Bounding Basis-Risk Using s-convex Orders on Beta-unimodal Distributions Claude Lefèvre, Stéphane Loisel, Pierre Montesinos April 2020
Framewok Cat model = 4 components OICA - Pierre Montesinos - April 2020 2 / 21
Framewok Cat model = 4 components Hazard: consists of a large number of catastrophe event scenarios that together provide a representation of possible loss-causing events, and an associated mod- eled rate of occurence for each. Let n be the number of stochastic events, and p i be the occurence rate of the event i , i = 1 , . . . , n . OICA - Pierre Montesinos - April 2020 2 / 21
Framewok Cat model = 4 components Hazard: catastrophe event scenarios + occurence rate Inventory: represents the exposure. In practice, the exposure is not perfectly known (uncertain even trajec- tory + uncertain exposure when the event occurs). Let Y i be a positive random variable representing the exposure of scenario i , i = 1 , . . . , n . Assume Y i ≤ m a.s., where for instance m stands for the total size of the portfolio. Sometimes, a i ≤ Y i ≤ b i a.s. OICA - Pierre Montesinos - April 2020 3 / 21
Framewok Cat model = 4 components Hazard: catastrophe event scenarios + occurence rate Inventory: uncertain exposure Vulnerability: provides the intensity of the catastro- phe event on the exposed portfolio. We use the damage ratio (aka destruction rate or loss proportion) repre- sented by S i ∼ B eta ( α i , β i ) , S i ⊥ Y i . The loss in scenario i is then X i = S i Y i . OICA - Pierre Montesinos - April 2020 4 / 21
Framewok Cat model = 4 components Hazard: catastrophe event scenarios + occurence rate Inventory: uncertain exposure Vulnerability: intensity = destruction rate S i ∼ B eta ( α i , β i ) , S i ⊥ Y i . Loss: translates the expected physical damage into monetary loss taking into account any insurance struc- tures. OICA - Pierre Montesinos - April 2020 5 / 21
Framewok Cat model = 4 components Hazard: catastrophe event scenarios + occurence rate Inventory: uncertain exposure Vulnerability: intensity = destruction rate S i ∼ B eta ( α i , β i ) , S i ⊥ Y i . Loss: translates the expected physical damage into monetary loss taking into account any insurance struc- tures. Index-based transaction: the index in scenario i takes the value c i . OICA - Pierre Montesinos - April 2020 5 / 21
Framewok Cat model = 4 components Hazard: catastrophe event scenarios + occurence rate Inventory: uncertain exposure Vulnerability: intensity = destruction rate S i ∼ B eta ( α i , β i ) , S i ⊥ Y i . Loss: translates the expected physical damage into monetary loss taking into account any insurance struc- tures. Index-based transaction: the index in scenario i takes the value c i . The final flow in scenario i is given by S i Y i − c i . OICA - Pierre Montesinos - April 2020 5 / 21
Framework Basis Risk The difference in payment between own losses incurred and a structured risk transfer mechanism to protect against these losses. OICA - Pierre Montesinos - April 2020 6 / 21
Framework Basis Risk The difference in payment between own losses incurred and a structured risk transfer mechanism to protect against these losses. Way to quantify Basis Risk in Index-based transactions OICA - Pierre Montesinos - April 2020 6 / 21
Framework Basis Risk The difference in payment between own losses incurred and a structured risk transfer mechanism to protect against these losses. Way to quantify Basis Risk in Index-based transactions Identification of worst case scenarios using s - convex orders Measurement of basis risk using penalty functions φ ! OICA - Pierre Montesinos - April 2020 6 / 21
Framework Basis Risk The difference in payment between own losses incurred and a structured risk transfer mechanism to protect against these losses. Way to quantify Basis Risk in Index-based transactions Identification of worst case scenarios using s - convex orders Measurement of basis risk using penalty functions φ ! In a nutshell: n BR ( s ) = � � �� X ( s ) � p i E φ i , max − c i , s = 2 , . . . i = 1 OICA - Pierre Montesinos - April 2020 6 / 21
Beta-unimodal Distributions Definition An R + -valued rv X has a continuous Beta-unimodal distribution if it has the product representation X = d SY , (1) where Y is a positive continuous rv, and S ∼ B eta ( α, β ) is a random contracting factor independent of Y . OICA - Pierre Montesinos - April 2020 7 / 21
Beta-unimodal Distributions Definition An R + -valued rv X has a continuous Beta-unimodal distribution if it has the product representation X = d SY , (1) where Y is a positive continuous rv, and S ∼ B eta ( α, β ) is a random contracting factor independent of Y . From Y to X : if Y ∼ F Y and if X is Beta-unimodal F X ( x ) = Γ( α + β ) ¯ x α ( I β φ )( x ) , (2) Γ( α ) where: 1 � ∞ Γ( β )( t − x ) β − 1 φ ( t ) dt is called - ( I β φ )( x ) = x the Weyl fractional-order integral operator, - φ ( t ) = ¯ F Y ( t ) t − α − β . OICA - Pierre Montesinos - April 2020 7 / 21
Beta-unimodal Distributions Definition An R + -valued rv X has a continuous Beta-unimodal distribution if it has the product representation X = d SY , (3) where Y is a positive continuous rv, and S ∼ B ( α, β ) is a random contracting factor independent of Y . From X to Y : if X ∼ F X and if X is Beta-unimodal Γ( α ) F Y ( x ) = ( − 1 ) n x α + β ¯ Γ( α + β )( I δ D n ψ )( x ) , (4) where: - δ ∈ [ 0 , 1 ] such as β + δ = n ∈ N , - ψ ( t ) = ¯ F X ( t ) t − α , - D n denotes the n -fold derivative operator. OICA - Pierre Montesinos - April 2020 8 / 21
Identification: s-convex orders Definition: s-convexity A function φ defined on S is said to be s − convex if the inequality [ x 0 , ..., x s ; φ ] ≥ 0 , holds for any choice of distinct points x 0 , ..., x s in S . [ x 0 , ..., x s ; φ ] denotes a divided difference of the function φ at the different points x 0 , ..., x s . OICA - Pierre Montesinos - April 2020 9 / 21
Identification: s-convex orders Definition: s-convexity A function φ defined on S is said to be s − convex if the inequality [ x 0 , ..., x s ; φ ] ≥ 0 , holds for any choice of distinct points x 0 , ..., x s in S . [ x 0 , ..., x s ; φ ] denotes a divided difference of the function φ at the different points x 0 , ..., x s . s differentiability condition: if φ ( s ) exists in S , then φ is s − convex if and only if φ ( s ) ≥ 0. - - - OICA - Pierre Montesinos - April 2020 9 / 21
Identification: s-convex orders Definition: s-convexity Definition: s-increasing convexity A function φ is said to be s − increasing convex on its domain S if and only if for all choices of k + 1 distincts points x 0 < x 1 < x k in S , we have [ x 0 , x 1 , ..., x k ; φ ] ≥ 0 , k = 2 , ..., s . We denote by U S s − icx the class of the s-increasing convex functions on S . - - - OICA - Pierre Montesinos - April 2020 10 / 21
Identification: s-convex orders Definition: s-convexity Definition: s-increasing convexity Definition: s-convex order Let X 1 and X 2 be two random variables that take on values in S . Then X 1 is said to be smaller than X 2 in the s − convex order, denoted by X 1 ≤ S s − cx X 2 if E [ φ ( X 1 )] ≤ E [ φ ( X 2 )] for all φ ∈ U S s − cx , (5) where U S s − cx is the class of all the s − convex functions φ : S → R . OICA - Pierre Montesinos - April 2020 11 / 21
Identification: s-convex extrema Definition: moment space We denote by B s ([ a , b ] , µ 1 , µ 2 , ..., µ s − 1 ) the moment space of all the random variables valued in [ a , b ] and with known s − 1 moments µ 1 , ..., µ s − 1 . OICA - Pierre Montesinos - April 2020 12 / 21
Identification: s-convex extrema Definition: moment space We denote by B s ([ a , b ] , µ 1 , µ 2 , ..., µ s − 1 ) the moment space of all the random variables valued in [ a , b ] and with known s − 1 moments µ 1 , ..., µ s − 1 . Theorem: s-convex extrema Let Y ∈ B s ([ a , b ] , µ 1 , µ 2 , ..., µ s − 1 ) . Within B s ([ a , b ] , µ 1 , µ 2 , ..., µ s − 1 ) , there exist two unique random variables Y ( s ) min and Y ( s ) max such that Y ( s ) min ≤ s − cx Y ≤ s − cx Y ( s ) max . Proof. See Denuit et al. (1999). s-convex extrema are actually extremal distributions built from the s-1 first moments of Y . OICA - Pierre Montesinos - April 2020 12 / 21
Identification: s-convex extrema 3-convex extremal distributions Let Y ∈ B 3 ([ a , b ] , µ 1 , µ 2 ) . µ 2 − µ 2 1 a with proba , − − − ( a − µ 1 ) 2 + µ 2 − µ 2 Y ( 3 ) 1 min = ( a − µ 1 ) 2 b with proba , ( a − µ 1 ) 2 + µ 2 − µ 2 1 and µ 1 − µ 2 − µ 2 ( b − µ 1 ) 2 1 with proba , ( b − µ 1 ) 2 + µ 2 − µ 2 b − µ 1 Y ( 3 ) 1 max = µ 2 − µ 2 1 b with proba . ( b − µ 1 ) 2 + µ 2 − µ 2 1 OICA - Pierre Montesinos - April 2020 13 / 21
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