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Financial and Actuarial Statistics: an Introduction, Second Edition Statistics SECOND EDITION SECOND EDITION Financial and Actuarial Statistics: An Introduction, Second Edition enables you to obtain the mathematical and statistical background required in the current financial and actuarial industries. It also advances the application and theory of ACTUARIAL STATISTICS statistics in modern financial and actuarial modeling. Like its predecessor, this second edition considers financial and actuarial modeling from a statistical point of view while adding a substantial amount of new material. FINANCIAL New to the Second Edition FINANCIAL • Nomenclature and notations standard to the actuarial field • 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 AND ACTUARIAL • A chapter on Markov chains and actuarial applications AND • 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 STATISTICS Pareto, lognormal, Weibull, and left truncated distributions • Expanded material on surplus models and ruin computations • Discussions of nonparametric prediction intervals, option pricing diagnostics, AN INTRODUCTION 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 unified approach to both financial and actuarial modeling through the use of general status structures. The authors define future time- BOROWIAK dependent financial actions in terms of a status structure that may be either SHAPIRO deterministic or stochastic. They show how deterministic status structures lead to DALE S. BOROWIAK classical interest and annuity models, investment pricing models, and aggregate claim models. They also employ stochastic status structures to develop financial and actuarial models, such as surplus models, life insurance, and life annuity ARNOLD F. SHAPIRO models. C8508 C8508_Cover.indd 1 10/8/13 8:53 AM FINANCIAL AND ACTUARIAL STATISTICS AN INTRODUCTION SECOND EDITION DALE S. BOROWIAK University of Akron Ohio, USA ARNOLD F. SHAPIRO Pennsylvania State University USA 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. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit- ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. 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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 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 Confidence Intervals ..............................................................44 2.1.3 Percentiles and Prediction Intervals ...................................45 2.1.4 Confidence 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 iii 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
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