GIRO Conference and Exhibition 2012
Juggling uncertainty the actuary’s part to play
20/09/2012
© 2012 The Actuarial Profession  www.actuaries.org.uk
GIRO Conference and Exhibition 2012
Individual Claim Development
Julien Saunier
© 2011 The Actuarial Profession  www.actuaries.org.uk
Agenda
• Context / Objectives / basic idea
• Methodology
• Simulation process
• Results
Context / Objectives / basic idea
• Innovative method for evaluating severity for long tail business
• Goal: To improve upon the default approach which consists of
applying Loss Development Factors calculated on aggregated
triangles to individual (large) losses
• Aim is therefore to estimate the ultimate value of each and
every individual claim.
• Basic idea: reproduce what we observe on “Closed” claims to
“Open” ones
Methodology – Initial step
• Trend data for inflation and hyper inflation (if necessary)
• Select claims which are “closed”. The definition of “closed” is
defined as :
– Case reserves less than X% of the incurred claims (ex
paid=96, incurred=100)
• For each “closed” claim calculate the ratio R which is its
ultimate value to its incurred value after n years of development.
Where S is the incurred value
• R is considered to be a random variable.
Methodology – Ratios to Ultimate
Claim 1
1
2
3
4
5
6
Claim 1
1
2
3
4
5
6
1
2
3
4
5
S2,1
S2,2
S2,3
S2,4
S2,5
6
Closed
Closed
1
2
3
4
5
R2,1
R2,2
R2,3
R2,4
R2,5
6
R 2,1 =
S 2,5
S 2,1
Methodology – Definition of thresholds
For all development
years
Ratio vs Incurred
30.000
Ratio to ultimate
25.000
20.000
15.000
10.000
5.000
0.000
0
500
1 000
1 500
2 000
2 500
Incurred
Dependency of R on amount of claim
3 000
Methodology – Definition of clustering
For all development
years
Ratio vs Incurred
4.500
Ratio to ultimate
4.000
3.500
3.000
2.500
2.000
1.500
1.000
0.500
0.000
0
500
1 000
1 500
2 000
2 500
Incurred
Clustering observed around R=0 and R=1
3 000
Definition of R ratios
Use R ratios observed on Closed claims and
apply them to Open ones
Consideration of claim size, development
year and clustering for ratios close to 0 and 1
For each open claim, conditional
simulation of its status
Status of open claims
Status
Claim size
Development
year
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• Stable: R around 1
• ‘sans-suite’ : R around 0
• Other : R<>0,1
• Definition of 3 bands of capital
• Maximum ratio
• Position of the claim also considered
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Status of Open claims
1
2
3
4
5
6
1
2
3
4
5
S2,1
S2,2
S2,3
S2,4
S2,5
S4,1
S4,2
S4,3
R=0
S 6,1
R=1
R<>0,1
R=0
R=1
R<>0,1
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Conditionally to
- claim size
- DY
Conditionally to
- claim size
- DY
For each Open claim we simulate conditionally its status
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Case where R is different from 0 and 1
• In this case, the goal is to find theoretical distributions which fit
observed ratios:
1
2
3
4
5
6
1
2
3
4
5
R2,1
R2,2
R 3,2
R4,2
R2,3
R2,4
R2,5
Fit
Fit
6
Fit
• Tests have been performed on French and Italian companies:
Regardless of the company and of the development year:
The distribution which best fits R is repeatedly the same:
Split Simple Pareto
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11
Case where R is different from 0 and 1
• The fact that Split Simple Pareto came out is worth noting:
– Splice of 2 different distribution: Power and Simple Pareto
– Corresponds to claims developing up or down
• This distribution has 3 parameters to be estimated using
conditional MLE with following elements:
– Studied interval: [min; lower band around 1] ∪ [upper band around 1; max ]
– Density:
β −1
 αβ
x
*   if 0 ≤ x ≤ θ

θ (α + β )  θ 
density ( x, α , β , θ ) = 
α +1
θ 
 αβ
θ (α + β ) *  x  if θ ≤ x ≤ ∞

– Negative likelihood:
β −1
 αβ
θ
αβ
 xi 
NLL = −∑ ln 
*   *10≤ xi ≤θ +
* 
θ (α + β )  xi
θ (α + β )  θ 
i =1
n
© 2011 The Actuarial Profession  www.actuaries.org.uk



α +1

1


*1θ ≤ xi ≤∞  + n * ln

 Cond.const 

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Case where R is different from 0 and 1
• If insufficient data, another way must be adopted to find
parameters:
– Market parameters
Possibility to apply this methodology on a market database
– If Pr(R≠0,1 / condition) is very low, possibility to force it to 0
it is generally true for the right part of the triangle
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13
Simulating Ultimate Claims - Example
Scheme with 100 simulations
Closed Claims
Open Claims
Pr(R=0 | ...) = 0.1
Pr(R=1 | ...) = 0.2
Pr(R<> 0,1 | ...) = 0.7
10 simulations
20 simulations
70 simulations
100 simulations
100 simulations
70 simulations ~
Split Simple Pareto
10 simulations = 0
100 simulations =
value when closed
20 simulations =
latest evaluation
Comparison with other approaches
Individual projection can turn out to be lighter in some cases
than aggregated methodologies
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Questions or comments?
Expressions of individual views by
members of The Actuarial Profession
and its staff are encouraged.
The views expressed in this presentation
are those of the presenter.
© 2011 The Actuarial Profession  www.actuaries.org.uk
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