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Introduction Basics The Bayes Premium Credibility Theory Ana-Iulia DOMNISAN 30th April 2009 Ana-Iulia DOMNISAN Credibility Theory Introduction Deﬁnition & Formula Basics A Simple Example The Bayes Premium Approaches Introduction Origins. Deﬁnition. Example. Approaches. Ana-Iulia DOMNISAN Credibility Theory Introduction Deﬁnition & Formula Basics A Simple Example The Bayes Premium Approaches Deﬁnition mathematical tool deals with the randomness of data helps predict future events or costs Ana-Iulia DOMNISAN Credibility Theory Introduction Deﬁnition & Formula Basics A Simple Example The Bayes Premium Approaches The First Formula basic formula for credibility weighted estimates Estimate = Z × [Observation] + 1 − Z × [Other Information] Z is the credibility assigned to the observation 1−Z is referred to as the complement of credibility 0≤Z ≤1 Ana-Iulia DOMNISAN Credibility Theory Introduction Deﬁnition & Formula Basics A Simple Example The Bayes Premium Approaches A Simple Example large population of drivers observed over a 5-year period. average driver has an annual frequency of 0.20 accidents per year. random driver has an annual frequency of 0.60 accidents per year. Question: What is the estimate of the expected future frequency rate for this driver? Ana-Iulia DOMNISAN Credibility Theory Introduction Deﬁnition & Formula Basics A Simple Example The Bayes Premium Approaches Solution Dilemma! 0.20, 0.60 or something in between? Best Solution: this driver’s Expected Future Accident Frequency = Z × 0.60 + 1 − Z ) × 0.20 Attention: anti-selection eﬀect! Ana-Iulia DOMNISAN Credibility Theory Introduction Deﬁnition & Formula Basics A Simple Example The Bayes Premium Approaches Approaches Classical Credibility: − limited ﬂuctuation credibility model. − homogenous risk classes. u B¨hlmann Credibility: − least squares credibility model. Bayesian Analysis: − formulas match those of B¨hlmann credibility estimation. u − linear weightening of current and prior information u with weights Z and (1 - Z) where Z is the B¨hlmann credibility. Ana-Iulia DOMNISAN Credibility Theory Introduction The Model Basics Individual Risk The Bayes Premium Premium Types The Model insurance company I insured risks, numbered i = 1, 2, . . . , I in a well-deﬁned insurance period, the risk i produces: − a number of claims Ni , (ν) − with claim sizes Yi , where ν = 1, 2, . . . , Ni , − which together give the aggregate claim amount Ni (ν) Xi = ν=1 Yi gross premium premium volume task: determine the pure risk premium Pi = E [Xi ]. Ana-Iulia DOMNISAN Credibility Theory Introduction The Model Basics Individual Risk The Bayes Premium Premium Types The Individual Risk black box Xj (j = 1, 2, . . . , n) denotes the claim amount during the time period j previous periods: X = (X1 , . . . , Xn ) present period: Xn+1 standard assumptions regarding the distribution function F (x) of the random variables Xj : − stationarity, − (conditional) independence. parameterisation: ϑ = ”risk proﬁle”, element of the set Θ. Ana-Iulia DOMNISAN Credibility Theory Introduction The Model Basics Individual Risk The Bayes Premium Premium Types From the Fictional to the True Premium The Two-urn Model The individual premium: P ind = µ Θ = E [ Xn+1 |X ] The collective premium: P coll = µ0 = Θ µ(ϑ) dU(ϑ) = E [ Xn+1 ] The Bayes premium: P Bayes = µ Θ = E [ µ Θ |X ] Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases Statistical Decision Theory observation vector X = (X1 , X2 , . . . , Xn ) distribution function Fϑ (x) = Pϑ [X ≤ x] GOAL: the value of a speciﬁc functional g (ϑ) of the parameter ϑ T (X) which: − depends only on X , − will estimate g (ϑ) ”as well as possible”. Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases Formulation of the Problem ϑ ∈ Θ: set of parameters, which contains the true value of ϑ T ∈ D: set of functions to which the estimator function must belong. D = {g (ϑ) : ϑ ∈ θ} loss function: L(ϑ, T (x)) risk function: RT (ϑ) := Eϑ [L(ϑ, T )] = Rn L(ϑ, T (x))dF(x) . Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases Bayes Risk, Bayes Estimator U(ϑ) a priori distribution for Θ. Bayes risk: R(T ) := Θ RT (ϑ)dU(ϑ). Bayes Estimator T : T := arg min R(T ). T∈ D Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases The Quadratic Loss correct individual premium µ(ϑ) Θ = the set of individual risk proﬁles ϑ. quadratic loss function L(ϑ, T (x)) = (µ(ϑ) − T (x))2 the Bayes estimator with respect to the quadratic loss function is given by µ Θ = E [µ(Θ)|X] Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases The Bayes Premium in Three Special Cases 1. The Poisson−Gamma Case 2. The Binomial−Beta Case 3. The Normal−Normal Case Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases 1. The Poisson−Gamma Case Motivation: F.Bichsel’s Problem − 1960’s in Switzerland, − Bonus-Malus System was created, − premium level based on horsepower of the car, − Task: construct a better risk premium adjusted to the indi- viduals risk proﬁle. − diﬀerences in individual numbers of claim. Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases Mathematical Modelling Nj claims during year j corresponding aggregate claim amount Xj . implicit assumption of Bichsel: − given: individual risk proﬁle ϑ, − E [Xj |Θ = ϑ] = C E [Nj |Θ = ϑ] holds the aggregate claim amount, − C - constant depending on the horsepower of the car, − E [Nj |Θ = ϑ] depends only on the driver. Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases Model Assumptions PG1: Conditionally, given Θ = ϑ, the Nj ’s (j = 1, 2, . . . , n) are independent and Poisson distributed with Poisson parameter ϑ: ϑk P(Nj = k|Θ = ϑ) = e −ϑ k! . PG2: Θ has a Gamma distribution with shape parameter γ and scale parameter β, the structural function has density: βγ u(ϑ) = Γ(γ) ϑγ−1 e −βϑ , ϑ ≥ 0. Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases The Claim Frequencies The individual claim frequency: F ind = E [Nn+1 |Θ] = Θ. The collective claim frequency: γ F coll = E [Θ] = β. Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases The Bayes Claim Frequency: Claim Frequency: γ + N• γ F Bayes = β + n = αN + (1 − α) β n 1 n where α = n+β , N = n j=1 Nj . The quadratic loss of F Bayes is: 2 2 E FBayes − Θ = 1-α E F coll − Θ 2 = αE N −Θ . Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases Remarks The quantities P ind , P coll and P Bayes are obtained by multiplication with C . The Bayes premium CF Bayes is a linear function of the observations (credibility premium) . F Bayes is an average of: − N = observed individual claim frequency and γ − E [Θ] = β = a priori expected claim frequency. Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases Remarks n α = n+β − credibility weight. great number of obervation years n leads to a large α . E Θ if β = large then α small. Var Θ Explanation of the quatradic loss: − the quadratic loss of F Bayes = (1 − α)· quadratic loss of F coll − the quadratic loss of F Bayes = α· quadratic loss of N Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases Estimators F coll is the estimator based only on the a priori knowledge from the collective, neglecting the individual claim experience. N is the estimator based only on the individual claims experience, neglecting the apriori knowledge. F Bayes combines both sources of information. Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases 2. The Binomial−Beta Case Motivation: − group life/accident insurance, − interesting: number of disability cases or disability frequency for a certain group. A few assumptions to simplify matters: − each member of the group has the same probability of disablement, − disabilities occur independently − disabled person leaves group. Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases Random Variables Nj = number of new disabilities occuring in the group in the year j = 1, 2, . . . Vj = number of (not disabled) members in the group at the beginning of year j = 1, 2, . . . Nj Xj = Vj observed disablement frequency in year j = 1, 2, . . . of interest is: Nn+1 Xn+1 = Vn+1 Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases Model Assumptions BB1: Conditionally, given Θ = ϑ, Nj (j = 1, 2, . . . , n) are independent and binomial distributed with : Vj P Nj = k|Θ = k ϑk (1 − ϑ)Vj −k . BB2: Θ has a Beta(a, b) distribution with a, b > 0, equivalently, the structural function has density: 1 u(ϑ) = B(a,b) ϑa−1 (1 − ϑ)b−1 , 0 ≤ ϑ ≤ 1. Γ(a)Γ(b) where B(a, b) = Γ(a+b) . Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases The Claim Frequencies The individual claim frequency: F ind = E [Xn+1 |Θ] = Θ. The collective claim frequency: a F coll = E [Θ] = a+b . Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases Bayes Claim Frequency: Claim Frequency: a+N• a F Bayes = a+b+N• = αN + (1 − α) a+b N• V• where N = V• , α= a+b+V• . The quadratic loss of F Bayes is: 2 2 E FBayes − Θ = 1-α E F coll − Θ 2 = αE N −Θ . Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases The Normal−Normal Case - Remarks no practical motivation insurance data is mostly not normally distributed sometimes useful for large portfolios consider an individual risk observation vector: X = (X1 , . . . , Xn ) Xj aggregate claim amount in the j − th year Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases Model Assumptions NN1: Conditionally, given Θ = ϑ, the Xj ’s (j = 1, 2, . . . , n) are independent and normally distributed with : Xj ∼ N ϑ, σ 2 . NN2: Θ ∼ N µ, τ 2 , and the structural function has the density: 1 ϑ − µ 2 u(ϑ) = √1 2π e− 2 ( τ ) . Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases The Premiums The individual Premium: P ind = E [Xn+1 |Θ] = Θ. The collective claim frequency: P coll = E [Θ] = µ. Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases Bayes Premium: Premium: τ 2 µ+σ 2 X• P Bayes = τ 2 +nσ 2 = αX + (1 − α)E [Θ] 1 n where X = n X• , α= 2 . n+ σ2 τ The quadratic loss of P Bayes is: 2 2 E PBayes − Θ = 1-α E P coll − Θ 2 = αE X −Θ . Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases Common Features the Bayes Premium is a linear function of the observations =⇒ a credibility premium P Bayes can be expressed as a weighted mean: P Bayes = α X + (1 − α) P coll . the weight α is given by: n α = n+κ , where κ is an appropiate constant. the quadratic loss of the Bayes premium is: 2 2 E PBayes − Θ = 1 - α E P coll − Θ 2 = αE X −Θ . Ana-Iulia DOMNISAN Credibility Theory Introduction Necessary Statistical Elements Basics Bayes Statistics The Bayes Premium Three Special Cases Thank You for Your Attention! Ana-Iulia DOMNISAN Credibility Theory

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