On a Portfolio Optimization Problem under GARCH Type Models and Solvency Constraints

This thesis discusses the portfolio optimization problem under solvency constraints, based on S. Asanga et al. ”Portfolio optimization under solvency constraints: a dynamical approach”[1]. Models of the gross return process and claim liability, and methods to solve the optimization problems are both borrowed from the paper. Under the background that insurance companies make profits mainly by investing the premiums and capital from shareholders on portfolio products, this thesis models the net loss of a company by the difference between gross return process of the investment and claim payments. The data used in this thesis are with the same period as the paper but downloaded from Yahoo, so there are subtle differences. The goal is to minimize the capital from shareholders under certain constraints built on three risk measures: ruin probability, conditional Value-at-Risk and expected policyholder deficit. We use univariate GARCH(1,1), Constant Conditional Correlation(CCC)-GARCH and Dynamic Conditional Correlation(DCC)-GARCH to model the gross return process considering volatility clustering and the covariance matrix of multiple assets in a portfolio. The claim’s payments are modeled by a lognormal distribution to guarantee the convexity of the optimization problems. The approach to solving the constrained optimization problems is nonlinear interior point method and analyzing efficient frontier. This thesis seeks to solve it using r, by trying package ’nloptr’ for constrained optimization, unfortunately, some functions have some bugs and others do not get a good result