Portfolio Optimization in Affine Models with Markov Switching

We consider a stochastic factor financial model where the asset price process and the process for the stochastic factor depend on an observable Markov chain and exhibit an affine structure. We are faced with a finite time investment horizon and derive optimal dynamic investment strategies that maximize the investor's expected utility from terminal wealth. To this aim we apply Merton's approach, as we are dealing with an incomplete market. Based on the semimartingale characterization of Markov chains we first derive the HJB equations, which in our case correspond to a system of coupled non-linear PDEs. Exploiting the affine structure of the model, we derive simple expressions for the solution in the case with no leverage, i.e. no correlation between the Brownian motions driving the asset price and the stochastic factor. In the presence of leverage we propose a separable ansatz, which leads to explicit solutions in this case as well. General verification results are also proved. The results are illustrated for the special case of a Markov modulated Heston model

A review of portfolio planning: Models and systems

In this chapter, we first provide an overview of a number of portfolio planning models which have been proposed and investigated over the last forty years. We revisit the mean-variance (M-V) model of Markowitz and the construction of the risk-return efficient frontier. A piecewise linear approximation of the problem through a reformulation involving diagonalisation of the quadratic form into a variable separable function is also considered. A few other models, such as, the Mean Absolute Deviation (MAD), the Weighted Goal Programming (WGP) and the Minimax (MM) model which use alternative metrics for risk are also introduced, compared and contrasted. Recently asymmetric measures of risk have gained in importance; we consider a generic representation and a number of alternative symmetric and asymmetric measures of risk which find use in the evaluation of portfolios. There are a number of modelling and computational considerations which have been introduced into practical portfolio planning problems. These include: (a) buy-in thresholds for assets, (b) restriction on the number of assets (cardinality constraints), (c) transaction roundlot restrictions. Practical portfolio models may also include (d) dedication of cashflow streams, and, (e) immunization which involves duration matching and convexity constraints. The modelling issues in respect of these features are discussed. Many of these features lead to discrete restrictions involving zero-one and general integer variables which make the resulting model a quadratic mixed-integer programming model (QMIP). The QMIP is a NP-hard problem; the algorithms and solution methods for this class of problems are also discussed. The issues of preparing the analytic data (financial datamarts) for this family of portfolio planning problems are examined. We finally present computational results which provide some indication of the state-of-the-art in the solution of portfolio optimisation problems

Mean-Variance Policy for Discrete-time Cone Constrained Markets: The Consistency in Efficiency and Minimum-Variance Signed Supermartingale Measure

The discrete-time mean-variance portfolio selection formulation, a representative of general dynamic mean-risk portfolio selection problems, does not satisfy time consistency in efficiency (TCIE) in general, i.e., a truncated pre-committed efficient policy may become inefficient when considering the corresponding truncated problem, thus stimulating investors' irrational investment behavior. We investigate analytically effects of portfolio constraints on time consistency of efficiency for convex cone constrained markets. More specifically, we derive the semi-analytical expressions for the pre-committed efficient mean-variance policy and the minimum-variance signed supermartingale measure (VSSM) and reveal their close relationship. Our analysis shows that the pre-committed discrete-time efficient mean-variance policy satisfies TCIE if and only if the conditional expectation of VSSM's density (with respect to the original probability measure) is nonnegative, or once the conditional expectation becomes negative, it remains at the same negative value until the terminal time. Our findings indicate that the property of time consistency in efficiency only depends on the basic market setting, including portfolio constraints, and this fact motivates us to establish a general solution framework in constructing TCIE dynamic portfolio selection problem formulations by introducing suitable portfolio constraints

Portfolio Selection with Narrow Framing: Probability Weighting Matters

This paper extends the model with narrow framing suggested by Barberis and Huang (2009) to also account for probability weighting and a convex-concave value function in the specification of cumulative prospect theory preferences on narrowly framed assets. We show that probability weighting is needed in order that investors reduce their holding of narrowly framed risky assets in the presence of negative skewness and high Sharpe ratios, which are typical characteristics of stock index returns. The model with framing and probability weighting can thus explain the stock participation puzzle under realistic assumptions on stock market returns. We also show that a convex-concave value function generates wealth effects that are consistent with empirical observations on stock market participation. Finally, we address the asset pricing implications of probability weighting in the model with narrow framing and show that in the case of negative skewness the equity premium of narrowly framed assets is much higher than when probability weighting is not taken into account.Narrow framing, cumulative prospect theory, probability weighting function,negative skewness, simulation methods

Extending the MAD Portfolio Optimization Model to Incorporate Downside Risk Aversion

The mathematical model of portfolio optimization is usually expected as a bicriteria optimization problem where a reasonable trade-off between expected rate of return risk is sought. In a classical Markowitz model the risk is measured by a variance, thus resulting in a quadratic programming model. As an alternative, the MAD model was proposed where risk is measured by (mean) absolute deviation instead of a variance. The MAD model is computationally attractive, since it is transformed into an easy to solve linear programming program. In this paper we present an extension to the MAD model allowing to account for downside risk aversion of an investor, and at the same time preserving simplicity and linearity of the original MAD model

Optimization in a Simulation Setting: Use of Function Approximation in Debt Strategy Analysis

The stochastic simulation model suggested by Bolder (2003) for the analysis of the federal government's debt-management strategy provides a wide variety of useful information. It does not, however, assist in determining an optimal debt-management strategy for the government in its current form. Including optimization in the debt-strategy model would be useful, since it could substantially broaden the range of policy questions that can be addressed. Finding such an optimal strategy is nonetheless complicated by two challenges. First, performing optimization with traditional techniques in a simulation setting is computationally intractable. Second, it is necessary to define precisely what one means by an "optimal" debt strategy. The authors detail a possible approach for addressing these two challenges. They address the first challenge by approximating the numerically computed objective function using a function-approximation technique. They consider the use of ordinary least squares, kernel regression, multivariate adaptive regression splines, and projection-pursuit regressions as approximation algorithms. The second challenge is addressed by proposing a wide range of possible government objective functions and examining them in the context of an illustrative example. The authors' view is that the approach permits debt and fiscal managers to address a number of policy questions that could not be fully addressed with the current stochastic simulation engine.Debt management; Econometric and statistical methods; Fiscal policy; Financial markets