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”
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Tsai, C. C., & Wu, A. D. (2019) · North American Actuarial Journal · doi:10.1080/10920277.2019.1614463 · Cited by 6
Allowance for Hidden Information by Heterogeneous Models and Applications to Insurance Rating
Pinquet, J. (1999) · Hübner international series on risk, insurance, and economic security · doi:10.1007/978-1-4615-4058-8_3 · Cited by 6
Lo, C. H., Fung, W. K., & Zhu, Z. Y. (2007) · Astin Bulletin · doi:10.1017/s0515036100014896 · Cited by 6
Credibility using a loss function from spline theory
Young, V. R. (1998) · Insurance Mathematics and Economics · doi:10.1016/s0167-6687(98)80013-9 · Cited by 6
Planning and control in insurance
van Gelder, H. (1982) · European Journal of Operational Research · doi:10.1016/0377-2217(82)90062-5 · Cited by 6
Norberg, R. (2004) · Encyclopedia of Actuarial Science · doi:10.1002/9780470012505.tac068 · Cited by 6
Novel credibility approaches for Lorenz curve and Gini coefficient estimation
Wen, L., Cao, R., & Zhang, Y. (2024) · Statistics · doi:10.1080/02331888.2024.2368246 · Cited by 5
A general optimal approach to Bühlmann credibility theory
Yan, Y., & Song, K. (2022) · Insurance Mathematics and Economics · doi:10.1016/j.insmatheco.2022.02.003 · Cited by 5
Modelling of Insurance Claim Count with Hurdle Distribution for Panel Data
Boucher, J., Denuit, M. M., & Guillén, M. (2008) · Birkhäuser Boston eBooks · doi:10.1007/978-0-8176-4626-4_4 · Cited by 5
Credibility models with dependence structure over risks and time horizon
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A CREDIBILITY APPROACH FOR COMBINING LIKELIHOODS OF GENERALIZED LINEAR MODELS
Christiansen, M. C., & Schinzinger, E. (2016) · Astin Bulletin · doi:10.1017/asb.2016.11 · Cited by 5
Credibility estimators with dependence structure over risks and time under balanced loss function
Zhang, Q., & Chen, P. (2018) · Statistica Neerlandica · doi:10.1111/stan.12125 · Cited by 5
Hong, L., & Martin, R. (2021) · Annals of Actuarial Science · doi:10.1017/s1748499521000117 · Cited by 5
Credibility theory based on winsorizing
Zhao, Q., & Poudyal, C. (2024) · European Actuarial Journal · doi:10.1007/s13385-024-00391-7 · Cited by 5
Multiple Bonus–Malus Scale Models for Insureds of Different Sizes
Boucher, J. (2022) · Risks · doi:10.3390/risks10080152 · Cited by 5
Ultimate ruin probability in discrete time with Bühlmann credibility premium adjustments
Trufin, J., & Loisel, S. (2013) · HAL (Le Centre pour la Communication Scientifique Directe) · Cited by 5
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Credibility Approximations for Bayesian Prediction of Second Moments
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Dynamic empirical Bayes models and their applications to longitudinal data analysis and prediction
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Valecký, J. (2017) · Prague Economic Papers · doi:10.18267/j.pep.621 · Cited by 5
Practical exponential smoothing credibility
Kremer, E. (1990) · Blätter der DGVFM · doi:10.1007/bf02858022 · Cited by 5
Rekursive SchÄtzverfahren in der KredibilitÄtstheorie
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Pitselis, G. (2013) · Insurance Mathematics and Economics · doi:10.1016/j.insmatheco.2013.01.008 · Cited by 4
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
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