Atomic portfolio selection MVSK utility optimization of global real estate securities

ATOMIC PORTFOLIO SELECTION: MVSK UTILITY OPTIMIZATION OF GLOBAL REAL ESTATE SECURITIES CHAN WEI-JIN, CALVIN LANZ B.Sc.. TABLE OF CONTENTS Page Acknowledgements ii Table of Contents iii S

ATOMIC PORTFOLIO SELECTION:

MVSK UTILITY OPTIMIZATION OF GLOBAL REAL ESTATE SECURITIES

CHAN WEI-JIN, CALVIN LANZ

(B.Sc Real Estate (Hons.), NUS)

A THESIS SUBMITTED FOR THE DEGREE OF M.SC (ESTATE MANAGEMENT)

DEPARTMENT OF REAL ESTATE NATIONAL UNIVERSITY OF SINGAPORE

2004

Acknowledgements

My sincerest gratitude extends to Professor Liow Kim Hiang for his invaluable guidance, advice and inspiration; to the referees from the fifth Maastricht-Cambridge

Real Estate Finance and Investment Symposium (June 2004) for their valuable

comments; to Dr Angelo Ranaldo (Associate Director, UBS Global Asset

Management) and also to two anonymous referees for their kind inputs

I dedicate this thesis to my family and to Samantha for their unequivocal support and

of course, to Him, for everything

TABLE OF CONTENTS

Page

Acknowledgements ii

Table of Contents iii

Summary v List of Tables viii

List of Figures viii

List of Appendixes viii

List of Abbreviations ix

CHAPTER ONE – INTRODUCTION 2.1 Rationale of Study 1

2.2 Research Objectives 5

2.3 Research Data 7 2.4 Research Methodology 8

2.5 Research Organization 9

CHAPTER TWO – LITERATURE REVIEW 2.1 Introduction 11

2.2 Portfolio Selection and Asset Pricing Theory 11

2.2.1 Capital Asset Pricing Model (CAPM) 17 2.2.2 Portfolio Optimization Theory and Techniques 18 2.3 Mean, Variance, Skewness and Kurtosis (MVSK) 24

2.3.1 Co-moments: Covariance, Coskewness and Cokurtosis 26 2.3.2 Higher-Moment CAPMs: MVS and MVSK Models 27 2.3.3 Evidence of Higher Moments in Asset Pricing 30 2.3.4 Portfolio Selection with Higher Moments 33 2.4 Atomic Portfolio Selection (APS) 45 2.5 Time-varying Alpha, Beta, Gamma and Delta 45 2.6 Summary 48

CHAPTER THREE – RESEARCH DATA 3.1 Introduction 49

3.2 Securitized Real Estate Markets 49 3.2.1 Securitized Real Estate Funds 54 3.3 Market Portfolios and Risk-free Rate 55 3.4 Descriptive Statistics 57

3.4.1 MVSK Analysis 59

3.4.2 Test of Normality: Jarque-Bera Statistics 61 3.5 Correlation and Covariance Analyses 65

3.6 Summary 68

CHAPTER FOUR – EMPIRICAL ANALYSIS & RESULTS I

4.3 Linear, Quadratic and Cubic Market Models 72

4.3.1 Generalized Method of Moments (GMM) Estimation 74

4.3.2 Covariance, Coskewness and Cokurtosis Estimates 76

4.3.3 Co-moment Conditions: Anti-moment, Bi-moments and Tri-moments 83

4.4 Theil Inequality Coefficients (TIC) Analysis 85

4.4.1 TIC Analysis Results of Simulated Risk Premia Forecasts 85

CHAPTER FIVE – EMPIRICAL ANALYSIS & RESULTS II

5.2 Time-varying Kalman Filter (KF) Methodology and Models 94

5.2.1 Time-varying KF Conditional Coefficients Results 96

5.3.1 Proposition I: Relative-Portfolio-Coskewness 101 5.3.2 Proposition II: Relative-Portfolio-Cokurtosis 102

5.3.4 MVSK 3-D Simulated Efficient Hyperplanes 108

CHAPTER SIX – CONCLUSION

Summary

With the recent nascent of the first real estate hedge fund in Asia in July 2004 in the

burgeoning hedge fund industry, more research is due for real estate securities in view

of global investors buying into the Asian real estate story This thesis addresses the growing interest in global real estate opportunities by analyzing higher-moment risk-return relationships and portfolio selection strategies for international real estate securities Specifically, this study determines whether higher moments are significantly priced and evaluates the time-varying higher-moment characteristics of real estate securities Finally, a higher-moment portfolio selection framework is suggested and the performance results are compared with the standard MV method

When asset returns exhibit high skewness and kurtosis over a certain threshold, they are referred to as extreme values This research approaches the portfolio selection

process from a return distributional moment perspective It expands the Capital Asset

Pricing Model (CAPM) of Sharpe (1964) from two moments to a four-moment

mean-variance-skewness-kurtosis (MVSK) framework By modeling higher-moment

CAPMs of global real estate securities in a GMM framework, which account for the

three co-moments of covariance, coskewness and cokurtosis, we empirically identify the significance of these moments and three co-moment conditions of anti-moment,

bi-moments and tri-moments in individual securities of a portfolio (atomic refers to

the individual securities of a portfolio with specific reference to each asset’s risk characteristics in terms of the distributional moments: specifically the covariance,

coskewness and cokurtosis)

Atomic risk premia are computed with considerations of the three co-moments with two reference markets proxied by MSCI world market and Datastream world real estate indexes (proxy to global real estate returns) With quadratic utility optimization

functions, we empirically perform Atomic Portfolio Selection (APS) as the process

that incorporates the atomic characteristics of covariance, coskewness and cokurtosis

of individual securities with a specified market portfolio in the suggested models, and the key identification of the three atomic co-moment conditions of each security

Additionally, we demonstrate a time-varying Kalman Filter analysis to determine the

stability in time of conditional co-moments We broadly generalize that when higher moments are significant, time-varying characteristics are mostly present in our sample, implying that higher moments are not stable over time Finally, the process of simulating optimized MVSK portfolios is demonstrated and performance benchmarks are compared The empirical results support higher-moment asset pricing in real estate securities and cokurtosis has greater explanatory power than coskewness Additionally, the APS methodology has wide applications in managing alternative investment strategies such as those of hedge funds that have a tendency for non-normality

We present three-dimensional simulated efficient hyperplanes of the MVSK models

with positive relative-portfolio-coskewness and negative relative-portfolio-cokurtosis

as constraints We find that when coskewness and cokurtosis are accounted for in addition to covariance, asset allocations do change significantly for portfolios with maximized returns for the MSCI world market However for minimized variance, asset allocations change albeit marginally (but important for large fund allocations),

implying that portfolio variance reduction dominates positive coskewness and negative relative-portfolio-cokurtosis utility constraints When non-risk adjusted returns are compared, the portfolio performance measurements are similar, however when risk-adjusted Sharpe ratios are considered, the suggested

relative-portfolio-models reveal to be a more conservative and safer approach for portfolio construction

This study provides an alternative methodology to account for higher moments in real estate securities and augments portfolio selection theory while incorporating the concepts from the extreme value theory This is the first comprehensive international study of higher moments of global real estate securities, while introducing a portfolio selection methodology in a four-moment framework It is envisaged that this research contributes to current literature by providing additional insights to managing risks that otherwise would not be captured by the standard MV method

List of Tables

Page Table 2.1 – Summary of Portfolio Selection and Optimization Studies 23

Table 2.3 – Summary of Time-varying Risk Studies 47

Table 3.2 – Description of Real Estate Securities Funds 54

Table 3.3 – Descriptive Statistics of Sample Data 58

Table 3.4 – Correlation Matrix of Monthly Returns 66

Table 3.5 – Covariance Matrix of Monthly Returns (%) 67

Table 4.1 – GMM Estimates for Linear Market Model (MSWD) 77

Table 4.2 – GMM Estimates for Linear Market Model (DSWR) 78

Table 4.3 – GMM Estimates for Quadratic Market Model (MSWD) 79

Table 4.4 – GMM Estimates for Quadratic Market Model (DSWR) 80

Table 4.5 – GMM Estimates for Cubic Market Model (MSWD) 81

Table 4.6 – GMM Estimates for Cubic Market Model (DSWR) 82

Table 4.10 – Atomic Risk Premia Summary Results 88

Table 5.1 – Time-varying KF Mean State Coefficients (MSWD) 98

Table 5.2 – Time-varying KF Mean State Coefficients (DSWR) 99

Table 5.3 – Portfolio Expected Returns and Volatility Estimates 105 Table 5.4 – Optimal Portfolio Allocations (Maximized Returns) 106 Table 5.5 – Optimal Portfolio Allocations (Minimized Variance) 106

List of Figures

Figure 3.1 – Price Index Graphs of 19 Securitized Real Estate Markets 50

Figure 3.2 – Price Index Graphs of Real Estate Securities Funds 55

Figure 3.3 – Price Index Graphs of Market Portfolios and Risk-free Rate 56

Figure 3.4 – Monthly Returns Histogram Graphs of Sample Data 62

Figure 5.2 – Sharpe MV Efficient Frontier (MSWD) 108 Figure 5.3 – Sharpe MV Efficient Frontier (DSWR) 108 Figure 5.4 – MVSK 3-D Simulated Efficient Hyperplanes (MSWD) 110 Figure 5.5 – MVSK 3-D Simulated Efficient Hyperplanes (DSWR) 111

List of Abbreviations

AIFD Alpine International Real Estate Equity Fund

APS Atomic Portfolio Selection

ARCH Autoregressive Conditional Heteroscedasticity

AUFD Alpine US Real Estate Equity Fund

CAPM Capital Asset Pricing Model

DGP Data Generating Process

DSWD DS World Market Index

DSWR DS World Real Estate Index

EREIT Equity Real Estate Investment Trusts

EVT Extreme Value Theory

GARCH Generalized Autoregressive Conditional Heteroscedasticity

GMM Generalized Method of Moments

GSFD Goldman Sachs International Real Estate Securities Class A Fund HEFD Henderson Horizon Pan-European Property Equities Fund

KL Kraus and Litzenberger

MSCI Morgan Stanley Capital International

MSWD MSCI World Market Index

MSWR MSCI World Real Estate Index

MVS Mean-Variance-Skewness

MVSK Mean-Variance-Skewness-Kurtosis

NAREIT National Association of Real Estate Investment Trusts

NCREIF National Council of Real Estate Investment Fiduciaries

REAS DS Real Estate Index: Asia

REAJ DS Real Estate Index: Asia ex-Japan

REAU DS Real Estate Index: Australia

RECH DS Real Estate Index: China

RECN DS Real Estate Index: Canada

REER DS Real Estate Index: Europe

REEU DS Real Estate Index: European Union

REEX DS Real Estate Index: Europe ex-UK

REFR DS Real Estate Index: France

REGE DS Real Estate Index: Germany

REHK DS Real Estate Index: Hong Kong

REIT DS Real Estate Index: Italy

REITs Real Estate Investment Trusts

REJP DS Real Estate Index: Japan

RENA DS Real Estate Index: North America

RENE DS Real Estate Index: Europe non-EU

RESE DS Real Estate Index: South East Asia

RESG DS Real Estate Index: Singapore

REUK DS Real Estate Index: United Kingdom

REUS DS Real Estate Index: United States of America

TIC Theil Inequality Coefficients

UCDM US-dollar 1-Month Certificate of Deposit

CHAPTER ONE – INTRODUCTION

Real estate is the world’s biggest business accounting for approximately 15 percent of

global gross domestic product (GDP) with assets of US$50 trillion compared with

US$30 trillion in equities markets (Bloomberg, 2004) This translates to trillions of dollars in assets under management by professional asset and fund managers predominantly for pension or retirement plans and for institutional clients With this enormous amount of equity in real estate related assets, the need for greater understanding of the risk to return characteristics of real estate assets is a growing concern, especially more so when real estate is treated as an alternative investment class that potentially offers more attractive returns, together with hedge funds, private equity, commodities and other derivatives, as compared to the more common investments of equities and bonds (Bloomberg, 2004)

Furthermore, it is estimated that there are currently a dozen pure real estate hedge

funds globally, with US$1.5 billion in investments, which predominantly invest in real

estate securities, in the entire rapidly expanding US$700 billion hedge fund industry

(Haughney, 2004) With the recent launch of Asia's first real estate hedge fund – LIM

Asia Alternative Real Estate Fund – in July 2004 (Bharwani, 2004), there is an

increasing need to further understand the unique risk to return characteristics of real

estate securities in terms of higher moments, since it is well known that hedge fund

strategies result in highly skewed and kurtotic return distributions Hence a research framework that incorporates higher moments, that is able to capture and account for the non-normality in asset returns, sheds further insights to the risk features of real

estate securities, and this information will be highly applicable and useful to the real estate fund management industry

Data from the National Association of Real Estate Investment Trusts (NAREIT) show that real estate investment trusts (REITs) have mushroomed into a sector worth US$240 billion from only US$12 billion a decade ago, giving funds plenty of opportunity to invest Existing real estate hedge funds have done fairly well recently

For example, Wesley Capital has had annualized returns of about 21% after fees and

expenses As more real estate hedge funds are introduced, a larger amount of institutional knowledge is required to make sound investments (Haughney, 2004) This growing interest in real estate hedge funds further justifies this study in global real estate securities

Some of the broad benefits and justifications of real estate as an investment class are: a) Predictable rental income streams;

b) Potential capital appreciation;

c) Enhanced portfolio diversification;

d) Inflation hedging or protection;

e) Attractive valuations and dividend yields;

f) Stable performance compared to the broad equity market;

g) Good fundamentals;

h) Intangible and other reasons

These provide the motivations for academics and practitioners to conduct greater real estate financial research to develop theories that help guide investment decisions in real estate

The n moment of a distribution is the expected value of the n power of the deviations from a fixed value The returns from any asset are usually described in

terms of four moments These are the mean (first moment), variance (second moment), skewness (third moment) and kurtosis (fourth moment) For example, if two

assets have the same expected return and variance, investors would view these as equivalent unless they have some knowledge about the skewness and/or kurtosis of returns With this additional information, investors would prefer the asset with the

highest positive skewness and lowest kurtosis since wealth-maximizing investors seek

above-target returns (positive skewness) while preferring returns with the least fluctuation or variation (low kurtosis)

Nevertheless, the majority of prior studies employ only mean and variance to characterize asset returns Additionally, a common assumption underlying these studies is that returns are normally distributed While this assumption is often made for convenience in theoretical models, it might be acceptable for returns over medium

to long horizons, such as quarterly or annual returns However, it is less appropriate for more frequently observed data (daily, weekly or monthly) as revealed by previous empirical findings Many studies of equity performance have found that a normal distribution does not adequately describe individual stock returns Instead the

distribution of returns often have fat-tails and is more peaked than would be expected

with a normal distribution (Brown and Matysiak, 2000) For example, Simkowitz and Beedies (1980), Singleton and Wingender (1986), and Badrinath and Chatterjee (1988) find evidence of skewness in individual stock returns as well as market indexes in US stock markets In real estate markets, as in other financial markets, some evidence in favour of skewness has been presented

With the increased allocation of US pension funds to global investments and an expansion in global market capitalization represented by Asian markets as well as the rise of China as a new economic giant, considerable attention has been given to various aspects of real estate performance in Asia and internationally As real estate securities returns are likely to have different risk-return profiles from the underlying stock and real estate markets, it is important to assess the fuller investment characteristics of real estate securities with respect to their risk measures and expected return determination Since real estate securities return volatility (measured by variance) is generally higher than the market, could additional risk factors such as

coskewness and cokurtosis better explain the distribution of real estate securities

returns? Additionally, Liow and Sim (2004) find that the majority of Asian real estate securities index returns are not normally distributed and that the main source of non-normality is kurtosis rather than skewness

The presence of skewness and kurtosis in real estate securities return distributions is reasonably documented However, little is known about the presence of the other systematic risk factors of coskewness and cokurtosis and, if any, their relevance in modeling asset pricing and the impact on risk premia estimation, and hence on the

portfolio selection process and the resulting effect on asset allocation In this study

we introduce the Atomic Portfolio Selection (APS) process, which is the portfolio selection process that incorporates the co-moments of covariance, coskewness and

cokurtosis of individual securities with reference to the market by modeling

higher-moment asset pricing models; and the key identification of atomic co-higher-moment conditions, i.e anti-moment, bi-moments and tri-moments of each security [the term

atomic is an adjective used to describe the risk characteristics of (a) the individual

securities of a portfolio, and (b) with specific reference to each security’s risk

components in terms of the three co-moments: covariance, coskewness and

cokurtosis The co-moment conditions are explained as follows: anti-moment refers to

either significant coskewness or cokurtosis but not both; bi-moments refers to both significant coskewness and cokurtosis; and tri-moments refers to all three significant

covariance, coskewness and cokurtosis] Thereafter, APS utilizes a MVSK utility optimization method to select the optimal portfolio, which includes the standard MV

optimization with two additional utility parameters: positive

relative-portfolio-coskewness and negative relative-portfolio-cokurtosis It is imperative to study

coskewness and cokurtosis as these could additionally explain non-normal returns in terms of distributional moments in relation to the market

There exists a literature on the incorporation of higher moments into risk premia (Kraus and Litzenberger, 1976; Friend and Westerfield, 1980; Sears and Wei, 1988; Hwang and Satchell, 1999, and others) It seems worthwhile to explore a higher-moment modeling strategy, if only to eliminate it as a potential explanation for real estate securities Therefore, throughout this study, we implicitly assume that higher moments of returns exist

The objective of this research is multifold and is broadly classified into two empirical analyses sections First, we investigate, over the last ten years, whether monthly returns of global real estate securities and their covariance, coskewness and cokurtosis with the market are significantly related by modeling higher-moment asset pricing models in a Generalized Method of Moments (GMM) framework Specifically the

goal is to determine whether the number of significant higher-order systematic risks supports the modeling of higher moments in global real estate markets in relation to the world market or world real estate market We then identify the co-moment

conditions of the sample for the purpose of asset-labelling (a term used to label assets

as anti-moment, bi-moments or tri-moments) for investment classification and decision-making, compute the Theil inequality coefficients (TIC) to determine the better forecasting models, estimate atomic risk premia and evaluate the differences in risk premia estimates with the traditional mean-variance (MV) method

In the next empirical analysis section, we apply the time-varying Kalman Filter (KF) methodology to test whether the significant higher-moment coefficients are time-varying This will be one of the first studies on time-varying higher moments In addition, we perform the mean-variance-skewness-kurtosis (MVSK) portfolio optimization process on global real estate securities This study is also one of the pioneering works to demonstrate the portfolio selection process in a four-moment framework Comparisons are made with four real estate securities funds to reveal greater insights to current real estate fund management practices Our results are expected to show how the portfolio selection process differs with the traditional MV method by presenting asset allocation changes when higher moments are accounted for Finally, we evaluate the portfolio performances and discuss how utility maximizing individuals should allocate their wealth among a set of assets that exhibit non-normality It is expected that this study will provide meaningful implications for portfolio construction of global real estate securities

1.3 Research Data

This research utilizes monthly returns from January 1994 to January 2004 (ten years)

of 19 global real estate securities indexes and four real estate securities funds obtained

from Datastream The 19 regions are Asia, Asia ex-Japan, Australia, Canada, China,

European Union (EU), Europe, Europe ex-EU, Europe ex-UK, France, Germany, Hong Kong (HK), Italy, Japan, North America, Singapore, South East Asia (SEA), United Kingdom (UK), and United States of America (US) The data covers the majority of real estate financial markets which satisfies the intention of this study to

provide a comprehensive overview of world real estate markets The four real estate

securities funds are the Alpine International Real Estate Equity Fund (AIFD), Alpine

US Real Estate Equity Fund (AUFD), Goldman Sachs International Real Estate Securities Class A Fund (GSFD) and the Henderson Horizon Pan European Property Equities Fund or simply the Henderson Property Fund (HEFD)

The two market portfolios used are the Morgan Stanley Capital International (MSCI) World Market Index (MSWD) and the Datastream (DS) World Real Estate Index (DSWR) (proxy to global real estate market returns), while the monthly US-dollar Certificate of Deposit is used to proxy the monthly risk-free return These time series are also obtained from Datastream

It should be noted that the German real estate securities price index might be calibrated differently from the other real estate indexes by Datastream We interpret the results in light of this Also we note that Global Property Research (GPR) data

is an alternative real estate securities database However Datastream is another

database that contains a rich source of real estate securities data and is available at the NUS library

To kick-start the research, the price index graphs of the sample data are analyzed and discussed Next, the distributional characteristics of the monthly returns are tabulated

in terms of the four moments and the Jarque-Bera (JB) test of normality is computed

We then specify higher-moment data generating processes (DGPs) that are in line

with higher-moment CAPMs (Hwang and Satchell, 1999) and Hansen’s (1982) GMM methodology is employed as the estimation procedure The higher-moment models

are the quadratic market model and the cubic market model In this study, we address these models as the mean-variance-skewness (MVS) and mean-variance-skewness-

kurtosis (MVSK) models respectively We compare the simulated risk premia

estimates utilizing TIC analysis and the atomic risk premia are then estimated from the chosen models

We extend the study by modeling time-varying KF models to test for time-varying

alpha (specific or unsystematic risk), beta (covariance), gamma (coskewness) and

delta (cokurtosis) To analyze the all-important asset allocation problem with higher

moments, we perform the APS process and compare our results with the traditional

MV framework The APS process encompasses the MVSK utility optimization

method, which includes two additional utility functions, namely the positive

relative-portfolio-coskewness (rCosp, a risk-preference parameter) and the negative

relative-portfolio-cokurtosis (rCokp, a risk-aversion parameter) Finally, to compare the simulated optimal portfolios, return-to-volatility and risk-adjusted Sharpe ratios are

computed The software packages used in this study are Eviews 4.0, MATLAB 6.5,

Origin 7.5 and Solver 5.0 in a Microsoft Excel 2000 environment

Chapter one introduces the broad framework of this study, giving the reader an encompassing overview of the motivations, objectives, data and methodologies Chapter two provides the relevant literature with specific attention to portfolio selection and asset pricing theory with higher moments Evidence of higher moments

all-in asset pricall-ing, higher-moment applications to portfolio selection and time-varyall-ing risk studies are also presented and discussed

Chapter three describes the research data in detail The full definitions and descriptions of the 19 economic regions and four real estate securities funds under study are provided We present, analyze and discuss the price index graphs, descriptive statistics, normality test results, return distribution histograms and correlation and covariance analyses of the sample data

To facilitate the reading of this thesis, the empirical investigations are presented in two separate instalments in Chapters four and five The research methodologies, empirical analyses, results and discussions are provided in that order alongside one another in each chapter Chapter four first explains the theoretical framework, followed by the specification of the higher-moment CAPMs The GMM estimates and co-moment conditions results are reported This is followed by the TIC analysis and the computation of atomic risk premia for the global real estate securities A discussion of the implications of the findings concludes the chapter

Chapter five presents the second instalment of the empirical investigation We introduce the chapter with the time-varying KF framework and report the conditional alpha, beta, gamma and delta coefficients results Next, the MVSK utility optimization method is discussed and demonstrated, and the simulated optimal portfolios are presented with a discussion The chapter concludes with a comparison

of portfolio performance measurements and a discussion of the implications of the empirical results

In Chapter six, the thesis is concluded by providing a summary of the main findings, a discussion on the research contributions in terms of both filling the knowledge gap and in practical industrial applications, the limitations of the research and recommendations for future studies

CHAPTER TWO – LITERATURE REVIEW

This chapter presents and expounds relevant past studies related to portfolio selection, asset pricing, portfolio optimization, higher moments and time-varying risks This literature review is segregated into four main sections – sections 2.2, 2.3, 2.4 and 2.5

In section 2.2, we broadly describe previous studies on portfolio selection, asset pricing theory and portfolio optimization In section 2.3, we specifically discuss studies of higher moments in asset pricing and portfolio selection A discussion on the

APS process is presented in section 2.4 and finally in section 2.5, the literature on

time-varying risks is presented Summaries of the relevant literature are provided in tabular format at the end of sections 2.2, 2.3 and 2.5 within this chapter for easy reference

Markowitz’s (1952) portfolio selection ideas have come to form the foundations of what is now popularly referred to as modern portfolio theory (MPT) Initially, MPT generated relatively little interest, but with time, the financial community strongly adopted his thesis, and now more than 50 years later, financial models based on those very same principles are constantly being revised and reinvented to incorporate all the new findings that result from that seminal work The MPT of Markowitz (1952) maximizes portfolio expected return subject to holding total portfolio variance below

a selected level Every stock in the market has a certain return It assumes that this

return has a normal distribution This means that the return distribution can be completely described by two terms The first term is the mean (expected return) and

the second is the variance (risk) of returns The variance is the square of the standard deviation, which is commonly referred to as volatility

In MPT, the covariance of returns are computed between any pair of stocks – those that move together will have positive covariances, those that move in opposite directions will have negative covariances If the expected return and variance of several stocks are known, a portfolio of these stocks can be put together that has a desired variance with a certain expected return An optimization software such as

Solver is used to pick the portfolio with the smallest variance for a certain expected

return or it could be used to find the highest expected return for a certain variance

Markowitz developed a method that computes the portfolio variance as the sum of the individual stock variances and covariances between pairs of stocks in the portfolio, weighted by the relative proportion of each stock in the portfolio This is correct from

a mathematical point of view, however, since all covariance terms between all stocks

must be known, this requires numerous calculations A portfolio with 100 different stocks would require more than 5000 covariance terms

An important outcome of the research generated due to the ideas formalized in MPT

is that today's investment professionals and investors are very different from those 50 years ago Not only are they more financially savvy and sophisticated, they are now armed with many more tools and concepts This allows both investment professionals

to better serve the needs of their clients, and investors to monitor and evaluate the performance of their investments Though widely applicable, MPT has had the most influence in the practice of portfolio management In its simplest form, MPT provides

a framework to construct and select portfolios based on the expected performance of the investments and the risk appetite of the investor MPT, also commonly referred to

as MV analysis, introduced a whole new terminology which now has become the

norm in the area of investment management

The theory of portfolio selection is a normative theory A normative theory is one that describes a standard or norm of behavior that investors should pursue in constructing

a portfolio, in contrast to a theory that is actually followed Asset pricing theory such

as the Capital Asset Pricing Model (CAPM) goes on to formalize the relationship that exist between asset returns and risk if investors constructed and selected portfolios

according to MV analysis In contrast to a normative theory, asset pricing theory is a

positive theory – a theory that hypothesizes how investors behave rather than how

investors should behave Based on that hypothesized behavior of investors, a model that provides the expected return (a key input into constructing portfolios based on

MV analysis) is derived and is called an asset pricing model

Together MPT and asset pricing theory provide a framework to specify and measure investment risk and to develop relationships between expected asset return and risk (and hence between risk and required return on an investment) However, it is critically important to understand that MPT is a theory that is independent of any theories about asset pricing As Fabozzi et al (2002) point out, that is, the validity of MPT does not rest on the validity of asset pricing theory

Conventional wisdom has always dictated not putting all your eggs in one basket In

more technical terms, this adage is addressing the benefits of diversification MPT

quantified the concept of diversification by introducing the statistical notion of covariance or correlation In essence, the adage means that putting all your money in investments that may underperform at the same time, i.e whose returns are highly correlated, is not a very prudent investment strategy – no matter how small the chance

is that any one single investment will underperform This is because if any one single investment underperforms, it is likely due to its high correlation with the other investments, that the other investments will also underperform, leading to the entire underperformance of the portfolio The concept of diversification is so intuitive that it has been continually applied to different areas within finance Indeed, numerous innovations within finance have either been an application of the concept of diversification or the introduction of new methods of obtaining improved estimates of the variances and covariances, thereby allowing for a more precise measure of diversification, and consequently, for a more precise measure of risk

MPT dictates that given estimates of the returns, volatilities and correlations of a set

of investments and constraints on investment choices (for example, maximum exposures and turnover constraints), it is possible to perform an optimization that results in the risk-return or MV efficient frontier This frontier is efficient because underlying every point on this frontier is a portfolio that results in the greatest possible expected return for that level of risk or results in the smallest possible risk for that level of expected return The portfolios that lie on the frontier make up the set of

efficient portfolios (in economic terms these are known as corner solutions in contrast

to interior solutions)

One of the most direct and widely used applications of MPT is asset allocation

Because the asset allocation decision is so important, almost all asset managers and financial advisors determine an optimal portfolio for their clients, be they institutional

or individual, by performing an asset allocation analysis using a set of asset classes They begin by selecting a set of asset classes e.g domestic large-cap and small-cap stocks, long-term bonds and international stocks To obtain estimates of the returns, volatilities and correlations, they generally start with the historical performance of the indexes representing these asset classes These estimates are used as inputs in the MV optimization which results in an efficient frontier Then, using some criterion (for

instance, using Monte Carlo simulations to compute the wealth distributions of the

candidate portfolios), they pick an optimal portfolio Finally, this portfolio is implemented using either index or actively managed funds

A number of approaches can be used to obtain estimates of the inputs that are used in

MV optimization and all approaches have their pros and cons The historical performance is the approach that is most commonly used as a guide for investment decisions If portfolio managers believe that the inputs based on the historical performance of an asset class are not a good reflection of the future expected performance of that asset class, they may objectively or subjectively alter the inputs Different portfolio managers may have different beliefs, in which case the alterations will be different The important thing here is that all alterations have theoretical justifications, which, in turn, ultimately leads to an optimal portfolio that closely aligns to the future expectations of the portfolio manager

When solving for the efficient portfolios, the differences in precision of the estimates should be explicitly incorporated into the analysis But MPT assumes that all estimates are as precise or imprecise, and therefore treats all assets equally Most commonly, practitioners of MV optimization incorporate their beliefs on the precision

of the estimates by imposing constraints on the maximum exposure of some asset classes in a portfolio The asset classes on which these constraints are imposed are generally those whose expected performances are either harder to estimate, or those whose performances are estimated less precisely The extent to which personal judgment that can be used to subjectively alter estimates obtained from historical data depends on our understanding of what factors influence the returns on assets, and what is their impact The political environment within and across countries, monetary and fiscal policies, consumer confidence and the business cycles of sectors and regions are some of the key factors that can assist in forming future expectations of the performance of asset classes

To summarize, it would be fair to say that using historical returns to estimate parameters that can be used as inputs to obtain the set of efficient portfolios depends

on whether the underlying economies giving rise to the observed outcomes of returns

are strong and stable Strength and stability of economies comes from political

stability and consistency in economic policies It is only after an economy has a lengthy and proven record of healthy and consistent performance under varying (political and economic) forces that impact free markets that historical performance of its markets can be seen as a fair indicator of their future performance

2.2.1 Capital Asset Pricing Model (CAPM)

There are a number of asset pricing theories formulated over the last 50 years such as

the Arbitrage Pricing Theory of Ross (1976), but in this study, we focus on asset pricing theories which are specifically based on return distributional moments Sharpe

(1964) devised another method of determining the expected return and variance of a portfolio This method assumes that the return of each stock is composed of two parts:

one part (denoted by beta) is dependent on the overall market's performance, and the other (denoted by alpha) is independent of the market So the expected return can be

written as:R=α+βm+ε, where R is the expected return, α and β are constants, m

is the market return and ε is a residual term In this equation, alpha and beta are constants that are different for each stock; the residual term is a random variable with

an expected value of zero When this formula is used, a portfolio with 100 stocks would only require 302 terms to fully describe its distribution (100 alphas, 100 betas,

100 residual variances, plus the market return and the market variance) Compared to more than 5000 terms of the Markowitz method, this is a big improvement

The CAPM has been one of the cornerstones of modern finance (Sharpe, 1964 and

Lintner, 1965) The CAPM is useful for two reasons First, it provides a benchmark

rate of return for evaluating existing securities (i.e looking for overpriced/underpriced assets) Second, it allows us to make an educated guess as to the initial price of a newly issued security Some of the assumptions include that there are many small

investors (small refers to their wealth relative to the economy); that no individual investor possesses enough wealth to influence the market (this is a standard price-

taking assumption present in most economic models); all investors are myopic (that

is, investors have very short time horizons Myopic strategies are rarely optimal, but it

is a useful simplification); investments are limited to stocks, bonds and a risk-free asset (i.e no derivatives); no taxes, brokerage fees, etc (this assumption simplifies the math); all investors are MV optimizers (i.e they choose their investments based on the Markowitz portfolio model); all investors have the same information and, hence, have the same estimates on mean, variance, covariance, etc (this is known as homogeneous expectations) The main results are that all investors will choose to hold the market portfolio; furthermore, the market portfolio will be on the efficient frontier and will be on the capital allocation line (the tangency to the efficient frontier) That

is, the market portfolio will be optimal; the risk premium (the difference between the market return and the risk-free return) will be proportional to its risk (variance) and the degree of risk-aversion in the economy

2.2.2 Portfolio Optimization Theory and Techniques

There are a host of methods for portfolio optimization The estimation and optimal portfolio selection using Bayesian methods is discussed These methods allow for a comparison to other optimization approaches where parameter uncertainty is either ignored or accommodated in an ad-hoc manner In the area of optimal portfolio asset allocation, Markowitz (1952) provides the foundation for the current theory of asset allocation He describes the task of asset allocation as having two stages The first stage starts with observation and experience and ends with beliefs about the future performances of available securities The second stage starts with the relevant beliefs and ends with the selection of a portfolio

Although Markowitz only deals with the second stage, he suggests that the first stage

should be based on a probabilistic model In a less well known part of Markowitz

(1952), he details a condition whereby MV efficient portfolios will not be optimal when an investor's utility is a function of more than two moments, i.e with all four moments: mean, variance, skewness and kurtosis (MVSK) Due to the inadequacy of

MV portfolio analysis, the possibility of the portfolio selection process with higher moments as an alternative in portfolio management practice is thus explored If investors’ utility functions are assumed to be of higher order than quadratic, portfolio analysis and risk management need to be extended to a higher-moment framework Asset pricing with higher moments has been a theme since the 1960s and most of the literature is based on market equilibrium models

In optimization theory, there are many ways to compute optimal portfolios such as: to minimize volatility (variance), minimize Value-at-Risk (VaR) or modified-VaR, minimize shortfall probability, minimize Value-of-Regret (VoR), minimize local negative correlation, minimize maximum drawdown, maximize returns, etc In modern portfolio management techniques, there are usually three commonly used

variants of asset allocation theory: strategic, tactical and dynamic Strategic asset

allocation is the optimal asset mixture which provides the best risk-return

performance taking into account investor risk tolerance and investment horizon

Tactical asset allocation is the process of adjusting the optimal mix to reflect changes

in forecasted economic conditions This represents a proactive management process

based on the forecasting ability of the manager Dynamic asset allocation is the

short-run adjustment of the portfolio's risk profile in response to market moves by using options and futures (also known as portfolio insurance) Dynamic allocation is an adaptive or reactive policy based on adapting to changes in market conditions The

problem with this descriptive approach is that it defines asset allocation as three distinct approaches when they are simply components of a more complex process

Portfolio management is a multi-step interrelated system that reflects these three variants as individual steps in a systematic approach to portfolio management Briefly

it is described as follows The first step is to screen investments for their risk-return characteristics Statistical measures such as variance, beta and Lower Partial Moment (LPM) are used to measure the amount of risk inherent in an investment The ranking

of assets by their risk-return statistics provides an initial screening of individual assets The next step is to adjust input data to reflect expected economic conditions While statistical measures may at first be estimated using historic data, the portfolio manager may wish to adjust the statistics to reflect future expectations and not just based on past information

For strategic and tactical optimization of portfolio funds, optimization may be carried out at a strategic level using general asset classes and it may be carried out at the tactical level with individual securities within a general asset class Optimization provides the individual asset allocations for portfolios with the best expected risk-

return performance Optimization algorithms can be set up to use different statistical

measures of risk (variance, beta, LPM, etc.) In the consideration of utility selection of the appropriate portfolio, optimization techniques only provide tradeoffs between risk and return There will be optimized high return-high risk portfolios, optimized medium return-medium risk portfolios, and optimized low return-low risk portfolios

At this point, the portfolio manager has to decide which portfolio will maximize the

economic satisfaction (utility) of the investor In other words, which portfolio will

best meet the needs of the investor The portfolio may also be manipulated to respond

to short term market conditions by using options and futures or to reflect forecasted

market conditions by switching into and out of the money (cash) markets

After the appropriate portfolio has been selected and purchased, then it has to be monitored to see whether it is providing the expected performance by the ongoing

evaluation of portfolio performance indicators Finally there should be a revision of

portfolio allocations to reflect changes in economic conditions and portfolio performance As noted above, the short term performance of the portfolio can be manipulated by switching to or from cash balances and by using options and futures However, with changing economic expectations and conditions, the allocations within the portfolio may have to be revised by going back to the first step

Statistical tools are used in conjunction with historic data to estimate future variability and return for an investment Since the optimization techniques maximize return and minimize risk, the selection of the measurement tool and the definition of risk are important Flawed estimates of return and risk lead to flawed optimization The most common statistical tools used to measure risk and return are as follows In measuring return, the geometric average and final wealth approaches are used and for the risk measure, variance is used The MV approach uses variance, which works best with normally distributed asset returns When return distributions are not normally distributed, estimation errors and other biases will result Using the variance to solve the portfolio problem (minimize risk and maximize return) is a very complex,

computationally intense, process The variance, however on its own, does not provide

a realistic description of investor behavior relative to risky investments

Beta is used as a risk measure and has been popular for some 20 over years It was originally developed in order to simplify the MPT process which is very computationally complex when the variance is used The beta simplifies the portfolio problem by relating an asset to a general market index This eliminates the need for correlation coefficients between all assets which makes the variance approach so difficult The original work in this area was done by Sharpe (1963, 1964) who shared the Nobel prize with Markowitz (1952, 1956, 1959) Later, he expanded this work into the CAPM, which was cited by the Nobel prize committee The beta is calculated

as a simple linear relationship between an asset and the general market index or a measure of an asset's variability relative to the general market index The line that describes this relationship is estimated by regression analysis This is usually called the volatility of the asset Beta is important because it is a measure of how much risk

an individual asset will add to a diversified portfolio

The variance of an asset is broken into two parts: variability due to the general market index and variability due to the unique characteristics due to the individual asset (variability that cannot be explained by the regression relationship) The risk due to

the market is called the systematic or the non-diversifiable risk since it is due to the

larger system (market) and cannot be diversified away The risk due to the individual

asset is called the unsystematic or diversifiable risk because it is not due to the larger

system and can be diversified away Because of this relationship, the beta can be used

as a risk measure only if the individual assets will be going into a well diversified (usually 15-20 assets) portfolio where the unsystematic risk is diversified away The beta should not be used to screen or rank assets unless the final portfolio will have

more than 15-20 assets The advantage of the beta is its simplicity which reduces complexity and improves its forecasting ability (Fabozzi et al., 2002)

Its disadvantages include: the beta is difficult to estimate because it assumes a linear relationship between the individual asset and the market It should be noted that a lot

of applications, compute the natural logarithms of the returns before the regression takes place This logarithmic transformation converts a large number of nonlinear relationships to linearity It also works best with normally distributed asset returns and

it provides a limited description of investor behavior relative to risky investments For further details, please refer to the following Table 2.1, which provides a summary of portfolio selection, asset pricing and optimization studies that have been reviewed in this thesis

Table 2.1 – Summary of Portfolio Selection and Optimization Studies

Diversification of Investments

Function subject to Linear Constraints

Analysis

Market Equilibrium under Conditions

of Risk

Selection of Risky Investments in Stock Portfolios and Capital Budgets

Mutual Fund Portfolio Selection

Uncertainty: The Continuous-Time Case

Dynamic Stochastic Programming

Selection and Revision

Accounting used in a Simple Portfolio Selection

Heuristic

Portfolio Analysis

Moment: Ex-Post Results

1992 Markowitz, Todd, Xu and Yamane Journal of Financial Engineering Fast Computation of Mean-Variance

Efficient Sets using Historical Covariances

1999 Markowitz, Schirripa and Tecotzky Journal of Portfolio Management A More Efficient Frontier

Mean-Variance Efficient Portfolio Weights

Multi-period Mean-Variance Formulation

Market Approach

Regime Shifts

Theory

Empirical Research in Economics, University of Zurich

A Note on Portfolio Selection under Various Risk Measures

Oertmann

and Applications

2003 Berkelaar, Kouwenberg and Post Forthcoming in Review of

Economics and Statistics

Optimal Portfolio Choice under Loss Aversion

While the mean is the average value of the series, obtained by adding up the series

and dividing the number of observations, the standard deviation, σ is a measure of

dispersion of spread in the series and is given by: ( ) /( 1)

N

i i

σ , where σˆ is an estimator for the standard deviation that is based

on the biased estimator for the variance (σˆ=σ (N−1)/N) The skewness of a symmetric distribution, such as the normal distribution, is zero Positive skewness means that the distribution has a long right tail and negative skewness implies that the

distribution has a long left tail A risk-averse investor does not like negative

skewness Kurtosis measures the peakedness or flatness or degree of fat-tails of the

distribution of the series Kurtosis, K is computed as: ∑

kurtosis is the value that is above three A risk-averse investor prefers a distribution

with low kurtosis (i.e returns not far from the mean) The non-normality of asset returns is a well established empirical regularity Whereas, non-normality of individual stock returns appears naturally, possibly because of time-varying parameters or rare, yet extreme realizations, for other assets, such as hedge funds, non-normality may be due to complex allocation strategies

A large body of finance literature has documented that stock returns are affected by skewness and/or kurtosis For example, Bekaert et al (1998) find that the majority of emerging country stock returns are not normally distributed The combination of skewness and kurtosis will contribute to different volatilities for different classes of investment (Brown and Matysiak, 2000) Similarly, prior real estate research has shown that the returns on individual properties and listed property securities are skewed For example, Young and Graff (1995) investigate the return distribution of individual properties in the Russell-NCREIF database They find evidence of time-varying heteroscedasticity and skewness over the period of the study In particular, for most years in the sample period (1980-1992), the returns on individual properties are negatively skewed Bond and Patel (2003) find that a large portion of property

company returns in the UK does exhibit skewness in the conditional return distribution

Many fields of modern science and engineering have to deal with events which are

rare but have significant consequences Extreme Value Theory (EVT) is considered to

provide the basis for the statistical modeling of such extremes The potential of EVT applied to financial problems has only been recognized recently Understanding the frequencies of rises or falls in financial markets is important in investment decision-making for investors The EVT, founded by the German mathematician, pacifist and anti-Nazi campaigner Emil Julius Gumbel who described the Gumbel distribution in the 1950s, is a probabilistic theory that provides some answers to the frequency of variations of returns beyond a given threshold Over a given period, EVT provides the type of extreme variation expected In line with EVT, highly skewed and kurtotic returns are representations of extreme values Hence, this thesis not only incorporates concepts from EVT but augments the theory by providing new methods to the management of extreme values in return distributions

2.3.1 Co-moments: Covariance, Coskewness and Cokurtosis

Co-moments are broadly defined either as the component of an asset’s moments related to the market portfolio’s corresponding moments; or as the covariance of one

asset’s return with the square (coskewness) or the cube (cokurtosis) of another asset’s

return or of the market’s return There are different definitions of coskewness and cokurtosis Hung et al (2004) analyzed higher co-moments in UK stock returns and define coskewness (cokurtosis) as the projection of a stock’s excess return on the

market excess return squared (cubed) While covariance is defined in the usual way, a

fortiori, following Hwang and Satchell (1999) and Ranaldo and Favre (2003), we

define in this thesis coskewness as the relationship between the asset’s excess return with the square of the unexpected systematic (market) return and cokurtosis as the relationship between the asset’s excess return with the cube of the unexpected

systematic (market) return

The focus so far has been mainly on applications in asset pricing, rather than on applications in portfolio selection From an asset pricing perspective, skewness and kurtosis of a given asset are also jointly analyzed with the skewness and kurtosis of the reference market Similar to the systematic risk or beta, some authors examine if there exists a systematic skewness and systematic kurtosis and, if any, whether they

are priced in asset prices Systematic skewness and systematic kurtosis are also called

coskewness and cokurtosis respectively by Christie-David and Chaudhry (2001)

Provided that the market has a positive skewness of returns, investors will prefer an asset with positive coskewness Cokurtosis measures the likelihood that extreme returns will jointly occur in a given asset and in the market and investors will prefer

an asset with low cokurtosis

2.3.2 Higher-Moment CAPMs: MVS and MVSK Models

One common characteristic of the models accounting for coskewness and cokurtosis

is to incorporate higher moments into the classical two-moment CAPM In the literature, two main approaches have been investigated, namely the three-moment and four-moment CAPMs As well as pricing the first co-moment of stock returns with the

market return (beta), Kraus and Litzenberger (1976) were the first to suggest that

higher moments may also be priced If market returns are not normal (but skewed or

leptokurtic), investors are also concerned about portfolio skewness and kurtosis If investors’ preferences contain portfolio skewness and kurtosis measures, each stock’s contribution to systematic skewness (coskewness) and systematic kurtosis (cokurtosis) may determine a stock’s relative attractiveness and therefore required return

Although a number of papers have tested higher-order pricing models for US stock data (Harvey and Siddique, 2000a,b; Dittmar, 2002; and Barone-Adesi et al., 2002) and for commodity contracts (Christie-David and Chaudhry, 2001), there has been little work outside the US (Hwang and Satchell, 1999; and Galagedera et al., 2002)

We expand the studies by analyzing higher moments in global real estate securities Kraus and Litzenberger (1976) and more recently Harvey and Siddique (2000a,b) developed asset pricing models incorporating coskewness terms while Dittmar (2002) developed models incorporating a cokurtosis term Loadings on market premium, premium squared and cubed can be estimated at any point in time from an extended time series regression

There is a collection of contributions, that focus on the implications of higher moments within an equilibrium context Kraus and Litzenberger (1976) provide an empirical implementation Further work in that area is by Friend and Westerfield (1980), Barone-Adesi (1985), Sears and Wei (1988), and more recently by Fang and Lai (1997), Harvey and Siddique (2000a,b), and Jurczenko and Maillet (2002) Shimaan (1993) studied the effect on the market-factor model of returns drawn from

an asymmetric student t-distribution Athayde and Flores (2001) focused on the

computation of the efficient frontier when several moments are present

Some empirical papers (for the UK included) have found that the CAPM is only

moderately significant once exposed to the Fama-French value and size factors (see

for example Chung et al., 2001) This is to say that once time series regressions are used to compute betas in cross-sectional regressions, these betas produce average slope coefficients (market risk premia) that are insignificant In contrast, Fama-French factors remain highly significant in explaining the cross-section of stock returns However, once the methodology of separating up and down markets is used and thus allocating a negative realized risk premium to the down markets, beta becomes highly significant in explaining the cross-section of returns for UK data Cross-sectional regression tests of the CAPM that ignore this methodology risk rejecting the CAPM when it might hold Furthermore the market beta in these cases remains significant when exposed to higher moments and Fama-French factors, and contributes to the explained variance One of the Fama-French factors, size, itself reacts differently to the experience of up and down markets (with different slope coefficients), in

particular, the size effect seems to manifest itself through anomalous higher returns for smaller stocks in the down markets The other factor, value, does not react like this

and reacts almost symmetrically across up and down markets Hence, the specification and results of higher-moment CAPMs should be viewed in light of these alternative forms that include size and value effects

Additionally, some authors examine determinants of the cross-section of portfolio returns from the CAPM beta and other strategies based on value and size It does so using two recent developments in the literature, firstly by using a test of CAPM which controls for the sign of realized market premia, and secondly, using higher-order asset pricing models that encompass systematic risks above the traditional CAPM beta

covariance (Harvey and Siddique, 2000a,b) Hung et al (2004) find that testing the CAPM by separating up and down markets yields a high significance for beta in

explaining the cross-section of stock returns Furthermore, this beta effect is robust

with respect to Fama-French size and value factors, a result that is contrary to much of asset return literature

So far, no real estate research has explored the joint pricing implications of covariance, coskewness and cokurtosis in global real estate securities Following broadly from Hwang and Satchell (1999), the traditional two-moment CAPM is

extended to the three-moment (quadratic market model or MVS model) and moment (cubic market model or MVSK model) CAPM, i.e less restrictive forms of

four-the traditional CAPM that accommodate systematic volatility (beta or covariance), systematic skewness (coskewness) and systematic kurtosis (cokurtosis) For the first time, this study estimates atomic risk premia values for the 19 real estate securities and four real estate securities funds, and graphically present the simulated risk premia forecast estimates Such analyses (not found in Hwang and Satchell, 1999) will allow

us to make some interesting observations and infer the general applicability of moment models in asset pricing and for portfolio design of global real estate securities

higher-2.3.3 Evidence of Higher Moments in Asset Pricing

Kraus and Litzenberger (1976) and Sears and Wei (1988) extend the classical CAPM

to incorporate the effect of skewness on portfolio evaluation and provide mixed results Barone-Adesi (1985) proposes a quadratic model to test a three-moment CAPM Homaifar and Graddy (1988) derive a higher-moment CAPM and test it using

principal component analysis, latent root regression and ordinary least squares Fang and Lai (1997), on the other hand, examine the effect of cokurtosis on asset prices using a four-moment CAPM Together, their results show that the expected excess rate of return is related to systematic variance, systematic skewness and systematic kurtosis Investors are generally compensated for taking high risk as measured by high systematic variance and systematic kurtosis Investors also forgo the expected returns for taking the benefit of increasing the skewness Additionally skewness and kurtosis cannot be diversified away by increasing the size of portfolios Thus non-diversified skewness and kurtosis play an important role in determining security valuations Harvey and Siddique (2000a,b) examine an extended CAPM that includes systematic skewness (coskewness) They find that the higher moment is priced and suggest a model incorporating skewness helps explain the cross-sectional variations of stock returns

Other finance researchers like Hwang and Satchell (1999), Christie-David and Chaudhry (2001), Berenyi (2002), Jurczenko and Maillet (2002), and Galagedera et

al (2002) propose the use of the cubic market model as a test for coskewness and cokurtosis Berenyi (2002) applies the four-moment CAPM to mutual and hedge fund data He shows that volatility is an insufficient measure for the risk of hedge funds and for medium risk-averse agents Christie-David and Chaudhry (2001) employ the four-moment CAPM on the futures markets They show that systematic skewness and systematic kurtosis increase the explanatory power of the return generating process of futures markets Hwang and Satchell (1999) investigate coskewness and cokurtosis in emerging markets Using a GMM approach, they show that systematic kurtosis explains the emerging markets returns better than systematic skewness Finally,