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Analytic Approach to Estimating the Required Surplus, Benchmark Profit, and Optimal Reinsurance Retention for an Insurance Enterprise
Title:  An Analytic Approach to Estimating the Required Surplus, Benchmark Profit, and Optimal Reinsurance Retention for an Insurance Enterprise. 
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Name(s): 
Boor, Joseph A. (Joseph Allen), author Born, Patricia, professor codirecting dissertation Case, Bettye Anne, professor codirecting dissertation Tang, Qihe, professor codirecting dissertation Rogachev, Grigory, university representative Okten, Giray, committee member Aldrovandi, Ettore, committee member Paris, Steve, committee member Department of Mathematics, degree granting department Florida State University, degree granting institution 

Type of Resource:  text  
Genre:  Text  
Issuance:  monographic  
Date Issued:  2012  
Publisher:  Florida State University  
Place of Publication:  Tallahassee, Florida  
Physical Form: 
computer online resource 

Extent:  1 online resource  
Language(s):  English  
Abstract/Description:  This paper presents an analysis of the capital needs, needed return on capital, and optimum reinsurance retention for insurance companies, all in the context where claims are either paid out or known with certainty within or soon after the policy period. Rather than focusing on how to estimate such values using Monte Carlo simulation, it focuses on closed form expressions and approximations for key quantities that are needed for such an analysis. Most of the analysis is also done using a distributionfree approach with respect to the loss severity distribution, so minimal or no assumptions surrounding the specific distribution are needed when analyzing the results. However, one key parameter, that is treated via an exhaustion of cases, involves the degree of parameter uncertainty, the number of separate lines of business involved. This is done for the no parameter uncertainty monoline compound Poisson distribution as well as situations involving (lognormal) severity parameter uncertainty, (gamma/negative binomial) count parameter uncertainty, the multiline compound Poisson case, and the compound Poisson scenario with parameter uncertainty, and especially parameter uncertainty correlated across the lines of business. It shows how the risk of extreme aggregate losses that is inherent in insurance operations may be understood (and, implicitly, managed) by performing various calculations using the loss severity distribution, and, where appropriate, key parameters driving the parameter uncertainty distributions. Formulas are developed that estimate the capital and surplus needs of a company(using the VaR approach), and therefore the profit needs of a company that involve tractable calculations. As part of that the process the benchmark loading for profit, reflecting both the needed financial support for the amount of capital to adequately secure to a given one year survival probability, and the amount needed to recompense investors for diversifiable risk is discussed. An analysis of whether or not the loading for diversifiable risk is needed is performed. Approximations to the needed values are performed using the moments of the capped severity distribution and analytic formulas from the frequency distribution as inputs into method of moments normal and lognormal approximations to the percentiles of the aggregate loss distribution. An analysis of the optimum reinsurance retention/policy limit is performed as well, with capped loss distribution/frequency distribution equations resulting from the relationship that the marginal profit (with respect to the loss cap) should be equal to the marginal expense and profit dollar loading with respect to the loss cap. Analytical expressions are developed for the optimum reinsurance retention. Approximations to the optimum retention based on the normal distribution were developed and their error analyzed in great detail. The results indicate that in the vast majority of practical scenarios, the normal distribution approximation to the optimum retention is acceptable. Also included in the paper is a brief comparison of the VaR (survival probability) and expected policyholder deficit (EPD) and TVaR approaches to surplus adequacy (which conclude that the VaR approach is superior for most property/casualty companies); a mathematical analysis of the propriety of insuring the upper limits of the loss distribution, which concludes that, even if unlimited funds were available to secure losses in capital and reinsurance, it would not be in the insured's best interest to do so. Further inclusions to date include a illustrative derivation of the generalized collective risk equation and a method for interpolating ``along'' a mathematical curve rather than directly using the values on the curve. As a prelude to a portion of the analysis, a theorem was proven indicating that in most practical situations, the n1st order derivatives of a suitable probability mass function at values L, when divided by the product of L and the nth order derivative, generate a quotient with a limit at infinity that is less than 1/n.  
Identifier:  FSU_migr_etd4726 (IID)  
Submitted Note:  A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor in Philosophy.  
Degree Awarded:  Summer Semester, 2012.  
Date of Defense:  April 19, 2012.  
Keywords:  funding level, required profit, required surplus, retention, survival probability of an insurance company  
Bibliography Note:  Includes bibliographical references.  
Advisory Committee:  Patricia Born, Professor CoDirecting Dissertation; Bettye Anne Case, Professor CoDirecting Dissertation; Qihe Tang, Professor CoDirecting Dissertation; Grigory Rogachev, University Representative; Giray Okten, Committee Member; Ettore Aldrovandi, Committee Member; Steve Paris, Committee Member.  
Subject(s):  Mathematics  
Persistent Link to This Record:  http://purl.flvc.org/fsu/fd/FSU_migr_etd4726  
Host Institution:  FSU 
Boor, J. A. (J. A. ). (2012). An Analytic Approach to Estimating the Required Surplus, Benchmark Profit, and Optimal Reinsurance Retention for an Insurance Enterprise. Retrieved from http://purl.flvc.org/fsu/fd/FSU_migr_etd4726