Credibility

Why it matters

A trucking company with 40 trucks asks for a price. Its own claims last year were lower than the industry average. Should the insurer charge it less? If the company had 4,000 trucks, its experience would clearly be reliable. With 40, one lucky year might be noise. Credibility theory answers the question "how much weight should this group's own data get?" It is used in group health and life pricing, workers' compensation, commercial auto, and experience rating everywhere.

The credibility formula

Every credibility method produces the same kind of estimate, a weighted average:

estimate = Z * (observed experience) + (1 - Z) * (manual or prior rate)

Z is the credibility factor, between 0 and 1. Z = 1 (full credibility) means trust the group's own data completely; Z = 0 means ignore it and charge the manual rate (the rate for the average risk). The methods differ in how they choose Z.

Limited fluctuation (classical) credibility

This older US method asks: how much data do we need so that the observed value is within k (say 5%) of its true mean with probability p (say 90%)? For claim counts that follow a Poisson distribution, the full credibility standard in number of claims is

n_full = (z / k)^2,   where z is the standard normal value with P(-z < Z < z) = p

For p = 90%, z = 1.645 and k = 5%:

n_full = (1.645 / 0.05)^2 = 1,082 claims     (the classic "1,082 rule")

With fewer claims, use the square-root rule:

Z = min(1, sqrt(n / n_full))

Worked example 1

A group had 400 claims. Its observed pure premium (losses per exposure) is 520; the manual pure premium is 480.

Z = sqrt(400 / 1082.2) = 0.608
estimate = 0.608 * 520 + 0.392 * 480 = 504.32

The group gets a rate between its own experience and the manual, closer to its own because it has a fair amount of data. Different standards exist for severity and for pure premium; the formulas add a term for the variability of claim sizes.

Buhlmann (greatest accuracy) credibility

The modern approach models two sources of variation:

• EPV, the expected value of the process variance: how much a single risk's results bounce around from year to year.
• VHM, the variance of the hypothetical means: how different the true average costs of different risks are.

Then

K = EPV / VHM          Z = n / (n + K)

where n is the number of observations (for example years of experience, or exposure units). If risks really differ a lot (large VHM), K is small and experience gets weight quickly. If results are very noisy (large EPV), K is large and you need much more data.

Worked example 2

For a portfolio of commercial policies, EPV = 8,000 and VHM = 200, so K = 40. A policy with 10 years of history gets Z = 10 / (10 + 40) = 0.2. With 40 years it would get 0.5; Z approaches 1 but never reaches it. The Buhlmann estimate is the best linear approximation to the Bayesian estimate, which is why it is called "greatest accuracy" credibility. The Buhlmann-Straub extension handles unequal exposure in different years.

def credibility_estimate(z, observed, manual):
    return z * observed + (1 - z) * manual

print(round(credibility_estimate(0.2, 1200, 1000), 2))  # 1040.0

Where the manual rate comes from

The "complement of credibility" (the 1 - Z part) should be an unbiased, relevant estimate: the industry rate, a larger related class, or last year's rate trended forward. The CAS literature on ratemaking discusses how to choose it. A poor complement weakens the whole estimate.

Common mistakes

• Treating Z as a probability. It is a weight, not the chance that the data is right.
• Mixing units. The classical standard of 1,082 is in claims, not policies or dollars.
• Using raw experience without adjustment. Losses should be developed to ultimate and trended before blending.
• Forgetting the cap. In the square-root rule, Z cannot exceed 1.

Quick recap

• Credibility estimate = Z * observed + (1 - Z) * manual.
• Limited fluctuation: n_full = (z/k)^2 claims (1,082 at 90%/5%), Z = min(1, sqrt(n/n_full)).
• Buhlmann: K = EPV/VHM, Z = n/(n + K).

Try it

In the Do tab, code both credibility methods and blend a group's experience with a manual rate.