Chapter 18 - credibility theory

Question 1

WHat is credibility theory

Answer 1

using actual experience and external information

actual is most relevant but may be unreliable if data is limited or include unusal alrge losses
extern data is more stable but may be less relevant

Question 2

explain bayesian and classical to calculating Z

Answer 2

Bayesian Credibility (Greatest Accuracy Credibility)
-Uses probability theory to calculate the best Z based on prior beliefs and observed data.

Classical Credibility (Limited Fluctuation Approach)
-Simpler, practical method for real-world use.
-Often used in markets like Lloyd’s where data is messy or limited.

Question 3

What does the standard for full credibility mean?

Answer 3

the min amount of data needed to assign full credibility.

Question 4

standard for full credibility frequency for poison and binomial/neg binomial

Answer 4

poisson - means var=mean
Q(y) = (1+P)/2
then using y -> nN >= (y/k)^2

if var is not equal to mean then use binomial or neg binomial
nN>= (y/k)^2(s.d^2/Mu)

Question 5

standard for full credibility severity

Answer 5

nx= (y/k)^2 x (sigmaX/MuX)^2

when (y/k)^2 is nN = full standard for frequency

use the distribution mentioned in question. e..g lognormal the use the equaion to calc mean and var from orange book. use the mu and sigma figures given in question to calculate the (sigmaX/MuX)^2 component
then multiply by the standard full frequency part.

Question 6

standard for full credibility aggregate loss

Answer 6

ns=nN(1+CVx^2)

if freq is not poisson then
credit for agg losses will be

ns=(y/k)^2(sigma/m…)

Question 7

Key info about partial credibility

Answer 7

Z= sqaure root of n/nN

Small samples (partial credibility) means larged s.d. and have higher variance than a full credibility estimate

We choose Z so that the variance of the credibility estimate equals the variance of an estimate based on fully credible data

Question 8

comparison of bayesian and classical credibility

Answer 8

classical square root n/nN where n is no of claims and nN is standard for full credibility.

Bayesian Z=n/(n+k) where k = E[S^2(thetha)]/var(m(theatha))
bayesian is based on the principle of minimizing the mean square error

classical and bayesian gives similar credibiliy weights. if Nn = either 7 or 8 times K larger
bayesian never reaches Z=1

for bayesian, you need known expected and variance estimates.
classical simple to work with and easier to explain to non actuaries

Question 9

What are the practical uses of complements of credibility(external data).

Answer 9

practical issues
-Data should be readily available.
-Easy to compute and explain.
-Balance accuracy with effort and cost.

competitive market issues
-Rates must be competitive.
-Avoid setting rates too high (lose customers) or too low (lose money).
-Rate should be unbiased (averages correct over many years) and accurate (low variance around true losses).

Regulatory issues
-Rates must be: not inadequate, not excessive, and not unfairly discriminatory.
-Complement should have a logical relationship to the class or individual’s losses to justify rates.

statistical issues
-Consider all sources of prediction error:
Process variance (random fluctuations)
Bias (predictor systematically over/underestimates)
Model or parameter error (if using models)
-Accuracy and independence of the complement are key:
Accurate complement improves final estimate.
independence from base statistic reduces compounded errors.
Best when errors in base and complement offset each other (negative correlation), though usually positively correlated.

Question 10

Desirable Qualities of a Complement

Answer 10

Accurate: Low variance around next year’s expected losses.
Unbiased: as a predictor of next year’s mean expected losses (that is, the differences between the predictor and the subsequent loss costs should average out near zero)
Independent: Not related to base statistic.
Available: Data can be obtained easily.
Easy to compute: Simple to process and explain.
Explainable: Logical connection to losses being estimated.

Extra considerations:
-Ease of explanation to managers/customers
-Time/cost to compute vs. accuracy gained
-Risk and soruces of errors in calculating the complement

Question 11

Bühlmann–Straub Credibility - what is the purpose

Answer 11

Purpose

The Bühlmann–Straub model is used to estimate a future claims ratio (or pure premium) for a risk when:

Different risks have different volumes of exposure, and

Past experience is partially credible.

It is a generalisation of the Bühlmann model that explicitly allows for varying exposure volumes.

Question 12

formula for Bühlmann–Straub Credibilit

Answer 12

si = insurance claims
vi = volume measure

xi = si/vi claims ratio

credibility factor zi= vi/(vi+theath/lamda))

Question 13

pratcical uses of credibility mdoels - experiecne rating plans

Answer 13

when designing experience rating plans: need to consider a few things.
- done frequenctly and simplicity
-plans usually include limit on impact of one claims (shouldnt increase premium massviely) this limit is called the swing
-big firms are self rate where very large insureds have lots of data so 100% credibility

pratcal models re simpler, include limits like swings caps, easier to use and explain
threotical models are complex, hard to explain and interpret.

Question 14

pratcial uses of credibility models where credibility and data considerations

Answer 14

Thus, two credibility-related problems emerge: (1) how to obtain more data or more reliable data (2) what is the most appropriate credibility complement?
–> increase volume by more years of data or wider geography
-but must ask if extra data is relevant and have risks changed over time

–> focus on more stable components like freq is more stable than seeverity. claim counts are more stable.

• use judgement when decideding the below, how to allow for large, teends, different opinions of the correct rate

-large claims - are thesesingle larg unsual, shoudl we spread cost .

-trends To what extent should an individual risk’s future premium be adjusted for claims trends that are an accepted feature of that line of business?

-goodness of fit - ie accuracy vs simplicity

-level of grouping vx accruacy

-stability of data - weightings based on numbers not amounts.

-use of partial premiums

Question 15

Practical considerations Issues to consider when using credibility theory in practice include: *

Answer 15

*
simplicity * * * * * * * *
visibility − consider imposing a maximum swing, or self-rating goodness of fit − ie accuracy versus simplicity level of grouping versus accuracy
source of data − more years / more locations / national data etc stability of data − eg weightings based on numbers, not amounts use of partial premiums
choice of credibility complement − accuracy / bias / independence from base data / availability / ease of calculation / relationship to risk
the need to use considerable judgement when considering how to allow for large claims, trends and differing opinions of the correct rate.