Mean-variance Optimization Model Research Articles

Robust portfolio optimization model for electronic coupon allocation

Currently, many e-commerce websites issue online/electronic coupons as an effective tool for promoting sales of various products and services. We focus on the problem of optimally allocating coupons to customers subject to a budget constraint on an e-commerce website. We apply a robust portfolio optimization model based on customer segmentation to the coupon allocation problem. We also validate the efficacy of our method through numerical experiments using actual data from randomly distributed coupons. Main contributions of our research are twofold. First, we handle six type of coupons, thereby making it extremely difficult to accurately estimate the difference in the effects of various coupons. Second, we demonstrate from detailed numerical results that the robust optimization model achieved larger uplifts of sales than did the commonly-used multiple-choice knapsack model and the conventional mean–variance optimization model. Our results open up great potential for robust portfolio optimization as an effective tool for practical coupon allocation.

Portfolio selection problem: main knowledge and models (A systematic review)

The challenge in portfolio optimization lies in creating a collection of assets that attains a target expected returnwhile mitigating risk. This problem is often framed as an optimization task, specifically Mean Variance Optimization (MVO). MVO involves formulating an objective function, typically quadratic, that is contingent on the composition of the portfolio, and linear constraints that represents the portfolio's asset allocation restriction. Several improvements have been proposed, such as adding constraints to MVO or using alternative risk measures. As a result, even though MVO model remains the most widely studied type of portfolio optimization, different types of portfolio optimization models, risk/return measurements and constraints have been suggested and used since its invention. In this work, we delve into the various risk and return measures, constraints, and mathematical models commonly used in portfolio optimization.We discuss the key risk measures employed in portfolio optimization, including the Sharpe ratio, beta, maximum drawdown and others, We explore the constraints commonly applied in portfolio optimization. Furthermore, we delve into the mathematical models utilized in portfolio optimization. Then, we emphasize the interplay between risk and return measures, constraints, and mathematical models in portfolio optimization. By providing a comprehensive overview of risk and return measures, constraints, and mathematical models, this work aims to enhance the understanding of portfolio optimization techniques and facilitate informed decisionmaking in the field of investment management. To illustrate different knowledge and models, several experiments were conducted on well-known real data portfolios.

Optimization of national grain imports to balance risk and return: a portfolio theory approach

Abstract Global grain trade plays a key role in food security. Many nations rely on imported grain to meet their dietary requirements. Grain imports may be at risk due to weather shocks, economic crises, or international conflicts. Countries aim to balance import risk with the expected return of their grain supplies. This research brings these dual objectives together in an innovative modern portfolio theory framework. Modern portfolio theory provides a set of concepts to formulate the trade-off between risk and expected return in national grain imports. Using Markowitz’s mean-variance optimization model, we identify opportunities to reduce risk in existing national grain import accounts, without increasing costs under realistic supply mass constraints of trade partners. Several major grain importers may be able to reduce risk in their grain imports without increasing cost, such as wheat imports in Egypt, maize imports in Vietnam, and rice imports in Saudi Arabia. However, some countries would indeed have to pay more to achieve more stable grain supplies, such as wheat imports in Turkey. This study provides a framework to quantify the different costs, benefits, and levels of risk in grain trade that can inform future research and decision-making.

The Marginal Abatement Cost of Antimicrobials for Dairy Cow Mastitis: A Bioeconomic Optimization Perspective

Maintaining udder health is the primary indication for antimicrobial use (AMU) in dairy production, and modulating this application is a key factor in decreasing AMU. Defining the optimal AMU and the associated practical rules is challenging since AMU interacts with many parameters. To define the trade-offs between decreased AMU, labor and economic performance, the bioeconomic stochastic simulation model DairyHealthSim (DHS)© was applied to dairy cow mastitis management and coupled to a mean variance optimization model and marginal abatement cost curve (MACC) analysis. The scenarios included three antimicrobial (AM) treatment strategies at dry-off, five types of general barn hygiene practices, five milking practices focused on parlor hygiene levels and three milk withdrawal strategies. The first part of economic results showed similar economic performances for the blanked dry-off strategy and selective strategy but demonstrated the trade-off between AMU reduction and farmers' workload. The second part of the results demonstrated the optimal value of the animal level of exposure to AM (ALEA). The MACC analysis showed that reducing ALEA below 1.5 was associated with a EUR 10,000 loss per unit of ALEA on average for the farmer. The results call for more integrative farm decision processes and bioeconomic reasoning to prompt efficient public interventions.

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.

A combinatorial optimization approach to scenario filtering in portfolio selection

Recent studies stressed the fact that covariance matrices computed from empirical financial time series appear to contain a high amount of noise. This makes the classical Markowitz Mean–Variance Optimization model unable to correctly evaluate the performance associated to selected portfolios. Since the Markowitz model is still one of the most used practitioner-oriented tool, several filtering methods have been proposed in the literature to overcome the problem. Among them, the two most promising ones refer to the Random Matrix Theory and to the Power Mapping strategy. The basic idea of these methods is to transform the estimated correlation matrix before applying the Mean–Variance Optimization model. However, experimental analysis shows that these two strategies are not always effective when applied to real financial datasets.In this paper we propose a new filtering method based on Quadratic Programming. We develop a Mixed Integer Quadratic Programming model, which is able to filter those observations that may affect the performance of the selected portfolio. We discuss the properties of this new model and test it on some real financial datasets. We compare the out-of-sample performance of our portfolios with the one of the portfolios provided by the two above mentioned alternative filtering methods giving evidence that our method outperforms them. Although our model can be solved efficiently with standard optimization solvers, the computational burden increases for large datasets. To solve also these problems, we propose a heuristic procedure, which, on the basis of our empirical results, shows to be both efficient and effective.

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.

The Black-Litterman Model: Extensions and Asset Allocation

The Black-Litterman Model (BLM), created by Fischer Black and Robert Litterman, is a sophisticated portfolio construction method that overcomes the problem of unintuitive, highly-concentrated portfolios, input-sensitivity, and estimation error maximization. The BLM uses a Bayesian approach to combine the subjective views of an investor regarding the expected returns of one or more assets with the market equilibrium vector of expected returns (the prior distribution) to form a new, mixed estimate of expected returns. The resulting new vector of returns (the posterior distribution), leads to intuitive portfolios with sensible portfolio weights. (Idzorek, 2004). BLM was first introduced in Black and Litterman (1990) and further explained in Black and Litterman (1991) and Black and Litterman (1992) It is an asset allocation model which has its roots in mean-variance (MV) optimization model and capital asset pricing model (CAPM). Model builds on MV optimization and CAPM by using a Bayesian framework that allows investors to incorporate their views on markets effectively into asset allocation process. Main contribution of BLM is that it enables investors to construct sensible portfolios without using unnecessary constraints which also reflect their views on markets. It is well known to both academics and practitioners that standard MV optimization is very sensitive to expected returns and often generates extreme portfolios (concentration in very few assets, large long and short positions). BLM overcomes these issues by choosing a neutral reference point, CAPM equilibrium. It also allows investors to express their views with varying confidence levels and integrate these views into CAPM prior by using a Bayesian framework. As a result, since its introduction BLM has been increasingly used among practitioners and allowed asset allocation decisions to be in more of a quantitative nature rather than qualitative because of the apparent shortcomings of MV optimization model.

Asset Allocation Effects of Industrial Index Portfolio with Changes in tau(τ) and lambda(λ) in the Black-Litterman's Inverse Optimization Model

본 연구에서는 Markowitz 평균-분산 최적화 모형의 자산배분 시 극단적 쏠림 현상의 한계를 극복하는 대안 모형으로 제시된 Black-Litterman 분석모형에 대한 실증분석을 수행하였다. 여기서 모형 모수 중 와 가 포트폴리오에 미치는 자산배분 효과를 알아보기 위해 이들 모수 값을 변화 시킴으로써 새롭게 조합된 수익률 벡터(NCRV)와 포트폴리오의 새로운 가중치 벡터(NWV)  의 변화와 Idzorek(2005)에 의해 제시된 암묵적 신뢰수준        의 변화를 분석하 였다. 본 연구의 기초데이터와 분석기간은 KOSPI와 KOSDAQ의 산업별 인덱스 일별자료를 이용하 여, 2015.01∼2017.12월까지를 분석기간으로 한 후 Jensen의  순으로 총 10개의 시장별 업종 지수 인덱스 포트폴리오를 구성한 후 이를 Black-Litterman 모형에 적용하였다. 본 연구의 연구방법은 사전확률분포와 사후확률분포를 가정 한 후 사전적으로 암묵적 균형초과 수익률(IEER)과 시장자본가중치(  )의 산출 후 시장전망 벡터 , 전망에 대한 He-Litterman(1999)의 불확실성 에 스캘링 팩더 를 적용하였다. 이 때 Idzorek(2005)의   , Satchell and Scowcroft (2000)의   , 그리고 재무론적 관점에서 무차익 완전시장과 위험중립측정의 가정 하에 새롭게 산출한   을 적용하였으며, 역사적 위험회피계수 값 6.8과 Idzorek(2005)의   을 적용한 후 상대적 및 절대적 전망에 대한 short 포지션 (underperform)과 long 포지션(outperform)의 공매도 및 매수 포지션의 변화량 증감을 살펴보고그 원인을 분석하였다. 본 연구의 시사점 및 분석결과는 다음과 같다. 먼저 시사점은 BL 모형에 재무적 관점의 새로운 와 역사적 위험회피도인 를 산출하였으며, 다양한 크기의 와 를 이용하여 크기가 증가함에 따른 포지션의 변화에 어떤 영향을 주는지를펴보았으며, 가 작을수록, 의 크기가 클수록 포트폴리오의 가중치 분포가 더 안정된다는 것을 발견하였다. 암묵적 신뢰수준은 와 가 커질 때 IEER의 확대로  이 증가함으로써 신뢰수준이 더 커진다는 것을 발견하였다. 또한 100%신뢰수준을 가정할 때, 의 변화는    에 영향을 미치지 않는 것을 알 수 있었다. 추후 연구과제로는 BL 모형을 적용한 가중치를 출발점으로 한위험예산(risk budget) 최적화 시 Markowitz와 동일가중(Equally weight)모형이 총위험공헌도 (TRC)와 한계위험공헌도(MRC)에 미치는 영향도를 비교·분석 해 볼 필요가 있다.In this paper, the Black-Litterman analytical model was applied to overcome the limitations of extreme deviations in the asset allocation of the Markowitz mean-variance optimization model. To analyze the asset allocation effect by λ and τ used on the portfolios analysis, we analyze the change of the newly combined return vector (NCRV) and the new weight vector (NWV) of the portfolio by changing these parameter values. For this purpose, We have used the returns which are composed as 10 industry index that is a superior position to Jensen s alpha. Using KOSPI and KOSDAQ industry indexes data from 2015.1 to 2017.12, we have applied that Black-Litterman Indverse Optimization Model. The analytical methods which is that applied to the uncertainty of posterior probability distribution is that τ(=0.025) is for Idzorek (2005), τ = 1.0 for Satchell and Scowroft (2000), and the newly calculated τ = 0.51 which was applied and applied to the analytical model along with the risk aversion coefficient λ = 6.8 which has generated from the historical returns. and also λ = 3.07 for Idzorek s method. The covariance matrix Ω of uncertainty is applied by He-Litterman method. The analysis results and implications of this study are as follows: First, as the τ applied to the Black-Literman model increased, the buying and selling positions were shown to be weighted further to change the asset allocation outcome, and the larger the λ in the same τ, the more stable the asset allocation result. The implicit confidence level was shown to increase as τ increased, and the confidence level was also greater when λ was greater. Future research issues will require financial institutions to explore the impact of asset allocation on risk, benchmark risks, and portfolio risks when they apply the Black-Litterman model to establish a strategic asset allocation risk budget.

Continuous control with Stacked Deep Dynamic Recurrent Reinforcement Learning for portfolio optimization

Recurrent reinforcement learning (RRL) techniques have been used to optimize asset trading systems and have achieved outstanding results. However, the majority of the previous work has been dedicated to systems with discrete action spaces. To address the challenge of continuous action and multi-dimensional state spaces, we propose the so called Stacked Deep Dynamic Recurrent Reinforcement Learning (SDDRRL) architecture to construct a real-time optimal portfolio. The algorithm captures the up-to-date market conditions and rebalances the portfolio accordingly. Under this general vision, Sharpe ratio, which is one of the most widely accepted measures of risk-adjusted returns, has been used as a performance metric. Additionally, the performance of most machine learning algorithms highly depends on their hyperparameter settings. Therefore, we equipped SDDRRL with the ability to find the best possible architecture topology using an automated Gaussian Process (GP) with Expected Improvement (EI) as an acquisition function. This allows us to select the best architectures that maximizes the total return while respecting the cardinality constraints. Finally, our system was trained and tested in an online manner for 20 successive rounds with data for ten selected stocks from different sectors of the S&P 500 from January 1st, 2013 to July 31st, 2017. The experiments reveal that the proposed SDDRRL achieves superior performance compared to three benchmarks: the rolling horizon Mean-Variance Optimization (MVO) model, the rolling horizon risk parity model, and the uniform buy-and-hold (UBAH) index.

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.

A similarity measure for the cardinality constrained frontier in the mean–variance optimization model

This paper proposes a new measure to find the cardinality constrained frontier in the mean–variance portfolio optimization problem. In previous research, assets belonging to the cardinality constrained portfolio change according to the desired level of expected return, so that the cardinality constraint can actually be violated if the fund manager wants to satisfy clients with different return requirements. We introduce a perceptual approach in the mean–variance cardinality constrained portfolio optimization problem by considering a novel similarity measure, which compares the cardinality constrained frontier with the unconstrained mean–variance frontier. We assume that the closer the cardinality constrained frontier to the mean–variance frontier, the more appealing it is for the decision maker. This makes the assets included in the portfolio invariant to any specific level of return, through focusing not on the optimal portfolio but on the optimal frontier.

A robust mean variance optimization model for economic dispatch with wind power integration and evenly distributed Pareto front generation

A robust mean variance optimization model for economic dispatch with wind power integration and evenly distributed Pareto front generation

Portfolio Optimization under Threshold Accepting: Further Evidence from a Frontier Market

Several improvements and alternatives to Mean Variance Optimization (MVO) have been suggested and used since its inception in 1952. The improvements have mostly included addition of constraints to the traditional MVO model, using alternative risk measures and using non risk-reward models. This paper seeks to compare MVO against the Threshold Accepting model, which is a general optimization model, in portfolio selection. Using data on 29 stocks in the Kenyan stock market we compare the relative performance of the two models using performance measures such as the Sharpe Ratio, Sortino Ratio and Information Ratio. We find that the Threshold Accepting (TA) model outperforms the Mean-Variance Optimization model though MVO yields similar results when we use monthly or weekly data but the latter is observed as a more consistent model. The TA model has portfolios with generally more superior risk-adjusted returns for the full period and during periods of high volatility in the stock market performance market. This observation implies that relatively more attention should be given to the TA model rather than relying entirely on the classical MVO approach.

Mean-Variance portfolio optimization when each asset has individual uncertain exit-time

The standard Markowitz Mean-Variance optimization model is a single-period portfolio selection approach where the exit-time (or the time-horizon) is deterministic. ‎In this paper we study the Mean-Variance portfolio selection problem ‎with ‎uncertain ‎exit-time ‎when ‎each ‎has ‎individual uncertain ‎xit-time‎, ‎which generalizes the Markowitz's model‎. ‎‎‎‎‎‎We provide some conditions under which the optimal portfolio of the generalized problem is independent of the exit-times distributions. Also, ‎‎it is shown that under some general circumstances, the sets of optimal portfolios‎ ‎in the generalized model and the standard model are the same‎.

A moment matching approach to log-normal portfolio optimization

We consider the problem where a manager aims to minimize the probability of his portfolio return falling below a threshold while keeping the expected return no worse than a target, under the assumption that stock returns are Log-Normally distributed. This assumption, common in the finance literature for daily and weekly returns, creates computational difficulties because the distribution of the portfolio return is difficult to estimate precisely. We approximate it with a single Log-Normal random variable using the Fenton–Wilkinson method and investigate an iterative, data-driven approximation to the problem. We propose a two-stage solution approach, where the first stage requires solving a classic mean-variance optimization model and the second step involves solving an unconstrained nonlinear problem with a smooth objective function. We suggest an iterative calibration method to improve the accuracy of the method and test its performance against a Generalized Pareto Distribution approximation. We also extend our results to the design of basket options.

Strategic Asset Allocation Of Credit Guarantors

<p>How to manage the portfolio of credit guarantors is important in practice and public policy, but has not been investigated well in the prior literature. We empirically compare four different approaches in managing credit guarantor portfolios. The four approaches are equal weighted, minimum variance, mean variance optimization and equal risk contribution methods. In terms of risk return ratio, the mean variance optimization model performs best in out-of-sample test. This result contrasts with previous findings against mean variance optimization. Our results are robust. The results do not change as the characteristics of guarantee portfolio vary.</p>

Fuzzy Portfolio Allocation Models Through a New Risk Measure and Fuzzy Sharpe Ratio

A new portfolio risk measure that is the uncertainty of portfolio fuzzy return is introduced in this paper. Beyond the well-known Sharpe ratio (i.e., the reward-to-variability ratio) in modern portfolio theory, we initiate the so-called fuzzy Sharpe ratio in the fuzzy modeling context. In addition to the introduction of the new risk measure, we also put forward the reward-to-uncertainty ratio to assess the portfolio performance in fuzzy modeling. Corresponding to two approaches based on ${\rm T}_{\rm M}$ and ${\rm T}_{\rm W}$ fuzzy arithmetic, two portfolio optimization models are formulated in which the uncertainty of portfolio fuzzy returns is minimized, while the fuzzy Sharpe ratio is maximized. These models are solved by the fuzzy approach or by the genetic algorithm (GA). Solutions of the two proposed models are shown to be dominant in terms of portfolio return uncertainty compared with those of the conventional mean-variance optimization (MVO) model used prevalently in the financial literature. In terms of portfolio performance evaluated by the fuzzy Sharpe ratio and the reward-to-uncertainty ratio, the model using ${\rm T}_{\rm W}$ fuzzy arithmetic results in higher performance portfolios than those obtained by both the MVO and the fuzzy model, which employs ${\rm T}_{\rm M}$ fuzzy arithmetic. We also find that using the fuzzy approach for solving multiobjective problems appears to achieve more optimal solutions than using GA, although GA can offer a series of well-diversified portfolio solutions diagrammed in a Pareto frontier.

Momentum and Markowitz: A Golden Combination

Mean-Variance Optimization (MVO) as introduced by Markowitz (1952) often presented as elegant but impractical theory. MVO an unstable and error-maximizing procedure (Michaud 1989), and is nearly always beaten by simple 1/N (DeMiguel, 2007). And to quote Ang (2014): Mean-variance weights perform horribly… The optimal mean-variance portfolio a complex function of estimated means, volatilities, and correlations of asset returns. There are many parameters to estimate. Optimized mean-variance portfolios can blow up when there are tiny errors in any of these inputs.... In our opinion, MVO a great concept, but previous studies were doomed to fail because they allowed for short-sales, and applied poorly specified estimation horizons. For example, Ang used a 60 month formation period for estimation of means and variances, while Asness (2012) clearly demonstrated that prices mean-revert at this time scale, where the best assets in the past often become the worst assets in the future. In this paper we apply short lookback periods (maximum of 12 months) to estimate MVO parameters in order to best harvest the momentum factor. In addition, we will introduce common-sense constraints, such as long-only portfolio weights, to stabilize the optimization. We also introduce a public implementation of Markowitz's Critical Line Algorithm (CLA) programmed in R to handle the case when the number of assets much larger than the number of lookback periods.We call our momentum-based, long-only MVO model Classical Asset Allocation (CAA) and compare its performance against the simple 1/N equal weighted portfolio using various global multi-asset universes over a century of data (Jan 1915-Dec 2014). At the risk of spoiling the ending, we demonstrate that CAA always beats the simple 1/N model by a wide margin.

An approach to improve mean-variance portfolio optimization model

In the classical mean-variance finance model where investors have a relative preference for risk versus return, the authors add a new factor – the average trading volume of shares of the portfolio’s security for a specified period of time measured as a percentage of its total float number of shares – that is used to quantify the portfolio’s components based on their potential price increase. The introduced trading volume factor cannot be incorporated into the classic model by increasing the portfolio’s expected return because the price and volume correlation is weak. The modified mean-variance optimization model shows that Markowitz’s portfolio can be improved. The generalized optimal portfolio problem is formulated as a multicriteria problem. The given examples demonstrate that the offered model can be used successfully in practice.