analysing the optimal level of leverage
Post on 07-Aug-2018
221 Views
Preview:
TRANSCRIPT
-
8/20/2019 Analysing the optimal level of leverage
1/135
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
-
8/20/2019 Analysing the optimal level of leverage
2/135
-
8/20/2019 Analysing the optimal level of leverage
3/135
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
-
8/20/2019 Analysing the optimal level of leverage
4/135
-
8/20/2019 Analysing the optimal level of leverage
5/135
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
-
8/20/2019 Analysing the optimal level of leverage
6/135
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.
-
8/20/2019 Analysing the optimal level of leverage
7/135
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
-
8/20/2019 Analysing the optimal level of leverage
8/135
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.
-
8/20/2019 Analysing the optimal level of leverage
9/135
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
-
8/20/2019 Analysing the optimal level of leverage
10/135
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
5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
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
6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Chapter 7: Conclusion and Future Work . . . . . . . . . . . . . . . . . . 107
7.1 Summary of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . 1087.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
-
8/20/2019 Analysing the optimal level of leverage
11/135
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
-
8/20/2019 Analysing the optimal level of leverage
12/135
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
-
8/20/2019 Analysing the optimal level of leverage
13/135
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
-
8/20/2019 Analysing the optimal level of leverage
14/135
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
-
8/20/2019 Analysing the optimal level of leverage
15/135
-
8/20/2019 Analysing the optimal level of leverage
16/135
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,
-
8/20/2019 Analysing the optimal level of leverage
17/135
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.
-
8/20/2019 Analysing the optimal level of leverage
18/135
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
-
8/20/2019 Analysing the optimal level of leverage
19/135
6 Chapter 1. Introduction
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
-
8/20/2019 Analysing the optimal level of leverage
20/135
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,
-
8/20/2019 Analysing the optimal level of leverage
21/135
8 Chapter 1. Introduction
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).
-
8/20/2019 Analysing the optimal level of leverage
22/135
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.
-
8/20/2019 Analysing the optimal level of leverage
23/135
10 Chapter 1. Introduction
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.
-
8/20/2019 Analysing the optimal level of leverage
24/135
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.
-
8/20/2019 Analysing the optimal level of leverage
25/135
-
8/20/2019 Analysing the optimal level of leverage
26/135
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
-
8/20/2019 Analysing the optimal level of leverage
27/135
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
-
8/20/2019 Analysing the optimal level of leverage
28/135
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-
-
8/20/2019 Analysing the optimal level of leverage
29/135
16 Chapter 2. Literature Review
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
-
8/20/2019 Analysing the optimal level of leverage
30/135
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
-
8/20/2019 Analysing the optimal level of leverage
31/135
18 Chapter 2. Literature Review
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-
-
8/20/2019 Analysing the optimal level of leverage
32/135
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):
-
8/20/2019 Analysing the optimal level of leverage
33/135
20 Chapter 2. Literature Review
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-
-
8/20/2019 Analysing the optimal level of leverage
34/135
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
-
8/20/2019 Analysing the optimal level of leverage
35/135
22 Chapter 2. Literature Review
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)
-
8/20/2019 Analysing the optimal level of leverage
36/135
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:
-
8/20/2019 Analysing the optimal level of leverage
37/135
24 Chapter 2. Literature Review
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)
-
8/20/2019 Analysing the optimal level of leverage
38/135
2.4. Geometric Mean, Leverage and End of Period Return 25
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
-
8/20/2019 Analysing the optimal level of leverage
39/135
26 Chapter 2. Literature Review
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)
-
8/20/2019 Analysing the optimal level of leverage
40/135
2.4. Geometric Mean, Leverage and End of Period Return 27
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
-
8/20/2019 Analysing the optimal level of leverage
41/135
28 Chapter 2. Literature Review
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
-
8/20/2019 Analysing the optimal level of leverage
42/135
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
-
8/20/2019 Analysing the optimal level of leverage
43/135
30 Chapter 2. Literature Review
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
-
8/20/2019 Analysing the optimal level of leverage
44/135
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
-
8/20/2019 Analysing the optimal level of leverage
45/135
32 Chapter 2. Literature Review
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:
-
8/20/2019 Analysing the optimal level of leverage
46/135
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
-
8/20/2019 Analysing the optimal level of leverage
47/135
34 Chapter 2. Literature Review
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.
-
8/20/2019 Analysing the optimal level of leverage
48/135
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
top related