Analysing the optimal level of leverage

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A thesis submitted for the degree of

PhD in Computational Finance

Analysing the optimal level of leverage in stock

markets using numerical methods and

agent-based modelling

Elton Felipe Sbruzzi

Thesis Advisor

Dr. Steve Phelps

Centre for Computational Finance and Economic Agents (CCFEA)

School of Computer Science and Electronic Engineering

University of Essex

November 2012


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Declaration

I hereby declare that this thesis is my own work, and has not been submitted for

the award of a higher degree elsewhere. Where other sources of information have

been used, they have been acknowledged.The work in this thesis has contributed to the following publications:

• Sbruzzi, E. and S. Phelps: 2011. Optimal Level of Leverage using NumericalMethods. In, 7th Conference of the Portuguese Finance Network, Aveiro,

Portugal.

• Sbruzzi, E. and S. Phelps: 2013. Testing leverage-based trading strategies

under an adaptive-expectations agent-based model. In, 12th International

Conference on Autonomous Agents and Multiagent Systems (AAMAS 2013).

Elton Felipe Sbruzzi

November 2012


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Abstract

Leverage offers the possibility of enhancing financial returns and, consequently, the

profit and the end of period wealth. Leverage is gaining importance and has been

widely adopted in the financial markets for two reasons. Firstly, brokers are in-terested in offering margins because they can charge higher transaction fees and

make profits from lending margins. Secondly, investors are also interested in taking

leverage because of the ability to enhance their individual returns. The motivation

of this thesis is that, even though leverage is gaining importance in modern invest-

ments, existing models in the literature models assume that the series of financial

returns are normally distributed. However, financial returns present high-level of

kurtosis and, hence, are not normally distributed. Thus, existing analytical models

underestimate extreme returns and consequently underestimate the risk of default.

I contribute to this field by proposing a new trading strategy that uses numerical

methods to calculate the optimal level of leverage instead of the existing analytical

models. The use of numerical methods allows me to relax the assumption of nor-

mally distributed returns, and hence minimises the risk of underestimate extreme

returns and the risk of default. I investigate whether the use of numerical meth-

ods leads to a more accurate optimal level of leverage than analytical models, and

if the use of the optimal level of leverage using numerical methods improves the

investment performance. In order to test the ability of the optimal level of lever-

age using numerical methods to improve the investment performance, I employ two

different approaches: back-testing and agent-based modelling. Back-testing allows

me to test the optimal level of leverage using numerical methods using empirical


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evidence, and agent-based modelling allows me to test the optimal level of leverage

using numerical methods in a totally controlled environment. The conclusions are

that the use of numerical methods leads to a more accurate optimal level of leverage

than analytical models; using daily historical data as an empirical evidence, the op-

timal level of leverage using numerical methods improves investment performance;

and in a totally controlled environment, the ability of the optimal level of leverage

to improve investment performance depends on the size of the market.


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Acknowledgments

Writing a thesis and getting a PhD is a life-changing opportunity. It requires the full

dedication of the candidate and the support of every one who is part of his life. For

this reason, I would like to express my gratitude to a number of people. Withouttheir help, support and patience, certainly this work could not have been done.

Firstly, I would like to thank the group of the most important people of my life,

my family; my parents, Mr. Arnaldo Felipe Sbruzzi and Mrs. Luciana de Souza

Sbruzzi; my wife, Mrs. Ana Paula Ribeiro Duarte Sbruzzi; and my son, the reason of

my life, who was born during the PhD period, Lúıs Felipe Duarte Sbruzzi. Without

them, nothing here would make much sense and I would not be able to reach any of

my objectives. They are unique for me and their love and support makes me strong

to reach my dreams.

Secondly, I would like to thank my supervisor Dr. Steve Phelps. His patience

with me was impressive. He provided me with an appropriate and supportive en-

vironment that allowed me to go further in my research. He offered me the chance

to express my ideas without fear of reprisal. He helped me to limit the scope of my

research and was receptive to correct the natural mistakes that I have made during

the research.

Thirdly, I would like to thank Mr. Davi Baccan. Mr. Baccan is a closed friend

and he has been pursuing a PhD in Computer Science at the University of Coimbra,

Portugal. Mr. Baccan has provided me with every necessary informal support in

many area of the research.

Fourthly, I would like to thank Cynthia Rocha. With her support, I could review


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and improve my writing, in a much clear and sophisticated fashion.

Fifthly, I would like to thank the Sfera Stock Analysis clients. It is the business

of my family, and this has more than 38k clients. The business funded my PhD

studies and my stay in the UK.

Sixthly, I would like to thank the staff of the Centre for Computational Finance

and Economic Agents of University of Essex. They have offered me every sup-

port that I have required and have provided me the sufficient infra-structure to do

my research. I had the opportunity to attend some important mini-courses in the

university which facilitated the research.

Finally, I would like to thank the University of Essex community and all my

friends from Colchester and in the UK. If I name each of them, certainly dozens of

pages would be necessary. Furthermore, there would be a risk of forgetting someone.

However, each one is very special and made my PhD experience more exciting and

amazing than I could have expected.


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Contents

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Summary of Research Contributions . . . . . . . . . . . . . . . . . . 81.4 Overview and Structure . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Chapter 2: Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Leverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Return Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Investor’s Objective Function . . . . . . . . . . . . . . . . . . . . . . 202.4 Geometric Mean, Leverage and End of Period Return . . . . . . . . . 23

2.4.1 Single Period Equity Return . . . . . . . . . . . . . . . . . . . 232.4.2 Geometric Mean and End of Period Return . . . . . . . . . . 25

2.5 Optimal Leverage Models . . . . . . . . . . . . . . . . . . . . . . . . 282.5.1 The Kelly Criterion . . . . . . . . . . . . . . . . . . . . . . . . 292.5.2 The Rotando and Thorp Model . . . . . . . . . . . . . . . . . 302.5.3 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . 32

2.6 Agent-Based Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 332.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Chapter 3: Proxy Vs Real Geometric Mean on Leverage . . . . . . . 41

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2.2 Single Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Chapter 4: Optimal Level of Leverage using Numerical Methods . . 554.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 60


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4.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Chapter 5: A Back-testing of the Optimal Level of Leverage using

Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2.2 Single Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . 765.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80


Chapter 6: Agent-Based Modelling of Optimal Leverage Model using

Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.2 Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.4.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97


Chapter 7: Conclusion and Future Work . . . . . . . . . . . . . . . . . . 107

7.1 Summary of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119


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List of Figures

3.1 The flow-chart of the sampling process of the proxy and the realgeometric mean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 The series of prices of UNH between 17/01/2001 and 04/08/2011. . . 45

3.3 The series of returns of UNH between 17/01/2001 and 04/08/2011. . 46

3.4 The cumulative equity returns of leveraged and unleveraged invest-ment in of UNH between 17/01/2001 and 04/08/2011. . . . . . . . . 47

3.5 Plot of the series of the proxy and the real geometric mean of theunleveraged investments. . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.6 Plot of the difference between the proxy and the real geometric meanof the unleveraged investments. . . . . . . . . . . . . . . . . . . . . . 49

3.7 The boxplot of the proxy of the geometric mean and the real geomet-ric mean of the returns of unleveraged investments. . . . . . . . . . . 50

3.8 Plot of the series of the proxy and the real geometric mean of theleveraged investments. . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.9 Plot of the difference of the series of the proxy and the real geometricmean of the leveraged investments. . . . . . . . . . . . . . . . . . . . 51

3.10 The boxplot of the proxy of the geometric mean and the real geomet-ric mean of the returns of leveraged investments. . . . . . . . . . . . . 51

4.1 The plot of the geometric means calculated using numerical methodsand geometric Brownian motion. . . . . . . . . . . . . . . . . . . . . 62

4.3 The boxplot of the geometric mean using numerical methods andbased on the geometric Brownian motion. . . . . . . . . . . . . . . . 63

4.2 The plot of the difference of the geometric means between numericalmethods and geometric Brownian motion. . . . . . . . . . . . . . . . 63

4.4 The plot of the level of leverage calculated using numerical methodsand geometric Brownian motion. . . . . . . . . . . . . . . . . . . . . 65

4.5 The plot of the difference of the level of leverages between numericalmethods and geometric Brownian motion. . . . . . . . . . . . . . . . 66

4.6 The boxplot of the level of leverage using numerical methods andbased on the geometric Brownian motion. . . . . . . . . . . . . . . . 67

5.1 The series of prices of NDAQ between 01/11/2002 and 19/10/2007. . 77

5.2 The series of returns of NDAQ between 01/11/2002 and 19/10/2007. 77

5.3 The series of the optimal level of leverage and of the geometric Brow-nian motion level of leverage of an investment in NDAQ between01/11/2002 and 19/10/2007. . . . . . . . . . . . . . . . . . . . . . . . 78


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2 List of Figures

5.4 The series of cumulative equity return of an investment in NDAQbetween 01/11/2002 and 19/10/2007 considering three different ap-proaches of leverage: unleveraged, optimally leveraged, and excessiveleveraged. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.5 Plot of the dynamic of the optimal level of leverage. . . . . . . . . . . 80

5.6 Plot of the average value of the optimal level of leverage per iteration. 815.7 The boxplot of the log of the end of period equity return of the

optimally leveraged vs. the geometric Brownian motion leveragedinvestment; the boxplot of the log of the end of period equity returnof the optimally leveraged vs. the unleveraged investment; and theboxplot of the log of the end of period equity return of the optimallyleveraged vs. the excessively levearaged investment. . . . . . . . . . . 84

6.1 The plot of the log of the end of period returns of leveraged agents,N=10 agents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.2 The plot of the log of the end of period returns of leveraged agents,N=10 agents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.3 The plot of the log of the end of period returns of leveraged agents,

N =100 agents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.4 The boxplot of the log of the end of period returns of leveraged agents,

N =100 agents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.5 The plot of the log of the end of period returns of leveraged agents,

N =1000 agents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.6 The plot of the log of the end of period returns of leveraged agents,

N =1000 agents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103


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List of Tables

3.1 Statistical analysis of the series of the proxy and the real geometricmean of the unleveraged investments. . . . . . . . . . . . . . . . . . . 48

3.2 Statistical test of the difference between the proxy of the geometricmean and the real geometric mean of unleveraged investments. . . . . 48

3.3 Statistical analysis of the series of the proxy and the real geometricmean of the leveraged investments. . . . . . . . . . . . . . . . . . . . 52

3.4 Statistical Test of the difference between the proxy of the geometricmean and the real geometric mean of leveraged investments. . . . . . 52

4.1 Statistical analysis of the geometric mean of using the geometricBrownian motion approach and the numerical methods to calculatethe optimal level of leverage. . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 Comparison between geometric Brownian motion approach and nu-merical methods of the geometric mean using the optimal level of

leverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 Statistical analysis of the optimal level of leverage of the use of thegeometric Brownian motion approach and the numerical methods. . . 64

4.4 Comparison between geometric Brownian motion approach and nu-merical methods of the optimal level of leverage. . . . . . . . . . . . . 64

5.1 Statistical analysis of the log of the end of period equity return of un-leveraged ln(EoP (1)), geoemtric Brownian motion leverage ln(EoP (lgbm)),optimally leveraged ln(EoP (l∗)) and excessively leveraged investmentln(EoP (10)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.2 T-Test of the difference between the log of the end of period equityreturn of optimally leveraged investment and geotmric Brownian mo-tion leveraged investment (GM (l∗) = GM (lgbm)); the difference be-tween the log of the end of period equity return of optimally lever-aged investment and unleveraged investment (GM (l∗) = GM (1));and the difference between the log of the end of period equity returnof optimally leveraged investment and excessive leveraged investment(GM (l∗) = GM (1)). . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.1 Statistical analysis of the log of the ends of period returns, N = 10agents. Opt = optimally leveraged, Unl = unleveraged and Exc =excessively leveraged. . . . . . . . . . . . . . . . . . . . . . . . . . . . 98


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List of Tables 1

6.2 T-test results between the log of the end of period return of theoptimally leveraged agent (Opt) and the unleveraged agent (Unl);and the optimally leveraged agent and the excessively leveraged agent(Exc). N = 10 agents. . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.3 Statistical analysis of the log of the ends of period returns, N = 10o

agents. Opt = optimally leveraged, Unl = unleveraged and Exc =excessively leveraged. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.4 T-test results between the log of the end of period return of optimallyleveraged agent (Opt) and unleveraged agent (Unl) and optimallyleveraged agent and excessively leveraged agent (Exc). N = 100 agents.100

6.5 Statistical analysis of the log of the ends of period returns, N = 1000agents. Opt = optimally leveraged, Unl = unleveraged and Exc =excessively leveraged. . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.6 T-test results between the log of the end of period return of optimallyleveraged agent (Opt) and unleveraged agent (Unl) and optimally

leveraged agent and excessively leveraged agent (Exc). N = 1000agents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102


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Chapter 1

Introduction

This thesis uses numerical methods to calculate the optimal level of leverage and

tests the ability of the optimal level of leverage using numerical methods to improve

individual’s investment performance. I assume that investment performance is the

end of period equity and equity is the amount of capital of the agent. For example,

if an investor with an initial equity of $1,000 opens a long position of 100 shares of

security A, which its price per share is $100, the leverage is 100 x $100 - $1,000 =

$9,000 and the level of leverage is 100 x $100 / $1,000 = 10.

This exposure expands the investor’s equity returns and can have a significant

impact of stock returns on equity returns. For example, suppose there is a scenario in

which two period returns of an investment in a single stock with a value of $100 and

with a stock returns of 8% and -10%. We also assume that the capital is reinvested

in the end of each period. If the level of leverage is 1, this means that the leverage is

0, and the equity increases to $108 in the first period, and, in the second period, the

equity decreases -10% to $97.20. Consequently, the cumulative equity return would

be -2.8%. However, if the level of leverage is 10, the equity increases to $180 [100*(1

+ 8%*10)] in the first period, and, in the second period, the equity decreases to $0

[180 * (1 - 10% * 10]. Consequently, the cumulative equity return would be -100%.

Hence, what dictates the equity return is the combination of stock return and the

level of leverage. Thus, in order to improve the individual investment performance,


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4 Chapter 1. Introduction

the appropriate use of leverage is as crucial as the appropriate selection of the stock.

In this chapter, I present the introduction of this thesis. This chapter is organized

as follows. In Section 1.1, I present the motivation of this thesis. In Section 1.2,

I present the research objectives. In Section 1.3, I describe the contributions of

this thesis. In Section 1.4, I present the outlines the general structure. Finally, in

Section 1.5, I present the publications originated from this thesis.

1.1 Motivation

In this section, I present the motivation of this thesis. Nowadays, leverage plays a

very important role in the financial markets. New instruments designed to facilitate

leveraged investments such as Contract for Difference (CFD) in conjunction with

technological advances in the electronic markets make leverage accessible for every

type of investor, from small individual investors up to big pension funds. CFD

is an informal contract between a broker and his client (Norman 2009). Using

CFD, broker controls the level of exposure of the client and is more interested in

allowing clients to take leverage than using stocks. For example, in 2007, according

to UK Financial Service Authority (FSA), 30% of the volume on the London Stock

Exchange was driven by leveraged investments (FSA 2007).

However, to the best of my knowledge, there are relatively few studies relating

leverage and Computational Finance. The existing papers analyse the implication

of leverage on the financial market as a whole (Geanakoplos 2009; Thurner, Farmer,

and Geanakoplos 2011). However, no previous research has analysed the implication

of leverage to an individual’s investment performance.

Computational Finance provides tools to experiment and to simulate hypotheses

regarding different aspects of finance. There is a considerable amount of research

on trading systems using Computational Finance tools (Lo and MacKinlay 1990;

LeBaron 2000; LeBaron 2002; LeBaron 2006; Martinez-Jaramillo and Tsang 2009;

Iori and Chiarella 2002), but such research has not taken into account leverage.


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1.1. Motivation 5

Studies considering leverage and individual investment performance in Computa-

tional Finance are relevant for three reasons.

Firstly, recent studies about leverage demonstrate that an excessive level of lever-

age is considered irrational behaviour (Geanakoplos 2009; Thurner, Farmer, and

Geanakoplos 2011). When agents obtain positive equity returns, they can be frus-

trated if the level of leverage is low because such returns could be higher if the

agent was more leveraged. This belief encourages the agent to increase their level

of leverage in the next period. However, this behaviour is based on a naive belief

rather than rational reasoning because there is no model to indicate whether the

increment of the level of leverage is appropriate.

Secondly, the equity return formula has two components: the stock return and

the level of leverage. Additionally, the level of leverage is linear in the equity return

formula (Peters 2010). This means that leverage has the ability to modify the

impact of the stock return on the equity return and, consequently, on an individual’s

investment performance. For example, a positive stock return can lead to negative

equity return if the level of leverage is negative.

Thirdly, leverage can lead the investor to ruin. For example, suppose that an

investor is 10 times leveraged and the stock return is -10%, the equity return is -

100%. Therefore, the end of period return is -100%, and then after the -100% equity

return, the investor has no more capital to invest and, consequently, would not be

able to open any position.

Leverage has the ability to impact the investment performance. However, if used

on an excessive level, leverage can lead investor to bankruptcy. Thus, if excessive

level of leverage is bad for investments, could leverage be good on some level? Is

there an optimal level of leverage? If yes, could it be measured?

Kelly (1956) demonstrated that in binomial games with positive expectation,

there is one specific value of leverage which maximises the geometric mean or the

end of period wealth. He proposed a model to obtain this value and this model was

called the Kelly criterion. Today, one of the possible uses of the Kelly criterion is


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to determine the optimal level of leverage on investments considering the possible

returns on different discrete scenarios. However, the Kelly criterion is not suitable

for determining the optimal level of leverage in continuous games when the number

of possible outcomes is unlimited, as with financial returns.

Rotando and Thorp (1992) propose the Kelly criterion version for continuous

gambling games. But the use of the Kelly criterion for continuous gambling games

is not straight forward. Continuous games can assume infinite number of outcomes

and the Kelly criterion was designed to assume a finite number of outcomes. Hence,

they assumed that the returns in the financial market are normally distributed and

such distribution should be transformed to quasi-normal distribution by cutting off

the tails in order to limit the number of outcomes and respect the Kelly criterion

original assumptions.

In order to obtain a more realistic approach, Peters (2010) introduces a new

model to obtain the optimal level of leverage using geometric Brownian motion. He

demonstrated that the optimal level of leverage is simply the estimation of log-return

divided by the variance of log-return. In this model, the objective is to estimate the

appropriate expected return and variance.

These three authors assume that the objective is to find the level of leverage

that maximises the geometric mean of the historical financial returns. Their models

yield a closed-form solution. In order to develop their models, they assume that

the historical financial returns are normally distributed, and ignore the existence of

fat-tails. However, Cont (2001) demonstrates that financial returns are not normally

distributed and exhibit fat-tails. This means that if analytical models that assume

the normal distribution are used to calculate the optimal level of leverage of a

historical financial returns, then they will underestimate the probability of extreme

returns and, consequently, underestimates the risk of default. The result is that it

could lead investors to use an excessive level of leverage which could be harmful to

the investor’s investment performance. For this reason, I argue that it is necessary

to develop models to optimize the level of leverage that relax the assumption of


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1.2. Research Objectives 7

normal distribution of the series of historical financial returns.

In the next section, I present the research objectives of this thesis.

1.2 Research Objectives

The research objectives of this thesis are to introduce a numerical method to cal-

culate the optimal level of leverage that allows the relaxation of the assumption of

normal distribution of the series of financial returns; and to test the ability of this

method to improve investment performance. Thus, the hypotheses of this research

are that using numerical methods, the geometric mean of the historical financial

returns using numerical methods is superior to the geometric mean of the historical

financial returns using analytical methods; and that the use of the optimal level of

leverage using numerical methods improves investment performance.

I use two different approaches in this thesis. In the first approach, in order to

test the hypothesis that the value of the optimal level of leverage is more appropriate

than using analytical methods, I assume that the agent’s objective is to maximise

the end of period return. Maximising the end of period return is the same as

maximising the geometric mean (Williams 1936, Latane 1959). Thus, similarly to

analytical models, I assume that the optimal level of leverage is the level of leverage

that maximises the geometric mean of the series of the historical financial returns.

Thus, the test is a T-test of difference between the series of the geometric mean

of the series of the leveraged historical financial returns calculated using numerical

methods and the series of the geometric mean of the series of the leveraged historical

financial returns calculated using an analytical model with a closed-form solution.

In the second approach, I compare the individual end of period equity return

under three different experimental treatments using an agent-based model of an

order-driven market. In the first treatment, the agent maintains her level of leverage

equal to 1. In the second treatment, the agent maintains her level of leverage equal

to the value calculated using numerical methods. Finally, in the third treatment,


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the agent maintains her level of leverage equal to 10.

I use two statistical tests in order to analyse the outcomes under these treat-

ments. The first test is a T-test of the difference of the series of the end of period

equity returns of optimally leveraged investment and the unleveraged investment.

The second test is a T-test of the difference of the series of the end of period equity

returns of optimally leveraged investment and the excessively leveraged investment.

In the next section, I summarise the contributions of this thesis.

1.3 Summary of Research Contributions

In this section, I describe the contributions of this thesis. The general contributions

of this thesis are the proposal of numerical methods to optimize the level of leverage

and the demonstration that the use of the optimal level of leverage using numerical

methods improves investment performance. The main advantage is that the use of

numerical methods allows the calculation of the optimal level of leverage directly

using the formula for the geometric mean. This allows me to relax the assumption

of the normal distribution of financial returns and, consequently, reduces the risk of

to underestimating extreme returns.

In order to do this, I assume that the investor’s objective is to maximise the end

of period equity return. Thus, this serves to measure investment performance, and

the end of period equity return can be represented by the geometric mean of the

equity returns (Williams 1936, Latane 1959).

There are three specific contributions of this research. The first contribution is

to show that a commonly-used proxy of the geometric mean, which is widely used

in the current literature, fails to accurately estimate leveraged returns. This proxy

uses the arithmetic average and the variance of the stock returns (Parramore and

Watsham 1997). However, this proxy assumes that returns can be characterised

by the first two moments and ignores the fat-tailed return distributions that are

observed in actual empirical data (Cont 2001).


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1.4. Overview and Structure 9

The second contribution is the introduction of a novel trading strategy which

uses numerical methods in order to calculate the optimal level of leverage. Previous

studies have proposed analytical models in order to calculate the optimal level of

leverage. However, these models assume that returns are normally distributed (Kelly

1952; Rotando and Thorp 1992, and Peters 2010). By using numerical methods in

order to optimise the level of leverage, I am able to avoid this assumption and hence

avoid under-estimating extreme returns (Cont 2001). Moreover, I show that my

trading strategy exhibits superior performance to a strategy which does not use

leverage, under an analysis which uses historical data from all components of the

S&P 500 index. By performing this back-testing, I am able to avoid making any

assumptions about the distribution of returns. This is in contrast to existing studies

which simply assume that returns are normally distributed, and do not evaluate their

performance using empirical data (Kelly 1952; Rotando and Thorp 1992, and Peters

2010).

Finally, the third contribution is the introduction of a leveraged component of

agents’ decision making into an agent-based model of a financial market. Previous

models have used expected utility optimisation as a framework for agents’ trading

decisions. In contrast, my model uses the aforementioned trading strategy in order

to decide the quantity of orders by numerically-optimising the level of leverage. By

so doing, I am able to use this agent-based model to evaluate the potential effect of

market impact on the performance of my leverage-based trading strategy. To the

best of my knowledge, this is the first attempt to systematically evaluate the effect

of market-impact on the performance of leverage-based trading.

In the next section, I present the structure of the thesis.

1.4 Overview and Structure

This thesis consists in seven chapters, including the current one. In Chapter 2, I

present the literature review of this thesis.


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In Chapter 3, I discuss the estimation of the returns on leverage. In the literature,

there is a widely used approximation of geometric mean that employs the arithmetic

average and the variance of the stock return (Parramore and Watsham 1997) but

ignores the kurtosis. I discuss the use of this formula for leveraged equity return

instead of stock return.

In Chapter 4, I propose a numerical method in order to calculate the optimal

level of leverage. The advantage of the use of numerical methods is that, differently

of the analytical models existing on the literature, the method proposed in this thesis

allows me to relax the assumption of normal distribution of the series of financial

returns and, consequently, undermine the risk of underestimation of extreme returns.

In Chapter 5, using the historical daily price of the components of the S&P 500

Index, I back-test the use of the optimal level of leverage using numerical methods. I

compare the end of period returns of the optimally leveraged investment to geomet-

ric Brownian motion leveraged, unleveraged and excessively leveraged investment,

respectively.

In Chapter 6, I employ an agent-based model in order to experiment the use of

the optimal level of leverage in a totally controlled environment. I introduce three

leveraged agents regarding their approach of leverage, the optimal level of leverage

using numerical methods, the unleveraged and the excessive level of leverage, in

an order-driven model proposed by Iori and Chiarella (2002). Then, I simulate it

regarding to three different sizes of market, 10, 100 and 1000 agents in the JASA

platform (Phelps 2007).

Finally, in Chapter 7, I summarize the main findings of this thesis and conclude.

I also discuss ideas for future research.

In the next chapter, I present the publications originated from this thesis.


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1.5. Publications 11

1.5 Publications

There are two publications from this thesis. Firstly, material from Chapter 4 has

been peer-reviewed and published as:

• Sbruzzi, E. and S. Phelps: 2011. Optimal Level of Leverage using NumericalMethods. In, 7th Conference of the Portuguese Finance Network, Aveiro,

Portugal.

Secondly, material from Chapter 6 has been submitted as:

• Sbruzzi, E. and S. Phelps: 2013. Testing leverage-based trading strategiesunder an adaptive-expectations agent-based model. In, 12th International

Conference on Autonomous Agents and Multiagent Systems (AAMAS 2013).

In the next chapter, I review the literature of the thesis.


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Chapter 2

Literature Review

In this chapter, I present the literature review of this thesis. Firstly, in Section 2.1,

I review the literature on leverage. Secondly, in Section 2.2, I review the literature

on models to estimate returns. Thirdly, in Section 2.3, I review the literature of the

investor’s objective function. Fourthly, in Section 2.4, I demonstrate how to relate

geometric mean, end of period return and level of leverage. Fifthly, In Section 2.5,

I present the existing models to optimize the level of leverage. Sixthly, in Section

2.6, I review the literature on the agent-based modelling. Finally, in Section 2.7, I

present the summary of the chapter.

2.1 Leverage

In this section, I present the literature review of leverage and its implication to the

individual investor’s performance and the financial market as a whole. Breuer (2002)

argues that leverage can be thought of as elasticity; indicating the responsiveness of

the value of equity to changes in the value of overall stocks:

kt = lt ∗ rt (2.1)

where kt is the equity return, lt is the level of leverage and rt is the stock return. Note

that stock return, rt, is different of equity return kt. Stock return is the percentage


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14 Chapter 2. Literature Review

difference of price along of two period time:

rt = P tP t−1

− 1

where P t is the price on the period of time t. Consequently, the impact of the stock

return on the equity return depends on the level of leverage of the position.

As changes in the value of equity are equal to changes in the stock portfolio,

leverage is conventionally defined as the ratio of assets to equity. Furthermore, by

assuming that the stock return expectation is approximately zero and that variance

is the representation of risk, Leverage increases the variance of the equity return

and, consequently, increase the risk of the equity return:

k2t = l2t ∗ r2t (2.2)

where k2 is the risk of equity return.

However, using some derivatives instruments such as futures, this incremental

risk associated to the leverage is off balance sheet and leads the regulators unin-formed about to the institution’s exposure to risk.

Breuer also argues that leverage can have two effects: (1) by definition it creates

and enhances the risk of default by market participants; and (2) it increases the

potential for rapid deleveraging - the unwinding of partially debt financed positions

by market participants - which can cause major disruptions in financial markets by

exaggerating market movements (Breuer 2002). The larger the leverage, the smaller

is the price change that may be needed to trigger an unwinding of the position.

While any unleveraged position would require similar actions, leveraged positions

may amplify this destabilizing mechanism and increase volatility more rapidly.

The increment of the stock volatility and its impact of the market are also

analysed by (Geanakoplos 2009; and Thurner, Farmer, and Geanakoplos 2011).

They demonstrate that leverage can become too high during boom times and too low

during bad times, i.e., leverage is cyclical. They suggest that this situation is caused


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2.1. Leverage 15

by the funds’ competition for new investors. During boom times, higher leverage

hedge funds show superior returns and these hedge funds use such returns to attract

new investors. This competition creates a bubble in leverage and, consequently, a

bubble in the financial markets.

Khandani and Lo (2007; 2008) demonstrate that the use of leverage was quite

common among quantitative equity market-neutral strategies, where the higher

leverage ratios were used by those managers engaged in high-frequency margin over-

flow strategies because those strategies exhibited the lowest volatilities and highest

Sharpe ratios.

Adrian and Shin (2010) point out the relationship between leverage and liquid-

ity; there is a positive correlation between leverage and the financial intermediaries

and commercial bank balance sheet. They demonstrate that, on one hand, the com-

mercial banks and financial intermediaries target a fixed leverage ratio; and, on the

other hand, for households, there is a strongly positive relationship between changes

in total assets value and changes in leverage. Furthermore, they state that leverage

is pro-cyclical. The adjustments of the leverage and price reinforce each other am-

plifying the financial cycle and, due to the size of the financial intermediaries, this

strongly influences the liquidity, possibly leading to unwinding as observed during

the period between the crisis in July and August 2007 (Khandani and Lo 2007).

Brunnermeier and Pedersen (2005) explore the predatory characteristics of the

combination of leverage, liquidity and the overreactions of strategies. The bear

market can force traders and financial intermediaries to liquidate their position to

cover their margin calls, and provided that others players have this information, it

can lead other players to trade in the same direction, resulting in a withdraw of

liquidity from the market. This activity makes liquidation costly and leads to price

overshooting.

According to Loeys and Panigirtzoglou (2005), leverage is considered an impor-

tant driver of how much the market reacts to news. Leverage increases volume

without a similar increment in the number of participants. This leads to the con-


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centration of the risk in fewer hands. Leverage also increases the probability of

bankruptcy and can force investors to close the position in hostile conditions, which

would force the price down, exacerbating volatility. Furthermore, volatility and

leverage are positively correlated and volatility causes leverage.

The conclusion that leverage and volatility are positively correlated is consistent

with the optimal leverage model for non-ergodicity as introduced by Peters (2010).

Using geometric Brownian motion, he demonstrates that the optimal level of leverage

is simply the estimation of log-return divided by variance of log-return. Hence, the

objective is to estimate the appropriate return and variance.

Domian, Racine and Wilson (2003) use a continuous time model to derive distri-

butions for leveraged portfolios over long periods. They demonstrate that expected

return is linear in leverage in a single-period CAPM model, but concave in the

continuous time model.

From the literature, leverage is a way to expand profitability. These charac-

teristics create the expectation to gain huge amount of money in shorter period.

Therefore, it can affect the investor’s behaviours with consequences to the financial

market. The point is that, together with the expansion of the profitability, there is

a natural increase in the probability of bankruptcy.

I assume that behaviour is irrational if the investor increase the level of leverage

without any mathematical optimization of the level of leverage dictating this incre-

ment. Hence, the combination of irrational behaviour and probability of bankruptcy

can be extremely harmful for the individual investor and, consequently, to the fi-

nancial market. If an individual agent is leveraged and in bankruptcy, there is a

probability that this agent does not repay its margin assumed. For this reason,

the counterpart of this position, the lender, can also have the consequences of this

situation. If the number of bankruptcy agent is low, the impact of this situation in

the financial system is low and, therefore, the chance of this to be absorbed by the

market is higher. However, if the number of bankruptcy agent high, the chance of

this to be absorbed by the market is low and, consequently, this can collapse the


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2.2. Return Estimation 17

financial market.

There is a competition for new investors and, particularly during boom times,

this competition is reinforced by the increase in the level of leverage. Thus, the

number of agents will increase their exposure and the level of leverage will increase

with the duration of the boom times. Naturally, this situation creates bubbles.

However, besides creating bubbles, this competition increases the impact of the

bearing market on the financial system sustainability, due to the huge number of

agents leveraged.

Although the literature on leverage focuses on the consequences to the market, it

is important to see the consequences of leverage to the individual investor. I assume

that the financial market is a result of the interaction of different agents. Hence,

instead of the contribution with techniques to use leverage in order to improve the

financial market, this thesis aims to contribute with techniques to the individual

investor to use leverage in their investment.

In the next section, I review the literature on the estimation of the returns.

2.2 Return Estimation

In this section, I review the literature on the appropriate estimator for the returns.

The reason is that in Chapter 3, I test the ability of the proxy of the geometric

mean to be used as an estimator of the real geometric mean of leveraged returns.

The debate about the appropriate estimator for returns has its origin in Williams

(1936), who demonstrated that speculators, in a multi-period framework, should be

concerned with the geometric mean instead of the arithmetic average. The main

argument is that arithmetic average does not consider the non-linearity of returns

present on exponential compound series, like financial series, which is only captured

by geometric mean (Latane 1959).

Latane (1959) and Williams (1936) argue that, empirically, the investor’s objec-

tive is to maximise the end of period wealth. Thus, the estimator has to be able to


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represent short-term returns which can be compounded to represent the long-term

returns.

However, due to its simplicity, the most widely used formula to estimate the

return is average arithmetic (AA) as described below:

µ = 1

n

nt=1

rt (2.3)

where AA is the arithmetic average, rt is the return obtained on time t.

However, there is an important issue that should be taken into account. This

formula does not consider the non-linearity of returns; that is, negative returns have

similar impacts to positive returns.

This is an issue because the end of period wealth is a consequence of the cu-

mulative return which is not linear. Thus, in order to capture the non-linearity,

the alternative formula to estimate the return is the geometric mean formula as

described below:

GM =

nt=1

(1 + rt)1

n − 1 (2.4)

where GM is the geometric mean of returns.

Thus we see that the arithmetic average is biased. Jacquier et al (2003) demon-

strate that, on one hand, this bias increases positively with time and it over-estimates

the end of period wealth. On the other hand, without the use of computer software,

the calculation of the geometric mean is not straight forward. In order to solve this

issue, a common approach is to relate the geometric mean to the arithmetic average

and the variance (Latane 1959). Accordingly, the proposed formula is:

GM µ − σ

2

2 (2.5)

where GM

is the approximation of the geometric mean and σ2 is the variance.

This approximation is widely used in the literature because it permits one to work

with the geometric mean in a closed-form solution. Consequently, it is straightfor-


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2.2. Return Estimation 19

ward to differentiate and obtain a closed solution of the model.

I demonstrate how the proxy of the geometric mean of the return, GM

, as shown

in Eq. (2.5), is obtained using Taylor approximation. Let’s define x = ln(1 + GM ).

Thus from Eq. (2.4):

x = 1

n

nt=1

ln(1 + rt)

Assuming −1 < rt ≤ 1 and using a Taylor expansion, we obtain:

x = 1

n

n

t=1

(rt

−

rt2

2 +

rt3

3 −

rt4

4 + ...)

Thus:

x = E (r)− E (r2)

2 +

E (r3)

3 − E (r

4)

4 + ...

where E (a) = 1n

nt=1 at. Assuming that rt is so small that if i > 2 then E (r

i) = 0:

x ≈ E (r) −E (r2)

2 (2.6)

Note that GM = ex − 1. Thus, using Taylor expansion and Eq. (2.6):

GM = x + x2

2! +

x3

3! +

x4

4! + ...

Assuming that if a + b > 2 then E (ra)E (rb) = 0 and E (ra)b = 0, then the proxy

of the geometric mean is:

GM ≈ x + x

2

2! (2.7)

Substituting Eq. (2.6) in Eq. (2.7):

GM ≈ E (r)− E (r

2)

2 +

(E (r) − E (r2)2

)2

2 (2.8)

Simplifying Eq. (2.8):


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GM ≈ E (r) − E (r

2) −E (r)22

(2.9)

Since the arithmetic average is µ = E (r), and the variance is σ2 = E (r2)−

E (r)2,

I obtain Eq.(2.5):

GM ≈ µ − σ

2

2

In order to obtain the result the closed-form solution Eq. (2.5), that the proxy of

the geometric mean is the arithmetic average minus half of the variance, the model

has three assumptions. The first assumption is that the returns are always greaterthan minus one and less or equal to one. This means that I assume that there is

no return less or equal to -100%, i.e., there is no probability of default. The second

assumption is that the stock returns are so small that the estimation of the return

power any value superior to two is zero. Finally, the third assumption is that if the

sum of the value of the power of the return is superior to two then the product of

the estimation of the return is also zero. The consequence of the second and the

third assumption is that the model assumes that the value of any statistical moment

superior to two is 0. Thus, the value of the skewness and the kurtosis is zero.

In the next section, I review the literature on the investor’s objectives.

2.3 Investor’s Objective Function

In this section, I review the literature on the investor’s objective function. The

reason is that, in Chapter 4, I assume that the investor’s objective is to use the

geometric mean approach in order to optimize the level of leverage.

Studies of optimal level of leverage are associated with debates of individual

investor’s objective function. In the literature, the investor’s objective function can

be divided in two different approaches: mean-variance (Markowitz 1952) and growth

optimal portfolio (Kelly 1956 and Latane 1959). Leverage is linear in the mean-


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2.3. Investor’s Objective Function 21

variance approach, and concave in the growth optimal portfolio approach. Due to

linearity of leverage, studies of leverage are irrelevant in the mean-variance approach,

because even when leverage is introduced into the parameters of the model, it does

not alter the model itself; i.e., the model is independent of leverage. However, due to

the non-linearity of leverage in the growth optimal portfolio approach, the study of

leverage is important because the introduction of leverage on the parameters alters

the results of the model.

There is a very important debate about the objectives of individual investors.

While Markowtiz (1952) developed the mean-variance method to determine the

optimal portfolio to the next period; Kelly (1956) and Latane (1959) argued that

the main objective of the individual investor is to maximise the end of period wealth;

hence, the investor should be concerned about the capital rate of growth which could

be measured using the geometric mean.

This debate continued during the 1960s and 1970s. On one hand, Breiman (1961)

argues that on a long sequence of trials, the game objectives are to minimise the

time to reach the target level of wealth, or to maximise the value of the end of

the period wealth. Breiman (1961) and Hakansson (1971a; 1971b) show that the

optimal strategy to attain both objectives is to maximise the expected value of the

log of the terminal wealth which is the same as maximising the geometric mean of

return.

On the other hand, Samuelson (1971; 1979) argues that geometric mean max-

imisation was only one among many investment rules and there is no rationale to

suppose that it is the best. He shows that the geometric mean rule leads to sub-

optimal expected utility and, because the end of period expected is the sum of the

utility for each period, the end of period wealth under geometric mean rule will also

be sub-optimal.

The debate was theoretical with each author advocating different rules by us-

ing complex mathematical models. The reason is that during that period, the

use of numerical methods was extremely difficult because of the primitive level


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of computer technology compared to today’s technology. The lack of technology

obstructed the debate for more than 20 years. However, during the 1990s and the

2000s, new authors such as MacLean, Ziemba, and Blazenko (1992), Ross (1999),

Hunt (2002), Leippold, Trojani, and Vanini (2004), Christensen (2005) and Estrada

(2010), restarted the debate, but at this time, using numerical methods to analyse,

simulate and compare different rules similar to the theory proposed in the 1970s.

Debates about investor objectives are associated with debates about the model

to measure the investment performance, risk-return or geometric mean. The mean-

variance approach is associated with the risk-return formula, and growth optimal

portfolio is associated with the geometric mean formula. The risk-return formula is:

RR = µ

σ (2.10)

where RR is the risk-return and µ and σ are the arithmetic average and the standard

deviation of the historical returns, respectively.

The alternative methodology to calculate the return is the geometric mean for-

mula:

GM =nt=1

(1 + rt)1

n − 1 (2.11)

A debate about the ideal methodology to measure average return has its origin in

Williams (1936), who demonstrates that speculators, in a multi-period framework,

should be concerned about the geometric mean instead of the arithmetic mean.

The main argument is that risk-return does not consider the non-linearity of returns

present on exponential compound series, like financial series, and such characteristics

are only captured by the geometric mean (Latane 1959).

In order to introduce leverage, Eq. (2.10) can be rearranged, and consequently

the leveraged risk-return formula (RRL) is:

RRL = lµlσ

= RR (2.12)


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2.4. Geometric Mean, Leverage and End of Period Return 23

where l is the level of leverage. Note that leverage is monotonic and linear in the

risk-return formula. For this reason there is no level of leverage that maximizes

the Eq. (2.12). Hence this formula is not able to be used to optimizes the level of

leverage. Rearranging Eq. (2.11), the geometric mean formula (GM(l)) is:

GM (l) =nt=1

(1 + rtl)1

n − 1 (2.13)

From Eq. (2.13), note that leverage is neither monotonic nor linear in the geo-

metric mean formula. For this reason, different models have been proposed to obtain

the level of leverage that optimize the geometric mean formula.

In the next section, I review the relation between geometric mean, end of period

return and level of leverage.

2.4 Geometric Mean, Leverage and End of Period

Return

In this section, I demonstrate how to relate the geometric mean, the end of period

return and the level of leverage. This section is divided in two subsections. In the

first subsection, I present the impact of the level of leverage in a single period equity

return. And, in the section subsection, I demonstrate the impact of the level of

leverage in the end of period return and on the geometric mean.

2.4.1 Single Period Equity Return

In this subsection, I demonstrate that the end of period return is dependent on

the stock return and on the level of leverage. Similar to Section 2.3, I assume that

instead of maximising stock returns, the investor’s objective is to maximise the end

of period equity which depends on the stock return and on the level of leverage.

The level of leverage is:


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lt = S tK t

(2.14)

where lt represents the level of leverage, S t represents the market value of the total

securities and K t represents the equity value. For simplicity, no transaction costs

are considered. The leverage is:

Lt = S t −K t (2.15)

where Lt is the leverage. The value of the portfolio of stocks is:

S t = P tQt (2.16)

where P t represents the price and Qt represents the quantity. I assume that the

level of leverage is represented by Eq. (2.14) and the market value of the position

is represented by Eq. (2.16). Thus, extending the model to two time periods, t and

t + 1, the stock return to the next period is represented by rt, insomuch:

P t+1 = P t(1 + rt+1) (2.17)

where P t+1 represents the price for the next period. Assuming that Qt+1 = Qt,

substituting Eq. (2.17) into Eq. (2.16) and rearranging:

S t+1 = S t(1 + rt+1) (2.18)

where the market value of the portfolio in the next period is S t+1. From Eq. (2.18),

the market value of the portfolio depends on the stock return. Assuming that the

leverage is fixed from time t to time t + 1, Lt+1 = Lt, substituting Eq. (2.18) into

Eq. (2.14) and rearranging :

K t+1 = K t(1 + rt+1lt+1) (2.19)


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where the equity for the next period is K t+1. From the Eq. (2.19), equity is depen-

dent on the stock return and on the level of leverage. The equity return is:

kt+1 = K t+1K t

− 1 (2.20)

where kt is the equity return at time t. Thus, substituting Eq. (2.19) in Eq. (2.21),

the equity return is:

kt+1(lt+1) = rt+1lt+1 (2.21)

Note that the equity return is the product of stock return and the level of lever-age. This means that the level of leverage is the impact of the stock return on

the equity return. Leverage has the ability to completely modify the impact of the

stock return because the level of leverage can assume negative values. For exam-

ple, assuming that the stock return is +1%, the position is short and the level of

leverage is 3. The short position means that the level of leverage can be rewritten

as -3. Thus, the equity return is -3%, i.e., even if the stock return is positive, +1%,

the equity return is negative -3%. Therefore, if the end of period equity return is

the consequence of cumulative single period equity return and the objective is to

maximise the end of period equity return, leverage is relevant because of its direct

impact on the single equity return.

In the next subsection, I demonstrate the impact of the level of leverage in the

end of period return and on the geometric mean.

2.4.2 Geometric Mean and End of Period Return

In this subsection, I demonstrate that the end of period return can be represented by

the geometric mean of the equity return and they are functions of the stock return

dynamic process and the level of leverage dynamic process. I also demonstrate that

what maximises the end of period equity is the same that maximises the geometric

mean of the equity returns. Initially, substituting Eq. (2.21) in Eq. (2.19), the next


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period’s equity is:

K t+1 = K t(1 + kt+1) (2.22)

From Eq. (2.22), the equity on the next period is a function of the equity return.

Proceeding similar calculation, the equity on the next period, t + 2, is:

K t+2 = K t+1(1 + kt+1) = K 0(1 + kt+1)(1 + kt+2) (2.23)

From Eq. (2.23), after two periods of time the equity is a function of the cumu-

lative equity returns over the time t and t + 1. Thus, applying similar calculationsuntil the end of period, t = T , and considering the initial time equal to 0, the end

of period equity is:

K T = K 0

T −1t=1

(1 + kt+1) (2.24)

From Eq. (2.24), the end of period equity is a function of the cumulative equity

returns. the geometric mean of is:

GM = [T −1t=1

(1 + kt+1)]1

T − 1 (2.25)

Substituting Eq. (2.25) in Eq. (2.24):

K T (GM ) = K 0(1 + GM )T (2.26)

Eq. (2.26) shows that the end of period equity is a function of the geometric

mean of the equity returns.

In order to verify if the maximisation of geometric mean of equity returns leads

the maximisation of end of period equity, I differentiate Eq. (2.26):

dK T

dGM = T K 0(1 + GM )T

−

1 > 0 (2.27)


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According to Eq. (2.27), the impact of the geometric mean on the end of period

equity is positive. Thus, the maximisation of the geometric mean of the returns

leads to the maximisation of the end of period equity.

According to Eq. (2.21), equity return is a function of the stock return and the

level of leverage. Thus, Substituting Eq. (2.21) in Eq. (2.26) and considering the

initial t − 1 = 0:

K T (GM (r, l)) = K 0(1 + GM (r, l))T (2.28)

From Eq. (2.28), the end of period equity is a function of the stock return

dynamic process and level of leverage dynamic process. The end of period return is:

K T (GM (r, l)) = (1 + GM (r, l))T − 1 (2.29)

Thus, from Eq. (2.29), the end of period return also depends on the series of

stock return r and level of leverage l.

Assuming that the stock return’s dynamic process is exogenous, i.e., out of the

investor’s control; the variable of control is the level of leverage. For this reason, in

order to maximise the geometric mean of the equity return, investors should to be

concerned on the level of leverage that maximises the geometric mean of the equity

return. Hence, the appropriate estimator of the geometric mean of leveraged returns

is relevant.

In Section 2.2, I stated that the proxy of geometric mean as Eq. (2.5) has three

assumptions. Firstly, the returns is greater than minus one and less or equal to one.

Secondly, the stock returns is so small that the estimation of the return power any

value superior to two is zero. Finally, the sum of the value of the power of the return

is superior to two then the product of the estimation of the return is also zero.

However, this proxy assumes that the returns are normally distributed and this

implicates that the value of the skewness and kurtosis is 0, i.e., extremes are barely

on the distribution. The key point is that leverage expands the impact of the stock


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return (Breuer 2002) and, naturally, increases the kurtosis. Thus, it contradicts

the assumptions of the advocators of the approximation for the geometric mean.

Therefore, the leverage characteristics could indicate that the approximation is not

appropriate for the equity return.

For example, if we expand our original example of two periods stock returns to

four periods stock return, the stock returns are +8%, -10%, +10% and +5%. Then,

if we assume that the level of leverage is 1, the arithmetic average, the proxy of

geometric mean and the real geometric mean of the equity returns are 3.25%, 2.84%

and 2.93%, respectively. Note that the value of the estimators are considerable

near to one another. Furthermore, this result suggests that the arithmetic average

overvalues the rate of return and the proxy of the geometeric mean undervalues the

rate of return, which agrees with the findings in Jacquer et al. (2003).

However, if we assume that the level of leverage is 10, the stock return remains

unchanged, but, the equity return are +80%, -100%, +100% and +50%. Conse-

quently, the arithmetic average, the proxy of geometric mean and the real geometric

mean of the equity returns are 32.50%, -8.63% and -100%, respectively. Note that

in this case, there is a considerable distance between the estimators. Furthermore,

this result demonstrates that the arithmetic average and the proxy of geometric

mean are not able to capture the real rate of return, which is -100%. The reason

for this failure is that both estimators do not capture the risk of default which is

increasing in leverage; i.e., the higher the level of leverage, the higher the risk of

default (Breuer 2002).

In the next section, I review the literature on optimal leverage models.

2.5 Optimal Leverage Models

In this section, I present the three existing models to optimize the level of leverage.

The first model is the Kelly criterion which assumes normal distribution of the

returns and discrete number of possible outcomes (Kelly 1956). The second model


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2.5. Optimal Leverage Models 29

is the Rotando and Thorp model which assumes normal distribution of the returns

and continuous number of possible outcomes (Rotando and Thorp 1992). Finally,

the third model is the Peters model which assumes normal distribution of the returns

and that price follows a geometric Brownian motion (Peters 2010).

This section is divided in three subsections. In the first subsection, I describe

the Kelly criterion. In the second subsection, I describe the Rotando and Thorp

model. Finally, in the third subsection, I describe the Peters model.

2.5.1 The Kelly Criterion

In this subsection, I describe the Kelly criterion. The importance of studies on

leverage is not recent. Kelly (1956) demonstrates that there is an optimal level of

leverage for binomial games such as coin tossing games. He assumes that the main

objective of the player is to maximise the trial’s expected value and, consequently,

the end of period cumulative return. He also demonstrates that if you have a positive

trial’s expected value, the size of your leverage depends upon two factors: the valueof your expected value and the probability of ruin. This model is called the Kelly

criterion. Originally, the Kelly criterion consists of a model to maximise returns on

a binomial game similar to coin tossing. The model is described below.

Suppose that on each trial the win probability is p > 1/2 and, consequently, the

probability of loss is q = 1− p 0 (2.30)

Assuming p > 1/2 > q and, consequently the expected value of the end of


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trials wealth E (X n) is positive, the objective is to find the amount of the available

capital which should be bet, Bi, and the amount of capital that should be saved

for later trials, X i−1. On one hand, if the player decides to bet all the capital, this

increases the probability of ruin. On the other hand, if he decides to bet the minimal

capital, the player reduces the probability to maximise the value of the end of trial’s

wealth. Therefore, there is some optimal level of leverage or optimal fraction, l ,

which balances the objective to maximise the expected value with the restriction to

minimise the probability of ruin.

In the coin-tossing game, since the gambling probability and the payoff at each

trial are the same, it is clear that the optimal level of leverage is the same for all

trials. This assumption of fixed leverage helps us to comprehend the purpose of the

Kelly criterion. Maximising the expected end of trial wealth is similar to maximising

the expected value of the growth rate coefficient or the geometric mean, GM k(l):

GM k(l) = E [log X nX 0

]1

n = p log(1 + l) + q log(1− l) (2.31)

GM k(l) = p

1 + l − q

1 − l (2.32)

GM k (l) = −l2 + 2l( p − q ) − 1

(1 − l2)2 (2.33)

where GM k(l) is the geometric mean approximation on the Kelly criterion and

0 ≤ l < 1. From Eq. (2.33), the function GM k(l) is concave in l , hence it can bemaximised. Furthermore, solving Eq. (2.32), the result is that the optimal level of

leverage is l∗ = p − q .In the next subsection, I describe the Rotando and Thorp model.

2.5.2 The Rotando and Thorp Model

In this subsection, I describe the Rotando and Thorp model. The Kelly criterion

cannot be directly used on investments. The main reason is that with games like coin


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2.5. Optimal Leverage Models 31

tossing there are a discrete number of possible outcomes, whereas with investments,

the number of possible outcomes is continuous. In order to use the Kelly criterion

in financial markets, the original model proposed in Eq. (2.31) should be modified

and adapted to represent continuous outcomes. This modification is proposed by

Rotando and Thorp (1992). They propose a new model incorporating continuous

outcomes, but respecting the Kelly’s original insight: the definition of the optimal

level of leverage in order to maximise the end of period wealth or the end of period

return, E [log(Xn/X 0)].

According to Rotando and Thorp (1992), trading financial securities can be

considered a continuous game. Thus, in order to model the possible outcomes, they

assume that financial returns are Normally distributed. However, the simple use of

the unaltered Normal probability distribution is inadequate because this distribution

allows an infinite range of possible returns. Therefore, they modify the standard

normal curve using a correction term for ”chopping off the tails” (Rotando and

Thorp 1992). This results in new parameters, h and α, which serves to maintain the

mean and the standard deviation at values similar to those of the standard Normal

distribution..

Similar to binomial games, the objective in this model is to find out the level of

leverage that maximises function GM rt(l), which in this case is:

GM rt(l) =

BA

log(1 + rl)dN (r) (2.34)

where GM rt is the geometric mean approximation if the Rotando and Thorp model.

GM rt(l) =

BA

log(1 + rl)[h + 1√

2πα2e−(r−µ)

2/2α2] (2.35)

where A = µ − 3σ and B = µ + 3σ.The demonstration of first-order and the second order conditions of Eq. (2.35)

is complicated. However, numerical methods can be used to obtain the value of l

that maximises the function GM rt(l). For example, simulating the model proposed


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for the 59 year period from 1926 to 1984, with the µ = 0.058 and σ = 0.216 by

using numerical methods, Rotando and Thorp (1992) found that the optimal level

of leverage is l∗ = 1.17.

In the next subsection, I describe the Peters model that uses the geometric

Brownian motion.

2.5.3 Geometric Brownian Motion

In this subsection, I describe the Peters model that uses the geometric Brownianmotion. Assuming that the price follows a geometric Brownian motion, and con-

sequently, that returns are log-normally distributed (Hull 2009), Peters proposes a

different model to optimize the level of leverage (Peters 2010). Suppose that the

price process is:

p(t) = p0[exp(µ

−σ2

2

)t + σ

∗W (t)] (2.36)

where the Wiener process W(t) is normally distributed.

Introducing the leverage, the estimated leveraged log-return and its variance are

respectively:

µl = lµ (2.37)

σ2l = l2σ2 (2.38)

where l is the level of leverage.

The estimated log-return of the leveraged investment to the next period is:

GM p(l) = E [log( pt+1)]

−log( pt) = (µl

−

σ2

2

) (2.39)

Substituting Eq. (2.37) and Eq. (2.38) into Eq. (2.39) gives:


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2.6. Agent-Based Modelling 33

GM p(l) = (lµ −l2σ2

2 ) (2.40)

In order to obtain the optimal level of leverage, Eq. (2.40) is differentiated with

respect to l and the result is set to zero. Then, the optimal level of leverage is:

l∗ = µ

σ2 (2.41)

where l∗ is the optimal level of leverage.

According to the Peters model, the optimal level of leverage is the estimated

log-return divided by its variance.One of the drawbacks of these models is that they assumes that the financial

returns is normally distributed and, consequently, ignores the presence of fat-tails

(Cont 2001). With the objective of relax this assumption, in Chapter 4, I propose

a numerical method in order to optimize the level of leverage. In the next section,

I review the literature on agent-based modelling.

2.6 Agent-Based Modelling

In this section I review the literature on agent-based modelling. The reason is that

in Chapter 6, I test the ability of the optimal level of leverage to improve investment

performance in an agent-based modelling experiment.

Agent-based modelling has the ability to go beyond the traditional economics

because it considers the market as a multi-agent system wherein each agent has

particular characteristics that differs from other agents, while the traditional eco-

nomics assume that every agent is similar and rational (Chakraborti et al. 2011).

The traditional economic approach with closed solutions was crucial to allow the

proposition of analytical models which would be unfeasible without the assumption

of similarity and rationality among the agents. However, advances in the technol-

ogy and in the computer platforms such as StarLogo (Resnick 1996), NetLogo (Sklar


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2007), Repast (Collier 2003), JABM (Java Agent-Based Modelling) (Phelps 2012)

and JASA (Java Auction Simulation API) (Phelps 2007) have enabled the progress

in numerical approaches and simulation instead of analytical approach in order to

model any aspects of different areas of research, for example, in the Biology and in

the Sociology (Chakraborti et al. 2011).

Naturally, agent-based modelling research has been proposed for Economics and

Finance. According to LeBaron (2000; 2006), the combination of agent-based models

and computational methods in finance creates a new sub-area called agent-based

computational finance. One use of agent-based computational finance is artificial

stock market which consists on a set of tools that intends to replicate artificially

the stock market (LeBaron 2002). This is a controlled environment which allows

one to run experiments using artificial agents in order to analyse a wide variety of

hypotheses from the literature of finance.

Different agent-based models have been proposed with the objective of inves-

tigating how different trading strategies may affect the dynamics of price, bid-ask

spreads, trading volume and volatility. The literature of agent-based modelling for

finance used in the thesis evolves along of the time as follows: Firstly, The proposal

of agent-based modelling for finance is not recent in the literature, its origins are

contained in Grossman (1976) who proposed a model with heterogeneous agents

that interact with each other. One of the most important debates in agent-based

modelling area is how to model the artificial agent, with its individual characteris-

tics, heterogeneity and behaviour (LeBaron 2000). Lettau (1997) proposes a very

simple model where agents should decide between a risk and risk free asset according

to personal utility function.

Secondly, Bak, Paczuski and Shubik (1996) propose a model which assumes that

orders are input according to a price line. They assume that there are N agents and

N/2 stocks. The agent is a buyer if he does not hold the stock, and a seller if he

holds a stock. Agents can only sell stock if they hold one, and they can only buy

stock if they don’t hold one. This means that there is no leverage in this model.


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2.6. Agent-Based Modelling 35

Agents advertise the value that they intend to buy (sell) the stock and the trade

occurs when the prices cross, i.e., the offer to buy is superior to the price to sell. At

this moment, the seller trades with the buyer that offers the highest price. In the

case of no trade, the prices are updated to the next round.

Thirdly, Maslov (2000) introduces a model with limit and market orders. This

is an important step towards the replication of the real financial market. In their

model there is no strategy pre-defined. Traders simply collect the last transaction

price, p(t) and aggregate a small variation on this price ∆. Thus, the order to sell

is above the current price, p(t) + ∆, and the order to buy is below the current price

p(t)−∆. In the first moment, there is no trade, because the highest order to buy isinferior to the lowest order to sell. However, in this model, agents are not allowed to

update or cancel their orders along of the rounds. Consequently, after some rounds,

the probability of having the order matched increases because the new current price

p(t + n) indicates one new order to sell p(t + n) + ∆ that could be inferior to the

original price to buy p(t)−∆.Fourthly, Lux and Marchesi (2001) propose that agents decide according to the

combination of three components. The first component is the fundamentalist. The

fundamentalist component uses the fundamental value of the security, For example,

in the case of stocks, the intrinsic value of the company is calculated using the

information in its balance sheet. The fundamentalist component indicates to sell

(buy) when the price is bigger (smaller) than the stock intrinsic value or fundamental

price.

The second component is the chartist. The chartist component uses the histori-

cal data instead of the balance sheet. The chartist component can be divided into

contrarians and trend followers. Contrarians are based on the existence of overre-

action on the stock returns (Lo and MacKinlay 1990). This indicates to buy (sell)

past losers (winners). Unlike the contrarians, trend-followers are based on the exis-

tence of trends on the financial s

Documents

Analysing the optimal level of leverage