Optimal Reinsurance with Positive Dependence Presenter: Wei Wei, - PowerPoint PPT Presentation

Outline Background Model Introduction Bivariate Case Multivariate Case Premium Constraint Conclusion Optimal Reinsurance with Positive Dependence Presenter: Wei Wei, Co-author: Jun Cai University of Waterloo Presentation in 46th ARC, Univ. of Conn. Aug 13, 2011

Outline Background Model Introduction Bivariate Case Multivariate Case Premium Constraint Conclusion Background 1 Model Introduction 2 Bivariate Case 3 Multivariate Case 4 Premium Constraint 5 Conclusion 6

Outline Background Model Introduction Bivariate Case Multivariate Case Premium Constraint Conclusion Classical Optimal Reinsurance Problem Statement of the Optimization Problem inf I ∈D ρ ( I ( X )) subject to π ( R ( X )) = p . R ( X ) – ceded risk, I ( X ) = X − R ( X ) – retained risk; Find a strategy to minimize the retained risk I ( X ). Ingredients of Optimization Problem π – the premium principle for reinsurance; ρ – risk measure as optimization criterion; D – admissible strategy class.

Outline Background Model Introduction Bivariate Case Multivariate Case Premium Constraint Conclusion Traditional risk measure: ρ ( X ) = E [ u ( X )] u ( x ) is a convex function. For example u ( x ) = x 2 – minimize variance; u ( x ) = e γ x – maximize utility of insurer’s wealth: u ( x ) = ( x − E [ X ]) 2 + – minimize semi-variance. Mean-Variance Premium Principle: E [ X ] = g ( π ( X ) , D X ) Expected value premium: π ( X ) = (1 + θ ) E [ X ]; Variance premium: π ( X ) = E [ X ] + β Var X ; Standard deviation premium: π ( X ) = E [ X ] + β D X .

Outline Background Model Introduction Bivariate Case Multivariate Case Premium Constraint Conclusion Solutions to Classical Optimizaition Problems Target – minimizing variance. Pure variance premium: quota reinsurance – R ( x ) = α x Expectation premium: stop loss reinsurance - R ( x ) = ( x − d ) + Mean-Variance premium: change loss reinsurance – R ( x ) = α ( x − d ) + . Reference: Borch 1969, Kaluszka 2001. Generalization of Classical Model Different premium principles, risk measures; Consider multiple risk instead of one-dimensional risk.

Outline Background Model Introduction Bivariate Case Multivariate Case Premium Constraint Conclusion Motivation A Practical Problem Consider an auto insurance policy covering two source of loss: vehicle damage and personal injury. Usually, different types of loss have to be reinsured separately. —How to make an optimal reinsurance arrangement? Modeling The risk is modeled by ( X 1 , X 2 ), X 1 , X 2 ≥ 0. The reinsurance strategy ( I 1 , I 2 ) is applied, i.e. For each X i , the insurer retains I i ( X i ). Objective: Minimize the total retained risk I 1 ( X 1 ) + I 2 ( X 2 ).

Outline Background Model Introduction Bivariate Case Multivariate Case Premium Constraint Conclusion Notations Statement of the Optimization Problem ( I 1 , I 2 ) ∈D E [ u ( I 1 ( X 1 ) + I 2 ( X 2 ))] subject to E [( I 1 ( X 1 ) + I 2 ( X 2 ))] = p . inf Expectation premium principle Convex risk measure: ρ ( X ) = E [ u ( X )] with convex u . Admissible Strategy Classes � � � I i ( x ) is non-decreasing in x ≥ 0 satisfying � D = ( I 1 , I 2 ) ; � 0 ≤ I i ( x ) ≤ x for i = 1 , 2 . � D p � = ( I 1 , I 2 ) ∈ D | E [( I 1 ( X 1 ) + I 2 ( X 2 ))] = p } ; ( I d 1 , I d 2 ) ∈ D p | I d i ( x ) = x ∧ d i , i = 1 , 2 D p � � = sl D p - global strategy class; D p sl - (bivariate) stop-loss strategy class.

Outline Background Model Introduction Bivariate Case Multivariate Case Premium Constraint Conclusion Comments on the Bivariate Model Individualized Strategy vs Global Strategy Global strategy I ( X 1 , X 2 ) = I ( X 1 + X 2 ) − → classical problem; Individualized Strategy I ( X 1 , X 2 ) = I 1 ( X 1 ) + I 2 ( X 2 ). Independent Case Heerwaarden et al (1989) has shown that: if X 1 and X 2 are independent, the optimal strategy has the stop loss form, i.e. ( I 1 ( x 1 ) , I 2 ( x 2 )) = ( x 1 ∧ d 1 , x 2 ∧ d 2 ). Ideas to Solve the Problem Under certain dependence structure, Show the optimality of bivariate stop loss strategy; Find out optimal solution among the stop loss strategy.

Outline Background Model Introduction Bivariate Case Multivariate Case Premium Constraint Conclusion Dependence Structure Definition: Stochastically Increasing X is stochastically increasing in Y , denoted as X ↑ SI Y , if P { X > x | Y = y 1 } ≤ P { X > x | Y = y 2 } , for any x , y 1 ≤ y 2 ; or equivalently, if X |{ Y = y 1 } ≤ st X |{ Y = y 2 } for any y 1 ≤ y 2 . Examples Independent or comonotonic random variables; Common shock: X 1 = Y 1 × Z , X 2 = Y 2 × Z . Random variables linked by typical copulas: such as Gaussian/Gumbel/Clayton copula with coefficient restriction.

Outline Background Model Introduction Bivariate Case Multivariate Case Premium Constraint Conclusion Optimization — D p vs D p sl Theorem 1 - Equivalence of Minimization in D p and D p sl If X 1 ↑ SI X 2 and X 2 ↑ SI X 1 , then for any ( I 1 , I 2 ) ∈ D p , there exists ( I d 1 , I d 2 ) ∈ D p sl such that E [ u ( I d 1 ( X 1 ) + I d 2 ( X 2 )) ≤ E [ u ( I 1 ( X 1 ) + I 2 ( X 2 ))] , for any convex function u ( x ). Application in Dynamic Model Consider a compound Poisson model: U ( I 1 , I 2 ) ( t ) = u + p t − � N ( t ) i =1 ( I 1 ( X 1 , i ) + I 2 ( X 2 , i )) , where ( X 1 , i , X 2 , i ) ∼ i . i . d . ( X 1 , X 2 ). Denote by φ ( I 1 , I 2 ) ( u ) the ruin probability of the surplus process U ( I 1 , I 2 ) ( t ), then there exists ( d 1 , d 2 ) ∈ D p sl such that φ ( d 1 , d 2 ) ( u ) ≤ φ ( I 1 , I 2 ) ( u ).

Outline Background Model Introduction Bivariate Case Multivariate Case Premium Constraint Conclusion The Premium Constraint The Curve Determined by Premium Constraint � d 1 � d 2 � � � � L = ( d 1 , d 2 ) F 1 ( x ) dx + F 2 ( x ) dx = p , d 1 , d 2 ≥ 0 � � 0 0 Properties of the Curve L On L , d 2 = L ( d 1 ) is a one-to-one mapping; ∂ d 1 = − F 1 ( d 1 ) L ( d 1 ) a convex function, with ∂ d 2 F 2 ( d 2 ) ; Denote the endpoints of L by ( d 1 , d 2 ) and ( d 2 , d 1 ). For simplicity, assume d 1 = d 2 = ∞ .

Outline Background Model Introduction Bivariate Case Multivariate Case Premium Constraint Conclusion Graph of L-Curve Figure: L-Curve and solution area

Outline Background Model Introduction Bivariate Case Multivariate Case Premium Constraint Conclusion Two Optimization Problems Problem Description Var [ I d 1 ( X 1 ) + I d 2 ( X 2 )] inf (1) ( d 1 , d 2 ) ∈D sl E [exp { s ( I d 1 ( X 1 ) + I d 2 ( X 2 )) } ] , s ∈ R . inf (2) ( d 1 , d 2 ) ∈D sl Explicit Solutions The solutions to (1) and (2) exist and are determined by: � E [( X 2 − d 2 ) − | X 1 > d 1 ] = E [( X 1 − d 1 ) − | X 2 > d 2 ] � d 1 � d 2 0 F 1 ( x ) dx + 0 F 2 ( x ) dx = p . � E [exp { s ( X 2 − d 2 ) − }| X 1 > d 1 ] = E [exp { s ( X 1 − d 1 ) − }| X 2 > d 2 ] � d 1 � d 2 0 F 1 ( x ) dx + 0 F 2 ( x ) dx = p .

Outline Background Model Introduction Bivariate Case Multivariate Case Premium Constraint Conclusion Multivariate Dependence Structure Definition: Positive Dependent through Stochastic Ordering Random vector X is said to be stochastically increasing in random variable Y , denoted as X ↑ SI Y , if X | Y = y 1 ≤ st X | Y = y 2 for any y 1 ≤ y 2 ; Random vector X is said to be positive dependent through stochastic ordering (PDS), if ( X i , i � = j ) ↑ SI X j for all j = 1 , 2 , · · · , n . Examples of Stochastically Increasing If X is linked by one of the following copulas, then X is PDS: The multivariate independence/comonotonicity copula; The multivariate Gaussian copula with nonnegative correlation matrix.

Outline Background Model Introduction Bivariate Case Multivariate Case Premium Constraint Conclusion Optimality of Stop Loss Strategy Multivariate Strategy Classes � � I i ( x ) is non-decreasing in x ≥ 0 satisfying � � M = I , � 0 ≤ I i ( x ) ≤ x for i = 1 , 2 , · · · , n � � � n � � ; M p M p � sl = { I ∈ M p | I i ( x ) = x ∧ d i } . = I ∈ M E [ I i ( X i )] = p � � � i =1 Theorem 2 - Generalization of Theorem 1 If X is PDS, then for any convex function u ( x ), � n � n � �� � �� � � inf I i ( X i ) = inf I i ( X i ) E u I ∈M p E u I ∈M p sl i =1 i =1

Outline Background Model Introduction Bivariate Case Multivariate Case Premium Constraint Conclusion Unbinding Constraint and Admissible Strategy Class Binding and Unbinding Constraints Assume expected value premium principle: � Binding: π ( � n ⇒ E [ � n i =1 I i ( X i )) = p 2 ⇐ i =1 I i ( X i )] = p , Unbinding: π ( � n ⇒ E [ � n i =1 I i ( X i )) ≤ p 2 ⇐ i =1 I i ( X i )] ≥ p . Admissible Strategy Classes � � n � � M ≥ p = � ; M ≥ p = { I ∈ M p | I i ( x ) = x ∧ d i } . I ∈ M E [ I i ( X i )] ≥ p � sl � � i =1 Clearly, M p ⊂ M ≥ p and M p sl ⊂ M ≥ p sl