Credibility theory calculator
How much should a risk's own claims experience count against the collective's? Edit the grid and watch the EBCT Model 1 machinery answer live: the variance components, the credibility factor Z, and every risk's blended premium.
Aggregate claims Xij (4 risks × 5 years)
| Risk | Yr 1 | Yr 2 | Yr 3 | Yr 4 | Yr 5 | X̄ᵢ | |
|---|---|---|---|---|---|---|---|
| 1 | 102.4 | ||||||
| 2 | 147.0 | ||||||
| 3 | 79.4 | ||||||
| 4 | 112.6 |
EBCT Model 1 estimates
Overall mean X̄110.35
E[s²(θ)] · within-risk variance82.10
V[m(θ)] · between-risk variance773.38
Credibility factor Z0.9792
Credibility premiums
Risk 1 · Z·102.4 + (1−Z)·110.4102.57
Risk 2 · Z·147.0 + (1−Z)·110.4146.24
Risk 3 · Z·79.4 + (1−Z)·110.480.04
Risk 4 · Z·112.6 + (1−Z)·110.4112.55
Z = n / (n + E[s²(θ)]/V[m(θ)]) with n the number of years. Each premium blends the risk’s own experience with the collective’s: make one risk wildly volatile and E[s²] rises, dragging Z down for everyone; spread the risk means apart and V[m] rises, pushing Z towards 1. More years always push Z up — experience earns credibility.
The two variances that decide everything
Empirical Bayes credibility (Model 1) treats each risk’s true mean as a draw from a collective, and the credibility premium Z·X̄ᵢ + (1−Z)·X̄ is the best linear estimate of it. Everything hinges on two variance components: E[s²(θ)], the noise within a risk’s own experience, and V[m(θ)], the genuine spread between risks. Z is just the ratio of signal to signal-plus-noise, scaled by the number of years. The standard exam traps are all mechanised here: the within variance uses divisor n−1, the between variance subtracts E[s²]/n, and a negative between-variance estimate means Z is set to zero, not negative.
Make it stick. Credibility sits in CS1 alongside the sampling theory in our statistical tables and the distributions cheat sheet. Memori is a flashcard app built by actuarial students, with a ready-made CS1 set in the shop. Join the beta.
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