 2 Sample based mean-variance MV (N 2 + 3N)/2 Momen Resricions 3 Minimum-variance MinV (N 2 + N)/2 4 Value-weighed VW 0 5 MacKinlay-Pásor MP (N 2 + 3N)/2 Bayesian Approach 6 Bayes-Sein BS (N 2 + 3N)/2 Porfolio Weigh Consrains 7 Mean-variance shor sale consrain MVsc (N 2 + 3N)/2 8 Minimum-variance shor sale consrain MinVsc (N 2 + N)/2 9 Bayes-Sein shor sale consrain BSsc (N 2 + 3N)/2 10 Mean-variance long-shor sale consrain MVlsc (N 2 + 3N)/2 Opimal Combinaions of Sraegies 11 Mean-variance and minimum-variance MV/MinV (N 2 + 3N)/2 12 Equally-weighed and minimum-variance EW/MinV (N 2 + N)/2 13 Equally-weighed and mean-variance EW/MV (N 2 + 3N)/2 14 Equally-weighed and mean-variance and minimum-variance Timing Sraegies EW/MV/MinV (N 2 + 3N)/2 15 Volailiy iming VT N 16 Reward-o-risk iming RRT 2N Second, prior research is updaed by also considering raher newly proposed sraegies such as he volailiy and reward-o-risk iming. Furhermore, differen heoreically appealing models such as he minimum-variance sraegy are included o serve as equal compeiors in he horse race beween opimized porfolio sraegies. Above all, he naive diversificaion approach is empirically very successful in erms of ou-of-sample porfolio performance alhough compleely lacking any inheren opimizaion and hus defines he benchmark o bea. Table 4.1 presens an overview of he opimized porfolio sraegies considered in his analysis. All models are assigned a class ha describes he underlying applied echnique of creaing he specific porfolios. However, his classificaion is no muually exclusive such ha models migh also be aribued o differen classes. Sill, he inenion is o provide a firs impression on he differen characerisics inheren in he models under consideraion. In addiion, Table 4.1 also displays he number of esimaed