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