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postgraduate thesis: Applications of comonotonicity in risksharing and optimal allocation
Title  Applications of comonotonicity in risksharing and optimal allocation 

Authors  
Advisors  Advisor(s):Cheung, KC 
Issue Date  2014 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Rong, Y. [戎軼安]. (2014). Applications of comonotonicity in risksharing and optimal allocation. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5334876 
Abstract  Over the past decades, researchers in economics, financial mathematics and actuarial science have introduced results to the concept of comonotonicity in their respective fields of interest. Comonotonicity is a very strong dependence structure and is very often mistaken as a dependence structure that is too extreme and unrealistic. However, the concept of comonotonicity is actually a useful tool for solving several research and practical problems in capital allocation, risk sharing and optimal allocation.
The first topic of this thesis is focused on the application of comonotonicity in optimal capital allocation. The Enterprise Risk Management process of a financial institution usually contains a procedure to allocate the total risk capital of the company into its different business units. Dhaene et al. (2012) proposed a unifying capital allocation framework by considering some general deviation measures. This general framework is extended to a more general optimization problem of minimizing separable convex function with a linear constraint and box constraints. A new approach of solving this constrained minimization problem explicitly by the concept of comonotonicity is developed. Instead of the traditional KuhnTucker theory, a method of expressing each convex function as the expected stoploss of some suitable random variable is used to solve the optimization problem. Then, some results in convex analysis with infimumconvolution are derived using the result of this new approach.
Next, Borch's theorem is revisited from the perspective of comonotonicity. The optimal solution to the Pareto optimal risksharing problem can be obtained by the Lagrangian method or variational arguments. Here, I propose a new method, which is based on a BreedenLitzanbeger type integral representation formula for increasing convex functions. It enables the transform of the objective function into a sum of mixtures of stoplosses. Necessary conditions for the existence of optimal solution are then discussed. The explicit solution obtained allows us to show that the risksharing problem is indeed a “pointwise” problem, and hence the value function can be obtained immediately using the notion of supremumconvolution in convex analysis.
In addition to the above classical risksharing and capital allocation problems, the problem of minimizing a separable convex objective subject to an ordering restriction is then studied. Best et al. (2000) proposed a pool adjacent violators algorithm to compute the optimal solution. Instead, we show that using the concept of comonotonicity and the technique of dynamic programming the solution can be derived in a recursive manner. By identifying the righthand derivative of the convex functions with distribution functions of some suitable random variables, we rewrite the objective function into a sum of expected deviations. This transformation and the fact that the expected deviation is a convex function enable us to solve the minimizing problem. 
Degree  Doctor of Philosophy 
Subject  Investments  Mathematical models Risk management  Mathematical models 
Dept/Program  Statistics and Actuarial Science 
Persistent Identifier  http://hdl.handle.net/10722/207205 
HKU Library Item ID  b5334876 
DC Field  Value  Language 

dc.contributor.advisor  Cheung, KC   
dc.contributor.author  Rong, Yian   
dc.contributor.author  戎軼安   
dc.date.accessioned  20141218T23:17:55Z   
dc.date.available  20141218T23:17:55Z   
dc.date.issued  2014   
dc.identifier.citation  Rong, Y. [戎軼安]. (2014). Applications of comonotonicity in risksharing and optimal allocation. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5334876   
dc.identifier.uri  http://hdl.handle.net/10722/207205   
dc.description.abstract  Over the past decades, researchers in economics, financial mathematics and actuarial science have introduced results to the concept of comonotonicity in their respective fields of interest. Comonotonicity is a very strong dependence structure and is very often mistaken as a dependence structure that is too extreme and unrealistic. However, the concept of comonotonicity is actually a useful tool for solving several research and practical problems in capital allocation, risk sharing and optimal allocation. The first topic of this thesis is focused on the application of comonotonicity in optimal capital allocation. The Enterprise Risk Management process of a financial institution usually contains a procedure to allocate the total risk capital of the company into its different business units. Dhaene et al. (2012) proposed a unifying capital allocation framework by considering some general deviation measures. This general framework is extended to a more general optimization problem of minimizing separable convex function with a linear constraint and box constraints. A new approach of solving this constrained minimization problem explicitly by the concept of comonotonicity is developed. Instead of the traditional KuhnTucker theory, a method of expressing each convex function as the expected stoploss of some suitable random variable is used to solve the optimization problem. Then, some results in convex analysis with infimumconvolution are derived using the result of this new approach. Next, Borch's theorem is revisited from the perspective of comonotonicity. The optimal solution to the Pareto optimal risksharing problem can be obtained by the Lagrangian method or variational arguments. Here, I propose a new method, which is based on a BreedenLitzanbeger type integral representation formula for increasing convex functions. It enables the transform of the objective function into a sum of mixtures of stoplosses. Necessary conditions for the existence of optimal solution are then discussed. The explicit solution obtained allows us to show that the risksharing problem is indeed a “pointwise” problem, and hence the value function can be obtained immediately using the notion of supremumconvolution in convex analysis. In addition to the above classical risksharing and capital allocation problems, the problem of minimizing a separable convex objective subject to an ordering restriction is then studied. Best et al. (2000) proposed a pool adjacent violators algorithm to compute the optimal solution. Instead, we show that using the concept of comonotonicity and the technique of dynamic programming the solution can be derived in a recursive manner. By identifying the righthand derivative of the convex functions with distribution functions of some suitable random variables, we rewrite the objective function into a sum of expected deviations. This transformation and the fact that the expected deviation is a convex function enable us to solve the minimizing problem.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  This work is licensed under a Creative Commons AttributionNonCommercialNoDerivatives 4.0 International License.   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.subject.lcsh  Investments  Mathematical models   
dc.subject.lcsh  Risk management  Mathematical models   
dc.title  Applications of comonotonicity in risksharing and optimal allocation   
dc.type  PG_Thesis   
dc.identifier.hkul  b5334876   
dc.description.thesisname  Doctor of Philosophy   
dc.description.thesislevel  Doctoral   
dc.description.thesisdiscipline  Statistics and Actuarial Science   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b5334876   