advancing the science of risk

Variance Advancing the Science of Risk 4350 North Fairfax Drive Suite 250 Arlington, Virginia 22203 www.variancejournal.org 2010 VOLUME 04 ISSUE 02 2010 121 Bootstrap Estimation of the Predictive Distributions of Reserves Using Paid and Incurred Claims by Huijuan Liu and Richard Verrall 136 Robustifying Reserving by Gary G. Venter and Dumaria R. Tampubolon 155 Prediction Error of the Future Claims Component of Premium Liabilities under the Loss Ratio Approach by Jackie Li 170 The Economics of Insurance Fraud Investigation: Evidence of a Nash Equilibrium by Stephen P. D’Arcy, Richard A. Derrig, and Herbert I. Weisberg 191 Optimal Layers for Catastrophe Reinsurance by Luyang Fu and C. K. “Stan” Khury VOLUME 04 ISSUE 02 VARIANCE Mission Variance is a peer-reviewed journal published by the Casualty Actuarial Society to disseminate work of interest to casualty actuaries worldwide. The focus of Variance is original practical and theoretical research in casualty actuarial science. Signicant survey or similar articles are also considered for publication. Membership in the Casualty Actuarial Society is not a prerequisite for submitting papers to the journal and submission by non-CAS members is encouraged. For more inFormation, visit the Cas Web site: WWW.CasaCt.org EntErprisE risk M anagEMEnt s yMposiuM March 14-16, 2011 Chicago, IL ratEMaking & p roduct M anagEMEnt (rpM) s EMinar March 20-22, 2011 New Orleans, LA cas spring MEEting May 15-18, 2011 Palm Beach, FL sEMinar on r EinsurancE June 6-7, 2011 Philadelphia, PA casualty loss r EsErvE sEMinar (clrs) September 15-18, 2011 Las Vegas, NV These upcoming CAS Continuing Education Opportunities will keep you up to date on the latest trends in actuarial science. cas profEssional E ducation calEndar Dates to Remember Table of Contents 115 A Note from the Editor by Roger W. Bovard 116 Contributors to this Issue 121 Bootstrap Estimation of the Predictive Distributions of Reserves Using Paid and Incurred Claims by Huijuan Liu and Richard Verrall This paper presents a bootstrap approach to estimate the prediction distributions of reserves produced by the Munich chain ladder (MCL) model. The MCL model was introduced by Quarg and Mack (2004) and takes into account both paid and incurred claims information. In order to produce bootstrap distributions, this paper addresses the application of bootstrapping methods to dependent data, with the consequence that correlations are considered. Numerical examples are provided to illustrate the algorithm and the prediction errors are compared for the new bootstrapping method applied to MCL and a more standard bootstrapping method applied to the chain-ladder technique. 136 Robustifying Reserving by Gary G. Venter and Dumaria R. Tampubolon Robust statistical procedures have a growing body of literature and have been applied to loss severity tting in actuarial applications. An introduction of robust methods for loss reserving is presented in this paper. In particular, following Tampubolon (2008), reserving models for a development triangle are compared based on the sensitivity of the reserve estimates to changes in individual data points. This measure of sensitivity is then related to the generalized degrees of freedom used by the model at each point. 155 Prediction Error of the Future Claims Component of Premium Liabilities under the Loss Ratio Approach by Jackie Li In this paper we construct a stochastic model and derive approximation formulae to estimate the standard error of prediction under the loss ratio approach of assessing premium liabilities. We focus on the future claims component of premium liabilities and examine the weighted and simple average loss ratio estimators. The resulting mean square error of prediction contains the process error component and the estimation error component, in which the former refers to future claims variability while the latter refers to the uncertainty in parameter estimation. We illustrate the application of our model to public liability data and simulated data. 114 CASUALTY ACTUARIAL SOCIETY VOLUME 4 / ISSUE 2 Variance Advancing the Science of Risk 170 The Economics of Insurance Fraud Investigation: Evidence of a Nash Equilibrium by Stephen P. D’Arcy, Richard A. Derrig, and Herbert I. Weisberg The behavior of competing insurance companies investigating insurance fraud follows one of several Nash Equilibria under which companies consider the claim savings, net of investigation cost, on a portion, or all, of the total claim. This behavior can reduce the effectiveness of investigations when two or more competing insurers are involved. Cost savings are reduced if the suboptimal equilibrium prevails, and may instead induce fraudulent claim behavior and lead to higher insurance premiums. Alternative cooperative and noncooperative arrangements are examined that could reduce or eliminate this potential inefciency. 191 Optimal Layers for Catastrophe Reinsurance by Luyang Fu and C. K. “Stan” Khury Insurers purchase catastrophe reinsurance primarily to reduce underwriting risk in any one experience period and thus enhance the stability of their income stream over time. Reinsurance comes at a cost and therefore it is important to maintain a balance between the perceived benet of buying catastrophe reinsurance and its cost. This study presents a methodology for determining the optimal catastrophe reinsurance layer by maximizing the risk-adjusted underwriting prot within a classical mean-variance framework. VOLUME 4 / ISSUE 2 CASUALTY ACTUARIAL SOCIETY 115 Variance Advancing the Science of Risk When I started off as an actuarial student, my mentors in life insurance introduced me to the technique known as “general reasoning.” One of the rst examples I encountered was a derivation for the present value (P) of an ordinary annuity certain. The general reasoning derivation goes as follows: The sum of $1 invested at rate i will yield annual income of i per year for n years (present value i*P) with the $1 principal still intact at the end of n years (present value v n ). Thus 1=iP+v n . From this expression, the well- known formula for P follows by simple algebra. I recall being impressed with the way this technique gets to the heart of the matter while bypassing the somewhat messier algebra associated with the mathematical derivation. There are a number of such general reasoning demonstrations in actuarial mathematics on the life side, but I have yet to encounter any in property and casualty insurance. However, I do occasionally encounter another technique for deriving useful results in an elegant manner while avoiding tedious algebraic manipulation. The technique I have in mind is transition to a higher level of abstraction. An example of this technique can be found in the very readable paper “Credibility Formulas of the Updating Type” by Jones and Gerber. The derivation in this paper is somewhat abstract, but accessible to most actuaries. The benet of abstraction is a smoother ow resulting in increased understanding of the subject matter. A reduction in the level of abstraction would probably force a substantial increase in algebraic manipulation, which is actually more distracting than illuminating. In an appendix to the paper, the authors present an alternate derivation that is even shorter, but more abstract. Readers familiar with linear algebra concepts are able to appreciate the shorter derivation. These observations are intended for readers as well as authors. It is the na- ture of those who author technical papers to build on other technical papers previously published. Thus, knowledge is advancing as well as increasing. As a practical matter, I encourage readers to invest time mastering some of the more advanced tools required to stay abreast. Jones, D.A., and H.U. Gerber, “Credibility Formulas of the Updating Type,” Transactions of the Society of Actuaries 27, 1975, pp. 31-46. Available for download from the Society of Actuaries Web Site: http://www.soa.org/library/research/transactions-of-society-of-actuaries/1975/ january/tsa75v274.pdf Roger Bovard A Note from the Editor 116 CASUALTY ACTUARIAL SOCIETY VOLUME 4 / ISSUE 2 Variance Advancing the Science of Risk Contributors to this Issue Stephen P. D’Arcy Stephen P. D’Arcy, FCAS, MAAA, Ph.D., is professor emeritus of - nance at the University of Illinois, a visiting lecturer at California State University Fullerton’s Mihaylo Col- lege of Business and Economics, and President of D’Arcy Risk Consult- ing, Inc. He is a past president of the Casualty Actuarial Society and of the American Risk and Insurance Asso- ciation. Luyang Fu Luyang Fu, FCAS, MAAA, is the di- rector of predictive modeling for State Auto Insurance Companies, where he leads the developments of personal lines pricing models, com- mercial lines underwriting models, and corporate DFA and ERM mod- els. Prior to joining State Auto, he served in various actuarial roles with both Grange Insurance and Bristol West Insurance. He holds a B.S. and an M.S. in economics from Renmin University of China, and a M.S. in nance and a doctorate in agricul- tural and consumer economics from University of Illinois at Urbana- Champaign. Richard A. Derrig Richard Derrig, Ph.D., is president of OPAL Consulting LLC, which provides research and regulatory support to the P&C insurance indus- try. For over 27 years, Dr. Derrig held various posts at the Massachu- setts Bureau of Automobile Insurers and the Massachusetts Bureau of In- surance Fraud. He has won several prizes, including the CAS Ratemak- ing Prize (1993 co-winner), the ARIA Prize (2003), the RIMS Edith F. Li- chota Award (1998), and ARIA’s Mehr Award (2005). Dr. Derrig has coedited three books on solvency and coauthored papers applying fuzzy set theory to insurance. Roger W. Bovard Editor in Chief Editorial Board EDITORS: Shawna S. Ackerman Avraham Adler Todd Bault Morgan Haire Bugbee Daniel A. Crifo Susan L. Cross Stephen P. D’Arcy Enrique de Alba Ryan M. Diehl Robert J. Finger Steven A. Gapp Emily Gilde Annette J. Goodreau Richard W. Gorvett David Handschke Philip E. Heckman Daniel D. Heyer John Huddleston Ali Ishaq Eric R. Keen Ravi Kumar ASSISTANT EDITORS: Joel E. Atkins Gary Blumsohn Frank H. Chang Clive L. Keatinge Dmitry E. Papush Christopher M. Steinbach Richard Fein Associate Editor— Peer Review Dale R. Edlefson Associate Editor— Copyediting Gary G. Venter Associate Editor— Development VOLUME 4 / ISSUE 2 CASUALTY ACTUARIAL SOCIETY 117 Variance Advancing the Science of Risk Contributors to this Issue Jackie Li Dr. Jackie Li is currently an assistant professor in the Division of Banking and Finance at Nanyang Business School (NBS), Nanyang Technologi- cal University (NTU), Singapore. Dr. Li obtained his Ph.D. degree in actu- arial studies from the University of Melbourne, Australia, and is a Fel- low of the Institute of Actuaries of Australia (FIAA). He has been lec- turing and tutoring various actuarial courses and his main research areas are stochastic loss reserving for prop- erty/casualty insurance and mortality projections. Before joining NBS, Dr. Li worked as an actuary in the areas of property/casualty insurance and superannuation. Huijuan Liu Dr. Huijuan Liu earned a Ph.D. at Cass Business School, London, in 2008. Her research was sponsored by Lloyd’s of London and she was su- pervised by Professor Richard Ver- rall. Together, they have recently published two papers in the ASTIN Bulletin. Dr. Liu now works for the Financial Services Authority in Lon- don. Yin Lawn Pierre Lepage Martin A. Lewis Xin Li Cunbo Liu Kevin Mahoney Donald F. Mango Leslie Marlo Stephen J. Mildenhall Christopher J. Monsour Roosevelt C. Mosley Jr. Mark W. Mulvaney Prakash Narayan Adam Niebrugge Darci Z. Noonan Jonathan Norton A. Scott Romito David Ruhm Theodore R. Shalack John Sopkowicz John Su James Tanser Neeza Thandi George W. Turner Jr. Trent R. Vaughn Cheng-Sheng Peter Wu Satoru Yonetani Navid Zarinejad Yingjie Zhang Alexandros Zimbidis COPY EDITORS: Nathan J. Babcock Laura Carstensen Hsiu-Mei Chang Andrew Samuel Golfin Jr. Mark Komiskey William E. Vogan C. K. “Stan” Khury C. K. ‘Stan’ Khury, FCAS, MAAA, CLU is a principal with Bass & Khury, an independent actuarial con- sulting rm located in Las Vegas, Nevada. Stan is a past president of the CAS and has written numerous papers and article in CAS publica- tions over a period spanning nearly 40 years. He provides a wide range of actuarial consulting services to in- surers, reinsurers, intermediaries, regulators, and law rms. CAS STAFF: Elizabeth A. Smith Manager of Publications Donna Royston Publication Production Coordinator Sonja Uyenco Desktop Publisher 118 CASUALTY ACTUARIAL SOCIETY VOLUME 4 / ISSUE 2 Variance Advancing the Science of Risk Dumaria R. Tampubolon Dumaria Rulina Tampubolon com- pleted her S1 degree (similar to an honors degree) in mathematics, ma- joring in statistics, at Institut Teknologi Bandung in Bandung, In- donesia. She holds a master of sci- ence degree in statistics from Monash University, Melbourne, Australia; and a Ph.D. degree in actuarial stud- ies from Macquarie University, Syd- ney, Australia. Her research interest is in general insurance and applied statistics. Currently she teaches at the Faculty of Mathematics and Nat- ural Sciences, Institut Teknologi Bandung. She can be reached at du- maria@math.itb.ac.id and drtampu- bolon@gmail.com. Mailing Address Variance 4350 North Fairfax Drive Suite 250 Arlington, Virginia 22203 staffeditor @ variancejournal.org For information on submitting papers visit: www.variancejournal.com Variance is published twice yearly by the Casualty Actuarial Society. Telephone: (703) 276- 3100; Fax: (703) 276-3108; E-mail: office@casact.org. Presorted Bound Printed Matter postage is paid at Baltimore, Maryland. Publications Mail Agreement No. 40035891. Return Undeliverable Canadian Addresses to PO Box 503, RPO West Beaver Creek, Richmond Hill, ON L4B 4R6. Postmaster: Send address changes to: Variance, 4350 North Fairfax Drive, Suite 250, Arlington, Virginia 22203. For permission to reprint material from Variance, please write to the editor in chief. Letters to the editor can be sent to staffeditor@variancejournal.org or the CAS Office. The Casualty Actuarial Society is not responsible for statements or opinions expressed in the articles, discussions, or letters printed in Variance. © 2010 Casualty Actuarial Society. Gary G. Venter Gary G. Venter, FCAS, CERA, ASA, MAAA, is head of economic capital modeling at Chartis and teaches graduate courses in actuarial science at Columbia University. His 35+ years in the insurance and reinsur- ance industry included stints at the Instrat group, which migrated from EW Payne through Sedgwick to Guy Carpenter; the Workers Compensa- tion Reinsurance Bureau; the Na- tional Council on Compensation In- surance; Prudential Reinsurance; and Fireman’s Fund. Gary has an under- graduate degree in mathematics and philosophy from UC-Berkeley and a master’s degree in mathematics from Stanford University. He has served on a number of CAS committees and is on the editorial team of several ac- tuarial journals. Contributors to this Issue Richard Verrall Richard Verrall has been at City Uni- versity since 1987. He is an Honor- ary Fellow of the Institute of Actuar- ies (1999), an Associate Editor of the British Actuarial Journal, the North American Actuarial Journal, and In- surance: Mathematics and Econom- ics, and a principle examiner for The Actuarial Profession (U.K.). Courses for industry include “Statistics for Insurance,” an introductory course aimed at non-specialists, such as un- derwriters, in the uses of statistics in risk assessment; “Stochastic Claims Reserving,” a specialist course for actuaries and statisticians on how to apply statistical methods to reserving for non-life companies; and “Bayes- ian Actuarial Models,” an introduc- tory course in Bayesian methods for premium rating and reserving. VOLUME 4 / ISSUE 2 CASUALTY ACTUARIAL SOCIETY 119 Variance Advancing the Science of Risk Contributors to this Issue Herbert I. Weisberg Herbert I. Weisberg received his Ph.D. in statistics from Harvard in 1970. He is the president of Correla- tion Research, Inc., a consulting rm specializing in litigation support and analysis of insurance fraud. He has published numerous articles and re- ports related to application and de- velopment of statistical methodolo- gy, and is a co-author of Statistical Methods for Comparative Studies: Techniques for Bias Reduction. Re- cently, his research has related to causal inference in statistics, draw- ing on and extending the burgeoning literature in this area. He has recently written a new book titled Bias and Causation: Models and Judgment for Valid Comparisons. 120 CASUALTY ACTUARIAL SOCIETY VOLUME 4 / ISSUE 2 Variance Advancing the Science of Risk [...]... those of the Mack method Since the purpose of the MCL method is to use more data to improve the estimation of the reserves, it is expected that the prediction errors should be lower than the Mack model This is 132 confirmed for these data by Table 5, which shows that the prediction error, as a percentage of the reserve, is lower for the MCL reserves than the prediction error of CL the reserves from the. .. preserve the required dependence Second, the correlation coefficient of paid and incurred claims is equal to the correlation coefficient of those residuals, as stated in Equations (2.6) and (2.10) Thus, in the case of the paid claims data, the triangles (which have the same dimensions) containing the residuals of the observed paid link ratios and the residuals of the ratios of incurred over paid (except the. .. derivative is just the reciprocal of the sum of the previous cumulatives, so the GDF for the cell is the quotient of its previous cumulative and the sum Thus these GDFs sum down a column to unity, so each development factor uses up a total GDF of 1.0 Essentially each factor uses 1 degree of freedom, agreeing with standard analysis The average GDF in a column is thus the reciprocal of the number of observations... link ratios and the ratios of incurred over paid claims and the inverse are all grouped together (Note here that an alternative approach would be to group three sets of residuals: the residuals of the paid and incurred link ratios and either the residuals of the paid over incurred ratios or the inverse This would produce the same results as grouping four sets of residuals, as the residuals of paid over... constant instead of factors to represent late development The effects of the diagonal dummies can also be seen, especially in the right of the triangle Now only one point has impact greater than 2, and one greater than 4 Table 6 shows the GDFs for the regression model For regression models the GDFs for the CASUALTY ACTUARIAL SOCIETY 143 Variance Advancing the Science of Risk Figure 2 Impact of regression... Unbiased estimates of the parameters In this section, we summarize the unbiased estimates of the parameters derived by Quarg and Mack (2004) For the paid data, estimates are required for the parameters of the development factors, the variances and also the correlation coefficient The estimates of the paid development factor parameters can be interpreted as weighted averP ages of the observed development... models of the U.S one-month treasury bill rates at monthly intervals Typical models postulate that the volatility of the rate is higher when the rate itself is higher Often the volatility is proposed to be proportional to the pth power of the rate The question is– what is p? One model, the CIR or Cox, Ingersoll, Ross model, assumes a p value of 0.5 Other models postulate p as 1 or even 1.5, and others... look at the generalized degrees of freedom (GDF) at each point This is defined as the derivative of the fitted value with respect to the observed value If this is near 1, the point’s initial degree of freedom has essentially been used up by the model The GDF is a measure of how much a point is able to pull the fitted value towards itself Part of the impact of a point is this power to influence the model,... statistic of interest Obviously, when bootstrapping the recursive MCL method, the residuals of the paid and incurred link ratios are required instead of the raw data The question arises of how to deal with these residuals in order to meet the requirement of not breaking the observed dependence between paid and incurred claims The answer is to group all four sets of residuals calculated in the MCL method,... development triangle used in Venter (2007a) Note that the first two accident years are developed all the way to the end of the triangle, at lag 11 Table 3 shows the impact of each cell on the reserve estimate using the usual CASUALTY ACTUARIAL SOCIETY 141 Variance Advancing the Science of Risk Figure 1 Impact of chain ladder by diagonal Table 4 GDFs of CL L0 AY0 AY1 AY2 AY3 AY4 AY5 AY6 AY7 AY8 AY9 AY10 . (2.10). Thus, in the case of the paid claims data, the triangles (which have the same dimensions) con- taining the residuals of the observed paid link ratios and the residuals of the ratios of incurred over. 115 Variance Advancing the Science of Risk When I started off as an actuarial student, my mentors in life insurance introduced me to the technique known as “general reasoning.” One of the rst. impressed with the way this technique gets to the heart of the matter while bypassing the somewhat messier algebra associated with the mathematical derivation. There are a number of such general