Mean-variance Portfolio Problem Research Articles

Open-loop equilibrium strategy for mean-variance Portfolio selection with investment constraints in a non-Markovian regime-switching jump-diffusion model

<p style='text-indent:20px;'>This paper is devoted to study the open-loop equilibrium strategy for a mean-variance portfolio problem with investment constraints in a non-Markovian regime-switching jump-diffusion model. Specially, the investment strategies are constrained in a closed convex cone and all coefficients in the model are stochastic processes adapted to the filtration generated by a Markov chain. First, we provide a necessary and sufficient condition for an equilibrium strategy, which involves a system of forward and backward stochastic differential equations (FBSDEs, for short). Second, by solving these FBSDEs, we obtain a feedback representation of the equilibrium strategy. Third, we prove a theorem ensuring the almost everywhere uniqueness of the equilibrium solution. Finally, the results are applied to solve an example of the Markovian regime-switching model.</p>

A mean-variance portfolio optimization approach for high-renewable energy hub

This paper proposes a high-renewable portfolio model of energy hub. In this model, geothermal-solar-wind multi-energy complementarities are fully explored based on electrolytic thermo-electrochemical effects of geothermal-to-hydrogen (GTH), which are converted, conditioned, and coupled through energy hub. The proposed high-renewable energy hub portfolio is an intractable optimization problem due to their inherent strong energy couplings and conflicted energy cost/risk. The original problem is thus characterized through the mean-variance approach to explicitly express the risk associated with the forecast uncertainties. The formulated mean-variance portfolio problem is subsequently modeled as a two-stage mixed-integer nonlinear programming (MINLP) stochastic programming to optimally determine appropriate energy generation, conversion, and storage candidates. Numerical studies on a community microgrid are implemented to verify the effectiveness and superiority of the proposed methodology over conventional wind-solar-battery scheme. Simulations results show that the portfolio cost can be reduced by at most 14.9% with a significantly higher operational flexibility.

A Constructive Approach to Existence of Equilibria in Time-Inconsistent Stochastic Control Problems

We extend the construction of equilibria for linear-quadratic and mean-variance portfolio problems available in the literature to a large class of mean-field time-inconsistent stochastic control problems in continuous time. Our approach relies on a time discretization of the control problem via n-person games, which are characterized via the maximum principle using Backward Stochastic Differential Equations (BSDEs). The existence of equilibria is proved by applying weak convergence arguments to the solutions of n-person games. A numerical implementation is provided by approximating n-person games using finite Markov chains.

Time-Consistent Mean-Variance Pairs-Trading Under Regime-Switching Cointegration

While cointegration models with constant parameters generate statistical arbitrage, the cointegration feature may change and even disappear due to regime shifts. This paper studies the time-consistent mean-variance portfolio problem in a Markov-modulated regime switching cointegration economy. We derive a novel closed-form solution to the equilibrium strategy. This analytical solution allows us to investigate statistical arbitrage with regime-switching pairs-trading rules. The presence of regime switching increases the risk of such trading strategies, especially near the switching times. Empirical analysis demonstrates the use of the derived formulas and shows the advantages of incorporating different market modes.

Robust Time-Inconsistent Stochastic Linear-Quadratic Control: An Open-Loop Approach

This paper studies stochastic linear-quadratic control problems for an ambiguity-adverse agent with a time-inconsistent objective. We allow the agent to incorporate disturbances into the state's drift or choose an alternative model among a set of models equivalent to the reference model, to reflect her ambiguity aversion on the drift coefficient of the state process. We adopt an innovative two-step equilibrium control approach to characterize the robust time-consistent controls and simultaneously preserve the preference order. Under a general framework allowing random parameters, we derive a sufficient condition for equilibrium controls using the forward-backward stochastic differential equation approach. We also provide analytical solutions to mean-variance portfolio problems for various settings. Our empirical studies confirm the improvement in the portfolio's performance in terms of Sharpe ratio by incorporating robustness.

Solving cardinality constrained mean-variance portfolio problems via MILP

Controlling the number of active assets (cardinality of the portfolio) in a mean-variance portfolio problem is practically important but computationally demanding. Such task is ordinarily a mixed integer quadratic programming (MIQP) problem. We propose a novel approach to reformulate the problem as a mixed integer linear programming (MILP) problem for which computer codes are readily available. For numerical tests, we find cardinality constrained minimum variance portfolios of stocks in S&P500. A significant gain in robustness and computational effort by our MILP approach relative to MIQP is reported. Similarly, our MILP approach also competes favorably against cardinality constrained portfolio optimization with risk measures CVaR and MASD. For illustrations, we depict portfolios in a portfolio map where cardinality provides a third criterion in addition to risk and return. Fast solution allows an interactive search for a desired portfolio.

On the Viability of Robo-Advising for Individual Investors

The development of robo-advising allows investors to receive advice on investments that were considered too small by traditional human advisers. In this paper, the viability of robo-advising for individual investors is discussed by investigating one of the basic strategies of portfolio management, diversification. We formulate a mean-variance portfolio problem with a constraint on investment size, and find that most of risk reduction from diversification is achieved even with a small portfolio. The consistency of our finding provides insight on the feasibility of robo-advising for all investors regardless of their level of wealth.

Mean-Field Maximum Principle for Optimal Control of Forward–Backward Stochastic Systems with Jumps and its Application to Mean-Variance Portfolio Problem

We study mean-field type optimal stochastic control problem for systems governed by mean-field controlled forward–backward stochastic differential equations with jump processes, in which the coefficients depend on the marginal law of the state process through its expected value. The control variable is allowed to enter both diffusion and jump coefficients. Moreover, the cost functional is also of mean-field type. Necessary conditions for optimal control for these systems in the form of maximum principle are established by means of convex perturbation techniques. As an application, time-inconsistent mean-variance portfolio selection mixed with a recursive utility functional optimization problem is discussed to illustrate the theoretical results.

Simplified mean-variance portfolio optimisation

We propose a simplified approach to mean-variance portfolio problems by changing their parametrisation from trading strategies to final positions. This allows us to treat, under a very mild no-arbitrage-type assumption, a whole range of quadratic optimisation problems by simple mathematical tools in a unified and model-independent way. We provide explicit formulas for optimal positions and values, connections between the solutions to the different problems, two-fund separation results, and explicit expressions for indifference values.

Simplified Mean-Variance Portfolio Optimisation

We propose a simplified approach to mean-variance portfolio problems by changing their parametrisation from trading strategies to final positions. This allows us to treat, under a very mild no-arbitrage-type assumption, a whole range of quadratic optimisation problems by simple mathematical tools in a unified and model-independent way. We provide explicit formulas for optimal positions and values, connections between the solutions to the different problems, two-fund separation results, and explicit expressions for indifference values.

Dynamic Mean-Variance Asset Allocation

We solve the dynamic mean-variance portfolio problem and derive its time-consistent solution using dynamic programming. Previous literature, in contrast, only determines either myopic or precommitment (committing to follow the initially optimal policy) solutions. We provide a fully analytical simple characterization of the dynamically optimal mean-variance portfolios within a general incomplete-market economy. We also identify a probability measure that incorporates intertemporal hedging demands and facilitates tractability. We illustrate this by easily computing portfolios explicitly under various stochastic investment opportunities. A calibration exercise shows that the mean variance hedging demands are economically significant.

Multistep predictions for multivariate GARCH models: Closed form solution and the value for portfolio management

In this paper we derive the closed form solution for multistep predictions of the conditional means and their covariances from multivariate ARMA-GARCH models. These are useful e.g. in mean variance portfolio analysis when the rebalancing frequency is lower than the data frequency. In this situation the conditional mean and covariance matrix of the sum of the higher frequency returns until the next rebalancing period is required as input in the mean variance portfolio problem. The closed form solution for this quantity is derived as well. We assess the empirical value of the result by evaluating and comparing the performance of quarterly and monthly rebalanced portfolios using monthly MSCI index data across a large set of ARMA-GARCH models. The results forcefully demonstrate the substantial value of multistep predictions for portfolio management.

Multi-horizon Markowitz portfolio performance appraisals: A general approach

This article extends the analysis of multi-horizon mean-variance portfolio analysis in the Morey and Morey [Mutual fund performance appraisals: a multi-horizon perspective with endogenous benchmarking. Omega 1999;27:241–58] article in several ways. First, instead of either proportionally contracting risk dimensions or proportionally expanding return dimensions, a more general efficiency measure simultaneously attempts to reduce risk and to expand return over all time periods. Second, a duality relation is established between this generalized multi-horizon efficiency measure and an indirect mean-variance utility function, underscoring the natural interpretation of this generalized efficiency measure in terms of investor's preferences. Furthermore, the need to properly apply time discounting in multi-horizon mean-variance portfolio problems is argued for. An empirical illustration based on the original mutual fund data set in Morey and Morey [Mutual fund performance appraisals: a multi-horizon perspective with endogenous benchmarking. Omega 1999;27:241–58] is added to contrast the new and the original approaches.

Multistep Predictions for Multivariate GARCH Models: Closed Form Solution and the Value for Portfolio Management

In this paper we derive the closed form solution for multistep predictions of the conditional means and covariances for multivariate GARCH models. These predictions are useful e.g. in mean variance portfolio analysis when the rebalancing frequency is lower than the data frequency. In this situation the conditional mean and the conditional covariance matrix of the cumulative higher frequency returns until the next rebalancing period are required as inputs in the mean variance portfolio problem. The closed form solution for this quantity is derived as well. We assess the empirical value of the result by evaluating and comparing the performance of quarterly and monthly rebalanced portfolios using monthly MSCI index data across a large set of GARCH models. The value of using correct multistep predictions is assessed by comparing the performance of the quarterly rebalanced portfolios based on the correct multistep predictions with the quarterly rebalanced portfolios incorrectly based on 1-step predictions and the monthly rebalanced portfolios. Using correct multistep predictions generally results in lower risk and higher returns. Furthermore the correctly computed quarterly rebalanced portfolios exhibit higher returns than monthly rebalanced portfolios. The empirical results thus forcefully demonstrate the substantial value of multistep predictions for portfolio management.

Mean-variance Hedging for Pricing European-type Contingent Claims with Transaction Costs

In this paper, a European-type contingent claim pricing problem with transaction costs is considered by a mean-variance hedging argument. The investor has to pay transaction costs which are proportional to the amount of stock transacted. The writer’s hedging object is to minimize the hedging risk, defined as the variance of hedging error at expiration, with a proper expected excess return level. At first, we consider the mean-variance hedging problem: for initial hedging wealth f, maximizing the excess expected return under the minimum hedging risk level V 0. On the other hand, we consider a mean-variance portfolio problem, which is to maximize the expected return with initial wealth 0 under the same risk level V 0. The minimum initial hedging wealth f, which can offset the difference of the maximum expected return of these two problems, is the writer’s price.

Mean Variance Portfolio and Hedging Strategies in Incomplete Markets

We address two mean-variance asset allocation and hedging problems in continuous-time incomplete markets. The resolution emphasizes the geometric properties of mean-variance portfolios and underlines the role of two specific portfolios: The minimum norm and the minimum variance portfolios. We extend the portfolio strategies of Bajeux-Besnainou and Portait (1998) to the incomplete market setting and provide a set of fictitious assets that can complement the market without being traded. Moreover, the fictitious assets can be chosen independently of the investor's level of risk aversion. As importantly, we show that the fictitious completion is irrelevent in mean-variance portfolio problems. Taking the orthogonal projection of fictitious portfolio distributions on the space of attainable distributions provides the required attainable portfolio. This approach is applied successfully in the imperfect hedging problem of Duffie and Richardson (1991) where the least favorable completion cannot be explicited.

The analytics of sensitivity analysis for mean-variance portfolio problems

This paper shows that all the efficient set mathematics for mean-variance (MV) portfolio problems and all the analytics of sensitivity analysis for MV portfolio problems follow from a general parametric quadratic programming problem, the Karush-Kuhn-Tucker or optimality conditions for it and the definitions of the expected return and variance of a portfolio. Then, for a change in the asset means, closed-form expressions are derived for an investor's optimal portfolio; multipliers; portfolio expansion path, the analog of the traditional minimum variance frontier; and the investor's security line (hyperplane), the analog of the traditional security market line (hyperplane).

The efficient set mathematics when mean-variance problems are subject to general linear constraints

In this paper we develop the efficient set mathematics for the case where mean-variance portfolio problems are subject to general linear constraints. We derive closed-form expressions for the optimal portfolio, the associated multipliers, and the portfolio's expected return and variance in each interval where the active set changes; identify where kinks on the efficient frontier occur; and show that securities plot on a security market hyperplane. The analysis extends our knowledge of portfolio theory, clarifies the zero-beta Capital Asset Pricing Model and the conditions under which securities plot on the Security Market Line (SML), and provides insight into the ambiguities associated with using the SML criterion to measure investment performance.

Risk, Human Capital, and the Investor's Portfolio

The twin theories of investment in human capital and management of marketable assets have developed along two remarkably independent paths. Models of investment in human capital have without exception ignored the closely related problem of managing portfolios. This has been possible only because until very recently the literature on human capital has largely avoided the important issue of uncertainty. Simultaneously, the literature on portfolio theory has essentially ignored human capital. Currently, no model of investment in risky marketable assets includes endogenous allocations to education.' In this paper, risky human capital is introduced into a two-period, mean-variance portfolio problem. As in Levhari and Weiss (1974), the basic theoretical model is simple, permitting easy identification of the various effects of uncertainty. Here, however, in contrast to previous work, the Opportunities for risky investment in education are introduced into a two-period, meanvariance portfolio problem. Uncertainties arising from expenditures on education have three sources: ambiguous inputs in the production of skills, a risky rate of depreciation or obsolescence of existing skills, and a stochastic future wage. Properties of efficient and optimal investments in securities and education are derived in detail with special emphasis on education.