Hierarchical Insurance Claims Modeling Travelers PASG (Predictive Analytics Study Group) Seminar

Hierarchical
Insurance Claims
Modeling
Hierarchical Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
joint work with Jed Frees, U of Wisconsin - Madison
Introduction
Travelers PASG (Predictive Analytics Study Group) Seminar
Tuesday, 12 April 2016
Data
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
Emiliano A. Valdez, PhD, FSA
Department of Mathematics
University of Connecticut
Storrs, Connecticut
page 1
A collection of work
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Frees and Valdez (2008), Hierarchical Insurance Claims
Modeling, Journal of the American Statistical Association,
Vol. 103, No. 484, pp. 1457-1469.
Introduction
Data
Frees, Shi and Valdez (2009), Actuarial Applications of a
Hierarchical Insurance Claims Model, ASTIN Bulletin, Vol.
39, No. 1, pp. 165-197.
The model
Model features
Model estimation
Covariates
Models of each component
Young, Valdez and Kohn (2009), Multivariate Probit
Models for Conditional Claim Types, Insurance:
Mathematics and Economics, Vol. 44, No. 2, pp. 214-228.
Antonio, Frees and Valdez (2010), A Multilevel Analysis of
Intercompany Claim Counts, ASTIN Bulletin, Vol. 40, No.
1, pp. 151-177.
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
page 2
Location of Singapore
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Introduction
Data
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
page 3
Car ownership in Singapore
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Owning and driving a car in Singapore is quite expensive.
Government has put up measures to manage car
ownership:
Introduction
Data
certificate of entitlement (COE)
vehicle quota system (VQS)
registration fees, road taxes (annual/semi-annual)
electronic road pricing (per use, driving in certain areas)
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
Every motor vehicle must have a valid insurance policy:
minimum required is coverage for personal injury to other
parties.
three major types available in the marketplace: third party,
third part + fire & theft, and comprehensive.
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
page 4
Basic data set-up
“Policyholder” i is followed over time t = 1, . . . , 9 years
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Unit of analysis “it” – a registered vehicle insured i over
time t (year)
Have available: exposure eit and covariates (explanatory
variables) xit
covariates often include age, gender, vehicle type, driving
history and so forth
Goal: understand how time t and covariates impact claims
yit .
Statistical methods viewpoint
basic regression set-up - almost every analyst is familiar
with:
part of the basic actuarial education curriculum
Introduction
Data
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
incorporating cross-sectional and time patterns is the
subject of longitudinal data analysis - a widely available
statistical methodology
page 5
Hierarchical
Insurance Claims
Modeling
More complex data set-up
Emiliano A. Valdez,
PhD, FSA
Some variations that might be encountered when
examining insurance company records
For each “it”, could have multiple claims, j = 0, 1, . . . , 5
For each claim yitj , possible to have one or a combination
of three (3) types of losses:
1
2
3
losses for injury to a party other than the insured yitj,1 “injury”;
losses for damages to the insured, including injury, property
damage, fire and theft yitj,2 - “own damage”; and
losses for property damage to a party other than the insured
yitj,3 - “third party property”.
Distribution for each claim is typically medium to long-tail
The full multivariate claim may not be observed:
Introduction
Data
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
Distribution of Claims, by Claim Type Observed
Value of M
1
2
3
4
5
6
Claim by Combination (y1 )
(y2 )
(y3 )
(y1 , y2 ) (y1 , y3 ) (y2 , y3 )
Number
102 17,216 2,899
68
18
3,176
Percentage
0.4
73.2
12.3
0.3
0.1
13.5
7
(y1 , y2 , y3 )
43
0.2
page 6
The hierarchical insurance claims model
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Traditional to predict or estimate insurance claims
distributions:
Introduction
Cost of Claims = Frequency × Severity
Data
The model
Model features
Joint density of the aggregate loss can be decomposed as:
Model estimation
Covariates
f (N, M, y)
= f (N) × f (M|N) × f (y|N, M)
Models of each component
Random effects NB model
Multimonial claim type
joint = frequency × conditional claim-type
× conditional severity.
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
This natural decomposition allows us to investigate and
model each component separately.
The fitted conditional
severity model
Conclusion
page 7
Model features
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Allows for risk rating factors to be used as explanatory
variables that predict both the frequency and the
multivariate severity components.
Introduction
Data
The model
Helps capture the long-tail nature of the claims distribution
through the GB2 distribution model.
Model features
Model estimation
Covariates
Models of each component
Provides for a “two-part” distribution of losses - when a
claim occurs, not necessary that all possible types of
losses are realized.
Allows to capture possible dependencies of claims among
the various types through a t-copula specification.
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
page 8
Literature on claims frequency/severity
There is large literature on modeling claims frequency and
severity
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Klugman, Panjer and Willmot (2004) - basics without
covariates
Kahane and Levy (JRI, 1975) - first to model joint
frequency/severity with covariates.
Coutts (1984) postulates that the frequency component is
more important to get right.
Many recent papers on frequency, e.g., Boucher and Denuit
(2006)
Introduction
Data
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Applications to motor insurance:
Multimonial claim type
Severity and copula
Brockman and Wright (1992) - good early overview.
Renshaw (1994) - uses GLM for both frequency and severity
with policyholder data.
Pinquet (1997, 1998) - uses the longitudinal nature of the
data, examining policyholders over time.
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
considered 2 lines of business: claims at fault and not at fault;
allowed correlation using a bivariate Poisson for frequency;
severity models used were lognormal and gamma.
Most other papers use grouped data, unlike our work.
page 9
Hierarchical
Insurance Claims
Modeling
Observable data
Model is calibrated with detailed, micro-level automobile
insurance records over eight years [1993 to 2000] of a
randomly selected Singapore insurer.
Emiliano A. Valdez,
PhD, FSA
Year 2001 data use for out-of-sample prediction
Information was extracted from the policy and claims files.
Unit of analysis - a registered vehicle insured i over time t
(year).
The observable data consist of
number of claims within a year: Nit , for
t = 1, . . . , Ti , i = 1, . . . , n
Introduction
Data
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
type of claim: Mitj for claim j = 1, . . . , Nit
the loss amount: yitjk for type k = 1, 2, 3.
exposure: eit
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
vehicle characteristics: described by the vector xit
Conclusion
The data available therefore consist of
{eit , xit , Nit , Mitj , yitjk } .
page 10
Risk factor rating system
Insurers adopt “risk factor rating system” in establishing
premiums for motor insurance.
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Some risk factors considered:
vehicle characteristics: make/brand/model, engine capacity,
year of make (or age of vehicle), price/value
driver characteristics: age, sex, occupation, driving
experience, claim history
other characteristics: what to be used for (private,
corporate, commercial, hire), type of coverage
Introduction
Data
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
The “no claims discount” (NCD) system:
Parameter estimates
The fitted frequency model
rewards for safe driving
discount upon renewal of policy ranging from 0 to 50%,
depending on the number of years of zero claims.
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
These risk factors/characteristics help explain the
heterogeneity among the individual policyholders.
page 11
Covariates
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Year: the calendar year - 1993-2000; treated as
continuous variable.
Vehicle Type: automotive (A) or others (O).
Introduction
Vehicle Age: in years, grouped into 6 categories -
The model
Data
Model features
0, 1-2, 3-5, 6-10, 11-15, <=16.
Model estimation
Covariates
Vehicle Capacity: in cubic capacity.
Models of each component
Random effects NB model
Multimonial claim type
Gender: male (M) or female (F).
Age: in years, grouped into 7 categories ages ≥21, 22-25, 26-35, 36-45, 46-55, 56-65, ≤66.
The NCD applicable for the calendar year - 0%, 10%,
20%, 30%, 40%, and 50%.
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
page 12
Random effects negative binomial count model
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Let λit = eit exp x0λ,it βλ be the conditional mean
parameter for the {it} observational unit, where
xλ,it is a subset of xit representing the variables needed for
frequency modeling.
Introduction
Data
The model
Negative binomial distribution model with parameters p
and r :
!
k +r −1 r
Pr(N = k |r , p) =
p (1 − p)k .
r −1
1
Here, σ = is the dispersion parameter and
r
p = pit is related to the mean through
1 − pit
= λit σ = eit exp(x0λ,it βλ )σ.
pit
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
page 13
Multinomial claim type
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Certain characteristics help describe the claims type.
To explain this feature, we use the multinomial logit of the
form
exp(Vm )
,
Pr(M = m) = P7
s=1 exp(Vs )
where Vm = Vit,m = x0M,it βM,m .
For our purposes, the covariates in xM,it do not depend on
the accident number j nor on the claim type m, but we do
allow the parameters to depend on type m.
Introduction
Data
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
Such has been proposed in Terza and Wilson (1990).
An alternative model to claim type, multivariate probit, was
considered in:
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
Young, Valdez and Kohn (2009)
page 14
Hierarchical
Insurance Claims
Modeling
Severity
We are particularly interested in accommodating the
long-tail nature of claims.
Emiliano A. Valdez,
PhD, FSA
We use the generalized beta of the second kind (GB2) for
each claim type with density
f (y ) =
exp (α1 z)
α +α ,
y |σ|B(α1 , α2 ) [1 + exp(z)] 1 2
where z = (ln y − µ)/σ.
Introduction
Data
The model
Model features
Model estimation
Covariates
Models of each component
µ is a location, σ is a scale and α1 and α2 are shape
parameters.
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
With four parameters, distribution has great flexibility for
fitting heavy tailed data.
Introduced by McDonald (1984), used in insurance loss
modeling by Cummins et al. (1990).
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
Many distributions useful for fitting long-tailed distributions
can be written as special or limiting cases of the GB2
distribution; see, for example, McDonald and Xu (1995).
page 15
GB2 Distribution
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Introduction
Data
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
Source: Klugman, Panjer and Willmot (2004), p. 72
page 16
Heavy-tailed regression models
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Loss Modeling - Actuaries have a wealth of knowledge on
fitting claims distributions. (Klugman, Panjer, Willmot,
2004) (Wiley)
Introduction
Data are often “heavy-tailed” (long-tailed, fat-tailed)
Data
Extreme values are likely to occur
The model
Extreme values are the most interesting - do not wish to
downplay their importance via transformation
Model estimation
Model features
Covariates
Models of each component
Random effects NB model
Studies of financial asset returns is another good example
Rachev et al. (2005) “Fat-Tailed and Skewed Asset Return
Distributions” (Wiley)
Healthcare expenditures - Typically skewed and fat-tailed
due to a few yet high-cost patients (Manning et al., 2005,
J. of Health Economics)
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
page 17
Hierarchical
Insurance Claims
Modeling
GB2 regression
Emiliano A. Valdez,
PhD, FSA
We allow scale and shape parameters to vary by type and
thus consider α1k , α2k and σk for k = 1, 2, 3.
Despite its prominence, there are relatively few
applications that use the GB2 in a regression context:
Introduction
Data
McDonald and Butler (1990) used the GB2 with regression
covariates to examine the duration of welfare spells.
Beirlant et al. (1998) demonstrated the usefulness of the
Burr XII distribution, a special case of the GB2 with α1 = 1,
in regression applications.
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
Sun et al. (2008) used the GB2 in a longitudinal data
context to forecast nursing home utilization.
We parameterize the location parameter as µik =
x0ik βk :
Thus, βk ,j = ∂ ln E (Y | x) /∂xj
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
Interpret the regression coefficients as proportional
changes.
page 18
Claim losses by type of claim
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Table 2.3. Summary Statistics of Claim Losses, by Type of Claim
Statistic
Third Party
Own Damage (y 2 )
Third Party
Injury (y 1 ) non-censored
all Property (y 3 )
Number
231
17,974
20,503
6,136
Mean
12,781.89
2,865.39 2,511.95
2,917.79
Standard Deviation
39,649.14
4,536.18 4,350.46
3,262.06
Median
1,700
1,637.40 1,303.20
1,972.08
Minimum
10
2
0
3
Maximum
336,596
367,183 367,183
56,156.51
Introduction
Data
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
page 19
Dependencies among claim types
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
We use a parametric copula (in particular, the t copula).
Suppressing the {i} subscript, we can express the joint
distribution of claims (y1 , y2 , y3 ) as
F(y1 , y2 , y3 ) = H (F1 (y1 ), F2 (y2 ), F3 (y3 )) .
Introduction
Data
The model
Model features
Model estimation
Here, the marginal distribution of yk is given by Fk (·) and
H(·) is the copula.
Modeling the joint distribution of the simultaneous
occurrence of the claim types, when an accident occurs,
provides the unique feature of our work.
Some references are: Frees and Valdez (1998), Nelsen
(1999).
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
page 20
Hierarchical
Insurance Claims
Modeling
The calendar year effect
Emiliano A. Valdez,
PhD, FSA
Introduction
Data
Count
0
1
2
3
4
5
Number
by Year
1993
91.5
7.9
0.5
0.1
4,976
Table 3.2. Number and Percentages of Claims, by Count and Year
Percentage by Year
1994 1995 1996
1997
1998
1999
2000
2001 Number
89.5
89.8
92.6
92.8
90.8
88.0
89.2
87.8 178,080
9.6
9.2
7.0
6.7
8.4
10.6
9.8
11.0
19,224
0.9
0.9
0.4
0.5
0.7
1.3
0.9
1.1
1,859
0.1
0.1
0.0
0.0
0.1
0.1
0.1
0.1
177
0.0
0.0
0.0
0.0
11
0.0
0.0
5,969 5,320 8,562 19,344 19,749 28,473 44,821 62,138 199,352
Percent
of Total
89.3
9.6
0.9
0.1
0.0
0.0
100.0
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
page 21
The effect of vehicle type and age
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Table 3.3. Number and Percentages of Claims, by Vehicle Type and Age
Percentage by Count
Count
Count Count Count Count Count
Percent
=0
=1
=2
=3
=4
=5
Number of Total
Vehicle Type
Other
88.6
10.1
1.1
0.1
0.0
0.0
43,891
22.0
89.5
9.5
0.9
0.1
0.0
155,461
78.0
Automobile
Vehicle Age (in years)
0
91.4
7.9
0.6
0.0
0.0
58,301
29.2
86.3
12.2
1.3
0.2
0.0
44,373
22.3
1
2
88.8
10.1
1.1
0.1
20,498
10.3
3 to 5
89.2
9.7
1.0
0.1
0.0
41,117
20.6
6 to 10
90.1
8.9
0.9
0.1
0.0
33,121
16.6
11 to 15
91.4
7.6
0.7
0.2
1,743
0.9
16 and older
89.9
8.5
1.5
199
0.1
Number
178,080 19,224 1,859
177
11
1 199,352
100.0
by Count
Introduction
Data
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
page 22
Hierarchical
Insurance Claims
Modeling
The effect of gender, age, NCD
Emiliano A. Valdez,
PhD, FSA
Table 3.4. Number and Percentages of Claims, by Gender, Age and NCD
for Automobile Policies
Percentage by Count
Count
Count Count Count Count Count
Percent
=0
=1
=2
=3
=4
=5
Number of Total
Gender
Female
89.7
Male
89.5
Person Age (in years)
21 and younger
86.9
22-25
85.5
26-35
88.0
36-45
90.1
46-55
90.4
56-65
90.7
66 and over
92.8
No Claims Discount (NCD)
0
87.7
10
87.8
20
89.1
30
89.1
40
89.8
50
91.0
Number
139,183
by Count
9.3
9.5
0.9
0.9
0.1
0.1
12.4
12.9
10.8
9.1
8.8
8.4
7.0
0.7
1.4
1.1
0.8
0.8
0.9
0.2
0.2
0.1
0.1
0.1
0.1
0.1
11.1
10.8
9.8
10.0
9.3
8.3
14,774
1.1
1.2
1.0
0.9
0.9
0.7
1,377
0.1
0.1
0.1
0.1
0.1
0.1
123
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
3
0.0
1
34,190
121,271
22.0
78.0
153
3,202
44,134
63,135
34,373
9,207
1,257
0.1
2.1
28.4
40.6
22.1
5.9
0.8
37,139
13,185
14,204
12,558
10,540
67,835
155,461
23.9
8.5
9.1
8.1
6.8
43.6
100.0
Introduction
Data
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
page 23
Comparing the alternative conditional claim type models
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Table 3.7. Comparison of Fit of Alternative Claim Type Models
Model Variables
Number of
-2 Log
Parameters Likelihood
AIC
BIC
Intercept Only
6
25,465.3 25,477.3 25,538.5
Automobile (A)
12
24,895.8 24,919.8 25,042.2
A and Gender
24
24,866.3 24,914.3 25,159.2
Year
12
25,315.6 25,339.6 25,462.0
Year1996
12
25,259.9 25,283.9 25,406.3
A and Year1996
18
24,730.6 24,766.6 24,950.3
VehAge2 (Old vs New)
12
25,396.5 25,420.5 25,542.9
VehAge2 and A
18
24,764.5 24,800.5 24,984.2
A, VehAge2 and Year1996
24
24,646.6 24,694.6 24,939.5
Introduction
Data
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
page 24
Hierarchical
Insurance Claims
Modeling
The fitted copula model
Table 3.8. Fitted Copula Model
Type of Copula
Independence
Normal copula
Parameter
Third Party Injury
σ1
1.316 (0.124)
1.320 (0.138)
α11
2.188 (1.482)
2.227 (1.671)
500.069 (455.832) 500.068 (408.440)
α12
βC,1,1 (intercept)
18.430 (2.139)
18.509 (4.684)
Own Damage
σ2
1.305 (0.031)
1.301 (0.022)
5.658 (1.123)
5.507 (0.783)
α21
α22
163.605 (42.021)
163.699 (22.404)
10.037 (1.009)
9.976 (0.576)
βC,2,1 (intercept)
βC,2,2 (VehAge2)
0.090 (0.025)
0.091 (0.025)
βC,2,3 (Year1996)
0.269 (0.035)
0.274 (0.035)
βC,2,4 (Age2)
0.107 (0.032)
0.125 (0.032)
βC,2,5 (Age3)
0.225 (0.064)
0.247 (0.064)
Third Party Property
σ3
0.846 (0.032)
0.853 (0.031)
0.597 (0.111)
0.544 (0.101)
α31
α32
1.381 (0.372)
1.534 (0.402)
βC,3,1 (intercept)
1.332 (0.136)
1.333 (0.140)
βC,3,2 (VehAge2)
-0.098 (0.043)
-0.091 (0.042)
βC,3,3 (Year1)
0.045 (0.011)
0.038 (0.011)
Copula
ρ12
0.018 (0.115)
ρ13
-0.066 (0.112)
ρ23
0.259 (0.024)
r
Model Fit Statistics
log-likelihood
-31,006.505
-30,955.351
number of parms
18
21
AIC
62,049.010
61,952.702
Note: Standard errors are in parenthesis.
t-copula
Emiliano A. Valdez,
PhD, FSA
1.320 (0.120)
2.239 (1.447)
500.054 (396.655)
18.543 (4.713)
1.302 (0.029)
5.532 (0.992)
170.382 (59.648)
10.106 (1.315)
0.091 (0.025)
0.274 (0.035)
0.125 (0.032)
0.247 (0.064)
0.853 (0.031)
0.544 (0.101)
1.534 (0.401)
1.333 (0.139)
-0.091 (0.042)
0.038 (0.011)
0.018 (0.115)
-0.066 (0.111)
0.259 (0.024)
193.055 (140.648)
Introduction
Data
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
-30,955.281
22
61,954.562
page 25
Concluding remarks
Model features
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Allows for covariates for the frequency, type and severity
components
Captures the long-tail nature of severity through the GB2.
Provides for a “two-part” distribution of losses - when a
claim occurs, not necessary that all possible types of losses
are realized.
Allows for possible dependencies among claims through a
copula
Introduction
Data
The model
Model features
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Allows for heterogeneity from the longitudinal nature of
policyholders (not claims)
Severity and copula
Parameter estimates
The fitted frequency model
Other applications
Could look at financial information from companies
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
Could examine health care expenditure
Compare companies’ performance using multilevel,
intercompany experience
page 26
Hierarchical
Insurance Claims
Modeling
Emiliano A. Valdez,
PhD, FSA
Introduction
Data
The model
Model features
- Thank you -
Model estimation
Covariates
Models of each component
Random effects NB model
Multimonial claim type
Severity and copula
Parameter estimates
The fitted frequency model
The fitted conditional claim
type model
The fitted conditional
severity model
Conclusion
page 27