Mean-variance Formulation Research Articles

A hybrid level-based learning swarm algorithm with mutation operator for solving large-scale cardinality-constrained portfolio optimization problems

We propose a hybrid variant of the level-based learning swarm optimizer (LLSO) for solving large-scale portfolio optimization problems. This solver fills the gap due to the inadequacy of the particle swarm optimization algorithm for high-dimensional instances. We aim to extend the classical mean-variance formulation by maximizing a modified version of the Sharpe ratio subject to cardinality, box, and budget constraints. The algorithm involves a projection operator to deal with these three constraints simultaneously. Further, we implicitly control transaction costs thanks to a rebalancing constraint handled by a suitable exact penalty function. In addition, we develop an ad hoc mutation operator to modify candidate exemplars in the highest level of the swarm. The experimental results, using three large-scale data sets, show that including this procedure improves the accuracy of the solutions. Then, a comparison with other variants of the LLSO algorithm and two state-of-the-art swarm optimizers points out the outstanding performance of the proposed solver in terms of exploration capabilities and solution quality. Finally, we assess the profitability of the portfolio allocation strategy in the last five years using an investable pool of 1119 constituents from the MSCI World Index.

Markowitz Mean-Variance Portfolio Selection and Optimization under a Behavioral Spectacle: New Empirical Evidence

This paper investigates the robustness of the conventional mean-variance (MV) optimization model by making two adjustments within the MV formulation. First, the portfolio selection based on a behavioral decision-making theory that encapsulates the MV statistics and investors psychology. The second aspect involves capturing the portfolio asset dependence structure through copula. Using the behavioral MV (BMV) and the copula behavioral MV (CBMV), the results show that stocks with lower behavioral scores outperform counterpart portfolios with higher behavioral scores. On the other hand, in the Forex market, the reverse is observed for the BMV approach, while the CBMV remains consistent.

Time consistent in efficiency dynamic mean–variance policy

The dynamic mean–variance formulation for a market with all risky assets is not time consistent in efficiency (TCIE), in the sense that the pre-committed policy, which is optimal for the entire investment horizon, could become mean–variance inefficient when facing a truncated time-horizon problem at an intermediate time instant. By removing the self-financing restriction and fully avoiding the irrational investment behaviours in intermediate time periods, we develop the TCIE dynamic mean–variance policy, which is TCIE. Comparing to the polices in the literature, such as the strictly dominating policy, the pre-committed policy and the time consistent policy, the TCIE policy can achieve better global investment performance.

Time-Consistent Investment and Reinsurance Strategies for Insurers Under Multi-Period Mean-Variance Formulation with Generalized Correlated Returns

The existing literature on investment and reinsurance is limited to the study of continuous-time problems, while discrete-time problems are always ignored by researchers. In this study, we first discuss a multi-period investment and reinsurance optimization problem under the classical mean-variance framework. When the asset returns with a serially correlated structure, the time-consistent investment and reinsurance strategies are acquired via backward induction. In addition, we propose an alternative time-consistent mean-variance optimization model that contrasts with the classical mean-variance model, and the corresponding optimal strategy and value function are also derived. We find that the investment and reinsurance strategies are both independent of the current wealth for the above two optimization problems, which coincides with the conclusion presented in the continuous-time problems. Most importantly, the above investment strategies with serially correlated structures are both conditional mean-based strategies, rather than unconditional ones. Finally, we compare the investment and reinsurance strategies suggested above based on the simulation approach, to shed light on which investment-reinsurance strategies are more suitable for insurers.

Multi-period mean-variance portfolio selection with practical constraints using heuristic genetic algorithms

Since Markowitz proposed the mean-variance (MV) formulation in 1952, it has been used to configure various portfolio selection problems. However Markowitz's solution is only for a single period. Multi-period portfolio selection problems have been studied for a long time but most solutions depend on various forms of utility function, which are unfamiliar to general investors. Some works have formulated the problems as MV models and solved them analytically in closed form subject to certain assumptions. Unlike analytical solutions, genetic algorithms (GA) are more flexible because they can solve problems without restrictive assumptions. The purpose of this paper is to formulate multi-period portfolio selection problems as MV models and solve them by GA. To illustrate the generality of our algorithm, we implement a program by Microsoft Visual Studio to solve a multi-period portfolio selection problem for which there exists no general analytical solution.

Multi-period mean-variance portfolio selection with practical constraints using heuristic genetic algorithms

Since Markowitz proposed the mean-variance (MV) formulation in 1952, it has been used to configure various portfolio selection problems. However Markowitz's solution is only for a single period. Multi-period portfolio selection problems have been studied for a long time but most solutions depend on various forms of utility function, which are unfamiliar to general investors. Some works have formulated the problems as MV models and solved them analytically in closed form subject to certain assumptions. Unlike analytical solutions, genetic algorithms (GA) are more flexible because they can solve problems without restrictive assumptions. The purpose of this paper is to formulate multi-period portfolio selection problems as MV models and solve them by GA. To illustrate the generality of our algorithm, we implement a program by Microsoft Visual Studio to solve a multi-period portfolio selection problem for which there exists no general analytical solution.

Multi-period mean-variance portfolio selection in a regime-switching market with a bankruptcy state and a state-dependent uncertain exit-time

In this paper, we study optimal multi-period portfolio selection problem with uncertain exit-time under mean-variance criterion in a Markovian regime-switching market. The market state space contains an absorbing state which represents the bankruptcy state. It is assumed that all random key parameters, i.e., asset returns, the recovery rate, and the exit-time depend on the current market state. Three common mean-variance formulations are considered, i.e., minimum variance formulation, maximum expected return formulation and the trade-off formulation. First, the problem with an uncertain exit-time is reformulated as a problem with a certain exit-time. Then, by applying the Lagrange duality method and the dynamic programming approach, the optimal multi-period portfolio strategies and the efficient frontier are derived in a closed form. Moreover, the conditions under which the aforementioned three problems are (mutually) equivalent are given. A numerical example is provided to illustrate the results.

Time-consistent investment and reinsurance strategies for insurers under multi-period mean-variance formulation with generalized correlated returns

The existing literature on investment and reinsurance is limited to the study of continuous-time problems, while discrete-time problems are always ignored by researchers. In this study, we first discuss a multi-period investment and reinsurance optimization problem under the classical mean-variance framework. When the asset returns with a serially correlated structure, the time-consistent investment and reinsurance strategies are acquired via backward induction. In addition, we propose an alternative time-consistent mean-variance optimization model that contrasts with the classical mean-variance model, and the corresponding optimal strategy and value function are also derived. We find that the investment and reinsurance strategies are both independent of the current wealth for the above two optimization problems, which coincides with the conclusion presented in the continuous-time problems. Most importantly, the above investment strategies with serially correlated structures are both conditional mean-based strategies, rather than unconditional ones. Finally, we compare the investment and reinsurance strategies suggested above based on the simulation approach, to shed light on which investment-reinsurance strategies are more suitable for insurers.

Taylor approximation and variance reduction for PDE-constrained optimal control under uncertainty

In this work we develop a scalable computational framework for the solution of PDE-constrained optimal control problems under high-dimensional uncertainty. Specifically, we consider a mean-variance formulation of the control objective and employ a Taylor expansion with respect to the uncertain parameter field either to directly approximate the control objective or as a control variate for variance reduction. The expressions for the mean and variance of the Taylor approximation are known analytically, although their evaluation requires efficient computation of the trace of the (preconditioned) Hessian of the control objective. We propose to estimate this trace by solving a generalized eigenvalue problem using a randomized algorithm that only requires the action of the Hessian on a small number of random directions. Then, the computational work does not depend on the nominal dimension of the uncertain parameter, but depends only on the effective dimension (i.e., the rank of the preconditioned Hessian), thus ensuring scalability to high-dimensional problems. Moreover, to increase the estimation accuracy of the mean and variance of the control objective by the Taylor approximation, we use it as a control variate for variance reduction, which results in considerable computational savings (several orders of magnitude) compared to a plain Monte Carlo method. In summary, our approach amounts to solving an optimal control problem constrained by the original PDE, the generalized eigenvalue equations at a small number of eigenfunctions, and a set of linearized PDEs that arise from the computation of the gradient and Hessian of the control objective with respect to the uncertain parameter. We use the Lagrangian formalism to derive expressions for the gradient with respect to the control and apply a gradient-based optimization method to solve the problem. We demonstrate the accuracy, efficiency, and scalability of the proposed computational method for two examples with high-dimensional uncertain parameters: subsurface flow in a porous medium modeled as an elliptic PDE, and turbulent jet flow modeled by the Reynolds-averaged Navier–Stokes equations coupled with a nonlinear advection-diffusion equation characterizing model uncertainty. In particular, for the latter more challenging example we show scalability of our algorithm up to one million parameters resulting from discretization of the uncertain parameter field.

A Markowitz Portfolio Approach to Options Trading

In this paper, we study the problem of option portfolio design under the Markowitz mean-variance framework. We extend the common practice of a pure-stock portfolio and include options in the design. The options returns are modeled statistically with first- and second-order moments, enriching the conventional delta-gamma approximation. The naive mean-variance formulation allows for a zero-risk design that, in a practical scenario with parameter estimation errors, is totally misleading and leads to bad results. This zero-risk fallacy can be circumvented with a more realistic robust formulation. Transaction cost is also considered in the formulation for a proper practical design. We propose an efficient BSUM-M-based algorithm to solve the optimization problem. The proposed algorithm can perform as well as the off-the-shelf solvers but with a much lower computational time—up to one order of magnitude lower. Numerical results based on real data are conducted and the performance is presented in terms of Sharpe ratio, cumulative profit and loss, drawdown, overall return over turnover, value at risk, expected shortfall, and certainty equivalent.

Time-Consistent Portfolio Policy for Asset-Liability Mean-Variance Model with State-Dependent Risk Aversion

In reality, when facing a multi-period asset-liability portfolio selection problem, the risk aversion attitude of a mean-variance investor may depend on the wealth level and liability level. Thus, in this paper, we propose a state-dependent risk aversion model for the investor, in which risk aversion is a linear function of current wealth level and current liability level. Due to the time inconsistency of the resulting multi-period asset-liability mean-variance model, we investigate its time-consistent portfolio policy by solving a nested mean-variance game formulation. We derive the analytical time-consistent portfolio policy, which takes a linear form of current wealth level and current liability level. We also analyze the influence of the risk aversion coefficients on the time-consistent portfolio policy and the investment performance via a numerical example.

Portfolio selection in a regime switching market with a bankruptcy state and an uncertain exit-time: multi-period mean–variance formulation

This paper studies three versions of the multi-period mean–variance portfolio selection problem, that are: minimum variance problem, maximum expected return problem and the trade-off problem, in a Markovian regime switching market where the exit-time is uncertain and exogenous. The underlying Markov chain contains an absorbing state, which denotes the bankruptcy state. When the Markov chain switches to this state, the investors only get a random fraction, known as the recovery rate, taking values in [0, 1] of their wealth. Asset returns, as well as recovery rate, depend on the market state. Dynamic programming and Lagrange duality method are used to derive analytical expressions for optimal investment strategies and the mean–variance efficient frontier. It is shown that portfolio selection models with no bankruptcy state and certain exit-time can be considered as special cases of our model. Some numerical examples are provided to demonstrate the effect of the recovery rate and exit-probabilities.

Time consistent behavioral portfolio policy for dynamic mean–variance formulation

When one considers an optimal portfolio policy under a mean-risk formulation, it is essential to correctly model investors’ risk aversion which may be time variant or even state dependent. In this paper, we propose a behavioral risk aversion model, in which risk aversion is a piecewise linear function of the current excess wealth level with a reference point at the discounted investment target (either surplus or shortage), to reflect a behavioral pattern with both house money and break-even effects. Due to the time inconsistency of the resulting multi-period mean–variance model with adaptive risk aversion, we investigate the time consistent behavioral portfolio policy by solving a nested mean–variance game formulation. We derive a semi-analytical time consistent behavioral portfolio policy which takes a piecewise linear feedback form of the current excess wealth level with respect to the discounted investment target. Finally, we extend the above results to time consistent behavioral portfolio selection for dynamic mean–variance formulation with a cone constraint.

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 example, demand for equipment that plants or harvests corn correlates with 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 reflecting demand volatility, 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&P 500 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. We give a complete characterization of the efficient frontier and quantify the improvement in risk-return trade-off achieved by the hedging strategy. Furthermore, we show that the hedging strategy is self-financing in the sense that the expected total wealth from both production and hedging stays nonnegative at all times. The online appendix is available at https://doi.org/10.1287/opre.2017.1597

Better than pre-committed optimal mean-variance policy in a jump diffusion market

Dynamic mean-variance investment model can not be solved by dynamic programming directly due to the nonseparable structure of variance minimization problem. Instead of adopting embedding scheme, Lagrangian duality approach or mean-variance hedging approach, we transfer the model into mean field mean-variance formulation and derive the explicit pre-committed optimal mean-variance policy in a jump diffusion market. Similar to multi-period setting, the pre-committed optimal mean-variance policy is not time consistent in efficiency. When the wealth level of the investor exceeds some pre-given level, following pre-committed optimal mean-variance policy leads to irrational investment behaviors. Thus, we propose a semi-self-financing revised policy, in which the investor is allowed to withdraw partial of his wealth out of the market. And show the revised policy has a better investment performance in the sense of achieving the same mean-variance pair as pre-committed policy and receiving a nonnegative free cash flow stream.

Asset Allocation Theory and Practice in Australian Investment Management

Asset allocation is the decision on how much of the investment portfolio to place in each of the broad asset classes (i.e., cash, fixed-interest securities, property, equities). It is a key decision area in the investment management industry, where professional investors manage pooled investments. This study sought to identify 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, and established that academics still believe in their relevance. An online survey among practitioners 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. Recommendations mostly revolving around improving the interaction between academia and industry are made that may help reduce the theory-practice dichotomy. Limitations of the study are discussed as well as future areas for related research.

Regime-dependent robust portfolio selection model

A multi-period mean-variance model is developed where the model parameters change according to a market with Markov random regime switching. Robust optimization is used to solve the problem which, in turn, gives the explicit expression of the efficient frontier of the mean-variance formulation. Furthermore, a numerical example based on real data from the Chinese stock market is provided to demonstrate the application of this paper.

On robust mean-variance portfolios

We derive closed-form portfolio rules for robust mean–variance portfolio optimization where the return vector is uncertain or the mean return vector is subject to estimation errors, both uncertainties being confined to an ellipsoidal uncertainty set. We consider different mean–variance formulations allowing short sales, and derive closed-form optimal portfolio rules in static and dynamic settings.

Store brand introduction in a two-echelon logistics system with a risk-averse retailer

We study a risk-averse retailer’s optimal decision of introducing her store brand product by using the mean–variance formulation. The effects of the substitution factor, the capital constraint, and the development cost are examined. Taking the product quantities as the decision variables, the risk deducted surplus of the store brand product and the substitution factor play a vital role in the retailer’s optimal policies. Both the capital constraint and the development cost reduce the mean–variance efficient solution set of the retailer and hence distort the risk management of the retailer. Some meaningful insights are generated.

Classical mean-variance model revisited: pseudo efficiency

Investigating the inverse problem of the classical Markowitz mean-variance formulation: Given a mean-variance pair, find initial investment levels and their corresponding portfolio policies such that the given mean-variance pair can be realized, we reveal that any mean-variance pair inside the reachable region can be achieved by multiple portfolio policies associated with different initial investment levels. Therefore, in the mean-variance world for a market of all risky assets, the common belief of monotonicity: ‘The larger you invest, the larger expected future wealth you can expect for a given risk (variance) level’ does not hold, which stimulates us to extend the classical two-objective mean-variance framework to an expanded three-objective framework: to maximize the mean and minimize the variance of the final wealth as well as to minimize the initial investment level. As a result, we eliminate from the policy candidate list the set of pseudo efficient policies that are efficient in the original mean-variance space, but inefficient in this newly introduced three-dimensional objective space.