advertisement

Incorporating Reinsurance and Risk Loads in Primary Insurance Pricing Presented By: David Appel, Ph.D. CAS Ratemaking Seminar March 11, 2004 Objectives 1. 2. 3. Briefly describe problems with direct ratemaking Outline methodology for incorporating reinsurance costs into rates Discuss allocation of profit based on residual risk Milliman USA Perspective is Bureau Ratemaking (but applicable to any direct ratemaking) Direct premiums, losses and expenses statewide (entire industry viewed as single aggregate company) Cost of Capital (i.e., fair return) = “r” – based on capital market data Degree of capitalization (i.e., leverage) = “P/S” ratio based on actual industry data for primary companies Milliman USA Ratemaking in Simplest Terms Direct Premium = Direct Losses + Expenses + r * (Dir Prem/(P/S)) or, Direct Premium = Direct Losses + Expenses [1-r/(P/S)] Given r and P/S, rates depend solely on direct losses and expenses Implicitly assumes all layers of loss distribution supported by same amount of capital Milliman USA Problems with Implicit Assumption Amount of capital based on observed P/S for primary companies Cost of capital based on observation in capital market (principally for primary companies) Primary company cost of capital and amount of capital depend on reinsurance availability Reinsurers (especially Cat reinsurers) have more supporting capital than primary insurers The cost of this additional capital must be reflected in rates Milliman USA Simple Example to Illustrate Problem Assume, without loss of generality: 1. 2. 3. 4. No expenses or investment income – premium depends solely on E[loss] and required profit Required return determined in capital markets Amount of capital (surplus) based on E[loss], and E[loss]/surplus ratio based on observed data E[loss] = $1.25 million, split 80%/20%, primary/reinsurer Milliman USA Illustration of Problem: Failure to Reflect Total Capital Supporting Insurance Transaction Current Procedure based on Direct Loss Expected Loss E[Loss]/Surplus Surplus Return on Equity Required Profit Premium Alternative Split into Primary and Reinsurance Layers $1,250,000 2 625,000 10% Primary $1,000,000 2 500,000 10% Reinsurance $250,000 0.5 500,000 10% 62,500 50,000 50,000 1,312,500 1,050,000 300,000 Milliman USA Setting Bureau Rates to Include Cost of Reinsurance Required Premium = Direct loss + expense + net cost of reinsurance = Direct loss + expense + reinsurer expense & profit However, because this is a bureau rate, there is no actual reinsurance contract; don’t know ceded premium and loss Therefore, to get reinsurer expense and profit, must estimate competitive market price for reinsurance Milliman USA Estimating a Competitive Reinsurance Premium Assume statewide reinsurance program Attachment point = 2x annual average loss Limit = 1 in 100 year event (99th %ile of loss distribution 10% quota share retained by primary co. within this layer Estimate ceded losses, given cat loss distribution and assumed program Add reinsurer expenses and profit to ceded loss Milliman USA Cat Reinsurance Premium Calculation of Reinsurance Cost Statewide Total Total (1) Hurricane Losses (2) Loss Adjustment Expense Factor (3) Hurricane Losses and Loss Expenses (1) x (2) (4) Percent Reinsured (5) Reinsured Losses and Loss Expenses (3) x (4) (6) Reinsurance Expense Factor (7) Reinsurance Loss+Expenses (5) / (6) (8) Reinsurance Expense Cost (7)-(5) (9) Reinsurance Premium to Surplus Ratio (10) Reinsurer Underwriting Return Percent of Surplus (11) Reinsurer Underwriting Return Percent of Premium (10) / (9) (12) Reinsurance Premium (7) / (1.000-(11)) (13) Reinsurance Profit Cost (12) - (7) (14) Direct Losses (15) Direct Losses and LAE (14) x (2) (16) Direct Variable Expense (Excl Reinsurance) (17) Direct Premium Including Reinsurance Cost ((15) + (12) + (8)) / (1.000-(16)) (18) Reinsurance Expense Cost as % of Direct Premium (8) / (17) (19) Reinsurance Profit Cost as % of Direct Premium (13) / (17) (20) Reinsurance Premium as % of Direct Premium (11) / (17) 31,909,123 1.154 36,823,128 0.511 18,823,846 0.90 20,915,385 2,091,538 0.37 15.4% 41.5% 35,752,015 14,836,630 48,447,610 55,908,541 32.00% 107,112,809 1.95% 13.85% 33.38% Milliman USA Statewide Premium Incl. Reinsurance Costs Statewide Total Total (1) Expected Value of Net Losses (2) Expected Value of Ceded Losses (3) Expected Value of All Losses (1)+(2) (4) Commission and Brokerage (5) Other Acquisition (6) General (7) Taxes Licenses and Fees (8) Deviations (9) Reinsurance Expense Cost (10) Reinsurance Profit Cost (11) Net Profit and Contingencies (12) Loss Adjustment Expense Factor (13) Total Indicated Premium ((3) x (12)) / (1-Sum[(4) to (11)]) (14) Total Indicated Underwriting Profit (11) x (13) 32,135,784 16,311,825 48,447,610 10.00% 5.00% 5.00% 2.00% 5.00% 1.95% 13.85% 5.00% 1.154 107,112,809 5,355,640 Milliman USA Allocating Reinsurance Costs and Profit Proportional to Risk Reinsurance costs vary by region, but… even after reinsurance, residual risk varies by region. Therefore, must allocate both reinsurance cost and profit relative to regional contribution to total risk. Identify risk metric Compute risk measure for each region Allocate costs proportional to risk Milliman USA Possible Risk Measures Variance of losses Standard deviation of losses Probability of ruin Expected policyholder deficit Any others you might like!! Milliman USA Allocation Using Standard Deviation Zone 1 Allocation of Primary Company Amounts (1) Standard Deviation of Net Losses (2) Allocation Percent [(1) / Sum(1)] (3) Expected Profit to Allocate (4) Expected Losses (5) Loss Adjustment Expense Factor (6) Expected Losses and Loss Expenses [(4) x (5)] (7) Expected Investment Income on Policy Reserves Percent (8) Underwriting Profit and Contingencies (3) - (6) x (7) (9) Variable Expense Percent Allocation of Reinsurer Amounts (10) Standard Deviation of Ceded Losses (11) Allocation Percent [(1) / Sum(1)] (12) Expected Profit to Allocate (13) Expected Ceded Losses (14) Loss Adjustment Expense Factor (15) Expected Losses and Loss Expenses [(4) x (5)] (16) Expected Investment Income on Policy Reserves Percent (17) Reinsurer Profit (12) - (15) x (16) (18) Reinsurer Expenses (Total (18) allocated with (13)) Summary of Expense Provisions (19) Indicated Premium [((6) + (8) + (17) + (18)) / (1.000 - (9))] (20) Underwriting Profit and Contingencies (Percent) (8) / (19) (21) Reinsurer Proift (Percent) (17) / (19) (22) Reinsurer Expenses (Percent) (18) / (19) Zone 2 Zone 3 Sum 59,635,323 85.2% 6,226,869 14,879,388 1.154 17,170,814 5.3% 5,322,087 6,998,894 10.0% 730,795 7,941,806 1.154 9,164,844 5.3% 247,872 3,371,907 4.8% 352,080 9,314,590 1.154 10,749,037 5.3% (214,319) 70,006,124 100.0% 7,309,744 32,135,784 1.154 37,084,695 5.3% 5,355,640 27.00% 27.00% 27.00% 27.00% 63,536,239 82.9% 13,127,070 14,283,493 1.154 16,483,151 5.3% 12,258,523 9,376,989 12.2% 1,937,357 1,648,536 1.154 1,902,411 5.3% 1,837,113 3,698,257 4.8% 764,088 379,796 1.154 438,284 5.3% 740,994 76,611,486 100.0% 15,828,515 16,311,825 1.154 18,823,846 5.3% 14,836,630 1,831,461 211,379 48,698 2,091,538 72,693,200 7.32% 18,306,328 1.35% 16,113,280 -1.33% 107,112,809 5.00% 16.86% 10.04% 4.60% 13.85% 2.52% 1.15% 0.30% 1.95% Milliman USA Summary of Profit Factors Under Alternative Allocations Summary - Reinsurance Cost and Profit by Zone Standard Deviation Method Variance Method Underwriting Profit and Contingencies Reinsurer Proift (Percent) Reinsurer Expenses (Percent) Zone 1 Zone 2 7.32% 1.35% 16.86% 10.04% 2.52% 1.15% T otal Profit plus Reinsurance Cost 26.70% 12.54% 3.57% 20.80% 8.14% 18.88% 2.37% -2.52% 1.55% 1.39% -3.70% 0.20% 0.33% 5.00% 13.85% 1.95% 29.39% 0.42% -3.17% 20.80% 6.11% 17.64% 2.54% 3.61% 8.89% 1.13% 1.70% 2.85% 0.30% 5.00% 13.85% 1.95% T otal Profit plus Reinsurance Cost 26.30% 13.63% 4.84% 20.80% Average Across Methods 27.46% 8.86% 1.75% 20.80% Underwriting Profit and Contingencies Reinsurer Proift (Percent) Reinsurer Expenses (Percent) T otal Profit plus Reinsurance Cost Probability of Ruin Method Underwriting Profit and Contingencies Reinsurer Proift (Percent) Reinsurer Expenses (Percent) Zone 3 -1.33% 4.60% 0.30% Sum 5.00% 13.85% 1.95% Milliman USA