Borrowing Credibility for a New Territory
An insurer is entering a new state and has no credible experience of its own. It borrows a prior for claim frequency from a large, mature book in a neighbouring state: prior mean 0.09 claims per policy-year, prior standard deviation 0.03. The actuary represents this as a gamma prior on the Poisson frequency parameter. Two years of experience in the new state are now available: 42 claims over 300 policy-years, with no rate changes and no material change in underwriting. The company must set next year's rates for a segment expected to generate 20 policy-years of exposure, and wants the probability that the segment produces no claims at all for a discount calculation.
1. Derive the posterior distribution, the Bühlmann credibility weight, and the credibility estimate of frequency, showing that the two approaches agree.
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Prior parameters. For a gamma(a, b) prior with mean a/b = 0.09 and variance a/b^2 = 0.0009:
b = mean/variance = 0.09/0.0009 = 100, a = mean x b = 9.
the prior carries an effective exposure of b = 100 policy-years.
Posterior (gamma-Poisson conjugacy). With 42 claims over 300 policy-years:
a' = 9 + 42 = 51, b' = 100 + 300 = 400.
Posterior mean = 51/400 = 0.1275. Posterior variance = 51/400^2 = 0.000319, sd = 0.0179 - a 40% reduction from the prior sd, which is the honest measure of how much was learned.
Credibility route. Z = exposure/(exposure + b) = 300/400 = 0.75. MLE = 42/300 = 0.14.
0.75 x 0.14 + 0.25 x 0.09 = 0.1050 + 0.0225 = 0.1275. Agreement is exact.
Read the weight carefully. Z = 0.75 looks reassuring, but it is only high because the borrowed prior was given the weight of 100 policy-years. The observed frequency is 56% above the prior mean, so the prior was materially wrong; the posterior is still 25% anchored to it.
2. Compute the predictive probability of zero claims for next year's 20 policy-years under (i) the posterior, and (ii) the prior alone, and explain the difference in pricing terms.
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Negative binomial predictive. For a gamma(a', b') posterior and Poisson exposure m, the predictive is NB with
P(N = 0) = (b'/(b' + m))^a'.
(i) Posterior: (400/420)^51 = (0.952381)^51. ln = -0.048790 x 51 = -2.4883; exp = 0.0830, i.e. about 8.3%.
(ii) Prior alone: (100/120)^9 = (0.833333)^9 = exp(9 x -0.18232) = 0.1938, about 19.4%.
A naive Poisson check: exp(-0.14 x 20) = exp(-2.8) = 0.061, and exp(-0.09 x 20) = 0.165.
Pricing interpretation. The prior says nearly one segment in five is claim-free; the data say fewer than one in twelve. A no-claim discount priced on the prior would be applied far too often and would be badly under-funded. Note also that beta (the posterior) matters more than the posterior mean here: the predictive is much more concentrated than either the prior predictive or a point-estimate Poisson because the uncertainty in lambda is carried through.
Use 8.3% for the discount calculation within this segment, but only after deciding whether the segment is representative of the 300 policy-years used to build the posterior.
3. The neighbouring state has a compulsory no-fault system and the new state does not, and the new state's 300 policy-years include a mid-period book transfer. How should these facts change your use of the posterior, and what is the cost of getting it wrong?
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The exchangeability assumption is the weak link, not the arithmetic. Gamma-Poisson credibility assumes the new state's lambda is a draw from the same population as the prior book. A different tort system changes claim reporting behaviour, not just claim occurrence, so the two books are plausibly not exchangeable and the prior variance of 0.0009 - already contradicted by an observation 56% above the prior mean - is the wrong measure of between-state spread.
What to do.
- Widening the prior variance is the conservative move: it raises the effective exposure denominator and lowers Z. With b doubled to 200, Z = 300/500 = 0.60 and the estimate becomes 0.60(0.14) + 0.40(0.09) = 0.120, roughly 6% below the conjugate answer.
- The mid-period book transfer means the 300 policy-years are not homogeneous; at minimum, test the pre- and post-transfer frequencies separately and, if they differ, either shorten the experience period or use a Bühlmann-Straub structure with exposure weights.
- Consider dropping the borrowed prior for the tort-sensitive component while retaining it for the stable component (e.g. weather-driven property claims).
Cost of getting it wrong. Frequency is a multiplicative driver of pure premium: the gap between 0.1275 and 0.140 is about 10% on frequency alone, and between 0.1275 and the prior-biased figure it can be larger. On a book being priced for the first time, an understated frequency compounds with each renewal and is not corrected until the credibility weight rises - by which time the rate level is embedded in the market.