q-Credibility OLIVIER LE COURTOIS EMLyon Business School First - PowerPoint PPT Presentation

q-Credibility OLIVIER LE COURTOIS EMLyon Business School

First Version Outline of the Talk Parametric approach Non-parametric approach Semi-parametric approach q-Credibility OLIVIER LE COURTOIS

First Version n q-Credibility that extends credibility theory to a quadratic setting. n Parametric Approach OLIVIER LE COURTOIS We want to solve the following program: min  2    ∑ ∑  α 0 , q + α i X i + i − E ( X n + 1 | θ )  . β i X 2    α 0 , q , { α i } , { β i } E  i = 1 i = 1

First Version Classic Notation Then, and, q-Credibility OLIVIER LE COURTOIS For hypothetical means: µ ( θ ) = E ( X | θ ) , µ = E ( µ ( θ )) . For the process variance: v ( θ ) = Var ( X | θ ) , v = E ( v ( θ )) . For the variance of hypothetical means: a = Var ( µ ( θ )) . Cov ( X i , X k ) = a , ∀ i ̸ = k , Cov ( X i , X i ) = Var ( X i ) = a + v .

First Version and also q-Credibility and Additional Notation OLIVIER LE COURTOIS and By analogy, we define Cov ( X 2 i , X k ) = b , ∀ i ̸ = k , Cov ( X 2 i , X i ) = b + g , Cov ( X 2 i , X 2 k ) = c , ∀ i ̸ = k , Cov ( X 2 i , X 2 i ) = Var ( X 2 i ) = c + h .

First Version and q-Credibility and q-Credibility Premium n OLIVIER LE COURTOIS where We have: P q = α 0 , q + Z q ¯ X + Y q X 2 , α 0 , q = µ ( 1 − Z q ) − Y q ( µ 2 + a + v ) , [ ] a ( nc + h ) − b ( nb + g ) Z q = ( na + v )( nc + h ) − ( nb + g ) 2 , n ( bv − ag ) Y q = ( na + v )( nc + h ) − ( nb + g ) 2 .

First Version q-Credibility Premium We also have: q-Credibility OLIVIER LE COURTOIS X − µ ) + Y q ( X 2 − ( µ 2 + v + a )) . P q = µ + Z q ( ¯

First Version Remark without any constraint on c or h , na n q-Credibility OLIVIER LE COURTOIS When b = g = 0, then Z q = na + v = a , Y q = 0 and α 0 , q = µ ( 1 − Z q ) , n + v so we recover the classic credibility case.

First Version n q-Credibility MSE The relative gain in MSE is measured by the quantity: va In the classic context, the formula reduces to yielding Mean Square Error OLIVIER LE COURTOIS n The Mean Square Error is computed as follows:  2    ∑ ∑ MSE q = E  α ∗ 0 , q + α ∗ i X i + β ∗ i − E ( X n + 1 | θ )  ,     i X 2 i = 1 i = 1 nv ( ac − b 2 ) + a ( hv − g 2 ) MSE q = n 2 ( ac − b 2 ) + n ( ah + cv − 2 bg ) + hv − g 2 . MSE = na + v . κ = MSE − MSE q .

First Version Conditional Poisson Case credibility theory. Then, and and also and q-Credibility OLIVIER LE COURTOIS Denote θ as λ and X as N and assume that N conditional on λ is Poisson distributed. µ = v = E ( λ ) and a = Var ( λ ) in classic b = a + E ( λ 3 ) − E ( λ 2 ) E ( λ ) , g = E ( λ ) + 2 E ( λ 2 ) , c = 2 b − a + Var ( λ 2 ) , h = E ( λ ) + 6 E ( λ 2 ) + 4 E ( λ 3 ) .

First Version Poisson-Gamma Case q-credibility reduces to classic credibility, with and and also q-Credibility OLIVIER LE COURTOIS Y q = 0 , Z q = Z , α 0 , q = α 0 = µ ( 1 − Z ) .

First Version . Then, q-Credibility and also and Poisson-Single Pareto Case OLIVIER LE COURTOIS Let λ be single Pareto-distributed with parameters ( η > 4 , χ ) . We ( ) 2 η − 1 and a = ηχ 2 ηχ ηχ know that µ = v = η − 2 − η − 1 η − 1 + 2 ηχ 2 ηχ b = ηχ 3 η − 3 − ηχ 2 ηχ g = η − 2 , η − 1 , η − 2 ( ) 2 c = 2 b − a + ηχ 4 ηχ 2 η − 4 − , η − 2 η − 1 + 6 ηχ 2 ηχ η − 2 + 4 ηχ 3 η − 3 . h =

First Version Poisson-Single Pareto Case Illustration q-Credibility OLIVIER LE COURTOIS The parameters of the single Pareto distribution are η = 5 and χ = 4 and we assume that 5 claims have been observed in the past n = 2 years. Then, µ = 5 and ¯ X = 2 . 5. According to classic credibility theory, P = 4.

First Version 8.5 q-Credibility 4.2259 4.1629 4.1314 P q 12.5 6.5 Poisson-Single Pareto Case X 2 (5,0) (4,1) (3,2) Number of claims distrib. According to the q-credibility approach, Illustration OLIVIER LE COURTOIS

First Version Poisson-Single Pareto Case Illustration We have: and so that q-Credibility OLIVIER LE COURTOIS MSE = 1 , MSE q = 0 . 9317 , κ = 6 . 83 %.

First Version and that of expected process variance is q-Credibility insured i . n n n Non-Parametric Approach 1 r OLIVIER LE COURTOIS The classic estimator of expected hypothetical means is n rn r ∑ ∑ µ = 1 ˆ X ij , i = 1 j = 1 ∑ ∑ ( X ij − ¯ ˆ v = X i ) 2 , r ( n − 1 ) i = 1 j = 1 ∑ where ¯ X i = 1 X ij is the empirical mean of past observations for j = 1

First Version Non-Parametric Approach q-Credibility X is the empirical mean of past observations for all insureds, n v OLIVIER LE COURTOIS 1 r The estimator of the variance of hypothetical means is ∑ X ) 2 − ˆ ( ¯ X i − ¯ ˆ a = r − 1 i = 1 where ¯ which is equal to ˆ µ .

First Version X 2 q-Credibility observations for a given insured i . X 2 n n i Non-Parametric Approach OLIVIER LE COURTOIS n The quadratic non-parametric estimators are given as follows. r 1 ∑ ∑ ( ) 2 ˆ h = ij − X 2 , r ( n − 1 ) i = 1 j = 1 ∑ i = 1 where X 2 ij is the empirical mean of past squared j = 1

First Version Non-Parametric Approach q-Credibility for all insureds. is the empirical mean of past squared observations ij X 2 n r rn where h OLIVIER LE COURTOIS r Then, X 2 1 ˆ ∑ ( ) 2 ˆ c = i − X 2 − n , r − 1 i = 1 ∑ ∑ X 2 = 1 i = 1 j = 1

First Version n q-Credibility g r 1 and Non-Parametric Approach OLIVIER LE COURTOIS Next, r 1 ∑ ∑ i )( X ij − ¯ ˆ g = ( X 2 ij − X 2 X i ) , r ( n − 1 ) i = 1 j = 1 ∑ X ) − ˆ ˆ i − X 2 )( ¯ X i − ¯ b = ( X 2 n . r − 1 i = 1

First Version 1 q-Credibility yellow submarines observed at each time. where each line is for a zone and each element is a number of 1 1 1 13 Non-Parametric Approach 10 6 2 1 Illustration OLIVIER LE COURTOIS Assume r = n = 3 and we have the following data:   X =  ,   

First Version Non-Parametric Approach Illustration According to classic credibility theory, the number of yellow submarines observed in each zone is as follows. and and q-Credibility OLIVIER LE COURTOIS P 1 ≈ 3 . 3932 , P 2 ≈ 6 . 4274 , P 3 ≈ 2 . 1795 .

First Version and q-Credibility P i Relative changes Non-Parametric Approach OLIVIER LE COURTOIS and According to the q-credibility approach, we have: Illustration P q , 1 ≈ 2 . 3890 , P q , 2 ≈ 6 . 2613 , P q , 3 ≈ 2 . 2928 . ( P q , i − P i ) i = 1 : 3 are respectively − 29 . 6 % , − 2 . 58 % , and 5 . 2 % .

First Version Non-Parametric Approach Illustration We have: and so that q-Credibility OLIVIER LE COURTOIS MSE = 3 . 1016 , MSE q = 2 . 7634 , κ = 10 . 9 %.

First Version Semi-Parametric Approach Here, we assume this conditional distribution to be of the Poisson type and X is denoted by N . for which i claims occurred. q-Credibility OLIVIER LE COURTOIS Situation where the distribution of X conditionally on θ is known. Each n i describes the number of insureds

First Version n i q-Credibility 1 n i . Then, Semi-Parametric Approach M n i OLIVIER LE COURTOIS i In the classic setting, we have: + ∞ + ∞ ∑ ∑ µ = ˜ ˜ v = = 1 in i , + ∞ ∑ i = 0 i = 0 i = 0 + ∞ ∑ where M = i = 0 + ∞ ∑ ( i − ˜ µ ) 2 n i + ∞ ∑ i = 0 ˜ a = − ˜ v = ( i − ˜ µ ) 2 n i − ˜ v . M − 1 + ∞ ∑ n i − 1 i = 0 i = 0

First Version and q-Credibility in i M i 2 n i M Semi-Parametric Approach 1 OLIVIER LE COURTOIS M In the q-credibility setting, we have: + ∞ ∑ ( 2 i 2 − i ) n i , ˜ g = 1 i = 0     + ∞ + ∞ + ∞ ∑ ∑ ∑  i 2 − 1 ˜ b =  i − 1  n i − ˜ g ,  M − 1 i = 0 i = 0 i = 0

First Version and q-Credibility 2 i 2 n i M Semi-Parametric Approach 1 OLIVIER LE COURTOIS M Further, + ∞ ∑ ( 4 i 3 − 6 i 2 + 3 i ) n i , ˜ h = 1 i = 0   + ∞ + ∞ ∑ ∑  i 2 − 1 n i − ˜ ˜ c = h .  M − 1 i = 0 i = 0

First Version Semi-Parametric Approach Illustration Assume we observed i n i 0 560 1 134 2 14 3 2 q-Credibility OLIVIER LE COURTOIS

First Version Semi-Parametric Approach q-Credibility According to the q-credibility approach: OLIVIER LE COURTOIS According to classic credibility theory: period for an insured who incurred i claims. We can estimate the expected future number of claims in the next Illustration P = [ 0 . 2359 0 . 2388 0 . 2417 0 . 2446 ] . P q = [ 0 . 2376 0 . 2266 0 . 2722 0 . 3743 ] .

First Version Semi-Parametric Approach Illustration We have: and so that q-Credibility OLIVIER LE COURTOIS MSE = 0 . 000681 , MSE q = 0 . 000585 , κ = 14 . 1 %.