What Did We Learn?
Conclusions from the Risk Premium Project
Presentation made at the
2005 Casualty Actuarial Society’s Ratemaking Seminar
March 2005
New Orleans, LA
Richard D. Phillips
Bruce A. Palmer Professor of Risk Management and Insurance
Georgia State University
Risk Premium Project
 Overall Research Objective
– Identify appropriate risk adjustments for insurer liabilities to
determine equilibrium prices for insurance
 Milestones
– Phase 1 – Literature Review
» Actuarial literature
» Finance literature
– Phase 2 – Analysis and Theoretical Conclusions
» Report CAS Forum Fall 2000
– Phase 3 – Empirical Research Complete
» Estimate equity cost of capital by-line of insurance
» Prices reflect allocations of capital
Theory Review: Prices in Perfect Capital Markets
Assets
Investments
Liabilities
SPi
Claims
SPV*(Li)
 With perfect and complete markets assumption
–
–
–
–
No need for capital (Modigliani-Miller)
No default risk (perfect enforceability)
Nothing to allocate
Pricing done under the risk neutral probability measure
Theory Review: Prices in Perfect Capital Markets with
Default Risk
Assets
Liabilities
S[Pi – Di] + S Claims
S[PV*(Li) – Di]
Frictional Costs of Capital
0
Equity
S
Investments
 With insurer default still no need to allocate capital (surplus is a
pooled asset)
– Allocate default costs ala
» Black and Cox (1976)
» Phillips, Cummins and Allen (1998)
– D = Equilibrium value of default option
D

e rt [( P  S )ra  L] f * (ra , L)dLdra

Theory Review: Prices in Perfect Capital Markets with
Default Risk and Capital Market Imperfections
Assets
Investments
Liabilities
S[Pi – Di + tSi] + S Claims
S[PV*(Li) – Di]
Frictional Costs of Capital StSi
Equity
S
 With market imperfections and default costs
– Allocate default cost by priority rule
– Allocate costs of capital to individual lines
– Frictional costs are related to capitalization, i.e., t = f(S)
Pricing Intermediated Risks: Summary
 With market imperfections, insurance prices should
reflect
1. Expected cash flow with adjustments for systematic risk
2. Production costs (expenses)
3. Default risk
4. Frictional capital costs
Project 1: Estimating Cost of Equity Capital for
Property-Liability Insurers
Forthcoming in Journal of Risk and Insurance
 Primary research questions
1. What is the cost of equity capital for P&L insurers using
a. Capital Asset Pricing Model
b. Multi-factor model proposed by Professors Fama and French
2. Are there significant differences in the cost of equity capital
across different lines of insurance?
Traditional Asset Pricing Model
 Dominant model has been Capital Asset Pricing Model
– CAPM cost of equity capital
E(ri)= rf + bi[E(rm) – rf]
– where
E(ri)
rf
E(rm)
bi
= expected return for firm i
= risk-free rate of interest
= expected return on market portfolio

Cov (ri , rm )

2
m

wi  i2   w j Cov (ri , r j )
i j
 m2
Failure of the CAPM? What Failure?
Note: Average annual returns vs. beta for 10 size-sorted stock portfolios.
Sample period 1947-1996.
Source: Cochrane, John H., 1999, “New Facts in Finance,” Economic Perspectives 23(3): 36-58.
Fama-French Multifactor Asset Pricing Model
 Fama-French 3 Factor Model
– Multi-factor asset pricing model
– Cost of capital estimate is
E(ri)= rf + bi[E(rm) – rf] + bs,iE(ps) + bv,iE(pv)
– where
E(ri)
rf
E(rm)
E(ps)
E(pv)
bi
bi
bi
= expected return for firm i
= risk-free rate of interest
= expected return on market portfolio
= expected market premium for firm size
= expected market premium for financial distress
= market beta to adjust for systematic portfolio risk
= beta to adjust for systematic risk associated with firm size
= beta to adjust for systematic associated with financial distress
Cost of Equity Capital Estimates for Pure Play U.S. Property
& Liability Insurers Using CAPM: 1997-2000
Year
1997
1998
1999
2000
Grand Total
Market Cap
Quartile
No. P&L
Insurers
Average b
Average
Sum b
Small
2
3
Big
Total
Small
2
3
Big
Total
Small
2
3
Big
Total
Small
2
3
Big
Total
21
21
21
22
85
18
19
19
19
75
19
19
19
19
76
18
18
18
19
73
309
0.646
0.861
0.709
0.820
0.760
0.632
0.687
0.652
0.917
0.723
0.570
0.616
0.642
0.690
0.629
0.316
0.654
0.642
0.712
0.583
0.677
0.893
1.144
0.809
0.932
0.944
0.926
0.908
0.811
0.999
0.911
0.812
0.677
0.736
0.746
0.743
0.631
0.763
0.696
0.817
0.728
0.836
Cost of Capital
b Sum b
12.3%
13.9%
12.0%
13.6%
11.2%
12.2%
10.8%
11.6%
12.1%
13.0%
Table shows average CAPM beta for firms which self-identify as property-liability insurers by listing their overall NAICS code as
524126 or 52413.
Cost of Equity Capital Estimates for Pure Play U.S. Property & Liability
Insurers Using Fama-French 3 Factor Model: 1997-2000
Year
1997
1998
1999
2000
Grand Total
Market Cap
Quartile
No. P&L
Insurers
bm
bs
bv
Sum b m
Sum b s
Sum b v
Small
2
3
Big
Total
Small
2
3
Big
Total
Small
2
3
Big
Total
Small
2
3
Big
Total
21
21
21
22
85
18
19
19
19
75
19
19
19
20
77
18
19
19
19
75
312
0.875
1.080
0.976
1.060
0.998
0.765
0.928
0.989
1.253
0.986
0.650
0.805
0.965
1.115
0.887
0.679
1.060
1.117
1.322
1.050
0.980
0.463
0.610
0.397
0.130
0.397
0.652
0.613
0.412
0.175
0.460
0.551
0.517
0.465
0.073
0.397
0.624
0.536
0.178
-0.174
0.286
0.386
0.624
0.743
0.668
0.673
0.677
0.538
0.735
0.850
0.983
0.779
0.453
0.633
0.953
1.023
0.769
0.972
0.990
1.041
1.183
1.048
0.813
1.189
1.205
0.978
1.080
1.112
1.091
0.999
0.992
1.238
1.080
0.735
0.651
1.023
1.108
0.882
1.069
1.038
1.053
1.167
1.082
1.040
0.766
0.785
0.093
-0.220
0.349
1.346
0.716
0.349
-0.018
0.589
1.055
0.751
0.528
0.150
0.615
0.742
0.635
0.441
0.105
0.477
0.503
1.086
0.834
0.608
0.355
0.716
1.328
0.765
0.982
0.778
0.958
0.723
0.578
1.244
1.056
0.902
1.385
1.059
1.277
1.180
1.223
0.942
Cost of Capital
b Sum b
18.4%
19.4%
18.9%
20.8%
17.8%
18.9%
20.3%
18.8%
21.8%
20.2%
Table shows average Fama French beta coefficient estimates for firms which self-identify as property-liability insurers by listing
their overall NAICS code as 524126 or 52413.
Recent Update: Results from 1999 - 2003
Year
1999
2000
2001
2002
2003
CAPM
Beta
0.830
0.963
0.537
0.403
0.349
0.616
Market
1.176
1.304
1.203
0.943
0.774
1.080
Risk-free Rate
Market Risk Premium
Size Risk Premium
Value Risk Premium
Fama-French
Size
1.755
1.413
1.772
1.512
1.414
1.573
Value
1.019
1.152
1.327
1.122
0.854
1.095
1.09%
7.59%
2.71%
4.68%
Cost of Equity Capital
CAPM
5.77%
FF3F
13.61%
Results are derived using Full Information Industry Beta methodology (details available in
Kaplan and Petersen 1998 and Cummins and Phillips 2005)
Which Asset Pricing Model Better Reflects Historical
Experience?
Cumlative Return Since Jan. 1, 1990
800%
P&L Insurers
700%
S&P 500
Index
Annualized
Compound
Return
Jan. 90–Dec. 03
300%
P&L
Insurers
16.7%
200%
S&P 500
9.8%
600%
500%
400%
100%
0%
-100%
01/90 01/92 01/94 01/96 01/98 01/00 01/02 01/04
The P&L index is the cumulative return on the market capitalization weighted portfolio of all
NYSE, AMEX, and NASDAQ firms with SIC code 6331.
Which Asset Pricing Model Better Reflects Historical
Experience?
Cumlative Return Since Jan. 1, 1990
800%
P&L Insurers
700%
Index
Annualized
Compound
Return
Jan. 90–Dec. 03
300%
P&L
Insurers
16.7%
200%
S&P 500
9.8%
P&L
Equally
Weighted
11.0%
S&P 500
600%
P&L Equally Weighted
500%
400%
100%
0%
-100%
01/90 01/92 01/94 01/96 01/98 01/00 01/02 01/04
The P&L index is the cumulative return on the market capitalization weighted portfolio of all
NYSE, AMEX, and NASDAQ firms with SIC code 6331.
Estimating Equity Cost of Capital by Line for CAPM
 Use Full-Information Industry Beta Methodology
– Kaplan and Peterson (1998)
– Firm specific betas are weighted average of betas from individual business
units
 Two steps in estimation
– 1. Estimate firm specific equity betas - bi
– 2. Impute full – information industry betas. For CAPM
n
bi   bf jw i, j  i
j1
where wi,j = percent of firm i’s participation in line/industry j
bi = firm i’s overall CAPM beta
bfj = full information industry beta for line/industry j
 Extend Full-Information Industry Beta method for Fama/French model
Summary Estimated Costs of Equity Capital by Line of Business Pairs
Dec. 2000: Panel Estimates from Market Value Weighted Regressions
Estimated Cost of Equity Capital
Line of Business
CAPM
Fama-French
Personal
11.7%
19.4%
Commercial
13.6%**
22.5%**
Automobile
11.3%
19.4%
Workers’ Comp
13.4%
16.5%
All other P&L lines
13.9%***1
23.0%**1
***, **, * - statistically significant at the 1, 5, and 10 percent levels, respectively.
1 – H : r
0
Auto = rAll Others
Project 2: Allocating the Costs of Capital
“Pricing Financially Intermediated Risks with Costly External
Finance: Evidence from the Insurance Industry”
by
J. David Cummins, Yijia Lin, and Richard D. Phillips

Primary research questions
1. Do insurer prices reflect capital allocation charges specific to the firm?
2. Are the implied capital allocations at least correlated with the method
proposed by Myers and Read?
3. What is the implied per unit cost of allocated capital?
Myers and Read 2001
 Myers and Read arrive at their suggested surplus
allocation formula
D

e rt [( P  S )ra  L] f * (ra , L)dLdra

– They ask: “How does the firm default option change for a
marginal increase in liability from line i?” I.e., what is
D ?
di 
Li
Myers and Read (2001) Result
di  d 
where
d
d  1

2
(si  s) 
[(



)

(



)
iL
L
iV
LV 
s
  

s = overall surplus-to-liability ratio of insurer
si = surplus-to-liability ratio for line of business i
 = overall volatility parameter of insurer
d = insolvency put per dollar of liabilities
di = default-to-liability ratio for line of business i
iL = covariance between losses line i and overall loss portfolio
L2 = volatility parameter for total losses
iV = covariance between losses for line i and firm assets
LV = covariance between assets and liabilities
– Using the equal priority rule, i.e., di = d, yields the surplus allocation formula
si  s 
1 d 1 d
( ) ( )[(iL  L2 )  (iV  LV )]
 s

Primary Empirical Predictions
 Hypothesis 1
– Price differences between lines of insurance reflect market
systematic risk differences
 Hypothesis 2
– Price for any line of business should be inversely related to firm
default risk
 Hypothesis 3
– Price differences across insurers within a given line of
insurance reflect firm specific capital allocation differences.
» Controlling for overall capital charges, lines requiring greater
capital within the firm will require a higher returns
• Empirically test if si is related to prices across insurers
s
Empirical Test: Dependent Variable
 Economic premium ratio
EPRij 
NPWij  DIVij  UEPij
( NLI
ij
–
–
–
–
–
–
NPWij
DIVij
UEXij
NLIij
LAEij
PVFi
 LAEij  * PVFi
= net premiums written line i insurer j
= PH dividends paid line i insurer j
= underwriting expenses paid line i insurer j
= net losses incurred line i insurer j
= net loss adjustment expenses incurred line i insurer j
= discount factor for payout tail line i
 We consider two aggregated lines groupings
– Long-tail/Liability lines vs. Short-tail/Property lines
Economic Premium Ratio Histograms
U.S. Property-Liability Insurers: 2000
Long-tail Lines
Short-tail Lines
40%
40%
35%
35%
30%
30%
25%
25%
20%
20%
15%
15%
10%
10%
5%
5%
0%
0%
0.1
0.5
0.9
1.3
Mean
Median
Std. Dev.
Num Obs.
1.7
2.1
= 1.22
= 1.09
= 0.59
= 1114
2.5
2.9
0.1
0.5
0.9
1.3
Mean
Median
Std. Dev.
Num Obs.
1.7
2.1
= 1.09
= 1.04
= 0.40
= 1331
2.5
2.9
Implementing Myers and Read Line Specific Surplus-toLiability Ratio
si  s 
1 d 1 d
( ) ( )[(iL  L2 )  (iV  LV )]
 s

 Volatilities and covariance parameters estimated from
– NAIC quarterly data 1991 – 2000
» Industry aggregate data
» Seasonally adjusted time series
» Discount losses incurred using risk-free term structure
– Asset return data for major asset classes
 Company specific values for
–
–
–
–
Line of business liability weights
Asset weights
Capital ratios
All adjusted to approximate market values
Summary Statistics: Overall and Line Specific Capitalization
Ratios: Year = 2000
Ave.
Std.
Dev.
P25
P50
P75
Overall Surplus-to-Liability Ratio
1.07
0.81
0.55
0.78
1.29
Property Surplus-to-Liability Ratio
1.30
1.40
0.56
0.85
1.48
Liability Surplus-to-Liability Ratio
1.34
1.08
0.65
0.95
1.61
Relative Property Surplus-to-Liability Ratio
1.15
0.43
0.91
1.06
1.25
Relative Liability Surplus-to-Liability Ratio
1.23
0.32
1.10
1.17
1.28
N=1114 for Property Lines
N=1331 for Liability Lines
Correlation Statistics: Year = 2000
Overall Surplus-to-Liability Ratio
Property Surplus-to-Liability Ratio
Liability Surplus-to-Liability Ratio
Relative Property Ratio
Relative Liability Ratio
N= 1040 observations
Overall Property Liability
Surplus- Surplus- SurplustototoRelative
Liability Liability Liability Property
Ratio
Ratio
Ratio
Ratio
1
0.805
1
0.886
0.862
1
0.153
0.594
0.344
1
0.117
0.346
0.478
0.552
Relative
Liability
Ratio
1
Average Economic Premium Ratio
Univariate Test 1: Price Inversely Related to
Default Risk
1.40
1.20
1.00
0.80
Property
0.60
Liability
0.40
A++,A+
A,A-
B++,B+
B,B-
C++,C+
C,C-
D
E,F
A.M. Best Rating
Chart shows average economic premium ratio by A.M. Best rating for all insurers 1997 – 2002.
Number of observations = 6681 for property and 8050 for liability.
Univariate Test 2: Price Positively Related to Insurer Overall
Capitalization
1.40
Average EPR
1.30
1.20
1.10
1.00
Liability Lines
0.90
Property Lines
0.80
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Log(Upper Limit of Overall Capitalization Ratio Decile)
Chart shows average economic premium ratio by overall capitalization deciles for all insurers
1997 – 2002. Number of observations = 6681 for property and 8050 for liability.
Univariate Test 3: Price Positively Related to Firm Specific
Internal Capital Allocation
1.40
Average EPR
1.30
1.20
1.10
Liability Lines
1.00
Property Lines
0.90
-0.5
0.0
0.5
1.0
1.5
2.0
Log(Upper Limit of Relative Line-Specifc Capitalization Ratio Decile)
Chart shows average economic premium ratio by line specific surplus-to-liability ratio relative to the
overall surplus-to-liability ratio by deciles for all insurers 1997 – 2002.
Number of observations = 6681 for property and 8050 for liability.
Multivariate Empirical Test 1
 Panel data of all U.S. P&L Insurers
– Time period: 1997 – 2002
 Empirical test: Price differences across firms
EPR ijt    b1  A.M.Best jt  b2
where EPRijt
A.M. Bestjt
sijt
sjt
Xjt
sijt
s jt
 b3s jt + 'X   j  t   jt
= Economic premium ratio line i for company j in year t
= Financial strength rating from A.M. Best company j year t
= Line i surplus-to-liability ratio for company j in year t
= Overall surplus-to-liability ratio for company j in year t
= Vector of other control variables for company j in year t
 Methodology: OLS and 2 way fixed effects
Regression Results: Price Differences Across Firms for
Property Lines of Business: All Insurers 1997 - 2002
Year Fixed Effects
Company Fixed Effects
Variable
Intercept
Rel. Property Surplus-Liability Ratio
Firm Surplus-Liability Ratio
Ind. if Best Rating is A or AInd. if Best Rating is B++ or B+
Ind. if Best Rating is B or lower
Property Line Growth Rate
Log(Book Value of Assets)
Adv. Exp. to Total Exp. Ratio
Single Firm Indicator
Part of publicly traded group
Part of a mutual group
2
R
No
No
1.571
0.110
-0.032
-0.041
-0.279
-0.689
-0.022
-0.248
0.107
-0.009
-0.067
2.89%
No
No
***
***
*
***
*
***
***
***
1.385
0.097
0.051
-0.026
-0.021
-0.244
-0.718
-0.015
-0.238
0.098
-0.008
-0.072
3.31%
Yes
No
***
***
***
***
**
***
***
***
1.675
0.109
-0.037
-0.043
-0.287
-0.949
-0.023
-0.213
0.109
-0.007
-0.068
3.54%
Yes
No
***
***
**
*
***
**
***
***
***
1.488
0.094
0.055
-0.031
-0.021
-0.251
-1.034
-0.015
-0.199
0.099
-0.006
-0.073
4.01%
Yes
Yes
***
***
***
*
***
***
***
***
***
0.105
-0.054
-0.104
-0.302
-1.859
-0.004
0.001
0.075
-0.089
0.062
51.62%
Note: 6681 Observations
***, **, * represents statistical significance at the 1, 5 and 10 percent level, respectively
Yes
Yes
***
0.106
0.032
*
-0.055
**
-0.110
*** -0.316
*** -1.867
-0.020
-0.014
0.075
**
-0.087
0.065
51.65%
***
*
*
**
***
***
**
Regression Results: Price Differences Across Firms for
Liability Lines of Business: All Insurers 1997 - 2002
Year Fixed Effects
Company Fixed Effects
Variable
Intercept
Rel. Liability Surplus-Liability Ratio
Firm Surplus-Liability Ratio
Ind. if Best Rating is A or AInd. if Best Rating is B++ or B+
Ind. if Best Rating is B or lower
Liability Growth Rate
Log(Book Value of Assets)
Adv. Exp. to Total Exp. Ratio
% NPW in Price Regulated Lines
Single Firm Indicator
Part of publicly traded group
Part of a mutual group
R2
No
No
1.671
0.101
-0.047
-0.119
-0.239
-1.665
-0.029
0.430
-0.147
0.040
-0.029
-0.089
5.45%
No
No
***
***
***
***
***
***
***
***
***
***
**
***
1.448
0.103
0.051
-0.040
-0.098
-0.208
-1.641
-0.020
0.408
-0.130
0.036
-0.028
-0.091
6.21%
Yes
No
***
***
***
***
***
***
***
***
***
***
***
**
***
1.689
0.103
-0.049
-0.119
-0.240
-1.495
-0.029
0.449
-0.152
0.040
-0.027
-0.088
5.96%
Yes
No
***
***
***
***
***
***
***
***
***
***
**
***
1.472
0.104
0.052
-0.042
-0.098
-0.209
-1.534
-0.020
0.433
-0.136
0.035
-0.026
-0.090
6.75%
***
***
***
***
***
***
***
***
***
***
***
**
***
Yes
Yes
Yes
Yes
0.126
-0.014
-0.048
-0.199
-2.303
0.027
-0.228
-0.214
-0.023
-0.062
0.044
49.20%
0.129
0.032
-0.014
-0.044
-0.187
-2.288
0.042
-0.254
-0.210
-0.023
-0.066
0.042
49.27%
Note: 8050 Observations
***, **, * represents statistical significance at the 1, 5 and 10 percent level, respectively
***
***
***
***
**
***
***
***
***
**
***
***
Economic Significance – A First Look
 Coefficient estimates can be used to derive the insurer’s
implied deadweight cost of allocated capital
– Discounted cash flow model of insurance premiums
Cummins and Phillips (2000)
 T r[S
Lt
E(rL)PV(L)t  1 
f t  1  PV(L)t  1]
P  

t



(rL)]t
(1  rf)t
[1  E(rL)]t 
t  1 [1  E
t  1
T
PV of future “taxes” on equity at date 0
PV loss payments
at RADR at date 0
Lt(1  vt)
P  
t
[1

E
(r
)]
t 1
L
T
Implied Equity Capital “Tax Rates”

Methodology
1. Estimate economic premium ratio using parameters for property of
liability lines of insurance
2. Assume we double the line-specific capitalization ratio
3. Determine the capital “tax rate” that would imply an increase in the
economic premium ratio equal to what was estimated from the reduced
form price regressions

Implied tax rates
–
–
–
Property: 49%
Liability: 43%
Average simulated MTR of P&L insurers was 27% in 2000 (Graham)
Risk Premium Project Conclusions
 Primary theoretical predictions
Conclusion I
– Both systematic and non-systematic risk are relevant factors
determining equilibrium prices for insurance
Conclusion II
– Multifactor asset pricing models empirically more successful than
CAPM
Conclusion III
– Theoretically appealing surplus allocation models now exist
Risk Premium Project Conclusions
 Primary empirical results
– Cost of equity capital for insurers
» CAPM vs. Multi-factor models
» In general FF3F model produces higher estimated costs of equity capital vs. the
CAPM
– Full information industry beta methodology
» In general cost of equity capital for property lines appears higher than liability lines
– Strong evidence prices vary across insurers as a function of
»
»
»
»
Overall default risk
Total capital charges
Internal allocation of those capital charges
Prices appear to be a function of a firm’s access to capital markets
– Limitations of competition and arbitrage
 Implementation issues
Suggested Reading
Butsic, Robert P, J. David Cummins, Richard A Derrig, and Richard D. Phillips, 2000, "The Risk
Premium Project (RPP): Phase I and II Report," Casualty Actuarial Society Forum, Fall 2000: 165230.
Cochrane, John H., 1999, “New Facts in Finance,” Economic Perspectives 23(3): 36-58.
Cummins, J. David and Richard D. Phillips, 2001, "Financial Pricing of Property-Liability Insurance," in
Georges Dionne, ed., Handbook of Insurance (Boston, MA: Kluwer Academic Publishers)
Cummins, J. David and Richard D. Phillips, 2005, “Estimating the Cost of Equity Capital for Property &
Liability Insurers,” forthcoming Journal of Risk and Insurance.
Cummins, J. David, Yijia Lin, and Richard D. Phillips, 2005, “Pricing Financially Intermediated Risks
with Costly External Finance: Evidence from the Insurance Industry,” Working Paper Georgia State
University, Atlanta, GA.
Derrig, Richard A.,and Elisha Orr, 2003, "Equity Risk Premium: Expectations Great and Small,"
presented at the 2003 Bowles Symposium, Georgia State University, Atlanta, GA.
Froot, Kenneth A., 2003, "Risk Management, Capital Budgeting and Capital Structure Policy for Insurers
and Reinsurers," Harvard Working Paper, Boston MA.
Suggested Reading
Froot, Kenneth A. and Jeremy C. Stein, 1998, "Risk Management, Capital Budgeting, and Capital
Structure Policy for Financial Institutions: An Integrated Approach," Journal of Financial
Economics 47: 55-82.
Kaplan, Paul D. and James D. Peterson, 1998, "Full-Information Industry Betas," Financial Management
27: 85-93.
Merton, Robert C., and Andre F. Perold, 1993, “Theory of Risk Capital in Financial Firms,” Journal of
Applied Corporate Finance 6:16-32.
Myers, Stewart C and James A. Read, Jr., 2001, "Capital Allocation for Insurance Companies," Journal of
Risk & Insurance 68: 545-580.
Phillips, Richard D., J. David Cummins, and Franklin Allen, 1998, "Financial Pricing of Insurance in the
Multiple Line Insurance Company," Journal of Risk and Insurance 65: 597-636.
Zanjani, George, 2002, “Pricing and Capital Allocation in Catastrophe Insurance,” Journal of Financial
Economics 65: 283-305.