Experience Rating Formula

[(# of claims in group) / (earned premium in group at present 0 yrs. claim-free rates)] / [(# of claims in class) / (earned premium in class at present 0 yrs. claim-free rates)]

Calculating the Mod

Mod = ZR + (1-Z) Z = credibility R = ratio of actual losses to expected losses

Calculating R

Years claim-free & R

1+ : 0

0 : 1 / (1 - e^^{-lambda}) where lambda = # of claims from class / earned car years of insureds in class

When to use a premium base for frequency

Hazam states that a premium base only eliminates maldistribution if:

1. High frequency territories are also high average premium territories.

2. Territorial (rate) differentials are proper.

Poisson formula

Pr(X = k) = (lambda^{k} * e^{-lambda}) / k!

Conclusions of paper

1. The experience of a single car for 1 year has significant and measurable credibility for experience rating.

2. Individual risk experience is more credible when there is more variance in loss experience within a risk class, which occurs in less refined risk classification systems.

3. The credibilities for varying years of experience should increase in proportion to the # of years of experience.

Credibility for 2 and 3 years of experience relative to 1 year

The credibility increases in proportion to the # of years only for low credibilities. The closer the credibilities for 2 and 3 years of experience are to 2 and 3 times the 1 year credibility, then the less variation in insured’s probability of an accident. This could be due to:

1. Less risks entering/exiting the portfolio.

2. Risk characteristics not changing much over time.

Bühlmann Credibility

Suppose X is a random variable with some distribution with parameter OMEGA, and OMEGA itself is a random variable with some distribution and additional parameters. In that case, the credibility of a sample of n observations from X is given by:

Z = n / (n+k)

n = # of claims in sample

k = E[Var(X|OMEGA)] / Var(E[X|OMEGA])