A risk management approach to capital allocation Khalil Said PhD - PowerPoint PPT Presentation

A risk management approach to capital allocation Khalil Said PhD supervisors: Mme Véronique Maume-Deschamps M. Didier Rullière Laboratoire de sciences actuarielle et financière (SAF) EA2429 Colloque Jeunes Probabilistes et Statisticiens Les Houches, April 19, 2016

Outline A risk management approach to capital allocation Introduction 1 Optimal allocation 2 Coherence properties 3 Discussion 4 Conclusion 5

Introcution Multivariate risk theory : Dependence modeling ; Multivariate ruin probabilities ; Multivariate risk measures... What is a capital allocation ? Euler and Shapley principles ([Tasche, 2007],[Denault, 2001]). Minimization of some ruin probabilities or multivariate risk indicators.

Introduction Optimal allocation Coherence properties Discussion Conclusion Bibliography What is a capital allocation ? F IGURE : What is a capital allocation ? Khalil Said - Les Houches, April 19, 2016 Colloque Jeunes Probabilistes et Statisticiens 4 / 20

Introduction Optimal allocation Multivariate risk indicators Coherence properties The allocation method Discussion Optimality conditions Conclusion Penalty functions Bibliography Optimal allocation 2 Multivariate risk indicators The allocation method Optimality conditions Penalty functions Khalil Said - Les Houches, April 19, 2016 Colloque Jeunes Probabilistes et Statisticiens 5 / 20

Multivariate risk framework We consider a vectorial risk process X p = ( X p 1 , . . . , X p d ) , where X p k corresponds to the losses of the k th business line during the p th period. k the reserve of the k th line at time p , so : We denote by R p p k , where u k ∈ R + is the initial capital of the k th � R p X l k = u k − l = 1 business line ; u = u 1 + · · · + u d is the initial capital of the group ; d is the number of business lines. u = { v = ( v 1 , . . . , v d ) ∈ [ 0 , u ] d , � d U d i = 1 v i = u } is the set of possible allocations of the initial capital u . u , then, � d ∀ i ∈ { 1 , . . . , d } let α i = u i i = 1 α i = 1 if ( u 1 , . . . , u d ) ∈ U d u . X k corresponds to the losses of the k th branch during one period ( n = 1 ) .

Optimal allocation Multivariate risk indicators Cénac et al. (2012) defined the two following multivariate risk indicators, for d risks and n periods, given penalty functions g k , k ∈ { 1 , . . . , d } : the indicator I :   d n � � g k ( R p I ( u 1 , . . . , u d ) = k ) 1 E 1 { R p k < 0 } 1 1 { � d  , j = 1 R p  j > 0 } k = 1 p = 1 the indicator J :   d n � � g k ( R p J ( u 1 , . . . , u d ) = k ) 1 E 1 { R p k < 0 } 1 1 { � d  , j = 1 R p  j < 0 } k = 1 p = 1 g k : R − → R + are C 1 , convex functions with g k ( 0 ) = 0 , g k ( x ) ≥ 0 , k = 1 , . . . , d , g k are decreasing functions on R − .

Optimal allocation Multivariate risk indicators F IGURE : Multivariate risk indicators

The allocation method Since the new regulation, such as Solvency 2, require a one year allocation strategy, in this paper we focus on a single period ( n = 1 ). Definition : Optimal allocation Let X be a non negative random vector of R d , u ∈ R + and u → R + a multivariate risk indicator associated to X and u . An K X : U d optimal allocation of the capital u for the risk vector X is defined by : ( u 1 , . . . , u d ) ∈ {K X ( v 1 , . . . , v d ) } . arg inf ( v 1 ,..., v d ) ∈U d u For risk indicators of the form K X ( v ) = E [ S ( X , v )] , with a scoring function S : R + d × R + d → R + , this definition can be seen as an extension in a multivariate framework of the concept of elicitability. For an initial capital u , and an optimal allocation minimizing the multivariate risk indicator I , we seek u ∗ ∈ R d + such that : I ( u ∗ ) = v 1 + ··· + v d = u I ( v ) , v ∈ R d inf + .

Introduction Optimal allocation Multivariate risk indicators Coherence properties The allocation method Discussion Optimality conditions Conclusion Penalty functions Bibliography Assumptions Assumptions H1 K X admits a unique minimum in U d u . In this case, we denote by A X 1 ,..., X d ( u ) = ( u 1 , . . . , u d ) the optimal allocation of u on the d risky branches in U d u . H2 The functions g k are differentiable and such that for all k ∈ { 1 , . . . , d } , g ′ k ( u k − X k ) admits a moment of order one, and ( X k , S ) has a joint density distribution denoted by f ( X k , S ) . H3 The d risks have the same penalty function g k = g , ∀ k ∈ { 1 , . . . , d } . The first assumption is verified when the indicator is strictly convex, this is particularly true if at least one function g k is strictly convex ; and the joint density f ( X k , S ) support contains [ 0 , u ] 2 . Khalil Said - Les Houches, April 19, 2016 Colloque Jeunes Probabilistes et Statisticiens 10 / 20

Optimality condition Under assumption H2, the risk indicators I and J are differentiable, � + ∞ d � g k ( v k − x ) f X k , S ( x , u ) dx + E [ g ′ ( ∇ I ( v )) i = i ( v i − X i ) 1 1 { S ≤ u } ] 1 { X i > v i } 1 v k k = 1 and, � + ∞ d � g k ( v k − x ) f X k , S ( x , u ) dx + E [ g ′ ( ∇ J ( v )) i = i ( v i − X i ) 1 1 { S ≥ u } ] . 1 { X i > v i } 1 v k k = 1 Under H1 and H2, using Lagrange multipliers, we obtain an optimality condition verified by the unique solution, E [ g ′ 1 { S ≤ u } ] = E [ g ′ 1 { S ≤ u } ] , ∀ ( i , j ) ∈ { 1 , . . . , d } 2 i ( u i − X i ) 1 j ( u j − X j ) 1 1 { X i > u i } 1 1 { X j > u j } 1

Penalty functions Ruin severity as penalty function A natural choice for penalty functions is the ruin severity : g k ( x ) = | x | . If the joint density f ( X k , S ) support contains [ 0 , u ] 2 , for at least one k ∈ { 1 , . . . , d } , our optimization problem has a unique solution. We may write the indicators as follows : d � � � I ( u 1 , . . . , u d ) = ( X k − u k ) + 1 E 1 { S ≤ u } , k = 1 and, d � � � J ( u 1 , . . . , u d ) = ( X k − u k ) + 1 E 1 { S ≥ u } . k = 1 The optimality condition : P ( X i > u i , S ≤ u ) = P ( X j > u j , S ≤ u ) , ∀ ( i , j ) ∈ { 1 , 2 , . . . , d } 2 . For the J indicator, this condition can be written : P ( X i > u i , S ≥ u ) = P ( X j > u j , S ≥ u ) , ∀ ( i , j ) ∈ { 1 , 2 , . . . , d } 2 .

Introduction Optimal allocation Coherence Coherence properties Other desirable properties Discussion Coherence of the optimal allocation Conclusion Bibliography Coherence properties 3 Coherence Other desirable properties Coherence of the optimal allocation Khalil Said - Les Houches, April 19, 2016 Colloque Jeunes Probabilistes et Statisticiens 13 / 20

Introduction Optimal allocation Coherence Coherence properties Other desirable properties Discussion Coherence of the optimal allocation Conclusion Bibliography Coherence Following Artzner et al. (1999) [Artzner et al., 1999] and Denault (2001)[Denault, 2001] we reformulate coherence axioms in a more general multivariate context. Definition : Coherent allocation A capital allocation ( u 1 , . . . , u d ) = A X 1 ,..., X d ( u ) of an initial capital u ∈ R + is coherent if it satisfies the following properties : 1. Full allocation : d � u i = u . i = 1 2. Riskless allocation : For a deterministic risk X = c , where the constant c ∈ R + : A X , X 1 ,..., X d ( u ) = ( c , A X 1 ,..., X d ( u − c )) . Khalil Said - Les Houches, April 19, 2016 Colloque Jeunes Probabilistes et Statisticiens 14 / 20

Definition : Coherent allocation 3. Symmetry : if ( X 1 , . . . , X i − 1 , X i , X i + 1 , . . . , X j − 1 , X j , X j + 1 , . . . , X d ) L = ( X 1 , . . . , X i − 1 , X j , X i + 1 , . . . , X j − 1 , X i , X j + 1 , . . . , X d ) , then u i = u j . 4. Sub-additivity : ∀ M ⊆ { 1 , . . . , d } , let ( u ∗ , u ∗ 1 , . . . , u ∗ r ) = A � i ∈ M X i , X j ∈{ 1 ,..., d }\ M ( u ) , where r = d − card ( M ) and ( u 1 , . . . , u d ) = A X 1 ,..., X d ( u ) : u ∗ ≤ � u i . i ∈ M 5. Comonotonic additivity : For r � d comonotonic risks, � A X ii ∈{ 1 ,..., d }\ CR , � k ∈ CR X k ( u ) = ( u i i ∈{ 1 ,..., d }\ CR , u k ) , k ∈ CR where CR denotes the set of the r comonotonic risk indexes.

Desirable properties Positive homogeneity An allocation is positively homogeneous, if for any α ∈ R + , it satisfies : A α X 1 ,...,α X d ( α u ) = α A X 1 ,..., X d ( u ) . Translation invariance An allocation is invariant by translation, if for all ( a 1 , . . . , a d ) ∈ R d such that u > � d k = 1 a k , it satisfies : � d � � A X 1 + a 1 ,..., X d + a d ( u ) = A X 1 ,..., X d u − + ( a 1 , . . . , a d ) . a k k = 1 Continuity An allocation is continuous, if for all i ∈ { 1 , . . . , d } : ǫ → 0 A X 1 ,..., ( 1 + ǫ ) X i ,..., X d ( u ) = A X 1 ,..., X i ,..., X d ( u ) . lim