## A Course in Credibility Theory and its ApplicationsThe topic of credibility theory has been for many years — and still is — one of our major interests. This interest has led us not only to many publications, but also has been the motivation for teaching many courses on this topic over more than 20 years. These courses have undergone considerable changes over time. What we present here, “A Course in Credibility Theory and its Applications”, is the ?nal product of this evolution. Credibility theory can be seen as the basic paradigm underlying the pricing of insurance products. It resides on the two fundamental concepts “individual risk” and “collective” and solves in a rigorous way the problem of how to analyse the information obtained from these sources to arrive at the “insurance premium”. The expression “credibility” was originally coined for the weight given to the experience from the “individual risk”. Credibility theory as a mathematical discipline borrows its methods from 2 many ?elds of mathematics, e. g. Bayesian statistics, L Hilbert space te- niques, least squares, and state space modelling to mention only the most important ones. However, credibility theory remains a lifeless topic if it is not linked closely with its applications. Only through these applications has cr- ibility won its status in insurance thinking. The present book aims to convey this dual aspect of credibility and to transmit the ?avour of the insurance applications also to those readers who are not directly involved in insurance activities. |

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### Contents

I | 1 |

II | 7 |

III | 8 |

IV | 9 |

V | 11 |

VI | 14 |

VII | 15 |

IX | 16 |

LV | 159 |

LVI | 162 |

LVII | 165 |

LVIII | 167 |

LIX | 169 |

LXI | 170 |

LXII | 173 |

LXIII | 174 |

X | 18 |

XI | 21 |

XIII | 31 |

XIV | 34 |

XV | 36 |

XVI | 38 |

XVII | 39 |

XVIII | 46 |

XIX | 47 |

XX | 49 |

XXI | 50 |

XXII | 55 |

XXIII | 56 |

XXIV | 58 |

XXV | 59 |

XXVI | 60 |

XXVII | 64 |

XXVIII | 67 |

XXIX | 71 |

XXX | 74 |

XXXI | 77 |

XXXIII | 79 |

XXXIV | 81 |

XXXV | 84 |

XXXVI | 86 |

XXXVIII | 91 |

XXXIX | 93 |

XL | 95 |

XLI | 97 |

XLII | 106 |

XLIII | 110 |

XLIV | 111 |

XLV | 113 |

XLVI | 117 |

XLVII | 125 |

XLVIII | 130 |

XLIX | 135 |

L | 136 |

LI | 143 |

LII | 145 |

LIII | 146 |

LIV | 148 |

LXIV | 177 |

LXV | 178 |

LXVI | 180 |

LXVII | 185 |

LXIX | 187 |

LXX | 189 |

LXXI | 193 |

LXXII | 199 |

LXXIII | 201 |

LXXIV | 202 |

LXXV | 205 |

LXXVI | 208 |

LXXVIII | 217 |

LXXIX | 219 |

LXXX | 220 |

LXXXI | 223 |

LXXXII | 226 |

LXXXIII | 230 |

LXXXIV | 238 |

LXXXV | 239 |

LXXXVI | 251 |

LXXXVII | 252 |

LXXXVIII | 253 |

LXXXIX | 255 |

XC | 262 |

XCI | 264 |

XCII | 275 |

XCIII | 276 |

XCIV | 277 |

XCV | 278 |

XCVI | 280 |

XCVII | 283 |

XCVIII | 287 |

XCIX | 293 |

C | 296 |

CI | 305 |

CII | 311 |

CIII | 314 |

CIV | 318 |

CV | 323 |

329 | |

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### Common terms and phrases

affine subspace aggregate claim amount assume average claim amount Bayes estimator Bayes premium Bayesian statistics Biihlmann-Straub model Buhlmann calculation Chapter claim frequency claim number claim sizes claims ratio coefficient of variation components conditionally consider contracts Corollary correct individual premium covariance covariance matrix cred credibility estimator based credibility formula credibility matrix credibility model credibility premium credibility weights data compression defined denote depends distribution function Example expected value FBayes Gamma distribution given hierarchical Hilbert space homogeneous credibility estimator horn individual risk profile Kalman Filter large claims loss matrix loss ratio multidimensional credibility negative binomial distribution normal equations notation number of claims observation vector observed claim Pareto distributed pBayes Poisson distributed Proof of Theorem pure risk premium Qn\n quadratic loss random variables recursive Remarks risk class risk groups Section statistics Straub model structural parameters Theorem Theorem 9.8 Total truncation point unbiased estimator variance whole portfolio