by Stephen P. D’Arcy and Richard W. Gorvett University of Illinois

Session C-5: ARIA Prize PaperCAS Spring Meeting May 2006

The Use of DFA to Determine Whether an Optimal Growth Rate Exists for a

Property-Liability Insurerby 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

• 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/

Objective of Paper

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

0

20

40

60

80

100

120

1 2 3 4 5 6 7 8 9 10 11 12

Age of Cohort (in Years)

Los

s R

atio

(%

)

Firm A

Firm B

Firm C

Firm D

Firm E

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)

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)

Multiple Regression Approach

Table 1 - Company Data (in Millions)

Year2001 Total Revenue Total Admitted Assets Market Value Policyholders' Surplus Net Written Premium(P/C) Operating RatioRatio of P/C

NWP/Revenue

Acceptance 176 440 70 129 93 1. 150 0. 529Allstate 28, 865 39,290 31, 723 13,796 21,991 0. 950 0. 762

AIG 61, 766 52, 458 216, 528 15, 362 14, 007 0. 910 0. 227Berkshire 37, 668 73, 400 94, 628 27, 103 11, 656 1. 025 0. 309

Chubb 7, 754 18, 218 14, 250 3, 526 5, 997 0. 990 0. 773Cincinnati Fin 2, 561 6, 873 7, 029 2, 530 2, 591 0. 888 1. 012

CNA 13, 203 32, 446 6, 919 6, 089 7, 663 1. 409 0. 580Hartford 15, 147 23, 997 16, 289 5, 804 5, 209 0. 971 0. 344HCC ins 505 904 1, 520 398 300 0. 925 0. 594

Ohio Casualty 1, 902 3, 830 1, 030 768 1, 472 0. 985 0. 774Progessive 7, 488 10, 391 7, 846 2, 641 7, 263 0. 917 0. 970SAFECO 6, 863 9, 658 4, 033 2, 280 4, 439 1. 103 0. 647Selective 1, 059 2, 209 590 519 927 0. 958 0. 875St.Paul 8, 943 22, 577 12, 257 4, 132 6, 136 1. 090 0. 686

United F&C 473 748 215 198 366 0. 960 0. 774

Companies in Sample

Table 2

Equation a S.E. b S.E c S.E. d S.E. R2

1 0. 88 0. 48 1. 01 0. 05 0. 16 0. 05 -2. 37 0. 35 0. 9382 0. 51 0. 49 0. 86 0. 06 0. 28 0. 06 -1. 99 0. 36 0. 932

Least squares linear regression Based on the experience of 15 companies over the period 1990-2001MV = Market Value PHS = Statutory Policyholders Surplus NWP =Net Written Premium CR=Combined Ratio OR = Operating Ratio

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

Regression Analysis

Table 3

Company a S.E. b S.E. c S.E d S.E. R2

Acceptance Ins. 447, 067 91, 359 1. 32 0. 30 0. 07 0. 21 -442, 500 80, 883 0. 969

Allstate -45, 842, 210 73, 592, 068 5. 86 2. 11 -3. 01 1. 87 60, 912, 024 73, 496, 027 0. 835

AIG 60, 741, 483 507, 769, 771 0. 10 7. 49 32. 87 13. 64 -282, 623, 500 528, 222, 750 0. 910

Berkshire 10, 310, 535 18, 250, 118 0. 48 0. 25 6. 61 0. 82 -5, 794, 440 15, 089, 729 0. 983

Chubb 565, 774 7, 472, 749 -2. 40 2. 67 4. 34 1. 69 -3, 209, 198 7, 346, 370 0. 861

Cincinnati 10, 802, 307 8, 247, 771 1. 83 0. 49 0. 89 0. 90 -10, 827, 410 8, 177, 836 0. 876

CNA -631, 440 2, 630, 205 0. 37 0. 17 0. 29 0. 22 2, 115, 101 1, 583, 555 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

HCC Ins. 465, 647 724, 190 3. 88 1. 71 -0. 23 1. 71 -720, 770 947, 986 0. 571

Ohio Casualty 3, 479, 368 951, 537 0. 91 0. 26 -0. 54 0. 39 -2, 201, 743 971, 712 0. 788

Progessive 13, 816, 591 13, 403, 390 5. 03 11. 60 -0. 34 3. 93 -15, 002, 413 13, 898, 864 0. 698

SAFECO -366, 248 4, 954, 621 2. 68 0. 82 -0. 89 0. 52 1, 475, 766 4, 565, 682 0. 791

Selective 1, 922, 264 1, 321, 201 0. 81 0. 54 0. 22 0. 51 -1, 819, 411 1, 282, 157 0. 786

St. Paul -5, 654, 781 6, 380, 946 1. 38 0. 56 0. 98 0. 90 3, 563, 808 7, 427, 642 0. 889United F&C 57, 765 180, 966 3. 07 0. 37 -1. 19 0. 24 -24, 077 167, 854 0. 899

All Companies 22, 701, 635 18, 248, 833 2. 13 0. 29 1. 57 0. 46 -23, 787, 168 17, 244, 028 0. 444

All Except AIG 1, 906, 580 3832821 1. 85 0. 06 0. 28 0. 10 -2, 076, 192 3, 623, 035 0. 900Least 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

MV=a+b*PHS+ c*NWP+d*CR

Dependent Variable: MV

Independent Variables: PHS, NWP, CR

Market Value Estimation

Results of regression for each company separately

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 illustrationTable 4Base Case

1 2 3 4 5 6 7 8 9 10

NI+22701635+2.13*PHS+ Standard Deviation

(Column 6) NI+1906580+1.85*PHS+ Standard Deviation

(Column 8)

Growth Rate 1.57*NWP-23787168*CR

(000) 0.28*NWP-2076192*CR

0% 55,234 13,239 68,956 1.057 236,706 17,621 134,442 17,968 0.6%

2.5% 52,252 10,547 78,531 1.060 242,633 19,941 128,908 20,171 1.2%

5% 48,632 7,243 89,079 1.063 248,091 24,181 121,853 23,745 3.0%

7.5% 44,059 3,012 100,661 1.069 252,180 30,556 112,394 28,896 15.2%

10% 38,277 -2,400 113,292 1.076 254,112 39,253 99,807 35,801 42.0%

12.5% 31,028 -9,247 127,027 1.085 253,178 50,543 83,376 44,672 76.8%

15% 22,117 -17,732 141,934 1.096 248,855 64,099 62,558 55,345 91.6%

Unacceptable Premium to

Surplus Ratio

Without AIG

Mean Values of 500 Simulations

PHS in 2007 (000)

NI from 2003-07

(000) NWP in

2007 (000) CR in 2007

All Companies

Figure 1 First Degree Stochastic Dominance

Cumulative Distribution Function

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Level of Wealth

Pro

bab

ility

F

G

Figure 2Second Degree Stochastic Dominance

Cumulative Distribution Function

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

Level of Wealth

Pro

bab

ilit

y

F

G

A

BG

F

A>B or A=B

Figure 3Histogram of Company Values

under Different Projected Growth RatesBase Case

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

-9 -7 -5 -3 -1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45

Each Unit is $10 Million

Fre

qu

ency

(O

ut o

f 500

Sim

ula

tion

s)

0% 2.50% 5% 7.50% 10%

Figure 4Commulative Distribution of Company Values

under Different Projected Growth RatesBase Case

0. 00

0. 10

0. 20

0. 30

0. 40

0. 50

0. 60

0. 70

0. 80

0. 90

1. 00

-9 -7 -5 -3 -1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45

Each Unit is $10 Million

Pro

bab

ility

0% 2.50% 5% 7.50% 10%

Table 5

F G

Intersection

Point Area of A Area of B

0% 2.5% 208 5.0 2,977.5 No0% 5.0% 222 106.5 5,824.0 No0% 7.5% 225 510.5 8,259.5 No0% 10.0% 231 1,433.5 10,153.5 No

2.5% 5.0% 232 157.0 2,904.5 No2.5% 7.5% 236 656.0 5,434.5 No2.5% 10.0% 241 1,790.0 7,537.0 No5.0% 7.5% 246 573.5 2,600.5 No5.0% 10.0% 249 1,872.5 4,871.5 No7.5% 10.0% 252 1,367.5 2,336.5 No

Test for Second Degree Stochastic Dominance

Base Case Second Degree

Stochastic Dominance

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 4Base Case

1 2 3 4 5 6 7 8 9 10

NI+22701635+2.13*PHS+

Standard Deviation

(Column 6)

NI+1906580+1

.85*PHS+

Standard Deviation

(Column 8)

Growth Rate

1.57*NWP-23787168*CR

(000) 0.28*NWP-2076192*CR

0% 55,234 13,239 68,956 1.057 236,706 17,621 134,442 17,968 0.6%

2.5% 52,252 10,547 78,531 1.060 242,633 19,941 128,908 20,171 1.2%

5% 48,632 7,243 89,079 1.063 248,091 24,181 121,853 23,745 3.0%

7.5% 44,059 3,012 100,661 1.069 252,180 30,556 112,394 28,896 15.2%

10% 38,277 -2,400 113,292 1.076 254,112 39,253 99,807 35,801 42.0%

12.5% 31,028 -9,247 127,027 1.085 253,178 50,543 83,376 44,672 76.8%

15% 22,117 -17,732 141,934 1.096 248,855 64,099 62,558 55,345 91.6%

Without AIG

Unacceptable Premium

toSurplus Ratio

Mean Values of 500 Simulations

PHS in 2007 (000)

NI from 2003-07

(000) NWP in

2007 (000) CR in 2007

All Companies

Comparative Statics

Initial state of the insurance market

Acuity of the aging phenomenon

Renewal rate

Starting interest rates

Initial state of the insurance market

1 6 7 10

NI+22701635+2.13*PHS+ Standard Deviation

(Column 6)

1.57*NWP-23787168*CR (000)

0% 236,706 17,621 0.6%2.5% 242,633 19,941 1.2%5% 248,091 24,181 3.0%

7.5% 252,180 30,556 15.2%10% 254,112 39,253 42.0%

12.5% 253,178 50,543 76.8%15% 248,855 64,099 91.6%0% 236,110 17,845 0.8%

2.5% 243,139 19,643 1.2%5% 249,874 23,248 3.6%

7.5% 255,422 29,275 14.0%10% 258,889 37,522 41.4%

12.5% 258,822 48,086 74.4%15% 254,685 61,214 93.6%

Mean Values of 500 Simulations

Unacceptable Premium to

Surplus Ratio

Mature Soft

Immature Soft

Table 6

Market condition

Growth Rate

All Companies

Acuity of the aging phenomenon

1 6 7 10

NI+22701635+2.13*PHS+ Standard Deviation

(Column 6)

Growth Rate 1.57*NWP-23787168*CR (000)

0% 236,706 17,621 0.6%2.5% 242,633 19,941 1.2%5% 248,091 24,181 3.0%

7.5% 252,180 30,556 15.2%10% 254,112 39,253 42.0%

12.5% 253,178 50,543 76.8%15% 248,855 64,099 91.6%0% 235,554 18,262 0.8%

2.5% 241,577 20,572 1.4%5% 247,183 24,776 3.6%

7.5% 251,537 31,036 15.4%10% 254,017 39,467 41.6%

12.5% 253,963 50,477 74.4%15% 250,949 63,703 90.2%

Mean Values of 500 Simulations

Unacceptable Premium to

Surplus Ratio

Base Case

Slower

Table 7

Different Acuities

All Companies

Renewal rate

1 6 7 10

NI+22701635+2.13*PHS+ Standard Deviation

(Column 6)

1.57*NWP-23787168*CR (000)

0% 236,706 17,621 0.6%2.5% 242,633 19,941 1.2%5% 248,091 24,181 3.0%

7.5% 252,180 30,556 15.2%10% 254,112 39,253 42.0%

12.5% 253,178 50,543 76.8%15% 248,855 64,099 91.6%0% 237,772 17,158 0.6%

2.5% 243,846 19,407 1.2%5% 249,685 23,407 2.6%

7.5% 254,381 29,517 11.4%10% 257,044 37,994 36.0%

12.5% 257,011 48,950 70.8%15% 253,712 62,356 89.8%

Mean Values of 500 Simulations

Unacceptable Premium to

Surplus Ratio

Base Case

Higher

Table 8

Renewal Rate

Growth Rate

All Companies

Starting interest ratesTable 9

1 6 7 10

NI+22701635+2.13*PHS+ Standard Deviation

(Column 6)

Growth Rate 1.57*NWP-23787168*CR (000)

0% 236,706 17,621 0.6%2.5% 242,633 19,941 1.2%5% 248,091 24,181 3.0%

7.5% 252,180 30,556 15.2%10% 254,112 39,253 42.0%

12.5% 253,178 50,543 76.8%15% 248,855 64,099 91.6%0% 235,758 16,344 0.2%

2.5% 246,686 17,583 1.0%5% 258,949 20,552 1.4%

7.5% 272,407 25,813 2.2%10% 286,621 33,423 11.0%

12.5% 301,110 43,815 34.8%15% 315,584 56,576 64.6%

Mean Values of 500 Simulations

Unacceptable Premium to

Surplus Ratio

Base Case

Lower

Interest Rate

All Companies

DFA Model CharacteristicsImplied 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

CaveatsModels 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