Financial and actuarial modeling is an everchanging field with an increased reliance on statistical techniques. This is seen in the changing of competency exams, especially at the upper levels, where topics include more statistical concepts and techniques. In the years since the first edition was published statistical techniques such as reliability measurement, simulation, regression, and Markov chain modeling have become more prominent. This influx in statistics has put an increased pressure on students to secure both strong mathematical and statistical backgrounds and the knowledge of statistical techniques in order to have successful careers. As in the first edition, this text approaches financial and actuarial model ing from a statistical point of view. The goal of this text is twofold. The first is to provide students and practitioners a source for required mathematical and statistical background. The second is to advance the application and theory of statistics in financial and actuarial modeling.
Financial and Actuarial Statistics: An Introduction, Second Edition enables you to obtain the mathematical and statistical background required in the current nancial and actuarial industries. It also advances the application and theory of statistics in modern nancial and actuarial modeling. Like its predecessor, this second edition considers nancial and actuarial modeling from a statistical point of view while adding a substantial amount of new material. New to the Second Edition • Nomenclature and notations standard to the actuarial eld • Excel™ exercises with solutions that demonstrate how to use Excel functions for statistical and actuarial computations • Problems dealing with standard probability and statistics theory, along with detailed equation links • A chapter on Markov chains and actuarial applications • Expanded discussions of simulation techniques and applications, such as investment pricing • Sections on the maximum likelihood approach to parameter estimation as well as asymptotic applications • Discussions of diagnostic procedures for nonnegative random variables and Pareto, lognormal, Weibull, and left truncated distributions • Expanded material on surplus models and ruin computations • Discussions of nonparametric prediction intervals, option pricing diagnostics, variance of the loss function associated with standard actuarial models, and Gompertz and Makeham distributions • Sections on the concept of actuarial statistics for a collection of stochastic status models The book presents a unied approach to both nancial and actuarial modeling through the use of general status structures. The authors dene future time- dependent nancial actions in terms of a status structure that may be either deterministic or stochastic. They show how deterministic status structures lead to classical interest and annuity models, investment pricing models, and aggregate claim models. They also employ stochastic status structures to develop nancial and actuarial models, such as surplus models, life insurance, and life annuity models. C8508 Statistics FINANCIAL AND ACTUARIAL STATISTICS FINANCIAL AND ACTUARIAL STATISTICS DALE S. BOROWIAK ARNOLD F. SHAPIRO BOROWIAK SHAPIRO AN INTRODUCTION SECOND EDITION SECOND EDITION C8508_Cover.indd 1 10/8/13 8:53 AM DALE S. BOROWIAK University of Akron Ohio, USA ARNOLD F. SHAPIRO Pennsylvania State University USA FINANCIAL AND ACTUARIAL STATISTICS AN INTRODUCTION SECOND EDITION CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2014 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20130923 International Standard Book Number-13: 978-0-203-91124-2 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com iii Contents Preface ix 1 Statistical Concepts 1 1.1 Probability 1 1.2 Random Variables 7 1.2.1 Discrete Random Variables 8 1.2.2 Continuous Random Variables 10 1.2.3 Mixed Random Variables 13 1.3 Expectations 14 1.4 Moment Generating Function 20 1.5 Survival Functions 22 1.6 Nonnegative Random Variables 25 1.6.1 Pareto Distribution 25 1.6.2 Lognormal Distribution 26 1.6.3 Weibull Distribution 26 1.6.4 Gompertz Distribution 27 1.6.5 Makeham Distribution 28 1.7 Conditional Distributions 29 1.8 Joint Distributions 31 Problems 36 Excel Problems 38 Solutions 38 2 Statistical Techniques 41 2.1 Sampling Distributions and Estimation 41 2.1.1 Point Estimation 42 2.1.2 Condence Intervals 44 2.1.3 Percentiles and Prediction Intervals 45 2.1.4 Condence and Prediction Sets 46 2.2 Sums of Independent Variables 49 2.3 Order Statistics and Empirical Prediction Intervals 54 2.4 Approximating Aggregate Distributions 57 2.4.1 Central Limit Theorem 57 2.4.2 Haldane Type A Approximation 61 2.4.3 Saddlepoint Approximation 62 2.5 Compound Aggregate Variables 65 2.5.1 Expectations of Compound Aggregate Variables 65 2.5.2 Limiting Distributions for Compound Aggregate Variables 66 2.6 Regression Modeling 70 iv Contents 2.6.1 Least Squares Estimation 71 2.6.2 Regression Model-Based Inference 74 2.7 Autoregressive Systems 75 2.8 Model Diagnostics 78 2.8.1 Probability Plotting 79 2.8.2 Generalized Least Squares Diagnostic 83 2.8.3 Interval Data Diagnostic 84 Problems 87 Excel Problems 88 Solutions 90 3 Financial Computational Models 93 3.1 Fixed Financial Rate Models 94 3.1.1 Financial Rate-Based Calculations 94 3.1.2 General Period Discrete Rate Models 99 3.1.3 Continuous-Rate Models 100 3.2 Fixed-Rate Annuities 101 3.2.1 Discrete Annuity Models 101 3.2.2 Continuous Annuity Models 104 3.3 Stochastic Rate Models 106 3.3.1 Discrete Stochastic Rate Model 106 3.3.2 Continuous Stochastic Rate Models 112 3.3.3 Discrete Stochastic Annuity Models 114 3.3.4 Continuous Stochastic Annuity Models 116 Problems 117 Excel Problems 119 Solutions 120 4 Deterministic Status Models 123 4.1 Basic Loss Model 123 4.1.1 Deterministic Loss Models 124 4.1.2 Stochastic Rate Models 126 4.2 Stochastic Loss Criterion 128 4.2.1 Risk Criteria 129 4.2.2 Percentile Criteria 130 4.3 Single-Risk Models 131 4.3.1 Insurance Pricing 131 4.3.2 Investment Pricing 135 4.3.3 Options Pricing 136 4.3.4 Option Pricing Diagnostics 139 4.4 Collective Aggregate Models 140 4.4.1 Fixed Number of Variables 141 4.4.2 Stochastic Number of Variables 143 4.4.3 Aggregate Stop-Loss Reinsurance and Dividends 145 4.5 Stochastic Surplus Model 148 vContents 4.5.1 Discrete Surplus Model 148 4.5.2 Continuous Surplus Model 152 Problems 155 Excel Problems 158 Solutions 159 5 Future Lifetime Random Variables and Life Tables 163 5.1 Continuous Future Lifetime 164 5.2 Discrete Future Lifetime 167 5.3 Force of Mortality 169 5.4 Fractional Ages 175 5.5 Select Future Lifetimes 177 5.6 Survivorship Groups 179 5.7 Life Models and Life Tables 182 5.8 Life Table Condence Sets and Prediction Intervals 185 5.9 Life Models and Life Table Parameters 187 5.9.1 Population Parameters 188 5.9.2 Aggregate Parameters 191 5.9.3 Fractional Age Adjustments 193 5.10 Select and Ultimate Life Tables 194 Problems 198 Excel Problems 200 Solutions 200 6 Stochastic Status Models 203 6.1 Stochastic Present Value Functions 204 6.2 Risk Evaluations 205 6.2.1 Continuous-Risk Calculations 205 6.2.2 Discrete Risk Calculations 206 6.2.3 Mixed Risk Calculations 207 6.3 Percentile Evaluations 208 6.4 Life Insurance 210 6.4.1 Types of Unit Benet Life Insurance 212 6.5 Life Annuities 215 6.5.1 Types of Unit Payment Life Annuities 217 6.5.2 Apportionable Annuities 220 6.6 Relating Risk Calculations 223 6.6.1 Relations among Insurance Expectations 223 6.6.2 Relations among Insurance and Annuity Expectations 225 6.6.3 Relations among Annuity Expectations 226 6.7 Actuarial Life Tables 227 6.8 Loss Models and Insurance Premiums 230 6.8.1 Unit Benet Premium Notation 232 6.8.2 Variance of the Loss Function 235 6.9 Reserves 237 vi Contents 6.9.1 Unit Benet Reserves Notations 240 6.9.2 Relations among Reserve Calculations 241 6.9.3 Survivorship Group Approach to Reserve Calculations 243 6.10 General Time Period Models 244 6.10.1 General Period Expectation 245 6.10.2 Relations among General Period Expectations 246 6.11 Expense Models and Computations 249 Problems 252 Excel Problems 254 Solutions 254 7 Advanced Stochastic Status Models 257 7.1 Multiple Future Lifetimes 257 7.1.1 Joint Life Status 258 7.1.2 Last Survivor Status 260 7.1.3 General Contingent Status 263 7.2 Multiple-Decrement Models 264 7.2.1 Continuous Multiple Decrements 264 7.2.2 Forces of Decrement 266 7.2.3 Discrete Multiple Decrements 268 7.2.4 Single-Decrement Probabilities 269 7.2.5 Uniformly Distributed Single-Decrement Rates 271 7.2.6 Single-Decrement Probability Bounds 273 7.2.7 Multiple-Decrement Life Tables 275 7.2.8 Single-Decrement Life Tables 278 7.2.9 Multiple-Decrement Computations 279 7.3 Pension Plans 280 7.3.1 Multiple-Decrement Benets 281 7.3.2 Pension Contributions 285 7.3.3 Future Salary-Based Benets and Contributions 287 7.3.4 Yearly Based Retirement Benets 288 Problems 290 Excel Problems 291 Solutions 292 8 Markov Chain Methods 295 8.1 Introduction to Markov Chains 296 8.2 Nonhomogeneous Stochastic Status Chains 297 8.2.1 Single-Decrement Chains 298 8.2.2 Actuarial Chains 299 8.2.3 Multiple-Decrement Chains 300 8.2.4 Multirisk Strata Chains 303 8.3 Homogeneous Stochastic Status Chains 307 viiContents 8.3.1 Expected Curtate Future Lifetime 309 8.3.2 Actuarial Chains 310 8.4 Survivorship Chains 312 8.4.1 Single-Decrement Models 313 8.4.2 Multiple-Decrement Models 314 8.4.3 Multirisk Strata Models 315 Problems 316 Excel Problems 317 Solutions 320 9 Scenario and Simulation Testing 323 9.1 Scenario Testing 323 9.1.1 Deterministic Status Scenarios 324 9.1.2 Stochastic Status Scenarios 325 9.1.3 Stochastic Rate Scenarios 328 9.2 Simulation Techniques 330 9.2.1 Bootstrap Sampling 331 9.2.2 Simulation Sampling 332 9.2.3 Simulation Probabilities 335 9.2.4 Simulation Prediction Intervals 337 9.3 Investment Pricing Applications 340 9.4 Stochastic Surplus Application 343 9.5 Future Directions in Simulation Analysis 344 Problems 346 Excel Problems 348 Solutions 350 10 Further Statistical Considerations 353 10.1 Mortality Adjustment Models 354 10.1.1 Linear Mortality Acceleration Models 355 10.1.2 Mean Mortality Acceleration Models 357 10.1.3 Survival-Based Mortality Acceleration Models 360 10.2 Mortality Trend Modeling 361 10.3 Actuarial Statistics 364 10.3.1 Normality-Based Prediction Intervals 365 10.3.2 Prediction Set-Based Prediction Intervals 366 10.3.3 Simulation-Based Prediction Intervals 368 10.4 Data Set Simplications 370 Problems 371 Excel Problems 371 Solutions 373 Appendix A: Excel Statistical Functions, Basic Mathematical Functions, and Add-Ins 375 viii Contents Appendix B: Acronyms and Principal Sections 377 References 379 ix Preface Financial and actuarial modeling is an ever-changing eld with an increased reliance on statistical techniques. This is seen in the changing of competency exams, especially at the upper levels, where topics include more statistical concepts and techniques. In the years since the rst edition was published statistical techniques such as reliability measurement, simulation, regres- sion, and Markov chain modeling have become more prominent. This inux in statistics has put an increased pressure on students to secure both strong mathematical and statistical backgrounds and the knowledge of statistical techniques in order to have successful careers. As in the rst edition, this text approaches nancial and actuarial model- ing from a statistical point of view. The goal of this text is twofold. The rst is to provide students and practitioners a source for required mathemati- cal and statistical background. The second is to advance the application and theory of statistics in nancial and actuarial modeling. This text presents a unied approach to both nancial and actuarial modeling through the utilization of general status structures. Future time- dependent nancial actions are dened in terms of a status structure that may be either deterministic or stochastic. Deterministic status structures lead to classical interest and annuity models, investment pricing models, and aggregate claim models. Stochastic status structures are used to develop nancial and actuarial models, such as surplus models, life insurance, and life annuity models. This edition is updated with the addition of nomenclature and notations standard to the actuarial eld. This is essential to the interchange of concepts and applications between actuarial, nancial, and statistical practitioners. Throughout this edition exercise problems have been added along with solu- tions listing detailed equation links. After each chapter a series of applica- tion problems listed as “Excel Problems,” along with solutions listing useful library functions, are newly included. Specic changes in this edition, listed by chapter, are now discussed. Chapter 1 from the rst edition is now split into two new chapters. Chapter 1 gives basic statistical theory and applications. Additional examples to help prepare students for the initial actuarial exams are also given along with a new section on nonnegative variables, namely, the Pareto, lognormal, and Weibull. Chapter 2 consists of statistical models and techniques includ- ing a new section on model diagnostics. Probability plotting, least squares, and interval data diagnostics are explored. In Chapter 4 the discussions of option pricing and stochastic surplus models are expanded. New discus- sions include option pricing diagnostics and upper and lower bounds on the probability of ruin for standard surplus models. Further, ruin computations [...]... include the typical discrete and continuous random variables, and the combinations of discrete and continuous variables, referred to as mixed random variables For a general discussion of random variables and corresponding properties we refer to Hogg and Tanis (2010, Chapters 3 and 4) and Rohatgi (1976, Chapter 2) In financial and actuarial modeling the time until a financial action occurs may be associated... discrete and continuous random variables that can be constructed 14 Financial and Actuarial Statistics: An Introduction There is a variation of the mixed-type random variable that utilizes both discrete and continuous random variables in defining the mpf This plays a part in insurance modeling, and an example of this type of random variable structure follows in Example 1.14 Example 1.14 A 1-year insurance... Concepts The modeling of financial and actuarial systems starts with the mathematical and statistical concepts of actions and associated variables There are two types of actions in financial and actuarial statistical modeling, referred to as nonstochastic or deterministic and stochastic Stochastic actions possess an associated probability structure and are described by statistical random variables Nonstochastic... (1.6) and (1.7) using (1.1) and utilizing disjoint sets These formulas have many applications, and we follow with two examples introducing two important actuarial multiple life structures 6 Financial and Actuarial Statistics: An Introduction A∪B A B A∩B Sample Space A Ac FIGURE 1.2 Venn diagram for rules (1.6) and (1.7) Example 1.5 In general nomenclature, we let (x) denote a life aged x Parties (x) and. .. empirical data Probabilistic and statistical aspects of such estimation must be accounted for in financial and actuarial models 1.3 Expectations The propensities of a random variable or a function of a random variable to take on particular outcomes are often important to financial and actuarial modeling The expectation is one method used to predict and assess outcomes of a random variable In general,... years) = 271 In the balance of this chapter we turn our attention to statistical topics useful to the financial and actuarial fields 1.2 Random Variables In financial and actuarial modeling there are two types of variables, stochastic and deterministic Deterministic variables lack any stochastic structure Random variables are variables that possess some stochastic structure Random variables include... practice many formulas used in the analysis of financial and actuarial actions are based on the ideas of conditioning and independence A clear understanding of these concepts aids in the mastery of future statistical, financial, and actuarial topics General properties and formulas of probability systems follow from the axioms of probability Two such properties frequently used in the application and development... variance of X is a central moment with r = 2 and is denoted by Var{X} = σ2, and after simplification the variance becomes σ2 = E{(X – μ)2} = E{X2} – μ2 (1.25) The existence of the second moment implies existence of the variance The standard deviation of X is σ = σ1/2 The variance and standard deviation of a random variable measure the dispersion or variability associated with the random variable and. .. specific random variables useful in actuarial and economic sciences and their distributions, namely, Pareto, lognormal, and Weibull, are discussed in Sections 1.6.1, 1.6.2, and 1.6.3, respectively The chapter ends with an introduction to conditional distributions in Section 1.7 and joint distributions of more than one random variable in Section 1.8 1.1 Probability This section presents a brief introduction. .. this chapter with a brief introduction to probability in Section 1.1 and then proceed to the various statistical topics Standard statistical concepts such as discrete, continuous, and mixed random variables and statistical distributions are discussed in Sections 1.2.1, 1.2.2, and 1.2.3 Expectations of random variables are introduced in Section 1.3, and moment generating functions and their applications . models, such as surplus models, life insurance, and life annuity models. C8508 Statistics FINANCIAL AND ACTUARIAL STATISTICS FINANCIAL AND ACTUARIAL STATISTICS DALE S. BOROWIAK ARNOLD F. SHAPIRO BOROWIAK SHAPIRO AN. with standard probability and statistics theory, along with detailed equation links • A chapter on Markov chains and actuarial applications • Expanded discussions of simulation techniques and. Financial and Actuarial Statistics: An Introduction, Second Edition enables you to obtain the mathematical and statistical background required in the current nancial and actuarial