credibility for excess (re)insurance casualty actuaries in reinsurance (care) 2007 david r. clark,...

Credibility for Excess (Re)insurance

Casualty Actuaries in Reinsurance (CARe) 2007David R. Clark, Vice PresidentMunich Reinsurance America, Inc.

2

Brief Review of Credibility Theory 3

Mashitz and Patrik Model 7

The Problem of Dependence 14

A Solution Using Relativities instead of Rates 17

Discussion on unresolved questions 21

Agenda

3


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?

4


22

2

22

2

BA

AB

BA

BA EstEstWeighted

kn

k

kn

nxWeighted 0

nx

x

22

Means alHypothetic of Variance

Variance Process Expected2

2

0

xk

Linear Approximation for Bayesian Credibility:

5


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…

BABA

BAAB

BABA

BABA EstEstWeighted

22 22

2

22

2

We will look at an alternative way of addressing the dependence between

experience and exposure rates.

6


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

7


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)

8


Credibility for Ground-Up Claim Counts:

mEE

mE

mE

m

nCountsWtdCred

m

ii

1

ni Actual number of claims in year “i”

m Number of years in the historical period

E(λ) A Priori expected number of annual claims

α 1/CVλ2 “shape” parameter of the prior gamma

distribution for the distribution of mean frequencies

9


Credibility for Excess Counts, when Severity is Known:

mqEqE

mqE

mqE

m

dnCountsWtdCred

m

ii

1

)(

d “deductible” or excess attachment point

ni(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 1/CVλ2 “shape” parameter of the prior gamma

distribution for the distribution of mean frequencies

q Probability that a ground-up loss would exceed “d”

10


kmqE

kqE

kmqE

mqE

m

dnCountsWtdCred

m

ii

1

)(

2

1

CV

Credibility for Excess Counts, when Severity is Unknown:

2222

1

qq CCCVCVk

The credibility constant “k” changes from α to the value:

11


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 ground-

up 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.

12


Historical OnLevel Count Premium Actual Expected

Period Premium LDF / LDF Counts n i Counts

1997 20,450,000 1.328 15,402,363 8 9.21998 20,850,000 1.402 14,870,773 5 8.91999 21,250,000 1.500 14,169,856 6 8.52000 21,700,000 1.632 13,299,850 9 8.02001 22,150,000 1.817 12,188,030 8 7.32002 22,600,000 2.094 10,790,701 12 6.52003 23,050,000 2.545 9,056,358 5 5.42004 23,500,000 3.394 6,924,300 5 4.22005 24,000,000 5.540 4,332,035 1 2.6

Total 199,550,000 101,034,267 59 60.6

Future Premium for 2007: 25,000,000 15.0 = λ

Sample for Including Growth and Development (relaxing assumption of constant exposure by year)

All numbers for illustration only

13


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.

kn

nyCredibilit

2

1

14


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?

15


Excess

Layer

Subject Premium * ELR

(but what is the source of the ELR?)

Treaty Limit

Treaty Retention

Exposure Rating “Layers” and overall loss:

16


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.

17


An Alternative Approach using Relativities instead of Rates:

The industry-based 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.

18


0%

100%

0 15,000X1 X2

F(X1)

F(X2)1-F(X1)

1-F(X2)

1

2

F1

F1p

X

X

Cumulative Distribution Function (CDF) for Severity

19


A full Bayesian model for excess layer counts:

Let N1 = # claims in lower (base) layer

p = probability of a loss in lower layer reaching higher layer

“survival ratio”

N2 = binomial random variable

E[N2|p] = N1·p

Var(N2|p) = N1·p·(1-p)

Let the survival ratio p have a prior beta distribution, with parameters ν and

ω.

f(p) = constant·pv·(1-p)ω E[p] = ν/(ν+ω)

20


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.

11

1

1

2

1

2

nn

n

n

n

n

n

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.

21


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.

22


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.

Thank you very much for your attention.

David R. Clark, Vice PresidentMunich 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.

credibility for excess (re)insurance casualty actuaries in reinsurance (care) 2007 david r. clark,...

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