Université de Montréal The Double Pareto-Lognormal Distribution and its applications in actuarial science and finance par Chuan Chuan Zhang Département de mathématiques et de statistique Faculté des arts et des sciences Mémoire présenté à la Faculté des études supérieures en vue de l’obtention du grade de Maître ès sciences (M.Sc.) en Mathématiques Orientation actuariat février 2015 (cid:2)c Chuan Chuan Zhang, 2014 Université de Montréal Faculté des études supérieures Ce mémoire intitulé The Double Pareto-Lognormal Distribution and its applications in actuarial science and finance présenté par Chuan Chuan Zhang a été évalué par un jury composé des personnes suivantes : Martin Bilodeau (président-rapporteur) Louis G. Doray (directeur de recherche) Maciej Augustyniak (membre du jury) Mémoire accepté le: Le 17 janvier 2015 iii SOMMAIRE Le but de ce mémoire de maîtrise est de décrire les propriétés de la loi double Pareto-lognormale, de montrer comment on peut introduire des variables explica- tives dans le modèle et de présenter son large potentiel d’applications dans le domaine de la science actuarielle et de la finance. Tout d’abord, nous donnons la définition de la loi double Pareto-lognormale et présentons certaines de ses propriétés basées sur les travaux de Reed et Jor- gensen (2004). Les paramètres peuvent être estimés en utilisant la méthode des moments ou le maximum de vraisemblance. Ensuite, nous ajoutons une variable explicative à notre modèle. La procédure d’estimation des paramètres de ce mo- dèleestégalementdiscutée. Troisièmement, desapplicationsnumériquesdenotre modèle sont illustrées et quelques tests statistiques utiles sont effectués. Mots-clés : Loi normale-Laplace, loi double Pareto-lognormale, estimation du maximum de vraisemblance, transformation de Box-Cox, variables explicatives, test d’ajustement. iv SUMMARY The purpose of this Master’s thesis is to describe the double Pareto-lognormal distribution, show how the model can be extended by introducing explanatory variables in the model and present its large potential of applications in actuarial science and finance. First, we give the definition of the double Pareto-lognormal distribution and present some of its properties based on the work of Reed and Jorgensen (2004). Theparameterscouldbeestimatedbyusingthemethodofmomentsormaximum likelihood. Next, we add an explanatory variable to our model. The procedure of estimation for this model is also discussed. Finally, some numerical applications of our model are illustrated and some useful statistical tests are conducted. Keywords: Normal-Laplace distribution, double Pareto-lognormal distribution, maximum likelihood estimation, Box-Cox transformation, explanatory variables, goodness-of-fit test. v CONTENTS Sommaire........................................................ iii Summary........................................................ iv List of Figures................................................... vii List of Tables.................................................... viii Acknowledgements............................................... 1 Introduction..................................................... 2 Chapter 1. The double Pareto-lognormal distribution........... 4 1.1. Moment generating function and cumulants....................... 4 1.2. The Normal-Laplace distribution.................................. 8 1.2.1. Genesis and definitions......................................... 8 1.2.2. Properties..................................................... 13 1.3. The double Pareto-lognormal distribution......................... 18 1.3.1. Definition of the double Pareto-lognormal distribution.......... 18 1.3.2. Properties..................................................... 21 Chapter 2. Estimation of parameters ........................... 27 2.1. Method of moments............................................... 27 2.2. Method of maximum likelihood................................... 30 Chapter 3. Double Pareto-lognormal distribution with covariates 34 3.1. The model........................................................ 34 3.2. Estimation........................................................ 36 Chapter 4. Numerical illustrations.............................. 38 4.1. Application of the model in finance ............................... 38 vi 4.1.1. Description of the data set..................................... 38 4.1.2. Fit of stock price returns....................................... 40 4.2. Application of the model in property and casualty insurance ...... 44 4.2.1. Description of the data set..................................... 44 4.2.2. Application of the model with a covariate...................... 49 4.2.2.1. Fit of the fire losses........................................ 49 4.2.2.2. Inclusion of an explanatory variable........................ 51 Chapter 5. Conclusion.......................................... 54 Bibliography..................................................... 56 Appendix A. The Box-Cox transformation...................... A-i A.1. Definition........................................................ A-i A.2. Estimation of λ ..................................................A-iii A.3. Numerical example...............................................A-iv Appendix B. Fit lognormal and inverse Gaussian to the fire loss claims............................................ B-i B.1. Fit lognormal distribution........................................ B-i B.2. Fit inverse Gaussian distribution .................................B-iii Appendix C. Code R and MATHEMATICA.................... C-i C.1. Code R .......................................................... C-i C.1.1. Application to daily logarithmic returns for BMO stock........ C-i C.1.2. Application to Danish fire insurance data......................C-iv C.2. Code MATHEMATICA for obtaining MME......................C-x vii LIST OF FIGURES 4.1 Histogram of daily stock price returns................................ 39 4.2 Histogram of daily logarithmic returns. .............................. 39 4.3 Normal-Laplace distribution fitted to daily logarithmic returns....... 42 4.4 Histogram of Danish fire loss claims greater than 100000. ............ 45 4.5 Histogram of log-transformed Danish fire loss claims.................. 45 4.6 Histogram of floor space.............................................. 46 4.7 Scatter plot: log-transformed fire losses vs floor space. ............... 47 4.8 Histogram of residuals. .............................................. 47 4.9 Normal Q-Q plot of residuals......................................... 48 4.10 Normal-Laplace fitted to log-transformed Danish fire loss claims...... 51 A.1 Estimation of λ......................................................A-v viii LIST OF TABLES 4.1 Statistical summary for daily price returns for BMO.................. 40 4.2 NL (using MME) fitted to the daily logarithmic returns.............. 41 4.3 NL (using MLE) fitted to the daily logarithmic returns............... 41 4.4 Chi-square test....................................................... 43 4.5 Statistical summary for Danish fire loss claims. ...................... 46 4.6 NL (using MME) fitted to the log-transformed data.................. 50 4.7 NL (using MLE) fitted to the log-transformed data................... 50 4.8 Chi-square test....................................................... 52 4.9 NL (using MLE) fitted to the log-transformed data with a covariate.. 53 A.1 Some traditional transformations in Box-Cox transformation.........A-ii B.1 Log-normal (using MME) fitted to the Danish fire loss data.......... B-ii B.2 Log-normal (using MLE) fitted to the Danish fire loss data...........B-ii B.3 Chi-square test.......................................................B-iii B.4 Inverse Gaussian (using MME) fitted to the Danish fire loss data.....B-iv B.5 Inverse Gaussian (using MLE) fitted to the Danish fire loss data......B-iv B.6 Chi-square test.......................................................B-v 1 ACKNOWLEDGEMENTS I would like to express my deep gratitude to Professor Louis G. Doray, my research supervisor, for his patient guidance, enthusiastic encouragement and useful critiques of this work. My grateful thanks are also extended to all the teachers and staff of the Department of Mathematics and Statistics for their extraordinary work that allowed me to complete my studies. I also wish to thank my parents for their support and encouragement throughout my studies. Finally, I would like to dedicate this thesis to the loving memory of my grandmother, who passed away last summer. INTRODUCTION The presence of extreme values in a sample is well-documented in many fields such as in insurance, finance, hydrology and geography, etc. The Gaussian model (normal distribution) might be relevant for the centre of a distribution, but not for the extreme values. Thus, the statistical analysis of extremes was developed for fitting parametric models to samples with extreme events and it is also key to many risk management problems related to insurance, reinsurance and finance. There are two standard parametric distributions in the field of extreme value analysis: one is the generalized extreme value distribution which is designed for a sample of extreme outcomes, and the other one is the generalized Pareto dis- tribution which plays an important role in modeling a sample of excesses over a high threshold (see Reiss and Thomas, 2007). Reed and Jorgensen (2004) introduce a new distribution named the double Pareto-lognormal distribution which exhibits Paretian (power law) behavior in both tails. This distribution has proved to be very useful in modeling the size distributions of various phenomena possibly with extreme events in a wide range of areas such as economics, finance and casualty and property insurance. In this thesis, we intend to show how the double Pareto-lognormal distribu- tion is derived and discuss its properties based on Reed and Jorgensen’s work; we will then try to extend the model by including explanatory variables and show its potential applications in insurance and finance with examples. This thesis consists of five chapters. In Chapter 1, we derive the normal-Laplace distribution based on the defini- tion of Reed (2004) and present its properties. Then the double Pareto-lognormal distribution can be defined as an exponentiated normal-Laplace distributed ran- dom variable; its properties such as its moment generating function, cumulative