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