INTRODUCTION: OVERVIEW OF GLMs AND OF THEIR ACTUARIAL APPLICATIONS

63 Pages·2005·1.45 MB·English

INTRODUCTION: OVERVIEW OF
GLMs AND OF THEIR
ACTUARIAL APPLICATIONS
GENERALIZED LINEAR MODELS
Michel DENUIT
[email protected]
Louvain School of Statistics, Biostatistics and Actuarial Science (LSBA)
UCL, Belgium
MichelDenuit,UNIL,Lausanne,2011-2012–p.1/63
Summary
1. Brief presentation of Generalized Linear Models (GLMs)
2. Overview of applications in actuarial science
- Analysis of claim frequencies and severities
- Graduation of death rates (life insurance) and
sickness rates (multistate modelling for generalized
life insurance)
- Rating by geographic area
- And even more...
MichelDenuit,UNIL,Lausanne,2011-2012–p.2/63
Technical vs commercial risk pricing
In non-life business, the pure premium is the expected
(cid:15)
cost of all the claims filed by policyholders during the
coverage period (under the assumption of the law of
large numbers) given observable characteristics.
The computation of the technical pure premium relies
(cid:15)
on a statistical model incorporating all the available
information about the risk.
The technical tariff aims to evaluate as accurately as
(cid:15)
possible the pure premium for each policyholder.
MichelDenuit,UNIL,Lausanne,2011-2012–p.3/63
Technical vs commercial risk pricing
Market premiums may differ from those computed by
(cid:15)
actuaries because, e.g., of regulation or of the position
of the company compared to its market competitors.
For instance, gender generally needs to be included in
(cid:15)
the technical tariff but has to be excluded from the
commercial premiums in the EU.
Thus, gender is a risk factor but not a rating factor.
(cid:15)
Here, we only discuss technical premiums incorporating
(cid:15)
all the relevant information to counteract adverse
selection and to monitor the premium transfers created
inside the portfolio by the application of the commercial
price list.
MichelDenuit,UNIL,Lausanne,2011-2012–p.4/63
Why classifying risks?
Each time a competitor uses an additional rating factor,
(cid:15)
the actuary has to refine the partition to avoid loosing
the best drivers with respect to this factor.
This explains why so many factors are used by
(cid:15)
insurance companies: this is not required by actuarial
theory but by competition among insurers, instead.
Insurance companies have to use a rating structure that
(cid:15)
matches the premiums for the risks as closely as the
rating structures used by competitors.
It is thus the competition between insurers that leads to
(cid:15)
more and more partitioned portfolios, and not actuarial
science.
MichelDenuit,UNIL,Lausanne,2011-2012–p.5/63
Why classifying risks?
This trend towards more risk classification often causes
(cid:15)
social disasters: bad drivers (or more precisely, drivers
sharing the characteristics of bad drivers) do not find a
coverage for a reasonable price, and are tempted to
drive without insurance.
Note also that even if a correlation exists between the
(cid:15)
rating factor and the risk covered by the insurer, there
may be no causal relationship between that factor and
risk.
Requiring that insurance companies establish such a
(cid:15)
causal relationship to be allowed to use a rating factor is
subject to debate.
MichelDenuit,UNIL,Lausanne,2011-2012–p.6/63
Generalized linear models (GLMs)
Multiple linear regression assumes
(cid:15)
p
2
Y = (cid:12) + (cid:12) x + (cid:15) with (cid:15) or(0; (cid:27) =w );
i 0 j ij i i i
(cid:24) N
j=1
X
p
2
(cid:27)
Y or (cid:12) + (cid:12) x ; :
i 0 j ij
, (cid:24) N 0 w 1
i
j=1
X
@ A
where w is a known weight associated to observation i.
i
In actuarial applications, a symmetric Normally
(cid:15)
distributed random variable with a fixed variance does
not adequately describe claim counts nor claim
amounts.
MichelDenuit,UNIL,Lausanne,2011-2012–p.7/63
Example: Claim numbers
0
0
0
0
6
0
0
0
0
5
0
0
0
0
4
y
c
Frequen 30000
0
0
0
0
2
0
0
0
0
1
0
0 1 2 3 4
Number of reported claims
MichelDenuit,UNIL,Lausanne,2011-2012–p.8/63
Example: Cost of claims
0
0
0
0
2
0
0
0
5
1
y
c
n
e
Frequ 0000
1
0
0
0
5
0
0e+00 1e+06 2e+06 3e+06 4e+06
Cost
MichelDenuit,UNIL,Lausanne,2011-2012–p.9/63
Example: Cost of claims on the log scale
0
0
0
6
0
0
0
5
0
0
0
4
y
c
n
e
u
q 0
e 0
Fr 30
0
0
0
2
0
0
0
1
0
2 4 6 8 10 12 14 16
Log. Cost
MichelDenuit,UNIL,Lausanne,2011-2012–p.10/63