Credibility for Excess (Re)insurance
Casualty Actuaries in Reinsurance (CARe) 2007
David R. Clark, Vice President
Munich Reinsurance America, Inc.

Agenda
Brief Review of Credibility Theory
Mashitz and Patrik Model
The Problem of Dependence
A Solution Using Relativities instead of Rates
Discussion on unresolved questions
14
17
3
7
21
2

Credibility for Excess (Re)insurance
The purpose of applying Credibility Theory:
Experience Rate = E[Loss | Account Loss Experience ]
Exposure Rate = E[Loss | External Information]
Final Rate = E[Loss | Account Loss Experience & External Information]
The question: How do we calculate the “best” expected loss E[Loss] given ALL of the information that is available to us?
3

Credibility for Excess (Re)insurance
Linear Approximation for Bayesian Credibility:

A




A
2

2

B

B
2



B




A
2

2

A

B
2


  x
2
 x
2 n k


 x
2
2

0

Weighted
 x
 n n
 k
 
0
 n k
 k
4

Credibility for Excess (Re)insurance
Two Key assumptions:
 The two estimates are UNBIASED
 The information in both estimates should be relevant for the contract being priced. This means we are “shooting at the right target” (see next slide).
 The two estimates are INDEPENDENT
 We can modify our formula if there is dependence…

A




2
A


2
B


2
B


2

A

B
A

B



B




2
A


2
A


2
B


2

A

B
A

B


We will look at an alternative way of addressing the dependence between experience and exposure rates.
5

Credibility for Excess (Re)insurance
Emmons Loses Gold Medal After Aiming at Wrong Target
Monday, August 23, 2004; Page D16
Matt Emmons was just focusing on staying calm. He wishes he had been more concerned with where he was shooting.
Emmons fired at the wrong target on his final shot, a simple mistake that cost the American a commanding lead in the 50-meter three-position rifle final and ruined his chance for a second gold medal.
Ahead after nine shots and needing only to get near the bull's-eye to win, Emmons fired at the target in Lane 3 while he was shooting in Lane 2. He had cross-fired -- an extremely rare mistake in elite competition -- and received a score of zero. That dropped Emmons to eighth place at 1,257.4 points and lifted Jia Zhanbo of China to the gold at 1,264.5.
"On that shot, I was just worrying about calming myself down and just breaking a good shot, and so I didn't even look at the number," said Emmons, 23. "I probably should have. I will from now on.“ © 2004 The Washington Post Company. Reprinted with permission
6

Credibility for Excess (Re)insurance
The 1990 paper by Mashitz & Patrik applies Bayesian Credibility to the problem of excess reinsurance treaty pricing.
Assumptions in the Mashitz & Patrik Model:
1) Restrict the credibility formula to frequency
2) Each risk (treaty) has claim counts distributed as Poisson, the Poisson means for all of the risks in the portfolio are distributed as Gamma
3) For a given risk, each historical year has the same volume of exposure (we will relax this assumption later)
7

Credibility for Excess (Re)insurance
Credibility for Ground-Up Claim Counts:
Cred Wtd Counts
 i m 

1 n i m



E
E

 
 m
 m
 



E



E
 

 m
 

 n i m
Actual number of claims in year “ i ”
Number of years in the historical period
E( λ) A Priori expected number of annual claims
α 1/CV
λ
2 “shape” parameter of the prior gamma distribution for the distribution of mean frequencies
8

Credibility for Excess (Re)insurance
Credibility for Excess Counts, when Severity is Known:
Cred Wtd Counts
 i m 

1 n i
d


 m E
E

 
 q

 q m
 m
 



E
 q



E

 q
 m
 

 d “deductible” or excess attachment point n i
(d) Actual number or claims above “ d ” in year “ i ” m Number of years in the historical period
E( λ) A Priori expected number of annual claims
α q
1/CV
λ
2 “shape” parameter of the prior gamma distribution for the distribution of mean frequencies
Probability that a groundup loss would exceed “ d ”
9

Credibility for Excess (Re)insurance
Credibility for Excess Counts, when Severity is Unknown:
Cred Wtd Counts
 i m 

1 n i
( d )


 m E
E

 
 q

 q m
 m
 k



E
 q



E k
 q
 m
 k


The credibility constant “ k ” changes from α to the value: k
 
CV

2
1

CV q
2 
C

2 
C q
2
  
1
CV

2
10

Credibility for Excess (Re)insurance
Observations:
 The credibility assigned to the experience is based on the expected counts, NOT based on the actual counts.
 When severity is known, the “k” in the Credibility = n/(n+k) is the same for groundup and excess counts.
 When the severity is not known with certainty, the credibility constant “k” is lower.
This means that we give more credibility weight to the experience when the severity distribution is uncertain.
11

Credibility for Excess (Re)insurance
Sample for Including Growth and Development (relaxing assumption of constant exposure by year)
Historical
Period
OnLevel
Premium
Count
LDF
1997
1998
1999
2000
2001
2002
2003
2004
2005
Total
20,450,000
20,850,000
21,250,000
21,700,000
22,150,000
22,600,000
23,050,000
23,500,000
24,000,000
199,550,000
1.328
1.402
1.500
1.632
1.817
2.094
2.545
3.394
5.540
Future Premium for 2007:
All numbers for illustration only
Premium
/ LDF
15,402,363
14,870,773
14,169,856
13,299,850
12,188,030
10,790,701
9,056,358
6,924,300
4,332,035
101,034,267
25,000,000
Actual
Counts n i
12
5
9
8
5
1
8
5
6
59
Expected
Counts
9.2
8.9
8.5
8.0
7.3
6.5
5.4
4.2
2.6
60.6
15.0
= λ
12

Credibility for Excess (Re)insurance
How should we set the credibility constant “k” in practice?
 Mashitz and Patrik recommend a survey of questions based on consistency of the historical business, data quality, etc.
 Adjust “process variance” based on variance of the historical counts.
 Practical Rule of thumb is that “k” represents the number of expected claims for which you would assign 50%/50% weights between the two methods.
Credibilit y

1
2
 n n
 k
13

Credibility for Excess (Re)insurance
The problem of Dependence between the two estimates:
Mashitz and Patrik consider an analogy with the application of credibility in primary insurance:
Primary Insurance:
Final Price = Actual Experience ·Z + Manual Rate·(1-Z)
Excess Reinsurance:
Final Price = Experience Rating·Z + Exposure Rating·(1-Z)
But does this analogy really hold?
14

Credibility for Excess (Re)insurance
Exposure Rating “Layers” and overall loss:
Treaty Limit
Treaty Retention
Excess
Layer
Subject Premium * ELR
(but what is the source of the ELR?)
15

Credibility for Excess (Re)insurance
If the Expected Loss Ratio (ELR) used in the exposure rating is based on account experience, then it is not truly an a priori ELR.
The reason for this is that in reinsurance we have subject premium, but not exposures and rates to calculate a true loss cost by layer.
Mark Cockroft (2004) describes this:
“…in the real world there are many instances when exposure and experience methods do interact already, blurring the credibility weighting”
This is where the original INDEPENDENCE assumption is violated.
16

Credibility for Excess (Re)insurance
An Alternative Approach using Relativities instead of Rates:
The industrybased severity distributions provide us with a means for “layering” losses, but they do not provide an absolute frequency to produce a rate. Instead, we typically base the exposure-rating on an ELR that is a ground-up experience rating. The “exposure-rate” therefore is already dependent on the excess experience.
An alternative is to select a base layer – considered to be 100% credible – and use layer relativities to estimate higher layers. The final relativity is a credibility weighted average of an experience relativity and a relativity from the industry severity distribution.
17

Credibility for Excess (Re)insurance
100%
F(X
2
)
F(X
1
)
Cumulative Distribution Function (CDF) for Severity
1-F(X
2
)
1-F(X
1
) p

1

1

F
F
 
X
2
 
1
0%
0
X
1
X
2
15,000
18

Credibility for Excess (Re)insurance
A full Bayesian model for excess layer counts:
Let N p
1
= # claims in lower (base) layer
= probability of a loss in lower layer reaching higher layer
“survival ratio”
N
2
= binomial random variable
E[N
2
|p] = N
1
·p
Var(N
2
|p) = N
1
·p·(1-p)
Let the survival ratio p have a prior beta distribution, with parameters ν and ω.
f(p) = constant ·p v ·(1-p) ω
E[p] = ν/(ν+ω)
19

Credibility for Excess (Re)insurance
The credibility-weighted average of the predictive survival ratio p is a linear average of the actual experience and the a priori expectation from the prior Beta distribution.

 
  n
2
 n
1


 n
2 n
1





  n
1
  n
1





 




  
   n
1


If we have an estimate of expected claims for the lower layer, we can then calculate an estimate of expected claims to the higher layer. The credibility is based on the number of claims in the lower layer, or equivalently, on the expected number of claims in the higher layer.
This procedure can be repeated for higher layers.
20

Credibility for Excess (Re)insurance
Unresolved Questions:
 Under what circumstances should multiple lines of business be combined?
 How to modify the credibility when only certain years are included in the
“selected” loss cost.
 Adjusting credibility when data quality is poor or possibly irrelevant
 How to simultaneously include the credibility in the development pattern with the credibility in the severity curve.
21

Credibility for Excess (Re)insurance
Select Bibliography:
Cockroft, Mark; Bayesian Credibility for Excess of Loss Reinsurance Rating ; GIRO
Conference 2004.
Dale, Andrew; Most Honourable Remembrance: The Life and Work of Thomas
Bayes; Springer 2003.
Mashitz, Isaac, and Gary Patrik; Credibility for Treaty Reinsurance Excess Pricing;
CAS Discussion Paper Program on Pricing, 1990.
Philbrick, Stephen; An Examination of Credibility Concepts ; CAS Proceedings,
1981.
Venter, Gary; Credibility Theory for Dummies ; CAS Forum, Winter 2003.
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Thank you very much for your attention.
David R. Clark, Vice President
Munich Reinsurance America, Inc.
© Copyright 2007 Munich Reinsurance America, Inc. All rights reserved. The Munich Re America name is a mark owned by Munich Reinsurance America, Inc.
The material in this presentation is provided for your information only, and is not permitted to be further distributed without the express written permission of Munich Reinsurance America. This material is not intended to be legal, underwriting, financial, or any other type of professional advice. Examples given are for illustrative purposes only.