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Lecture 3- Credibility 2

Lecture 3- Credibility 2 - Credibility II ActSCi 432/832 1...

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Credibility II ActSCi 432/832 1 / 22
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Greatest Accuracy Credibility Theory (GACT) Bayesian approach developed by H. B¨uhlmann Suppose we have a risk r.v. X that depends on a fixed but unknown parameter θ . (Here θ can be a vector in general) For example, θ represents the risk level of each policy holder (p.h.) in the rating class, and value of θ varies by p.h. In practice we can never pinpoint the true θ for a p.h. The perfect premium for a single p.h. whose risk level is θ is E ( X | θ ) = Z x f ( x | θ ) dx , (1) However, θ is never known in practice, and this quantity does not help much. We set μ ( θ ) := E ( X | θ ) and call it the hypothetical mean. 2 / 22
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Similarly we define the process variance v ( θ ): v ( θ ) := Var ( X | θ ) = E ( X 2 | θ ) - [ E ( X | θ )] 2 (2) With no past history, the best premium we get is the expected unconditional mean loss μ := E Θ [ μ (Θ)] = Z μ ( θ ) π ( θ ) d θ (3) This is called the collective premium and suitable for a new p.h. with no history. 3 / 22
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Incorporating history Let us now assume there is past data: x = ( x 1 , ..., x n ) In auto insurance, this can be translated as the record of a driver for the last n years, with x i being the loss amount in year i With this past record we can determine more personalized premium based on the predictive random variable x n +1 | x x n +1 | x can be translated as the loss for the next year (year n + 1) given the past n -year records. The premium for next year then is the mean of the predictive r.v.: E ( X n +1 | x ) , (4) a.k.a. the Bayesian premium. 4 / 22
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Bayesian framework: A quick review Consider an iid sample X = ( X 1 , ..., X n ) from conditional density f ( x | θ ) Prior density of Θ is denoted by π ( θ ) Then we can write Joint density of X and Θ Marginal density of X Posterior density of Θ | X Predictive density of X n +1 | X 5 / 22
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Example (Good driver and Bad driver) The world consists of good drivers (75% of the population) and bad drivers (25%). In any given year, good drivers have zero claims with prob 0.7, one claim with prob 0.2, and two claims with prob 0.1. Bad drivers have zero, one, or two claims with probabilities 0.5, 0.3, and 0.2, respectively. Suppose that for a particular policy holder we have x 1 = 0 and x 2 = 1 . Determine: 1 The collective premium for X 3 2 The posterior distribution of Θ | ( X 1 = 0 , X 2 = 1) 3 The predictive distribution of X 3 | ( X 1 = 0 , X 2 = 1) 4 The Bayesian premium for X 3 5 The marginal distribution of X 1 and X 2 6 The joint distribution of X 1 , X 2 and X 3 6 / 22
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