# Forward and inverse portfolio problems with a range of risk measures

Maximizing investment utilities and modelling investors’ risk preferences are central problems in various applications of financial risk management including investment science and portfolio theory in particular. In general, investors need to solve forward portfolio optimization problems to maximize investment utilities and solve inverse portfolio optimization problems to identify risk preferences. A forward portfolio optimization problem requests to minimize the risk of a portfolio subject to a budget constraint to obtain optimal trading strategies in terms of investment weights among financial assets. In fact, the forward portfolio optimization problem can be solved with a range of risk measures, such as CVaR and drawdown measures. In this thesis, the benchmark-based CVaR-deviation and the benchmark-based drawdown as new possible risk measures are firstly introduced and the forward portfolio optimization problem is also solved with both new risk measures. Additionally, deviation measures are also possible risk measures, which is a family of risk measures containing MAD and CVaR-deviation as special cases. Traditionally, as a drawback, the forward portfolio optimization problems with deviation measures can only be solved separately by these special cases. Thus, to eliminate the drawback, this thesis develops a new general framework to deal with the forward problem based on the common features of the deviation measure family. As a result, a general linear program for the minimization of general deviation measures is derived, subject to a linear constraint, and minimization of an arbitrary positively homogeneous convex functional, whose dual set is given by linear inequalities, possibly involving auxiliary variables. This allows to reduce to linear programming for individual and cooperative portfolio optimization problems with arbitrary deviation measures whose risk envelopes are given by a finite number of linear constraints. In contrast, an inverse portfolio optimization problem requests to determine the investor’s risk preference that can be represented as a certain class of risk measures which makes the given portfolio optimal. Usually, the inverse portfolio optimization problems are static. In this thesis, a dynamic inverse problem called robust inverse portfolio problem is firstly introduced. The robust inverse is solved with the mixed CVaR measure, whose result is relatively stable to represent the risk over shift-time periods. In addition, the inverse portfolio optimization problem with drawdown measure is also firstly addressed in this thesis. All these problems are demonstrated with examples.

## History

## Supervisor(s)

Bogdan Grechuk## Date of award

2022-11-27## Author affiliation

School of Computing and Mathematical Sciences## Awarding institution

University of Leicester## Qualification level

- Doctoral

## Qualification name

- PhD