Recap (1.2.1 in EB)
• Property insurance is economic responsibility
for incidents such as fires and accidents
passed on to an insurer against a fee
• The contract, known as a policy, releases
claims when such events occur
• A central quantity is the total claim X amassed
during a certain period of time (typically a
year)
Overview pricing (1.2.2 in EB)
Premium
Individual
Claim
Insurance
company
0 (no event above deductible ), with probabilit y 1 - p
Total claim  
 X (event above deductible ) with probabilit y p
Due to the law of large numbers the insurance company is cabable of estimating
the expected claim amount
Distribution of X, estimated with claims data
E( X )  p 
all x
Risk premium
sf (s)ds  (1  p)0  
Pu
Expected claim amount given an event
•Probability of claim,
Expected consequence of claim
•Estimated with claim frequency
•We are interested in the distribution of the claim frequency
•The premium charged is the risk premium inflated with a loading (overhead and margin)
Control (1.2.3 in EB)
•Companies are obliged to aside funds to cover future obligations
•Suppose a portfolio consists of J policies with claims X1,…,XJ
•The total claim is then
  X 1  ...  X J
Portfolio claim size
E( )
•We are interested in
as well as its distribution

•Regulators demand sufficient funds to cover
with high probability
•The mathematical formulation is in term of q , which is the solution of the equation
Pr{  q }  

where is small for example 1%
•The amount q is known as the solvency capital or reserve
Insurance works because risk can be
diversified away through size (3.2.4 EB)
•The core idea of insurance is risk spread on many units
•Assume that policy risks X1,…,XJ are stochastically independent
•Mean and variance for the portfolio total are then
E (  )   1  ...   J and var(  )   1  ...   J
and  j  E ( X j ) and  j  sd ( X j ). Introduce
1
1
2
  ( 1  ...   J ) and   ( 1  ...   J )
J
J
which is average expectation and variance. Then
E (  )  J  and sd (  )  J  so that
sd (  )  / 

E(  )
J
•The coefficient of variation (shows extent of variability in relation to the mean)
approaches 0 as J grows large (law of large numbers)
•Insurance risk can be diversified away through size
•Insurance portfolios are still not risk-free because
•of uncertainty in underlying models
•risks may be dependent
How are random variables sampled?
Inversion (2.3.2 in EB):
•Let F(x) be a strictly increasing distribution function with inverse X  F 1 (U ) and let
1
1
X  F (U ) or X  F (1  U ), U ~ Uniform
•Consider the specification on the left for which U=F(X)
•Note that
Pr( X  x)  Pr( F ( X )  F ( x))
 Pr(U  F ( x))  F ( x)
since Pr(U  u )  u
1
U[0,1]
F(x)
u1
x1
Outline of the course
|
Basic concepts and introduction
How is claim frequency modelled?
How is claim reserving modelled?
How is claim size modelled?
How is pricing done?
Solvency
Credibility theory
Reinsurance
Repetition
Models treated
Poisson, Compound Poisson,
Poisson regression, negative
binomial model
Delay modelling, chain ladder
Gamma distribution, log-normal
distribution, Pareto distribution,
Weibull distribution
Binomial models
Monte Carlo simulation
Buhlmann Straub
Curriculum
EB 1.2, 2.3.1, 2.3.2, 3.2, 3.3
Duration in
lectures
1
EB 8.2, 8.3, 8.4
EB 8.5, Note
2-3
1-2
EB 9
EB 10
EB 10, Note
EB 10
EB 10
2-3
1
1-2
1
1
1
7
Course literature
Curriculum:
Chapter 1.2, 2.3.1, 2.3.2, 2.5, 3.2, 3.3 in EB
Chapter 8,9,10 in EB
Note on Chain Ladder
Lecture notes by NFH
Exercises
The following book will be used (EB):
Computation and Modelling in Insurance and Finance, Erik Bølviken,
Cambridge University Press (2013)
•Additions to the list above may occur during the course
•Final curriculum will be posted on the course web site in due time
Assignment must be approved to be able to participate
in exame
8
Overview of this session
The Poisson model (Section 8.2 EB)
Some important notions and some practice too
Examples of claim frequencies
Random intensities (Section 8.3 EB)
9
Poisson
Introduction
Some notions
Examples
Random intensities
•
•
Actuarial modelling in general insurance is broken down on claim
frequency and claim size
This is natural due to definition of risk premium:
E( X )  p 
all x
Risk premium
sf (s)ds  (1  p)0  
Pu
Expected claim amount given an event
•Probability of claim,
Expected consequence of claim
•Estimated with claim frequency
•
•
•
The Poisson distribution is often used in modelling the distribution of claim
numbers
The parameter is lambda = muh*T (single policy) and lambda = J*muh*T
(portfolios
The modelling can be made more sophisticated by extending the model for
muh, either by making muh stochastic or by linking muh to explanatory
variables
10
The world of Poisson (Chapter 8.2)
Poisson
Some notions
Examples
Number of claims
Ik
Ik-1
t0=0
tk-2
tk-1
Random intensities
Ik+1
tk
tk+1
tk=T
•What is rare can be described mathematically by cutting a given time period T into K
small pieces of equal length h=T/K
•On short intervals the chance of more than one incident is remote
•Assuming no more than 1 event per interval the count for the entire period is
N=I1+...+IK ,where Ij is either 0 or 1 for j=1,...,K
•If p=Pr(Ik=1) is equal for all k and events are independent, this is an ordinary Bernoulli
series
Pr( N  n) 
K!
p n (1  p) K n , for n  0,1,..., K
n!( K  n)!
•Assume that p is proportional to h and set
p  h where 
is an intensity which applies per time unit
11
The world of Poisson
Pr( N  n) 
K!
p n (1  p ) K  n
n!( K  n)!
Some notions
Examples
Random intensities
K!
 T   T 


 1 

n!( K  n)!  K  
K 
n
Poisson
K n
( T ) n K ( K  1)    ( K  n  1)  T 
1

1



n!
Kn
K   T  n

1 

K 

K
1
 e T
K 
K 
1
K 
( T )  T
 Pr( N  n) 
e
K 
n!
n
In the limit N is Poisson distributed with parameter
  T
12
Poisson
The world of Poisson
Some notions
Examples
Random intensities
•Let us proceed removing the zero/one restriction on Ik. A more flexible specification is
Pr( I k  0)  1  h  o(h), Pr( I k  1)  h  o(h),
Pr(I k  1)  o(h)
Where o(h) signifies a mathematical expression for which
o( h)
 0 as h  0
h
It is verified in Section 8.6 that o(h) does not count in the limit
Consider a portfolio with J policies. There are now J independent processes in
parallel and if
the total number
 j is the intensity of policy j and Ik
of claims in period k, then
J
Pr(I k  0)   (1   j )
j 1
No claims


Pr(I k 1)   i h (1   j h)
i 1 
j i

J
and
Claims policy i only
13
Poisson
The world of Poisson
Some notions
Examples
Random intensities
•Both quanities simplify when the products are calculated and the powers of h identified
J 3 3
Pr(I k  0) 
 (1   h)  (1   h)(1   h)(1   h)
j
1
2
3
j 1
 (1  1h   2 h   2 1h 2 )(1  3 h)
 1  1h   2 h   2 1h 2  3 h(1  1h   2 h   2 1h 2 )
 1  1h   2 h  3 h  o(h)
J
Pr(I k  1)  (  j )h  o(h)
j 1
•It follows that the portfolio number of claims N is Poisson distributed with parameter
  (1  ...   J )T  JT , where   (1  ...   J ) / J
•When claim intensities vary over the portfolio, only their average counts
14
Poisson
When the intensity varies over time
•A time varying function
   (t ) handles the mathematics. The binary
variables I1,...Ik are now based on different intensities
1 ,....,  K
Some notions
Examples
Random intensities
where k   (tk ) for k  1,..., K
•When I1,...Ik are added to the total count N, this is the same issue as if K different
policies apply on an interval of length h. In other words, N must still be Poisson, now
with parameter
K
T
k 1
0
  h k    (t )dt as h  0
where the limit is how integrals are defined. The Poisson parameter for N can also be
written
  T where
T
1
    (t )dt ,
T0
And the introduction of a time-varying function
doesn’t change things
 (t )
much. A time average
takes over from a constant


15
Poisson
The Poisson distribution
Some notions
Examples
Random intensities
•Claim numbers, N for policies and N for portfolios, are Poisson distributed with
parameters
  T
Policy level
The intensity
and
  JT
Portfolio level
is an average over time and policies.
Poisson models have useful operational properties. Mean, standard deviation and
skewness are
E( N )  ,
sd ( N )  
and
skew( ) 
1

The sums of independent Poisson variables must remain Poisson, if N1,...,NJ are
independent and Poisson with parameters
then
1 ,...,
J
Ν  N1  ...  N J ~ Poisson (1  ...  J )
16
Poisson
Some notions
Client
Examples
Random intensities
Policies and claims
Policy
Insurable object
(risk)
Claim
Insurance cover
Cover element
/claim type
Poisson
Some notions
Car insurance client
Examples
Random intensities
Car insurance policy
Policies and claims
Insurable object
(risk), car
Claim
Third part liability
Insurance cover third party liability
Legal aid
Driver and passenger acident
Fire
Theft from vehicle
Insurance cover partial hull
Theft of vehicle
Rescue
Insurance cover hull
Accessories mounted rigidly
Own vehicle damage
Rental car
Poisson
Some notes on the different insurance covers on the previous slide:
Some notions
Third part liability is a mandatory cover dictated by Norwegian law that covers damages
Examples
on third part vehicles, propterty and person. Some insurance companies
provide
Random intensities
additional coverage, as legal aid and driver and passenger
accident insurance.
Partial Hull covers everything that the third part liability covers. In addition, partial hull covers damages on own
vehicle caused by fire, glass rupture, theft and vandalism in association with theft. Partial hull also includes rescue.
Partial hull does not cover damage on own vehicle caused by collision or landing in the ditch. Therefore, partial hull is
a more affordable cover than the Hull cover. Partial hull also cover salvage, home transport and help associated with
disruptions in production, accidents or disease.
Hull covers everything that partial hull covers. In addition, Hull covers damages on own vehicle in a collision,
overturn, landing in a ditch or other sudden and unforeseen damage as for example fire, glass rupture, theft or
vandalism. Hull may also be extended to cover rental car.
Some notes on some important concepts in insurance:
What is bonus?
Bonus is a reward for claim-free driving. For every claim-free year you obtain a reduction in the insurance premium
in relation to the basis premium. This continues until 75% reduction is obtained.
What is deductible?
The deductible is the amount the policy holder is responsible for when a claim occurs.
Does the deductible impact the insurance premium?
Yes, by selecting a higher deductible than the default deductible, the insurance premium may be significantly
reduced. The higher deductible selected, the lower the insurance premium.
How is the deductible taken into account when a claim is disbursed?
The insurance company calculates the total claim amount caused by a damage entitled to disbursement. What you
get from the insurance company is then the calculated total claim amount minus the selected deductible.
19
Poisson
Some notions
Key ratios – claim frequency
Examples
Random intensities
•The graph shows claim frequency for all covers for motor insurance
•Notice seasonal variations, due to changing weather condition throughout the years
Claim frequency all covers motor
35,00
30,00
25,00
20,00
15,00
10,00
5,00
2012 D
2012 O
2012 A
2012 J
2012 A
2012 F
2011 +
2011 N
2011 S
2011 J
2011 M
2011 M
2011 J
2010 D
2010 O
2010 A
2010 J
2010 A
2010 F
2009 +
2009 N
2009 S
2009 J
2009 M
2009 M
2009 J
0,00
20
Poisson
Some notions
Key ratios – claim severity
Examples
Random intensities
•The graph shows claim severity for all covers for motor insurance
Average cost all covers motor
30 000
25 000
20 000
15 000
10 000
5 000
2009 J
2009 M
2009 M
2009 J
2009 S
2009 N
2009 +
2010 F
2010 A
2010 J
2010 A
2010 O
2010 D
2011 J
2011 M
2011 M
2011 J
2011 S
2011 N
2011 +
2012 F
2012 A
2012 J
2012 A
2012 O
2012 D
0
21
Poisson
Some notions
Key ratios – pure premium
Examples
Random intensities
•The graph shows pure premium for all covers for motor insurance
Pure premium all covers motor
6 000
5 000
4 000
3 000
2 000
1 000
2009 J
2009 M
2009 M
2009 J
2009 S
2009 N
2009 +
2010 F
2010 A
2010 J
2010 A
2010 O
2010 D
2011 J
2011 M
2011 M
2011 J
2011 S
2011 N
2011 +
2012 F
2012 A
2012 J
2012 A
2012 O
2012 D
0
22
Poisson
Some notions
Key ratios – pure premium
Examples
Random intensities
•The graph shows loss ratio for all covers for motor insurance
Loss ratio all covers motor
140,00
120,00
100,00
80,00
60,00
40,00
20,00
2009 J
2009 M
2009 M
2009 J
2009 S
2009 N
2009 +
2010 F
2010 A
2010 J
2010 A
2010 O
2010 D
2011 J
2011 M
2011 M
2011 J
2011 S
2011 N
2011 +
2012 F
2012 A
2012 J
2012 A
2012 O
2012 D
0,00
23
Poisson
Key ratios – claim frequency TPL
and hull
Some notions
Examples
Random intensities
•The graph shows claim frequency for third part liability and hull for motor
insurance
Claim frequency hull motor
35,00
30,00
25,00
20,00
15,00
10,00
5,00
24
2012 D
2012 O
2012 J
2012 A
2012 F
2012 A
2011 +
2011 N
2011 J
2011 S
2011 M
2011 J
2011 M
2010 D
2010 O
2010 J
2010 A
2010 F
2010 A
2009 +
2009 N
2009 J
2009 S
2009 M
2009 J
2009 M
0,00
Poisson
Key ratios – claim frequency and
claim severity
Some notions
Examples
Random intensities
•The graph shows claim severity for third part liability and hull for
motor insurance
Average cost third part liability motor
70 000
Average cost hull motor
25 000
60 000
20 000
50 000
40 000
15 000
30 000
10 000
20 000
5 000
10 000
0
2009 J
2009 M
2009 M
2009 J
2009 S
2009 N
2009 +
2010 F
2010 A
2010 J
2010 A
2010 O
2010 D
2011 J
2011 M
2011 M
2011 J
2011 S
2011 N
2011 +
2012 F
2012 A
2012 J
2012 A
2012 O
2012 D
2009 J
2009 M
2009 M
2009 J
2009 S
2009 N
2009 +
2010 F
2010 A
2010 J
2010 A
2010 O
2010 D
2011 J
2011 M
2011 M
2011 J
2011 S
2011 N
2011 +
2012 F
2012 A
2012 J
2012 A
2012 O
2012 D
0
25
Poisson
Some notions
Random intensities (Chapter 8.3)
•
•
•
Driver ability, personal risk averseness,

This randomeness can be managed by making
a stochastic variable
This extension may serve to capture uncertainty affecting all policy holders jointly, as well, such
as altering weather conditions
The models are conditional ones of the form
N |  ~ Poisson ( T )
and
Ν |  ~ Poisson ( JT )
Policy level
•
Random intensities
How  varies over the portfolio can partially be described by observables such as age or sex
of the individual (treated in Chapter 8.4)
There are however factors that have impact on the risk which the company can’t know much
about
–
•
•
Examples
Let
Portfolio level
  E (  ) and   sd(  ) and recall that E ( N |  )  var( N |  )  T
which by double rules in Section 6.3 imply
E ( N )  E (T )  T
•
and var ( N )  E (T )  var( T )  T   2T 2
Now E(N)<var(N) and N is no longer Poisson distributed
26
Poisson
Some notions
The rule of double variance
Examples
Random intensities
Let X and Y be arbitrary random variables for which
 ( x)  E (Y | x)
and
 2  var(Y | x)
Then we have the important identities
  E (Y )  E{ ( X )}
var(Y )  E{ 2 ( X )}  var{ ( X )}
and
Rule of double expectation
Rule of double variance
Recall rule of double expectation
E ( E (Y | x)) 
 ( E (Y | x)) f
X
( x)dx 
all x

  yf
all y all x
X ,Y
(x, y)dxdy 
  yf
Y|X
(y|x)dy f X ( x)dx
all x all y
 yf
all y
all x
X ,Y
(x, y)dxdy 
 yf
Y
( y )dy  E (Y )
all y
27
Poisson
wikipedia tells us how the rule of double
variance can be proved
Some notions
Examples
Random intensities
Poisson
Some notions
The rule of double variance
Examples
Random intensities
Var(Y) will now be proved from the rule of double expectation. Introduce
^
Y   ( x)
^
E(Y )  E(Y)
and note that
which is simply the rule of double expectation. Clearly
^
^
^
^
^
^
(Y   )  ((Y  Y )  (Y   ))  (Y  Y )  (Y   )  2(Y  Y )(Y   ).
2
2
2
2
Passing expectations over this equality yields
var(Y )  B1  B2  2B3
where
^
^
^
^
B1  E (Y  Y ) , B2  E (Y   ) , B3  E (Y  Y )(Y   ),
2
2
which will be handled separately. First note that
^
 ( x)  E{(Y   ( x)) | x}  E{(Y  Y ) 2 | x}^
2
2
and by the rule of double expectation applied to
^
E{ ( x)}  E{(Y  Y ) 2  B1.
2
The second term makes use of the fact that
double expectation so that
(Y  Y ) 2
^
bythe
E (rule
Y ) of
29
Poisson
Some notions
The rule of double variance
Examples
Random intensities
^
B2  var(Y )  var{ ( x)}.
The final term B3 makes use of the rule of double expectation once again which yields
B3  E{c( X )}
where
^
^
^
^
c( X )  E{(Y  Y )(Y   ) | x}  E{(Y  Y ) | x}(Y   )
^
^
^
^
^
 {E (Y | x)  Y )}(Y   )  {Y  Y )}(Y   )  0
^
And B =0. The second equality is true because the factor
is
(Y   )
fixed by X. Collecting the expression for B1, B2 and B3 proves the double variance
formula
3
30
Poisson
Random intensities

Specific models for

Some notions
Examples
Random intensities
are handled through the mixing relationship
Pr( N  n)   Pr( N  n |  ) g (  )d   Pr( N  n | i ) Pr(   i )
i
0
Gamma models are traditional choices for
(  ) below
andgdetailed
Estimates of  and 
can be obtained from historical data without specifying
( claims
)
. Let n1,...,nn gbe
from n policy holders and T1,...,TJ their exposure to risk.^ The intensity
if individual j is then estimated as  j
.
 j  n j / Tj
Uncertainty is huge. One solution is
^
n
^
   wj  j
where w j 
j 1
Tj
T
i 1
and
n
^
2 
^
 w (
j 1
j
(1.5)
n
i
^
j
  )2  c
n
1   w2j
j 1
^
wh ere
c
(n  1) 
n
T
i 1
(1.6)
i
Both estimates are unbiased. See Section 8.6 for details. 10.5 returns to this.
31
Poisson
The negative binomial model
Some notions
Examples
The most commonly applied model for muh is the Gamma distribution.
It is then assumed that
Random intensities
  G where G ~ Gamma(  )
Here Gamma(  ) is the standard Gamma distribution with mean one, and
fluctuates around  with uncertainty controlled by .Specifically
E (  )   and sd (  )   / 
Since sd(  )  0
emerges in the limit.

as   , the pure Poisson model with fixed intensity
The closed form of the density function of N is given by
Pr( N  n) 
( n   )
p (1  p) n
(n  1)( )
where p 

  T
nbin(  , .)Mean,
for n=0,1,.... This is the negative binomial distribution to be denoted
standard deviation and skewness are
E ( N )  T , sd ( N )  T (1  T /  ) , skew(N) 
1  2T/ 
T (1  T /  )
(1.9)
Where E(N) and sd(N) follow from (1.3) when
   / is inserted.
Note that if N1,...,NJ are iid then N1+...+NJ is nbin (convolution property).
32
Poisson
Some notions
Fitting the negative binomial
Examples
Random intensities
Moment estimation using (1.5) and (1.6) is simplest technically. The estimate of is simply

^
 which yields
in (1.5), and
invoke (1.8) right
 for
^
^
^
^
  / 
^
2
^
2
so
that



/

.
^
^
If   0 , interpret it as an infinite  or a pure Poisson model.
Likelihood estimation: the log likelihood function follows by inserting nj for n in (1.9) and
adding the logarithm for all j. This leads to the criterion
n
L( ,  )   log( n j   )  n{log( ( ))   log(  )} 
j 1
n
n
j 1
j
log(  )  (n j   ) log(   T j )
where constant factors not depending on
and
 omitted.
have been
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