A.3. Bailey & Simon  PPV Credibility
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Merit rating
 # full years since most recent accident / licensed date
 A, X, Y, B = 3+, 2, 1, 0 years
 A+X = experience for 2+ accidentfree years
 Premium = Base x Merit Factor x Other Rating Variables

Experience rating formula
 credibility weighted LR = Z x actual LR + (1  Z) x expected LR
 Experience Mod = R + (1  Z), Z = credibility, R = ratio of actual to expected losses
 Z depends not only on the volume of data but also on amount of variation within the class

B&S Method to Calculate Mod
 use frequency to premium instead of LR = same as assuming constant severity
 (# claims with rating) / (onlevel EP for rating at present “B” rates) divided by
 (# claims in total for class) / (class total onlevel EP at present “B” rates)

Bailey & Simon methodology to calculate R
 infer or estimate what R would’ve been for the prior term (1 year)
 either risk is A, X, Y so had no claim therefore R = 0, Mod = 1  Z
 or risk is B and we approximate R as 1 / (1  e^{λ}) using car years instead of premium, with λ = # claims / earned car years
 plug Mod and R in the formula to solve for Z

Credibility vs number of years claims free
 if an individual insured's chance for an accident remained constant year to year and
 if there were no risks leaving or entering the class then
 credibility for experience would vary in proportion of the amount of years

Maldistribution (exposure correlation)
 B&S are specifically concerned with correlation between territory variable and merit rating variable, i.e. high frequency territories producing more X, Y, B and higher premium
 Once we correct for it by using premium as a base for frequency instead of exposure:
 high frequency territories are also high rate territories
 territorial differential are proper (e.g. same loss ratio across all territories)

Paper conclusions
 experience for one car for one year has significant and measurable credibility
 in a highly refined PPV rating classification, credibility would be low
 the credibility for varying years of experience should increase in proportion to the # years of experience

Bühlmann credibility
 suppose X is a random variable with distribution parameter Θ
 Θ itself is a random variable with some distribution
 credibility of sample of n is Z = n / (n + k)
 k = E[Var(XΘ)] / Var[E(XΘ)] = exp value of process var / var of hypothetical means
 if X ~ Poisson(λ) and λ ~ Normal(μ,σ^{2}) then k = E[λ] / Var[λ] = μ / σ^{2}

Poisson Formula
Pr(X = k) = λ^{k}e^{k} / k!