## Estimating the Probability Distribution of Insurance Payments

```/*-------------------------------------------------------------------------

SAS Sample Library

Name: hcdmex01.sas
Description: Example program from SAS/ETS High Performance
Procedures User's Guide, The HPCDM Procedure
Title: Estimating the Probability Distribution of Insurance Payments
Product: SAS High Performance Econometrics Software
Keys: Compound Distribution Modeling,
Aggregate Loss Modeling
PROC: HPCDM
Notes:

---------------------------------------------------------------------------*/

/* Simulate data for losses incurred by several policyholders
in some period of time */
data losses(keep=policyholderId age gender carType annualMiles
education carSafety income noloss lossamount);
call streaminit(12345);
array cx{5} age gender carType annualMiles education;
array cbeta{6} _TEMPORARY_ (1 -0.75 1 0.6 -1 -0.25);

array sx{3} carType carSafety income;
array sbeta{4} _TEMPORARY_ (3.5 1.5 -0.8 0.6);

alpha = 1/3; theta = 1/alpha;
Sigma = 1;
do policyholderId=1 to 5000;
/* simulate policyholder and vehicle attributes */
age = MAX(int(rand('NORMAL', 35, 15)),16)/50;

if (rand('UNIFORM') < 0.5) then gender = 1; * female;
else gender = 2; * male;

if (rand('UNIFORM') < 0.7) then carType = 1; * sedan;
else carType = 2; * SUV;

annualMiles = MAX(1000, int(rand('NORMAL', 12000, 5000)))/5000;

educationLevel = rand('UNIFORM');
if (educationLevel < 0.5) then education = 1; *high school graduate;
else if (educationLevel < 0.85) then education = 2; *college graduate;
else education = 3; *advanced degree;

carSafety = rand('UNIFORM'); /* scaled to be between 0 & 1 */

income = MAX(15000,int(rand('NORMAL', education*30000, 50000)))/100000;

/* simulate number of losses incurred by this policyholder */
cxbeta = cbeta(1);
do i=1 to dim(cx);
cxbeta = cxbeta + cx(i) * cbeta(i+1);
end;
Mu = exp(cxbeta);
p = theta/(Mu+theta);
numloss = rand('NEGB',p,theta);

/* simulate severity of each of the losses */
if (numloss > 0) then do;
noloss = 0;
do iloss=1 to numloss;
Mu = sbeta(1);
do i=1 to dim(sx);
Mu = Mu + sx(i) * sbeta(i+1);
end;
lossamount = exp(Mu) * rand('LOGNORMAL')**Sigma;
output;
end;
end;
else do;
noloss = 1;
lossamount = .;
output;
end;
end;
run;

/* Aggregate number of losses for each policyholder */
data losscounts(keep=age gender carType annualMiles education numloss);
set losses;
by policyholderId;
retain numloss 0;
if (noloss ne 1) then
numloss = numloss + 1;
if (last.policyholderId) then do;
output;
numloss = 0;
end;
run;

/* Fit count regression model for the number of losses */
proc countreg data=losscounts;
model numloss = age gender carType annualMiles education / dist=negbin;
store work.countregmodel;
run;

/* Fit severity scale regression model for the loss severity */
proc severity data=losses plots=none outest=work.sevregest;
loss lossamount;
scalemodel carType carSafety income;
dist logn;
run;

/* Generate the scenario data set for multiple policyholders */
data groupOfPolicies(keep=policyholderId age gender carType annualMiles
education carSafety income
lowDeductible highDeductible limit annualLimit);
call streaminit(67897);

do policyholderId=1 to 5;
age = MAX(int(rand('NORMAL', 35, 15)),16)/50;

if (rand('UNIFORM') < 0.5) then gender = 1; * female;
else gender = 2; * male;

if (rand('UNIFORM') < 0.7) then carType = 1; * sedan;
else carType = 2; * SUV;

annualMiles = MAX(1000, int(rand('NORMAL', 12000, 5000)))/5000;

educationLevel = rand('UNIFORM');
if (educationLevel < 0.5) then education = 1; *high school graduate;
else if (educationLevel < 0.85) then education = 2; *college graduate;
else education = 3; *advanced degree;

carSafety = rand('UNIFORM'); /* scaled to be between 0 & 1 */

income = MAX(15000,int(rand('NORMAL', education*30000, 50000)))/100000;

lowDeductible = 100*(1+floor(rand('UNIFORM')*5));
highDeductible = lowDeductible + 500*(1+floor(rand('UNIFORM')*2));
limit = 2500*(1+floor(rand('UNIFORM')*3));
annualLimit = 10000*(1+floor(rand('UNIFORM')*2));

output;
end;
run;

/* Simulate the aggregate loss distribution and aggregate adjusted
loss distribution for the scenario with multiple policyholders */
proc hpcdm data=groupOfPolicies nreplicates=10000 seed=13579 print=all
countstore=work.countregmodel severityest=work.sevregest
plots=(edf pdf) nperturbedSamples=50
severitymodel logn;

if (_sev_ <= lowDeductible) then
amountPaid = 0;
else do;
if (_sev_ <= highDeductible) then
amountPaid = highDeductible *
(_sev_-lowDeductible)/(highDeductible-lowDeductible);
else
amountPaid = MIN(_sev_, limit); /* imposes per-loss payment limit */
end;
run;

/* Simulate the aggregate loss distribution and aggregate adjusted
loss distribution for the modified set of policy provisions */
proc hpcdm data=groupOfPolicies nreplicates=10000 seed=13579 print=all
countstore=work.countregmodel severityest=work.sevregest
plots=none nperturbedSamples=50
severitymodel logn;

if (_sev_ <= lowDeductible) then
amountPaid = 0;
else do;
if (_sev_ <= highDeductible) then
amountPaid = highDeductible *
(_sev_-lowDeductible)/(highDeductible-lowDeductible);
else
amountPaid = MIN(_sev_, limit); /* imposes per-loss payment limit */

/* impose policyholder's annual limit */
amountPaid = MIN(amountPaid, MAX(0,annualLimit - _cumadjsevforobs_));

/* impose group's annual limit */
amountPaid = MIN(amountPaid, MAX(0,15000 - _cumadjsev_));
end;
run;

```