Portfolio Risk

AN EMPIRICAL STUDY TO DERIVE AN ASYMMETRY-ADJUSTED RISK ESTIMATE

A MASTER THESIS BY

Jacob F. B. Jensen [410189]

Kristian R. S. Pedersen [402046]

Supervisor Otto Friedrichsen

Department of Business Studies

SPRING 2013

MSC. FINANCE & INTL. BUSINESS

AARHUS UNIVERSITY

BUSINESS AND SOCIAL SCIENCES

Acknowledgements

A special thank to those who have provided insightful discussion and valuable

advice. Otto Friedrichsen (Formuepleje), Rikke Gunnergaard (Nordeal Wealth

Management), Jean-Guy Simonato (HEC Montréal), Andreas Gra�und (Nykredit

Markets), Tine Arhøj (Danske Commodities), and Paul Keller (Quality America

Inc.)

i

�A measure of dispersion is needed. Remeber the old story about the

mathematician who believed an average by itself was an adequate

description of a process and drowned in a stream with an average depth

of two inches.�

(Elton & Gruber, pp. 46-47)

ii

Contents

I. Introduction 1

1. Research Design 2

1.1. A Primer on Portfolio Theory . . . . . . . . . . . . . . . . . . . . . . 2

1.2. Purpose and Problem Statement . . . . . . . . . . . . . . . . . . . . 3

1.3. Delimitations and Assumptions . . . . . . . . . . . . . . . . . . . . . 4

1.4. Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5. Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2. Literature Review 8

II. Data 11

3. Data Overview 12

3.1. Performance Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2. Statistical Characteristics . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.1. Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.2. Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.3. Commodities . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.4. Currencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.5. Real Estate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.6. Equally-weigthed Index . . . . . . . . . . . . . . . . . . . . . . 23

3.3. Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.1. Correlation in Asset Classes . . . . . . . . . . . . . . . . . . . 26

3.3.2. Cross-sectional Correlation . . . . . . . . . . . . . . . . . . . . 27

iii

3.3.3. Conditional Correlation . . . . . . . . . . . . . . . . . . . . . 28

III. Portfolio Theory 32

4. Portfolio Optimization 33

4.1. Mean-Variance Optimization . . . . . . . . . . . . . . . . . . . . . . . 33

4.1.1. The E�cient Frontier . . . . . . . . . . . . . . . . . . . . . . . 35

IV.Non-normality 41

5. Higher Moment Orders 42

5.1. Moments of the Distribution . . . . . . . . . . . . . . . . . . . . . . . 42

5.1.1. Test for Normality . . . . . . . . . . . . . . . . . . . . . . . . 44

6. Expansions 48

6.1. Cornish-Fisher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.2. Gram-Charlier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.3. Non-normality E�ects on Quantile Estimation . . . . . . . . . . . . . 52

V. Empirical Results 56

7. Risk Comparison 57

7.1. Allocation Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7.2. Con�dence Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.3. Implementation of Cornish-Fisher and Gram-Charlier . . . . . . . . . 61

7.4. Optimizing with Cornish-Fisher and Gram-Charlier Standard Devia-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8. Rolling-period Optimization 71

8.1. Rolling-period Methodology . . . . . . . . . . . . . . . . . . . . . . . 71

8.2. Rolling-period Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . 73

8.3. Rolling-period Quantile Approximation . . . . . . . . . . . . . . . . . 76

8.4. Roling-period Standard Deviations . . . . . . . . . . . . . . . . . . . 78

iv

8.5. Rolling-period Performance . . . . . . . . . . . . . . . . . . . . . . . 81

9. Drawbacks 83

9.1. Domain of Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

9.2. Analysis Critique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

VI.Concluding Remarks 92

10.Conclusion 93

11.Implications and Further Research 97

VII.Appendix 105

v

List of Figures

3.1. Historical development of stock indicies for strategy/regional and sector . . 16

3.2. Historical development of the bond indices . . . . . . . . . . . . . . . . . 19

3.3. Historical development of the commodity indices . . . . . . . . . . . . . . 21

3.4. Historical development for currencies . . . . . . . . . . . . . . . . . . . . 22

3.5. Historical development of REITs . . . . . . . . . . . . . . . . . . . . . . 23

3.6. Historical development of the asset classi�cations . . . . . . . . . . . . . 24

4.1. MVO e�cient frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2. E�cient portfolios allocation . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3. MVO e�cient frontier with OP, and MVP . . . . . . . . . . . . . . . . . 38

5.1. The return distribution of the OP �tted to the normal curve . . . . . . . 42

6.1. Cornish-Fisher, Gram-Charlier, and normal quantiles in skewed distributions 53

6.2. Cornish-Fisher, Gram-Charlier, and normal quantiles in distributions with

excess kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.3. Left skewed and leptokurtic distribution, skewness = -1.5, kurtosis = 6 54

7.1. MVO optimization with and without portfolio restrictions . . . . . . . . . 59

7.2. Cornish-Fisher, Gram-Charlier, and normal quantiles for MVP and OP . . 62

7.3. Cornish-Fisher, Gram-Charlier, and MVO at the 95% con�dence level . . . 66

7.4. Cornish-Fisher, Gram-Charlier, and MVO at the 98% con�dence level . . . 67

8.1. Rolling-period method . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

8.2. Skewness from the MVO portfolios, MVP and OP . . . . . . . . . . . . . 73

8.3. Kurtosis from MVO portfolio MVP and OP . . . . . . . . . . . . . . . . 74

8.4. MVO Jarque-Bera barometer . . . . . . . . . . . . . . . . . . . . . . . . 75

8.5. Cornish-Fisher and Gram-Charlier quantiles at 95% con�dence level . . . . 77

vi

8.6. Cornish-Fisher and Gram-Charlier quantiles at 98% con�dence level . . . . 77

8.7. Cornish-Fisher, Gram-Charlier, and MVO standard deviations at the 98%

con�dence level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

8.8. Cornish-Fisher, Gram-Charlier, and MVO standard deviations in the �-

nancial crisis of 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8.9. Cornish-Fisher RTV before versus after optimization . . . . . . . . . . . . 81

9.1. Monotonic and non-monotonic functions . . . . . . . . . . . . . . . . . . 84

9.2. Non-monotone distribution function . . . . . . . . . . . . . . . . . . . 84

9.3. Gram-Charlier density function with negative values . . . . . . . . . . . . 85

9.4. Gram-Charlier with excess kurtosis above 3 . . . . . . . . . . . . . . . . 87

9.5. Cornish-Fisher with excess kurtosis above 7 . . . . . . . . . . . . . . . . 88

9.6. Gram-Charlier with negative excess kurtosis . . . . . . . . . . . . . . . . 88

9.7. Cornish-Fisher with negative excess kurtosis . . . . . . . . . . . . . . . . 89

vii

List of Tables

3.1. Descriptive statistics for stocks . . . . . . . . . . . . . . . . . . . . . . . 17

3.2. Descriptive statistics for bonds . . . . . . . . . . . . . . . . . . . . . . . 20

3.3. Descriptive statistics for commodities . . . . . . . . . . . . . . . . . . . 21

3.4. Descriptive statistics for currencies . . . . . . . . . . . . . . . . . . . . . 22

3.5. Descriptive statistics for REITs . . . . . . . . . . . . . . . . . . . . . . . 23

3.6. Descriptive statistics for each asset category . . . . . . . . . . . . . . . . 25

4.1. MVP and OP portfolio allocation weights . . . . . . . . . . . . . . . . . 39

5.1. Test for normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

7.1. Portfolio allocation weights . . . . . . . . . . . . . . . . . . . . . . . . . 58

7.2. Percentiles at 95% and 98% con�dence intervals . . . . . . . . . . . . . . 62

7.3. Standard deviations at 95% and 98% con�dence intervals . . . . . . . . . 62

9.1. Valid kurtosis and skewness pairs for the Cornish-Fisher expansion . . . . 86

9.2. Valid kurtosis and skewness pairs for the Gram-Charlier expansion . . . . 86

viii

Part I.

Introduction

1

1. Research Design

1.1. A Primer on Portfolio Theory

The typical Markowitz framework that has dominated the portfolio allocation theory

for more than half a century assumes the return distribution is normally distributed.

This assumption is grounded by the Gaussian distributions sole reliance on the

�rst two moments, mean and variance. However, as Mandelbrot (1963) states; 'the

empirical distributions for price changes are usually too "peaked" to be relative to

samples from Gaussian populations ' [1, p. 394]. Further research supports these

�ndings through investigation of multiple time-periods [2, 3, 4].

Failing to account for the distributional characteristics of the return series can

therefore cause serious implications in risk management and faulty allocate the assets

in the portfolio [5, 6].

Considerable evidence has since shown that investor preferences go beyond mean

and variance to higher moments such as skewness and kurtosis. The concern in

particular regards the downside risk which in recent years has caused signi�cant

losses to the vast majority of investors active in the �nancial markets. Skewness and

kurtosis enables the investor to more correctly quantify the downside risk exposure

and has therefore become important considerations in asset allocations [7, 8].

The �nancial crisis of 2008 has led many investors to look for ways that approx-

imates an alternative risk estimation tool that captures the true risk exposure the

Markowitz mean-variance optimization fails to quantify [9]. Several statistical mod-

els have been made to account for fat tails and approximate the investor's true loss

exposure. To mention a few of the most well-known are the Lévy stable hypoth-

esis [1], the student's t-distribution [10] and the mixture-of-Gaussian-distributions

hypothesis [11]. None of them has though been recognized as the ultimate risk esti-

2

mation tool, as they have been found to be unstable, which implies, the distribution

shape changes at various time horizons and do not obey scaling relations [9].

Even though the mean-variance theory illustrates these obvious risk estimation

errors it is still the most widely used method by practitioners and �nancial insti-

tutions. Many �nancial institution simply increase the quantile multiple associated

with normal law in order to account for asymmetry. As an example, the multiple

associated to a threshold of 5% is -1.645 for the standard normal distribution. In

order to consider the asymmetry in the historical return distributions some �nancial

institutions merely use a multiple equal to 2 or 3. This methodology is however not

scienti�c and well-grounded [12].

This dissertation aims to examine whether expansions, such as the Cornish-Fisher

and Gram-Charlier approximations are more useful to de�ne the quantile multiple,

and derive a more precise historical risk estimate in the context of portfolio opti-

mization.

1.2. Purpose and Problem Statement

The author's interest in examining the downside risk exposure was spurred by the

�nancial crisis of 2008. Both �nancial institutions and investors incurred massive

losses and the e�ect of diversi�cation seemed non-existent as several asset categories

across industries and geographical areas plunged almost simultaneously causing in-

creased asymmetry in the return distributions. The limited teachings of portfolio

optimization at the FIB master's program and complete exclusion of how to adjust

the risk measure in the presence of asymmetry, has motivated the authors to ex-

tent their knowledge regarding these subjects. In addition, it has been discovered

that �nancial institutions such as Nordea, Saxo Bank and several others base their

portfolio optimization on the Markowitz framework, which further motivates the au-

thors to investigate the mean-variance theory's ability to quantify the risk measure

imposed on the investor in �nancial distressed periods12.

1http://www.saxoprivatbank.dk/kapitalforvaltning/produkter/aktivallokering.aspx2The Nordea e-mail correspondance can be found in appendix N

Hence, the main objective of the dissertation is to investigate:

Whether the classical mean-variance optimization theory underestimate the risk

exposure in the presence of asymmetric return distributions?

In reaching the main objective the following research questions will be investi-

gated:

� How the statistical characteristics of the selected return series have developed

over the estimation period?

� How is the risk estimation adjusted to adhere asymmetric return distribution?

� How does asymmetric return distribution a�ect the estimation of portfolio

risk?

� How much does mean-variance optimization underestimate the risk estimation

during asymmetric return distribution?

� How will a changing correlation pattern a�ect the degree of asymmetry and

risk estimation?

1.3. Delimitations and Assumptions

The authors have enforced a number of delimitations and assumptions in order to

focus solely on the questions raised in the problem statement.

The authors will not investigate the various holdings of each index, therefore each

index is assumed to properly re�ect what the title states. The indices are assumed be

representative of the global �nancial development for the entire estimation period.

In addition, test for heteroscedasticity and autocorrelation in the return series will

not be performed and is therefore assumed not to impact the risk estimation results.

It is important to stress that to make volatility of national indices comparable,

local currency is used as a measure as it removes the increased volatility from cur-

rency �uctuations [13, p. 455]. However, as this dissertation focus on the return

4

characteristics of the �nancial series it is assumed that the all currency exposure is

hedged and will therefore not impact total return with increased volatility.

In order to illustrate that several alternatives exist that adjust the risk estima-

tion for asymmetry and hence provide di�ering results, the dissertation shows two

expansions; Cornish-Fisher and Gram-Charlier. The authors will only numerically

show the di�erence between the models, and what main factors in�uence the risk

estimation. It will not be attempted to analytically explain the derivation of the

risk estimates or rank the expansions. Additionally, other expansions or alternative

distribution functions will not be accounted for, but will only be mentioned periph-

erally in the proper context.

Besides these general assumptions and limitations the reader will be noti�ed in

the proper context when more speci�c assumptions and imitations appear in order

to ease overall understanding.

1.4. Method

In advance of the analysis the authors will mention some considerations on how the

research questions will be accomplished and what empirical and theoretical frame-

work will be utilized. Hence, this section describes the method in order to increase

the readers understanding of foundations of the dissertation.

The dissertation utilizes back testing over the historical period from January 1st

2001 to October 1st 2012. The �nancial series is based on total return and consist of

33 assets spread over 5 asset categories; stocks, bonds, real estate, commodities and

currencies. These asset categories are chosen to illustrate several di�erent invest-

ment areas, and the assets within the categories are assumed to ultimately re�ect

a global diversi�ed portfolio. All data have been extracted from Thomson Reuters

DataStream. In addition, all estimates will be denominated in annual terms to ease

the readers understanding.

5

The portfolio allocation model is built on the Markowitz (1952) [14] framework,

whereas the expansion is derived from the studies by Cornish-Fisher(1937) [15],

Simonato(2011) [16], Johnson and Kotz (1970) [17] and Millard (2012) [18].

Initially, the Markowitz mean-variance framework will be used to construct e�cient

portfolios based on the empirical data that serves to quantify the risk estimate using

only the �rst two moments.

Then, the traditional mean-variance theory will be expanded to account for higher

moment orders with the expansion of Cornish-Fisher and Gram-Charlier.

This methodology enables the comparison of risk estimates between the original

mean-variance theory and the expansions. The risk estimates will initially be com-

pared when adjusting the portfolios derived from mean-variance optimization for

asymmetry, and secondly the risk estimate will be compared when adjusting each

index for asymmetry prior to the optimization. Further elaboration on the method-

ology and practical application in Excel will be described in the proper context to

ease the readers understanding.

1.5. Structure

The structure of the dissertation does not follow the traditional structure of �nancial

papers. Instead, initially the data material will be represented and characterized.

This enables the practical application of the data material as soon as theory has

been represented. The reader will therefore immediately observe the e�ect of ap-

plying the theory. The dissertation structure will thus create a natural �ow as one

section naturally leads to the next. However, all sections and chapters required in

the traditional thesis structure are accounted for in this dissertation.

Overall the dissertation can be divided into seven main parts and 11 chapters.

Part I will present the research design in chapter 1 and gives a literature overview

of the expansion in chapter 2.

Part II presents the data and statistical characteristics, and further performs a

6

correlation analysis in chapter 3.

Part III will present the portfolio theory and incorporate the data from part II

in chapter 4.

Part IV investigates non-normally. In chapter 5 the higher moment orders theory

will be presented and applied to the empirical data. Following this conclusion, the

theory behind the Cornish-Fisher and Gram-Charlier expansion will be explained

and illustrated in chapter 6.

Part V Presents the empirical results. Initially, a comparison of the risk estimates

will be analyzed in chapter 7. Chapter 8 follows with a rolling-period optimization

analysis and chapter 9 provides the reader with certain drawbacks and limitations

of the expansions.

Part VI will initially sum up the main �ndings in chapter 10 and �nally suggest

topics for further studies in chapter 11.

Part VII provides documentation in the appendix.

7

2. Literature Review

The purpose of this chapter is to introduce the reader to the usage of the included

models. This will be done through presentation of the empirical literature and �nd-

ings on the expansion models Cornish-Fisher and Gram-Charlier.

The Gram-Charlier expansion is an approximate density function of the normal

density function. Various studies in empirical �nance have used the Gram-Charlier

expansion to overcome the restrictions imposed by normality. The studies �nd

that if a probability distribution function is approximately close to being normal

distributed, it can be approximated by the Gram-Charlier expansion [19, 20]. In

�nancial forecasting Jondeau & Rockinger (2001) �nd in a GARCH study that the

Gram-Charlier expansions are useful to model densities which deviate from normal-

ity [21]. Another GARCH study by Gallant & Tauchen (1989) transform density

functions by using Gram-Charlier expansion to describe deviations from normal-

ity [22]. Del Brio, Ñíguez & Perote (2009) introduce a new family of multivari-

ate distributions based on the Gram-Charlier and Edgeworth expansions to obtain

well-de�ned densities. They conclude that the distributions capture the skewness

and kurtosis often seen in �nancial return distribution [23]. Another study by Del

Brio & Perote (2012) compares the two alternative estimation methods, maximum

likelihood and methods of moments, for estimating the density function underlying

�nancial returns speci�ed in terms of the Gram-Charlier expansion. They show that

the method of moments applied to Gram-Charlier densities serves as an accurate

tool for forecasting [24].

In option pricing Knight and Satchell (1997) develop a Gram-Charlier-based option

pricing model employing the �rst four moments of the return distribution [25]. In

addition to this, the Gram-Charlier expansion has been used to �t risk-neutral asset

8

price distributions to the implied volatility smile, ensuring an arbitrage-free interpo-

lation of implied volatilities across exercise prices [20]. Dufresne & Chateau (2012)

derive Gram-Charlier based formulas for European option prices and conclude that

the expansion provides better estimates on return series that are signi�cantly af-

fected by skewness and kurtosis varying from normality [26].

In risk management the Gram-Charlier expansions have often been used to Value-

at-Risk computations. Polanski & Stoja (2010) for instance investigate several fore-

casting Value-at-Risk models and �nd that the Gram-Charlier-based model outper-

forms other empirical constant and time-varying higher-moments models [27]. A

recent study by Jean-Guy Simonato (2011) compares the Johnson distribution to

the Gram-Charlier expansion by computing expected shortfall and Value-at-Risk.

Simonato �nds that the Johnson approach can yield superior approximate risk mea-

sures that are more robust to all input combinations, as well as being more accurate

on average, than the Gram-Charlier and Cornish-Fisher expansions [16]. Christof-

fersen & Goncalves (2005) however argue that the expected shortfall computation

with the Gram-Charlier expansion involves serious problems .

The Cornish-Fisher expansion is, as the Gram-Charlier expansion, an approxi-

mate density function of the normal density function. A study by Pichler & Selitsch

(1999) compares �ve di�erent approaches to calculate Value-at-Risk on portfolios

that include options: variance-covariance, Johnson distributions, and three Cornish-

Fisher approximations based on the second, fourth and sixth order. They conclude

that the sixth Cornish-Fisher approximation provides most accurate results [29].

Another study on Value-at-Risk for an option-included portfolio evaluates four dif-

ferenct methods for speed and accuracy [30]. Mina and Ulmer (1999) conclude that

the Cornish-Fisher expansion is extremely fast, but at the same time it is less robust

than the other methods due to its unacceptable yield of results in one of four sample

portfolios. This becomes even more signi�cant when the distribution increasingly

departs from normality.

A Value-at-Risk paper by Jaschke (2002) focuses on the Cornish-Fisher properties

in the context of monotonicity. He argues that these assumptions make the expan-

sion undesirable and di�cult to use. However, in several practical situations, the

9

Cornish-Fisher expansion provides satisfactory accuracy in addition to being very

fast to compute- when compared to other methods like numerical Fourier inversion

[31]. Zangari (1996) infers that the Cornish-Fisher expansion o�ers an improve-

ment over normal Value-at-Risk estimates for a portfolio that include a government

bond and an option [32]. A study conducted by Boudt, Peterson & Croux (2008)

introduces new estimators for expected shortfall and Value-at-Risk consistent with

the Cornish-Fisher Value-at-Risk estimator from [32]. They �nd that for moderate

values of skewness and kurtosis, their modi�ed expected shortfall and Value-at-Risk

are better estimators than the normal expected shortfall [33]. Favre and Geleano

(2002) expand the traditional Value-at-Risk measure by using the Cornish-Fisher

expansion to compute the left tail of the return distribution, and thereby modify-

ing the Value-at-Risk to include skewness and kurtosis. Through empirical tests

they conclude that if �nancial assets have negative skewness and/or positive excess

kurtosis, the modi�ed Value-at-Risk will be higher than the normal Value-at-Risk

computation [34].

The purpose of this chapter is to shed some light on the literature on the two

expansions. The authors will not attempt to replicate these studies and thus refrain

from elaboration. The literature review serve as an overview and concludes that

much of the literature concerning Cornish-Fisher and Gram-Charlier focus on option

pricing and Value-at-Risk approximation studies. To the authors' knowledge, the

empirical work on quantifying asymmetry-adjusted risk estimation in the form of

standard deviations has yet to be identi�ed.

10

Part II.

Data

11

3. Data Overview

This chapter presents the empirical data that is used in the dissertation along with

the selection process. It will also describe the fundamental return series calculations.

The data will be applied to the theory as the dissertation progresses.

All data has been extracted from the �nancial database, Thomson Reuters DataS-

tream, which enables access to a vast array of various �nancial assets. In order to

fully capture the nature of the return distribution and asset correlation pattern over

multiple asset classi�cations, it is attempted to create a diversi�ed portfolio that

replicates several investment opportunities on a global scale. The construction of

such a global portfolio based on single asset selection is too comprehensive a task

for the scope of this dissertation and thus, indices re�ecting the various investment

vehicles based on region, industry, and investment strategy are chosen. Only indices

that a liquid, and have available data for the entire period are selected.

The data range from the 1st of January 2001 to the 1st of October 2012 and is

based on daily total return. The period is chosen based on the availability of com-

plete empirical data series and to ensure that the �ndings of this dissertation are

as current and up-to-date as possible. It is also the objective of the dissertation to

acquire as much data before the �nancial crisis in order to highlight the di�ering

nature of the return series in this period.

The total return feature in Thomson Reuters DataStream ensures the theoretical

growth in value over a speci�ed period where it is assumed that all cash distribu-

tions, such as dividends, are re-invested to purchase additional units of an asset at

the closing price. Total return does however not account for any investment fees,

taxes or other costs of that nature. For this dissertation these costs are assumed

12

insigni�cant and therefore, they will not a�ect the total return of the indices.

All data is obtained in the local currency to minimize the currency e�ect. One

year is assumed to represent 260 trading days [13].

Rather than prioritizing a data frequency analysis it has been assessed that the

usage daily historical returns in order to capture the true dispersion will �t the aim

of this dissertation. Therefore, outliers have not been removed because of the impor-

tance of portraying the most correct picture of the return distribution. Further, it

is the aim of this dissertation to investigate the risk-estimate di�erence between the

true distribution and the normal distribution, it is imperative to have the highest

frequency of data where none of the returns have been removed in order to fully

understand the distribution parameters and quantify the exposure.

Additionally, evidence generally suggests that distributions of daily returns are

more fat-tailed relative to the a normal distribution compared to for example monthly

returns [35, 36]. Consequently, using daily returns makes the distributions more

prone to asymmetry and therefore enables the authors to emphasize the importance

making the adjustment in order to derive a more proper risk estimate.

As mentioned, the data extracted from Thomson Reuters DataStream is based

on total returns. For the duration of this dissertation let St be the price of an asset

at time t and let rt be the natural logarithm of the return, de�ned by1

rt = ln(St/St−1) (3.1)

Log returns are the most popular and commonly used returns when examin-

ing �nancial series, especially when investigating the Mean-Variance Optimization

(MVO) properties [37]. Therefore this study is restricted to this de�nition of daily

returns. The log return is hereafter simply referred to as return. Further, the returns

are nominal and not adjusted for in�ation or currency movements.

1In Excel, the continuously compounded return is multiplied by 100 to arrive at percent

13

3.1. Performance Measure

A useful concept to determine the performance of the portfolio or index is the Sharpe

Ratio [38]. This performance measure is useful when comparing multiple investment

opportunities, because it shows the pay-o� between risk and return. It assumes that

the investor holds two assets; the risk-free rate and the risky portfolio. The Sharpe

ratio divides excess return over the sample period by the standard deviation over

the same period [39, 40, p. 567]. The excess return is found by subtracting the

risk-free rate from the expected return of the portfolio.

SharpesRatio =rP − rfσP

(3.2)

Sharpe's ratio is also known as the Reward-to-variability ratio (RTV) and will be

used interchangeably throughout the dissertation. The Sharpe ratio will be an essen-

tial part of the portfolio optimization in chapter 4 as it is used to maximize the slope

of the Capital Market Line (CML) and �nd the portfolio with the optimal RTV [41].

The risk-free rate used to calculate Sharpe's ratio is based on a 10-year German

government bond ranging from 2002 to 2012, with an average annual rate of 3.62%2.

This period re�ects the back-testing period and the German government bond is

assumed to be both liquid and stable enough to present a risk-free investment alter-

native. A risk-free rate more representative of the current �nancial situation could

have been chosen. However, as the purpose is to investigate the risk error estimation

over the observed period, and not to �nd the optimal allocation going forward, the

relatively high 10-year German risk-free rate is found more suitable.

Due to the scope of this dissertation no further investigation of alternative per-

formance measures, such as for example Treynor's ratio (1965) or Jensen's alpha

(1968) will be examined [42][43]. It is assumed that the Sharpe ratio represents the

appropriate risk measure for this dissertation.

Sharpe's ratio will hence be used in the next section to evaluate and compare the

risk and return relationship of the various indices.

2http://sdw.ecb.europa.eu

3.2. Statistical Characteristics

The data can be divided in the following �ve asset classi�cations that are assumed to

represent an ample amount of investment opportunities. Each section will start by

stating the reasoning behind choosing the various indices. For each asset category,

the index date is 1.1.2001, and all �gures and tables are from own creation. To get

an overview and elaboration of the various indices within each category please see

appendix A .

3.2.1. Stocks

For the stock indices category it is attempted to create a wide selection of indices

that benchmark various regions, industry sectors and strategies. This is done in or-

der to capture su�cient investment choices and investigate the correlation pattern

and diversi�cation potential, across continents, industries, and strategies.

It should be noted that some of the indices will overlap. By capturing the three

aspects mentioned for a diversi�ed stock index portfolio, it will be close to impossible

to create a portfolio where the indices are completely independent of each other. The

inclusion of some of the same securities in multiple indices, as for example with MSCI

World growth and S&P 500, will therefore a�ect the correlation pattern of the stock

indices and reduce the diversi�cation potential.

Thus, it is attempted to limit the occurrence of dependency by choosing a wide

selection of indices that maximize the diversi�cation e�ect. As technology merges

the world economy, it is di�cult to clearly de�ne isolated areas and industries that

will not have certain interdependencies [44]. This is also why this dissertation does

not have sole focus on stock portfolios; rather it attempts to include multiple asset

categories to investigate the risk measure in a well-diversi�ed portfolio across asset

categories.

The geographical region based index is divided into four categories in order to

portray the largest economies; America, Europe, Asia and Japan. The benchmark

indices for these are S&P 500, MSCI Europe, FTSE 100, MSCI Japan, and MSCI

15

Asia Paci�c excluding Japan. Besides these it is also attempted to �nd a proxy for

emerging economies which is represented by MSCI Emerging Markets. Benchmark

indices representing strategy are MSCI world growth and MSCI world value. There

are many strategies for security investment. That being said, it is the assumption of

this dissertation that the two above-mentioned strategies are the most eligible and

recognized3. The region and strategy benchmark index can be seen in �gure 3.1.

MSCI provides ten sector indices that are assumed to proxy all industry aspects of

the economy.

By observing daily stock return since January 2001, it is evident that after a couple

of dismal years of return, almost all index have experienced remarkable growth up

until 2008. In the region-based indices, especially MSCI Emerging Markets and

MSCI Asia Paci�c have enjoyed tremendous growth. In the MSCI industry sector-

based index it is especially materials and energy-related stocks that have provided

investors with extra high returns.

Figure 3.1.: Historical development of stock indicies for strategy/regional and sector

In 2008 this upward trend came to a bright stop and sudden steep decline. All

3http://www.northerntrust.com

16

indices experienced declining returns in the following year dropping the total return

for certain indices below January 2001 initial starting value. The four high perform-

ers of the years before 2008 were also the ones experiencing the greatest decline.

Since 2009 most indices, except MSCI Japan that exited October 2012 with a -34.5%

return, have slowly rebounded. Again, Emerging Markets, Asia paci�c, materials

and energy have performed superior reaching pre-�nancial highs, ending October

2012 with respectively 411.77%, 363.53%, 274.98%, and 266.75% total return.

It can be assumed from the data selection that no matter to what region, strategy

or sector the investment is allocated, stocks have moved in the overall same trend.

A few stocks outperform the others but with accompanied high volatility. Holding

a portfolio which only contained these benchmark indices would therefore not have

gained much from a diversi�cation aspect. Especially the immense sudden drop in

2008 a�ecting all benchmark indices will be interesting to investigate closer as this

might challenge the assumption of stocks returns being normally distributed. The

correlation pattern will be further analyzed in section 3.3.

Return σ Min Max Sharpe ratio

FTSE 100 2.87 20.84 -9.27 9.38 -0.04

MSCI World Industrials 3.75 19.55 -7.70 6.74 0.01

MSCI World Consumer Disc. 3.55 18.81 -6.98 12.41 0.00

MSCI World Consumer Staples 7.45 12.91 -5.34 7.62 0.30

MSCI World Energy 8.30 25.28 -13.66 13.59 0.19

MSCI World Financials -0.40 24.10 -10.15 11.47 -0.17

MSCI World Health 3.23 15.41 -6.38 10.00 -0.03

MSCI World IT -0.60 25.06 -8.05 9.97 -0.17

MSCI World Materials 8.56 23.88 -10.72 9.56 0.21

MSCI World Telecomm. 1.08 19.46 -7.58 9.90 -0.13

MSCI World Utilities 4.18 16.00 -7.85 11.97 0.03

MSCI Japan -3.58 22.88 -10.44 13.06 -0.31

MSCI AC Asia Pac. ex Japan 10.92 21.60 -9.41 9.52 0.34

MSCI Emerging Markets 11.97 21.23 -9.96 10.07 0.39

S&P 500 2.59 21.44 -9.46 10.96 -0.05

MSCI World Growth 1.00 16.85 -7.36. 8.99 -0.16

MSCI World Value 2.05 18.17 -7.57 8.45 -0.09

MSCI Europe 3.25 24.34 -10.18 10.76 -0.02

Table 3.1.: Descriptive statistics for stocksNote: return, sigma, min, and max are denoted in percentages

17

The average annual return during the entire period has been positive for all but

MSCI Japan, MSCI World IT and MSCI World Financials. MSCI World Energy is

one of the best performing indices but also accompanies a high volatility of 25.27%.

Furthermore, this index has a experienced the largest volatility in a single day price

movement with a decrease of -13.66%. MSCI Emerging Markets and MSCI Asia

Paci�c have performed best, annually returning 11.97% and 10.92%.

It is not surprising that MSCI Emerging Markets and MSCI Asia Paci�c have the

highest Sharpe ratio when observing the annual standard deviation compared to

the annual return. Many of the indices have generated negative Sharpe ratios. This

implies that the investor would actually have been better o� investing in the risk-free

rate. However, none of the indices stand out with exceptionally high Sharpe ratios

which probably can be attributed to the volatile period during 2007-2009 resulting in

fairly high standard deviations across the board. MSCI consumer staples have been

the least volatile investment with a standard deviation of 12.91%. Overall it can be

seen that all stock indices experienced severe negative impacts in 2008 causing all

of them to plummet with high speed. The investor may therefore be in�icted with

a downside risk which the normal distribution will not be able to capture.

3.2.2. Bonds

The indices chosen as benchmark for bonds are based on maturity, region and gov-

ernment versus corporate bonds. The U.S. Treasury STRIPs represent 1-, 5- and

10-year maturity. This will re�ect the di�ering volatility over multiple maturity

periods. Bank of America Merrill/Lynch provides the benchmark bond index for

Europe and a global bond index. As mentioned in the stock index selection, this

division will probably cause some of same correlation bias in the dataset. Bank

of America Merrill/Lynch additionally provide the benchmark index for U.S. cor-

porate bonds which is likely to be more correlated with the stock indices, and to

some degree provide opposite return movement of the more secure and less volatile

government bond alternatives.

Again there is a close to in�nite array of debt obligations to choose from, but it is

the attempt of this dissertation to concentrate the array of options to well-known

18

and rather secure options from Europe and America. It is assumed that the average

investor selects bonds as a safer aspect of the portfolio and therefore will refrain

from choosing low-grade bonds with high yields and high volatility.

The selected bond indices had an overall upward trend during the estimation pe-

riod, all �nishing with a total return above the initial value. Long-term bond have

produced a signi�cantly higher total return. However, as there is no free lunch,

the volatility for the longer term bonds has also been signi�cantly higher. Bank of

America/Merrill Lynch U.S. Corporate bonds experienced a drop in mid-2008 while

STRIPs and EU Government bonds increased. As the investors' faith to corporate

stocks and corporate bonds were tested in 2008 more low-risk investments such as

short-term government bonds seem to have been preferred. In order to kick-start an

economy after a crisis, lowering the interest rate is often a powerful monetary policy,

as it will encourage spending instead of saving. This is exactly what the Federal

Reserve in the U.S. did, which can be seen on the leveling out of the 1-year STRIP

total return. The U.S. 1-year Treasury STRIP has given close to no return since

20084.

Figure 3.2.: Historical development of the bond indices

4http://www.nytimes.com/2008/12/17/business/economy/17fed.html?_r=0

19

Return σ Min Max Sharpe ratio

U.S. Treasury 1-year STRIP 2.63 0.95 0.45 0.49 -1.03

U.S. Treasury 5-year STRIP 6.29 5.23 -2.05 2.42 0.51

U.S. Treasury 10-year STRIP 8.53 10.13 -3.13 4.36 0.48

BOFA ML EU Gvt. 4.96 3.80 -0.98 1.39 0.35

BOFA ML Global Gvt. 6.62 7.12 -1.90 3.72 0.42

BOFA ML US Corp. 6.82 5.36 -2.31 1.97 0.60

Table 3.2.: Descriptive statistics for bondsNote: return, sigma, min, and max are denoted in percentages

The �nancial crisis sparked a �ight to safety as investors sought liquid and low

volatility investments5. Nevertheless, the crisis did also a�ect the longer-term bonds

as they have shown to be increasingly more volatile since 2008. This could have se-

vere impact on the portfolio risk estimate, as bonds are assumed to be the safe

alternative in the portfolio and hence induce the diversi�cation e�ect. The table

above veri�es that long-term maturity bonds have both higher standard deviation

and the largest one-day price movement. European government bonds have been

slightly less volatile than global government bonds, which is also why the largest

economy in Europe, Germany, is deliberately chosen as the risk-free rate.

Besides the 1-year STRIP, the chosen bond indices have been a strong investment

alternative during the estimation period, all displaying positive Sharpe ratios su-

perior to stocks. Consequently, this characteristic will have great impact on the

portfolio allocation in chapter 4.

3.2.3. Commodities

S&P GSCI Commodity Index and TR/Je�eries CRB represent a global benchmark

of multiple commodities, whereas S&P GSCI Gold index only represents the com-

modity of gold. It is interesting to the scope of this dissertation to include gold to

observe the gold craze over the last �ve years and its e�ect on portfolio optimization

results, as it assumable will o�er diversi�cation potential compared to the especially

stocks and REITs.

The commodity indices have steadily increased since 2001 only su�ering a minor

drop in 2006, before peaking in mid-2008. Similar to stocks and certain bonds,

5http://themoneyupdate.com/tag/safe-investments

20

commodities plunged during the crisis. However, contrary to the two commodity

indices, gold only decreased slightly before increasing tremendously, and exiting Oc-

tober 2012 with a staggering 600% total return during the estimation period. This

unprecedented growth in gold will have a signi�cant impact on how much is allocated

to the S&P Gold index when �nding the optimal portfolio in chapter 4, assumed no

restriction are imposed.

Figure 3.3.: Historical development of the commodity indices

Returning impressive 15.08% p.a. and a fairly low standard deviation, S&P GSCI

Gold index has by far the highest Sharpe ratio of 0.603 compared to -0.077 for S&P

GSCI Commodity index and 0.108 for TR/Je�eries CRB index. The two multiple

commodity indices seem to correlate positively. However, the return and volatility

suggests TR/Je�eries CRB index as the better choice over the observation period.

Return σ Min Max Sharpe ratio

S&P GSCI Commodity 1.68 25.35 -9.71 7.22 -0.08

TR/Je�eries CRB 5.65 18.75 -6.88 5.75 0.11

S&P GSCI Gold 15.08 19.02 -7.54 8.59 0.60

Table 3.3.: Descriptive statistics for commodities

Note: return, sigma, min, and max are denoted in percentages

3.2.4. Currencies

The British Pound (GBP), the Japanese Yen (JPN) and the American Dollar (USD)

is chosen in accordance with the region division in the stock section, due to the fact

that these currencies represent three of the world's most in�uential economies.

21

Figure 3.4.: Historical development for currencies

Holding in currencies over estimation period has not been pro�table. Only the

Japanese Yen has provided the investor with a minor total return of 0.42% p.a. over

a 12-year period. Holding U.S. Dollars and British Pounds would have resulted in

total return of -28% and -21%. Nonetheless, currencies do possess some hedging

possibilities as the Yen and Dollar increased in mid-2008, where it is observed that

many of the other asset groups dropped in value. Still, holding large amount of

currencies does however not seem to be favorable long-term due to especially time-

value of money. None of the currencies results in a Sharpe ratio above zero.

Return σ Min Max Sharpe ratio

GBP -2.11 7.95 -3.14 2.74 -0.72

JPN 0.42 12.36 -3.85 5.78 -0.26

USD -2.74 10.37 -4.62 3.86 -0.61

Table 3.4.: Descriptive statistics for currenciesNote: return, sigma, min, and max are denoted in percentages

3.2.5. Real Estate

Real Estate Investment Trusts, REITs, from UK, U.S., and developed Asia are

assumed to be a reliable representation of the global real-estate market over the

estimation period. FTSE EPRA/NAREITS is speci�c indices that represent trends

in real estate equities in the three selected regions.

Unprecedented appreciation and sub-prime lending has caused many to blame real

estate as the asset classi�cation that in�icted the �nancial crisis in 2008. It is

observed that much of the civilized world enjoyed very high returns on the real

estate market, as all three REITs reached above 300% total return in 2007. When

22

the housing bubble imploded, these REITs lost almost all their value over the next

two years. UK REITs for example ended in -50% compared to the initial index value

in 2001.

As the US REITs seems to climb again, the UK REITs have just recently regained

its same value from 2001, and the Asian housing market looks rather stagnating.

After 2009 the three markets does not seem to correlate as much as before which

opens up for diversi�cation potentials.

Figure 3.5.: Historical development of REITs

Even with an immense -21.69% single day price drop, the US REITs have annually

returned 9.98%. This is followed with a very high volatility of 34.52%, but the return

is enough to adjust for this increased risk and provide the highest Sharpe ratio, where

only the UK REIT has a negative ratio.

Return σ Min Max Sharpe ratio

FTSE ESPR/NAREIT US 9.98 34.52 -21.69 16.85 0.18

FTSE ESPR/NAREIT UK 1.39 25.84 -10.43 10.19 -0.09

FTSE ESPR/NAREIT Dev. Asia 4.63 25.24 -11.84 9.79 0.04

Table 3.5.: Descriptive statistics for REITsNote: return, sigma, min, and max are denoted in percentages

3.2.6. Equally-weigthed Index

It can be concluded by observing the charts that many of the asset indices tend to

correlate positive within their classi�cation, consequently leading to limited diversi-

�cation potential. The equally-weighted index presents the �ve major asset groups

total return movement, and emphasize the importance of diversifying across multi-

ple asset classi�cations. As asset group moves opposite at various times it can be

23

assumed the correlations are negative and thereby favorable for reducing the risk of

a portfolio.

Figure 3.6.: Historical development of the asset classi�cations

As the equally-weighted stock index had a negative trend in the �rst couple

of years, commodities and REITS increased steadily. It is not until after 2003

that all asset classi�cation, besides currencies, began increasing intensely. Whereas

bonds steadily increased throughout the period, stocks, commodities, and REITS

all peaked in the period 2007-2008 followed by a dramatic plunge. However, the

sudden drop did not happen at the exact same time. REITs were the �rst, followed

by stocks and �nally commodities. These extreme negative returns across asset

categories may challenge the normal distribution's ability to yield the proper risk

estimate.

The �ve asset groups have periods where the correlation pattern tends to be nega-

tive based on the equally-weighted index chart, which will increase the diversi�cation

e�ect and lower the overall risk of the portfolio. However, it can also be observed

that across asset classi�cations there are multiple periods where the correlation is

positive. When the overall trend for the asset groups is increasing, this property is

marginalized, but when all asset groups suddenly plunge, as they did in 2007-2008,

the diversi�cation aspect disappears and the asymmetric extreme negative returns

will challenge the properties of the mean-variance theory to provide a correct risk

estimate.

24

Return σ Min Max Sharpe ratio

Currencies -1.48 10.23 -4.62 2.74 -0.53

Bonds 5.98 5.43 -3.13 4.36 0.22

Commodities 7.47 21.04 -9.17 8.59 0.21

REITs 5.33 28.53 -21.69 16.85 0.05

Stocks 3.90 20.43 -13.66 13.59 0.02

Table 3.6.: Descriptive statistics for each asset category

Note: return, sigma, min, and max are denoted in percentages

Over the entire period bonds produced the best risk/return relationship, with a

Sharpe ratio of 0.222. Currencies had a quite low volatility, but still had a negative

total return of -1.48%. Commodities provide the highest return with 7.47% followed

by REITs and stocks with 5.33% and 3.90%. Of all the asset groups REITs have

shown to be the most volatile with a standard deviation of 28.55%.

The equally-weighted index provides a good overview over the movement of the

di�erent asset classi�cation, but it should be noted that it is not a complete rep-

resentation of how these broad asset groups correlate. The various indices within

the asset groups represent di�erent maturity, regions, and industries and should in

practice not be equally weighted. The S&P Gold index will for example bias the

overall commodity outlook positively over the entire period, because of the return

characteristics gold possessed over the last �ve years.

Therefore, to examine the interdependence of all the indices across asset classi�ca-

tions more closely, a correlation analysis will follow next.

3.3. Correlation Analysis

The e�ectiveness of the portfolio diversi�cation depends on the correlation between

the asset returns [40, p. 152]. The correlation coe�cient is obtained by dividing

the covariance between asset X and asset Y by the standard deviation of each

asset. The correlation coe�cient represents the direction and the strength of the

relationship between asset X and Y. The correlation coe�cient is given by

ρ =ΣNi=1

(Xi − X

) (Yi − Y

)σXσY

(3.3)

The possible coe�cient value ranges from -1 to +1. The signs of the coe�cient

25

indicate the direction of the relationship: a positive coe�cient represents a positive

linear relationship between asset X and Y, while a negative coe�cient represents a

negative linear relationship between asset X and Y. The strength of the relationship

is indicated by the absolute value of the coe�cient: a coe�cient close to 1 indicates

a strong linear relationship, while a coe�cient close to zero indicates a weaker linear

relationship [45].

In appendix C the correlation matrix is presented with a 95% and 99% signi�-

cance level. This is done using r as an estimator in hypothesis testing for the true

correlation coe�cient ρ6, assuming X and Y are normally distributed [46, p. 462].

3.3.1. Correlation in Asset Classes

In general, the vast majority of the correlation coe�cients are statistically signi�-

cant. The correlation in stock indices is positive and statistically signi�cant, with

coe�cients ranging from 0.13 for S&P 500 and MSCI Japan, and 0.95 for MSCI

World Value and MSCI World Growth. This con�rms the authors' expectations

from section 3.2.1. The relatively low correlation between S&P 500 and MSCI Japan

indicates a Japanese business cycle that has somewhat been disconnected from the

S&P 500 �uctuations [13, p. 461]. On the other hand, a possible explanation for the

high correlation in stocks could be that as the �nancial stock markets grow more

interconnected, the correlation between the stocks will converge to one [44].

The bond indices include both the near risk-free Treasury STRIPs as well as the

riskier BOFAMerrill-Lynch Corporate bond index. The correlation pattern in bonds

is very high, and thus the hypothesis of no correlation is rejected. As a consequence,

investing solely in bonds does not o�er a great diversi�cation e�ect regardless of re-

gion, bond classi�cation or maturity. Further, it is expected that without any weight

restriction on asset classi�cations in the portfolio, the bonds will dominate the port-

folio due to their attractive risk-adjusted return over the observed period which will

be examined in chapter 4. Consequently, this causes limited diversi�cation across

6H0 : ρ= 0, is tested by a two-tailed test shown below at both a 95% and 99% con�dence level,with n = 3074. The sample correlation coe�cient is denoted by r, which is the estimate ofρ,also referred to as the Pearson product moment correlation t(n−2) =

r√(1−r2)(n−2)

[46, pp. 461-3].

26

multiple asset classi�cations.

The highest correlation in the matrix is found within commodities, where the S&P

GSCI Commodity index and TR/Je�eries CRB index has a correlation of 0.954. This

may indicate a high concentration of the same commodities in the indices. Adding

to the fact that S&P GSCI Gold index has a very attractive risk-adjusted return,

the gold index will assumable be favored above the other two commodity indices in

the portfolio optimization in chapter 4.

The REIT indices have been extremely volatile, which were con�rmed in section

3.2.5. Additionally, the correlation is generally moderately positive and statistically

signi�cant ranging from 0.11 for US and Asia, and 0.42 for UK and Asia. The

relatively lower correlation for US and Asia is caused by recent uptrends in the US

REIT index, whereas the latter has very attractive risk-adjust return properties.

It can be concluded that allocating assets in one asset category only o�ers a slight

diversi�cation potential as the assets tend to be highly correlated. This highlights

the importance of allocation across asset categories.

3.3.2. Cross-sectional Correlation

In this section, the correlation pattern of stocks will be compared with the other

asset categories with the purpose of investigating the potential for diversi�cation

across the chosen categories. Stocks will be used because it is argued that they will

serve as su�cient evidence to illustrate the possible increased diversi�cation poten-

tial when mixing asset categories in the portfolio.

The correlation for currencies and stocks illustrate a moderate negative correla-

tion, but also several statistically insigni�cant coe�cients. The moderate negative

correlation in some of the cases may prove bene�cial when creating e�cient portfo-

lios in chapter 4, even though currencies have shown low risk-adjusted returns.

The somewhat same trend is evident for bonds and stocks. Compared to cur-

rencies, bonds have shown very attractive risk-adjusted returns which make them

27

favorable to currencies. Further, there is evidence from the correlation analysis of

a negative relationship between the Asian stock indices and US/EU bonds. This

relationship is favorable to the risk reduction of the portfolio, holding all else equal.

From an empirical point of view, gold has been seen as a useful diversi�cation tool

for stock-based investors due to its low or even negative correlation with stocks [47].

This fact is con�rmed by the high performance of the gold index over the last �ve

years, indicating that when the �nancial crisis incurred, the investor sought gold as

safe investment alternative to stocks.

Finally, the particularly volatile REITs are highly correlated to stocks, which

con�ne the diversi�cation e�ect, and assumedly, make these risky indices slightly

unattractive when performing portfolio optimization. However, the MSCI Japan

and the US REIT index coincide with the bonds versus stock conclusion, showing

statistically insigni�cant correlation, emphasizing a weak relationship.

It can be concluded that allocating across multiple asset categories may o�er

diversi�cation potential as the correlation patterns are not as highly correlated as

within each asset category.

The correlation analysis presented above assumes a linear relationship between X

and Y, and the assumption of normal distributed returns. The latter property will

be put to the test in section 5.1.1. As a consequence of the possible detour from

the assumption of normality, the conclusions from the correlation analysis should be

used with caution as is may produce a faulty risk metric and asset allocation.

The next section will investigate the correlation between asset categories in posi-

tive and negative return periods.

3.3.3. Conditional Correlation

The correlation analysis above assumes a linear relationship between variables and

that the correlation between assets is constant over time. In order to hedge poten-

tial loss, the investor holding a diversi�ed portfolio prefers a negative correlation

pattern when the market is characterized by a downward trend. However, if the

overall �nancial market shows an upward trend, the investor would here prefer a

positive correlation pattern between portfolio assets. To analyze this, each asset's

return distribution is subdivided into positive and negative arrays and compared

to a benchmark index. In appendix D the vertical axis represents the benchmark

index in the matrix.

ρ+ =ΣNi=1

(X+i − X+

) (Yi − Y

)σ+XσY

(3.4)

X+i denotes the positive observations andYi denotes the benchmark average return

for the assets. Equation 3.4 is adjusted to analyze the correlation pattern during

positive and negative return periods.

Examining the matrix in appendix D it is apparent that the correlation in pos-

itive and negative periods can diverge signi�cantly. It is especially interesting to

observe the S&P Gold index, as it in both stocks and REITs illustrates positive

correlation patterns in positive return periods and negative correlation patterns in

negative return periods. However, these coe�cients are not statistically signi�cant.

Conversely, other benchmarks such as MSCI World Value and S&P 500 show

statistically signi�cant negative coe�cients to the gold index. This correlation, as

mentioned above, is favorable to the investor. These stock indices emphasize the

conclusion from section 3.3.2 that the gold index is a useful diversi�cation tool for

the equity-based investor. This also strengthens the assumption that gold will be a

very advantageous asset when generating an e�cient portfolio.

Benchmarks in stocks, currencies, commodities and REITs to The BOFA Merrill

Lynch U.S. Corporate bond index illustrate the somewhat same trend as for the

S&P Gold index. The statistical signi�cance of the coe�cients is not conclusive,

though. The correlation to the UK REIT illustrates statistically signi�cant positive

correlation patterns in positive time-periods and negative correlation patterns in

dismal time-periods. Ceteris paribus, this correlation is favorable to the investor

as both assets tend to move in opposite direction when either illustrates negative

returns. TR/Je�eries commodity index and GBP currency index to the BOFA ML

U.S. Corporate bond index signals favorable correlation as both benchmarks return

tend to go in the opposite direction when that bond produce negative returns. Espe-

cially TF/Je�eries commodity index illustrates statistically signi�cant coe�cients,

which could be used as a hedging tool in a portfolio with the U.S. Corporate bond

index. MSCI Emerging Markets and U.S. Corporate bond index shows the same

hedging properties, however, the correlation is found to be statistically insigni�cant.

Comparing the linear correlation and conditional correlation it can be argued

that the correlation is not constant over time. A recent study on correlation across

a number of international stock market indices �nds empirical evidence that the

correlation between sampled index returns is changing over time [48]. Another em-

pirical study �nds that the correlation of monthly excess returns for seven major

countries in the period 1960-1990 is unstable. The hypothesis of conditional con-

stant correlation was also rejected [49].

Research has further indicated that when extreme and sudden unforeseeable changes

stress the �nancial market, correlation across asset classes will diverge towards one.

Compared with the overall globalization of the �nancial market as already men-

tioned, the diversi�cation e�ect in these stressed �nancial situations will be greatly

reduced [50].

It can be concluded that the assumption of constant correlation over an observed

period will not properly re�ect the true diversi�cation e�ect and the risk estimate

should be used with caution. By having an estimation period dating back to 2001

and ending in 2012, the global �nancial crisis of 2008 will greatly a�ect the correla-

tion pattern of the asset categories. It is therefore found necessary for the purpose

of this dissertation to create a rolling-period optimization model that captures the

changing correlation pattern over time and shows a changing risk estimate from a

changing correlation pattern.

This chapter has served to characterize the statistical properties of the selected

return indices. This data analysis has created the foundation of the dissertation and

30

will be applied to the theory once it is presented. The next chapter is the funda-

mentals of portfolio theory, and it will illustrate how the indices will be allocated

based on the statistical characteristics.

31

Part III.

Portfolio Theory

32

4. Portfolio Optimization

In �nance, portfolio theory is based on the subset that all investment opportunities

are more or less accompanied by a certain level of risk, which makes the expected

return unknown to the investor. By holding multiple assets that do not correlate

in perfect synchrony, it is possible for the investor to diversify the risk compared to

holding a single asset. To achieve the optimal reward to risk combination, deter-

mined by the investors risk aversion, the investment can be allocated among various

asset categories such as the ones presented in the data analysis.

4.1. Mean-Variance Optimization

Creating e�cient asset allocation strategies was introduced by Harry M. Markowitz.

Markowitz's Mean-Variance Optimization (MVO) theory has been the standard for

e�cient portfolio selection for more than half a century. Markowitz showed how

an investor could construct optimal portfolios by the return distribution's historical

mean and variance. The essential part of the framework was that an investor could

minimize portfolio risk by combining risky assets and achieve the diversi�cation ben-

e�ts [14, 51]. In the following section measures of return, variance and co-variance

are de�ned.

The literature describes multiple methods on how to conduct portfolio optimiza-

tion. This dissertation utilizes the approach for MVO described in 'Financial Mod-

eling' by Simon Benninga (2008), 'Modern Portfolio Theory' by Elton and Gruber

(1995), and 'Aktie investering - Teori og paktisk anvendelse' by Christensen and

Pedersen (2003) [52][45][53].

33

Initially, the measures for return and risk can be seen in equation 4.1 and 4.2 [45,

pp. 55-60]. The expected return of a portfolio, RP , is given by

RP =N∑i=1

(XiRi

)(4.1)

Where Xi and Ri, represent the fraction of asset i invested in the portfolio, and

the expected return of asset i respectively. Furthermore, Xi must sum to one. The

second measure is the variance, which is a measure of how much the returns deviate

from the expected return of the portfolio, which is given by

σ2P =

N∑i=1

X2i σ

2i +

N∑i=1

N∑j=1

XiXjσij (4.2)

Again, Xi must sum to one, and σij denotes the co-variance of asset i and j.

The covariance term indicates how the asset moves together. A negative covariance

means that the assets tend to move in opposite directions and vice versa.

The covariance is given by

σij = σiσjρij (4.3)

σ denotes the standard deviation for asset i and j, which is simply found by taking

the square root of the variance, and the correlation coe�cient, ρij, has already been

de�ned in section 3.3.

Via the use of the MVO framework it is possible to reduce the risk in a portfolio

by combining the assets. This diversi�cation e�ect and minimization of risk are

best achieved if the co-variation and correlation between the assets are negative.

However, the total risk contains systematic and unsystematic risk. It is pointed

out that it is only theoretically possible to eliminate the unsystematic part of the

risk by diversi�cation when increasing the number of assets in the portfolio. The

systematic risk cannot be eliminated by diversi�cation as it is determined by the

external �nancial environment where the investor has no control [40, 54, p. 167; p.

481].

Using the two measures of return and risk, it is possible to construct a portfolio

34

that seeks to allocate assets in order to optimize the return at a given level of risk.

So far, the variance has been used to quantify the risk of the portfolio. However,

to get the risk term in the same dimension as the expected return, the variance is

squared. Henceforth, the risk term is dimensioned as standard deviation.

The optimization process resulting from the mean and standard deviations, will

construct an e�cient frontier of portfolios, which are investigated in the next section.

4.1.1. The E�cient Frontier

All portfolios on the e�cient frontier satisfy the criteria of having the highest ex-

pected return for a given risk level, or the lowest possible risk for a given expected

return. These portfolios are mathematically constructed by solving the minimization

problem given below with Excel Solver

Min(σ2P

)= ΣN

i=1X2i σ

2i + ΣN

i=1ΣNj=1XiXjσij (4.4)

s.t. ΣNi=1Xi = 1

0 ≥ Xi ≤ 1

¯RP = c

The top two limitations ensure that the portfolio weights sums to one, and that

the portfolio weights cannot be negative, i.e. short-sale restrictions are imposed.

The bottom restriction ensures a minimization of the portfolio variance for a given

return, denoted by c [52]. In iterative steps the constant c is increased, and the

minimization problem is solved accordingly, resulting in an envelope curve of e�cient

portfolios, see �gure 4.1 below.

35

Figure 4.1.: MVO e�cient frontier

Note: The grey dot at the far left is not actually tangent to the e�cient frontier, as stated in the text

The e�cient frontier illustrates that by combining the assets, an investor is able

to obtain a better risk to return results, relative to holding a single asset. As stated

above, the rationality for a better return-to-risk result is due to the correlation of

the portfolio's assets, which enable the investor to obtain a diversi�cation bene�t

[53]. Figure 4.1 above further illustrates where the indices are placed in the return to

risk relationship analyzed in chapter 3. Whereas bonds and currencies have proven

less volatile; stock, commodities and REITs indicated that they increasingly more

risky alternatives. Observing the e�cient frontier the portfolio with minimum risk

actually shows a standard deviation below that of the 1-year STRIP, which is the

least risky asset in the observed period. This emphasizes the diversi�cation bene�t

as this portfolio combination contains the 1-year STRIP, but also MSCI World

industry indices and currencies.

The allocation of the e�cient portfolios can be seen in �gure 4.2. In the iterative

process of increasing the constant c, the e�cient portfolios are initially constituted

by currencies and bond indices due to their low volatility. Moving further along

the e�cient frontier, the e�cient portfolios tend to be dominated by the S&P Gold

index to satisfy the higher return criteria.

36

Figure 4.2.: E�cient portfolios allocation

Note: The x-axis represents each portfolio return, given by c and accordingly solved by the minimization problem.

The y-axis shows the weigth of each asset

Observing the e�cient frontier in �gure 4.1 there are especially two of the portfolio

combinations which are interesting in terms of further examination; the Minimum-

Variance Portfolio (MVP) and the Optimal Portfolio (OP).

The MVP is that combination of assets which results in the lowest possible risk.

To �nd the MVP the minimization formula aforementioned is used without the last

restriction.

The OP can be found in many ways. One example is to determine the risk pro�le

of the investor and by marginal utilization determine the trade-o� between increased

returns and the investors risk aversion [45]. This method is however very subjective

so instead it is assumed the investor will always try to increase the return at a given

level of risk.

The OP can therefore be explained by the linear relationship between the e�cient

frontier and the risk-free rate, known as the CML mentioned in section 3.1.

rP = rf +rP − rfσP

σP (4.5)

The CML is determined by the risk-free rate and the standard deviation of the

portfolio multiplied by Sharpe's ratio [53].

When Sharpe's ratio is maximized in the formula, the CML will tangent the e�cient

frontier at a point known as the tangent portfolio or in this case the OP. The OP

assumes all wealth is invested in the risky asset. It is possible to move up or down

37

the CML by either levering the OP or allocating more in the risk-free rate.

To �nd the OP, which yields the highest return relative to its risk, the following

maximization problem is solved

Max (RTV ) =rP − rfσP

(4.6)

s.t. ΣNi=1Xi = 1

0 ≥ Xi ≤ 1

The �gure below illustrates the MVP, OP and CML based on MVO from the

empirical data.

Figure 4.3.: MVO e�cient frontier with OP, and MVP

The MVP has a return of 2.51% at a risk level of 0.85% standard deviation p.a.

whereas the OP has a 7.71% return with 4.40% standard deviation p.a. As MVP

only tries to reduce volatility, RTV actually becomes negative as MVP return falls

below the risk-free rate.

The positions the MVO has selected in the various indices can be seen in table 4.1.

Holding these weights stationary over the estimation period would result in the risk

and return estimates mentioned above. The MVP of course targets the asset with

lowest volatility, and therefore allocates almost 94% to the 1-year STRIP. The rest

is allocated in the currencies and insigni�cant small weights are spread across few

38

stock indices. Any portfolio below MVP is not an interesting point for examination

because the investor is induced with higher risk for less return.

Minimum-Variance Portfolio % Optimal Portfolio %

Currency 3.51 Currency 0.00

GBP 0.60 Bonds 73.97

JPN 0.09 US Treasury 5-year STRIP 16.76

USD 2.82 BOFA ML EU Gvt. 22.01

Bonds 93.83 BOFA ML US Corp. 35.20

US Treasury 1-year STRIP 93.83

Commodities 0.62 Commodities 11.61

TR/Je�eries CRB 0.62 S&P GSCI Gold 11.61

S&P GSCI Gold 0.06

REITs 0.00 REITs 2.37

Stocks 2.05 FTSE EPRA/NAREIT US 2.37

MSCI World Consumer Staples 0.46

MSCI World IT 0.19 Stocks 14.42

MSCI World Telecomm. 0.27 MSCI Emerging Markets 6.14

MSCI Emerging Markets 0.24 MSCI World Consumer Staples 5.91

S&P 500 0.52

MSCI Europe 0.37

Std. deviation 0.85 Std. deviation 4.40

Expected return 2.51 Expected return 7.71

RTV -1.31 RTV 0.93

Table 4.1.: MVP and OP portfolio allocation weights

Note: return, sigma, min, and max are denoted in percentages

The OP is more interesting to examine. As analyzed in section 3.2.2, the return

and risk characteristics of bonds have made them a favorable asset category to hold

for the investor. With low volatility and reasonable high return it means that,

the MVO allocates 74% of bonds to the OP, divided between 5-year STRIPs, U.S.

Corporate bonds, and EU Government bonds. S&P Gold represents 12% whereas

the rest of the weights is divided with 2.4% U.S. REITs, 5.9% MSCI consumer

staples, and 6.14% MSCI Emerging Markets.

The characteristics of bonds over the estimation period make the MVP and

OP questionable to whether they actually provide a well-diversi�ed portfolio, and

thereby present a valid picture of an investor's allocation preferences. Imposing re-

striction on the asset categories and single indices is therefore assumed to create a

more correct picture of an investor's portfolio allocation and risk-aversion. This is

39

done in order to present a more real-life estimate of the risk measure di�erence if the

empirical return distributions break the assumption of being normally distributed.

The restrictions will be imposed and elaborated in chapter 7.

This chapter has shown the data applied to the MVO theory in order to explain

and show the derived risk estimation. Both portfolios will be analyzed in the dis-

sertation in order to investigate, how the risk minimization focus in the MVP or

the return-to-variability maximization focus in the OP, will impact the degree of

asymmetry in the portfolio. In the next chapter the higher moment characteristics

of the empirical return series will be closely examined for non-normality.

40

Part IV.

Non-normality

41

5. Higher Moment Orders

5.1. Moments of the Distribution

It has so far been assumed that the return distribution is symmetrical around the

mean. However, solely relying on the �rst two moments of the return distribution;

the mean and standard deviations, will not account for the occurrence of fat-tailed

return distributions.

Figure 5.1.: The return distribution of the OP �tted to the normal curve

Observing the return distribution for the OP in �gure 5.1 derived from the MVO,

it exempli�es the normal distribution inability to capture the real loss exposure to

the investor.

Solely relying on the �rst two moments implicitly means that the return distribu-

tion is assumed to form a symmetric bell-shaped curve known as the Gaussian or

normal distribution [55].

The bell curve is shaped around the mean making returns on both the left and

right side equally probable. To describe the probability of these returns, the stan-

dard deviation is used. For instance, the probability of getting a return within one

42

standard deviation of the mean is 68.26% whereas a return within two standard

deviations is 95.45%, and �nally 99.73% with three standard deviations. Thus, the

higher a standard deviation the more dispersed returns are from the mean [51].

These properties of the normal distribution leave very small probabilities for ex-

treme returns. Therefore, the sole focus of the �rst two moments and meaningless

small probabilities has been the subject for heavy critique over the last couple of

years. It is approximated that empirical extreme events occur 10 times more often

than what the normal distribution is able to predict [2, 9, 56].

This is exempli�ed with the S&P 500 where the normal distribution estimates a

0.13% probability of returning less than -15.64% of S&P 500 a month over an 85-

year period. The 0.13% probability corresponds to less than two months of the 1,025

total months. However, examining the data shows ten periods where the monthly

return is below -15.64%. Therfore, the normal distribution is not well suited to �t

the occurrence of extreme returns in the S&P 500 index as it underestimates the

probability of it happening [51]. The MVO framework assumes a symmetric bell-

shaped curve, and this implies that the framework is not well suited for asset classes

with asymmetric return distributions which are often found to be present in many

�nancial return series [1].

In order to better capture the returns beyond the bell curve, here follow an in-

troduction of the third and fourth moments. The third moment is skewness and

is a statistical measure that describes how symmetrical the returns are distributed

around the mean. Skewness di�erent from zero indicates that the distribution is

asymmetric. In the normal distribution, skewness equals zero. Skewness is de�ned

as

Skewness =ΣNi=1 (xi − x)3

σ3(5.1)

Positive skewness indicates that the distribution is right-skewed, meaning the ma-

jority of observations is found on the left side of the mean causing a long tail going

right. Negative skewness is left-skewed and results in the tail going left. Aside from

high returns and lower standard deviation, investors naturally have preference for

return distributions with positive skewness.

43

The fourth moment is the kurtosis and is a measure of the relationship between

the tails and the peakedness of the distribution. The kurtosis is relative to the

normal distribution. The normal distribution assumes a kurtosis of three, but if the

distribution in question is higher than this, it is known as being leptokurtic. This

means positive excess kurtosis, which increases the likelihood of extreme returns and

consequently fatter tails.

A kurtosis below three indicates negative excess kurtosis and the distribution are

then called platykurtic. Kurtosis and excess kurtosis is de�ned as

Kurtosis =ΣNi=1 (xi − x)4

σ4Excess kurtosis =

ΣNi=1 (xi − x)4

σ4− 3 (5.2)

By not accounting for skewness and excess kurtosis in the dataset, the MVO will

provide a faulty risk estimate. Hence, the next section will determine the level of

asymmetry in the empirical indices in order to assess the e�ect on the portfolio risk

estimates.

5.1.1. Test for Normality

Various tests exist that examines non-normality in the sample distribution. The

most commonly known is the Jarque-Bera test (JB) that utilizes the third and

fourth moment in order to determine the degree of asymmetry [57]. Other popular

tests for normality is the two goodness of �t tests, Shapiro-Wilk and Kolmogorov-

Smirnov, which compares the theoretical distribution function with the empirical

distribution function. Because JB, contrary to the two goodness of �t test, can

be applied to a wider range of sample sizes, it is considered superior and therefore

su�cient for the purpose of testing the empirical dataset, used in this dissertation,

for normality [58].

The test asymptotically follows the chi-squared distribution with two degrees of

freedom [59]:

JB = n

[Skewness2

6+

(Kurtosis− 3)2

24

]v χ2

k=2 (5.3)

44

The critical value for the JB test is 5.99 at a 95% con�dence level. Hence, if the

return distribution results in a JB value above 5.99, the null-hypothesis of normality

is rejected.

Table 5.1 displays the higher moment characteristics of the index return distribu-

tions in the dataset. It can be observed that the majority of indices have left-skewed

distributions and several indicates leptokurtic properties. Especially the more risky

assets like stock, REITs, and commodities illustrate high kurtosis coe�cients. This

can be tied to the extreme decreases these asset classes experienced in 2008. This

causes the rejection of normality by the JB-test in all but one of the indices. Only

the 5-year U.S. Treasury STRIP ful�lls the criteria for the returns being normally

distributed at the 95% con�dence level.

45

Skewness Kurtosis JB Probability

GBP -0.21 3.25 29.9 0.00

JPN 0.31 4.02 182.2 0.00

USD -0.10 2.55 31.1 0.00

U.S. Treasury 1-year STRIP 0.28 6.52 1,630.0 0.00

U.S. Treasury 5-year STRIP -0.07 2.92 3.7 0.16

U.S. Treasury 10-year STRIP -0.10 1.96 142.6 0.00

BOFA ML EU Gvt. -0.07 1.60 253.4 0.00

BOFA ML Global Gvt. Index 0.23 3.32 39.5 0.00

BOFA ML US Corp. -0.38 2.43 117.3 0.00

S&P GSCI Commodity -0.28 2.50 72.4 0.00

TR/Je�eries CRB -0.31 2.82 54.6 0.00

S&P GSCI Gold -0.21 4.46 293.6 0.00

FTSE EPRA/NAREIT US -0.22 15.58 20,299,4 0.00

FTSE EPRA/NAREIT UK -0.30 5.98 1,180.4 0.00

FTSE EPRA/NAREIT Dev. Asia -0.73 7.41 2,758.9 0.00

FTSE 100 -0.14 6.28 1,390.0 0.00

MSCI World Industrials -0.37 5.11 639.2 0.00

MSCI World Consumer Disc. 0.12 7.85 3,018.3 0.00

MSCI World Consumer Staples -0.35 8.92 4,545.8 0.00

MSCI World Energy -0.57 9.44 5,484.6 0.00

MSCI World Financials -0.10 9.02 4,636.1 0.00

MSCI World Health -0.24 8.25 3,558.2 0.00

MSCI World IT 0.14 4.57 324.8 0.00

MSCI World Materials -0.48 7.39 2,583.3 0.00

MSCI World Telecomm. 0.01 5.20 622.2 0.00

MSCI Utilities -0.14 12.87 12,485.8 0.00

MSCI Japan -0.32 7.16 2,265.2 0.00

MSCI AC Asia Pac. ex. Japan -0.53 6.44 1,660.1 0.00

MSCI Emerging Markets -0.50 7.77 3,045.7 0.00

S&P 500 -0.17 8.25 3,539.6 0.00

MSCI World Growth -0.25 7.00 2,085.2 0.00

MSCI World Value -0.24 6.99 2,068.0 0.00

MSCI Europe -0.09 6.14 1,270.4 0.00

Table 5.1.: Test for normality

Many of the indices actually have kurtosis below three. As just stated, this is

known as platykurtic distributions and will have a thinner tail e�ect than the normal

distribution because of less dispersed observations [46, p. 33]. These characteristics

will, however, also reject the assumption of normality. Hence, the MVO theory will

in these cases overestimate the risk exposure provided skewness is zero.

It should be noted that skewness often tends to be limited when the number of

assets increase [60, p. 24]. The emphasis should therefore be put on the properties

46

of kurtosis. Kurtosis was additionally found to have the greatest impact on the

JB-test exceeding the critical value for normality.

The index that has incurred the most extreme returns, of which the normal dis-

tribution would have failed to account for, is the FTSE EPRA/NAREIT U.S. with

a kurtosis of 15.58. Combining this fact with a skewness of -0.217, the JB-test

strongly rejects the hypothesis of normally distributed returns. The majority of the

returns are placed on the right side of the mean but there is a tail-e�ect going left

with several extreme negative returns. By not taking these higher moment charac-

teristics of the FTSE EPRA/NAREIT U.S. into account, the normal distribution

will underestimate the probability of extreme loss and provide a deceivingly lower

risk approximation for that particular index. To see the actual distributions of the

indices �tted in normal distribution see appendix B.

The empirical literature strengthens the �ndings in the dataset, as it has revealed

that empirical �nancial return series often appear skewed and fat-tailed [1]. Some of

the index return series only show minor �ights from normality, such as the British

pound and the U.S. dollar. The true risk estimate will in these instances only slightly

deviate from what the standard deviation provides. As the empirical distributions

gets increasingly skewed and leptokurtic, the true risk approximation moves further

away from the MVO theoretical estimate. This fact emphasizes the importance of

adjusting for skewness and kurtosis in order to obtain the true risk of the portfolio

over the historical period.

Consequently, it will be attempted in section 7.4 to adjust the return series for

non-normality by including the higher moments and thereby provide a more correct

risk measure.

47

6. Expansions

As the MVO theory only uses the �rst two moments, the expansions in this chapter

provide a way to approximate the quantiles of an unknown distribution based on

the �rst four moments. Hence, this will result in an asymmetry corrected risk esti-

mate for the optimized portfolio. In addition, the latter section of the chapter will

highlight how asymmetry a�ect the quantile multiple estimation.

It is possible to calculate quantile approximations of the standardized distribu-

tion and to consider these approximations to the corresponding quantiles of the

actual distribution. Hence, when the normality assumption breaks, the idea is to

correct the discrepancies arising from normal quantiles by including skewness and

kurtosis in the calculation. Basically, these expansions are an approximate relation

between the percentiles of a distribution and its moments [12]. When returns are

normally distributed, it makes it possible to estimate the quantile of the distribution

corresponding to the threshold. This implies the random variable X follows

X ∼ N(µ, σ2

)(6.1)

The variable can therefore be transformed to the standard normal variable

X = µ+ zασ (6.2)

Where zα is the threshold of probability for risk estimation. However, as the

dataset shows, it often occurs that �nancial series tend to be skewed and leptokurtic.

Equation 6.2 therefore proves insu�cient when estimating the correct risk exposure

to the investor when these two properties are present in the unknown distribution.

The Cornish-Fisher and Gram-Charlier expansions include all of the �rst four mo-

48

ments of the target distribution, and thereby they derive an approximate distribution

and adjusted quantile [16]. In the expansions the terms are polynomial functions

of the appropriate unit normal quantile with coe�cients that are functions of the

moments of the target distribution. This leads to an analytic approximation of the

quantile as long as the moments of the distribution are known [12].

Due to the vast extent of derivation for these approximations, the derivations have

not been included in the dissertation. Only the main formulas will be explained and

the expansions will be assessed numerically. Details on the derivation of Cornish-

Fisher expansion can be seen in [17] and details on the Gram-Charlier expansion

can be seen in [16]. Chapter 9 will further investigate the drawbacks and validity to

the use of Cornish-Fisher and Gram-Charlier approximations on the empirical data.

The approach for determining adjusted risk estimates from the approximations

is based on the literature by Simonato Jean-Guy (2011), Peter Zangari (1996),

Amédée-Manesme & Barthélémy (2012), and Didier Millard (2012)[16][32][12][18].

6.1. Cornish-Fisher

Often used in Value-at-Risk theory, the Cornish-Fisher expansion is a mathematical

expression and the outcome of the 'Moments and Cumulants in the Speci�cation of

Distributions ' by E.A. Cornish and R.A. Fisher (1938) [15]. It approximates the

quantiles of random variables based on the �rst four moments [18]. As stated above,

the overall idea of Cornish-Fisher expansion is to adjust the discrepancies arising

from normal quantiles.

The Cornish-Fisher transformed quantile function can be written as [16]

yCF = µ+ σφ−1CF (p, S, K) (6.3)

Where yCF is the estimated value at the threshold from the sample distribution

and φ−1CF is the Cornish-Fisher adjusted quantile of a standard normal random vari-

able evaluated at p. S represents skewness and K represents excess kurtosis.

In its original form from 1938 the Cornish-Fisher expansion can be written as

49

xα = µx+1

2

(σ2x + 1

)φ−1N +

1

6

[(φ−1N

)2 − 1]S+

1

24

[(φ−1N

)3 − 3φ−1N

]K− 1

36

[2(φ−1N

)3 − 5φ−1N

]S2

(6.4)

Where φ−1N is the standardized variable zα evaluated at p.

By standardizing the sample distribution, the term xα becomes φ−1CF = xα−µ

σ. This

implies the mean is zero and the variance is equal to one. Consequently, the �rst

term of equation 6.4 drops out and becomes the Cornish-Fisher adjusted quantile

[17, Ch. 12]:

φ−1CF = φ−1

N +1

6

[(φ−1N

)2 − 1]S +

1

24

[(φ−1N

)3 − 3φ−1N

]K − 1

36

[2(φ−1N

)3 − 5φ−1N

]S2

(6.5)

φ−1CF adjusted from the standard variable zα given values for S and K must now be

multiplied by the sample distribution's standard deviation and added to the mean,

as given in equation 6.3 in order to obtain the threshold [16]. As this dissertation

does not aim to obtain a value-at-risk measure, the adjusted risk measure from the

Cornish-Fisher expansion that will be interesting to observe is illustrated in equation

6.6.

σCF = σφ−1CF (6.6)

This measure allows the investor to obtain a risk approximation adjusted for an

asymmetric return distribution and compare it with the MVO standard deviation

over the historical estimation period at the chosen percentile.

6.2. Gram-Charlier

The Gram-Charlier expansion is as the Cornish-Fisher expansion an approximation

of an unknown sample distribution based on the �rst four moments in order to ob-

tain the adjusted quantile. The Gram-Charlier distribution has a density that is a

polynomial times a normal density function [26].

50

As in Cornish-Fisher it is possible to derive the transformed quantile function in

equation 6.7 based on the third and fourth moments by

yGC = µ+ φ−1GC (p, S, K) (6.7)

S represents skewness and K represents excess kurtosis. As equation 6.7 shows, it

also requires the adjusted quantile φ−1GC for the Gram-Charlier approximate density

function. Obtaining this quantity can be done by using the cumulative distribution

function that corresponds to the Gram-Charlier approximate density function. The

expression for the Gram-Charlier approximate cumulative distribution is given below

φGC (k , S, K) = φN −S

6

[fN(k2 − 1

)]− (K − 3)

24

[fNk

(k2 − 3

)](6.8)

evaluated at k1. Where the standard normal distribution is given by

φN =1

σ√

2Πe

−(x−µ)2

2σ2 (6.9)

And the density function is given by

fN =1

2

[1 + erf

(x− µ√

2σ2

)](6.10)

The distribution φGC must then be inverted numerically to obtain the Gram-

Charlier quantile φ−1GC . With a numerical search, φ−1

GC can then be computed using

Goal Seek in Excel by solving k for φGC (k, S, K) = p, where p is the signi�cance

level.

As in Cornish-Fisher, the adjusted risk measure from the Gram-Charlier expansion

that will be interesting to observe is illustrated in equation 6.11.

σGC = σφ−1GC (6.11)

1To see the derivation of equation 6.8 please refer to Appendix B in the article by Jean-GuySimonato

51

It has been shown how it possible to adjust the risk estimation for asymmetry.

Hence, the next section will illustrate the impact of asymmetric return and how

the Cornish-Fisher and Gram-Charlier expansions result in alternative quantile es-

timates compared to the standard normal quantile.

6.3. Non-normality E�ects on Quantile Estimation

The adjusted risk measure from the expansions seeks to estimate a multiple, φ−1x ,

associated to the standard normal distribution, in order to take into account the

skewness and excess kurtosis. The e�ect of non-normality is illustrated below in a

cumulative distribution function. The �gure to the left illustrates the cumulative

distribution function, and the �gure to the right enlarges the left tail.

For a left skewed distribution, the smallest approximate quantiles that the expan-

sions derive are lower than the standard normal quantiles, which ultimately result in

a higher standard deviation compared to the standard normal distribution deviation.

This is illustrated in �gure 6.1a, where a skewness of -1, results in φ−1CF0.05

= −1.910.

φ−1GC0.05

= −1.99 compared to φN0.05 = −1.645. The opposite is true for a right-

skewed distribution, where the smallest percentiles lead to quantiles greater than

the standard quantiles, and which result in a lower standard deviation compared to

the normal distribution estimate. This is evident in �gure 6.1b, where a skewness of

0.5 results in φ−1CF0.05

= −1.498, and φ−1GC0.05

= −1.522 compared to φN0.05 = −1.645

52

(a) Left skewed distribution: skewness = -1, kurtosis = 3

(b) Right skewed distribution: skewness = 0.5, kurtosis = 3

Figure 6.1.: Cornish-Fisher, Gram-Charlier, and normal quantiles in skewed distributions

Figure 6.2a illustrates a leptokurtic distribution that leads to lower quantiles for

both expansions compared to the normal standard quantile. However, at the 5th

percentile, the Cornish-Fisher expansion results in a larger quantile estimate com-

pared to the normal.

A platykurtic distribution leads to higher Cornish-Fisher and Gram-Charlier quan-

tiles at the lowest percentiles, which is illustrated in �gure 6.2b. However, this

conclusion is not decisive at the 5th percentile, as Cornish-Fisher result in smaller

quantiles compared to the standard normal ones.

53

(a) Leptokurtic distribution, skewness = 0, kurtosis = 4

(b) Playtokurtic distribution, skewness = 0, kurtosis = 2.5

Figure 6.2.: Cornish-Fisher, Gram-Charlier, and normal quantiles in distributions with

excess kurtosis

Figure 6.3 illustrates that the greatest adjustment to the standard normal quan-

tiles will occur in the case of left skewed and leptokurtic distribution. Also, as

the asymmetry increases, the cumulative distribution function derived from the ex-

pansions deviates more from normality. Besides having fat-tails, the curve grows

increasingly steep, hence, �tting the function to the characteristics of the return

series.

Figure 6.3.: Left skewed and leptokurtic distribution, skewness = -1.5, kurtosis = 6

It can be concluded from the analysis that skewness and kurtosis coe�cients have

a large impact on the approximated quantile. It can be seen that the expansions

54

provide alternative estimates of the true risk measure the investor encounters as

the expansions capture the fat-tailed characteristics of the asymmetric distribution.

This is particular evident at the lowest quantiles, and they will consequently result

in a higher standard deviation compared to the estimate arising from the mean-

variance theory. Especially the Cornish-Fisher expansion will approximate lower

quantiles at the lowest percentiles whereas the opposite happens at the higher per-

centiles, which makes the choice of con�dence interval extremely important.

It has been shown that the risk estimates derived from the expansion di�er from

each other at various percentiles and at various degrees of asymmetry. This will

consequently mean that MVO will either underestimate or overestimate the true ex-

posure from the speci�c estimation period. Simonato (2011) also �nds that the two

expansions di�er dependent on the con�dence level and degree of asymmetry. His

results coincide with the results illustrated above, as he likewise �nds Cornish-Fisher

to provide lower quantiles at the lowest percentiles compared to Gram-Charlier [16].

This dissertation serves to portray alternative risk measure to what MVO derives,

so the authors will not explain from an analytical perspective why they di�er. Thus,

the conclusion is solely drawn from numerical investigation and examples.

In the next part of the dissertation, the empirical data will be applied to the three

models in order to investigate and quantify the di�ering risk estimates.

55

Part V.

Empirical Results

56

7. Risk Comparison

In chapter 5 it was established that the empirical data showed asymmetric character-

istics. Thus, the purpose of this chapter is to investigate and quantify the potential

risk estimation di�erence after applying the Cornish-Fisher and Gram-Charlier ex-

pansions. This analysis serves to identify the error in risk estimation that MVO

yields over the estimation period.

Initially, the standard deviations of the OP and the MVP resulting from MVO will

be compared with the two expansion estimates of the portfolios' standard deviations

following the correction of asymmetry in the return distributions. As the allocation

remains the same, this comparison will focus solely on the risk estimate. This is

done in order to quantify and highlight the di�erence of the standard deviations in

the presence of asymmetry.

Secondly, the expansions will be applied on every asset return series, in order to

obtain standard deviations adjusted for asymmetric returns, hence, creating a new

frontier with alternative risk measure and asset allocations. This will lead to the cor-

rect standard deviations for every index, and thereby result in asymmetry-adjusted

optimized portfolios. Additionally, it will enable both the comparison of risk esti-

mation and the performance of the optimized portfolios with the ones resulting from

MVO.

Finally, a rolling-period optimization model is introduced which will investigate

the changing asymmetry in the returns over the estimation period. This model

serves to highlight the impact of dissimilarities in the risk exposure during various

time periods. Further, it will enable the isolation of the �nancial crisis in 2008 as

57

this period is assumed to present increasing asymmetry in the return series and ad-

ditionally lead to decreasing diversi�cation potential due to unfavorable correlation

patterns.

Before the implementation of the expansion commence, it was necessary to impose

restrictions to the weights of the asset holdings of the portfolio in order to properly

re�ect a �real-life� investor. The reasoning for the choice of con�dence level will

additionally be described below.

7.1. Allocation Restrictions

So far, no restrictions have been imposed on how much to allocate in each asset

class or in a single index. It was observed in chapter 4 that MVO favored bonds

due to their low volatility and fairly high return. As this dissertation attempts to

investigate the di�erence in risk exposure over a historical estimation period to a

'real-life' investor, it is not assumed holding more than 70% in bonds would have

properly re�ected this.

Therefore, it is found necessary to impose certain restrictions that limit the maxi-

mum allocation weights to each asset class and each index. Refraining from entering

utilization theory or any subjective allocation methods, the imposed restrictions in

table 7.1 are assumed constant over the entire observation period

Category Maximum weight

Currencies 10 %

Bonds 30 %

Commodities 10 %

REITs 10 %

Stocks 40 %

Single index 10 %

Table 7.1.: Portfolio allocation weights

Note: Weights are inspired by the AXA Group

The pro�le represented in table 7.1 ensures that each asset class will be represented

in the portfolio and that the portfolio will be diversi�ed among several asset groups

and indices. This will also enable the possibility to discover the impact of all asset

58

classes assumable diverging towards one in �nancially stressed periods.

The majority allocation to a single asset class is in stocks with 40% followed by

bonds. The rest of the asset categories represent only 10% each of the allocation.

Additionally, the investor could maximum allocate 10% to each index.

Compared to portfolio optimization without restrictions, imposing restrictions will

force the allocation of more risky assets than just bonds in the portfolio. These index

return series are often characterized with higher volatility and return distribution

that are negatively skewed and leptokurtic as seen in chapter 5. Consequently, this

will increase the standard deviation and presence of asymmetric returns in the MVP

and OP, thereby underestimating the risk exposure over the historical estimation

period from MVO. However, the inclusion of more risky assets will also increase the

expected return, especially for the MVP.

Figure 7.1.: MVO optimization with and without portfolio restrictions

Observing �gure 7.1, the e�cient frontier with restrictions has moved to the right,

thus increasing the standard deviation from 0.85% to 6.50% p.a. for the MVP and

from 4.40% to 8.03% p.a. for the OP. The expected return has also increased but

not proportionally. Without restrictions the MVP had an expected return of 2.51%,

whereas the OP had 7.71% expected annual return. With restrictions the MVP now

has an expected return of 5.00% and the OP has an 8.27% expected return. The

investor is therefore punished by the restrictions with more risk per unit of return.

This is also illustrated in the Sharpe ratios. The slope of the OP's CML has de-

59

creased from 0.93 to 0.58. However, the MVP now has a positive Sharpe's ratio of

0.21 compared to a negative ratio when no restrictions were imposed.

Because restrictions have been imposed, the span of e�cient portfolios between

the MVP and OP have been reduced. The table in appendix E illustrates the

holdings of the MVP and OP following the imposed restrictions.

The distribution characteristics of the restricted portfolios are also, as stated,

deviating more from normality. As the return distributions of the MVP and OP

without restriction was slightly left-skewed and with low kurtosis, the MVP and OP

with restrictions now yields skewness of -0.74 and -0.34 and kurtosis of 8.49 and

7.65. The MVP is actually constructed by indices with return series that are more

asymmetric than the OP as the Jarque-Bera test results in 2826.3 for OP and 4135.4

for MVP. However, both portfolios clearly reject normality. These characteristics

will greatly in�uence the MVO ability to provide a correct risk estimate to the in-

vestor.

An important aspect of using either Cornish-Fisher or Gram-Charlier is that the

standard deviation will be expressed at a certain con�dence level. Therefore, it is

important to choose the levels prior to utilizing these expansions. The next section

will therefore elaborate on the chosen con�dence intervals for this research.

7.2. Con�dence Level

This dissertation is concerned with the downside risk exposure over the historical

estimation period and therefore; it is only interesting to observe the left tail of

the return distribution. Henceforth, when standard deviation or risk estimate is

mentioned, it is assumed to represent the returns to the left side of the mean.

Also, it is attempted to provide a better risk estimate with the expansions that

capture the fat-tailed characteristics of the asymmetric series, as it is often here

MVO fails to yield valid estimates of loss exposure. As both Cornish-Fisher and

Gram-Charlier require the standard variable z to be determined at a signi�cance

level, the one-sided 95% and 98% con�dence interval is chosen corresponding to the

60

2.5th and 1st percentiles. The choice of con�dence interval will have great impact

on the risk estimation as can be seen from �gure 6.3 in section 6.3 on page 54. As

a result, all standard deviations going forward will be expressed at either the 1st

or 2.5th percentiles, equaling -2.33 and -1.96 for the quantile of the standardized

normal distribution. An additional reason for choosing these con�dence intervals

is that the Cornish-Fisher expansion has indicated non-monotonic properties at

certain skewness and kurtosis pairs. This will be further elaborated and examined

in chapter 9. With the restrictions imposed and con�dence interval chosen, the next

section will analyze the implementation of the expansions on the MVO portfolios

and compare the risk estimation di�erence.

7.3. Implementation of Cornish-Fisher and

Gram-Charlier

Cornish-Fisher and Gram-Charlier expansions are implemented in order to investi-

gate the con�icting risk estimates that result from the return distribution charac-

teristics of MVP and OP. Initially, referring back to chapter 6, the approximated

quantiles will be illustrated and then applied to the standard deviation measure.

Figure 7.2 illustrates the approximated quantiles of the MVP and OP compared to

the standard normal quantiles. Both portfolio return distributions exhibit leptokur-

tic and left skewed characteristics that form fat-tails to the left. Consequently, this

underestimates the true risk over the historical estimation period when using MVO.

It can be seen from the �gures that the expansions capture the fat-tails of the distri-

butions and approximate a lower quantile for both the 1st and the 2.5th percentiles.

However, it can be observed from the MVP and OP that Cornish-Fisher approxi-

mates a lower quantile at the lowest percentiles as asymmetry increases which was

indicated by the Jarque-Bera test in section 5.1.1. The Gram-Charlier expansion

seems to converge with the normal distribution at the very lowest percentiles, how-

ever, still illustrating notable di�erent quantiles approximations at the two selected

percentiles. In appendix I, a numerical investigation of the skewness and kurtosis

pairs' impact on the quantile approximations shows these features.

61

(a) MVP: Skewness = -0.74, kurtosis =8.49

(b) OP: Skewness = -0.34, kurtosis = 7.65

Figure 7.2.: Cornish-Fisher, Gram-Charlier, and normal quantiles for MVP and OP

The quantiles resulting from the equation 6.5 on page 50 and 6.8 on page 51 are

shown below.

Method Mean-Variance Cornish-Fisher Gram-Charlier

Percentile (%) 0.01 0.025 0.01 0.025 0.01 0.025

Quantile (MVP) -2.33 -1.96 -3.96 -2.61 -2.89 -2.41

Quantile (OP) -2.33 -1.96 -3.64 -2.43 -2.62 -2.15

Table 7.2.: Percentiles at 95% and 98% con�dence intervals

Having calculated the adjusted quantiles it is now possible to �nd the correspond-

ing standard deviations by equation 6.6 and 6.11.

Method Mean-Variance Cornish-Fisher Gram-Charlier

Percentile (%) 0.01 0.025 0.01 0.025 0.01 0.025

Std. deviation (MVP) 15.15 12.74 25.74 16.97 18.79 15.67

Std. deviation (OP) 18.71 15.74 29.23 19.51 21.04 17.26

Table 7.3.: Standard deviations at 95% and 98% con�dence intervals

By overlooking negative skewness and excess kurtosis found for MVP and OP, it

is evident that the MVO underestimates the risk exposure to the investor at both

con�dence intervals considerably.

In table 7.3 it can be observed that the Cornish-Fisher expansion obtains higher

standard deviations for the MVP at both percentiles. At the 95% con�dence level,

MVO underestimates the volatility with 33.20% and at the 98% con�dence level

62

MVO underestimates the exposure of negative extreme returns with 69.90%. The

Gram-Charlier expansion also �nds the MVO to underestimate the risk but to a

lesser extent. At the 95% con�dence interval MVO underestimate left-tail exposure

with 23.00% and at the 98% con�dence level, a 24.03% underestimation.

The OP illustrates the same pattern. At the 95% con�dence level, the standard

deviation of MVO is underestimated by 23.95% according to the Cornish-Fisher ex-

pansion, and 9.66% according to the Gram-Charlier approximations. At the 98%

con�dence level Cornish-Fisher �nds the standard deviation to be 56.23% under-

estimated, and Gram-Charlier estimates it to 12.45%. As stated, the increased

asymmetric properties in the return series seem to have a greater impact on the

Cornish-Fisher approximation of standard deviations at the lower percentiles for

both MVP and OP.

As the return remains constant, the increased risk estimate by the expansions will

also negatively in�uence the Sharpe's ratio. Without adjusting MVP, the reward-

to-variability ratio yielded 0.11 at the 95% con�dence level and 0.09 at the 98%

con�dence level. The Sharpe ratio will naturally decrease as the standard devia-

tion is expressed at higher levels, and as a consequence, this ratio only provides

value when comparing the relative changes between the models. After adjusting for

asymmetry with the Cornish-Fisher expansion, Sharpes ratio for the MVP is now

respectively 0.08 and 0.05 at the 95% and 98% con�dence level. For MVP adjusted

with the Gram-Charlier expansion, the Sharpe ratio is now 0.09 at the 95% con�-

dence level and 0.07 at the 98% con�dence level. For the OP without adjustment the

Sharpe ratio yields 0.30 at the 95% con�dence level and 0.25 at the 98% con�dence

level. Adjusting for asymmetry with Cornish-Fisher the ratio now decreases to 0.24

at the 95% level and to 0.16 at the 98% level. For the Gram-Charlier adjusted OP,

Sharpe's ratio is now 0.27 and 0.22 for each level of con�dence.

By adjusting for asymmetry the risk adjusted return to investor is much lower

than what MVO predicts. This naturally occurs as for every unit of return the in-

vestor is actually exposed to a much higher volatility due to the asymmetric nature

of return characteristics of MVP and OP.

Even though, the expansions yield di�erent risk estimations, it is evident that

both expansions �nd the MVO estimation of volatility to be underestimated for

both the MVP and OP. However, the Cornish-Fisher expansion seems to weigh the

e�ect of asymmetry to a larger extent than Gram-Charlier. This feature is mag-

ni�ed at the lowest percentiles where Cornish-Fisher results in risk estimation way

above both MVO and Gram-Charlier. Due to the fact that this dissertation will

only investigate the di�ering quantile approximations numerically, it does not seek

to investigate the analytical di�erence between the models. It serves to show that

several models exist which derive alternative risk measures as their treatment of

asymmetry varies. In order to illustrate MVO's inability to capture the asymmetric

and extreme returns, the �gure in appendix H shows an example of the empirical

return distribution of the OP with restrictions �tted to the normal bell-shaped curve.

This section showed the risk estimate an investor using MVO over the observation

period would actually have had when holding the MVP and OP with their return

distribution characteristics. In particular, it shows how important it is to consider

asymmetry in order to properly assess the real risk. In the presence of left-skewed

and leptokurtic distribution, the MVO underestimates the true risk exposure to the

investor holding these portfolios considerably.

The portfolio optimization in this section utilized the standard deviations from

the empirical indices. However, it was experienced that almost every index exhib-

ited left-skewed and leptokurtic characteristics. Hence, the next section will adjust

each index for asymmetry by the Cornish-Fisher and Gram-Charlier expansions be-

fore portfolio optimization, and thereby derive both alternative risk estimation and

performance measure.

7.4. Optimizing with Cornish-Fisher and

Gram-Charlier Standard Deviations

The previous section investigated and compared the risk exposure incurred by the

investor from the portfolio return distribution resulting from MVO. As most of the

indices show return characteristics deviating from normality, this section will initially

adjust each index for higher moment order by the Cornish-Fisher and Gram-Charlier

64

expansions. As the quantile, will di�er compared to the standard normal quantile it

will result in di�erent standard deviations for each index and hence, an alternative

allocation. This adjustment will then be utilized to construct MVP and OP with

portfolio optimization in order to investigate the risk estimate compared to the un-

adjusted MVO portfolios and the adjusted MVO portfolios.

Figure 7.3 and 7.4 illustrates the e�cient frontier optimized with standard devi-

ation at the 95% and 98% con�dence levels. The MVO optimization is identical to

the one in �gure 7.1 on page 59 just moved further to the right.

Hence, at the 95% con�dence level it results in a standard deviation for MVP at

12.73% and OP at 15.75%. After correcting the indices for skewness and kurtosis,

the e�cient frontier resulting from optimization shows increased risk exposure. The

standard deviation of MVP resulting from the Cornish-Fisher expansion adjusted

indices is now 13.68% and for the OP it is 16.52%. The standard deviations resulting

from the Gram-Charlier adjusted indices are even higher for the MVP with 13.78%

and 17.10% for the OP. Compared to MVO unadjusted portfolios, adjusting the

indices by the Cornish-Fisher expansions leads to an increased risk estimation at

the 95% con�dence level of 7.46% for the MVP and 4.89% for the OP. The same

applies to Gram-Charlier resulting in increased standard deviations of 8.25% for

MVP and 8.57% for the OP.

The risk clearly increased after adjusting the indices, but the expected return has

not moved proportionally. For MVO the expected return of MVP is 5.00%, whereas

the optimization with the Cornish-Fisher and Gram-Charlier adjusted indices results

in 5.28% and 4.97% returns, respectively. For the OP, MVO outperforms with 8.27%

to Cornish-Fisher at 8.11% and Gram-Charlier at 8.16%. Consequently, maintaining

the approximate same level of return relative to an increased standard deviation

results in lower Sharpe ratios. This is true for all but MVP Sharpe ratio between

MVO and Cornish-Fisher. In this case, the reward-to-variability actually increases

from 0.11 for MVO to 0.12 for Cornish-Fisher at the 95% con�dence level.

65

Figure 7.3.: Cornish-Fisher, Gram-Charlier, and MVO at the 95% con�dence level

As can be seen by comparing �gure 7.3, and �gure 7.4 seen on the next page, the

e�cient frontiers are not clustered that close to each other at the 98% con�dence

level. The MVO's inability to capture the probability of negative returns at the

lowest percentiles widens the gap between the three e�cient frontiers and highlights

the attention of models to the lower percentiles.

At the 98% con�dence interval, the MVO estimates standard deviation to be

15.14% for MVP and 18.72% for the OP. After the adjustments of the indices,

Cornish-Fisher yields an MVP with standard deviation of 19.87% and an OP of

23.14%. This corresponds to risk estimation di�erence of 31.24% for the MVP and

23.61% for the OP. The Gram-Charlier adjusted indices optimization yields standard

deviations of 16.70% for the MVP and 20.83% for the OP, corresponding to 10.30%

di�erence for the MVP and 11.27% for the OP.

The standard deviations derived from the optimization of the adjusted indices are

signi�cantly higher than the MVO standard deviations at the 98% con�dence level.

Another important observation from the �gures is that the risk estimation resulting

from Cornish-Fisher compared to Gram-Charlier at the 98% con�dence level is much

higher. At the 95% con�dence level, optimization from the Gram-Charlier adjusted

indices resulted in higher standard deviations than Cornish-Fisher whereas the op-

posite occurs at the 98% level. It clearly illustrates the attention Cornish-Fisher

has on capturing the fat-tails and therefore, the impact of excess kurtosis. Figure

6.3 illustrated the case where Gram-Charlier estimates lower quantiles at higher

66

percentiles and moreover where the shift occurs as the percentiles drop. So, it is

extremely important to identify the con�dence level as the risk estimate from the

three models varies greatly depending on the level of asymmetry.

Figure 7.4.: Cornish-Fisher, Gram-Charlier, and MVO at the 98% con�dence level

When observing the return, the MVO retains the same return for the MVP and OP

at the 98% con�dence interval, whereas the allocation di�ers at the 98% con�dence

level for both expansions, and so, the expected return changes. Again, the Sharpe

ratio is negatively impacted by the increasing risk estimate by the optimization with

adjusted indices, as the ratio decreases relatively more at the 98% con�dence level

compared to the 95% level. The Sharpe ratio will naturally decrease as the stan-

dard deviation is expressed at higher levels. So, this ratio only provides value when

comparing the relative changes between the models. The Cornish-Fisher adjusted

optimization actually increases return relative to risk and again outperforms the risk

adjusted return for the MVP compared to the MVO MVP, by 0.10 to 0.09. Other

than that, the return and Sharpe ratios have decreased for all portfolios. For detail

see appendix F, and appendix G.

It can be observed that by adjusting the indices for higher moment orders, an al-

ternative portfolio is constructed that presents a higher, but more valid, risk estimate

to the investor. However, when comparing the standard deviations of the adjusted

indices optimization portfolios with the MVO's asymmetry-adjusted portfolios in

section 7.3, it can observed that it is actually possible to decrease the volatility. The

67

adjusted standard deviation from the Cornish-Fisher expansion at the 2.5th per-

centile resulted in 16.97% for the MVP and 19.51% for the OP in MVO. However,

optimizing with the Cornish-Fisher adjusted indices results in standard deviations

at the same percentile of 13.68% for MVP and 16.52% for the OP. That is in fact

a decrease of volatility by -19.39% for the MVP and -15.33% for the OP. At the

1st percentile this results in an even greater volatility decrease of -22.80% for the

MVP and -20.83% for the OP. The Gram-Charlier adjusted indices also enable the

reduction of risk after the correction to the true standard deviation for the MVO

portfolios. At the 2.5th percentile the Gram-Charlier expansion approximates the

standard deviation of the MVP to be 15.67% and the OP to be 17.26%. By adjust-

ing the indices prior to optimization this leads to standard deviation for the MVP

of 13.78% and 17.10%. Hence, this leads to a risk reduction of the MVP by -8.25%

and OP of -0.93%. At the 1st percentile this leads to a risk reduction of the MVP

of -11.12% and OP of -1.00%.

In addition, even though the Sharpe ratios have slightly decreased at both con�-

dence levels for all but one of the adjusted indices optimization, they still outperform

the Sharpe ratio's after correcting the MVO portfolios for asymmetry. The increase

in volatility resulting from the adjusted indices optimization is relatively less than

the asymmetry-adjusted volatility resulting from the MVO portfolios.

The increased risk estimates for both portfolios is a result of the quantile approx-

imation of each index from the Cornish-Fisher and Gram-Charlier expansions. As

most of the indices are left-skewed and leptokurtic, the adjusted quantile multiple

will be lower than the standard quantiles. Further, the standard deviations will

increase for both the indices and the optimized portfolios. However, it also implies

that when asymmetry is right-skewed, the quantiles increase on the left side, while

the standard deviations decrease. Appendix I illustrates numerically how the quan-

tile multiple for both expansions at the 2.5th and 1st percentiles with various pairs

of skewness and kurtosis.

Consequently, the optimization, disregarding the correlation pattern and holding

68

the standard deviation approximately the same, favors the least left-skewed and

low excess kurtosis indices, because the quantile multiple will be higher. This fact

can be observed from the portfolio allocation in appendix F, and appendix G. As

an example, the unadjusted OP from MVO allocates 10% to U.S. REITs. After

adjusting for an incredibly high kurtosis of 15.58 and derive the adjusted standard

deviation by the Cornish-Fisher expansion at the 95% con�dence level, the portfolio

now only allocates 4% to U.S. REITs. At both the 95% and 98% con�dence level,

approximately 24-30% of the MVP's portfolio allocation di�ers between MVO and

the adjusted indices allocation. For the OP, only 20% of the allocation is di�erent.

It can be concluded that by adjusting the empirical index series with approx-

imated quantiles derived from the higher moment orders, portfolio optimization

with adjusted indices will initially yield a higher risk estimate and lower reward-to-

variability compared to the unadjusted MVO portfolios. However, when adjusting

the MVO portfolios for asymmetry after optimization and deriving the adjusted

standard deviation at the 1st and 2.5th percentiles, it is actually possible to reduce

the overall risk of the portfolios by correcting each index for asymmetry prior to

the optimization thereby gaining an improved reward-to-variability ratio. Again, it

is made evident that the Cornish-Fisher expansion's focus on the lowest percentiles

greatly lowers the corrected quantiles that are to be multiplied on the standard de-

viation. As stated and further con�rmed in appendix I, the level of excess kurtosis

and which percentile is in question immensely in�uences the quantile multiple for

the Cornish-Fisher compared to both MVO and Gram-Charlier. Compared to the

Gram-Charlier expansion, the risk reduction e�ect for the Cornish-Fisher expansion

is also found superior between the adjusted index optimization and the MVO port-

folios adjusted for asymmetry.

So far, the portfolio optimizations have been executed over the entire estimation

period. However, assuming constant correlations over such a long period will most

likely hold some challenges. Hence, the next chapter will present a rolling-period

optimization model that enables the investigation of the risk estimate by chang-

ing return characteristics of the empirical series and changing correlation pattern

69

between the asset categories.

70

8. Rolling-period Optimization

In section 3.3 it was emphasized that that the correlation pattern has empirically

shown not to be constant over time as it has, so far, been assumed in the portfolio

optimization. In the data analysis it became obvious that the �nancial turmoil of

2008 caused almost all asset categories to plummet. This is supported by Lore-

tan and English (2000) who discover signi�cantly changing correlations following a

distressed global �nancial market in�icted with unpredictable events [50].

Assuming a constant correlation pattern over the entire period neglects the chang-

ing risk exposure to the investor because the diversi�cation e�ects become dismal.

As a consequence, it was found necessary to construct a rolling-period optimiza-

tion model that captures these correlation changes. Furthermore, this model will

enable the study of the e�ect of the true volatility exposure at various time peri-

ods. Moreover, it enables the study of the impact that varying asymmetry will have

on the risk and performance estimation arising from the MVO, Cornish-Fisher and

Gram-Charlier.

8.1. Rolling-period Methodology

The algorithm is constructed to optimize the portfolios based on a one-year period

(260 days) and then rolling the period one day and optimize for a 260 days window

yet again. Figure 8.1 illustrates the method.

71

Figure 8.1.: Rolling-period method

Where i = i ... n.

Portfolio i will therefore in t0 illustrate the risk and performance of the opti-

mized portfolio over a 260 days observation period going forward. This entails that

when interpreting the graphs, it is assumed the investor will be at time t0 and hold

the speci�c allocation over the next 260 days to gain that speci�c risk and return

characteristics. 260 days is assumed to be ample observations in constructing the

optimized portfolios and challenge the normal distribution assumptions. The model

will result in 2814 portfolios over the full estimation period from which it is possible

to derive the �rst four moments.

The optimization approach is similar to the analysis over the full estimation pe-

riod. Initially the MVP and OP for MVO will be constructed and the risk estimate

will be adjusted using the Cornish-Fisher and Gram-Charlier approximations. From

this rolling-period optimization it is also possible to analyze the skewness and kur-

tosis arising from the MVO in the two e�cient portfolios and obtaining focus on

the risk estimate when the return characteristics changes. Secondly, the indices will

again be adjusted with the Cornish-Fisher and Gram-Charlier expansion prior to

optimization in order to analyze asymmetry in each index and how it will a�ect risk

and performance.

Hence, the model will be executed on the one-sided left tail returns at the 95%

and 98% con�dence level. Ultimately, this leads to six rolling-period optimization

models; each consisting of 2,814 iterations. As the execution time of each iteration

ranges from 30-45 seconds on a 3.00 GHz computer, the full execution of the models

72

is a time-consuming task and will take 23-35 hours for each model to be completed.

Excel VBA has been utilized to perform the iterations and the code can be found

in appendix O, and on the CD.

8.2. Rolling-period Asymmetry

Initially, the potential deviation from normality arising from the MVO portfolios

will be analyzed. The rolling periods will enable an investigation of when the risk

estimations of MVO deviate and to what degree.

Figure 8.2 illustrates the skewness in the return distribution of the MVP and the

OP resulting from MVO. As can be observed, based on 260 days observations, both

portfolios show left-skewed properties for the vast majority. The skewness spans from

approximately -1.3 to 0.4, where the red line indicates normality. These negative

skewness characteristics are as already stated, not favorable to the investor. The

reason is that as left tails will emerge from which the standard normal distribution

not will be able to properly estimate the risk exposure.

Comparing the skewness between the MVP and the OP, it tends to be the OP

resulting in the most negative skewed return distributions. However, as the MVP

attempts to minimize standard deviations and the OP attempts to maximize the

reward-to-variability, the skewness coe�cient between them alternates in deviating

furthest from normality depending on the �nancial situation in that time period.

Figure 8.2.: Skewness from the MVO portfolios, MVP and OP

Figure 8.3 illustrates the kurtosis coe�cient for the MVP and OP resulting from

73

unadjusted MVO. The red line indicates normality. Again, the OP tends to have

a slightly higher kurtosis in most periods. However, these results do not coincide

with skewness and kurtosis found for the MVP and OP over the entire period. The

MVP actually showed both more skewed and leptokurtic compared to the trend

found in the rolling window. This con�rms the importance of not having a constant

estimation period as the investor's risk exposure that arises from asymmetry is

constantly changing over time.

As can be seen, the return distributions are actually for the most portfolios below

three. It indicates platykurtic distribution properties and thereby negative excess

kurtosis. MVO will in these cases, ceteris paribus, actually overestimate the risk

exposure to the investor. However, skewness and kurtosis must be analyzed together

in order to get a correct picture of the risk exposure. The time periods in�icting the

portfolios with most kurtosis are therefore similar to a period with high negative

skewness. Especially, the period from 2008 shows immense high kurtosis. This high

kurtosis and negative skewness indicate an extreme event, causing the assets in the

portfolio to drop tremendously.

Figure 8.3.: Kurtosis from MVO portfolio MVP and OP

In �gure 8.4 a Jarque-Bera barometer is constructed to illustrate the relation-

ship of the skewness and kurtosis pairs from each portfolio and the e�ect on the

normality assumption. The red line is the 5.99 threshold for the 95% signi�cance

level. Anything above the critical value indicates the return distributions that de-

viate from normally distributed returns. Depending on the coe�cients of skewness

and kurtosis, the barometer shows to what extent the portfolio return distributions

deviate from normality.

It can be concluded that the Jarque-Bera test seems very sensitive to any presence

74

of asymmetry as only very few portfolios do not lead to values above 5.99 at the 95%

con�dence level. The true risk exposure for the investor will in most of the cases

therefore by MVO, be faulty estimated. Only with the investigation of skewness

and kurtosis from �gure 8.2 and 8.3, it can be concluded this faulty estimation most

of the time will occur on the left tails, and thereby underestimating the probability

of loss. The period standing out is portfolio 1,771 which ranges from October 15th

2007 to October 10th 2008. This portfolio showed skewness coe�cients of -0.2 and

kurtosis coe�cients of 12.60 leading to a Jarque-Bera value of 1000.04. The window

two day prior is portfolio 1,769, which only showed limited �ight from normality

with skewness of -0.74 and kurtosis of 2.59 resulting in a Jarque-Bera value of 25.42.

In portfolio 1,770 it can be seen that the asymmetry slowly started to increase as

the Jarque-Bera reached 98.74 from skewness of -1 and kurtosis of 5.24. Portfolio

1771 therefore indicates the occurrence of extreme outswings in the returns of the

indices. Observing the indices it can be seen that volatility on October 10th was

very high across almost all asset categories . To to mention a few indices across

three asset categories: the S&P commodity index dropped -8.65%, the FTSE dev.

Asia index fell -11.84, and the FTSE 100 stock index plunged -9.27%. The CBOE

volatility index hit a record high that Friday afternoon in October 20081.

By imposing the restrictions the portfolios cannot allocate the majority to the less

volatile index such as bonds. It serves to shows how several asset categories starts

correlating, hence decreasing the diversi�cation e�ect and increasing asymmetry in

the portfolios.

Figure 8.4.: MVO Jarque-Bera barometer

The Jarque-Bera barometer can be used to indicate the presence of asymmetry.1http://money.cnn.com/2008/10/10/markets/markets_newyork/index.htm?postversion=20081010119

75

As seen, the drop in returns across asset categories happens from one day to the

other, so it will be di�cult to safeguard the investor from this.

As aforementioned the rolling-period shows that the degree of deviation from

normality is very di�erent depending on the �nancial situation in that time period.

Therefore, observing asymmetry over the entire period neglects to show the periods

where the investor is increasingly exposed to loss than MVO can predict. The next

section derives the corresponding quantile approximations when adjusting for this

asymmetry in order to compare them with normal standard z-values of -1.96 and

-2.33.

8.3. Rolling-period Quantile Approximation

The return characteristics showed asymmetry in almost all of the constructed portfo-

lios. Consequently, this will lead to an alternative quantile approximation. As most

of the asymmetry showed either left-skewed or leptokurtic properties, the quantile

approximation is likely to be lower than the standard normal quantile. The OP

quantile approximations from the rolling-period optimizations can be seen in �gure

8.5 and �gure 8.6 at the 2.5th and 1st percentiles. The blue line indicates the stan-

dard normal quantile z.

At the 95% con�dence interval, the critical value is -1.96. For the most part, the

quantile approximations (that arise from both Cornish-Fisher and Gram-Charlier)

leads to lower quantiles than what MVO predicts. It seems that Gram-Charlier �nds

the quantiles to be lower at the 95% con�dence level and deviate more from the

standard normal quantile as discovered in section 7.4. Contrary to Gram-Charlier,

Cornish-Fisher actually in many cases approximates higher quantiles than MVO,

which is a consequence of the low kurtosis coe�cients seen in �gure 8.3. Skewness

therefore seems to have a larger impact on the Gram-Charlier approximations at

the 95% con�dence level which can also be seen in appendix I.

76

Figure 8.5.: Cornish-Fisher and Gram-Charlier quantiles at 95% con�dence level

At the 98% con�dence level, Gram-Charlier again tops Cornish-Fisher in most of

the periods and produce lower quantile multiples. However, while Gram-Charlier

quantiles tend to be quite static right above -2.33, Cornish-Fisher approximations

prove much more volatile at the lowest percentile and di�er extensively from nor-

mality.

Figure 8.6.: Cornish-Fisher and Gram-Charlier quantiles at 98% con�dence level

As could be expected from the asymmetry analysis, one period stands out from the

other when observing the entire period. The �nancial crisis in 2008 and particular

October 10th, shows quantile multiples signi�cantly below normality quantiles.

At the 95% con�dence interval both expansions derive quantiles much lower than

-1.96. The impact of high kurtosis and low skewness drives the Cornish-Fisher quan-

tile multiple beyond the Gram-Charlier multiple. Both expansions would ultimately

in 2008 have had an immense impact of the true risk estimation over that speci�c

77

period at the 95% con�dence level.

At the 98% con�dence level the underestimation would have been even greater for

the Cornish-Fisher approximation as the quantiles during this period is estimated

over twice the value of the standard normal quantile. What is interesting is that the

Gram-Charlier expansion completely neglects to account for the high asymmetry at

the 98% con�dence.

The analysis of the quantile multiple-estimation must be compared with the anal-

ysis of asymmetry in section 8.2. The quantile approximations show the importance

of choosing the right con�dence interval for the investor because the expansion

exposes very di�erent risk estimates at the lowest percentiles. Cornish-Fisher ap-

proximates lower quantiles at the lowest percentiles, whereas Gram-Charlier derives

lower quantiles at the higher percentiles. Naturally, this is dependent on the level

of asymmetry as the expansion is impacted by higher moment in di�erent ways.

Gram-Charlier shows high sensitivity to negative skewness which results in lower

quantile approximations even though negative excess kurtosis is present. Contrary,

the Cornish-Fisher expansion derives higher quantiles than MVO at these character-

istics. As a result, the Cornish-Fisher proves more sensitive to excess kurtosis and

also to skewness but to a lesser degree. The impact of kurtosis on Cornish-Fisher

is so great that it even produced lower quantile estimations than Gram-Charlier at

the 95% con�dence level portfolios starting October 10th.

The quantile estimation for the expansions was only for the OP. To see the quantile

estimation for the MVP at the 95% and 98% con�dence interval please see appendix

J. The same conclusion can be applied to the MVP portfolios; however, the quantile

approximations seem to be slightly less volatile due to the sole aim of minimizing

the standard deviations.

8.4. Roling-period Standard Deviations

The quantile approximations from the expansions will ultimately lead to a more

correct standard deviation of the MVP and OP than MVO at both percentiles.

78

With the rolling-period optimization it is now possible to show the adjusted risk to

the investor over multiple periods and how it changes with the degree of asymmetry.

Figure 8.7 illustrates the MVO, Cornish-Fisher and Gram-Charlier at the 1st

percentile standard deviation of the OP. Until 2008 the risk estimation between the

models replicates the characteristics of the quantile estimation from �gure 8.6 by

not deviating signi�cantly from normality. However, with emergence of the �nancial

crisis, the standard deviations of all the methods almost tripled. As can be seen from

the empirical indices and appendix P, prices plummet across multiple asset categories

causing the expected return for the OP to drop. Hence, the asymmetry increased

due to unfavorable correlations, and the risk estimation got severely underestimated

by the MVO theory. October 10th 2008, the Cornish-Fisher expansion estimates

standard deviations of 49.32% annually compared to MVO standard deviations of

24.23% annually. The underestimation over this period shows that by not accounting

for asymmetry the risk is underestimated by more than 100%.

The period captured by the square shows the di�erence in the true risk estimation

the investor incurred during the crisis.

Figure 8.7.: Cornish-Fisher, Gram-Charlier, and MVO standard deviations at the 98%

con�dence level

Figure 8.8 illustrates the caption of the square in �gure 8.7. It interesting to

note that the risk estimation di�erence resulting from the expansions is highest in

the �ve month following portfolio 1771 on October 10th 2008. This period matches

the steep peak in kurtosis. This indicates that as the sudden drop in returns will

cause a period with extremely high kurtosis and that the deviation from normality

becomes excessive. However, as the trend continues and returns start leveling, the

distribution �nds its way back to normality. The Gram-Charlier expansion only

79

results in risk estimates slightly higher than MVO standard deviations, which again

highlights some di�culties at the lowest percentiles.

The standard deviations resulting from MVP at the 98% con�dence interval and

the OP and MVP at the 95% can be seen in appendix L, and appendix K. The

main di�erence from con�dence interval has already been discussed in the quantile

approximation. However, the standard deviations indicate less volatility for the

MVPs.

Figure 8.8.: Cornish-Fisher, Gram-Charlier, and MVO standard deviations in the �nancial

crisis of 2008

It is during these extreme loss exposure periods that diversi�cation must provide a

safety to the investor. However, as the empirical indices indicate, the crisis a�ected

almost all asset categories causing the asymmetry of the optimized portfolios to

increase, hence, providing no safe haven for the investor. This means that the risk

estimations resulting from MVO are deceivingly low and will not portray a reliable

estimate of the crisis.

This section has shown how the risk estimation resulting from MVO provides

faulty standard deviations in the presence of asymmetry and in most cases greatly

underestimates the exposure in the period of observation. Hence, the Cornish-Fisher

and Gram-Charlier expansions have shown to account for these higher moment or-

ders that captures these non-normal return characteristics and provide an alternative

risk estimation.

The next section will adjust the standard deviations of each index for asymme-

try and compare the optimization results with the asymmetry-adjusted portfolios

80

resulting from MVO optimization in order to study the risk and performance devel-

opment.

8.5. Rolling-period Performance

It was experienced in section 7.4 that it is actually possible to derive a lower standard

deviation when adjusting the indices with the expansions prior to the optimization

compared to the asymmetry-adjusted portfolios resulting fromMVO. However, when

observing the rolling RTV in �gure 8.9 it shows a very interesting di�erence. From

2001 and until the �nancial crisis in 2008, the MVO portfolios corrected for asym-

metry with the Cornish-Fisher expansion actually yields higher RTV than indices

corrected by Cornish-Fisher prior to the optimization at the 98% con�dence interval.

This contradicts the prior �ndings. However, entering the crisis the roles switch and

the portfolios with indices adjusted prior to optimization yield higher RTVs. This

is also the case for Cornish-Fisher OP at the 95% con�dence interval which can be

seen in appendix M. Appendix M also illustrates the OP in the case with Gram-

Charlie. However, the di�erence in RTV is not as conclusive as with Cornish-Fisher,

because Gram-Charlier yields RTV's very similar to MVO. The di�erence is due to

the Cornish-Fisher expansions' ability or inability to capture fat tail at the lowest

percentiles and the already discussed attention to asymmetry.

Figure 8.9.: Cornish-Fisher RTV before versus after optimization

Optimally, the investor should keep the optimization with unadjusted indices

but correct the optimized portfolios with the expansions to re�ect the true risk es-

timates. Then, when the distribution characteristics resulting from the portfolio

becomes highly left-skewed and leptokurtic, the optimization should be constructed

81

with the adjusted indices. The Cornish-Fisher quantiles from the adjusted indices

will in �nancially distressed periods become very low and increase the standard de-

viations of the indices with the greatest asymmetry. Hence, the optimization will

not choose these indices as their risk estimates will outweigh their performance.

It can be concluded from the rolling-period optimization that asymmetry changes

over time and that the investor would have been exposed to risk estimates that would

deviate signi�cantly from normality if the standard MVO is utilized. The investor's

true risk estimate constantly changes from under- to over-estimated. However, as the

portfolio distributions become increasingly left-skewed and leptokurtic, the Jarque-

Bera barometer increases simultaneously, and the alternative risk estimates resulting

from Cornish-Fisher and Gram-Charlier deviate more from the MVO estimate and

derives signi�cantly higher risk exposure. It is therefore essential for the investor

to observe and adjust the return characteristics for both optimized portfolios; es-

pecially when they start to grow increasingly asymmetric, because it will increase

the probability of deriving a faulty risk estimate by MVO. Adjusting the indices

for asymmetry prior to optimization in �nancially distressed periods and following

extreme events would help reduce the overall risk exposure for each portfolio.

As has become evident, Cornish-Fisher and Gram-Charlier provide di�erent ap-

proximations depending on the con�dence level and the degree of asymmetry. This

di�erence is, however, only approached numerically. It shows that more than one

model in the �nancial theory seeks to approximate a more proper left-tail risk expo-

sure estimate. Thus, the next chapter will discuss the drawbacks of each expansion

in order to show the potential limitations before swearing to either of the models.

82

9. Drawbacks

As seen in the dissertation, the Cornish-Fisher has a rather quick and straightfor-

ward application in order to adjust for asymmetry in the empirical distribution and

to derive a more appropriate risk estimate. However, the strength of the Cornish-

Fisher expansion is very dependent of the domain of validity. This means that the

expansion fails to generate valid quantile or density functions for some skewness and

kurtosis pairs which may generate non-monotone quantile functions [12, 61, 18, 16].

The non-monotonic issue occurs when the transformation is not bijective. This

implies that the derivative of φ−1CF relative to zα is non-null [12]. If the transformation

is not bijective, the order of the quantiles of the distribution will not be preserved.

This can be tested by using skewness and kurtosis as inputs to

S2

9− 4

(K

8− S2

6

)(1− K

8− 5S2

36

)≤ 0 (9.1)

The �gures below illustrate the case of non-monotonocity. The left-sided chart

is monotone as the Y value does not decrease or is subject to a value higher than

the previous for an increasingly higher X value. The chart at the right breaks the

monotone function property by �rst increasing and then decreasing, i.e. subject to

a smaller Y value for an the same X value.

83

(a) Monotonic (b) Non-montonic

Figure 9.1.: Monotonic and non-monotonic functions

By breaking the monotonic property, the density function will compute incoherent

risk computations for the required con�dence levels; for instance would an investor

get a higher risk measure at the 95% con�dence level than at the 98% con�dence

level. This is illustrated in the �gure below with a skewness of 0.8 and a kurtosis of

2.

Figure 9.2.: Non-monotone distribution functionNote: Skewness = 0.8, kurtosis = 2

Given the skewness and kurtosis parameters the bijective test yields a posi-

tive value, resulting in a non-monotone distribution function. Non-monotonocity

arises as the polynomials inherent in the transformation are non-monotonic. Cher-

nozhukow et al (2010) show how rearrangement can solve the monotonic assump-

tion's issue that can lead to important shortcomings mentioned above [61].

An important drawback of the Gram-Charlier expansion is that it fails to generate

valid quantiles for some skewness and kurtosis pairs [21]. This is a consequence of

84

being a polynomial approximation. It may therefore generate densities with negative

values, and it may suggest multiple solutions [16]. By yielding negative values the

distribution function is theoretically unappealing and the density function can be

multimodal. The dotted line in the �gure below represents the 2.5th percentile where

the left-sided distribution function intersects this percentile three times. Ultimately

this could result in three di�erent standard deviations. Both points are illustrated

in the �gure below.

Figure 9.3.: Gram-Charlier density function with negative values

Note: Skewness = -1.7, kurtosis = 7

Futhermore, in a study from 1952 by Barton & Dennis, conditions on the param-

eters that guarantee positive densities are obtained through a numerical method.

They conclude that there does not seem to be an easy and analytic characterization

of skewness and kurtosis for which the density will take positive values[62]. The

positivity of the density function is of course crucial for the expansion to be appli-

cable [21].

To fully comprehend the domain of valid skewness and kurtosis parameters, the

next section will show for which skewness and kurtosis pairs the expansions yield

valid quantiles.

85

9.1. Domain of Validity

Returning to the issue of non-monotonicity, table 9.1 below speci�cally illustrates for

which skewness and kurtosis pair the Cornish-Fisher expansion will yield monotone

quantile functions. Outside of this set, the Cornish-Fisher expansion provides non-

monotone quantiles in either tail of the distributions. As the table shows, these

failures can occur for moderate kurtosis below four, but also for large kurtosis values,

with skewness around minus or plus one.

Kurtosis

0 1 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Skew

ness

-1.5

-1 * * * * * * * * * *

-0.5 * * * * * * * * * * * * * *

-0 * * * * * * * * * * * * * * *

0.5 * * * * * * * * * * * * * *

1 * * * * * * * * * *

1.5

Table 9.1.: Valid kurtosis and skewness pairs for the Cornish-Fisher expansion

Note: * marks the monotone quantile function.

Following the numerical approach suggested by Barton and Dennis, it is possible

to acquire Gram-Charlier multiples [62]. The �gure below shows the skewness and

kurtosis pairs that generate valid Gram-Charlier expansions. The pairs highlighted

with a star provide densities that have non-negative values, as required. Outside

this set, the Gram-Charlier expansion provides densities with negative values. As

with the Cornish-Fisher expansion, these failures can occur for moderate kurtosis

values below four, but also for large kurtosis values with skewness around minus or

plus one.

Kurtosis

0 1 2 2.5 3 3.5 4 4.5 5 5.5 6 7 8 9 10

Skew

ness

-1.5

-1 *

-0.5 * * * * * *

-0 * * * * * * *

0.5 * * * * * *

1 *

1.5

Table 9.2.: Valid kurtosis and skewness pairs for the Gram-Charlier expansion

Note: * marks the monotone quantile function.

86

The conclusion from the charts shows that the domain of validity for Cornish-

Fisher is more �exible relative to Gram-Charlier. The �ndings of the domain of

validity is con�rmed in Simonato (2011) who �nds approximately the same locus of

control for valid skewness and kurtosis pairs [16]. The validity of various skewness

and kurtosis pairs is very limited for both expansions though. The Gram-Charlier

expansion tends to be more applicable when the return characteristics only deviate

slightly from normality. The entire estimation period yielded kurtosis coe�cients

above 7 for both the MVP and OP after imposing restrictions. Thus, this makes the

risk estimation derived from Gram-Charlier invalid as it falls beyond the domain of

control and produce negative density values. Cornish-Fisher would, however, have

been able to calculate a proper risk approximation as it falls with the domain of

validity.

Reviewing the skewness and kurtosis pairs over the entire period therefore enables

the investigation of when the expansions' retain the domain of validity and the in-

vestor can rely on the risk estimation. The shaded area in �gure 9.4 below illustrates

the period where the empirical kurtosis of the OP breaches the frontier of what the

Gram-Charlier approximation can handle. As the shaded area represents the period

where the �nancial crisis peaked and the return characteristics deviate signi�cantly

from normality, Gram-Charlier does not, even though it derived a higher risk ex-

posure estimate than MVO, show a conclusive risk estimate to the investor which

captures the extreme outliers in that period.

Figure 9.4.: Gram-Charlier with excess kurtosis above 3

The same applies to �gure 9.5 that shows the Cornish-Fisher risk estimation of the

OP. Cornish-Fisher fails at kurtosis coe�cient above 10, illustrated by the shaded

87

area. However, Cornish-Fisher shows increasingly robust at the higher kurtosis

coe�cient and it only fails to derive a valid risk estimate in very few periods. It can

be argued that these are the most important periods as this is where the investor

would inherit the largest loss.

Figure 9.5.: Cornish-Fisher with excess kurtosis above 7

At the lowest kurtosis pairs seen in �gure 8.3 which caused the return distributions

of OP to be platykurtic, the expansions also fails to provide valid quantile approxi-

mations. Observing �gure 9.6 and �gure 9.7, however, indicates that Gram-Charlier

is superior at the negative excess kurtosis values. Still, in these periods the return

characteristics will be fully captured by the normal distribution and the MVO will

provide a superior risk alternative, holding all else equal. Adding skewness to the

equation might alter this conclusion, though.

Figure 9.6.: Gram-Charlier with negative excess kurtosis

88

Figure 9.7.: Cornish-Fisher with negative excess kurtosis

These drawbacks of the Cornish-Fisher quantile function and Gram-Charlier den-

sity function will question the validity of the risk estimates derived from the various

portfolio characteristics due to possible breach of the domain of validity. There-

fore, when the �nancial markets become highly stressed and returns tend to portray

increasing asymmetry it is the recommendation of this dissertation to utilize the

Cornish-Fisher expansion due to the fact that it will account for higher excess kur-

tosis coe�cients.

9.2. Analysis Critique

Following the analysis in this dissertation there are important elements and condi-

tions that should be kept in mind when interpreting the outcome.

The empirical data used as foundation for the analysis only covers a limited range

of assets in each category. It has been attempted to replicate a global portfolio that

represents the entire range of investment choices. However, replicating the entire

spectrum is close to impossible. Including exotic investment choices and �nancial

instruments in the portfolio would most likely have provided alternative conclusions

when examining the return characteristics and deriving the risk estimate.

The returns used in the dissertation are based on daily observation. However,

using less frequent observations have shown distributions that deviate less from nor-

mality. Hence using for example monthly returns would most likely have lead risk

estimation di�erences smaller than found in this dissertation.

89

The allocation restriction in the analysis was based on the authors' subjectivity

and the reasons have been mentioned in that section. However, changing these re-

striction would lead to entirely di�erent risk estimations. It is the recommendation

of this dissertation to adjust the restrictions based on the asymmetry of the vari-

ous indices, as it will help reduce the error in risk estimation resulting from MVO.

Also, the restriction could be changed based on the past economic outlook and the

degree of risk-aversion by the investor. Changing the allocations would have lead

to alternative estimations, but as the data analysis showed, all indices are more or

less asymmetric, so it would only have been the degree of risk estimation error that

would have altered.

The sole focus on the left-sided quantile lead to an approximation of the standard

deviation at either 1st or the 2.5th percentiles. This was done in order to keep

focus on the exposure to loss incurred by the investor. However, the expansions

actually produce quantile approximations on both sides of the mean. As a result,

they do not share the symmetrical characteristics of the MVO standard deviations.

Utilizing the Sharpe ratio as a measure of performance therefor proves problematic

is it assumes symmetrical properties. Hence, comparing performance ratios with

the ones resulting from the Cornish-Fisher and Gram-Charlier expansions should be

done carefully. Based on the characteristics of the asymmetry the ratio will therefore

over- or underestimate the performance of the index or portfolio. Thus, the authors

recommend using the Sharpe ratio only to comparable methods, or alternatively

utilizing the expansions as a Value-at-Risk estimate.

The choice of con�dence interval has also been concluded to have an immense

impact on the level of risk estimation by each model. It is up to the individual

investor or investment fund to choose the respective con�dence interval that can

convince them will re�ect the proper risk exposure.

The construction of the rolling-period optimization window was performed based

on 260 days observations and then moved one day forward over the entire estimation

period. This was assumed to present a proper population size in order to charac-

90

terize the return distributions parameters, and moving the period one day forward

would enable the caption of all sudden changes. However, the length of the rolling

window and iterations of one day may be subject to change and as a result it could

alter the past risk outlook and population parameters. Choosing a longer window

may decrease the asymmetry characteristics and hence, the risk estimate. On the

other hand, moving the period more than one day forward may overlook important

aspects of the distribution. Furthermore, as the length of the estimation period

changes, the entire observation period and the rolling window may not be directly

comparable.

Finally, the expansion does not provide the ultimate risk estimation tool for the

investor over an historical period observation. It indicates that risk estimates pro-

vided by MVO have shortcomings in the presence of asymmetric returns within the

domain of control. The standard deviation is not the ultimate risk estimate and con-

sequently, it is the recommendation of this dissertation that multiple risk estimation

proxies before swearing to a single measure are applied.

91

Part VI.

Concluding Remarks

92

10. Conclusion

The dissertation focuses on the third and fourth moment order and challenges the

assumptions of the classical mean-variance optimization. The authors proposes the

Cornish-Fisher and Gram-Charlier expansion as two possible methods to account

for higher moment orders and incorporate them in the risk estimation. According

to the two expansions, the classical mean-variance theory underestimates the risk

estimation to the greatest extend in the presence of leptokurtic and left skewed re-

turn distributions.

Based on daily total return data for 33 assets over �ve categories from January 1st

2001 to October 1st 2012, the data shows that almost all indices have experienced

increased volatility due to the �nancial crisis of 2008. The statistical characteristics

of the indices illustrated particularly how bonds proved to be a favorable investment

choice as this category has provided low volatility and satisfactory returns. Curren-

cies did not o�er much return, whereas real estate, stock and commodities indices

yielded periods with high returns but accompanied with high volatility.

The correlation analysis showed how assets tend to correlate within the categories.

However, the diversi�cation e�ect was found greater across asset categories. It

can also be seen from the return charts that correlation does not seem constant

over the estimation period. Especially, stocks, real estate and commodities seemed

to increasingly positively correlate in �nancial distressed periods. The conditional

correlation proved this suspicion as is it showed correlation patterns to be changing

depending on periods with positive or negative returns.

Additional, it was seen that all but one of the �nancial series over the estimation

period proved to be left skewed and leptokurtic, causing substantial departures from

normality shown with a Jarque-Bera test.

93

It was shown in section 6.3, that when the normality assumption breaks, it is

possible with the Cornish-Fisher and Gram-Charlier expansion to calculate quantile

approximations of the standardized distribution by including higher moment orders,

and to consider these approximations to the corresponding quantiles of the actual

distribution. With the transformed quantile function approach it is possible to de-

rive the adjusted downside standard deviation at various percentiles. It has been

shown numerically in section 6.3, that the risk estimates derived from the expansion

di�ers at various percentiles from each other and at various degrees of asymmetry.

Cornish-Fisher leads to lower quantiles at the lowest percentiles whereas Gram-

Charlier yields lower quantiles at the higher percentiles during the left-skewed and

leptokurtic return characteristics. These results lead the authors to investigate the

risk estimate at the 95% and 98% con�dence level corresponding to the 2.5th and

1st percentiles.

The Mean-Variance Optimization (MVO) allocated over 70% to bonds in both the

Optimal Portfolio (OP) and Minimum -Variance Portfolio (MVP), because of the

favorable low risk characteristics and reasonable returns. So, it was found necessary

to impose certain restrictions before implementing the expansions. Additionally,

this insured the inclusion of all asset categories in the portfolio. Consequently,

this resulted in the MVP and OP respectively has skewness of -0.74 and -0.34 and

kurtosis of 8.49 and 7.65. Ultimately, these characteristics resulted in the MVO to

considerably underestimate the downside risk exposure to the investor at both the

95% and 98% con�dence levels.

At the 95% con�dence level, MVO estimates the MVP standard deviation to be

12.74%. According to the Cornish-Fisher expansion the correct standard deviation

is 16.97% at the 95% con�dence level thereby resulting in a 33.20% underestimation

by the MVO. At the 98% con�dence level the underestimation increases to 69.90% as

Cornish-�sher estimates standard deviation of 25.74% and the MVO only estimates

the standard deviation to be 15.15%. The Gram-Charlier expansion �nds the under-

estimation of risk exposure to a lesser extent. At the 2.5th percentile Gram-Charlier

estimate the standard deviation to 15.67%. Consequently, MVO underestimate left-

94

tail exposure with 23.00% annually. At the 1st percentile Gram-Charlier �nds the

standard deviation to be 18.79%, thus causing a 24.03% underestimation for the

MVP.

The OP shows the same pattern. At the 95% con�dence level for the standard de-

viation of MVO is underestimated by 23.95% with Cornish-Fisher, and 9.66% with

Gram-Charlier approximations. At the 98% con�dence level Cornish-Fisher �nds

the standard deviation to be 56.23% underestimated and Gram-Charlier estimates

it to 12.45% underestimation of the risk exposure. It can be concluded that the

increased asymmetric properties in the return series seem to have a greater impact

on the Cornish-Fisher approximation of standard deviations at the lower percentiles

for both MVP and OP. Even though the expansions yields di�erent risk estimations,

it is evident from that they both �nd the MVO estimation volatility to be underes-

timated for both the MVP and OP.

By adjusting the empirical index series prior to the portfolio optimization, it

was possible to derive a lower risk estimate than simply adjusting the optimized

portfolios based on unadjusted indices.

The adjusted standard deviation of the MVP and the OP with the Cornish-Fisher

expansion at the 2.5th percentile resulted in 16.97% and 19.51%. However, optimiz-

ing with the Cornish-Fisher adjusted indices results in standard deviations at the

same percentile of 13.68% for MVP and 16.52% for the OP. That is in fact a decrease

in volatility by -19.39% for the MVP and -15.33% for the OP. At the 1st percentile

this results in even greater volatility decrease of -22.80% for MVP and -20.83% for

the OP. The Gram-Charlier adjusted indices also enable the reduction of risk after

the correction to the true standard deviation for the MVO portfolios. At the 2.5th

percentile the Gram-Charlier expansion approximates the standard deviation of the

MVP to be 15.67% and the OP to be 17.26%. By adjusting the indices prior to

optimization this leads to standard deviation for the MVP of 13.78% and 17.10%.

Hence, this leads to a risk reduction of the MVP by -8.25% and OP of -0.93%. At

the 1st percentile this leads to a risk reduction for the MVP of -11.12% and OP of

-1.00%.

95

In order to account for changing correlation patterns and investigate the e�ect

on risk estimation from the degree of asymmetry, a rolling-period optimization was

constructed.

It can be concluded from this analysis that the asymmetry is changing over time

causing extensive departures from normality at certain periods. The investors true

historical risk estimate constantly changes from under- to overestimated. However,

as the portfolio distributions becomes increasingly left skewed and leptokurtic, it

consequently increases the value of the Jarque-Bera barometer. This causes the

adjusted risk estimates resulting from Cornish-Fisher and Gram-Charlier to deviate

increasingly from the MVO estimate and yield signi�cantly higher exposure. Espe-

cially, the �nancial crisis of 2008 caused a period with extremely high asymmetry,

resulting in greatly deviating risk estimates.

Finally, before using the expansion some precautions must be taken that might

question the validity of the results. Certain pairs of skewness and kurtosis, will for

the Cornish-Fisher expansion result in non-monotone quantile functions. For the

Gram-Charlier expansion certain pairs will generate densities with negative values.

The domain of validity was shown to be more �exible for the Cornish-Fisher ex-

pansion, as it would account for distribution with kurtosis coe�cient up to 10 and

skewness down to -1. The Gram-Charlier expansion was found to only produce valid

estimates at kurtosis coe�cient up to 5.5 and skewness down to -1.

The results conclude that several models exist that derive alternative risk measures

as their treatment of asymmetry varies. The authors recommend not proclaiming

one model as the ultimate risk estimation tool before further research is conducted

and more asymmetry-adjusting models is analyzed compared.

96

11. Implications and Further

Research

This dissertation investigates the application of alternative risk estimation in pres-

ence of higher moment orders in the return distribution. The analysis inspires to

further investigations and this chapter presents implications and suggestions for fur-

ther research .

It has been pointed out that Sharpe's ratio as the performance measure may not

be suitable when utilizing Cornish-Fisher or Gram-Charlier expansion. Instead, a

performance measure constructed to have a sole focus on the left tail of the distri-

bution could be used. Sortino's ratio is almost identical to Sharpe's ratio. Sortino's

ratio measure excess return in the numerator based on the expected return and risk-

free rate like Sharpe's ratio. However, in the denominator Sortino only penalizes the

return below the expected return thereby deriving a downside standard deviation.

Sortino ratio =rP − rf

downside σP=

rP − rf√1N

Σr < r (r − r)2(11.1)

However, Sortino's ratio has limitation such as the variations of the downside de-

viation calculations which will vary according to what the investor utilize1.

The issue of non-monotonocity for the Cornish-Fisher expansion signi�cantly lim-

its the domain of validity. This causes the usability and risk estimations to be

questionable. However, as aforementioned, Chernozhukow et al (2010) shows the

rearrangement that overcomes the issue of non-monotonicity [61]. Thus, further

1http://articles.economictimes.indiatimes.com/2012-07-30/news/32942194_1_sharpe-ratio-pension-research-institute-risk-free-rate

97

studies that investigate and implement the rearrangement is needed.

This study examined two alternative risk estimation tools. Simonato (2011)

showed the Johnson distribution ability to account for many skewness and kurtosis

pairs beyond the locus of control of both Cornish-Fisher and Gram-Charlier [16] . In

order to assess the accuracy and robustness of the expansions used in the disserta-

tion, future �elds of research could incorporate the implementation and comparison

of other downside risk estimation tools on various �nancial series. Especially, the

Johnson distribution would have been interesting to apply on the empirical dataset.

By constructing the rolling-period optimization the issue of changing correlation

patterns is addressed. It has been observed asset categories starts to correlate to-

wards one in �nancial distressed periods, but it has also been assumed the global

market gets increasingly correlated on a regular basis. Further research is needed in

order to fully investigate the correlation pattern trends to address the future e�ect

of diversi�cation as it will impact the portfolio risk estimation negatively.

Finally, this dissertation numerically examines two expansions, not just to show

the shortfalls of MVO theory, but also to convey that several models exist that

attempts to capture the true risk imposed on the investor. Simonato (2011) found

varying risk measures depending on the degree of asymmetry and choice of con�dence

interval, and the same conclusion can be applied to this dissertation. Therefore, in

order to quantify this di�erence and explain how the expansion functions treat these

factors, a more analytical approach must be taken in further studies.

98

Bibliography

[1] B. Mandelbrot. The variation of certain speculative prices, 1963.

[2] T. Bollerslev. A conditionally heteroskedastic time series model for speculative

prices and rates of return. Review of Economics and Statistics, 69(3):542, 1987.

[3] A. Gabrielsen, P. Zagaglia, A. Kirchner, and Z. Liu. Forecasting value-at-risk

with time-varying variance, skewness and kurtosis in an exponential weighted

moving average framework, 2012-06-06 2012.

[4] Campbell R. Harvey and Akhtar Siddique. Conditional skewness in asset pricing

tests. The Journal of Finance, 55(3):1263, 2000.

[5] Eric Jondeau and Michael Rockinger. Conditional volatility, skewness, and kur-

tosis: existence, persistence, and comovements. Journal of Economic Dynamics

and Control, 27(10):1699, 2003.

[6] Q. Sun and Y. Yan. Skewness persistence with optimal portfolio selection.

Journal of Banking and Finance, 27(6):1111, 2003.

[7] Campbell R Harvey, John C Liechty, Merrill W Liechty, and Peter Muller.

Portfolio selection with higher moments. Quantitative Finance, 10(5):469, 2010.

[8] A. J. Patton. On the out-of-sample importance of skewness and asymmetric

dependence for asset allocation. Journal of Financial Econometrics, 2(1):130,

2004.

[9] James X Xiong and Thomas M Idzorek. The impact of skewness and fat tails

on the asset allocation decision. Financial Analysts Journal, 67(2):23, 2011.

99

[10] Robert C. Blattberg and Nicholas J. Gonedes. A comparison of the stable

and student distributions as statistical models for stock prices. The Journal of

Business, 47(2):244, 1974.

[11] Peter K. Clark. A subordinated stochastic process model with �nite variance

for speculative prices. Econometrica, 41(1):135, 1973.

[12] C. O. Amedee-Manesme and F. Barthelemy. Cornish-�sher expansion for real

estate value at risk. Working Paper, 2012.

[13] B. Solnik and D. McLeavey. International investments. Addison-Wesley,

Boston, 5. edition edition, 2004.

[14] H. M. Markowitz. Portfolio selection. Journal of Finance, 7(1):77�91, 03 1952.

[15] E. Cornish A. and R. A. Fisher. Moments and cumulants in the speci�cation

of distributions. Revue de l'Institut International de Statistique / Review of the

International Statistical Institute, 5(4):307, 1938.

[16] J. G. Simonato. The performance of johnson distributions for computing value

at risk and expected shortfall. Journal of Derivatives, 19(1):7, 2011.

[17] N. L. Johnson and S. Kotz. Continuous Univariate Distributions 1 + 2.

Houghton Mi�in, Boston, 1970.

[18] D. Millard. A user's guide to the cornish �sher expansion. Working Paper,

2012.

[19] R. C. Boston. Evaluation of gram-charlier coe�cients. Electronics Letters - LA

English, 7(17), 1971.

[20] E. Schloegl. Option pricing where the underlying assets follow a gram-charlier

density of arbitrary order. Journal of Economic Dynamics and Control, 37(3),

3 2013.

[21] Eric Jondeau and Michael Rockinger. Gram-charlier densities. Journal of Eco-

nomic Dynamics and Control, 25(10):1457, 2001.

100

[22] A. Ronald Gallant and George Tauchen. Seminonparametric estimation of

conditionally constrained heterogeneous processes: Asset pricing applications.

Econometrica, 57(5):1091�1120, Sep. 1989.

[23] Esther B. Del-Brio, Trino-Manuel Niguez, and Javier Perote. Gram-charlier

densities: a multivariate approach. Quantitative Finance, 9(7), 2009.

[24] Esther B. Del-Brio and Javier Perote. Gram-charlier densities: Maximum like-

lihood versus the method of moments. Insurance: Mathematics and Economics,

51(3), 11 2012.

[25] J. L. Knight and S. E. Satchell. Pricing derivatives written on assets with arbi-

trary skewness and kurtosis. Birkbeck College, University of London, Working

paper (Birkbeck College. Institute for Financial Research), 1997.

[26] D. Dufresne and J: P. Chateau. Gram-ccharlier processes and equity-indexed

annuities. Melbourne, Centre for Actuarial Studies, Research paper series -

Centre for Actuarial Studies, Faculty of Economics; Commerce, the University

of Melbourne, 2012.

[27] Arnold Polanski and Evarist Stoja. Incorporating higher moments into value-

at-risk forecasting. Journal of Forecasting, 29(6):523, 2010.

[28] Peter Christo�ersen and S. Goncalves. Estimation risk in �nancial risk man-

agement. The Journal of Risk, 7(3):1, 2005.

[29] P. Pichler and K. Selitsch. A comparison of analytical var methodologies for

portfolios that include options. 1999.

[30] J. Mina and A. Ulmer. Delta-gamma four ways. RiskMetric Monitor, 1999.

[31] S. R. Jaschke. The cornish-�sher expansion in the context of delta-gamma-

normal approximations. Journal of Risk, 2002.

[32] P. Zangari. A var methodology for portfolios that include options. RiskMetric

Monitor, 1996.

101

[33] K. Boudt, B. Peterson, and C. Croux. Estimation and decomposition of down-

side risk for portfolios with non-normal returns. Journal of Risk, 11(2):79,

2008.

[34] Laurent Favre and Jose-Antonio Galeano. Mean-modi�ed value-at-risk opti-

mization with hedge funds. The Journal of Alternative Investments - LA En-

glish, 5(2):21, 2002.

[35] Stephen J. Brown and Jerold B. Warner. Using daily stock returns. Journal of

Financial Economics, 14(1):3, 1985.

[36] E. F. Fama. E�cient capital markets - review of theory and empirical work.

The Journal of Finance, 25(2):340, 1970.

[37] J. Toeyli. Essays on return distributions. Helsinki University of Technology,

2002.

[38] J. Y. Campbell and L. M. Viceira. Strategic asset allocation: portfolio choice

for long-term investors. Oxford University Press, Oxford, 2002.

[39] W. F. Sharpe. Mutual fund performance. The Journal of Business, 39(1, Part

2: Supplement on Security Prices):119�138, Jan. 1966.

[40] Z. Bodie, A. Kane, and A. J. Marcus. Essentials of investments. Irwin McGraw-

Hill, Boston, Mass., 3. edition edition, 1998.

[41] L. M. Ronald. Portfolio performance: Illustrations from morningstar. Journal

of Education for Business, 77(4):226, 2002.

[42] J. L. Treynor. How to rate management of investment funds. Harvard Business

Review - LA English, 43(1):63, 1965.

[43] Michael C. Jensen. The performance of mutual funds in the period 1945-1964.

The Journal of Finance, 23(2, Papers and Proceedings of the Twenty-Sixth An-

nual Meeting of the American Finance Association Washington, D.C. December

28-30, 1967):389�416, May 1968.

102

[44] P. Chiou. Bene�ts of international diversi�cation with investment constraints:

An over-time perspective. Journal of Multinational Financial Management,

19(2):93�110, 4 2009.

[45] E. J. Elton and M. J. Gruber. Modern portfolio theory and investment analysis.

John Wiley, New York, 5. edition edition, 1995.

[46] A. D. Aczel. Complete business statistics. Irwin McGraw-Hill, Boston, Mass.,

4. ed. edition, 1999.

[47] Tully E. Lucey B. M., Poti V. International portfolio formation, skewness and

the role of gold. IIIS Discussion Paper, No. 30., 2004.

[48] C. A. Ball and W. N. Torous. Stochastic correlation across international stock

markets. Journal of Empirical Finance, 7(3):373, 2000.

[49] F. Longin and B. Solnik. Is the correlation in international equity returns

constant: 1960-1990? Journal of International Money and Finance, 14(1):3,

1995.

[50] M. Loretan and W. B. English. Evaluating "correlation breakdowns" during

periods of market volatility, 01 2000.

[51] Morningstar. Asset allocation optimization methodology. 2011.

[52] S. Benninga. Financial Modeling. MIT Press, Cambridge, MA, 3. ed. edition,

2008.

[53] M. Christensen and F. Pedersen. Aktieinvestering - teori og praktisk anvendelse,

2003.

[54] John Hull. Options, futures, and other derivatives. Pearson, Boston, 8. ed.,

global ed. edition, 2012.

[55] K. Sydsaeter and P. Hammond. Essential mathematics for economic analysis.

Financial Times/Prentice Hall, Harlow, UK, 2. edition, 4. printing edition, 2007.

[56] R. Kitt and J. Kalda. Leptokurtic portfolio theory. The European Physical

Journal B, 50(1):141, 2006.

103

[57] Carlos M. Jarque and Anil K. Bera. A test for normality of observations and

regression residuals. International Statistical Review / Revue Internationale de

Statistique, 55(2):163�172, Aug. 1987.

[58] D. Oztuna, E. Tuccar, and A. H. Elhan. Investigation of four di�erent normality

tests in terms of type 1 error rate and power under di�erent distributions.

Turkish Journal of Medical Sciences, 36(3):171, 2006.

[59] H. M. Park. Univariate analysis and normality test using sas, stata, and spss.

The University Information Technology Services (UITS) Center for Statistical

and Mathematical Computing, Indiana University, (Working Paper), 2008.

[60] John Knight and Stephen E. Satchell. Return distributions in �nance.

Butterworth-Heinemann, Oxford, 2001.

[61] V. Chernozhukov, I. Fernandez-Val, and A. Galichon. Rearranging edgeworth-

cornish-�sher expansions. Economic Theory, 42(2):419, 2010.

[62] D. E. Barton and K. E. Dennis. The conditions under which gram-charlier and

edgeworth curves are positive de�nite and unimodal. Biometrika, 39(3/4):425�

427, Dec. 1952.

104

Part VII.

Appendix

105