Stable Portfolio Design Using Bayesian Change Point Models and Geometric Shape Factors

Abstract

The goal of this thesis is to design stable portfolios that protect an investor from severe drawdowns in financial markets as they have occurred during market crises in the last decades. For that two new mathematical concepts are introduced. First, a new approach for the Bayesian change point (BCP) algorithm is presented that does not only react on changes within the trend but also the variance of the data under consideration. For the prior distributions the data input is assumed to follow a normal distribution and the mean of the data is assumed to follow a normal distribution as well. In comparison to already existing implementations the variance is modelled using a generalized inverse Gaussian (GIG) distribution (the N-NGIG model). The posterior distributions are derived analytically and combined with the product partition model (PPM) to infer on the unknown mean and variance by taking into consideration possible changes within the underlying dynamics. Second, geometric shape factors (GSF) as a novel approach on examining the state of a multivariate investment universe are presented. They are obtained by describing the investment universe as an image of the feasible set. The GSF are defined to be the area, centre, orientation and eccentricity of that image. This provides us with information about the optimization benefit (area), the expected performance per unit of risk (centre), the direction of the risk premium (orientation) and the correlation between risk and performance (eccentricity) of the investment universe. As such the GSF are readily understandable and useable in a portfolio optimization context. Additionally, it is shown how the BCP algorithm can be used to calculate the covariance matrix of the investment universe in order to calculate the GSF such that they do also respect changes within the underlying dynamics. These concepts are the building blocks for the design of the stable portfolios that are presented within the last part of the thesis. For equities the BCP algorithm is used to design a dynamically hedged signal portfolio. For non-equities the orientation of the GSF is used to design a dynamically optimized portfolio that adjusts its objectives based on the direction of the risk premium. Finally, a combination of these two portfolios is presented in order to outline a generic concept to manage real-world investment portfolios.