The DOI matches its Crossref record; 3 of the 3 references it cites were found in a registry or an index; none of the 10 sources its claims cite is checked for support yet.
Experience Rating and Credibility
Open accessAbstract
Classical statistics deals with the following standard problem of estimation: Given: random variables X1, X2 … Xn independent, identically distributed, and observations x1, X2 … xn, Estimate: parameter (or function thereof) of the distribution function common to all Xi. It is not surprising that the “classical actuary” has mostly been involved in solving the actuarial equivalent of this problem in insurance, namely Given: risks R1R2 … Rn no contagion, homogeneous group, Find: the proper (common) rate for all risks in the given class. There have, of course, always been actuaries who have questioned the assumptions of independence (no contagion) and/or identical distribution (homogeneity). As long as ratemaking is considered equivalent to the determination of the mean, there seem to be no additional difficulties if the hypothesis of independence is dropped. But is there a way to drop the condition of homogeneity (identical distribution)?
Overview
AI snapshot
Written from the paper’s full text.
TL;DR
Bühlmann reviews seven papers on credibility and experience rating, arguing that experience rating gradually adjusts an initial rate during coverage and that Bayesian and credibility methods can handle heterogeneous risks. He also offers an equilibrium justification for the posterior mean and derives a distribution-free credibility formula.
Key findings
- Experience rating is described as the gradual alteration of an initially fixed rate during the risk-coverage period, implemented through devices such as profit participation, no-claims discounts, bonus-malus scales, sliding-scale premiums or commissions, and reinsurance arrangements.
- Among the reviewed papers, Bohman finds that maximum business volume is generally not reached at the lowest possible premiums; Benckert tests tariff-class adequacy using a lognormal distribution for individual claim amounts; Johansen treats risk parameters as random variables to identify accident-prone risks; and Lundberg proves consistency of a Bayes estimate from a Poisson process while showing that estimates based on time of absence are not consistent.
- Franckx derives the credibility formula from least squares without assuming a distribution for individual risks, though still depending on the prior distribution of risk parameters; Derron computes credibility improvement for a Pareto-type distribution when the largest claim is excluded.
- Bühlmann proposes equilibrium across experience-rated subclasses as a justification for using the posterior mean, and derives a credibility relation with a factor of the form \(b = n/(n+k)\), noting that it makes no assumption about the individual-risk distribution or the structural distribution.
Methods
- The text is a discussion/report on seven submitted papers plus the author’s own technical remarks; it does not present an empirical dataset or sample.
- It reviews sequential statistical methods, Bayes procedures, and credibility techniques, then develops mathematical arguments using conditional expectation, least-squares linear approximation, and an equilibrium postulate.
- The main analysis is conceptual and mathematical, including variance-component expressions and a distribution-free credibility derivation.
Limitations
- The paper states that classical sequential methods such as Wald’s Sequential Probability Ratio Test cannot go beyond testing an entire tariff class and fail for the individual-risk case; it also notes that only one submitted paper specifically addresses credibility techniques, and that direct estimation of \(E[\sigma^2(\theta)]\) and \(\mathrm{Var}[\mu(\theta)]\) from a priori observations has not yet been attempted.
- It observes that Bichsel treats random changes in individual risks over time, but suggests that trend-type changes remain untreated.
- Evident from the text: the work is a commentary/report rather than an empirical study, and some formulas are incomplete in the provided text.
Model-written summary. Check it against the paper before relying on it.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
10 Sources not found · 4 Own interpretation
In the paper
“It is noteworthy that the classical sequential methods, such as those developed by Abraham Wald, do not permit to go beyond the testing of a whole tariff class.”
What the sources say
- Not matchedcites Abraham Wald
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Not matched
In the paper
“His main result, namely that the maximum volume is generally not achieved at the lowest possible premium rates, is certainly of great practical value.”
What the sources say
- Not matchedcites Harald Bohman
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Not matched
- Probably not a substantial claim. 11
- It may state two separate claims as one. 1
In the paper
“His main result consists in the proof of the consistency of the Bayes estimate derived from a Poisson process.”
What the sources say
- Not matchedcites Ove Lundberg
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
- Not matched
In the paper
“Derron computes the credibility improvement for a Pareto-type distribution if the largest claim is excluded.”
What the sources say
- Not matchedcites Marcel Derron
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
- Not matched
In the paper
“This advantage, by the way, is not bought at the expense of an increasingly complicated formula, a fact that for practical purposes will certainly be appreciated as well.”
What the sources say
- Not matchedcites Paul Johansen
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The above relation is exactly Kolmogoroff's definition of the conditional expectation”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
- The authors state this as an assumption: Each class of risks with equal observed risk performance should pay its own way (equilibrium postulate).
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“It is worthwhile to note here that the credibility formula used by our American colleagues is nothing but a linearization of the above estimator function”
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
In the paper
“It seems to me that in doing so he is the first author to prove the credibility relation independent of the distribution function which governs the individual risks (but still dependent on the prior distribution of the risk parameter or parameters).”
What the sources say
- Not matchedcites Edouard Franckx
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
- Not matched
In the paper
“Ove Lundberg seems to have been the first actuary to realize the importance of Bayes procedures for experience rating.”
What the sources say
- Not matchedcites Ove Lundberg
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
In the paper
“I mean that Marcel Derron has very clearly shown to us the importance of truncating or curtailing our basic data to get more reliable information out of them.”
What the sources say
- Not matchedcites Marcel Derron
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
- Not matched
In the paper
“In contrast to the Bayes estimate, Ove Lundberg shows that any estimate based on the time of absence is not consistent”
What the sources say
- Not matchedcites Ove Lundberg
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
boosted- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
- Not matched
In the paper
“Wald's Sequential Probability Ratio Test can be of help in case a), but unfortunately fails to solve the problem in case b).”
What the sources say
- Not matchedcites Abraham Wald
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plainconditionaln = 2- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The hypothesis of independence and identical distribution of the observational random variables of the same individual risk could easily be dropped.”
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: fails — The paper may give no reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
- The authors state this as an assumption: The observational random variables X_i for the same individual risk are independent and identically distributed (homogeneity in time, but not in mass).
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“To get a classification of risks inside a tariff class, it is necessary to treat the risk parameters no longer as constants, but as random variables.”
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
The authors state this as an assumption.
The observational random variables X_i for the same individual risk are independent and identically distributed (homogeneity in time, but not in mass).
“As is usually done I assume the Xt's to be independent and identically distributed (homogeneity in time, but not in mass).”
The authors state this as an assumption.
Each class of risks with equal observed risk performance should pay its own way (equilibrium postulate).
“In other words it is postulated that each class of risks with equal observed risk performance should pay its own way.”
The paper relies on this without support.
All actuarial work so far has dropped homogeneity in mass but still required homogeneity in time of the individual risk.
“All actuarial work so far has, at least according to my knowledge, been done by dropping homogeneity in mass but by still requiring homogeneity in time of the individual risk”
Citations
Cites
On random processes and their application to sickness and accident statistics
Lundberg, O. (1940) · Almqvist & Wiksell eBooks · Cited by 175
A generalization of the collective theory of risk in regard to fluctuating basic-probabilities
Ammeter, H. (1948) · Scandinavian Actuarial Journal · doi:10.1080/03461238.1948.10404899 · Cited by 102
Einige Bemerkungen zur Bonusfrage in der Kraftfahrversicherung
Philipson, C. (1963) · Blätter der DGVFM · doi:10.1007/bf02809297
Research works printed but not linked: 1
- 1964, Eine Bemerkung zu Bichsels Herleitung der bedingten zukünftigen Schadenhäufigkeit einer Polya-Verteilung
Cited by
Sorted by citations
Items 401–448 of 448
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Quantile Credibility Models under Common Effect Structure
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Hazewinkel, M. (1997) · Springer eBooks · doi:10.1007/978-94-015-1288-6_3
Goulet, V. (2008) · Encyclopedia of Quantitative Risk Analysis and Assessment · doi:10.1002/9780470061596.risk0342
Parameter Shrinkage for Age-Period-Cohort Modeling of Opioid Mortality RatesPreprint
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Klugman, S. A. (2005) · Encyclopedia of Statistical Sciences · doi:10.1002/0471667196.ess6046.pub2
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Klugman, S. A. (2015) · Wiley StatsRef: Statistics Reference Online · doi:10.1002/9781118445112.stat00023.pub2
Jewell, W. S. (2004) · Encyclopedia of Actuarial Science · doi:10.1002/9780470012505.tab005
Theory and Applications of Generalized Linear Models in Insurance
Zhou, J. (2011) · Spectrum Research Repository (Concordia University)
Estimation of structural parameters in balanced Bühlmann credibility model with correlation risk
Yang, Y., & Wang, L. (2022) · Communication in Statistics- Theory and Methods · doi:10.1080/03610926.2022.2150057
The Balanced Quantile Credibility Model with Two-Level Common Effects
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An R Package For Crossed Classification Credibility Model: Application Regarding Non-Performing Loan
Altan, S. T., Özdemir, M., & Ebegıl, M. (2023) · The Journal of International Scientific Researches · doi:10.23834/isrjournal.1253490
An Empirical Study to Determine Rice Insurance Premium
Tac, N. V., & Khải, H. V. (2023) · Advances in finance, accounting, and economics book series · doi:10.4018/978-1-6684-8613-9.ch017
A Geometric Process Credibility ModelPreprint
Lam, Y., Chan, J., & Choy, S. T. B. (2024) · SSRN Electronic Journal · doi:10.2139/ssrn.4754622
Özen, Y. (2024) · Akademisyen Kitabevi eBooks · doi:10.37609/akya.3167
Özen, Y. (2024) · Akademisyen Kitabevi eBooks · doi:10.37609/akya.3166
On Quantile Credibility Estimators Under An Equal Correlation Structure Over Risks
Arafa, G., Metiri, F., Sadoun, A., et al. (2024) · Statistics Optimization & Information Computing · doi:10.19139/soic-2310-5070-2004
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Credibility Theory Under the Least Squared Relative Loss Function
Yong, Y., Zhang, Y., & Zhu, X. M. (2024) · Applied Stochastic Models in Business and Industry · doi:10.1002/asmb.2913
Tian, X. (2025) · Asian Journal of Law and Economics · doi:10.1515/ajle-2025-0037 · Cited by 1
Abed, M., & Najafabadi, A. T. P. (2025) · Annals of Actuarial Science · doi:10.1017/s1748499525000065
Bonus-Malus System: Statistical Foundation
Gómez–Déniz, E. (2025) · Springer eBooks · doi:10.1007/978-3-662-69359-9_75
A novel credibility-based estimation approach for multivariate esscher premiums
Wen, L., Liu, W., & Zhang, Y. (2025) · Applied Mathematical Modelling · doi:10.1016/j.apm.2025.116539
Zhao, Q., & Poudyal, C. (2025) · Annals of Actuarial Science · doi:10.1017/s1748499525100201
Multivariate credibility with application to cross-selling financial services products
Thuring, F. (2012) · City Research Online (City University London)
Inference in the Hierarchical Credibility Model
Klugman, S. A. (1987) · Springer eBooks · doi:10.1007/978-94-009-4796-2_7
Of the credibility of mapping and benchmarking credit risk estimates for internal rating systems
Oung, V. (2008) · Elsevier eBooks · doi:10.1016/b978-075068158-2.50010-x
Model stochastycznej zależności liczby szkód i wartości pojedynczej szkody
Sawicka, J. (2013) · RePEc: Research Papers in Economics
Improving Marginal Expected Shortfall estimates using credibility methods
Wen, L. (2026) · Finance research letters · doi:10.1016/j.frl.2026.109490
Generalized laplace approximation and its application to credibility theory
Oh, R., Park, K., Joo, W., & Ahn, J. Y. (2026) · Scandinavian Actuarial Journal · doi:10.1080/03461238.2026.2629286
An Age Grouping Framework for Multi-Population Mortality Modeling
Câmpeanu, C., & Meng, Y. (2026) · Risks · doi:10.3390/risks14030059
Stratégies tarifaires en assurance automobile : optimisation et expérimentation
Nader, R. B. (2016) · theses.fr (ABES) · doi:10.70675/b3b3afd3z352ez4c6dz9d8dz26a5bd621946
Uncertainty-Based Credibility and Its Applications
Parodi, P., & Bonche, S. (2009) · Variance · doi:10.66573/001c.141780
The Credibility of the Overall Rate Indication: Making the Theory Work
Boor, J. (2016) · Variance · doi:10.66573/001c.120864
Credibility theory using fuzzy numbers
Gao, D., & Ren, J. (2024) · Variance · doi:10.66573/001c.94735
Interval Estimation of the Credibility Factor
Gau, W.-C., Gangopadhyay, A. K., & Han, Z. (2008) · Variance · doi:10.66573/001c.141987
A new credibility estimation of process variance with applications
Wen, L., & Liu, W. (2026) · Astin Bulletin · doi:10.1017/asb.2026.10094
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Functional analysis of loss-development patterns in P&C insurance
Charpentier, A., Guo, Q., & Ludkovski, M. (2026) · Insurance Mathematics and Economics · doi:10.1016/j.insmatheco.2026.103284
“Small insurers display the greatest volatility, with wider envelopes across nearly all lags, consistent with classical credibility theory (B¨uhlmann, 1967).”
A Data Science Approach for Evaluating Experience Variables in Insurance Risk Modeling
Hu, N. (2026) · IEEE International Conference on Information Reuse and Integration (IRI) · doi:10.1109/iri69576.2026.00060
Diop, M. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.7441664
Zagadnienia kalkulacji składki zaufania na podstawie łącznej wartości i liczby szkód
Sawicka, J. (2011) · Prace Naukowe Uniwersytetu Ekonomicznego we Wrocławiu
Optimisation de la prise en compte de la sinistralité dans la tarification Automoteur Agricole
Osseni, Z. (2014) · INRIA a CCSD electronic archive server
Deriving Credibility Premiums Under Different Bayesian Methodology
Déeniz, E. G. (2008) · Birkhäuser Boston eBooks · doi:10.1007/978-0-8176-4626-4_16
Frees, E. W. (2004) · Cambridge University Press eBooks · doi:10.1017/cbo9780511790928.008
Essais en économétrie de l'assurance non-vie
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Metadata
- Authors
- Bühlmann, H.
- Corresponding author
- Bühlmann, Hans
- Venue
- ASTIN Bulletin
- Publisher
- Cambridge University Press (CUP)
- Published
- 1967-07
- Volume
- 4
- Issue
- 3
- Pages
- 199-207
- ISSN
- 0515-0361 (ISSN-L)
- 1783-1350
- OpenAlex
- W2400614507
- Semantic Scholar
- 6e0d5024a35bc27c601c4b19050490a1d3fa7e8d
- Cited by
- 438
- References
- 2
- Citations by year
- 2026: 12 · 2025: 7 · 2024: 19 · 2023: 6 · 2022: 15 · 2021: 4 · 2020: 16 · 2019: 13 · 2018: 13 · 2017: 17 · 2016: 15 · 2015: 19 · 2014: 22 · 2013: 17 · 2012: 18
- FWCI
- 2.66
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Astin Bulletin (publishedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- Topics
- Social Sciences › Decision Sciences › Management Science and Operations Research › Probability and Risk Models
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Insurance and Financial Risk Management
- Keywords
- experience rating, credibility theory, ratemaking, insurance premium estimation
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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