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UNIVERSITY OF SÃO PAULO

SCHOOL OF ECONOMICS, BUSINESS ADMINISTRATION AND ACCOUNTING DEPARTAMENT OF BUSINESS ADMINISTRATION

GRADUATE PROGRAM IN BUSINESS ADMINISTRATION

DISCRETE TIME

PORTFOLIO ANALYSIS

Fernando Hideki Kato

fkato@sti.com.br

Thesis presented to the Business Administration

Department of the School of Economics,

Business Administration and Accounting of

University of São Paulo as a partial requisite for

the degree of Master of Science.

Area of concentration:

Quantitative Methods and Informatics

Advisor:

Prof. Dr. José de Oliveira Siqueira

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© 2004 Fernando Hideki Kato

Original title: Análise de Carteiras em Tempo Discreto

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ABSTRACT

In this thesis, Markowitz’s portfolio selection model will be extended by means of a discrete time analysis and more realistic hypotheses.

A finite tensor product of Erlang densities will be used to approximate the multivariate probability density function of the single-period discrete returns of dependent assets. The Erlang is a particular case of the Gamma distribution. A finite mixture can generate multimodal asymmetric densities and the tensor product generalizes this concept to higher dimensions. Assuming that the multivariate density was independent and identically distributed (i.i.d.) in the past, the approximation can be calibrated with historical data using the maximum likelihood criterion. This is a large-scale optimization problem, but with a special structure.

Assuming that this multivariate density will be i.i.d. in the future, then the density of the discrete returns of a portfolio of assets with nonnegative weights will be a finite mixture of Erlang densities. The risk will be calculated with the Downside Risk measure, which is convex for certain parameters, is not based on quantiles, does not cause risk underestimation and makes the single and multiperiod optimization problems convex.

The discrete return is a multiplicative random variable along the time. The multiperiod distribution of the discrete returns of a sequence of T portfolios will be a finite mixture of Meijer G distributions. After a change of the distribution to the average compound, it is possible to calculate the risk and the return, which will lead to the multiperiod efficient frontier, where each point represents one or more ordered sequences of

T portfolios. The portfolios of each sequence must be calculated from the future to the present, keeping the expected return at the desired level, which can be a function of time. A dynamic asset allocation strategy is to redo the calculations at each period, using new available information.

If the time horizon tends to infinite, then the efficient frontier, in the average compound probability measure, will tend to only one point, given by the Kelly’s portfolio, whatever the risk measure is.

To select one among several portfolio optimization models, it is necessary to compare their relative performances. The efficient frontier of each model must be plotted in its respective graph. As the weights of the assets of the portfolios on these curves are known, it is possible to plot all curves in the same graph. For a given expected return, the efficient portfolios of the models can be calculated, and the realized returns and their differences along a backtest can be compared.

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RESUMO

Nesta dissertação, o modelo de seleção de carteiras de Markowitz será estendido com uma análise em tempo discreto e hipóteses mais realísticas.

Um produto tensorial finito de densidades Erlang será usado para aproximar a densidade de probabilidade multivariada dos retornos discretos uniperiódicos de ativos dependentes. A Erlang é um caso particular da distribuição Gama. Uma mistura finita pode gerar densidades multimodais não-simétricas e o produto tensorial generaliza este conceito para dimensões maiores. Assumindo que a densidade multivariada foi independente e identicamente distribuída (i.i.d.) no passado, a aproximação pode ser calibrada com dados históricos usando o critério da máxima verossimilhança. Este é um problema de otimização em larga escala, mas com uma estrutura especial.

Assumindo que esta densidade multivariada será i.i.d. no futuro, então a densidade dos retornos discretos de uma carteira de ativos com pesos não-negativos será uma mistura finita de densidades Erlang. O risco será calculado com a medida Downside Risk, que é convexa para determinados parâmetros, não é baseada em quantis, não causa a subestimação do risco e torna os problemas de otimização uni e multiperiódico convexos.

O retorno discreto é uma variável aleatória multiplicativa ao longo do tempo. A distribuição multiperiódica dos retornos discretos de uma seqüência de T carteiras será uma mistura finita de distribuições Meijer G. Após uma mudança na medida de probabilidade para a composta média, é possível calcular o risco e o retorno, que levará à fronteira eficiente multiperiódica, na qual cada ponto representa uma ou mais seqüências ordenadas de T carteiras. As carteiras de cada seqüência devem ser calculadas do futuro para o presente, mantendo o retorno esperado no nível desejado, o qual pode ser função do tempo. Uma estratégia de alocação dinâmica de ativos é refazer os cálculos a cada período, usando as novas informações disponíveis.

Se o horizonte de tempo tender a infinito, então a fronteira eficiente, na medida de probabilidade composta média, tenderá a um único ponto, dado pela carteira de Kelly, qualquer que seja a medida de risco.

Para selecionar um dentre vários modelos de otimização de carteira, é necessário comparar seus desempenhos relativos. A fronteira eficiente de cada modelo deve ser traçada em seu respectivo gráfico. Como os pesos dos ativos das carteiras sobre estas curvas são conhecidos, é possível traçar todas as curvas em um mesmo gráfico. Para um dado retorno esperado, as carteiras eficientes dos modelos podem ser calculadas, e os retornos realizados e suas diferenças ao longo de um backtest podem ser comparados.

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This work is dedicated to my parents, Osamu and Alice, who

always strived to give me the best education possible.

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THESIS COMMITEE

Prof. Dr. José de Oliveira Siqueira

Universidade de São Paulo

Faculdade de Economia, Administração e Contabilidade

Departamento de Administração

Prof. Dr. Adolpho Walter Pimazoni Canton

Universidade de São Paulo

Faculdade de Economia, Administração e Contabilidade

Departamento de Administração

Prof. Dr. Vladimir Belitsky

Universidade de São Paulo

Instituto de Matemática e Estatística

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AKNOWLEDGMENTS

Thanks

To my advisor, Prof. Dr. José de Oliveira Siqueira, for the patience and disposition to understand this work.

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CONTENTS

1. INTRODUCTION __________________________________________________________________ 1

1.1. MOTIVATION AND IMPORTANCE OF THE SUBJECT... 1

1.2. OBJECTIVES... 1

1.3. RESEARCH QUESTION... 2

1.4. METHODOLOGY AND ADOPTED HYPOTHESES... 2

1.5. ORGANIZATION OF THE THESIS... 3

1.6. ABBREVIATIONS... 4

1.7. GLOSSARY... 4

1.8. MATHEMATICAL NOTATION... 5

2. LITERATURE REVIEW _____________________________________________________________ 7 2.1. INTRODUCTION... 7

2.2. RISK... 7

2.3. SINGLE-PERIOD CAPITAL ALLOCATION... 8

2.4. RISK MEASURES... 9

3. DISTRIBUTIONS, TRANSFORMS AND SPECIAL FUNCTIONS __________________________ 11 3.1. INTRODUCTION... 11

3.2. RELATED FUNCTIONS... 12

3.2.1. Particular cases ... 14

3.2.2. Properties... 14

3.3. THE ERLANG DISTRIBUTION AND ITS RELATED DISTRIBUTIONS... 15

3.3.1. Properties... 15

3.4. MIXTURE AND TENSOR PRODUCT... 16

3.5. CALIBRATION OF MIXTURE AND TENSOR PRODUCT... 18

3.6. NUMBER OF SAMPLES AND SAMPLING ERROR... 26

3.7. THE LAPLACE AND MELLIN TRANSFORMS... 26

3.7.1. Particular cases ... 28

3.7.2. Properties of the Mellin transform... 29

3.8. THE MEIJER G AND FOX H FUNCTIONS... 30

3.8.1. Auxiliary notations to the Meijer G and Fox H functions ... 33

3.8.2. Particular cases ... 33

3.8.3. Properties, integrals and transforms ... 34

3.8.4. Meijer G and Fox H distributions... 36

3.9. NUMERICAL EVALUATION OF THE MEIJER G FUNCTION... 36

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3.11. POSITIVE LINEAR COMBINATION... 45

3.12. MULTIPLICATIVE CONVOLUTION... 48

3.13. POWER... 48

3.14. EXPECTED VALUE... 50

3.15. DOWNSIDE RISK... 52

3.16. AUXILIARY LEMMAS... 56

3.17. CONCLUSIONS... 58

4. RISK ____________________________________________________________________________ 60 4.1. INTRODUCTION... 60

4.2. PROBLEMS OF RISK MEASURES WITH GENERAL DISTRIBUTIONS... 60

4.3. COHERENT MEASURES OF RISK... 62

4.4. DOWNSIDE RISK... 63

4.4.1. Convexity ... 65

4.5. DEFAULT... 67

4.6. CONCLUSIONS... 69

5. SINGLE-PERIOD PORTFOLIO OPTIMIZATION _______________________________________ 70 5.1. INTRODUCTION... 70

5.2. ADDITIVE CONVOLUTION... 70

5.3. POSITIVE LINEAR COMBINATION... 71

5.4. EFFICIENT FRONTIER... 72

5.4.1. Multivariate distribution of the returns of the assets ... 73

5.4.2. Distribution of the returns of the portfolio ... 74

5.4.3. Expected return... 75

5.4.4. Downside Risk ... 75

5.4.5. Statement of the optimization problem ... 76

5.5. CONCLUSIONS... 77

6. MULTIPERIOD PORTFOLIO OPTIMIZATION_________________________________________ 78 6.1. INTRODUCTION... 78

6.2. WORLDS OF DISCRETE AND CONTINUOUS RETURNS... 79

6.3. MULTIPLICATIVE RANDOM VARIABLE... 81

6.4. MULTIPERIOD DISTRIBUTION OF THE PORTFOLIO RETURNS... 82

6.5. MULTIPLICATIVE CONVOLUTION... 83

6.6. EXPECTATION AND VARIANCE OF MULTIPLICATIVE RANDOM VARIABLES AS A FUNCTION OF TIME... 83

6.7. AVERAGE COMPOUND DISTRIBUTION... 85

6.7.1. Power utility function ... 90

6.8. AVERAGE COMPOUND DISTRIBUTION WITH LONGEVITY... 91

6.8.1. Concavity of the expected value... 93

6.8.2. Convexity of the Downside Risk... 95

6.9. REBALANCING CALCULATED BY DYNAMIC PROGRAMMING... 95

6.10. EFFICIENT FRONTIER... 96

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6.10.2. Distribution of the returns of the portfolio ... 97

6.10.3. Average compound distribution ... 98

6.10.4. Expected value ... 99

6.10.5. Downside Risk ... 99

6.10.6. Statement of the optimization problem ... 99

6.11. A DYNAMIC STRATEGY... 101

6.12. CONCLUSIONS... 101

7. INFINITE-PERIOD PORTFOLIO OPTIMIZATION _____________________________________ 103 7.1. INTRODUCTION... 103

7.2. LIMIT OF THE AVERAGE COMPOUND DISTRIBUTION... 103

7.2.1. Logarithmic utility function ... 106

7.3. KELLY’S CRITERION... 106

7.4. EFFICIENT FRONTIER... 110

7.5. DERIVATIVES OF KELLY’S CRITERION... 111

7.6. CONCLUSIONS... 112

8. SELECTION OF A PORTFOLIO OPTIMIZATION MODEL______________________________ 114 8.1. INTRODUCTION... 114

8.2. MOTIVATION... 115

8.3. METHODOLOGY... 115

8.4. CONCLUSIONS... 117

9. CONCLUSIONS _________________________________________________________________ 119 9.1. CONCLUSIONS... 119

9.2. ORIGINAL CONTRIBUTIONS... 121

9.3. LIMITATIONS AND SUGGESTIONS FOR FUTURE RESEARCH... 122

APPENDIX A – MATHEMATICA NOTEBOOKS __________________________________________ 124 EXAMPLE 3.5.1 ... 124

EXAMPLE 3.5.2 ... 125

EXAMPLE 3.9.1 ... 127

EXAMPLE 3.10.1 ... 128

EXAMPLE 3.11.1 ... 129

EXAMPLE 6.7.1 ... 130

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1. INTRODUCTION

1.1. Motivation and importance of the subject

This thesis was motivated by the existence of some criticisms in the theory of capital allocation, also called portfolio selection or portfolio optimization.

Although the large number of historical asset quotes enables a good approximation of the multivariate probability density function of the returns, in Markowitz’s (1952) portfolio selection, only the means, variances and covariances of the rates of return of the assets are used. The use of the variance-covariance matrix assumes a multivariate normal (or, more generically, an elliptical distribution) for the rates of return. But the normal distribution allows rates of return smaller than 100%, which is not possible for asset quotes.

The definition of risk as standard deviation penalizes assets with long tails to the right. Markowitz (1991a, p.194) recognized this problem and proposed the semivariance as an alternative and preferable risk measure (Markowitz, 1991a, p.374).

Considering that the time period is one day, then investors usually have time horizon greater than one period. But the existing multiperiod capital allocation models are based on utility function, use the standard deviation as risk measure or assume that the time horizon is infinite. These are assumptions that simplify the calculation. Finance textbooks do not even mention that the discrete return is a multiplicative variable along the time, a fact of utmost importance in the multiperiod analysis.

Having available several capital allocation models, an investor must select one of them before investing his money. It is necessary to define a criterion for the selection of one among several models.

These are important problems, easily recognized by anyone in the Finance area. The desire to obtain a mathematically more exact and computationally feasible model motivated the author’s research on this subject and resulted in this work.

1.2. Objectives

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The secondary objectives are: i) to extend the single-period analysis to the multiperiod and infinite-period cases; and ii) to propose a procedure to compare the relative performances among portfolio selection models.

1.3. Research question

During its conception, this thesis generated the following research questions.

1. How to obtain an approximation of the multivariate density of dependent random variables? 2. How to calculate the density of the discrete returns of a portfolio from the multivariate density

of the discrete returns of its assets?

3. How to define the risk measure to take advantage of the density of the discrete returns of the portfolio?

4. How to calculate the density of the discrete returns of a sequence of portfolios from their single-period densities?

5. Which criterion to use to compare the performances of the proposed models and of the existing ones?

The statement of the research question joins these questions.

œ How to extend Markowitz’s portfolio selection model with the hypotheses that the discrete returns of dependent assets must be nonnegative, the risk must be calculated over the undesirable returns and the investment horizon must be greater than one period; and how to compare the performances of portfolio selection models?

1.4. Methodology and adopted hypotheses

This thesis is a proposition of new capital allocation models and a backtest procedure to compare them to the existing ones. It is worth to remind that new models can never be considered totally new because they are always an evolution of prior models. Model creation is considered an exploratory qualitative research that, at first, does not require numerical data gathering. All theoretical chapters of this thesis are part of the methodology.

In this thesis, the problem of portfolio optimization will be studied with the hypothesis that the time is discrete. The assumptions or simplifying hypotheses assumed in this work are given in the following.

œ The investor is rational (he prefers more return and less risk). œ There is no transaction costs.

œ The weights of the assets of the portfolios are nonnegative reals.

œ There is zero probability of the return of an asset being smaller or equal to zero (that is, of the rate of return being smaller or equal to 100% ).

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œ The time is discrete (in some parts of this work, the time will be considered continuous). œ The distributions are independent and identically distributed (i.i.d.) along the time.

The independent variables are the prices of the assets quoted in stock exchanges. From them, the returns are obtained. The investor’s time horizon and his desired level of expected return or risk are also independent variables. The dependent variables are the weights of the assets in the infinite portfolios that can be formed. The most important portfolios are the ones on the efficient frontier and, in particular, the one with the desired level of expected return or risk.

1.5. Organization of the thesis

As it will be seen along the chapters, the analysis in discrete time allows a clear separation of the steps involved in the portfolio optimization.

In Chapter 2, a literature review about the existing models of portfolio optimization and some related topics will be done.

The mathematical part of this thesis is concentrated in Chapter 3. This will avoid the interruption of the financial reasoning in the subsequent chapters. The Erlang distribution, its related functions and distributions, and the concepts of mixture and tensor product will be presented. The maximum likelihood criterion will be used in the calibration of the parameters of the multivariate density represented by a finite tensor product of Erlangs. The curse of dimensionality will appear in this optimization problem. The Meijer G and Fox H special functions will be presented, as well as a formula for the numerical evaluation of a particular case. The additive and multiplicative convolutions will be presented. Some properties involving the Erlang and Meijer G densities will be demonstrated. The Downside Risk will be calculated for some distributions. An interesting result that will be shown is the behavior of the variance and the standard deviation of a multiplicative random variable as a function of continuous time.

Some risk measures will be shown in Chapter 4. The coherent properties of the risk measures will be discussed. The measure chosen for this thesis is the Downside Risk. Although it does not satisfy all “coherent” properties, it does not cause the problem of either under or overestimation of the “undesirable” or “bad” returns, and it makes convex the optimization problems involving the construction of the efficient frontier. The risk of default due to levered portfolios will be discussed.

The single-period portfolio optimization will be treated in Chapter 5. The discrete return is an additive variable at each instant of time. If the multivariate density of the returns of the assets is a finite tensor product of Erlangs, then the density of the returns of a portfolio with positive weights will be a finite mixture of Erlangs. The Downside Risk of these portfolios will be calculated.

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expected return and the Downside Risk will be calculated. The allocation strategy for the desired time horizon and a dynamic strategy will be discussed.

In Chapter 7, it will be assumed that the investor’s time horizon tends to infinite. It will be shown that the Kelly’s criterion can be obtained with help of the world of continuous returns or with properties of the Mellin transform. When the time horizon tends to infinite, the average compound distribution of constant-weight portfolios will tend to a Dirac Delta. The efficient frontier, if it exists, will be only one point, and the risk measure will become irrelevant for a financial decision.

In Chapter 8, a criterion will be proposed to help a rational investor to select one among several portfolio optimization models. The efficient frontiers of the models must be plotted in the same graph, a level of return must be fixed and a backtest must be done. The temporal series of the realized returns and their differences will be used for the comparison of the relative performances among the models.

In Chapter 9, the conclusions, original contributions, limitations and suggestions for future works will be presented.

The software Mathematica® version 5.0, from Wolfram Research, was used in this work.

1.6. Abbreviations

i.i.d. independent and identically distributed apud in (indirect citation)

et al. and others

s.t. subject to (the following constraints)

p. page

N. number

V. volume

CRRA Constant Relative Risk Aversion HARA Hyperbolic Absolute Risk Aversion CAPM Capital Asset Pricing Model VaR Value-at-Risk

CVaR Conditional Value-at-Risk

1.7. Glossary

Return Price at period t over the price at period t1. Rate of return Variation of price over the original price. Discrete return Return over a discrete period.

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World of discrete returns or multiplicative world

World where the returns are capitalized over discrete periods.

World of continuous returns or additive world

World where the continuous returns at different instants can be added.

Additive variable Variable that has the sum operation well defined. Multiplicative variable Variable that has the product operation well defined.

Additive convolution Mathematical operation used to obtain the density of the sum of two independent variables.

Multiplicative or scale convolution

Mathematical operation used to obtain the density of the product of two independent variables.

Risk measure Function that measures the undesired or risky events of a variable. Downside Risk An example of risk measure.

1.8. Mathematical notation

ˆ Set of the complex numbers. † Set of the real numbers.

\

x x 0

^

‰

† † Set of the positive real numbers.

\x x 0^

✂ ‰ p

† † Set of the nonnegative real numbers.

\..., 2, 1,0,1,2,...^

‡ Set of the integer numbers.

\1, 2,...^

Š Set of the positive integer numbers.

\0,1, 2,...^

Š Set of the nonnegative integer numbers.

ˆ†

D A subset of the real numbers.

t

P Price of an asset or portfolio at time t.

1

t t

t

P R

P

Return of an asset or portfolio at time t.

1

t t

r R Rate of return of an asset or portfolio at time t. 1

f f

R r Return of the risk-free asset (with null variance).

f

r Rate of return of the risk-free asset (with null variance).

i

w Weight of asset i of the portfolio.

t

W Wealth of the investor at time t.

X Random variable.

f x

:

X Random variable X that follows the density f x .

t

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...

t ¸ ¸ ¸1444442444443

t

X X X X

Product of t independent copies of the random variable X .

< >

E X E f x Expected value of the random variable X that follows the density

f x .

< >

V X V f x Variance of the random variable X that follows the density f x .

< >X f x

R R Risk of the random variable X that follows the density f x .

<

>

E U X Expected value of the utility function of the random variable X .

<X1,X2,...,XD>

Transposed vector with D random variables.

ln

R% R Continuous return in the world of continuous returns is the natural logarithm of the discrete return.

R

R

f R e f e

✞ ✞

% % Density of the continuous return.

1 2

f € f x Additive convolution. Density of the sum of the independent random variables X1 and X2.

1 2

f  f x Multiplicative or scale convolution. Density of the product of the independent random variables X1 and X2.

^n

f x

Power. Density of the n-th power of the random variable X .

...

n

n

f x f f x

€ €

1444442444443 Additive convolution of n equal densities.

...

n

n

f x f f x

 

1444442444443 Multiplicative or scale convolution of n equal densities.

\

;

^

f x ¶¶lL F s L f x s The Laplace transform of f x is F s .

\

;

^

f x ¶¶¶MlF s M f x s The Mellin transform of f x is F s .

( )

n n

n

f x

f x

x

s

s n-th derivative of f x .

Re z Real part of z.

1

i Square root of 1 .

Œ . Sufficient conditions for the validity of an expression. W End of proof (Halmos square).

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2. LITERATURE REVIEW

2.1. Introduction

The number of papers and books with topics related to the portfolio selection or optimization is too big for a wide literature review. In this chapter, only a few of the most important topics will be mentioned. For a good historical survey, see Bernstein (1996).

2.2. Risk

Risk is a concept with several definitions. According to a paper of Borch (1969) about utility, the “Risk Theory” had its origin with the work of Tetens (1789), who studied orderings depending on the mean and the mean deviation of distribution functions.

Fischer (1906, p.409, apud Grootveld and Hallerbach, 1999) is an author who considered “the chance of earnings falling below the interest-paying line”.

According to Levy and Sarnat (1984, p.236), numerous economists have identified investment risk with the dispersion of returns. Keynes, for example, identified the risk involved in an investment with the possible deviations from the average return. According to Keynes (1937, apud Levy and Sarnat, 1984), an individual who invests in an asset whose returns have a widely dispersed distribution must be given a premium to compensate him for the risk taken.

Like Keynes, Hicks (1946, apud Levy and Sarnat, 1984) also identified the variance of returns with risk. Hicks emphasized the fact that the greater the dispersion of returns (for a given level of expectation), the less attractive is the investment. He also emphasized that when returns are uncertain, the third moment of the distributions, the index of asymmetry or the skewness, may also be a significant factor affecting investors’ decisions.

Although Marschak (1938, apud Levy and Sarnat, 1984) had said that decision making under conditions of risk should reflect all moments of the distribution, he also noticed that in many cases, two moments, “the mathematical expectation and the coefficient of variation”, would suffice. In other words, Marschak, too, identified (in some cases) investment risk with the variance, or rather, the coefficient of variation (standard deviation divided by the mean).

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Therefore, they suggested that investors should measure risk solely on the basis of that possibility: “Of all possible questions which the investor may ask, the most important one, it appears to us, is concerned with the probability of actual yield being less than zero, that is, with the probability of loss. This is the essence of risk.”. These two authors developed a quantitative index of risk affected both by the probability of getting a result less than zero, and by the size of the possible loss. Thus, they emphasized the negative segment of the probability distribution, and according to their model, a larger dispersion per se does not necessarily involve a greater risk.

Baumol (1963, apud Levy and Sarnat, 1984) is another author who argued that variance per se does not indicate risk. According to this author, risk mainly reflects the possibility that the random variable may have extremely low values. If the expected return from an investment is high relative to its standard deviation, Baumol suggested that the spread between the expected return and k times the standard deviation

(E< >RkT) should be taken as the risk index, since the probability of a random variable of having a value

lower by k standard deviations than its mean is bounded (by Chebyshev’s Inequality) by 1/k2.

A survey done by Petty, Scott and Bird (1975, apud Park and Sharp-Bette, 1990, p.380) indicated that people in business view risk primarily associated with the probability of not achieving a target return. Almost 40% of the corporate executives interviewed described risk in this manner. In other words, management is more concerned about negative variation than about the total variation of possible investment outcomes. The second most popular definition of risk that the survey found is related to variation in returns, which is equivalent to variance as a measure of risk.

Bernstein (1996) made a long historical research about risk, showing its evolution from the ancient Greeks until current days. Pedersen and Satchell (1998) gave examples of several risk measures.

2.3. Single-period capital allocation

Markowitz, in his classic paper (1952), studied the portfolio selection in a return versus risk plane (more specifically, in the variance of returns versus expected return plane). The first great contribution of Markowitz was having noticed that the risk of a portfolio is not the sum of the risks of the component assets. The second was having specified a good criterion for the choice of portfolios. Using statistical concepts, Markowitz calculated the return and the risk of unlevered portfolios, and plotted a region or compact in this graph. The estimators of the means, variances and correlations of the rates of return of the component assets are usually obtained from historical (past) returns. Notice that making inferences to the future, based on historical data, requires the hypothesis that the future multivariate density will be the same as the past one.

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A closed formula for the efficient frontier is not known for the case in which the weights of the assets must be nonnegative. Markowitz (1991a, p.154, 316) created the method of the critical line to solve this problem numerically. He also mentioned that Wolfe (1957, apud Markowitz, 1991a, p.337) created a variant of the Simplex method capable of calculating the minimum of a quadratic function with positive semidefinite matrix subject to linear constraints.

Markowitz (2000, p.11, 39) calls the model in which the weights can be negative “Black’s model”. In this case, the efficient frontier is a hyperbola in the expected rate of return versus standard deviation plane, and it is a parabola in the expected rate of return versus variance plane. The equation of the frontier was obtained by Merton (1972) and can also be found in Ingersoll, Jr. (1987, p.82) and in Huang and Litzenberger (1988, p.63). By the two-fund separation theorem, with two portfolios on this frontier, it is possible to obtain all the others.

With the inclusion of a risk-free asset, the efficient frontier becomes the semiline that starts at the risk-free asset and is tangent to the original efficient frontier in the expected return versus standard deviation plane. This is the one-fund separation theorem.

2.4. Risk measures

There are several mathematical operationalizations, definitions or measures for the concept of risk. Some of them will be presented in the following, considering that X is a random variable of rate of return type, p x is its probability density function, P x Prob Xbx is its accumulated probability density function, N is its mean, B is a specified quantile (ex.: 5%), b is a given rate of return and dP x is the notation of the Lebesgue-Stieltjes integral.

Variance (Markowitz, 1952).

< > 2 2

V T x N dP x

¨

X (2.1)

Standard deviation.

1 2 2

2 x dP x

T T N

 ¬­

ž ­

ž

žž ­­

­­

žŸ

¨

® (2.2)

Semivariance (Markowitz, 1991a, p.193).

< > \ ^2

1 1

max ,0

n

b i

i

S b x

n

œ

X (2.3)

< > 2

b b

S b x dP x

✏✒✑

¨

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Probability of loss.

< >

0

0

L dP x

✓✒✔

¨

X (2.5)

Value-at-Risk (Artzner et al., 1999; Delbaen, 2000; Acerbi and Tasche, 2001a).

< > inf

\

|

^

VaRX x P x B (2.6)

Value-at-Risk (Acerbi and Tasche, 2001b).

< > sup

\

|

^

VaRX x P x bB (2.7)

The definition of Value-at-Risk (VaR) is nontrivial in the cases in which there are Dirac Delta distributions or in which the density is zero in an interval right after the rate of return of the B quantile. In this last case, there are an inferior and a superior abscissa for the same quantile. The superior abscissa is used by definitions (2.6) and (2.7). The negative sign is due to the fact that the random variable is of rate of return type.

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3. DISTRIBUTIONS, TRANSFORMS AND SPECIAL FUNCTIONS

3.1. Introduction

The mathematical concepts necessary for this thesis are concentrated in this chapter. This will avoid the interruption of the financial reasoning in the following chapters.

In this work, random variables of the return type will be used, which will have to be added and multiplied. Given the densities of two independent random variables X and Y, it will be necessary to determine the densities of XY and of XY. This will be done with the additive and multiplicative convolutions, by means of the Laplace and Mellin transforms. It will also be necessary to determine the density of w1Xw2Y , where w1p0, w2p0 and w1w21, when X and Y are dependent, and find a way of representing this dependence.

A random variable is called additive if it has the sum operation well defined, that is, it is possible to add two variables of the same type. A random variable is called multiplicative if it has the multiplication operation well defined, that is, it is possible to multiply two variables, resulting in another with physical or economic sense.

A mixture of distributions is a weighted sum of distributions and it allows a better adjustment of observed data. The sum of the weights must be the unity. Some weights can be negative, but in problems of adjustment, they can be forced to be nonnegative because guaranteeing the nonnegativity of the density is a difficult problem. It is simpler to use nonnegative weights and increase the number of parcels in the mixture. The tensor product is an extension of this concept to several dimensions. It allows the representation of the multivariate density of dependent random variables by means of the weighted sum of products of univariate densities. The separability of the functions helps the calculation of the integrals that arise when it is necessary to obtain the density of linear combination of dependent random variables.

After having defined the family of the density of the mixture or the tensor product, it is necessary to adjust or calibrate the weights and parameters of the component densities according to an adjustment criterion. In this thesis, the maximum likelihood criterion will be used. The opposite of the likelihood is a convex function over the weights of the mixture and tensor product. But this function is, in general, nonconvex over the parameters of the component densities.

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becomes the Erlang distribution, and if this parameter is equal to 1, then the Exponential distribution is obtained. The mixture of Erlangs has important properties for this thesis.

The sum of two or more independent random variables that follow an Erlang or a finite mixture of Erlangs results in a random variable that follows a finite mixture of Erlangs. The finite mixture of Erlangs is a distribution closed to the additive convolution. If the multivariate density of dependent random variables is a finite tensor product of Erlangs, then the density of a positive linear combination of these variables will be a finite mixture of Erlangs.

The product of two or more independent random variables that follow Erlangs generates a random variable that follows a distribution based on a particular case of the Meijer G function. Since the Erlang is a particular case of this class of Meijer G distributions, this last one is closed to the multiplicative (or scale) convolution.

The Meijer G function is a particular case of the Fox H function, which is the most general special function currently known. The Fox H can represent several elementary functions, other special functions and definite integrals. The Meijer G function is used internally by softwares such as Mathematica® and Maple® in the calculation of algebraic integration.

3.2. Related functions

Before presenting the Erlang distribution, the integral transforms and the special functions, it is necessary to define the Gamma function and some related functions.

Definition 3.2.1. Gamma function. The Gamma function (:ˆ\0, 1, 2,...^lˆ is defined by

1

0

z t

z t e dt

✘✙✘

(

¨

, for Re z 0, (3.1)

and by the property

z z 1 z

(

( , for Re z b0, zŠ\0, 1, 2,...^. (3.2)

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5 ✚

4 ✚

3 ✚

2 ✚

1 1 2 3 4 5

x

10

9

8

7

✚ 6 ✚ 5 ✚ 4 ✚ 3

2

✚ 1

1 2 3 4 5 6 7 8 9 10

x✢

Figure 3.2.1. Gamma function. Graph of the Gamma function for real arguments. The poles occur in 0 , 1

, 2 , … .

Definition 3.2.2. Superior incomplete Gamma function. The superior incomplete Gamma function

\ ^

: 0, 1, 2,... ✣✤

( ˆ q† lˆ is defined by

1

, z t

a

z a t e dt

✦✧✦

(

¨

, for Re z 0, (3.3)

and by the property

1,

,

a z

z a e a

z a

z

(

( , for Re z b0, zŠ\0, 1, 2,...^. (3.4)

Definition 3.2.3. Inferior incomplete Gamma function. The inferior incomplete Gamma function

\ ^

: 0, 1, 2,...

H ˆ q†✩✪ lˆ is defined as

, ,

z a z z a

H ( ( . (3.5)

Definition 3.2.4. Psi function. The derivative of the natural logarithm of the Gamma function is called Psi function Z:ˆ\0, 1, 2,...^lˆ and is defined as

ln z

d

z z

dz z

Z ( (a

( . (3.6)

Definition 3.2.5. PolyGama function. The high order derivatives of the Psi function are represented by the PolyGama function Z( )n :ˆ\0, 1, 2,...^lˆ, which is defined as

( )

,

n n

n

d

PolyGamma n z z z

dz

Z Z

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where n0,1,....

Definition 3.2.6. Generalized Zeta function. The Generalized Zeta function [:ˆ ˆq lˆ is defined as

0

, s

i i a

s a a i

[

✫ ✬

œ

. (3.8)

3.2.1. Particular cases

Some particular cases of the previously defined functions will be given in the following. 1

2 Q

 ¬­ ž

(ž ­žŸ ®­ (3.9)

3

2 2

Q

 ¬­ ž

(ž ­žŸ ®­ (3.10)

Euler constant H Z 1 0.57721566... (3.11)

3.2.2. Properties

Some properties of the defined functions will be given in the following.

n (n 1)!, n {1, 2,...}, 0! 1

( ‰ (3.12)

1 , 0, 1, 2,...

x x x x

( ( v (3.13)

2 1 1

2 2

2

x

x x Q x

✯  ¬­

ž

( ( žžŸ ­­ (

® (3.14)

12 12 1

0

2 , 1, 2,...

m m mx

j

j

mx m x m

m

Q

 ¬­ ž

(



( žžŸ ­­® (3.15)

1

0 1

, {0,1, 2,...}, 0, 1, 2,...

n n

j

p p p

n p jq n

q q q q

 ¬­  ¬­

ž ž

( žŸž ®­­­ (žžŸ ®­­­



‰ v (3.16)

< > 1

0

1 1 1 , 0, 1, 2,...

i

x x i x i x

Z Z

✴ ✵

œ

v (3.17)

1 1 1 1 ... 1 , {1, 2,...}

2 3 1

n n

n

Z Z ‰

(3.18)

1

( )n 1n ! 1, , {1,2,...}, Re 0

z n n z n z

Z [

‰ (3.19)

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3.3. The Erlang distribution and its related distributions

Definition 3.3.1. Gamma probability density function. The Gamma probability density function, with a scale parameter a and a shape parameter b , is the function Hba: ✸ ✸

✹ l ✹

† † defined as

1

( )

b b a

b ax

a x x

b e

H

( , (3.20)

where a0, b0 and xp0.

The notation Hba x was chosen for saving horizontal space in the text and for being analogous to

the notation of the Bernstein-Bézier basis function.

Definition 3.3.2. Erlang probability density function. If the shape parameter of the Gamma density is a positive integer, then it is called Erlang and it is the function Hba: ✸ ✸

✹ l ✹

† † defined as

1

( )

b b a

b ax

a x x

b e

H

( , (3.21)

where a0, b‰{1, 2,...}and xp0.

Definition 3.3.3. Exponential probability density function. If b1, then the Exponential density is obtained, which is the function fa: ✻ ✻

✼ l ✼

† † defined as

1

a

a ax

a

f x x

e

H

, (3.22)

where a0 and xp0.

Definition 3.3.4. Generalized Gamma probability density function. The Generalized Gamma density is the function Hb ca, : ✽ ✽

✾ l ✾

† † defined as

1

,

( ) c

b bc a

b c

ax

c a x x

b e

H

( , (3.23)

where a0, b0, cv0 and xp0.

The Generalized Gamma distribution was discovered several times by researchers from different areas along the 20th century; see Hegyi (1999). It has as particular cases distributions such as Gamma, Erlang, Exponential, Chi-Square, Chi, Maxwell-Boltzmann, Rayleigh, Generalized Normal, Half Normal, Weibull, the most known particular case of the Lévy and the Inverse Generalized Gamma. It has as limit cases the Lognormal and the Pareto.

During the development of this thesis, these and some other interesting distributions were studied but they will not be defined here because they are not going to be used.

3.3.1. Properties

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1 , 0

a a

b x b x

H M H M

M

(3.24)

1

a

b a a

b b

x

a x x

x

H

HH

s

s (3.25)

1

a

b a a

b b

x b

x x

a a

H

H H

s

s (3.26)

1 0

1 1

, {1, 2,...}, 0

y b

a a

b j

j

x dx y b

a

H M H M M

M M

❃ ❄

‰

œ

¨

(3.27)

a b

b

E x

a

H (3.28)

2

a b

b

V x

a

H (3.29)

0

ln x Hba x dx Z b ln a

¨

(3.30)

1 2

2

( 1) ( 1) !

... , {0,1,2,...},

ax n n n

n ax n

n

e nx n n x n

x e dx x c n c

a a a a

❆ ❆

 ¬­

ž ­

žžžŸ ­­ ‰ ‰

®

¨

† (3.31)

1 1 2 3

1

1

, ( 1) ( 1)( 2) ... ( 1)!

, {1, 2,...}, 0

b t x b b b

x b

i i

b x t e dt e x b x b b x b

b H x b x

❈✙❈ ❈ ❈ ❈ ❈

(

( ‰ p

¨

œ

(3.32)

3.4. Mixture and tensor product

A mixture is a univariate distribution built from the weighted sum of distributions. The weights must have a unity sum and can be negative, but like any distribution, the density of the mixture must be nonnegative on its domain. If the number of parcels is finite, then the mixture is called finite mixture. Usually, but not always, the component distributions are from the same family.

Definition 3.4.1. Finite mixture. The finite mixture is the probability distribution whose density is the function f : ❊

l

D R given by

1

; , ,... ,

n

i i i i i

f x p f x R M

œ

(3.33)

where n‰Š is the number of parcels in the mixture such that 1b dn ; pi‰†, for 1b bi n, are

weights such that 1 1

n i ip

œ

; and fi: ■

l

D R are probability densities with parameters R Mi, ,...i , for

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The concept of mixture can be extended to multiple dimensions by means of the weighted sum of multivariate distributions. A particular case of multivariate mixture is the tensor product, which is a multivariate distribution built from the weighted sum of the product of D univariate distributions, where D is the number of random variables. If D1, then the tensor product reduces to a mixture.

Definition 3.4.2. Finite tensor product. The finite tensor product is the probability distribution whose density is the function m: D1q q... DDl†❑▲ given by

1 1 1 1 1 1 1 1

1

1 ,..., 1, ,..., 1 1, ,..., 1, ,..., , ,..., , ,..., , ,..., 1 1

,..., ... ; , ,... ... ; , ,... ,

D

D D D D D D D

D

n n

D i i i i i i i i D i i D D i i D i i

i i

m x x p f x R M f x R M

▼ ▼

œ œ

(3.34)

where D‰Š is the number of random variables; 1bnd d, for 1b bd D; pi1,...,iD ‰†, for 1b bid nd

and 1b bd D, are weights such that

1

1 1

,..., 1 1

... 1

D D D

n n i i i i

p

◆ ◆

œ œ

;

and fd i, ,...,1 iD: d

P

l

D R are univariate probability densities with parameters Rd i, ,...,1 iD,Md i, ,...,1 iD,..., for 1b bid nd and 1b bd D.

In this thesis, the finite mixture and the finite tensor product of Erlangs will be used. Since many weights will be equal to zero, a more compact notation is preferable, which will be given in the following. Definition 3.4.3. Alternative definition for the finite tensor product of Erlangs. The finite tensor product of Erlangs is the probability distribution whose density m:

D

❘ l ❘

† † is given by

,1 ,

,1 ,

1 1

1

,..., i ... i D

i i D

n

a a

D i b b D

i

m x x pH x H x

œ

, (3.35)

where D‰Š is the number of random variables; n‰Š is the number of parcels such that 1b dn ;

i

p ‰†, for 1b bi n, are weights such that 1 1

n i ip

œ

; and ,

,

i d i d

a d b x

H are Erlang densities such that 0

d

x p , ai d, 0 and bi d, ‰{1, 2,...}, for all i and d, 1b bi n, 1b bd D.

If the scale parameters of the Erlangs are constant for each random variable, that is, ai d, ad, for all

i and , 1d b bi n, 1b bd D, then it is possible to define the finite tensor product of Erlangs with constant scale parameters. This will be done as follows.

Definition 3.4.4. Finite tensor product of Erlangs with constant scale parameters. The finite tensor product of Erlangs with constant scale parameters is the probability distribution whose density

: D

m ❯ ❯

❱ l ❱

† † is given by

1

,1 ,

1 1

1

,..., ... D

i i D

n

a a

D i b b D

i

m x x pH x H x

œ

, (3.36)

where D‰Š is the number of random variables; n‰Š is the number of parcels such that 1b dn ;

i

p ‰†, for 1b bi n, are weights such that 1 1

n i ip

œ

; and d,

i d

a d b x

H are Erlang densities such that 0

d

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The finite tensor product of Erlangs with constant scale parameters will be used in this work to approximate the single-period multivariate density of the dependent returns of the assets.

Since the finite mixture of Erlangs can have an arbitrary number of parcels, then it can approximate, with arbitrary precision, any continuous density defined over a closed interval of nonnegative reals. This is a theorem analogous to Weierstrass’, whose proof can be found in Powell (1981, p.61).

3.5. Calibration of mixture and tensor product

Given a probability distribution function and a set of observations or samples, a common task in Statistics is the adjustment or calibration of the distribution parameters so that it can represent the samples. This can be done for the uni and multivariate cases. It is necessary to define an adjustment criterion. There are several available criteria such as the minimization of the mean squared error and the method of moments. In this thesis, the maximum likelihood criterion will be used. The greater the likelihood, the better the adjustment of the density to the samples. A necessary hypothesis for the use of this criterion is that there must be independence among the samples, though it is possible to occur dependence among the variables in the multivariate case.

A result relative to finite mixtures and tensor products is that if the parameters of the distributions are constant, that is, only the weights are variables in the optimization problem, then the likelihood function is concave over them.

Theorem 3.5.1. Concavity of the likelihood function of a mixture of distributions with fixed parameters. Let x x1, 2,...,xS, where Sp1, be independent unidimensional samples and let a mixture with

density f :Dl†❨❩ be given by

1

; , ,...

n

i i i i i

f x p f x R M

œ

,

where 1b b dn ; pi‰†, for 1b bi n, are weights such that 1 1

n i ip

œ

; and fi:Dl†❪❫ are densities

with parameters , ,...R Mi i , for 1b bi n. If these parameters are constant, then the likelihood function

1 1 1 1 1

1 1

,..., ; ,... , ,..., , ,..., , ,..., ,... ; , ,...

S n

n S n n n i i s i i

i s

L p p x x f f R R M M p f x R M



œ

is concave over the weights pi.

Proof. Let

< > < >

1 1,1, 1,2,..., 1,n , 2 2,1, 2,2,..., 2,n

v p p p v p p p

❵ ❵

and

< >

1 2 1,1 2,1 1,2 2,2 1, 2,

3 , ,..., 3,1, 3,2,..., 3,

2 2 2 2

n n

n

v v p p p p p p

v  ¡ ¯° p p p

¡ °

¢ ±

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1 1 1,

1 1

2 2 2,

1 1

3 3 3,

1 1 ; , , , ,... ; , ,... , ; , , , ,... ; , ,... , ; , , , ,... ; , ,... . S n

s i i i i i s i i i

s S n

s i i i i i s i i

i s

S n

s i i i i i s i i

i s

L L v x f p f x

L L v x f p f x

L L v x f p f x

R M R M

R M R M

R M R M

❜ ❜ ❜ ❜ ❜ ❜

œ



œ



œ



The terms f xi s;N Ti, i,... are constant. Since the vectors v1 and v2 are arbitrary, it is sufficient to prove that 1 2 3 2 L L

L p (3.37)

or that (because the natural logarithm is an increasing function)

1 2

3 1 2

ln ln

ln ln

2

L L

L p L L . (3.38)

Any vector of type kv1 (1 k v) , 02 b bk 1, can be attained by successive bisections of the interval between v1 and v2. Removing the logarithm of (3.38), it is necessary to prove that

3 1 2 L p L L

or

1, 2,

1, 2,

1 1 1

1 1 1

; , ,... ; , ,... ; , ,...

2

S n S n S n

i i

i s i i i i s i i i i s i i

i i i

s s s

p p

f x R M p f x R M p f x R M

❝ ❝ ❝

❝ ❝ ❝

 ¬ ¬

ž ­­ž ­­

p žžž ­­­žžž ­­­

Ÿ ®Ÿ ®

œ

œ

œ







or 1, 2, 1 1 1 1, 2, 1 1 1 ; , ,... ; , ,... 2 ; , ,... ; , ,... n n

i i s i i i i s i i S

i i

s

S n n

i i s i i i i s i i

i i

s

p f x p f x

p f x p f x

R M R M

R M R M

❞ ❞ ❞ ❞ ❞ ❞ p  ¬­ ¬­ ž ­ž ­ ž ­ž ­ ž ­ž ­ ž ž Ÿ ®Ÿ ®

œ

œ



œ

œ



. (3.39)

For each index s, there are two nonnegative numbers at each side of (3.39). Since the arithmetic mean of nonnegative numbers is always greater or equal to the geometric mean, the following inequality is true. 1, 2, 1 1 1, 2, 1 1 ; , ,... ; , ,... ; , ,... ; , ,... 2 n n

i i s i i i i s i i n n

i i

i i s i i i i s i i

i i

p f x p f x

p f x p f x

R M R M

R M R M

❡ ❡

❡ ❡

 ¬ ¬

­ ­

ž ­ž ­

p žžž ­­žžž ­­

Ÿ ®Ÿ ®

œ

œ

œ

œ

.

Multiplying the S terms, the signal of the inequality is kept and the expression (3.39) is obtained. W Corollary 3.5.1. Concavity of the likelihood function of the tensor product of distributions with fixed parameters. Let x1<x1,1,...,x1,D> , x2<x2,1,...,x2,D> , ..., xS <xS,1,...,xS D, >

❢ ❢ ❢

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independ-ent samples with Dp1 random variables; and let a tensor product with a density m: D1q q... DDl†❣❤ be

given by

1 1 1 1 1 1 1 1

1

1 ,..., 1, ,..., 1 1, ,..., 1, ,..., , ,..., , ,..., , ,..., 1 1

,..., ... ; , ,... ... ; , ,...

D

D D D D D D D

D

n n

D i i i i i i i i D i i D D i i D i i

i i

m x x p f x R M f x R M

✐ ✐

œ œ

,

where 1bnd b d, for 1b bd D; pi1,...,iD ‰†, for 1b bid nd and 1b bd D, are weights such that

1

1 1

,..., 1 1

... 1

D D D

n n i i i i

p

❥ ❥

œ œ

;

and fd i, ,...,1 iD : d

l

D R , for 1b bid nd and 1b bd D, are univariate probability densities with parameters

1 1

, ,...,D, , ,...,D,...

d i i d i i

R M . If these parameters are constant, then the likelihood function

1 1 1 1 1

1 1 1 1 1 1 1 1

,..., , ,..., , ,..., , ,...,

,..., 1, ,..., ,1 1, ,..., 1, ,..., , ,..., , , ,..., , ,..., 1 1

1

; , , , ,...

... ; , ,... ... ; , ,...

D D D D

D

D D D D D D D

D

i i s i i i i i i i i i S n n

i i i i s i i i i D i i s D D i i D i i i i

s

L p x f

p f x f x

R M

R M R M

♠ ♠

œ œ



is concave over the weights pi1,...,iD .

Proof. For each index s, the product f1, ,...,i1 iD xs,1;R1, ,...,i1 iD,M1, ,...,i1 iD,... ... fD i, ,...,1 iD xs D, ;RD i, ,...,1 iD,MD i, ,...,1 iD,... is a constant number and the proof is analogous to the previous theorem’s.

W The previous corollary and theorem can be proved with Jensen’s inequality; see Boyd and Vandenberghe (2004, p.77, 79).

The choice of the distribution and parameters to be optimized is crucial to the computational feasibility of the likelihood maximization. The choice of a multivariate density in the form of tensor product leads to an explosion of the number of variables to be optimized. This effect is known as “curse of dimensionality”. But the fact that the opposite of the likelihood is a convex function over the nonnegative weights of the mixture alleviates the computational effort problem. The opposite of the likelihood is, in general, nonconvex over the parameters of the distributions and, hence, it is desirable the use of the smallest number of parameters possible.

The weights of the mixture or of the tensor product can be negative. However, in the optimization problems of this thesis where they appear as optimization variables, the weights will be forced to be nonnegative because guaranteeing the nonnegativity of the density is a difficult problem. It is simpler to use nonnegative weights and increase the number of parcels in the mixture or in the tensor product.

In the following, it will be discussed some variations of how the optimization problem can be stated for the unidimensional case, that is, for the mixture of densities.

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function density equal to 1. The space of the nonnegative weights is convex. Changing the signal of the likelihood function makes it convex over the weights. Hence, this is a convex optimization problem.

1 1

min ; , , , ,... ; , ,... i

S n

i s i i i i i s i i

p

i s

L p x f N T p f x N T



œ

1 . .

0 , 1 1

i n

i i

s t

p i n

p

p b b

œ

The advantage of stating convex problems is due to the solution being unique (not considering flat plateau regions) and to the existence of efficient algorithms to solve them.

Optimization problem 3.5.2. Calibration of a positive finite mixture of Erlang distributions. The scale parameters are not necessarily the same and can vary from 0 until a given maximum value amax. The shape parameters are integers between 1 and a given bmax.

max

,

1 1

min , i

i i

S b a

i i i i s

a p

i s

L a p pH x



œ

max

max max

max

1 . .

0 , 1

0 , 1 1

i i b

i i

s t

a a i b

p i b

p

q

b b b

p b b

œ

In the previous optimization problem, the scale and shape parameters have a maximum value. But by fixing the maximum value of b, the scale parameter a will be optimized automatically. The scale parameters can be forced to be equal in all parcels of the mixture.

Optimization problem 3.5.3. Calibration of a positive finite mixture of Erlang distributions with constant scale parameters. The scale parameters of all components of the mixture have the same value. The shape parameters are integers between 1 and a given bmax.

max

,

1 1 min ,

i

S b a

i i i s

a p

i s

L a p pH x

r

r



œ

max

max

1 . .

0 0 , 1

1

i b

i i

s t

a

p i b

p

s

p b b

œ

Imagem

Figure 3.2.1.  Gamma function.  Graph  of the  Gamma  function  for real  arguments.  The  poles  occur  in  0 ,  1 ,  2 , … .
Figure  3.5.1.  Calibration  of  a  finite  mixture  of  Erlang  distributions  with  constant  scale  parameters.
Figure  3.5.2.  Calibration  of  a  bivariate  finite  tensor  product  of  Erlangs  with  constant  scale parameters
Figure 3.11.1. Positive linear combination of two random variables.
+7

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O teor ideal de aditivo necessário para aperfeiçoar as propriedades do concreto é encontrado através de diversos testes experimentais, uma vez que a dosagem de aditivo está

• Conselho Nacional de Combate à Pirataria e Delitos contra a Propriedade Intelectual (CNCP-MJ) e as instituições da sociedade que o compõe e atuam em projetos que o INPI

A Conferência das Nações Unidas sobre o Meio Ambiente Humano, sediada em Estocolmo, Suécia, em 1972, representa um marco na inserção da questão ambiental no debate

Recognizing the ability of metaheuristic approaches to solve optimization problems, more precisely regarding the portfolio optimization of electricity market players, and analysing

a) Nada, me sinto totalmente preparado. b) A inclusão de uma matéria/componente curricular que lide com algumas questões, tais como o processo da morte e do morrer. c)