Markowitz Mean-variance Optimization Research Articles

Optimizing Portfolio Allocation in the Indian Defense Sector: A Python-Based Approach

This paper presents a detailed analysis of portfolio construction strategies aimed at maximizing risk-adjusted returns for investors, utilizing Python-based methodologies. By leveraging historical data on stock returns and risks, a meticulous selection process identified 5 stocks with superior risk-return profiles, outperforming the dataset's average measures with higher returns and lower risks. The selected stocks from the defense industry formed the foundation of an optimal portfolio designed to minimize sector-specific risks while maximizing growth potential. Through rigorous portfolio optimization techniques, including Markowitz's mean-variance optimization, an optimal portfolio allocation was identified with a return of 62.40% and volatility of 27.74%. Insights from the study provide valuable perspectives for investors seeking realistic and sustainable wealth accumulation strategies. The research also highlights future avenues for exploration, such as integrating alternative asset classes, incorporating behavioral finance insights, and leveraging advancements in risk management technologies. This study contributes to advancing portfolio theory and offers practical guidance for investment decision-making.This study demonstrates the effectiveness of Python in financial analysis and portfolio optimization while furthering the theory of portfolios and providing valuable guidance for making investment decisions. Keywords: Returns, Risk, Volatility, Optimal Portfolio Construction, Defense Sector, Markowitz Portfolio Theory,

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.

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.

Capital allocation and RORAC optimization under solvency 2 standard formula

Solvency II Directive 2009/138/EC requires an insurance and reinsurance undertakings assessment of a Solvency Capital Requirement by means of the so-called “Standard Formula” or by means of partial or full internal models. Focusing on the first approach, the bottom-up aggregation formula proposed by the regulator allows for a capital reduction due to the diversification effect, according to the typical subadditivity property of risk measures. However, once the overall capital has been assessed no specific allocation formula is provided or required in order to evaluate the contribution of each risk source on the overall Solvency Capital Requirement. The aim of this paper is twofold. First, we provide a closed formula for capital allocation fully compliant with the Solvency II Capital Requirement assessed by means of the Standard Formula. The solution enables a top-down approach to assess the allocated Solvency Capital Requirement among the risks considered in the Solvency II multilevel aggregation scheme; we demonstrate that the allocation formula adopted is consistent with the Euler allocation principle. Second, a solution is found as a result of an optimum capital allocation problem based on a Return On Risk Adjusted Capital measure; we establish the equivalence between the Return On Risk Adjusted Capital optimization, when the risk adjusted capital is calculated according to the Standard Formula, and the Markowitz mean-variance optimization.

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.

Macro Asset Allocation with Social Impact Investments

Using a unique dataset of 50 listed companies that meet the majority of the OECD requirements for social impact investments, we construct a social impact finance stock index and investigate how investing in social impact firms can contribute to portfolio risk-return performance. We build portfolios with three different methodologies (naïve, Markowitz mean-variance optimization, GARCH-copula model), and we study the performance in terms of returns, Sharpe ratio, utility, and forecast premium based on a constant relative risk aversion function for investors with different levels of risk aversion. Consistent with the idea that social impact investment can improve portfolio risk-return performance, the results of our macro asset allocation analysis show the importance of a large fraction of investor portfolios’ stake committed to social impact investments.

Life Cycle Goal Achievement or Portfolio Volatility Reduction?

This paper is concerned with the use of currently available technology to provide individuals, financial advisors and pension fund financial planners with detailed prospective financial plans tailored to an individual's financial goals and obligations. By taking account of all prospective cash flows of an individual, including servicing current liabilities, and simultaneously optimizing prospective spending, saving, asset allocation, tax, insurance, etc. using dynamic stochastic optimization, the paper addresses the question of the title by comparing the results of such a goal-based fully dynamic strategy with representative current best practices of the financial advisory industry. These include piecemeal fixed allocation portfolios for specific goals, target-date retirement funds and fixed real income post-retirement financial products, all using Markowitz mean variance optimization based on the very general goal of minimizing portfolio volatility for a specific portfolio expected return over a finite horizon. Making use of the same data and market calibrated Monte Carlo stochastic simulation for all the alternative portfolio strategies, we find that flexibility turns out to be of key importance to individuals for both portfolio and spending decisions. The performance of the adaptive dynamic goal-based portfolio strategy is found to be far superior to all the industry’s Markowitz-based approaches. Superiority is measured here by the certainty equivalent increase in expected utility of individual lifetime consumption (gamma) and the extra initial capital required by an individual to put the dominated strategy on the same expected utility footing as the optimal dynamic strategy (initial capital gap). These empirical results should put paid to the commonly held view amongst finance professionals that the extra complexity of holistic dynamic stochastic models is not worth the marginal extra value obtained from their employment. We hope that such approaches implemented in currently available technologies will rapidly find acceptance by individuals, financial advisors and pension funds to the genuine benefit of individual investors.

Predictive blends: Fundamental Indexing meets Markowitz

Abstract When constructing a portfolio of stocks, do you turn a blind eye to the firms’ future outlooks based on careful consideration of companies’ fundamentals, or do you ignore the stocks’ correlation structures which ensure the best diversification? The Fundamental Indexing (FI) and Markowitz mean-variance optimization (MVO) approaches are complementary but, until now, have been considered separately in the portfolio choice literature. Using data on S&P 500 constituents, we evaluate a novel portfolio construction technique that utilizes the benefits of both approaches. Relying on the idea of forecast averaging, we propose to blend the two previously mentioned techniques to provide investors with a clear binocular vision. The out-of-sample results of the blended portfolios attest to their superior performance when compared to common market benchmarks, and to portfolios constructed solely based on the FI or MVO methods. In pursuit of the optimal blend between the two distinct portfolio construction techniques, MVO and FI, we find that the ratio of market capitalization to GDP, being a leading indicator for an overpriced market, demonstrates remarkably advantageous properties. Our superior results cannot be explained by classic asset pricing models.

Comment on: Kritzman, M. 2006, 'Are Optimizers Error Maximizers?'

Kritzman (2006) asserts that the widely cited Michaud (1989) study characterizing the impact of estimation error on Markowitz (1952) mean-variance (MV) optimization as “error maximization” is hype. His paper consists of two examples of MV portfolio optimizations: an eight-country index asset allocation with near identical returns and four-indices with very different estimates. In spite of the absence of standard references in estimation error, authors continue to cite the paper. In this brief note we demonstrate that even in these stylized and unrealistic examples, Kritzman’s MV optimized portfolios perform on average worse than equal weighting out-of-sample. A MV optimization worse than equal weighting has little practical investment value or interest. The impact of optimizer error maximization properly measured appears alive and very well.

Predictive Blends: Fundamental Indexing Meets Markowitz

When constructing a portfolio of stocks, do you turn a blind eye to the firms’ future outlooks based on careful consideration of companies’ fundamentals, or do you ignore the stocks’ correlation structures which ensure the best diversification? The Fundamental Indexing (FI) and Markowitz mean-variance optimization (MVO) approaches are complementary but, until now, have been considered separately in the portfolio choice literature. Using data on S&P 500 constituents, we evaluate a novel portfolio construction technique that utilizes the benefits of both approaches. Relying on the idea of forecast averaging, we propose to blend the two previously mentioned techniques to provide investors with a clear binocular vision. The out-of-sample results of the blended portfolios attest to their superior performance when compared to common market benchmarks, and to portfolios constructed solely based on the FI or MVO methods. In pursuit of the optimal blend between the two distinct portfolio construction techniques, MVO and FI, we find that the ratio of market capitalization to GDP, being a leading indicator for an overpriced market, demonstrates remarkably advantageous properties. Our superior results cannot be explained by classic asset pricing models.

A new method for better portfolio investment: A case of the Korean stock market

In this study, a method is devised to estimate a correlation matrix capable of constructing a well-diversified portfolio by the Markowitz mean-variance (MV) optimization function (MVOF), after which evidence is presented to empirically prove that the proposed method effectively reduces the sensitivity of portfolio output caused by the error of input variables, such as the mean and standard deviation of stocks in a portfolio. The proposed method removes the property of a market factor included in the sample correlation matrix through random matrix theory. The results demonstrate the comparative advantage of the proposed method in effectively reducing the sensitivity on both the estimation error and the prediction error from the mean and standard deviation. In particular, this comparative advantage is dependent on the striking reduction of portfolio risk gained by constructing the well-diversified portfolio. The proposed method also achieves high investment performance in the risk-return domain, and is particularly stronger in the unstable situation of either a market crash or a higher-risk portfolio. Consequently, this study offers new insight into how to enhance the practical applicability of the MVOF by controlling the property of the market factor in the sample correlation matrix.

A CVaR Scenario-Based Framework for Minimizing Downside Risk in Multi-Asset Class Portfolios

Multi-asset class (MAC) portfolios can be composed of investments in equities, fixed income, commodities, foreign exchange, credit, derivatives, and alternatives such as real estate and private equity. The return for such nonlinear portfolios is asymmetric with significant tail risk. The traditional Markowitz mean–variance optimization (MVO) framework, which linearizes all the assets in the portfolio and uses the standard deviation of return as a measure of risk, does not always accurately measure the risk for such portfolios. The authors consider a scenario-based conditional value at risk (CVaR) approach for minimizing the downside risk of an existing portfolio with MAC overlays. The approach consists of two phases: Phase 1 uses Monte-Carlo simulations to generate the asset return scenarios, and Phase 2 incorporates the return scenarios in a scenario-based convex optimization model to generate the overlay holdings. The authors illustrate the methodology in two examples involving the hedging of an equity portfolio with index puts and the hedging of a callable bond portfolio with interest rate caps. They compare the CVaR approach with parametric MVO approaches that linearize all the instruments in the MAC portfolio and show that the CVaR approach generates portfolios with better downside risk statistics; and further, it selects hedges that produce more attractive risk decompositions and stress test numbers—tools commonly used by risk managers to evaluate the quality of hedges.

Portfolio Selection via Subset Resampling

As the cornerstone of the modern portfolio theory, Markowitz's mean-variance optimization is a major model adopted in portfolio management. However, the estimation errors in its input parameters substantially deteriorate its performance in practice. Specifically, loss could be huge when the number of assets for investment is not much smaller than the sample size of historical data. To hasten the applicability of Markowitz's portfolio optimization to large portfolios, in this paper, we propose a new portfolio strategy via subset resampling. Through resampling subsets of the original large universe of assets, we construct the associated subset portfolios with more accurately estimated parameters without requiring additional data. By aggregating a number of constructed subset portfolios, we attain a well-diversified portfolio of all assets. To investigate its performance, we first analyze its corresponding efficient frontiers by simulation, provide analysis on the hyperparameter selection, and then empirically compare its out-of-sample performance with those of various competing strategies on diversified datasets. Experimental results corroborate that the proposed portfolio strategy has marked superiority in extensive evaluation criteria.

Two-faced property of a market factor in asset pricing and diversification effect

This study empirically investigates the test hypothesis that a market factor acting as a representative common factor in the pricing models has a negative influence on constructing a well-diversified portfolio from the Markowitz mean–variance optimization function (MVOF). We use the comparative correlation matrix (C-CM) method to control a single eigenvalue among all eigenvalues included in the sample correlation matrix (S-CM), through the random matrix theory (RMT). In particular, this study observes the effect of the largest eigenvalue that has the property of the market factor. According to the results, the largest eigenvalue has the highest explanatory power on the stock return changes. The C-CM without the largest eigenvalue in the S-CM constructs a more diversified portfolio capable of improving the practical applicability of the MVOF. Moreover, the more diversified portfolio constructed from this C-CM has better out-of-sample performance in the future period. These results support the test hypothesis for the two-faced property of the market factor, defined by the largest eigenvalue.

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.

Effects of common factors on stock correlation networks and portfolio diversification

This study empirically investigates the effects of common factors on the connectivity of the network among stocks and on the distribution of the investment weights for stocks. The network is defined as a stock correlation network from the minimal spanning tree (MST), and portfolio is defined as an efficient portfolio from the Markowitz mean-variance (MV) optimization function (MVOF). For these research goals, we devise a method using the comparative correlation matrix (C-CM), which does not have the property of a single common factor included in the sample correlation matrix (S-CM). The results reveal that common factors clearly affect the changes of connectivity among stocks in the networks, and that their influence is much greater on stocks with many links to other stocks in the network. Further, common factors significantly affect the determination of the investment weight's distribution for stocks from the MVOF. In particular, among the common factors, a market factor plays a dominant role in both structuring the network among stocks and in constructing the well-diversified portfolio. In addition, the devised method of the C-CM without the property of the market factor in the S-CM plays a crucial role in constructing a more diversified portfolio with better out-of-sample performance in the future period. These results are robust in both the Korean and the U.S. stocks markets.

Markowitz's mean-variance optimization with investment and constrained reinsurance

This paper deals with the optimal investment-reinsurance strategy for an insurer under the criterion of mean-variance. The risk process is the diffusion approximation of a compound Poisson process and the insurer can invest its wealth into a financial market consisting of one risk-free asset and one risky asset, while short-selling of the risky asset is prohibited. On the side of reinsurance, we require that the proportion of insurer's retained risk belong to $[0, 1]$, is adopted. According to the dynamic programming in stochastic optimal control, the resulting Hamilton-Jacobi-Bellman (HJB) equation may not admit a classical solution. In this paper, we construct a viscosity solution for the HJB equation, and based on this solution we find closed form expressions of efficient strategy and efficient frontier when the expected terminal wealth is greater than a certain level. For other possible expected returns, we apply numerical methods to analyse the efficient frontier. Several numerical examples and comparisons between models with constrained and unconstrained proportional reinsurance are provided to illustrate our results.

Lifecycle Goal Achievement or Portfolio Volatility Reduction?

This article is concerned with the use of currently available technology to offer individuals, financial advisors, and pension fund financial planners detailed prospective financial plans tailored to an individual’s financial goals and obligations. By taking account of all an individual’s prospective cash flows, including servicing current liabilities, and simultaneously optimizing prospective spending, saving, asset allocation, tax, and insurance, etc. using dynamic stochastic optimization, the authors compare the results of their goal-based fully dynamic strategy with the financial advisory industry’s representative current best practices. These include piecemeal fixed-allocation portfolios for specific goals, target-date retirement funds, and fixed real-income post-retirement financial products, all using Markowitz mean-variance optimization based on the very general goal of minimizing portfolio volatility for a specific portfolio expected return over a finite horizon. Making use of the same data and marketcalibrated Monte Carlo stochastic simulation for all the alternative portfolio strategies, the authors find that flexibility is of key importance for both individual portfolio and spending decisions. The authors measure superiority by the certainty-equivalent increase in expected utility of individual lifetime consumption (gamma) and the extra initial capital required by an individual to put the dominated strategy on the same expected-utility footing as the optimal dynamic strategy (initial capital gap). They find that the adaptive dynamic goal-based portfolio strategy’s performance is far superior to all the industry’s Markowitz-based approaches. These empirical results should put paid to the commonly held view that the extra complexity of holistic dynamic stochastic models is not worth the marginal extra value obtained from their use.

Morningstar vs. Michaud Optimization

Classical linear constrained Markowitz (1952, 1959) mean-variance (MV) optimization has been the standard for defining portfolio optimality for more than fifty years. However, Markowitz efficient portfolios are known in practical application to be unstable and highly sensitive to estimation error in risk-return inputs. Michaud optimization (1998, 2008a, 2008b) is a U.S. patented generalization of linear constrained Markowitz MV efficiency that uses modern statistical resampling technology to address estimation error and instability in portfolio optimization. The Morningstar® Encorr® software also features a portfolio optimization procedure that uses the terms “resampling” and “resampled frontiers.” In this report we discuss the similarities and differences of the two methods and illustrate the results using identical inputs and portfolio optimality criteria. We show that the procedures are fundamentally different and the results typically very dissimilar. While the Michaud portfolios are investment intuitive, stable, and well diversified across the entire efficient frontier the Morningstar portfolios are often inconsistent with sensible perceptions of diversification and generally reflect serious limitations as alternatives to MV optimization limitations. The lack of theoretical framework for the procedure and the non-uniqueness of the solutions defeats Morningstar claims of superior investment value relative to Markowitz or Michaud optimality.

Evaluating New Methodologies in Asset and Risk Allocation

Risk parity strategies favor bonds over equities, but they do not necessarily outperform a static portfolio of 60% equity/40% bonds. Portfolio optimization is evolving beyond traditional Markowitz mean–variance optimization to incorporate the higher moments of skewness and kurtosis, which are applicable to alternative investments.