A.3. An Actuarial Note on the Credibility of Experience of a Single Private Passenger Car

Question 1

Experience Rating Formula

Answer 1

Mod = ZR + (1-Z)

Z = credibility

R = ratio of actual loss experience to expected loss experience

Question 2

Calculating the Mod

Answer 2

[(# of claims with rating) / (on-level EP for rating at present ’B’ rates)]/[(# of claims in total for class) / (class total on-level EP at present ’B’ rates)]

Question 3

Calculating R

Answer 3

Years claim-free R

1+ 0
0 1/[1−e^(−λ)]

where λ = (# of claims from class) / (earned car years of insureds in class)

Question 4

When to use a premium base for frequency

Answer 4

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.

Question 5

Poisson formula

Answer 5

Pr(X = k) = [ λ^k*e^(−λ) ]/ k!

Question 6

Conclusions of paper

Answer 6

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.

Question 7

Credibility for 2 and 3 years of experience relative to 1

year

Answer 7

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.

Question 8

Bühlmann Credibility

Answer 8

Suppose X is a random variable with some distribution with parameter Θ, and Θ 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|Θ)] / Var(E[X|Θ])