Dynamic Portfolio Selection by Augmenting the Asset Space

1995

Dynamic programming solutions for optimal portfolios in which the solution for the portfolio vector of risky assets is constant were solved by Merton in continuous time and by Hakansson and others in discrete time. There is no case with a closed form solution where this vector of risky asset holdings changes dynamically. This paper derives such solutions for the first time, and is thus a dynamic dynamic-programming solution as opposed to a static dynamic-programming solution for this vector. The solution is valid when there is a set of basis assets whose excess expected return is linear in the state vector, whose variance-covariance matrix is time-dependent and for which the interest rate is a quadratic function of the state vector.

Applied Mathematics and Optimization, 1999

This paper develops a continuous time portfolio optimization model where the mean returns of individual securities or asset categories are explicitly affected by underlying economic factors such as dividend yields, a firm's return on equity, interest rates, and unemployment rates. In particular, the factors are Gaussian processes, and the drift coefficients for the securities are affine functions of these factors. We employ methods of risk-sensitive control theory, thereby using an infinite horizon objective that is natural and features the long run expected growth rate, the asymptotic variance, and a single risk-aversion parameter. Even with constraints on the admissible trading strategies, it is shown that the optimal trading strategy has a simple characterization in terms of the factor levels. For particular factor levels, the optimal trading positions can be obtained as the solution of a quadratic program. The optimal objective value, as a function of the risk-aversion parameter, is shown to be the solution of a partial differential equation. A simple asset allocation example, featuring a Vasicek-type interest rate which affects a stock index and also serves as a second investment opportunity, provides some additional insight about the risk-sensitive criterion in the context of dynamic asset management.

Mathematics and Computers in Simulation, 2009

In mean-variance (M-V) analysis, an investor with a holding period [0,T] operates in a two-dimensional space-one is the mean and the other is the variance. At time 0, he/she evaluates alternative portfolios based on their means and variances, and holds a combination of the market portfolio (e.g., an index fund) and the risk-free asset to maximize his/her expected utility at time T. In our continuous-time model, we operate in a three-dimensional space-the first is the spot rate, the second is the expected return on the risky asset (e.g., an index fund), and the third is time. At various times over [0,T], we determine, for each combination of the spot rate and expected return, the optimum fractions invested in the risky and risk-free assets to maximize our expected utility at time T. Hence, unlike those static M-V models, our dynamic model allows investors to trade at any time in response to changes in the market conditions and the length of their holding period. Our results show that (1) the optimum fraction y*(t) in the risky asset increases as the expected return increases but decreases as the spot rate increases; (2) y*(t) decreases as the holding period shortens; and (3) y*(t) decreases as the risk aversion parameter-γ is larger.

Journal of Economic Dynamics and Control, 2003

This paper is dedicated to the problem of dynamic portfolio optimization for the case when the number of decision periods is large and new information about market arrives during each such period. We propose the family of adaptive portfolio selection policies which rebalance the current portfolio during each decision period by adopting portfolio from a speciÿed family with the best performance on the past data. In the absence of transaction costs the general conditions are found under which this policy yields asymptotically the same performance as the best portfolio from the same family constructed with the full knowledge of the future. These results are extended for the case of nonzero transaction costs by introducing a class of threshold portfolio optimization policies which rebalance current portfolio only when its performance di ers from performance of the best portfolio by a given threshold. The value of this threshold is adapted to the changing market conditions. We show that it is possible to select a sequence of threshold values in such a way that the asymptotic in uence of transaction costs on portfolio performance is negligible and overall portfolio performance is asymptotically the same as the performance of portfolio with the perfect knowledge of the future. We do not assume neither speciÿc probabilistic structure of the market data nor their stationarity. Our theory is illustrated by numerical experiments with real data. Finally, we discuss the relevance of our results in the context of high performance computing.

In real investment situations, one desires to only minimize downside risk or portfolio loss without affecting the upside potentials. This can be accomplished by mean semi-variance optimization but not by mean variance. In the Black-Scholes setting, this paper proposes for the very practical yet intractable dynamic mean semi-variance portfolio optimization problem, an almost analytical solution. It proceeds by reducing the multi-dimensional portfolio selection problem to a one-dimensional optimization problem, which is then expressed in terms of the normal density, leading to a very simple and efficient numerical algorithm. A numerical comparison of the efficient frontier for the mean variance and semi-variance portfolio optimization problem is presented.

2013

In this paper we propose a portfolio choice problem under the hypothesis of Markovian returns. In particular, we assume stable Paretian distributed returns which imports a more flexible environment rather than the traditional Gaussian modeling. Therefore under these assumptions we perform an ex-post analysis to investigate the real benefit of our approach and draw some remarkable conclusions.

Journal of Computational …, 2009

The classical Markowitz approach to portfolio selection is compromised by two major shortcomings. First, there is considerable model risk with respect to the distribution of asset returns. Especially mean returns are notoriously difficult to estimate. Moreover, the Markowitz approach is static in that it does not account for the possibility of portfolio rebalancing within the investment horizon. We propose a robust dynamic portfolio optimization model to overcome both shortcomings. The model arises from an infinite-dimensional min-max framework. The objective is to minimize the worst-case portfolio variance over a family of dynamic investment strategies subject to a return target constraint. The worst-case variance is evaluated with respect to a set of conceivable return distributions. We develop a quantitative approach to approximate this intractable problem by a tractable one and report on numerical experiments.

Frontiers in Applied Mathematics and Statistics

In this article we provide a short survey on continuous-time portfolio selection. We explain the pioneering contribution of Merton and the use of dynamic programming. Then, we discuss Bismut's application of the Pontryagin maximum principle to portfolio selection and the dual martingale approach. We also explain two models with potential applicability to practice: life-cycle models with explicit consideration of retirement and models with protection against decline in spending power.

Mathematical Finance, 2008

This paper formulates and studies a general continuous-time behavioral portfolio selection model under Kahneman and Tversky's (cumulative) prospect theory, featuring S-shaped utility (value) functions and probability distortions. Unlike the conventional expected utility maximization model, such a behavioral model could be easily misformulated (a.k.a. ill-posed) if its different components do not coordinate well with each other. Certain classes of an ill-posed model are identified. A systematic approach, which is fundamentally different from the ones employed for the utility model, is developed to solve a well-posed model, assuming a complete market and general Itô processes for asset prices. The optimal terminal wealth positions, derived in fairly explicit forms, possess surprisingly simple structure reminiscent of a gambling policy betting on a good state of the world while accepting a fixed, known loss in case of a bad one. An example with a two-piece CRRA utility is presented to illustrate the general results obtained, and is solved completely for all admissible parameters. The effect of the behavioral criterion on the risky allocations is finally discussed.

2004

In this paper, we develop a dynamic investment model maximizing the expected “utility” of excess return relative to a benchmark portfolio. Following the standard setting of continuous time framework for the securities market developed by Merton (1971) and others, we obtain an explicit formula for the optimal portfolio policy for a utility function of Hyperbolic Absolute Risk Aversion (HARA). Unlike the constant proportion portfolio strategies, the resulting optimal portfolio is state dependent, namely a function of time, the return on the benchmark portfolio, and the return on the investment portfolio itself. It is shown that the optimal return is equal to the return on the benchmark portfolio plus a value added return related to the growth optimum portfolio on a risk adjusted basis. As a special case, the optimal portfolio is identical to the benchmark portfolio, if the utility function is of Constant Relative Risk Aversion (CRRA).