Seriatim Claim Valuation
from
Detailed Process Models
Philip E. Heckman
Aon Worldwide Resources, Inc.
Presented to the
Casualty Actuarial Society
Casualty Loss Reserve Seminar
September 14, 1999
Data Requirements
Minimal:
Claim identifier
Valuation date
Status: open/closed
Claim dates
Accident
Report
Closing
Indemnity paid
Loss adjustment expense paid
Policy limit.
Optional:
Policy dates
Inception
Retroactive
Expiration
Indemnity reserve
Loss adjustment expense reserve
Deductible amount
Reopen indicator
Reopen date
Legal status
Attorney involvement
Trial indicator
Jury indicator.
Reinsurance information
Treaty year
Treaty type
Layers.
Process Diagram
Default Model Hypotheses
Variable Conditionals Distribution Remarks
Report lag Accident date Gamma Right truncated
Payment lag Report date Gamma Right censored
Indicators
CNP Accident date Logistic Details difficult
CWE|CWP Report lag to verify.
CWIE|CWI Payment lag
Expense on Accident date Lognormal Logs of lag
CWE Report lag variables
Payment lag usually better.
Payment lag
Indemnity Accident date Lognormal Right censored Report lag at policy limits.
Payment lag
Expense on Accident date Lognormal Full list of
CWI Report lag conditionals
Payment lag usually
Indemnity (log) not needed.
Random Variables
XT = I(CWP| tA, r, s) ·
{ I(CWI|CWP, tA, r, s) · [ XI| tA, r ,s +
I(CWIE|CWI, tA, r, s) · XE|XI , tA, r, s ] +
I(CWE|CWP, tA, r, s) · XE|XI =0, tA, r, s } ,
where
XT = total payments for indemnity and expense;
I( · ) = indicator functions, modeled as Bernoulli variables with linear logits;
XI = indemnity payments conditional on accident date, lag variables, censored at policy limit U
= min[U, exp( aI + bI · tA + cI · ln(r) +
dI · ln(s) + N(0, sI2))] ;
XE|XI = expense payments on CWI's
= exp( aEI + bEI · tA + cEI · ln(r) + dEI · ln(s) + eEI · ln(XI) + N(0, sEI2));
XE|XI =0 = expense payments on CNI's
= exp( aE + bE · tA + cE · ln(r) +
dE · ln(s) + N(0, sE2) ) ;
tA = accident date, always a given number;
r = report lag from accident date ~ G(ar , br ),
a gamma variable truncated on the right at the valuation lag;
s = payment lag from report date ~ G(as , bs ),
a gamma variable censored on the right at the valuation lag.
PROFESSIONAL LIABILITY: LAGS AND DECISION VARIABLES
PARAMETER ESTIMATES
REPORT LAGS, RIGHT-TRUNCATED GAMMA
6017 CASES INPUT; 2 PARAMETERS
PARAMETER SUMMARY GAMMA
ESTIMATE STANDARD ERROR PARAMETERS
Exponent -.27296 .01607 .761
Scale -6.38551 .02649 .001716/day
Mean 443.5 days
CORRELATION MATRIX
1 2
1 1.0000 .0003
2 .0003 1.0000
PAYMENT LAGS, RIGHT-CENSORED GAMMA WITH SCALE REGRESSION
6217 CASES INPUT; 3 PARAMETERS
DLOSS= 40381.23241 AIC= 80768.46481
PARAMETER SUMMARY GAMMA
ESTIMATE STANDARD ERROR PARAMETERS
Exponent .38538 .01757 1.47
Scale -6.36792 .03514 .001311/day
Trend .01457 .00408 .0146/year
Mean 1121.3 days
CORRELATION MATRIX
1 2 3
1 1.0000 .6291 .1209
2 .6291 1.0000 .0001
3 .1209 .0001 1.0000
Pr(CWP|Report Lag, Payment Lag), LOGIT REGRESSION
5151 CASES INPUT; 3 PARAMETERS 1575 PAID.
DLOSS= 2751.54487 AIC= 5509.08974
PARAMETER SUMMARY
ESTIMATE STANDARD ERROR
1 -2.26002 .08778 (intercept)
2 .00054 .00008 (report lag)
3 .00152 .00009 (payment lag)
CORRELATION MATRIX
0 1 2
1 1.0000 -.5037 -.8450
2 -.5037 1.0000 .0000
3 -.8450 .0000 1.0000
PARAMETER ESTIMATES (continued)
Pr(CWI|CWP,Report Lag, Payment Lag), LOGIT REGRESSION
1575 CASES INPUT; 3 PARAMETERS 1039 CWI.
DLOSS= 914.00373 AIC= 1834.00745
PARAMETER SUMMARY
ESTIMATE STANDARD ERROR
1 2.18960 .13933 (intercept)
2 -.00118 .00014 (report lag)
3 -.00086 .00010 (payment lag)
CORRELATION MATRIX
0 1 2
1 1.0000 -.4377 -.7998
2 -.4377 1.0000 .0000
3 -.7998 .0000 1.0000
Pr(CWIE|CWI,Report Lag, Payment Lag), LOGIT REGRESSION
1039 CASES INPUT; 3 PARAMETERS 532 CWIE.
DLOSS= 546.68885 AIC= 1099.37769
PARAMETER SUMMARY
ESTIMATE STANDARD ERROR
1 -2.42237 .24407 (intercept)
2 .00126 .00025 (report lag)
3 .00220 .00025 (payment lag)
CORRELATION MATRIX
0 1 2
1 1.0000 -.5470 -.8943
2 -.5470 1.0000 .0000
3 -.8943 .0000 1.0000
SEVERITY ANALYSIS
PARAMETER ESTIMATES
Indemnity
General Linear Models Procedure
Dependent Variable: LINP
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 2 350.6169 175.3084 68.96 0.0001
Error 1112 2826.7321 2.5420
Total 1114 3177.3490
R-Square C.V. Root MSE LINP Mean
0.110349 15.99358 1.594373 9.96883
Source DF Type I SS Mean Square F Value Pr > F
RPFC 1 39.2509 39.2509 15.44 0.0001
LPL 1 311.3659 311.3659 122.49 0.0001
Source DF Type III SS Mean Square F Value Pr > F
RPFC 1 102.9343 102.9343 40.49 0.0001
LPL 1 311.3659 311.3659 122.49 0.0001
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
INTERCEPT 6.622558 18.15 0.0001 0.364849
RPFC 0.000322615 6.36 0.0001 0.000050
LPL 0.618344 11.07 0.0001 0.055870
Simulation Schematic
Opening queries.
Open claim file.
Open parameter file.
Options.
Layers.
Summary Periods.
Discount Rate.
Random number seeds.
Reported/IBNR.
End options.
End opening queries.
Simulation procedure.
Initialize seedstream 1.
Load claim file to capacity (repeat to EOF).
Initialize seedstream 2.
Open subcase summary file.
Subcase loop (stored claims).
Simulate parameter values
(stream 2, same for same case).
Claim loop (stored claims).
If open claim, simulate outcome (stream 1).
For all claims, simulate IBNR
counts and outcomes (stream 1).
Add to subcase summary.
End claim loop.
Write to subcase summary file.
End subcase loop.
Close subcase summary file.
End claim file load loop.
End simulation procedure.
Simulation Schematic (continued).
Collate subcases.
Case loop.
Read in subcase summary files.
Add to case summary.
Output case summary.
End case loop.
Close and delete subcase files.
End collate.
End schematic.
IBNR Simulation
Pr(C=c) = p(1 - p)c ,
E[C] = (1-p)/p,
p = probability for reporting
before valuation date.
Inverse Cumulative:
C = floor[ ln U/ ln(1-p)];
U ~ uniform(0, 1).
SAMPLE SIMULATION OUTPUT
PROFESSIONAL LIABILITY CLAIMS-MADE BY REPORT YEAR
INDEMNITY
0-1M 1M-6M 6M-11M OVER 11M TOTAL
------
1984 789,147 0 0 0 789,147
1985 263,710 103,861 102,041 40,663 510,274
1986 9,019,368 3,746,050 646,961 485,907 13,898,287
1987 17,463,443 12,799,066 4,295,600 20,591,956 55,150,065
1988 22,299,627 17,444,216 3,717,174 13,617,153 57,078,170
1989 24,798,133 27,911,787 6,172,546 9,875,022 68,757,488
1990 11,543,181 4,814,687 808,652 6,763 17,173,284
------
TOT 86,176,609 66,819,667 15,742,974 44,617,465 213,356,715
ALAE
0-1M 1M-6M 6M-11M OVER 11M TOTAL
------
1984 69,180 0 0 0 69,180
1985 73,818 848 625 249 75,540
1986 1,828,163 180,433 14,117 4,743 2,027,457
1987 4,002,306 577,122 193,326 222,853 4,995,608
1988 5,050,339 569,154 62,131 35,131 5,716,755
1989 4,527,334 624,868 83,968 110,853 5,347,024
1990 2,371,179 177,830 19,427 25 2,568,462
------
TOT 17,922,318 2,130,257 373,594 373,854 20,800,024
ULTIMATE CLAIM COUNTS
0-1M 1M-6M 6M-11M OVER 11M TOTAL
------
1984 1.00 0.00 0.00 0.00 1.00
1985 5.96 0.02 0.00 0.02 6.00
1986 62.06 3.76 0.12 0.06 66.00
1987 148.49 7.29 0.63 0.59 157.00
1988 185.69 10.02 0.84 0.45 197.00
1989 208.90 12.24 1.00 0.86 223.00
1990 139.43 3.31 0.22 0.04 143.00
------
TOT 751.53 36.63 2.82 2.02 793.00
Future Improvements to the System
· Single estimation program
· Incorporating parameter uncertainty
· Recoding the simulation into Fortran or C
· Introducing Bayesian priors into the estimation