An empirical study of stock portfolios based on
diversification and innovative measures of risks.
Master Thesis - Final Report
Supervised by Pr. Dr. Sornette
Chair of Entrepreneurial Risks at ETH Zurich
Thibaut Simon
10 February 2010
Abstract
When a measure of risks such as variance does not take into account the fact that
distribution of returns are non-Gaussian and exhibit non-linear dependencies,
it is ineffective to generate portfolios with standard optimization procedures.
This study proposes to explore new measures of risks based on series and new
concepts of level of risks. This empirical study back-tests these innovative ideas
on the long run and evaluates them with random portfolios analyses. Persistent performance and risk management are achieved with the use of Maximum
DrawDown of returns and levels of diversification.
Contents
1 Introduction
3
2 Classical approach of Portfolio construction
2.1 Markowitz, Mean-Variance . . . . . . . . . . . . . . . . . . . . .
2.2 Different estimates of the Covariance matrix to improve MeanVariance approach . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Shrinkage . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Re-sampling . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Reverse engineering . . . . . . . . . . . . . . . . . . . . .
2.3 CAPM and extensions . . . . . . . . . . . . . . . . . . . . . . . .
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3 Genetic algorithm
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3.1 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Focus on the generation of a new population . . . . . . . . . . . 14
3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Random Portfolios
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4.1 First attempt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Second attempt . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Innovative idea on risks
5.1 Measures of risks and dependencies
5.1.1 Coherent Measure of risk .
5.1.2 Dependencies . . . . . . . .
5.1.3 Some measure of risks . . .
5.2 Level of risks and diversification .
5.3 A winner mix . . . . . . . . . . . .
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6 Simulation
6.1 Methodology . . . . . .
6.1.1 Data . . . . . . .
6.1.2 Test . . . . . . .
6.1.3 Set of strategies .
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6.2
Results . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Tables . . . . . . . . . . . . . . . . . .
6.2.2 Wealth evolution - first interpretation
6.2.3 Random Portfolios - Luck or skill? . .
6.2.4 Sharpe Ratio . . . . . . . . . . . . . .
6.2.5 Transaction costs overview . . . . . .
6.2.6 Impact of the diversification - concrete
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7 What’s next?
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7.1 Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.2 To go further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
8 Annexes
75
8.1 Maximum DrawDown - Matlab Code . . . . . . . . . . . . . . . . 75
8.2 Average Maximum DrawDown - Matlab Code . . . . . . . . . . . 76
2
Chapter 1
Introduction
Asset allocation is today a topic of real importance. All investors want to
invest in the winning combination of assets. This combination should give them
the maximum level of return for the level of risk they are able to take. The
problematic of asset allocation can be found across all industries and sectors
of activity. For example, an oil company will have the choice of investing in
different fields with different techniques for different products. How to choose
the weight each project should have in my investment portfolio? Answering this
question requires to take into account expected returns of projects as well as
risks of failure, or delay that could penalize future returns. Actually, investing
in several assets means trade-offs. However, making a decision is hard because
investors have no idea about what will happen and what the consequences of
their choice will be? Thus the idea of portfolio optimization is to develop a
decision-tool to help investors to rationalize their choice.
A lot of researches have been done in the area of asset allocation and portfolio
optimization and construction. However, the topic is still widely open. Why is
it still the case? Harry Markowitz created the modern portfolio theory based
on historical distributions of returns considered to be Gaussian. As described
in my thesis, this theory never performed well. However, the Markowitz’ idea
of minimizing risks for a given level of return is still widely accepted as a good
start for a new theory. In the 1950’s means of computation were limited but
Markowitz developed a quadratic programming equation that could be easily
solvable. Today new measures of risks and dependencies between assets can be
used, since computation and optimization of complex problem is more convenient. These new measures allow to create new strategies that perform better
than the traditional one. I use a genetic algorithm (see chapter 3) to solve optimization problems. The performance of these strategies is evaluated through
random portfolios and Sharpe ratio analyses. This study covers stock portfolios taken in a universe composed of 28 stocks from the CAC40. The question
answered in my thesis is:
3
How to achieve persistent return for different levels of risks?
This report will help people to start from scratch in portfolio management
and optimization to understand the problematic of portfolio construction and
evaluation. Hence, this thesis will start by explaining the classical approach, its
limits and extensions before describing the method and tools to construct and
evaluate portfolios. From the development of random portfolios emerges the
innovative idea of separating measures of risks and levels of risks. It is tested
in the chapter presenting the simulation. My thesis ends with some insight and
way to continue it.
4
Chapter 2
Classical approach of
Portfolio construction
Portfolio construction aim at producing portfolios minimizing risks for investors and maximizing their wealth. Portfolios are constructed from a basis
of selected assets. Determining the universe of stocks that seems interesting to
get in a portfolio is done by financial analyst. Hence a basic set of stocks is
produced, the question is: how to weight each of these assets? That is what
is about portfolio theory. Keep in mind, that a good theory, should produce
well diversified portfolio, to decrease risks and volatility of the portfolio. However it should still be able to increase wealth of investors more than a random
allocation could. This chapter is about the state of the art on the traditional
approach of investment.
2.1
Markowitz, Mean-Variance
Harry Markovitz is considered as the father of modern portfolio theory [16].
He developed during the 50’s a theory based on the assumption that the utility
function of investors is quadratic. This theory tells that investors want to minimize their risks for a given level or return. In this quadratic framework the risk
is modelled by the standard deviation of the portfolio. At this time this was
a good model, because this problem has an analytical solution which allowed
people to use it with the small computation means of this age.
This approach never gave fully satisfaction, due to a lack of empirical evidence. In practice some limitations have been identified: returns don’t follow a
normal distribution and samples are too noisy to provide good Covariance estimates. Different solutions emerged to improve covariance’s estimates. I present
three of them in the following section.
5
Minimize wT Vw
Subject to :
wT µ = r0
∀i, wi ≥ 0
n
X
wi = 1
i=1
Where:
• w : portfolio weights
• V : covariance matrix
• r0 : the desired level of expected return of the portfolio
• µ : vector of expected returns
Figure 2.1: Formulation of Mean-Variance optimization
Figure 2.2: Instability of the EFFICIENT FRONTIER moving at each periods.
Efficient frontiers from 28 assets over 12 periods of 6 months.
6
Figure 2.3: DEVIATION of a portfolio from the EFFICIENT FRONTIER.
Same frontiers as figure 2.2 with the evolution of a portfolio taken on the first
efficient frontier. A portfolio done in the Mean-Variance frameworks become
rapidly inefficient.
7
2.2
Different estimates of the Covariance matrix
to improve Mean-Variance approach
Covariance matrix can be estimate with the sample covariance matrix. However this simple estimator does not work much for portfolio construction, since
samples are not sufficiently big and are to noisy to give a good estimate. Actually, sample covariance estimates are really sensitive to outliers. Nevertheless
this estimator is unbiased.
2.2.1
Shrinkage
The shrinkage method allows to provide a better estimation of the covariance
matrix when the number of assets p is bigger than the size of the sample n. In
the case p > n, the sample covariance matrix become singular. The shrinkage
method is an alternative. It consists to evaluate through a cross validation,
the shrinking parameter. The shrinking parameter is the weight determining
the trade-off between the sample covariance estimator and the diagonal target
matrix. This operation regularizes the covariance matrix.
b=
S
k
TF
+ (1 −
k
T )S
• k : estimator of the optimal shrinkage constant
• T : number of observations
• F : single index covariance matrix
• S : sample covariance matrix
Figure 2.4: SHRINKAGE Method - a trade off between sample covariance and
single index
2.2.2
Re-sampling
The idea is to expand sample by creating samples from the original sample.
This method is well known in statistics under the name of bootstrap method.
It allows to reduce the variance uncertainty while increasing the bias on the
measure. However this technique seems to give some results, if we believe M.
Michaud who is the developer of this technique to portfolios construction. According to him, the re-sampling decrease the sensitivity of optimization to noise.
2.2.3
Reverse engineering
This method consists in assuming that the market portfolio is on the efficient
frontier. The aim is to correct the covariance matrix estimate with the help of
8
the market efficient portfolio. This idea is theoretically pleasant. However it
has a lack of empirical use, since market portfolio is hard to estimate.
Figure 2.5: REVERSE ENGINEERING - Measure of distance from the sample
to the reverse engineering estimate. The factor alpha corresponds to the tradeoff between the bias and the variance of our estimation
Figure 2.6: REVERSE ENGINEERING - Minimizing the distance of the reverse
engineering estimate to the sample estimate under the condition that the market
efficient portfolio belows to the efficient frontier
2.3
CAPM and extensions
Capital Asset Pricing Model Sharpe, Lintner and Mossin developed the
CAPM theory in the 60’s [18]. This model complements the Markovitz portfolio
theory by adding the possibility to lend or borrow money at a risk free rate.
Main assumptions are taken from the Markovitz portfolio theory: agents are risk
averse and maximize their utility, stocks returns are normally distributed, perfect markets, informational efficiency, supply and demand equilibrium. It gives
a clear relation between risk and return. The Capital Asset Pricing Model takes
into account two risks: the unique risk which can be cancelled by diversification,
and the market risk which corresponds to the risk of being on the market. The
expected risk premium on stock is equal to r −rf where rf is the risk free rate of
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return and r the expected return of the stock. Sharpe established the following
relationship between risk and return:
r − rf = β(rm − rf )
Beta represents the contribution of the market risk to the non-diversifiable stock
risk. Beta is the relative covariance of stock to market returns. Beta measures
the dependence between the stock and the market, weighted by the ratio of the
stock volatility by the market volatility.
βi =
Cov(ri ,rm )
V ar(rm )
This model is linear, so easily understandable. However this theory is not fully
useful since the market portfolio can only be approximate ( Roll critique [17]).
These critics lead researchers to improve this model.
3-factors model based on size of firms and book-to-value Eugene Fama
and Kenneth French extended the CAPM model by adding two new factors: firm
size and the book-to-market equity [12]. With this improvement, the model becomes multi-linear. The cross-section of expected returns is better explained
with this theory. However this model is built on empirical study and the mechanisms are not deeply understood.
Our preliminary work on economic fundamentals suggests that highBE/ME1 firms tend to be persistently poor earners relative to lowBE/ME firms. Similarly, small firms have a long period of poor
earnings during the 1980s not shared with big firms. The systematic
patterns in fundamentals give us some hope that size and book-tomarket equity proxy for risk factors in returns, related to relative
earning prospects, that are rationally priced in expected returns. [12]
4-factors model based on momentum strategies This model developed
by John Cahart aims at taking into consideration the effect of time when pricing
stocks. [8] Researchers detected a momentum effect from data. Momentum
strategies are founded on empirical and behavioural finance. They benefit from
an inefficiency of the market, with a lag generated by investors. Stocks with the
highest return remain good choice for about 6 months, and the opposite with
low returns stocks[13, 1]. The fact of adding a one year momentum in stocks
returns improves the prediction power of the model.
”First, note the relatively high variance of the SMB2 , HML3 , and
PR1YR4 zero-investment portfolios and their low correlations with
each other and the market proxies. This suggests the 4-factor model
1 ME
refers to size of the firm. It is equal to the market value. BE refers to the book value
of the firm. BE/ME is called the book-to market equity.
2 factor of size
3 factor of book-to-market equity
4 factor referring to the one year momentum in stock returns.
10
can explain sizeable time-series variation.” ”I find that the 4-factor
model substantially improves on the average pricing errors of the
CAPM and the 3-factor model”[8]
2-factors model based on market concentration This model has been
developed by Malevergne, Santa-Clara and Sornette in their papers Professor
Zipf goes to Wall Street (2009)[15]. It is based on the Zipf distribution of
firm size. The idea is that the market portfolio is concentrated into very few
companies (about 20) due to the heavy-tailed shape of the Zipf distribution.
This market concentration can be measured with the Herfindahl Index, that
will be used widely in the following sections. This concentration leads to a
new factor of systematic risk. The difference between the equally weighted and
value-weighted market portfolios is used as a proxy of the Zipf factor, and allows
to take this new risk into account in the pricing model. This idea of market
concentration influenced my work.
Application of these models These models are used to evaluate the performance of investment strategies. The idea is to compare the average returns of
the portfolio against portfolios sharing the same factors. Then, it remains the
question of the persistence of these results. Are managers lucky or truly skilled?
[11]. This question is fundamental. The random portfolio theory of P. Burns,
is preferred in the following for portfolios performance evaluation.
11
Chapter 3
Genetic algorithm
Genetic algorithm allows to optimize all kinds of functions. Genetic algorithms are part of evolutionary algorithm [14, 9]. They directly come from the
Darwin’s theory of evolution. This algorithm reproduces the selection process
working in the nature. Generation after generation, the population adapts itself to the environment. In the case of a stable environment, GA produces an
optimization solver which converges. The evolution of the population is done
through small change coming from the mix of parents or through mutation. It
is really convenient to use, once parameters that regularize performance of the
optimization, are understood. Its name and the associated vocable come from
genetics. The process of reproduction is actually a model of what is going on in
our cells. The genotype is evolved across generations through selection, crossover and mutation processes, which converge to a final population answering
the problem. I use this algorithm to do the final simulation of this study.
3.1
Composition
The genetic algorithm is composed of 4 main steps :
1. initialisation of the population
2. evaluation of the population
3. selection
4. reproduction
The steps 2,3 and 4 are in a loop which stops when a stopping criteria is reached.
The reproduction step is composed of three possibilities:
1. elite count
2. crossover
3. mutation
12
Figure 3.1 represents the reproduction process.
Below is published the description of the genetic algorithm used in my work.
This algorithm exists already in the GA toolbox of Matlab. Here are the options
and properties used to do optimization of portfolio. In order to obtain the
attending result, it is good to understand, how each function works, and to
have a global view of the process.
Population Options:
Population is composed of portfolios represented as vectors of weights.
Population size specifies how many individuals there are in each generation. With a large population size, the genetic algorithm searches the
solution space more thoroughly, thereby reducing the chance that the
algorithm will return a local minimum that is not a global minimum.
Here you can choose between 100 and 1000.
Initial population specifies an initial population for the genetic algorithm. It is set as random with the constraint : ∀i, 0 ≤ wi . The
other constraint of normality is taken into account in the evaluation
function.It is not an optimization constraint, because the answer of
the problem is a vector representing the proportion of the different
assets. Hence the portfolio is normalized, only during the evaluation
of the portfolio. At the end, the final vector of weights is normalized
to further use. This astuteness avoids to use constraint options of
the GA toolbox. It saves computation time.
Evaluation function: The evaluation function quantifies the objective of the
optimization. It allows to rank portfolios in function of the distance to
the solution of the optimization problem. These ranks are used in the
selection process.
Selection function: It allows to select parents for reproduction. The Stochastic uniform selection function lays out a line in which each parent corresponds to a section of the line of length proportional to its scaled value.
The algorithm moves along the line in steps of equal size. At each step,
the algorithm allocates a parent from the section it lands on. The first
step is a uniform random number less than the step size.
Reproduction:
Crossover fraction specifies the fraction of the next generation other
than elite children, which are produced by crossover.
Elite count specifies the number of individuals that are guaranteed to
survive to the next generation.
13
Mutation uses the Gaussian mutation function, which adds a random
number taken from a Gaussian distribution with mean 0 to each
entry of the parent vector.
Crossover uses the scattered crossover function, which creates a random
binary vector and selects the genes where the vector is a 1 from the
first parent, and the genes where the vector is a 0 from the second
parent, and combines the genes to form the child.
Stopping Criteria Options: The algorithm runs until the cumulative change
in the fitness function value over Stall generations is less than or equal to
Function Tolerance. Here the number of generation is limited to 200 and
stall limit generation can be chosen between 10 and 50.
3.2
Focus on the generation of a new population
The process of the generation of the new population is composed of three
operations: direct transfer from the old to the new generation, mix of two parents and mutation of one guy. Figure 3.1 explains the process and gives the
ventilation between the three operations. Parameters of the algorithm determine its performance feature. Large population (over 300) produces a result
more pertinent in less generations, but increase the computation. Moreover a
small crossover rate increases the number of mutation and the chance to find
local minimum. However the convergence of the population will become slower
and slower. The crossover decreases the variance of the population by making
chromosome more and more identical(figure 3.2). The crossover operation look
for global minimum. In opposition, the mutation operation, which add some
random change in the population, tend to diversify the population. In the algorithm that I used in this study, the Gaussian mutation function adds a random
number taken from a Gaussian distribution with mean 0 to each entry of the
parent vector(figure 3.3). This operation helps in the convergence process to
find the local minimum.
14
Figure 3.1: REPRODUCTION PROCESS using elite count, crossover and mutation. In this example the crossover rate is 0.8 and the elite count is 2
Figure 3.2: A view of the scattered CROSSOVER function process. A random
vector of bit is generated of the length equal the number of assets in the portfolio.
A zero means we select the weight of parent 1 and a one the weight of parent 2
15
Figure 3.3: Gaussian MUTATION function adds a random number taken from
a Gaussian distribution with mean 0 to each entry of the parent vector.
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3.3
Applications
With a good parametrization this algorithm performs well in optimization
of portfolios. First, this GA will be used to construct portfolios following a
strategy. In this case, the population is a population of random portfolios.
These random portfolios will be generation after generation evolved until they
optimized the fitness function. The fitness function is the objective function
describing the investment strategies. For instance, a strategy could be to reduce
the variance of portfolios returns while keeping a diversification of 10 stocks. All
the condition have to be integrate in the objective function (diversification,...).
The second use of this algorithm, is the creation of set of random portfolios.
Since the initial population and the search process are random, the GA allows
to create such random portfolios meeting the needed constraints. These two
applications are used for simulation in this study.
17
Chapter 4
Random Portfolios
Introduction Random portfolios technique is about statistical simulation of
portfolios. Random portfolios constitute a benchmark. This benchmark can
be used to evaluate other portfolios. Papers [7, 6] explain how to use this
technique for testing trading strategies, assessing the skill exhibited by funds,
and implementing investment mandates. Random Portfolios are an application
of Monte Carlo simulation. The result of a random portfolios simulation is the
possibility to rank a portfolio against random ones. The main difficulty of this,
is to compare what is comparable. In this sense, the way random portfolios are
generated is of terrible importance. The first section of this chapter is a naive
approach of random portfolios. I explain why it does not work. The second
section presents a good procedure to obtain results from random portfolios. I
choose to present this party in a narrative way to present the dynamic of my
approach.
Hypothesis Random portfolios are used as a statistical test. The hypothesis
defining the p-value are:
H0 : Strategy is not better than random portfolios
H1 : Strategy is better than random portfolios
Hence a p-value of ten percent means that the strategy performs better than 90
percents of random portfolios.
4.1
First attempt
Use of the function random I generated random portfolios from the simplest method I knew : the function random. Each weight is assigned to a random
double between 0 and 1. Then the portfolio vector is normalized to respond to
the constraint: sum of weights equal to 1. I generated 1000 portfolios. Then
I applied them to a period of 6 months of data returns. I obtain a Gaussian
18
distribution of returns of the 1000 portfolios. These results are represented figure 4.1. The test period is 6 months. The 6-month mean returns is about five
percent and the standard deviation two percent. Figure 4.2 are plotted the returns of the thousand portfolios sorted from the smallest return to the highest.
It allows to compare your strategy to random portfolios. Hence the p-value of
your strategy is directly readable on this graphic, as soon as you know your return on the selected periods of data and universe. My first attempt to simulate
random portfolios gave me surprising results. Every returns of the strategies
that I tested, were above or below the simulated random portfolios. Hence their
p-value were respectively 0 or 1.
Figure 4.1: Histogram of the 1000 RANDOM PORTFOLIOS generated with
the random and the normalize function. Returns are computed on a period of
6 months. The universe of study is composed on 28 stocks of the CAC40 over
6 months.
What’s wrong? I firstly check my code, then recheck, then I changed several
times my data without any success. After such a defeat, you need to stand
back, in order to do the good analyze of the problem. When looking at the
portfolio things become clear. They were too diversified in comparison of the
strategy I was looking at. The average Herfindahl index was close to 1/27 and
the variance was really low. My universe was constituted of 28 assets. Hence
the conclusion was clear, my random portfolios were almost equally weighted
portfolios. I needed to change the way I generate random portfolios, to add
more diverse kind of portfolios.
Idea Starting from the idea of comparing what is comparable, a simple idea
is to compare portfolio of the same level of risks. As it is shown in the section
about ”New Strategies”, I think that in a universe constituted of about the same
kind of stocks (here 28 stocks from the CAC40), a good indicator of the level of
risks is the level of diversification of the portfolio. The Herfindahl Index, which
is a concentration index, is a good estimator of diversity. Hence the problem,
19
Figure 4.2: Graphics of RANDOM PORTFOLIOS sorted by increasing returns.
The 1000 random portfolios were generated with the random and the normalize
function. Returns are computed on a period of 6 months. The universe of study
is composed on 28 stocks of the CAC40 over 6 months..
is to generate random portfolios of the same Herfindahl index as the studied
portfolio. This could not be done using only the random function.
4.2
Second attempt
Using the Genetic Algorithm The second attempt consisted in creating
random portfolios with a constrained level of diversification. The Genetic Algorithm(GA) developed and parametrized to do portfolio optimization revealed
itself a good way of creating random portfolios. The fitness function of the GA
has to be equal to the constraint |Σwi2 − Hportf olio |. Then the GA select a
random portfolio which satisfies to the constraint of the level of diversification.
Hence the GA has to be ran thousand times to create the population of thousand random portfolios. It is a computationally time consuming to do, but it
gives the expected result. 1
Result of the experience Figure 4.3 and 4.4 you can observe that the new
repartition of random portfolios in term of returns is wider. The range of returns
is now from - 20 percent et + 30 percent. Now the calculated p-value are between
0 and 1. The evaluation of strategies is possible. The p-value represent the rank
in percentage of the strategy against the random portfolios. However one p-value
for one period is not enough to decide whether the strategy is good or whether
the result is due to luck.
1 A computationally cheaper way to generate random portfolios with a Herfindahl Constraint is to use hyper-spherical coordinate
20
Figure 4.3: Histogram of the 1000 RANDOM PORTFOLIOS generated with a
Genetic Algorithm at a Herfindahl Index of 1/3. Same period and universe as
figure 4.1. We can observe that the range of returns is much wider than this of
figure 4.1
Figure 4.4: Graphics of RANDOM PORTFOLIOS sorted by increasing returns.
The 1000 random portfolios were generated with a Genetic Algorithm at a
Herfindahl Index of 1/3. Same period and universe as figure 4.2. This graphics
can be used to read the p-value of your strategy on this particular period and
universe if its herfindahl index is 1/3.
21
4.3
Conclusion
Random Portfolios technique should be used carefully. Always remind this
sentence of good sense: compare what is comparable. Figure 4.5 represent pvalues for different Herfindahl index value. Random portfolios are specific of a
universe. Based on a same level of risks, the use of random portfolios and pvalue is a really powerful tool to assess the performance of a strategy. However
this evaluation should be done over a significant amount of periods of time to
ensure the validity of the result against luck. The use of statistic to create
interval of confidence around the mean p-value observed should give insight on
the fundamental question skilled or not? One idea to increase the validity of
the p-value test would be to cut a big period on different scales. For example,
cutting a period of 6 months in 60 periods of 3 months to compute more pvalues. However this cut could have an impact on the pertinence of result by
uncorrelating the test and the strategy studied.
Figure 4.5: RANDOM PORTFOLIOS return for different level of the Herfindahl
Index. Each line corresponds to a different p-value. The universe of portfolios
is composed of 28 stocks of the CAC40. Six months of daily returns are used to
evaluate portfolios returns. Thousand random portfolios were generated with a
Genetic Algorithm for the 28 levels of the Herfindahl index ( 1,1/2,1/3...1/28)
22
Chapter 5
Innovative idea on risks
The creation of new strategies should be based on new measure of risks. The
use of the variance or standard deviation as a measure of risks could not any
more agreed as valid by researchers. The general idea of investment strategy
should still minimize risks and maximize returns of the portfolio. Computation is not any more a problem with computer of today. The use of genetic
algorithm allows strategies based on non-linear measures of risks. With the
genetic algorithm, there is no need for a n-th improvement of the estimation
of the covariance matrix, because optimization problems don’t need to be exposed under a quadratic form to be solve. My innovative idea is to separate the
measure of risks to be minimized and the level of risks taken. The level of risks
represents the profile of risk wanted by investors. The measure of risks will give
information to select stocks in a safe way.
5.1
Measures of risks and dependencies
5.1.1
Coherent Measure of risk
In the paper ”Coherent Measure of risk” [2], the authors develop a theory
explaining what properties should have a good measure of risk in portfolio management.
”A risk measure satisfying the four axioms of translation invariance,
subadditivity, positive homogeneity and monotonicity is called coherent.” [2]
Translation invariance If a ∈ R and X ∈ L then ρ(a + X) ≤ ρ(X) − a
The value a is just adding cash to your portfolio X, which acts like an
insurance: the risk of X + a is less than the risk of Z, and the difference is
exactly the added cash a. In particular, if a = ρ(Z) then ρ(Z + ρ(Z)) = 0.
Sub-additivity If X1 , X2 ∈ L, then ρ(X1 + X2 ) ≤ ρ(X1 ) + ρ(X2 )
The risk of two portfolios together cannot get any worse than adding
23
the two risks separately: this is the diversification principle.
Positive homogeneity If α ≥ 0 and X ∈ L then ρ(αX) = αρ(X)
Loosely speaking, if you double your portfolio then you double your risk.
Monotonicity If X1 , X2 ∈ L and X1 ≤ X2 , then ρ(X1 ) ≤ ρ(X2 )
That is, if portfolio X2 has better values than portfolio X1 under all
scenarios then the risk of X2 should be bigger than the risk of X1 : more
profit, more risk.
5.1.2
Dependencies
Measures of risks are correlated to the universe of the study. Dependencies
are included in the measure of risks. Properties of dependencies depend on
the measure of risks used. For example, minimizing the variance of a portfolio
is not the same as minimizing the weighted some of variances of single assets
corresponding to the same portfolio, because variance is non-linear (actually,
it is quadratic). Minimizing a measure of risks on a whole portfolio takes into
consideration the dependencies between assets. This compensation ,between
assets of a portfolio, has a positive impact on the property of the portfolio, if
the measure of risks is coherent. Hence dependencies are taken into account.
5.1.3
Some measure of risks
Below are listed some measure of risks and their main interests. These measures are used in the section Simulation, to create portfolio. They are all well
known, except the Maximum DrawDown of returns for which I think to be
the creator. ”If Maximum DrawDown of stock prices is the speed, Maximum
Drawdown of returns is the acceleration.”
Variance: That is the most commonly used measure of risk in portfolio optimization. It is convenient to calculate and represent. One problem of the
variance is that extreme risks are not taken into account.
Moment of even higher order (4,6,8): Moment of order 4, 6, 8 takes into
account extreme risks. They indeed characterize fat-tailed distribution.
Maximum DrawDown of stock price: The maximum loss from peak to valley. The Maximum DrawDown has some nice property. It is invariant by
translation , it is homogeneous and convex. Thus the MDD is a coherent
measure as the volatility.
Average Maximum DrawDown of stock price: It takes the average from
the Maximum DrawDown of a time-series of stock prices. See the code of
the function in annexe
Maximum DrawDown of returns: Like the Maximum DrawDown of stock
prices, but based on the series of returns. In comparison with physics,
24
Figure 5.1: Illustration of two MAXIMUM DRAWDOWNS
it represents the extreme acceleration of the system. As it will be shown
in the part simulation, the MDD of returns performs well in portfolio
optimization.
Value at Risk: The VaR measures the loss of a portfolio for a given quantile.
It is currently controversial since it does not take into account extreme
risks. Moreover this measure is not coherent for a non-Gaussian distribution.
Conditional Value at Risk: the second name of C-VaR is the expected shortfall. This measure is known as a coherent measure of risks. It benefits
from the criticism of the VaR. It is more sensitive to the shape of the loss
distribution. Extreme risks are taken into account, because it focuses on
worst scenario quantiles.
Semi-variance: This risk measure aims at measuring the variance for the negative returns. The logic is that variance of positive returns is an opportunity, but variance of negative returns represents a risk.
Deviation to the median: This risk measure is a kind of standard deviation.
It takes into account the asymmetry of the distribution of returns.
5.2
Level of risks and diversification
Introduction Risks are often associated with diversification. Using coherent measure of risks implies that risks is diminishing with diversification (subadditivity property). A portfolio containing stocks of one company, will loose
everything if the company die. However if the portfolio contains stocks of two
companies in equal proportion, and one company dies, the portfolio will still
have the value from the other company.
25
An easy example to understand the impact of diversification
Given A and B two independent companies,
P(A=1) the probability that the company A survives until next year,
P(A=0) the probability that the company A dies before next year,
the value of the company remains equal to 1 during the year in case of survival,
the value of the company remains equal to 0 in case of death,
Π1 the portfolio containing only stocks of A for a total value of 100
Π2 the portfolio containing stocks A and B in equal proportion for a total
value of 100
P(A=1)=P(B=1)=0,99
P(A=0)=P(B=0)=0,01
The expected value of portfolios at the end of the year is the same in both case:
E(Π1 ) = 100 × (1 × P (A = 1) + 0 × P (A = 0)) = 0, 99
E(Π2 ) = 50 × (1 × P (A = 1) + 0 × P (A = 0)) + 50 × (1 × P (B =
1) + 0 × P (B = 0)) = 0, 99
E(Π1 ) = E(Π2 )
However, the extreme risk that the portfolio is equal to 0 is:
P (Π1 = 0) = P (A = 0) = 0.01
P (Π2 = 0) = P (A ∩ B = 0) = P (A = 0) × P (B = 0) = 0.0001
This example shows a reduction by hundred of the risk of loosing everything.
Note that for non-independent companies, P (A∩B = 0) = P (A = 0)×P (B = 0)
is not valuable any-more. However the result P (Π2 = 0) ≤ P (Π1 = 0) is still
true. In conclusion, diversification gives benefit in lowering extreme risks.
Herfindahl Index The Herfindahl index is a measure of diversification. It
is used in economics, to study the concentration of a market. If it’s value is
close to zero, it means that the market shares are distributed among a lot of
companies. In opposition, if it’s value is close to one, the market share are
concentrated in few companies.This indicator allows to select the number of
companies wanted in your portfolio during the process of optimisation. For
instance, if you want about
P55 different assets under the assumption of equally
weighted portfolio, H = i=0 wi2 = 15 . As it is shown in the figure 5.2, the
risks is decreasing with the increase of the diversification of portfolios. In this
example, the minimum return of simulated portfolios goes from -17 percent to
2 percent when the average number of stocks in portfolios goes from 28 to 1.
The same is true of the maximum return of the portfolios. Over the period, it is
divided by 6 from 30 percent to 5 percent. Small Herfindahl Index implies small
range of variation and lower risks. Moreover the Herfindahl index is quite stable
with the evolution of a portfolio. As you can see on figure 5.3, variations of the
26
Herfindahl index are not significant over a period of 122 days of evolution. It is
true with other value of the Herfindahl index and other period of time.
To be short, ”High degree of diversification implies small range of performance
available. Low degree of diversification implies high range of performance
available”
Figure 5.2: ENVELOP of returns taken by 1000 random portfolios calculated
for H0 = [1, 1/2 1/3, 1/4, 1/5, ..., 1/28] over a period of 121 days. H0 is the
value of the Herfindahl Index the first day of the period.
27
Figure 5.3: VARIATION of the HERFINDAHL Index of a portfolio over a
period of 122 days. H0 is equal to 0.1 The second graphic is a zoom of the first
one. We can observe that the variation of the Herfindahl Index are really small.
Var(H) about equal to 10−7
5.3
A winner mix
Combining Diversification and Innovative risks measures The conclusion of the first two paragraphs is that there is various definitions of risks and
that the level of risks could be thought in term of level of diversification. My
idea, to create new strategy, is to minimize a measure of risks and to set the
level of risks by choosing a value of diversification trough the use of the Herfindahl index. Since the distribution of returns is unstable over time, I think that
measure of risks should not come from value related to the distribution. In
opposition, the measure of risks should come from time series statistics, like the
Maximum DrawDown. The mix of measure of risks and diversification should
lead to make safer and more profitable portfolios.
Active versus Passive Portfolio Management The process of asset management is complex and multiple. A good asset manager should take into consideration the allocation between class of assets. This mean the ventilation of
the portfolio between cash, bonds, stocks and other. It should select the good
assets in each class. This mean doing, for example, the right stocks or bonds
picking. It should also apply a market-timing strategy. This mean adapting
the allocation strategy to market cycle (bear and bull markets). A ideally good
asset manager should be able to do these three tasks in order to benefit of ac28
tive or dynamic portfolio management. A lot of studies show that funds don’t
beat the market [11]. Hence the question of active management versus passive
management become pertinent. In the following, I choose to study a light active
management: adjustment of the portfolio every 6 months.
29
Chapter 6
Simulation
Aim of the simulation This simulation looks at the impact of new measure
of risks in portfolio construction. The main question is: Is it possible to create
new allocation strategies, for a given level of risks, that give persistent and
consistent results? I developed for the purpose of this study, an optimization
tool and an analysis tool to perform this simulation, using Matlab and a genetic
algorithm. Moreover the analysis of performance will estimate the p-values from
random portfolios analysis, to compare the different strategies.
6.1
6.1.1
Methodology
Data
Stocks returns The data used in this simulation are 28 stocks of the CAC
40 from the 17/01/2003 to the 03/08/2009. Data come from the website Yahoo
Finance. Returns on stocks are calculated from daily adjusted-closing prices.
The adjusted-closing price takes into account changes due to dividend and split
of the stock price. The large number of stocks has been chosen to create realistic
portfolios. You can observe figure 6.1, the evolution of the CAC 40 during the
period used. It corresponds to the end of the 2001 crisis, followed by the subprime bubble, the crash of the stocks market in 2008, followed by the increase
of 2009.
Survival and Look-ahead Bias In empirical finance, results can be biased
due to a survival effect coming from the selection of the data. The survival
bias corresponds to the fact, that simulation is done from data of stocks which
are still alive. Hence it forgot enterprise that died during the time set of the
simulation. It has been shown that the survival bias could increase return of a
strategy during a simulation [5, 10]. I choose only 28 assets from the CAC40
instead of 40. Is my study subject to survival bias? First, no enterprise of the
CAC40 died during the time of the simulation. Enterprise of the CAC40 are
30
Figure 6.1: Evolution of the CAC40.
generally big enough to survive. However some enterprises disappear due to
mergers and acquisition (GDF-Suez), or appear due to new regulation for example Suez environment detached from Suez or GDF detached from EDF. It did
not increase the survival bias since these enterprises are not under-performing
or over-performing the market. Second I did not include every assets due to
a lack of quality of certain data where the adjusted price was not taking into
account split. In conclusion I think that my data are not impacted a lot by
survival bias. Concerning the look-ahead bias, my procedure of test will use a
period to create the portfolio, which will be test on the period after. Since I
did not use future information in the past, my study should not be impacted
by this bias (see figure 6.2). However it is necessary to be introduced to the
look-ahead benchmark bias which appears in case of benchmark comparison.
It comes from the fact that the constitution of the benchmark is permanently
evolving and that the information of this constitution is not easily available and
usable. [10]
6.1.2
Test
Strategy The main idea of this simulation is to test the idea of level of risks
based on the Herfindahl index for investment in stocks in a limited universe. The
strategies of investment will take into account a constraint of diversification and
a measure of risks that will be minimized. All these strategies will be compared
to the equally weighted portfolio. This will provide a base of comparison. The
strategies will be back tested with the data described below. The portfolios
will be constructed from the last year of data and kept during the 6 following
31
Periods
period
period
period
period
period
period
period
period
period
period
period
period
1
2
3
4
5
6
7
8
9
10
11
12
Training data
from
to
17/01/2003 16/01/2004
04/07/2003 02/07/2004
19/12/2003 17/12/2004
04/06/2004 03/06/2005
19/11/2004 18/11/2005
06/05/2005 05/05/2005
21/10/2005 20/10/2006
07/04/2007 06/04/2007
22/09/2006 21/09/2007
09/03/2007 07/03/2008
24/08/2007 22/08/2008
11/02/2008 11/02/2009
Test
from
19/01/2004
05/07/2004
20/12/2004
06/06/2005
21/11/2005
08/05/2006
23/10/2006
09/04/2007
24/09/2007
10/03/2008
25/08/2008
12/02/2009
data
to
02/07/2004
17/12/2004
03/06/2005
18/11/2005
05/05/2006
20/10/2006
06/04/2007
21/09/2007
07/03/2008
22/08/2008
11/02/2009
03/08/2009
Figure 6.2: This table contains the different PERIODS OF DATA used during
the simulation. Training data are the data used to create a portfolio. Test data
are the data used to evaluate the portfolio. Training data contained 260 days
and test data 120 days.
months.
Optimization The optimization will be realized with a tool developed in
Matlab, using a genetic algorithm. The optimization tool cut the data into
periods on which it performs in serial the optimization of all the strategies. The
result of this optimisation is stored in one file to be analysed next. It contains
the portfolio selected for the next 6-months period and data of this test period
in order to enable the analysis later on.
Evaluation of performance The analysis is done by an other tool developed
in Matlab. It allows to look at each of the strategies for each sub-period and
to compare them. It proceeds to the evaluation of transaction costs, of the
wealth, of the evolution of the strategies over all the periods. Moreover random
portfolios p-values and Sharpe ratios of strategies will be estimated from test
data at each sub-period.
6.1.3
Set of strategies
Here are the different sets of strategies analysed. The optimal portfolio for the
period will be calculated by the optimization software on the precedent period.
w represents the vector of portfolio’s weights of dimension n the number of
assets. R represents the matrix of returns n × m with n the number of assets
and m the number of trading days of the period. V represents the matrix of
stocks’ price.
32
w∗ optimal portfolio such as F (R.w∗ ) + Hc(w∗ ) is minimum with F the risk
measure and Hc the diversification constraint. No short selling is allowed:
n
X
∀i, 0 ≤ wi ≤ 1. The other constraint on weights is:
wi = 1.
i=1
Minimum Variance Strategy
1. min var(R.w) + |1 − h(w)|
2. min var(R.w) + | 21 − h(w)|
3. min var(R.w) + | 13 − h(w)|
4. min var(R.w) + | 14 − h(w)|
5. min var(R.w) + | 15 − h(w)|
6. min var(R.w) + | 16 − h(w)|
7. min var(R.w) + | 17 − h(w)|
8. min var(R.w) + | 18 − h(w)|
9. min var(R.w) + | 19 − h(w)|
1
10. min var(R.w) + | 10
− h(w)|
11. min var(R.w)
1
12. min | 28
− h(w)|
Minimum Moment of order 4 Strategy
1. min µ4 (R.w) + |1 − h(w)|
2. min µ4 (R.w) + | 12 − h(w)|
3. min µ4 (R.w) + | 13 − h(w)|
4. min µ4 (R.w) + | 14 − h(w)|
5. min µ4 (R.w) + | 15 − h(w)|
6. min µ4 (R.w) + | 16 − h(w)|
7. min µ4 (R.w) + | 17 − h(w)|
8. min µ4 (R.w) + | 18 − h(w)|
9. min µ4 (R.w) + | 19 − h(w)|
1
10. min µ4 (R.w) + | 10
− h(w)|
11. min µ4 (R.w)
1
12. min | 28
− h(w)|
Minimum Maximum DrawDown of returns Strategy
1. min M DD(R.w) + |1 − h(w)|
2. min M DD(R.w) + | 21 − h(w)|
3. min M DD(R.w) + | 31 − h(w)|
33
4. min M DD(R.w) + | 41 − h(w)|
5. min M DD(R.w) + | 51 − h(w)|
6. min M DD(R.w) + | 61 − h(w)|
7. min M DD(R.w) + | 71 − h(w)|
8. min M DD(R.w) + | 81 − h(w)|
9. min M DD(R.w) + | 91 − h(w)|
1
− h(w)|
10. min M DD(R.w) + | 10
11. min M DD(R.w)
1
12. min | 28
− h(w)|
Minimum Maximum DrawDown of values Strategy
1. min M DD(V.w) + |1 − h(w)|
2. min M DD(V.w) + | 21 − h(w)|
3. min M DD(V.w) + | 13 − h(w)|
4. min M DD(V.w) + | 14 − h(w)|
5. min M DD(V.w) + | 15 − h(w)|
6. min M DD(V.w) + | 16 − h(w)|
7. min M DD(V.w) + | 17 − h(w)|
8. min M DD(V.w) + | 18 − h(w)|
9. min M DD(V.w) + | 19 − h(w)|
1
10. min M DD(V.w) + | 10
− h(w)|
11. min M DD(V.w)
1
− h(w)|
12. min | 28
Minimum Average Maximum DrawDown of values Strategy
1. min AverageM DD(V.w) + |1 − h(w)|
2. min AverageM DD(V.w) + | 12 − h(w)|
3. min AverageM DD(V.w) + | 13 − h(w)|
4. min AverageM DD(V.w) + | 14 − h(w)|
5. min AverageM DD(V.w) + | 15 − h(w)|
6. min AverageM DD(V.w) + | 16 − h(w)|
7. min AverageM DD(V.w) + | 17 − h(w)|
8. min AverageM DD(V.w) + | 18 − h(w)|
9. min AverageM DD(V.w) + | 19 − h(w)|
1
10. min AverageM DD(V.w) + | 10
− h(w)|
11. min AverageM DD(V.w)
1
12. min | 28
− h(w)|
34
6.2
Results
6.2.1
Tables
These tables represent indicators resulting from the simulation. The data
were subset in 12 test periods from the 19/01/2004 to the 03/08/2009. New
portfolios are generated at each sub-period.
Herfindahl
Variance
µ4
MDD(r)
MDD(v)
A-MDD(v)
1
180
221
231
139
95
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
175
123
141
205
115
140
150
141
174
122
251
122
123
128
98
127
87,8
146
133
127
101
125
150
134
109
126
114
133
130
118
141
124
144
128
127
107
89
139
122
119
105
85
137
120
108
free
133
131
129
134
116
1
28
112
112
112
112
112
Figure 6.3: END WEALTH of the different strategies at the date of 03/08/2009.
Starting value of 100 the 19/01/2004.
Herfindahl
Variance
µ4
MDD(r)
MDD(v)
A-MDD(v)
1
165
204
216
111
64
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
145
93
112
179
85
112
123
117
152
98
222
92
101
106
72
103
59
125
114
105
77
99
133
116
88
105
91
117
114
99
119
101
129
113
111
86
69
126
109
103
86
65
125
107
94
free
124
121
116
117
100
1
28
108
108
108
108
108
Figure 6.4: END WEALTH including TRANSACTION COSTS of 1 percent of
the total value traded. Starting value of 100 the 19/01/2004.
Herfindahl
Variance
µ4
MDD(r)
MDD(v)
A-MDD(v)
1
0,38
0,35
0,36
0,40
0,52
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
0,38
0,46
0,45
0,29
0,47
0,40
0,41
0,38
0,35
0,45
0,24
0,46
0,41
0,46
0,51
0,45
0,51
0,38
0,41
0,43
0,56
0,43
0,35
0,41
0,45
0,43
0,51
0,37
0,40
0,44
0,37
0,45
0,36
0,40
0,40
0,41
0,64
0,35
0,43
0,44
0,48
0,60
0,36
0,43
0,50
Figure 6.5: Average RANDOM PORTFOLIOS P-VALUES of the different
strategies for the period from 19/01/2004 to 03/08/2009.
35
free
0,32
0,31
0,34
0,40
0,43
1
28
0,49
0,50
0,50
0,49
0,49
Herfindahl
Variance
µ4
MDD(r)
MDD(v)
A-MDD(v)
1
0,82
1,01
0,85
0,65
0,22
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
0,74
0,56
0,60
1,08
0,53
0,98
0,71
1,12
1,03
0,75
1,28
0,48
1,01
0,75
0,60
0,78
0,55
1,25
1,09
0,89
0,68
0,67
1,27
1,08
0,85
0,92
0,61
1,16
1,05
0,89
1,12
0,73
1,23
1,07
1,02
1,03
0,37
1,20
0,96
0,99
0,83
0,45
1,21
0,96
0,75
free
1,34
1,32
1,24
0,97
0,88
1
28
0,73
0,73
0,73
0,73
0,73
Figure 6.6: Average yearly SHARPE RATIO of the different strategies for the
period from 19/01/2004 to 03/08/2009.
Ranks
Variance
µ4
MDD(r)
MDD(v)
A-MDD(v)
Wealth generated
2
4
1
3
5
Sharpe ratio
3
5
1
2
4
Random Portfolios
3
5
1
3
4
Overall Performance
3
5
1
3
4
Figure 6.7: Result summary. Numbers represent the rank of strategies against
each other.
36
6.2.2
Wealth evolution - first interpretation
Some stylized facts on result A first comparison of the wealth generated by
each set of strategies during the back-testing period give an idea on performance.
Figure 6.3 exhibits the value of portfolios at the end of the 12 periods of the
simulation for every set of strategies. For each strategy new portfolios are
generated at the beginning of each period based on the previous year of data.
The starting value (19/01/2004) is 100. The end of the simulation is 03/08/2009.
At this date, the equally weighted portfolios achieved a performance of 12.2
percent. The equally weighted portfolio strategy represents the performance
of the market. It is used as a benchmark. Looking at the generated wealth,
strategies based on Maximum DrawDown of returns and of values have results
above the equally weighted portfolios for every level of diversification simulated.
These two measures of risks seem to contain above the average information
about the capacity to grow of portfolios. Their power of prediction leads to high
returns. Moreover, figures 6.10 and 6.11 represent the evolution of their wealth.
These graphics confirm the fact that portfolios based on Maximum DrawDown
of returns and values perform at every level of diversification more than the
benchmark on the long run. Set of strategies based on Variance and 4th Moment
don’t give such strong robustness and persistence on results. They have high
performance with portfolios poorly diversified (1-4 stocks) and low performance
against the benchmark with diversified portfolios(5-10 stocks). Finally strategy
based on Average Maximum DrawDown of values seem to provide no interests.
Robustness to diversification A measure of risks, which could produce
good performance whatever its level of diversification, provide a good base for an
investment strategy. It proves its capacity to predict which stocks will produce
high performance over the next period. The reliability of the measure of risks
should be proved for different configuration of portfolios. Hence the robustness
to diversification is a good indication of reliability. Figure 6.13 shows results
of the table 6.3 under a Box plot form. Maximum DrawDown of returns box
is smaller and above the others. It means that in average its performance is
above the other strategy. Moreover its performance is robust to diversification
since the size of the box is small. It means that strategies based on Maximum
DrawDown on return provide, in the long run, high returns for every level of
diversification. Strategy based on Variance provide more risky investment since
the Box of results is wide. It means that this measure of risks is really sensible
to diversification. It could have low or high unexpected results. Are these
results the fruit of luck? This question needs a random portfolios analysis to
be answered. Finally, strategies based on Average-Maximum-DrawDown on
value produce robustness to diversification for bad returns. Its box is small, but
located below the others with a small mean. Hence this measure should not be
used (same conclusion for the 4th Moment measure of risks).
Stock picking The aim of an investment strategy in stocks is to select stocks
that produce high returns and have low volatility. Stock picking is based on
37
qualitative and quantitative analysis. The study of the different set of strategies shows for some measures of risks a non-robustness. However, 4th Moment
and Variance strategies show extremely high return for portfolios composed of 1
to 4 assets. One explanation could be luck! Nevertheless, an other explanation
could be that selecting 1 to 4 stocks over 28, require less information and competency than selecting 10 good stocks. This could explain the fact that Variance
and 4th Moment measures of risks could lead to select some good stocks, but not
be able to select one entire portfolio. Moreover, portfolios construct without
constraint of diversification (Herfindahl level free) seems to perform better and
lead to portfolio of lower volatility than portfolio of equivalent degree of diversification. The constraint impacts negatively the level of information contained
by the measure of risks.
Profile of risks In the universe of commercial finance, assets’ managers have
to care about the profile of their investors, to offer them, product adapted to
their risk acceptance. I think that based on the result of figures 6.3 and 6.13,
3 profiles emerged. These profile take their sense in a fund which would use a
measure of risks robust to diversification in a small universe of large stocks like
the CAC40.
• The speculative profile composed of 1 or 2 stocks.
• The risky profile composed from 3 to 5 stocks.
• The secure profile composed by the optimisation without any constraint.
38
Figure 6.8: Wealth evolution of VARIANCE based strategies from 19/01/2004
to 03/08/2009.
Figure 6.9: Wealth evolution of 4th MOMENT based strategies from 19/01/2004
to 03/08/2009.
39
Figure 6.10: Wealth evolution of MAXIMUM DRAWDOWN of RETURNS
based strategies from 19/01/2004 to 03/08/2009.
Figure 6.11: Wealth evolution of MAXIMUM DRAWDOWN of VALUES based
strategies from 19/01/2004 to 03/08/2009.
40
Figure 6.12: Wealth evolution of Average MAXIMUM DRAWDOWN of VALUES based strategies from 19/01/2004 to 03/08/2009.
41
Figure 6.13: Comparison of the strategies in term of WEALTH at the end of the
simulation from 19/01/2004 to 03/08/2009. All Herfindahl index mixed. The
red horizontal line represent the median. The boundaries of the box represent
the 25 and 75 quantile. The whiskers represent the minimum and maximum
values. Maximum values corresponds generally to high Hefindahl index (look
at the table 6.3 to see the corresponding values plotted here). This plot shows
that strategies based on different measures of risks have different responses to
diversification.
42
6.2.3
Random Portfolios - Luck or skill?
Consistency of results Random portfolios allow to rank a strategy. In this
study, the p-value is calculated for each sub-strategy and each sub-period. Results are plotted in graphics 6.14, 6.15, 6.16, 6.17, 6.18. It is interesting to
observe how variable is the p-value for each strategy. A consistent strategy
should have a high mean p-value and a small range of variation in order to show
consistency. On the maps plotted below the 3D graphics, dark area represents
low p-value. Hence a map with a high dark coverage along the 2 axes: strategy
and period; means that the measure of risks is robust to diversification and gives
persistent value added during time.
4th Moment This set of strategies does not show any robustness to diversification. The median p-value is really different for each sub-strategy(level
of diversification). Box of the box-plot are big. Hence the p-value of each
sub-strategy is very different for different periods. It means that this measure of risks don’t provide many information. The map seems completely
random implying no consistency in results.
Variances This set of strategies does not show any regularity of results. The
map seems random. However p-value are in averaged smaller than 0.5
, meaning that they are performing better than the expected value of
someone acting randomly(see table 6.5 for the average p-values over subperiods). Is it due to luck? Sub-strategy 4 and 11 are especially good in
this simulation.
MDD of returns This set of strategies shows a good robustness to diversification. Boxes of sub-strategies are quite similar. dark area on the map
are predominant. P-values are almost the same for each sub-strategies.
The map reveals a periodicity. For example, p-values are bad for all substrategies at the period 10 and 12. The mean and median p-values are
low for most of the periods meaning that strategies based on Maximum
DrawDown of returns provide a real plus to luck. This strategy seems
consistent.
MDD of values This set of strategies shows a good robustness to diversification. Like the Maximum DrawDown of returns, dark area is predominant
with a periodicity of bad p-value. Median p-value is quite good (less than
0.5) for all the sub-strategies but less than those of Maximum DrawDown
of returns. On the boxplot graphic, sub-strategy 2 (with 2 stocks) is
particularly good. However the difference of p-value for different period
implies a strong correlation of the performance to market condition.
Average MDD of values This set of strategies shows a kind of robustness to
diversification, however the bright trend is dominant. P-values are high
meaning no skill at all. This measure of risks does not provide information
of prediction. It is not effective at selecting interesting stocks.
43
Figure 6.14: p-values analysis for 4th MOMENT based strategies. Results come
from a random portfolios analysis with 1000 random portfolios of the same
Herfindahl index as the analysed portfolio. Each box corresponds to the pvalues of the 12 sub-periods for one level of Herfindahl Index. In the box,
the red horizontal line represent the median p-value over the 12 periods. The
boundaries of the box represent the 25 and 75 quantiles. The whiskers represent the minimum and maximum p-values. The map represents p-values for a
given sub-period and sub-strategies(the 12 sub-strategies correspond to the 12
different levels of Herfindahl index).
44
Figure 6.15: p-values analysis for VARIANCE based strategies. Results come
from a random portfolios analysis with 1000 random portfolios of the same
Herfindahl index as the analysed portfolio. Each box corresponds to the pvalues of the 12 sub-periods for one level of Herfindahl Index. In the box,
the red horizontal line represent the median p-value over the 12 periods. The
boundaries of the box represent the 25 and 75 quantiles. The whiskers represent the minimum and maximum p-values. The map represents p-values for a
given sub-period and sub-strategies(the 12 sub-strategies correspond to the 12
different levels of Herfindahl index).
45
Figure 6.16: p-values analysis for MAXIMUM DRAWDOWN of RETURNS
based strategies. Results come from a random portfolios analysis with 1000
random portfolios of the same Herfindahl index as the analysed portfolio. Each
box corresponds to the p-values of the 12 sub-periods for one level of Herfindahl
Index. In the box, the red horizontal line represent the median p-value over the
12 periods. The boundaries of the box represent the 25 and 75 quantiles. The
whiskers represent the minimum and maximum p-values. The map represents pvalues for a given sub-period and sub-strategies(the 12 sub-strategies correspond
to the 12 different levels of Herfindahl index).
46
Figure 6.17: p-values analysis for MAXIMUM DRAWDOWN of VALUES based
strategies. Results come from a random portfolios analysis with 1000 random
portfolios of the same Herfindahl index as the analysed portfolio. Each box
corresponds to the p-values of the 12 sub-periods for one level of Herfindahl
Index. In the box, the red horizontal line represent the median p-value over the
12 periods. The boundaries of the box represent the 25 and 75 quantiles. The
whiskers represent the minimum and maximum p-values. The map represents pvalues for a given sub-period and sub-strategies(the 12 sub-strategies correspond
to the 12 different levels of Herfindahl index).
47
Figure 6.18: p-values analysis for AVERAGE MAXIMUM DRAWDOWN of
VALUES based strategies. Results come from a random portfolios analysis with
1000 random portfolios of the same Herfindahl index as the analysed portfolio.
Each box corresponds to the p-values of the 12 sub-periods for one level of
Herfindahl Index. In the box, the red horizontal line represent the median pvalue over the 12 periods. The boundaries of the box represent the 25 and 75
quantiles. The whiskers represent the minimum and maximum p-values. The
map represents p-values for a given sub-period and sub-strategies(the 12 substrategies correspond to the 12 different levels of Herfindahl index).
48
Asymmetry of performance As observed in the previous paragraph, Maximum DrawDown of returns and values especially show a sensibility to the period in performance. Figures 6.19,6.21,6.20 represent the performance of substrategies against 10 percent best random portfolios (p-value of 0.1). The market
period 11 decreases a lot and the market period 12 increases strongly. These
graphics confirm the asymmetry of performance of strategy based on measure
of risks, especially in the case of the Maximum DrawDown. This asymmetry
comes from the ability of these strategies to select stocks that perform the most
in bear market, but to select the ones that perform the less in extremely growing
market. The leverage effect can be an explanation of this assymetry. This effect
corresponds to a negative correlation between past returns and future volatility
[4]. Moreover minimizing a measure of risks necessary means minimizing the
extreme variation. Hence these measures failed to pick stocks that will grow
strongly. However this effect is interesting to create defensive strategy for bear
market.
Remark on the equally weighted portfolio The equally weighted portfolio
has a p-value of 0.5 since there is only one equally weighted portfolio by universe.
It represents a good estimator of the 0.5 p-value random portfolios for every level
of diversification. Hence the equally weighted portfolio is a good benchmark of
performance, because it represents the expected return of a random strategy.
49
Figure 6.19: Return of strategies based on MAXIMUM DRAWDOWN of RETURNS, against random portfolios of p-value of 0.1 . In Period 11, the market
decreases and in period 12, it increases
50
Figure 6.20: Return of strategies based on 4th MOMENT, against random portfolios of p-value of 0.1 In Period 11, the market decreases and in period 12, it
increases
51
Figure 6.21: Return of strategies based on VARIANCE, against random portfolios of p-value of 0.1 In Period 11, the market decreases and in period 12, it
increases
52
6.2.4
Sharpe Ratio
The Sharpe Ratio is defined as the ratio of the mean excess-year return and the
yearly standard deviation. Excess returns correspond to the series of portfolio
returns minus the risk free rate. The risk free rate of return is estimated with
the US-3-months-treasury bills. This ratio gives an information on how big is
the performance in comparison with its variability. It is a classical indicator
of performance computed in most studies. A ratio above one is considered as
good.
Analysis of the strategies The comparison of these results confirms the excellent performance of portfolios based on the Maximum DrawDown of returns.
These portfolios get in average a Sharpe ratio above one when they contain more
than two stocks (see table 6.6). Almost all levels of diversification give better
result than the equi-weighted portfolio. Portfolios based on the 4th Moment
give particularly bad Sharpe ratios, often below the equi-weighted portfolio.
Surprisingly, portfolios based on Variance are not the best. This is a proof
that the measure of the one-year variance is not a good predictor to select stocks
that provide small variance. Indeed Variance portfolios generate more wealth
than Maximum DrawDown of Values portfolios. When in the mean time, the
average Sharpe ratio of Maximum DrawDown of Values portfolios is higher (see
table 6.7).
Graphics representing Sharpe ratio for sub-periods and sub-strategies show
that its value is strongly dependent on the period, and moreover on the market
performance of the period (see figures 6.22, 6.23, 6.24, 6.25,6.26).
Relationship between Sharpe Ratio and Random Portfolios There
is no relationship between these two indicators. However results from these
two indicators go in the same direction. The assumptions under the Sharpe
ratios are the effectiveness of variance as a reliable-in-sample measure of risks.
Since there is no contradiction between results with random portfolios, it could
be true. However Sharpe ratio is a absolute measure of performance strongly
linked to the market performance. In opposition, the random portfolio p-value
is a relative measure of performance, which is uncorrelated to the market, but
depends on the universe of the study. Hence these indicators are complementary.
53
Figure 6.22: Yearly SHARPE ratio analysis for VARIANCE based strategies.
Computed for 12 sub-strategies and 12 sub-periods.
54
Figure 6.23: Yearly SHARPE ratio analysis for µ4 based strategies. Computed
for 12 sub-strategies and 12 sub-periods.
55
Figure 6.24: Yearly SHARPE ratio analysis for MAXIMUM DRAWDOWN
of RETURNS based strategies. Computed for 12 sub-strategies and 12 subperiods.
56
Figure 6.25: Yearly SHARPE ratio analysis for MAXIMUM DRAWDOWN of
VALUES based strategies. Computed for 12 sub-strategies and 12 sub-periods.
57
Figure 6.26: Yearly SHARPE ratio analysis for AVERAGE MAXIMUM
DRAWDOWN of VALUES based strategies. Computed for 12 sub-strategies
and 12 sub-periods.
58
6.2.5
Transaction costs overview
Transaction costs are an important issue for investors. If a strategy is more
expensive than it adds value, this strategy is useless. Here, transaction costs are
evaluated in function of the total amount traded. At each end of sub-periods,
portfolios are reconfigured, due to the investment strategy. These changes in
value are summed up to give the total amount traded, see figures 6.27, 6.28,
6.29, 6.30, 6.31. Then transaction costs are a fraction of this amount. Note
that I choose to exclude the creation cost of the first portfolio corresponding
to the cost of the first period. The total amount traded increases, if a strategy
performs well. Hence the cost of a good strategy is usually high. As shown
in figures 6.27, 6.28, 6.29, this proposition is true with an exception: the 1
stock strategy. This exception comes from the fact that the strategy one, select
the same stock during consecutive periods. In this case, transaction costs are
equal to zero. The most diversified is a portfolio, the less change in portfolio
weights there is, hence the less costly is the strategy. Observe in figures quoted
above, that the equally weighted portfolio is the one, the most diversified and
the cheapest. In order to conclude, the table 6.4 gives an idea of the impact
of transaction costs in strategies. In this example, transaction costs are equal
to one percent of the total amount traded. It is for sure, excessive in the case
of hedge funds, but represents more or less the reality for individual investors.
Comparing table 6.3 and 6.4, it is obvious, that the number of strategies underperforming the equally weighted portfolio, increases a lot if transaction costs are
counted. For strategies based on Maximum DrawDown of returns, it goes from
0 to 1, for strategies based on Variance from 3 to 5 and for strategies based on
µ4 from 3 to 8. It gives here a clear advantage for strategies based on Maximum
DrawDown of returns.
59
Figure 6.27: Total transaction costs of the simulation for 4th MOMENT based
strategies
Figure 6.28: Total transaction costs of the simulation for VARIANCE based
strategies
60
Figure 6.29: Total transaction costs of the simulation for MAXIMUM DRAWDOWN of RETURNS based strategies
Figure 6.30: Total transaction costs of the simulation for MAXIMUM DRAWDOWN of VALUES based strategies
61
Figure 6.31: Total transaction costs of the simulation for AVERAGE MAXIMUM DRAWDOWN of VALUES based strategies
62
Figure 6.32: How many times a strategy was the best of a period in the simulation for VARIANCE based strategies
6.2.6
Impact of the diversification - concrete cases
Diversification has a positive and a negative impact. The positive impact
is the one that leads to lower risks by inhibition of unique risks in portfolio.
The unique risk is a specific company risk. This company risk is diluted by
other companies. Hence a diversified portfolio should only support market risks.
The negative impact is the one that leads portfolio to loose opportunities of
growth. Imagine the case where few stocks are performing highly above the
market. For example, in the case of a universe containing 28 stocks and during
a period where only 2 stocks have positive performance, if a 10-stocks portfolio
is created, it will at least contain 8 bad stocks. Figures 6.32, 6.33, 6.34, 6.35,
6.36 represent the number of times, that a strategy earns the highest return
on a period. Results clearly show that exceptional performance comes from
poorly diversified portfolios. In conclusion, there is a trade-off to do between
exceptional performance and low risks.
63
Figure 6.33: How many times a strategy was the best of a period in the simulation for 4th MOMENT based strategies
Figure 6.34: How many times a strategy was the best of a period in the simulation for MAXIMUM DRAWDOWN of RETURNS based strategies
64
Figure 6.35: How many times a strategy was the best of a period in the simulation for MAXIMUM DRAWDOWN of VALUES based strategies
Figure 6.36: How many times a strategy was the best of a period in the simulation for AVERAGE MAXIMUM DRAWDOWN of VALUES based strategies
65
Chapter 7
What’s next?
7.1
Recommendation
A winning strategy From this study, I would recommend to use the one year
Maximum DrawDown of returns as a measure of risks for portfolio optimization.
This measure is coherent and provides good information to select stocks of your
portfolios for the next 6 months. This recommendation is valuable for a small
universe of investment (about 40 stocks) of big capitalization since the study
was based on CAC. I think it could be extended to a similar universe such
as DOW INDUSTRIALS, DAX, FTSE or SMI. The Maximum DrawDown of
returns measures how brutal are changes of trend of stocks from peak to valley.
This leads to select stocks the most stable. Moreover the minimization of this
measure for a portfolio will lead to find anti-correlate stocks in period of tough
market change. This ensure the robustness of these portfolio in crisis time.
As seen before, the level of diversification of the portfolio drives the overall
range of performance. Thus, the idea is to use diversification as a measure
of the level of risks inherent in a strategy. Logically a portfolio with high
diversification will offer less performance with a same level of skill than a poorly
diversified portfolio. Hence the optimization process and the measure of risks
used to construct portfolios should be robust to diversification. As observed, in
the evaluation of performance of strategies with random portfolios, Maximum
DrawDown of returns is robust to diversification since mean p-values are good
for all the levels of diversification. From the result of the back-testing, from
17/01/2003 to 03/08/2009, I identify three profiles of investors: a speculative
profile with 1-2 stocks, a risky profile with 3-5 stocks and a neutral profile
with no constraint of diversification. The back-testing on this stressed period
reveals a propensity of Maximum DrawDown of returns strategies to get high
performance in bear market and average performance in bull market. Thus it
ensures safety of the investment in time crisis, even for a speculative profile, and
good performance in the long run.
66
A good back-testing A back-testing is relevant when applied to a period
long enough to contain consecutive bull and bear markets. Indeed, it enables
investors to stress test their investment strategies. Moreover, the evaluation of
p-values, with random portfolios gives a really deep understanding of the performance of a strategy over time. It is important to understand the performance
of a strategy in detail to see if it provides value-added. From a study of random portfolios and after many comparisons, I suggest that the equally weighted
portfolio, which represents the expected value of investors acting randomly, is a
reliable benchmark of performance. Actually, comparing the wealth produced
by the equally weighted portfolio and a strategy is equivalent to compare the
strategy with a 50 percent p-value random portfolio.
7.2
To go further
Exploring all my ideas would have required more time. Some of them are listed
below to go further in my study of investment strategy:
Impact of time This study have been done using one year of training data
and 6 months of holding in portfolio. Varying these durations may lead
to other conclusions for the different indicators.
Impact of the universe The universe of study was quite small. What if it
integrates hundred of stocks of emerging markets?
Maximum DrawDown in multi-asset optimization What will be the result of an optimisation of the Maximum DrawDowns of returns on a mixed
universe of bonds and stocks.
Applying re-sampling for strategies based on moment of orders 2 and 4
Re-sampling may lead to better results on this kind of measure based on
returns.
Combining market timing tool to asset allocation Some risk measures have
better result in bear market, other in bull market. Predicting these market trends and adapting the optimized measure should produce exceptional
performance.
Using hyper-spherical constraints in portfolio optimization and random
portfolios generation. It should significantly decrease the time of computation.
Short selling short selling is not studied in this study, because I don’t believe that it is a position very useful for long run investment. However it
could be interesting to observe its effects on measures such as Maximum
DrawDown.
67
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
3.1
3.2
3.3
4.1
Formulation of Mean-Variance optimization . . . . . . . . . . . .
Instability of the EFFICIENT FRONTIER moving at each periods. Efficient frontiers from 28 assets over 12 periods of 6 months.
DEVIATION of a portfolio from the EFFICIENT FRONTIER.
Same frontiers as figure 2.2 with the evolution of a portfolio taken
on the first efficient frontier. A portfolio done in the MeanVariance frameworks become rapidly inefficient. . . . . . . . . . .
SHRINKAGE Method - a trade off between sample covariance
and single index . . . . . . . . . . . . . . . . . . . . . . . . . . .
REVERSE ENGINEERING - Measure of distance from the sample to the reverse engineering estimate. The factor alpha corresponds to the trade-off between the bias and the variance of our
estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
REVERSE ENGINEERING - Minimizing the distance of the reverse engineering estimate to the sample estimate under the condition that the market efficient portfolio belows to the efficient
frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
REPRODUCTION PROCESS using elite count, crossover and
mutation. In this example the crossover rate is 0.8 and the elite
count is 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A view of the scattered CROSSOVER function process. A random vector of bit is generated of the length equal the number
of assets in the portfolio. A zero means we select the weight of
parent 1 and a one the weight of parent 2 . . . . . . . . . . . . .
Gaussian MUTATION function adds a random number taken
from a Gaussian distribution with mean 0 to each entry of the
parent vector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Histogram of the 1000 RANDOM PORTFOLIOS generated with
the random and the normalize function. Returns are computed
on a period of 6 months. The universe of study is composed on
28 stocks of the CAC40 over 6 months. . . . . . . . . . . . . . . .
68
6
6
7
8
9
9
15
15
16
19
4.2
4.3
4.4
4.5
5.1
5.2
5.3
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Graphics of RANDOM PORTFOLIOS sorted by increasing returns. The 1000 random portfolios were generated with the random and the normalize function. Returns are computed on a
period of 6 months. The universe of study is composed on 28
stocks of the CAC40 over 6 months.. . . . . . . . . . . . . . . . .
Histogram of the 1000 RANDOM PORTFOLIOS generated with
a Genetic Algorithm at a Herfindahl Index of 1/3. Same period
and universe as figure 4.1. We can observe that the range of
returns is much wider than this of figure 4.1 . . . . . . . . . . . .
Graphics of RANDOM PORTFOLIOS sorted by increasing returns. The 1000 random portfolios were generated with a Genetic Algorithm at a Herfindahl Index of 1/3. Same period and
universe as figure 4.2. This graphics can be used to read the pvalue of your strategy on this particular period and universe if
its herfindahl index is 1/3. . . . . . . . . . . . . . . . . . . . . . .
RANDOM PORTFOLIOS return for different level of the Herfindahl Index. Each line corresponds to a different p-value. The universe of portfolios is composed of 28 stocks of the CAC40. Six
months of daily returns are used to evaluate portfolios returns.
Thousand random portfolios were generated with a Genetic Algorithm for the 28 levels of the Herfindahl index ( 1,1/2,1/3...1/28)
20
21
21
22
Illustration of two MAXIMUM DRAWDOWNS . . . . . . . . . . 25
ENVELOP of returns taken by 1000 random portfolios calculated
for H0 = [1, 1/2 1/3, 1/4, 1/5, ..., 1/28] over a period of 121 days.
H0 is the value of the Herfindahl Index the first day of the period. 27
VARIATION of the HERFINDAHL Index of a portfolio over a
period of 122 days. H0 is equal to 0.1 The second graphic is a
zoom of the first one. We can observe that the variation of the
Herfindahl Index are really small. Var(H) about equal to 10−7 . 28
Evolution of the CAC40. . . . . . . . . . . . . . . . . . . . . . . .
This table contains the different PERIODS OF DATA used during the simulation. Training data are the data used to create a
portfolio. Test data are the data used to evaluate the portfolio.
Training data contained 260 days and test data 120 days. . . . .
END WEALTH of the different strategies at the date of 03/08/2009.
Starting value of 100 the 19/01/2004. . . . . . . . . . . . . . . .
END WEALTH including TRANSACTION COSTS of 1 percent
of the total value traded. Starting value of 100 the 19/01/2004.
Average RANDOM PORTFOLIOS P-VALUES of the different
strategies for the period from 19/01/2004 to 03/08/2009. . . . .
Average yearly SHARPE RATIO of the different strategies for
the period from 19/01/2004 to 03/08/2009. . . . . . . . . . . . .
Result summary. Numbers represent the rank of strategies against
each other. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
31
32
35
35
35
36
36
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
Wealth evolution of VARIANCE based strategies from 19/01/2004
to 03/08/2009. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wealth evolution of 4th MOMENT based strategies from 19/01/2004
to 03/08/2009. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wealth evolution of MAXIMUM DRAWDOWN of RETURNS
based strategies from 19/01/2004 to 03/08/2009. . . . . . . . . .
Wealth evolution of MAXIMUM DRAWDOWN of VALUES based
strategies from 19/01/2004 to 03/08/2009. . . . . . . . . . . . . .
Wealth evolution of Average MAXIMUM DRAWDOWN of VALUES based strategies from 19/01/2004 to 03/08/2009. . . . . . .
Comparison of the strategies in term of WEALTH at the end of
the simulation from 19/01/2004 to 03/08/2009. All Herfindahl
index mixed. The red horizontal line represent the median. The
boundaries of the box represent the 25 and 75 quantile. The
whiskers represent the minimum and maximum values. Maximum values corresponds generally to high Hefindahl index (look
at the table 6.3 to see the corresponding values plotted here).
This plot shows that strategies based on different measures of
risks have different responses to diversification. . . . . . . . . . .
p-values analysis for 4th MOMENT based strategies. Results
come from a random portfolios analysis with 1000 random portfolios of the same Herfindahl index as the analysed portfolio. Each
box corresponds to the p-values of the 12 sub-periods for one level
of Herfindahl Index. In the box, the red horizontal line represent
the median p-value over the 12 periods. The boundaries of the
box represent the 25 and 75 quantiles. The whiskers represent the
minimum and maximum p-values. The map represents p-values
for a given sub-period and sub-strategies(the 12 sub-strategies
correspond to the 12 different levels of Herfindahl index). . . . .
p-values analysis for VARIANCE based strategies. Results come
from a random portfolios analysis with 1000 random portfolios of
the same Herfindahl index as the analysed portfolio. Each box
corresponds to the p-values of the 12 sub-periods for one level of
Herfindahl Index. In the box, the red horizontal line represent
the median p-value over the 12 periods. The boundaries of the
box represent the 25 and 75 quantiles. The whiskers represent the
minimum and maximum p-values. The map represents p-values
for a given sub-period and sub-strategies(the 12 sub-strategies
correspond to the 12 different levels of Herfindahl index). . . . .
70
39
39
40
40
41
42
44
45
6.16 p-values analysis for MAXIMUM DRAWDOWN of RETURNS
based strategies. Results come from a random portfolios analysis with 1000 random portfolios of the same Herfindahl index
as the analysed portfolio. Each box corresponds to the p-values
of the 12 sub-periods for one level of Herfindahl Index. In the
box, the red horizontal line represent the median p-value over
the 12 periods. The boundaries of the box represent the 25 and
75 quantiles. The whiskers represent the minimum and maximum
p-values. The map represents p-values for a given sub-period and
sub-strategies(the 12 sub-strategies correspond to the 12 different
levels of Herfindahl index). . . . . . . . . . . . . . . . . . . . . .
6.17 p-values analysis for MAXIMUM DRAWDOWN of VALUES based
strategies. Results come from a random portfolios analysis with
1000 random portfolios of the same Herfindahl index as the analysed portfolio. Each box corresponds to the p-values of the 12
sub-periods for one level of Herfindahl Index. In the box, the
red horizontal line represent the median p-value over the 12 periods. The boundaries of the box represent the 25 and 75 quantiles. The whiskers represent the minimum and maximum pvalues. The map represents p-values for a given sub-period and
sub-strategies(the 12 sub-strategies correspond to the 12 different
levels of Herfindahl index). . . . . . . . . . . . . . . . . . . . . .
6.18 p-values analysis for AVERAGE MAXIMUM DRAWDOWN of
VALUES based strategies. Results come from a random portfolios analysis with 1000 random portfolios of the same Herfindahl
index as the analysed portfolio. Each box corresponds to the
p-values of the 12 sub-periods for one level of Herfindahl Index.
In the box, the red horizontal line represent the median p-value
over the 12 periods. The boundaries of the box represent the
25 and 75 quantiles. The whiskers represent the minimum and
maximum p-values. The map represents p-values for a given subperiod and sub-strategies(the 12 sub-strategies correspond to the
12 different levels of Herfindahl index). . . . . . . . . . . . . . . .
6.19 Return of strategies based on MAXIMUM DRAWDOWN of RETURNS, against random portfolios of p-value of 0.1 . In Period
11, the market decreases and in period 12, it increases . . . . . .
6.20 Return of strategies based on 4th MOMENT, against random portfolios of p-value of 0.1 In Period 11, the market decreases and in
period 12, it increases . . . . . . . . . . . . . . . . . . . . . . . .
6.21 Return of strategies based on VARIANCE, against random portfolios of p-value of 0.1 In Period 11, the market decreases and in
period 12, it increases . . . . . . . . . . . . . . . . . . . . . . . .
6.22 Yearly SHARPE ratio analysis for VARIANCE based strategies.
Computed for 12 sub-strategies and 12 sub-periods. . . . . . . . .
6.23 Yearly SHARPE ratio analysis for µ4 based strategies. Computed
for 12 sub-strategies and 12 sub-periods. . . . . . . . . . . . . . .
71
46
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50
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6.24 Yearly SHARPE ratio analysis for MAXIMUM DRAWDOWN of
RETURNS based strategies. Computed for 12 sub-strategies and
12 sub-periods. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.25 Yearly SHARPE ratio analysis for MAXIMUM DRAWDOWN of
VALUES based strategies. Computed for 12 sub-strategies and
12 sub-periods. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.26 Yearly SHARPE ratio analysis for AVERAGE MAXIMUM DRAWDOWN of VALUES based strategies. Computed for 12 substrategies and 12 sub-periods. . . . . . . . . . . . . . . . . . . . .
6.27 Total transaction costs of the simulation for 4th MOMENT based
strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.28 Total transaction costs of the simulation for VARIANCE based
strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.29 Total transaction costs of the simulation for MAXIMUM DRAWDOWN of RETURNS based strategies . . . . . . . . . . . . . . .
6.30 Total transaction costs of the simulation for MAXIMUM DRAWDOWN of VALUES based strategies . . . . . . . . . . . . . . . .
6.31 Total transaction costs of the simulation for AVERAGE MAXIMUM DRAWDOWN of VALUES based strategies . . . . . . . .
6.32 How many times a strategy was the best of a period in the simulation for VARIANCE based strategies . . . . . . . . . . . . . .
6.33 How many times a strategy was the best of a period in the simulation for 4th MOMENT based strategies . . . . . . . . . . . . .
6.34 How many times a strategy was the best of a period in the simulation for MAXIMUM DRAWDOWN of RETURNS based strategies
6.35 How many times a strategy was the best of a period in the simulation for MAXIMUM DRAWDOWN of VALUES based strategies
6.36 How many times a strategy was the best of a period in the simulation for AVERAGE MAXIMUM DRAWDOWN of VALUES
based strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
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[9] M.M. Carhart. On persistence in mutual fund performance. The Journal
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74
Chapter 8
Annexes
8.1
Maximum DrawDown - Matlab Code
75
8.2
Average Maximum DrawDown - Matlab Code
76
Master Thesis -Software Documentation
Thibaut Simon
02/12/2009
Contents
1 Introduction
2
2 Optimization tool
2.1 How to use it? First use. . . . . . . . . . . . . . . . . .
2.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Input Data . . . . . . . . . . . . . . . . . . . .
2.2.2 Output Data . . . . . . . . . . . . . . . . . . .
2.3 Options . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Genetic Algorithm Options . . . . . . . . . . .
2.3.2 Duration of training data and test data for the
construction . . . . . . . . . . . . . . . . . . . .
2.3.3 Fitness function for the portfolio construction .
. . . . . .
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portfolio
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3 Analysis tool
3.1 User Interface . . . . . . . . . .
3.1.1 Aspect . . . . . . . . . .
3.1.2 Possibilities . . . . . . .
3.2 Data . . . . . . . . . . . . . . .
3.2.1 Input Data . . . . . . .
3.2.2 Output Data . . . . . .
3.3 What do you want to analyse?
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4 To go further
10
4.1 What could be done to improve these tools? . . . . . . . . . . . . 10
4.2 Tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.3 FAQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.3.1 How to change fitness functions? . . . . . . . . . . . . . . 11
4.3.2 How to add a functionality to the analysis or optimization
tool? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5 Annexes
13
5.1 Optimization tool figures . . . . . . . . . . . . . . . . . . . . . . 14
5.2 Analysis tool figures . . . . . . . . . . . . . . . . . . . . . . . . . 21
1
Chapter 1
Introduction
This work is part of my thesis on ”Portfolio construction : How to achieve
better strategy through the use of Genetic Algorithm and non-linear objective
function based on new measures of risks and dependencies?”. During my study
of this subject I needed to solve optimization problem to construct portfolio and
then analyse their performance. That’s why I did two tools, with user-friendly
interfaces, to help me in my work. These tools were at first really basic without
user-interface, but soon they became quite complex, due to the multiplication
of functions. Hence I decided to create a user interface that allows other people
than me, to use it easily in a future work. For this reason, I chose to do it in
MATLAB. I will start with the optimization tool which makes the input data
used by the analysis tool. Then, I will explain how to use the analysis tool to
exploit the result of the optimization tool. Finally, I will tale you what could
be done to enhance the tool and how to do it.
2
Chapter 2
Optimization tool
The optimization tool allows to create portfolio from different objective functions. These objective functions correspond to allocation strategies. The optimization is done by a genetic algorithm coming from a library of Matlab. You
can choose the precision of the genetic algorithm by setting up different options.
You can as well choose the length of the data you use to create your portfolio
and the duration you keep your portfolio before doing another one. The optimiser use a file containing as data: the date of trading, the adjusted price of the
stocks at the closure (see the section data below). This tool have the advantage
to create portfolios from 12 different strategies over the length of data you want.
Then you will be able to analyse these portfolio with the analysis tool.
The data I will use to present this tool are : 28 stocks coming from the
french index CAC40 over about 1500 days from 2003.
2.1
How to use it? First use.
You will need to: first, launch Matlab, second, set the current directory to
the ”optimization directory”, then just write ”optimization” in the command
window (see figure 5.1). You can observe the window represented in figure 5.2.
If you want to create your first portfolio, just check the fitness you want
to use and click on the button ”launch computation”. The program will run
the optimization with the default value for the time-window and the genetic
algorithm. You can follow the evolution of the process in the command window
where you will see the period and the fitness currently being calculated. Result
of these optimizations are saved into the Matlab file ”portfolios.mat”. This file
is the file that will be required by the analysis tool. In order to understand how
to choose the different parameters, I invite you to read the following parts.
3
2.2
Data
The aim of this section is to give you the ability to put the data you want into
the application. In order to be read by the program and produce the expected
result, data should respect a format for the input data. The paragraph on the
output data is done to understand what is stored in ”portfolios.mat” file.
2.2.1
Input Data
The input data come originally from .cls file found in yahoo finance (see figure
5.5). This yahoo file contains a lot of field. For our purpose we need to keep only
the adjusted close price and the date of each price (see figure 5.6). At the end all
should be gathered in a unique file that will be import into Matlab and saved as
”data.mat”. The file ”data.mat” is a structure containing the numerical array
”data” and the text array ”textdata”. These two array are necessary for the
good execution of the program. Figure 5.3 you can observe the numerical array
”data” and figure 5.4 the text array ”textdata”. In this file, data should be
sorted from the most recent to oldest, like in the file provided by Yahoo finance.
Each column corresponds to a stock. The first column correspond to the date
of each price in text.The two last columns contains the reference index (here
the CAC40) and the risk free rate (here the 13 weeks treasury bills). Look at
the screen-shots below to get a better view of these file and to understand the
procedure to follow in order to use your own data.
2.2.2
Output Data
The tool optimization produce the file ”portfolios.mat” which is a Matlab
data file containing the number of periods, the name of the fitness function, and
the portfolios structure. The portfolios structure store the portfolio weight, the
training data of the period, the test data to evaluate the portfolio over the next
days. And other data useful to analyse the portfolio as the reference index data
and risk free rate data over the period of test. You can see the details on the
figure 5.11. This file can be read next by the analysis tool to plot the interesting
figures and compare them.
2.3
2.3.1
Options
Genetic Algorithm Options
This program use the Genetic Algorithm of Matlab. In the interface you
can change three options that allow you to make more precise or quicker the
optimisation. Choices available here are the size of the population, the crossover rate and the stall limit generation. However, it is possible to change other
options more fundamental by entering into the code of opt portfolio3.m and
changing gaoptimset settings. Look in the help of Matlab for more details
about the GA toolbox. Here is the list of options I used for the GA.
4
Population Options:
Population type , the individuals in the population are represented with
the data type double.
Population size specifies how many individuals there are in each generation. With a large population size, the genetic algorithm searches
the solution space more thoroughly, thereby reducing the chance that
the algorithm will return a local minimum that is not a global minimum. Here you can choose between 100 and 1000 with the slider
button.
Initial population specifies an initial population for the genetic algorithm. It is set as random.
Selection function: the Stochastic uniform selection function lays out a line
in which each parent corresponds to a section of the line of length proportional to its scaled value. The algorithm moves along the line in steps of
equal size. At each step, the algorithm allocates a parent from the section
it lands on. The first step is a uniform random number less than the step
size.
Reproduction:
Crossover fraction specifies the fraction of the next generation, other
than elite children, that are produced by crossover.
Elite count specifies the number of individuals that are guaranteed to
survive to the next generation.
Mutation Options: the Gaussian mutation function adds a random number
taken from a Gaussian distribution with mean 0 to each entry of the parent
vector.
Crossover Options: The scattered crossover function creates a random binary
vector and selects the genes where the vector is a 1 from the first parent,
and the genes where the vector is a 0 from the second parent, and combines
the genes to form the child.
Stopping Criteria Options: The algorithm runs until the cumulative change
in the fitness function value over Stall generations is less than or equal to
Function Tolerance. Here the number of generation is limited to 200 and
stall limit generation can be chosen between 10 and 50 with the slider
button.
5
2.3.2
Duration of training data and test data for the portfolio construction
In order to study the influence of time, it is interesting to variate the amount
of data you use to create your portfolio and the time you keep it. This is possible
to do with the two sliders button of the panel window parameters. If you increase
the holding duration of your portfolio, you decrease the final number of periods,
hence you decrease the number of portfolios to calculate.
2.3.3
Fitness function for the portfolio construction
The fitness function can be choose by checking the name in the user interface.
If you need to create or study new fitness function, you will have to enter a
little bit in the code. The fitness function is the mathematical expression of
your strategy of risks mitigation. In order to implement your own innovative
strategy, you will need to create your own fitness function.The fitness function is
then used by the genetic algorithm to evaluate each portfolio. Fitness functions
are stored into the files fitness1.m, fitness2.m, ..., fitness12.m. Look at the last
chapter for more details.
6
Chapter 3
Analysis tool
The analysis tool has been developed to analyse portfolios coming from the
tool optimization. It allow you to get a deep understanding of the performance
of each portfolio against the others. As you will see in this section, this interface
allow you to analyse the portfolios under different perspectives.
3.1
User Interface
The analysis tool use a Matlab user interface that you can launch by entering analyse ui in the command window, once you have set the correct current
directory.
3.1.1
Aspect
Figure 5.12, you can observe a screen-shot of the application. On the left,
you choose what you want to observe and on the right, graphics are plotted.
3.1.2
Possibilities
Choices available for analysis are :
• Specific Fitness function
• Particular Period
• Global evolution of portfolios
• Which strategy for which period?
• What about transaction cost?
• Efficient frontier evolution
• Synthesis
7
A click on the button of your choice will plot the corresponding graphics into
the right panel. If you look on the bottom left panel,you will see a command
panel where you can change parameters of the graphics and data plotted. I
advise you to try the different possibilities which have all been selected for their
pertinence in portfolio analysis.
3.2
3.2.1
Data
Input Data
The tool optimization produce the file ”portfolios.mat” which is a Matlab
data file containing the number of periods, the name of the fitness function,
and the portfolios structure. The portfolios structure store the portfolio weight,
the training data of the period, the test data to evaluate the portfolio over the
next days. And other data useful to analyse the portfolio as the reference index
data and risk free rate data over the period of test. You can see the details on
the figure 5.11. This file is the input data used by the analysis tool. Note that
fitness index and periods index in the portfolios structure should go from one
to n without discontinuities in order to works.
3.2.2
Output Data
No output data for the moment. Export to excel and image of graphics could
be produce in further development.
3.3
What do you want to analyse?
Specific Fitness function Study a specific fitness function. A fitness function
correspond to an investment strategy. Each strategy lead to different
portfolio. You can choose witch function you want to analyse in the popup menu. Observe the evolution of the composition at each period. Look
at the evolution in value of the strategy. You can also look at the return
on each period to see when the portfolio perform the best.(see figure 5.12)
Particular Period Study all the different investment strategy for a selected
period. You can select the period you want to analyse in the pop-up
menu. For each period you will be able to compare the composition of the
portfolios, the evolution of the value during the period and the returns
over the period.(see figure 5.13)
Global evolution of portfolios Plot the evolution of portfolios along all the
periods. Thus you can have a global view of the performance. The starting
value of portfolios is 100.(see figure 5.14)
8
Which strategy for which period? You would like to know which strategy
perform the best, or the most often. Trough the different plots, look at
the distribution of best strategies, or the repartition along periods of the
best strategy. Portfolios are ranked by rate of returns over the periods.
Moreover you can check the composition of best portfolio to see if they
are diversified or not.(see figure 5.15)
What about transaction cost? It is interesting to have a look at the transaction costs. They are calculated in absolute value. They correspond to
the change in value of each asset to get the new portfolio. Compare the
cumulative costs of each strategy. Look which periods incur costs.(see
figure 5.16)
Efficient frontier evolution This section show the evolution of the efficient
frontier over time. It is done to show how big are the change of the markets.
The efficient frontier is calculated from Markovitz portfolio theory with
the help of a quadratic solver. Often change are important and put some
discredit on the pertinence of the Markovitz theory.(see figure 5.17)
Synthesis Finally you need some figures to figure out how diversified is the
portfolio with the Herfindahl index, what is the end value, the end value
tacking into account the transaction costs or the overall returns. Transaction costs depend of who you are, hence you can choose the level of
transaction costs you want to apply in the pop-up menu.(see figure 5.18)
9
Chapter 4
To go further
This program has been developed in a limited amount of time. A lot can
be done to simplify its use. I will propose some idea of what can be done to
improve the utility and the easiness of use. This program has been done in the
aim of constructing portfolios and analysing their performance and behaviour
for my master thesis. However I decided as it was really powerful for me, to
make it accessible and available for further use. It explains at some points, why
it is not always as functional as it should be or as I would like it to be.
4.1
What could be done to improve these tools?
Here are some idea to improve the functionality of this application. I will try
to give some hints to develop these ideas.
Compare strategy with different duration
Plot the evolution of the market index
Plot the Sharpe ratio
Export graphics to excel
Improve the process of data collection which is not automatic
Add a system to modify the fitness function easily
4.2
Tips
Tip 1: The user interface as been done using MATLAB GUIDE, which allows
to create nice interface. To launch this tool, write ”guide” in the command
window and load the optimization file ”optimization.fig”
10
Tip 2: The optimisation tool uses the Genetic Algorithm toolbox of Matlab.
In order to get a wider view on the possibilities of GA in optimisation, you
should go looking in the help of the GA toolbox.
Tip 3: The analysis tool call 7 functions called analyse1.m, analyse2.m ....
Each of these functions correspond to one choice of analysis user interface. If
you want to modify one of these functions, you will see, that they are built in
the same way. First, extract the interesting data from the structures portfolios.
Second, make the calculation needed. Finally plot the interesting variables into
the interface with the use of handles.
4.3
4.3.1
FAQ
How to change fitness functions?
The fitness function is the mathematical expression of your strategy of risks
mitigation. In order to implement your own innovative strategy, you will need
to create your own fitness function.The fitness function is then used by the
genetic algorithm to evaluate each portfolio. Fitness functions are stored into
the files fitness1.m, fitness2.m, ..., fitness12.m. In order to create your own
fitness function, the easiest way, is to modify an existing one. Input variables of
fitness functions are the portfolio weights w and the data of returns returnsdata.
You can basically evaluate what you want without worrying about linearity
and so one. Each fitness function start by the command normalize(w) which
allows to keep the condition sum of weight equal to one. Then add z= .. what
you want to minimize. Names of fitness functions are stored into the code file
optimization.m in the variable fit name choice that you can change to put your
own name. See figure 4.1 for an example of fitness function.
4.3.2
How to add a functionality to the analysis or optimization tool?
These two applications can be modified. For example, it is possible to add
a functionality to the Analysis tool. It will require to modify the interface file
analysis-ui.fig and the m-file analysis-ui.m. In the interface file, you need for
example to add a button for your functionality. This new button will have
to be programmed in the m-file. A callback function is used to execute an
action following a click on the new button. I advice you to create a new m-file
analysis8.m and to make your function of analysis inside, on the model of the
other analysis1.m ... files. See tip 3 for more information.
11
Figure 4.1: Fitness function code in Matlab - minimization of the single action
maxdrawdown for a specific level of the Herfindahl index
12
Chapter 5
Annexes
The figures referred before are grouped in this section. The optimization tool
screen-shots represent the process of importing new data for the tool. Figures
of the analysis section represent the different possibilities of analysis.
13
List of Figures
4.1
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.1
Fitness function code in Matlab - minimization of the single action maxdrawdown for a specific level of the Herfindahl index . .
Command window . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimization interface . . . . . . . . . . . . . . . . . . . . . . . .
Numerical array ”data” in Matlab . . . . . . . . . . . . . . . . .
Text array ”textdata” in Matlab . . . . . . . . . . . . . . . . . .
Data file downloaded in yahoo finance . . . . . . . . . . . . . . .
Yahoo file treated by a VBA macro to keep only dates and prices
The file which should be imported to Matlab . . . . . . . . . . .
Import window 1 in Matlab . . . . . . . . . . . . . . . . . . . . .
Import window 2 in Matlab . . . . . . . . . . . . . . . . . . . . .
Save window in matlab to produce ”data.mat” which is needed
by the program . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Details of data contained in the portfolios structure . . . . . . . .
Study of a specific fitness function . . . . . . . . . . . . . . . . .
Comparison of the different strategies for a specific period . . . .
Overview of the evolution of the strategies . . . . . . . . . . . . .
Which strategy perform the best? . . . . . . . . . . . . . . . . . .
Study of the transaction cost . . . . . . . . . . . . . . . . . . . .
Plot the evolution of efficient frontier . . . . . . . . . . . . . . . .
Synthesis of the performance of the different strategies . . . . . .
Optimization tool figures
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Figure 5.1: Command window
Figure 5.2: Optimization interface
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Figure 5.3: Numerical array ”data” in Matlab
Figure 5.4: Text array ”textdata” in Matlab
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Figure 5.5: Data file downloaded in yahoo finance
Figure 5.6: Yahoo file treated by a VBA macro to keep only dates and prices
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Figure 5.7: The file which should be imported to Matlab
Figure 5.8: Import window 1 in Matlab
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Figure 5.9: Import window 2 in Matlab
Figure 5.10: Save window in matlab to produce ”data.mat” which is needed by
the program
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Figure 5.11: Details of data contained in the portfolios structure
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5.2
Analysis tool figures
Figure 5.12: Study of a specific fitness function
Figure 5.13: Comparison of the different strategies for a specific period
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Figure 5.14: Overview of the evolution of the strategies
Figure 5.15: Which strategy perform the best?
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Figure 5.16: Study of the transaction cost
Figure 5.17: Plot the evolution of efficient frontier
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Figure 5.18: Synthesis of the performance of the different strategies
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