Session C-5: ARIA Prize Paper CAS Spring Meeting May 2006 The Use of DFA to Determine Whether an Optimal Growth Rate Exists for a Property-Liability Insurer by Stephen P. D’Arcy and Richard W. Gorvett University of Illinois Published in the Journal of Risk and Insurance, December, 2004 Overview Introduction Dynamic Financial Analysis Aging Phenomenon Market Value of P-L Insurance Company Optimal Growth Rate Analysis of Results Dynamic Financial Analysis An approach to modeling insurance companies Solvency testing Ratings DFA models also allow managers to test various operational strategies Objective of Paper • Utilize a DFA model to determine the optimal growth rate based on - mean-variance efficiency - stochastic dominance - constraints of leverage • Based on the latest version of a public access DFA model (DynaMo3) http://www.pinnacleactuaries.com/ Aging Phenomenon • New business has a very high loss ratio, often in excess of the initial premium • The loss ratio then declines with each renewal cycle to the profitable point • Longer-term business has an even lower loss ratio, making it very profitable • A P-L insurer’s growth rate has a significant effect on profitability Automobile Insurance Loss Ratios by Age of Cohort 120 Loss Ratio (%) 100 80 Firm A Firm B 60 Firm C Firm D Firm E 40 20 0 1 2 3 4 5 6 7 8 Age of Cohort (in Years) 9 10 11 12 Market Value of P-L Insurance Company • Determining the market value of a hypothetical property-liability insurer is not a simple task. • Only a few P-L insurers are stand-alone companies that are publicly traded, allowing the market value of the firm to be observed Approaches to Determine Company Value • Fama-French model (three factor model) r - Rf = beta x ( Km - Rf ) + bs x SMB + bv x HML + alpha SMB - small [cap] minus big HML - high [book/price] minus low • CAPM • Multiple Regression (our method) Multiple Regression Approach The market value of an insurer is measured by - Policyholders’ Surplus - Net Written Premium (the size of the book of business) - Combined Ratio and Operating Ratio (profitability) Companies in Sample Table 1 - Company Data (in Millions ) Year2001 Acceptance Allstate AIG Berkshire Chubb Cincinnati Fin CNA Hartford HCC ins Ohio Casualty Progessive SAFECO Selective St.Paul United F&C Total Revenue 176 28, 865 61, 766 37, 668 7, 754 2, 561 13, 203 15, 147 505 1, 902 7, 488 6, 863 1, 059 8, 943 473 Total Admitted Assets 440 39,290 52, 458 73, 400 18, 218 6, 873 32, 446 23, 997 904 3, 830 10, 391 9, 658 2, 209 22, 577 748 Market Value 70 31, 723 216, 528 94, 628 14, 250 7, 029 6, 919 16, 289 1, 520 1, 030 7, 846 4, 033 590 12, 257 215 Policyholders' Surplus Net Written Premium(P/C) Operating Ratio 129 13,796 15, 362 27, 103 3, 526 2, 530 6, 089 5, 804 398 768 2, 641 2, 280 519 4, 132 198 93 21,991 14, 007 11, 656 5, 997 2, 591 7, 663 5, 209 300 1, 472 7, 263 4, 439 927 6, 136 366 1. 150 0. 950 0. 910 1. 025 0. 990 0. 888 1. 409 0. 971 0. 925 0. 985 0. 917 1. 103 0. 958 1. 090 0. 960 Ratio of P/C NWP/Revenue 0. 529 0. 762 0. 227 0. 309 0. 773 1. 012 0. 580 0. 344 0. 594 0. 774 0. 970 0. 647 0. 875 0. 686 0. 774 Regression Analysis Table 2 Market Value Estimation Equation 1: LN(MV)=a+b*LN(PHS)+ c*LN(NWP)+d*CR Equation 2: LN(MV)=a+b*LN(PHS)+ c*LN(NWP)+d*OR Equation 1 2 a 0. 88 0. 51 S.E. 0. 48 0. 49 b 1. 01 0. 86 S.E 0. 05 0. 06 c 0. 16 0. 28 S.E. 0. 05 0. 06 d - 2. 37 - 1. 99 Least squares linear regression Based on the experience of 15 companies over the period 1990-2001 MV = Market Value PHS = Statutory Policyholders Surplus NWP =Net Written Premium CR=Combined Ratio OR = Operating Ratio S.E. 0. 35 0. 36 R2 0. 938 0. 932 Results of regression for each company separately Table 3 Market Value Estimation MV=a+b*PHS+ c*NWP+d*CR Dependent Variable: MV Independent Variables: PHS, NWP, CR Company a S.E. b S.E. c S.E d S.E. R2 447, 067 91, 359 1. 32 0. 30 0. 07 0. 21 - 442, 500 80, 883 0. 969 - 45, 842, 210 60, 741, 483 10, 310, 535 73, 592, 068 507, 769, 771 18, 250, 118 5. 86 0. 10 0. 48 2. 11 7. 49 0. 25 - 3. 01 1. 87 32. 87 13. 64 6. 61 0. 82 60, 912, 024 - 282, 623, 500 - 5, 794, 440 73, 496, 027 528, 222, 750 15, 089, 729 0. 835 0. 910 0. 983 CNA 565, 774 10, 802, 307 - 631, 440 7, 472, 749 8, 247, 771 2, 630, 205 - 2. 40 1. 83 0. 37 2. 67 0. 49 0. 17 4. 34 0. 89 0. 29 1. 69 0. 90 0. 22 - 3, 209, 198 - 10, 827, 410 2, 115, 101 7, 346, 370 8, 177, 836 1, 583, 555 0. 861 0. 876 0. 768 Hartford 42, 030, 565 271, 590, 558 - 0. 34 4. 90 - 1. 16 13. 35 - 18, 526, 345 147, 906, 538 0. 030 465, 647 3, 479, 368 724, 190 951, 537 3. 88 0. 91 1. 71 0. 26 - 0. 23 - 0. 54 1. 71 0. 39 - 720, 770 - 2, 201, 743 947, 986 971, 712 0. 571 0. 788 13, 816, 591 - 366, 248 1, 922, 264 13, 403, 390 4, 954, 621 1, 321, 201 5. 03 2. 68 0. 81 11. 60 - 0. 34 0. 82 - 0. 89 0. 54 0. 22 3. 93 0. 52 0. 51 - 15, 002, 413 1, 475, 766 - 1, 819, 411 13, 898, 864 4, 565, 682 1, 282, 157 0. 698 0. 791 0. 786 - 5, 654, 781 57, 765 22, 701, 635 6, 380, 946 180, 966 18, 248, 833 1. 38 3. 07 2. 13 0. 56 0. 37 0. 29 0. 98 - 1. 19 1. 57 0. 90 0. 24 0. 46 3, 563, 808 - 24, 077 - 23, 787, 168 7, 427, 642 167, 854 17, 244, 028 0. 889 0. 899 0. 444 1, 906, 580 3832821 1. 85 0. 06 Least squares linear regression Based on the experience of each company over the period 1990-2001 MV = Market Value PHS = Statutory Policyholders Surplus NWP =Net Written Premium CR=Combined Ratio 0. 28 0. 10 - 2, 076, 192 3, 623, 035 0. 900 Acceptance Ins. Allstate AIG Berkshire Chubb Cincinnati HCC Ins. Ohio Casualty Progessive SAFECO Selective St. Paul United F&C All Companies All Except AIG Optimal Growth Rate Target Metric Net income over the projection period plus the terminal value of the company at the end of the five-year period Sensitivity Test • Assume several different growth rates within the range of reasonable values • Mean-Variance analysis • First-degree stochastic dominance • Second -degree stochastic dominance Mean-Variance illustration Table 4 Mean Values of 500 Simulations Base Case 1 2 3 4 5 6 7 All Companies Growth Rate 0% 2.5% 5% 7.5% 10% 12.5% 15% PHS in 2007 (000) NI from 2003-07 NWP in (000) 2007 (000) 55,234 13,239 52,252 10,547 48,632 7,243 44,059 3,012 38,277 -2,400 31,028 -9,247 22,117 -17,732 68,956 78,531 89,079 100,661 113,292 127,027 141,934 CR in 2007 1.057 1.060 1.063 1.069 1.076 1.085 1.096 8 9 10 Without AIG Standard Deviation Standard Deviation Unacceptable NI+22701635+2.13*PHS+ (Column 6) NI+1906580+1.85*PHS+ (Column 8) 1.57*NWP-23787168*CR Premium to (000) 0.28*NWP-2076192*CR Surplus Ratio 236,706 242,633 248,091 252,180 254,112 253,178 248,855 17,621 19,941 24,181 30,556 39,253 50,543 64,099 134,442 128,908 121,853 112,394 99,807 83,376 62,558 17,968 20,171 23,745 28,896 35,801 44,672 55,345 0.6% 1.2% 3.0% 15.2% 42.0% 76.8% 91.6% Figure 1 First Degree Stochastic Dominance Cumulative Distribution Function 1.00 0.90 0.80 Probability 0.70 0.60 0.50 0.40 G 0.30 0.20 F 0.10 0.00 Level of Wealth Figure 2 Second Degree Stochastic Dominance Cumulative Distribution Function F 1.00 0.90 B G 0.80 Probability 0.70 0.60 0.50 0.40 0.30 G 0.20 0.10 A F 0.00 Level of Wealth A>B or A=B Figure 3 Histogram of Company Values under Different Projected Growth Rates Base Case 190 180 170 160 150 Frequency (Out of 500 Simulations) 140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 -9 -7 -5 -3 -1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 Each Unit is $10 Million 0% 2.50% 5% 7.50% 10% 29 31 33 35 37 39 41 43 45 Figure 4 Commulative Distribution of Company Values under Different Projected Growth Rates Base Case 1. 00 0. 90 0. 80 Probability 0. 70 0. 60 0. 50 0. 40 0. 30 0. 20 0. 10 0. 00 -9 -7 -5 -3 -1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 Each Unit is $10 Million 0% 2.50% 5% 7.50% 10% 29 31 33 35 37 39 41 43 45 Table 5 Test for Second Degree Stochastic Dominance Area of B Second Degree Stochastic Dominance 2,977.5 5,824.0 8,259.5 10,153.5 2,904.5 5,434.5 7,537.0 2,600.5 4,871.5 2,336.5 No No No No No No No No No No Base Case F G 0% 2.5% 5.0% 7.5% 10.0% 5.0% 7.5% 10.0% 7.5% 10.0% 10.0% 0% 0% 0% 2.5% 2.5% 2.5% 5.0% 5.0% 7.5% Intersection Point Area of A 208 222 225 231 232 236 241 246 249 252 5.0 106.5 510.5 1,433.5 157.0 656.0 1,790.0 573.5 1,872.5 1,367.5 Operating Constraints • The optimal growth rate cannot be determined based on – mean-variance analysis – first- or second-degree stochastic dominance • Impact of adding constraints Constraining Premium-to-Surplus Ratios The proportion of outcomes that lead to unacceptable premium-to-surplus levels can be added as a constraint in the maximization process. Table 4 Mean Values of 500 Simulations Base Case 1 2 3 4 5 6 7 All Companies PHS in 2007 Growth Rate (000) 0% 2.5% 5% 7.5% 10% 12.5% 15% NI from 2003-07 (000) NWP in 2007 (000) 55,234 13,239 52,252 10,547 48,632 7,243 44,059 3,012 38,277 -2,400 31,028 -9,247 22,117 -17,732 68,956 78,531 89,079 100,661 113,292 127,027 141,934 CR in 2007 1.057 1.060 1.063 1.069 1.076 1.085 1.096 8 9 10 Without AIG Standard Standard NI+22701635 Deviation NI+1906580+1 Deviation Unacceptab +2.13*PHS+ (Column 6) .85*PHS+ (Column 8) le Premium 1.57*NWPto 23787168*CR 0.28*NWPSurplus (000) 2076192*CR Ratio 236,706 242,633 248,091 252,180 254,112 253,178 248,855 17,621 19,941 24,181 30,556 39,253 50,543 64,099 134,442 128,908 121,853 112,394 99,807 83,376 62,558 17,968 20,171 23,745 28,896 35,801 44,672 55,345 0.6% 1.2% 3.0% 15.2% 42.0% 76.8% 91.6% Comparative Statics Initial state of the insurance market Acuity of the aging phenomenon Renewal rate Starting interest rates Initial state of the insurance market Table 6 1 Mean Values of 500 Simulations 6 7 10 All Companies Market condition NI+22701635+2.13*PHS+ Growth Rate 0% 2.5% Mature Soft 5% 7.5% 10% 12.5% 15% 0% 2.5% Immature Soft 5% 7.5% 10% 12.5% 15% Standard Deviation (Column 6) 1.57*NWP-23787168*CR (000) 236,706 242,633 248,091 252,180 254,112 253,178 248,855 236,110 243,139 249,874 255,422 258,889 258,822 254,685 17,621 19,941 24,181 30,556 39,253 50,543 64,099 17,845 19,643 23,248 29,275 37,522 48,086 61,214 Unacceptable Premium to Surplus Ratio 0.6% 1.2% 3.0% 15.2% 42.0% 76.8% 91.6% 0.8% 1.2% 3.6% 14.0% 41.4% 74.4% 93.6% Acuity of the aging phenomenon Table 7 1 Mean Values of 500 Simulations 6 7 10 All Companies Different Acuities NI+22701635+2.13*PHS+ Standard Deviation (Column 6) Growth Rate 1.57*NWP-23787168*CR (000) 0% 2.5% Base Case 5% 7.5% 10% 12.5% 15% 0% 2.5% 5% Slower 7.5% 10% 12.5% 15% 236,706 242,633 248,091 252,180 254,112 253,178 248,855 235,554 241,577 247,183 251,537 254,017 253,963 250,949 17,621 19,941 24,181 30,556 39,253 50,543 64,099 18,262 20,572 24,776 31,036 39,467 50,477 63,703 Unacceptable Premium to Surplus Ratio 0.6% 1.2% 3.0% 15.2% 42.0% 76.8% 91.6% 0.8% 1.4% 3.6% 15.4% 41.6% 74.4% 90.2% Renewal rate Table 8 1 Mean Values of 500 Simulations 6 7 10 All Companies Renewal Rate NI+22701635+2.13*PHS+ Growth Rate 0% 2.5% Base Case 5% 7.5% 10% 12.5% 15% 0% 2.5% 5% Higher 7.5% 10% 12.5% 15% Standard Deviation (Column 6) 1.57*NWP-23787168*CR (000) 236,706 242,633 248,091 252,180 254,112 253,178 248,855 237,772 243,846 249,685 254,381 257,044 257,011 253,712 17,621 19,941 24,181 30,556 39,253 50,543 64,099 17,158 19,407 23,407 29,517 37,994 48,950 62,356 Unacceptable Premium to Surplus Ratio 0.6% 1.2% 3.0% 15.2% 42.0% 76.8% 91.6% 0.6% 1.2% 2.6% 11.4% 36.0% 70.8% 89.8% Starting interest rates Table 9 1 Mean Values of 500 Simulations 6 7 10 All Companies Interest Rate NI+22701635+2.13*PHS+ Growth Rate 0% 2.5% Base Case 5% 7.5% 10% 12.5% 15% 0% 2.5% 5% Lower 7.5% 10% 12.5% 15% Standard Deviation (Column 6) 1.57*NWP-23787168*CR (000) 236,706 242,633 248,091 252,180 254,112 253,178 248,855 235,758 246,686 258,949 272,407 286,621 301,110 315,584 17,621 19,941 24,181 30,556 39,253 50,543 64,099 16,344 17,583 20,552 25,813 33,423 43,815 56,576 Unacceptable Premium to Surplus Ratio 0.6% 1.2% 3.0% 15.2% 42.0% 76.8% 91.6% 0.2% 1.0% 1.4% 2.2% 11.0% 34.8% 64.6% DFA Model Characteristics Implied rate change variable depends on - current market condition (mature hard, immature soft, mature soft and immature hard) - targeted growth rate - rate change impacts profitability Potential impact on persistency (renewal rate) - rate changes could impact persistency - effect could vary by age of business Managing growth rates - DFA program uses constant growth rate - managers likely to vary growth target based on market conditions - need to modify DFA program Caveats Models are simplified versions of reality This DFA model deals with quantifiable risk only Excludes the following risks - A line of business being socialized - Management fraud - Catastrophic risks other than historical patterns Conclusions Increasing the growth rate reduced statutory policyholders’ surplus and current net income, but increased both the future market value of the insurer and the volatility of results The optimal growth rate for the modeled insurer varied from zero to 7.5 percent Growth rates of 10 percent or higher generated unacceptable premium to surplus ratios too frequently Low initial interest rates increased the incentive for growth High initial interest rates lowered the optimal growth rate Varying the other key parameters did not affect the optimal growth rate significantly