Comparison of Risk Allocation Methods - Bohra-Weist DFAIC Distributions

This purpose of this study is to compare the results of several risk allocation methods for a realistic insurance company example. The basis for the study is the fitted loss distributions of Bohra and Weist (2001), which were derived from the hypothetical data for DFA Insurance Company (DFAIC). This hypothetical data was distributed by the Casualty Actuarial Society's Committee on Dynamic Financial Analysis, as part of its 2001 call for papers. In addition, Ruhm and Mango (2003) utilized these fitted distributions to produce 2000 simulated loss scenarios for DFAIC. This detailed simulation data was included in a spreadsheet that accompanied their paper. The Ruhm-Mango simulation data is also an important source of input for this study. All of the data, analysis and results for the study" are shown on the accompanying Excel Workbook ("bohra-weist data.xls"). The actual Bohra-Weist fitted distributions are shown on the "Data for Study" sheet. This sheet also provides some explanatory notes regarding the Ruhm-Mango simulation data. The actual Ruhm-Mango simulation data, sorted in ascending order, is shown on several different sheets, including the "RMK Capital Consumption" sheet. In general, the calculations for each of the individual methods are displayed on a separate sheet of the Workbook. There is also a "summary" sheet that summarizes the resulting allocation and pricing for each method. In order to focus on differences in the allocation results for the various methods, each of the methods has been "calibrated" to the same overall corporate premium level. This overall premium amount is $1,242,777, which represents a total risk loading of $100,000. This total corporate risk load could be based on a financial pricing model (such as the Fama-French 3-Factor Model), or it could simply be based on a judgmental ROE.or combined ratio goal that has been set by the Board of Directors. In this study, we have also ignored complications caused by existing reserves, loss discounting, and long-tailed payouts, t In other words, we are assuming that these loss distributions apply to a start-up insurance company with no existing reserves. We are also implicitly ignoring differences in the duration of payments for the various lines. Some of the methods in this study also require a specific value for the policyholders' surplus of DFAIC. We have assumed that surplus at the time of writing is $900,000. This implies a premium-to-surplus ratio of $1,242,777 / $900,000 = 1.38. The expected return-on-equity (ROE) for the company, ignoring investment income, is $100,000 / $900,000 = 11.1%. Many of the methods in this study 2 directly determine the capital cost allocation for each of the subject lines of business. For these methods, the resulting premium by line is then determined by the following formula: Premium = Expected Loss + Pro-Rata Allocation of Risk Load x $100,000 Note that the expected loss in this formula is undiscounted. Also, there is no provision made for underwriting and loss adjustment expenses. The remaining methods in this study 3 directly determine the premium for each of the subject lines of business. For these methods, the total premium is "calibrated" at $1,242,777. The corresponding capital cost allocation by line is then determined according to the following formula: Capital Cost Allocation = (Premium - Expected Loss) / $100,000 The remainder of this paper will provide explanatory notes for each of the methods, followed by short summary, of the observations and results of the study.