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Cited by 448Probability and Risk ModelsProbability and Statistical ResearchInsurance and Financial Risk Management

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 access
Bühlmann, Hans·1967·ASTIN Bulletin
DOI 10.1017/s0515036100008989
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Abstract

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

Crossref · 10.1017/s0515036100008989

Cites

3 of 3 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Verification

Claims the paper makes

0 of 10 cited sources checked · 0 supported · 0 flagged

  • Premisecitation
    § 3. Report on Papers Submitted

    Classical sequential methods, such as those developed by Abraham Wald, do not permit going beyond the testing of a whole tariff class.

    “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.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Abraham Wald

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Premisecitation
    § 3. Report on Papers Submitted

    The maximum volume of business is generally not achieved at the lowest possible premium rates.

    “His main result, namely that the maximum volume is generally not achieved at the lowest possible premium rates, is certainly of great practical value.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Harald Bohman

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim. 11
  • It may state two separate claims as one. 1

  • Probably not a substantial claim.

    Premisecitation
    § 3. Report on Papers Submitted

    Lundberg's main result consists in the proof of the consistency of the Bayes estimate derived from a Poisson process.

    “His main result consists in the proof of the consistency of the Bayes estimate derived from a Poisson process.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Ove Lundberg

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim.

    Premisecitation
    § 3. Report on Papers Submitted

    Derron computes the credibility improvement for a Pareto-type distribution if the largest claim is excluded.

    “Derron computes the credibility improvement for a Pareto-type distribution if the largest claim is excluded.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Marcel Derron

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Interpretationnone given
    § 3. Report on Papers Submitted

    The advantage gained by assuming the risk parameter to be a random variable is not bought at the expense of an increasingly complicated formula.

    “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.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Paul Johansen

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim.

    Interpretationreasoning
    § 4. Equilibrium in an experience rated portfolio

    The equilibrium condition for any subset X' is exactly Kolmogoroff's definition of the conditional expectation.

    “The above relation is exactly Kolmogoroff's definition of the conditional expectation”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Each class of risks with equal observed risk performance should pay its own way (equilibrium postulate).

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

    Interpretationnone given
    § 5. A Distribution Free Credibility Formula

    The credibility formula used by American colleagues is nothing but a linearization of the estimator function E[μ(θ)|X1...Xn].

    “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”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim.

    Premisecitation
    § 3. Report on Papers Submitted

    Franckx 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).

    “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).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Edouard Franckx

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim. The paper’s conclusions may not depend on it.

    Premisecitation
    § 3. Report on Papers Submitted

    Ove Lundberg seems to have been the first actuary to realize the importance of Bayes procedures for experience rating.

    “Ove Lundberg seems to have been the first actuary to realize the importance of Bayes procedures for experience rating.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Ove Lundberg

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim.

    Interpretationcitation
    § 3. Report on Papers Submitted

    Derron has shown the importance of truncating or curtailing basic data to get more reliable information out of them.

    “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.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Marcel Derron

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim.

    Premisecitation
    § 3. Report on Papers Submitted

    Any estimate based on the time of absence is not consistent.

    “In contrast to the Bayes estimate, Ove Lundberg shows that any estimate based on the time of absence is not consistent”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Ove Lundberg

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • It may state two separate claims as one.

    Interpretationreasoning
    § 3. Report on Papers Submitted

    Wald's Sequential Probability Ratio Test can help in case a) (rating a risk with no or dubious observational data) but fails to solve the problem in case b) (rating a risk that cannot be grouped into a homogeneous class except one too small for significant statistical inference).

    “Wald's Sequential Probability Ratio Test can be of help in case a), but unfortunately fails to solve the problem in case b).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Abraham Wald

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.

    Interpretationnone given
    § 5. A Distribution Free Credibility Formula

    The hypothesis of independence and identical distribution of the observational random variables of the same individual risk could easily be dropped.

    “The hypothesis of independence and identical distribution of the observational random variables of the same individual risk could easily be dropped.”
    Quote in the paperNot 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).

  • Probably not a substantial claim.

    Interpretationreasoning
    § 3. Report on Papers Submitted

    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.

    “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.”
    Quote in the paperSubstantial claim
12 of 12

  • 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”
3 of 3

Citations

Cites

Flagged first, then by citations

Research works printed but not linked: 1

  1. 1964, Eine Bemerkung zu Bichsels Herleitung der bedingten zukünftigen Schadenhäufigkeit einer Polya-Verteilung

Cited by

Sorted by citations

Related

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
DOI
10.1017/s0515036100008989
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
Licence
www.cambridge.org/core/terms (unspecified, 17226 d)
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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