Non-concave Utility Portfolio Optimization And Robust Risk Aggregation

Portfolio selection and risk management are two key issues in mathematical finance and actuarial science.In the area of portfolio selection,several recent developments,including the application of behavioral finance,participating life insurance contracts,principal-agent conflicts and incentive schemes in hedge funds,lead to a general(nonconcave)utility of the decision maker and hence various obstacles in solving optimal wealth and portfolio.In the area of risk management,the marginal distributions of all assets are usually available,while the dependence structure among them is difficult to obtain and calculate.This thesis is dedicated to addressing four relevant topics in nonconcave utility portfolio optimization and robust risk aggregation.We consider a weighted S-shaped utility optimization problem of the insurance company and the policyholder with a participating contract in pension fund management.We establish two categories of concave envelopes to solve the highly non-concave optimization problem and obtain the optimal wealth.In addition,we propose a numerical integration by substitution technique to deal with the corresponding implicit integration problem.Numerically,we compare our results with the traditional objective and show that the utilities of two counterparts can be simultaneously improved by switching into the weighted objective and appropriately amending the contract.We study a principal's constraint problem in the principal-agent model of two general S-shaped utilities without explicit expressions.This problem is essentially a complicated double S-shaped utility optimization problem.We propose a novel classification approach by asymptotic analysis.First,we classify the optimal wealth into “One-side-loss Case”and “Option Case”.More importantly,we find a division reservation utility such that the two cases can be separated.Hence,one can predetermine the feature of the optimal wealth directly by the size of the reservation utility.We illustrate in application to asset management that the key factor resulting in different profit-sharing and risk-taking choices is the size of the reservation utility.We formulate a central-planned portfolio selection problem in hedge fund management.We obtain the closed-form Pareto optimal portfolio,divide the optimal portfolio into three terms(Merton term,Aggressive term and Conservative term)and propose an asymptotic analysis approach.Next,we prove that the collection of Pareto points of a single contract is a strictly decreasing and strictly concave frontier.Furthermore,we discover a way of Pareto improvement by simultaneously adding the investor's utility into the manager's objective and increasing the manager's incentive rate.We investigate the problem of quantile aggregation with dependence uncertainty.It is a long-existing problem(Fréchet problem)in probability theory and does not admit an analytical expression for general cases.Using inf-convolution inequalities of quantilebased risk measures,we establish new analytical bounds for quantile aggregation which we call convolution bounds.We prove that convolution bounds are sharp in many relevant cases.In fact,the convolution bound provides a unified analytical approximation formula for this fundamental problem,and is genuinely the best one available.Convolution bounds enjoy other advantages,including derivation for the distribution of the sum variable,interpretability on the extremal dependence structure,analytical tractability and computational convenience.As a fundamental finding,the convolution bound has wide applications in other fields.We illustrate a few applications in operations research(assembly line crew scheduling).