Mean-variance Formulation Research Articles

Production with Risk Hedging Optimal Policy and Efficient Frontier

Demand for many products may depend on the price of a tradable asset or on the economy in general. For instance, demand for equipment that plants or harvests corns correlates with the fluctuations of the corn price on the commodity market; and discount stores experienced increased sales revenue during the last recession. Thus, we model demand as a stochastic process with two components: in addition to the usual Gaussian component due to forecast errors, there is a drift component taking the form of a function of a tradable asset price. (In the case of dependence on the general economy, the asset price can be a broad market index, such as the S&P500 index.) With this demand model, we study the one-time production quantity decision along with a real-time risk-hedging strategy over a given planning horizon (the production cycle). Pursuing a mean-variance formulation, we derive the optimal solution to both production and hedging decisions. In addition, we give a complete characterization of the efficient frontier, and quantify the improvement in risk-return tradeoff achieved by the hedging strategy.

Asset Allocation: Analysis of Theory and Practice in the Australian Investment Management Industry

Asset allocation is the decision on how much of the investment portfolio to place in each of the broad asset classes (e.g. cash, fixed interest securities, property, equities). It is a key decision area in the investment management industry, where professional investors manage pooled investments. The present research examined any dichotomy between theory and practice of asset allocation in the Australian investment management industry. It surveyed the available body of research on asset allocation consisting of Modern Portfolio Theory from the seminal Markowitz mean-variance formulation to subsequent research strands. It utilised a combination of qualitative and quantitative methods to examine the level of awareness and usage of asset allocation theories and theory-based methods among investment management industry practitioners. The present research established that while there is a high level of awareness, there is a low level of usage of asset allocation theory and theory-based methods in the industry. Structural Equation Modelling identified perceived usefulness as having the strongest influence on usage. Recommendations mostly revolving around improving the interaction between academe and industry are made that may help reduce the theory-practice dichotomy. Limitations of the present research are discussed as well as future areas for related research.

Mean-variance type controls involving a hidden Markov chain: models and numerical approximation

Motivated by applications arising in networked systems, this work examines controlled regime-switching systems that stem from a mean-variance formulation. A main point is that the switching process is a hidden Markov chain. An additional piece of information, namely, a noisy observation of switching process corrupted by white noise is available. We focus on minimizing the variance subject to a fixed terminal expectation. Using the Wonham filter, we convert the partially observed system to a completely observable one first. Since closed-form solutions are virtually impossible be obtained, a Markov chain approximation method is used to devise a computational scheme. Convergence of the algorithm is obtained. A numerical example is provided to demonstrate the results.

Use of DEA cross-efficiency evaluation in portfolio selection: An application to Korean stock market

We propose a way of using DEA cross-efficiency evaluation in portfolio selection. While cross efficiency is an approach developed for peer evaluation, we improve its use in portfolio selection. In addition to (average) cross-efficiency scores, we suggest to examine the variations of cross-efficiencies, and to incorporate two statistics of cross-efficiencies into the mean-variance formulation of portfolio selection. Two benefits are attained by our proposed approach. One is selection of portfolios well-diversified in terms of their performance on multiple evaluation criteria, and the other is alleviation of the so-called “ganging together” phenomenon of DEA cross-efficiency evaluation in portfolio selection. We apply the proposed approach to stock portfolio selection in the Korean stock market, and demonstrate that the proposed approach can be a promising tool for stock portfolio selection by showing that the selected portfolio yields higher risk-adjusted returns than other benchmark portfolios for a 9-year sample period from 2002 to 2011.

Portfolio optimization with uncertain exit time in infinite-time horizon

In this paper, we study infinite-period mean-variance formulations for portfolio selections with an uncertain exit time. We employ the convergence control method together with the dynamic programming algorithm to derive analytical expressions for the optimal portfolio policy and the mean-variance efficient frontier under certain conditions. We illustrate these results by an numerical example.

Supply chain risk analysis with mean-variance models: a technical review

Pioneered by Nobel laureate Harry Markowitz in the 1950s, the mean-variance (MV) formulation is a fundamental theory for risk management in finance. Over the past decades, there is a growing popularity of applying this ground breaking theory in analyzing stochastic supply chain management problems. Nowadays, there is no doubt that the mean-variance (MV) theory is a well-proven approach for conducting risk analysis in stochastic supply chain operational models. In view of the growing importance of MV approach in supply chain management, we review a selection of related papers in the literature that focus on MV analytical models. By classifying the literature into three major areas, namely, single-echelon problems, multi-echelon supply chain problems, and supply chain problems with information updating, we derive insights into the current state of knowledge in each area and identify some associated challenges with a discussion of some specific models. We also suggest future research directions on topics such as information asymmetry, supply networks, and boundedly rational agents, etc. In conclusion, this paper provides up-to-date information which helps both academicians and practitioners to better understand the development of MV models for supply chain risk analysis.

Optimal multi-period mean–variance policy under no-shorting constraint

We consider in this paper the mean–variance formulation in multi-period portfolio selection under no-shorting constraint. Recognizing the structure of a piecewise quadratic value function, we prove that the optimal portfolio policy is piecewise linear with respect to the current wealth level, and derive the semi-analytical expression of the piecewise quadratic value function. One prominent feature of our findings is the identification of a deterministic time-varying threshold for the wealth process and its implications for market settings. We also generalize our results in the mean–variance formulation to utility maximization with no-shorting constraint.

Composition of robust equity portfolios

Robust portfolios resolve the sensitivity issue identified as a concern in implementing mean–variance analysis. Because robust approaches are not widely used in practice due to a limited understanding regarding the portfolios constructed from these methods, we present an analysis of the composition of robust equity portfolios. We find that compared to the Markowitz mean–variance formulation, robust optimization formulations form portfolios that contain a fewer number of stocks, avoid large exposure to individual stocks, have higher portfolio beta, and show low correlation between weight and beta of the stocks composing the portfolio. These properties are also found for global minimum-variance portfolios.

Optimal investment policy in the time consistent mean–variance formulation

As a necessary requirement for multi-period risk measure, time consistency can be examined from two aspects: dynamic risk measure and optimal investment policy. In this paper, we first study the relationship between the time consistency of dynamic risk measure and the time consistency of optimal investment policy and obtain the following conclusions: if the dynamic risk mapping is time consistent and monotone, then the corresponding optimal investment policy satisfies the time consistency requirements; however, if the dynamic risk mapping is time consistent but not monotone, then the time consistency requirements of an optimal investment policy will no longer be satisfied. Since the variance operator does not satisfy the smoothing property, the optimal investment policy derived from the existing multi-period mean–variance model is not time consistent. To overcome this shortcoming, we propose the notation of a separable expected conditional mapping and then construct a time consistent dynamic mean–variance model. We prove that the optimal investment policy derived from our model is time consistent. Moreover, for two cases with or without a riskless asset, we obtain the time consistent analytical optimal investment policy and the mean–variance efficient frontier of the new model with the self-financing constraint. Finally, numerical results illustrate the flexibility and superiority of our multi-period mean–variance model and the optimal investment policy over those in the literature.

Optimal Pricing, Modularity, and Return Policy Under Mass Customization

Mass customization (MC) is a pertinent industrial practice. Different from the non-MC products, consumer return for MC products is typically prohibited. MC retailers can thus gain significant competitive advantages by offering a consumer return policy. By constructing an analytical model with both demand and return uncertainties, we study in this paper the optimal policy with three dimensional decisions on pricing, consumer return, and level of modularity under a mean-variance formulation. Structural properties of the model are revealed, and the closed-form solutions for the optimal decisions are derived. An extensive sensitivity analysis is subsequently conducted to explore how the risk sensitivity, demand uncertainty, return uncertainty, and other important parameters affect the optimal decisions and profits. A few counterintuitive findings are obtained, and important insights are generated.

Market Timing Strategy in Dynamic Portfolio Selection: A Mean-Variance Formulation

Investment could be costly for several reasons. The most significant contributor, undoubtedly, goes to bad market timing. Investors thus have to consider market timing strategies, i.e., to strategically shift the funds completely between risky and risk free assets after analyzing market conditions and carrying out predictions. Market friction, such as management fees charged by fund managers, makes market timing more necessary as cost and benefit of investment may change the dominance relationship during the investment process. Thus, smart investors do not always invest their wealth in the market and they do so only in these time periods with sufficiently good investment opportunities (market condition). Motivated by such common phenomena, this paper investigates market timing strategies for a market with correlated returns and management fees under a dynamic mean-variance framework. More specifically, for a finite time horizon with correlated returns and a form of management fee which is a concave increasing function of both the investment time periods and the initial wealth, we offer an analytical solution to help investors to decide best time cardinality (the total number of time periods to invest), to identify market timing (when to stay in the market) and to decide the optimal investment policy when investing. Our results reveal that the optimal market timing strategy (whether investing in risky assets or not in a given time period) depends entirely on the realization of an adaptive process, which is dictated by the modified first and second order conditional moments of the excess returns of the risky assets.

Optimal Multiperiod Mean-Variance Policy Under No-Shorting Constraint

We consider in this paper the mean-variance formulation in multi-period portfolio selection under no-shorting constraint. Recognizing the structure of a piecewise quadratic value function, we prove that the optimal portfolio policy is piecewise linear with respect to the current wealth level, and derive the semi-analytical expression of the piecewise quadratic value function. One prominent feature of our findings is the identification of a deterministic time-varying threshold for the wealth process and its implications for market settings. We also generalize our results in the mean-variance formulation to utility maximization under no-shorting constraint.

Hybrid Approaches and Dimensionality Reduction for Portfolio Selection with Cardinality Constraints

A novel memetic algorithm that combines evolutionary algorithms, quadratic programming, and specially devised pruning heuristics is proposed for the selection of cardinality-constrained optimal portfolios. The framework used is the standard Markowitz mean-variance formulation for portfolio optimization with constraints of practical interest, such as minimum and maximum investments per asset and/or on groups of assets. Imposing limits on the number of different assets that can be included in the investment transforms portfolio selection into an NP-complete mixed-integer quadratic optimization problem that is difficult to solve by standard methods. An implementation of the algorithm that employs a genetic algorithm with a set representation, an appropriately defined mutation operator and Random Assortment Recombination for crossover (RAR-GA) is compared with implementations using Simulated Annealing (SA) and various Estimation of Distribution Algorithms (EDAs). An empirical investigation of the performance of the portfolios selected with these different methods using financial data shows that RAR-GA and SA are superior to the implementations with EDAs in terms of both accuracy and efficiency. The use of pruning heuristics that effectively reduce the dimensionality of the problem by identifying and eliminating from the universe of investment assets that are not expected to appear in the optimal portfolio leads to significant improvements in performance and makes EDAs competitive with RAR-GA and SA.

Equity home bias in Australian superannuation funds

Equity home bias for Australia superannuation funds is examined under a model that reflects observed decision processes. The mix of Australian and international equities is evaluated as a two-asset choice under the influence of legacy, an objective function that trades off expected returns against portfolio risk and peer risk, and under expectations that are formed adaptively and allow for taxation differences. The model closely replicates the observed equity mix, particularly relative to more traditional mean-variance formulations. The main implication is that home bias may be better explained under models that reflect industry practices and allow for various commingled influences. JEL classifications G11, G23

A generalized multi-period mean–variance portfolio optimization with Markov switching parameters

In this paper, we deal with a generalized multi-period mean–variance portfolio selection problem with market parameters subject to Markov random regime switchings. Problems of this kind have been recently considered in the literature for control over bankruptcy, for cases in which there are no jumps in market parameters (see [Zhu, S. S., Li, D., & Wang, S. Y. (2004). Risk control over bankruptcy in dynamic portfolio selection: A generalized mean variance formulation. IEEE Transactions on Automatic Control, 49, 447–457]). We present necessary and sufficient conditions for obtaining an optimal control policy for this Markovian generalized multi-period mean–variance problem, based on a set of interconnected Riccati difference equations, and on a set of other recursive equations. Some closed formulas are also derived for two special cases, extending some previous results in the literature. We apply the results to a numerical example with real data for risk control over bankruptcy in a dynamic portfolio selection problem with Markov jumps selection problem.

Multi-period mean-variance portfolio optimization with markov switching parameters

In this paper we deal with a multi-period mean-variance portfolio selection problem with the market parameters subject to Markov random regime switching. We analytically derive an optimal control policy for this mean-variance formulation in a closed form. Such a policy is obtained from a set of interconnected Riccati difference equations. Additionally, an explicit expression for the efficient frontier corresponding to this control law is identified and numerical examples are presented.

Multi-period portfolio selection for asset-liability management with uncertain investment horizon

It is often the case that some unexpected event may force an investor to terminate her investment and leave the market. We consider in this paper the mean-variance formulation of multi-period portfolio optimization for asset-liability management with an uncertain investment horizon. Under the assumption that exit time follows a given distribution, the problem under investigation with uncertain investment horizon can be translated into one with deterministic exit time. By making use of the embedding technique of Li and Ng (2000), we derive an analytical optimal strategy and an analytical expression of the mean-variance efficient frontier for the mean-variance formulation of the problem.

Generalized Mean-Variance Portfolio Selection Model with Regime Switching

In this paper we deal with a generalized multi-period mean-variance portfolio selection problem with the market parameters subject to Markov random regime switchings. We present necessary and sufficient conditions for obtaining an optimal control policy for this Markovian generalized multi-period mean-variance model, based on a recursive procedure. The analytical solution of our model provides the base for the solution of a great variety of mean-variance formulations.

Optimality of Financial Planning Clients' Strategic Asset Allocations

In this paper, the optimality of Australian financial planning clients’ asset allocations are analysed using the mean-variance formulation of the Modern Portfolio Theory. The asset allocations recommended by financial planning groups are examined. The mean-variance characteristics of the various asset classes are derived from historical indices, using last 21 years data and last 5 years data. The return-risk values of the recommended portfolios are determined and a simple method of iso-risk maximum return calculation using the Excel Solver command is utilised to determine the corresponding optimal portfolios. The recommended portfolios were found to have expected returns that are around 8% and 32% below optimal returns based on last 21 years data and last 5 years data, respectively.

Risk Mitigation in Newsvendor Networks: Resource Diversification, Flexibility, Sharing, and Hedging

This paper studies how judicious resource allocation in networks mitigates risk. Theory is presented for general utility functions and mean-variance formulations and is illustrated with networks featuring resource diversification, flexibility (e.g., inventory substitution), and sharing (commonality). In contrast to single-resource settings, risk-averse newsvendors may invest more in networks than risk-neutral newsvendors: some resources and even total spending may exceed risk-neutral levels. With normally distributed demand, risk-averse newsvendors change resource levels roughly proportionally to demand variance, while risk-neutral agents adjust only proportionally to standard deviation. Two effects explain this operational hedge and suggest rules of thumb for strategic placement of safety capacity and inventory in networks: (1) Risk pooling suggests rebalancing capacity toward inexpensive resources that serve lower-profit variance markets. This highlights the role of profit variance (instead of demand variance) in risk-averse network investment. (2) Ex post revenue maximization suggests rebalancing capacity toward substitutable flexible but away from shared capacity when markets differ in profitability. Capacity imbalance and allocation flexibility thus mitigate profit risk and truly are operational hedges.