Mean-variance Efficient Frontier Research Articles

Computing cardinality constrained portfolio selection efficient frontiers via closest correlation matrices

In this paper, we demonstrate a completely new approach for computing cardinality constrained mean-variance efficient frontiers. By cardinality constrained, it is meant that if there is to be investment in a security, it is to be of at least some minimum amount (a buyin threshold), and that there is also a specification on the number of securities to be held in a portfolio (called a cardinality constraint). Whereas the usual strategy, as such problems are NP-hard, is to take the original exact problem and apply heuristics to solve, in this paper the strategy is to perturb the original problem and then apply exact procedures to solve. The advantages of the approach are that the perturbations are tiny, they are only applied to the problem’s correlation matrix, and they allow for the accurate computation of cardinality constrained efficient frontiers in problems with up to at least 1000 securities in remarkably little time. Moreover, the simplicity of the approach is such that it can be inserted into existing portfolio management systems without requiring any re-training beyond what a typical portfolio analyst would already know.11The authors have benefited from associations with Markus Hirschberger and Sebastian Utz in this research.

A Study on the Criteria for Investment in the Bond Market

This paper uses relevant information to illustrate some of the factors that influence investors' choice of bonds: by analyzing a cross-section of firms to conclude that insurance companies with stronger regulatory capital constraints tend to seek higher yields; according to the findings of a study into the added value of inflation-linked bonds in a portfolio, inflation-linked bonds widen the mean-variance efficient frontier for investors holding nominal bonds and stocks by simulating credit spreads that are more like those actually seen in the market and modeling leverage as a mean-reversion; and the findings indicate that market liquidity is neutrally or favorably impacted by portfolio trading transparency. This was established by using a controlled experiment to investigate the impact of transparency on the liquidity of the BBB bond market. In addition, some suggestions are put forward for some related investors, and the future research direction is also proposed at the end of the article.

Non-contour efficient fronts for identifying most preferred portfolios in sustainability investing

The paper focuses on investors whose strength of interest in sustainability issues (such as environmental, social, and governance) causes ESG to become a third criterion alongside risk and return in portfolio selection. This causes the efficient frontier to become an efficient surface. This means that an investor’s optimal portfolio is no longer the point of most preferred risk/return tradeoff on the mean-variance (M-V) efficient frontier, but is the point of most preferred risk/return/ESG tradeoff on the investor’s M-V-ESG efficient surface. However, to find such a point requires non-trivial ESG integration which is the name given to the process of integrating ESG into the portfolio construction process after screening. With the third objective transporting the problem into 3D-space, it is difficult to search the efficient surface in any kind of comprehensive fashion using M-V based or other bi-criterion techniques as this is akin to a 2-dimensional being trying to view a 3-dimensional object. To remedy the situation, the paper proposes a tri-criterion approach that computes efficient surfaces and special non-contour curves (called NC-efficient fronts in the paper) that are stretched across the efficient surface so as to dragnet it for the points of best ESG integration within it. Using the methodology and data from the S&P500, the paper conducts computational tests on problems with up to 500 securities and under different constraint conditions so as to know what to expect from the new approach over a range of situations.

A proposal for measuring efficiency losses of asset management companies: Frontier-based approach

The performance of funds by asset management companies needs to be based on an objective benchmark. Studies related to the topic focus on measuring performance using a discrete form that cannot capture precise total efficiency losses. In this study, we propose a continuous approach to compare the performance of funds by taking advantage of the mean-variance efficient frontier with consideration of multiple risk levels. In an empirical analysis, our proposed method is applied to asset management companies with respect to open-end funds. For comparison, we use the output of an averaged Sharpe index. Because averaging the Sharpe index is inevitable for multiple risk levels, this method of calculation causes a loss of efficiency information. Hence, the proposed method has a continuous form of measuring performance, and the results of the two methods demonstrate significantly different patterns.

How Does Risk Hedging Impact Operations? Insights from a Price-Setting Newsvendor Model

Financial asset price movement impacts product demand and thus influences operational decisions of a firm. We develop and solve a general model that integrates financial risk hedging into a price-setting newsvendor. The optimal hedging strategy is found analytically, which leads to an explicit objective function for optimization of pricing and service levels. We find that, in general, the presence of hedging reduces the optimal price. It also reduces the optimal service level when the asset price trend positively impacts product demand ("asset price benefits demand"), while it may increase the optimal service level by a small margin when the impact is negative ("asset price hurts demand"). We construct the mean-variance efficient frontier that characterizes the risk-return trade-off, and we quantify the risk reduction achieved by the hedging strategy. Our numerical case study using real data of Ford Motor Company shows that the markdown in price and decrease in service level are small under our model, and the hedging strategy substantially reduces risk without materially reducing operational profit.

Mean–Variance Hedging for Production Planning with Multiple Products

We study production planning in a multi‐product setting, in which demand for each product depends on multiple financial assets (such as commodities, market indices, etc). In addition to the production quantity decision at the beginning of the planning horizon, there is also a real‐time hedging decision throughout the horizon; and we optimize both decisions jointly. With a mean–variance problem formulation, we first derive the optimal hedging strategy, given the production quantities. This leads to an explicit objective function with which bounds on optimal production quantities are identified. Thus, optimization of the production policies can be readily solved numerically as a static minimization problem. This way, we are able to give a complete characterization of the mean–variance efficient frontier, and quantify the contribution of the hedging strategy by the variance reduction it achieves. Furthermore, the model and results are extended to allow dynamic production control that tracks the demand rates.

Reconciling mean-variance portfolio theory with non-Gaussian returns

Mean-variance portfolio theory remains frequently used as an investment rationale because of its simplicity, its closed-form solution, and the availability of well-performing robust estimators. At the same time, it is also frequently rejected on the grounds that it ignores the higher moments of non-Gaussian returns. However, higher-moment portfolios are associated with many different objective functions, are numerically more complex, and exacerbate estimation risk. In this paper, we reconcile mean-variance portfolio theory with non-Gaussian returns by identifying, among all portfolios on the mean-variance efficient frontier, the one that optimizes a chosen higher-moment criterion. Numerical simulations and an empirical analysis show, for three higher-moment objective functions and adjusting for transaction costs, that the proposed portfolio strikes a favorable tradeoff between specification and estimation error. Specifically, in terms of out-of-sample Sharpe ratio and higher moments, it outperforms the global-optimal portfolio, and also the global-minimum-variance portfolio except when using monthly returns for which estimation error is more prominent.

Mean-Variance Portfolio Analysis for Continuous Compound Return

We consider the dynamic mean-variance portfolio optimization problem in a continuous-time frame- work. Our objective is to maximize the expected continuous compound return and minimize its variance. We derive the optimal investment strategy and present a closed form mean-variance efficient frontier. We also show that the mean-variance optimization is equivalent to the CRRA utility maximization. The connection between the mean-variance analysis and the Kelly criterion is discussed, and we show that the mean-variance optimal investment strategy is indeed the fractional Kelly criterion.

Model Selection with Transaction Costs

Failing to account for transaction costs materially impacts inferences drawn when evaluating asset pricing models, biasing tests in favor of those employing high cost factors. Ignoring transaction costs, the Hou, Xue, and Zhang (2015) q-factor model and the Barillas and Shanken (2018) six-factor model models have high maximum squared Sharpe ratios and small alphas across 120 anomalies. They do not, however, come close to spanning the achievable mean-variance efficient frontier. Accounting for transaction costs, the Fama and French (2015, 2018) five-factor model has a significantly higher squared Sharpe ratio than either of these alternative models, while variations employing cash profitability perform better still. More generally, these results highlight the importance of incorporating real-world concerns into financial research.

Practical Implementation of the Kelly Criterion: Optimal Growth Rate, Number of Trades, and Rebalancing Frequency for Equity Portfolios

We develop a general framework to apply the Kelly criterion to the stock market data, and consequently, to portfolio optimization. Under few conditions, using Monte Carlo simulations with different scenarios we prove that the Kelly criterion beats any other approach in many aspects. In particular, it maximizes the expected growth rate and the median of the terminal wealth. We also show that, under a normal distribution of returns, the Kelly criterion has the best performance in the long run. Next, we optimize a portfolio with the Kelly criterion with no leverage and no short selling conditions and show that this portfolio lays in the mean-variance efficient frontier and has higher expected return and higher variance, although it is less diversified, respect to the tangent portfolio optimized under the Markowitz approach. Finally, we implement a dynamic strategy applied on the European stock market data and compare the results between the tangent and the optimal Kelly portfolios. In a dynamic setting, the rolling Kelly portfolio outperforms competitors particularly in the case of rebalanced portfolios optimized with a 2-years window width.

Second order of stochastic dominance efficiency vs mean variance efficiency

In this paper, we compare two of the main paradigms of portfolio theory: mean variance analysis and expected utility. In particular, we show empirically that mean variance efficient portfolios are typically sub-optimal for non satiable and risk averse investors. We illustrate that the second order stochastic dominance (SSD) efficient set is the solution of a multi-objective optimization problem. We further show that the market portfolio is not necessarily a solution to this optimization problem. We also conduct an empirical analysis, examining the ex ante and ex post performance of SSD and mean variance efficient portfolios, using a bootstrap approach. In an ex ante analysis, we compare empirical moments, the level of diversification and set distances of mean variance and SSD efficient sets. We also show that the global minimum variance (GMV) portfolio and the part of the mean variance efficient frontier (MVEF) composed of highly diversified portfolios is second order stochastically dominated. This result also provides a possible alternative explanation for the diversification puzzle. Conducting an ex post analysis, we construct second order stochastic dominating strategies that outperform the GMV portfolio in terms of wealth and various other performance measures, producing a positive ex post opportunity cost.

A machine learning efficient frontier

We propose a simple approach to bridge between portfolio theory and machine learning. The outcome is an out-of-sample machine learning efficient frontier based on two assets, high risk and low risk. By rotating between the two assets, we show that the proposed frontier dominates the mean–variance efficient frontier out-of-sample. Our results, therefore, shed important light on the appeal of machine learning into portfolio selection under estimation risk.

The Effects of Multi-Slice Helical Computed Tomography and Phlebography on Edema Regression Nursing of Lower Limbs Varicosis

Objective: To explore the effects of multi-slice helical computed tomography (MSCT) and phlebography on edema regression nursing of lower limbs varicosis, and to analyze the values of MSCT and phlebography on the diagnosis and treatment of lower limb varicosis. Methods: A total of 193 patients with lower limb varicosis treated in the Vascular Surgery Department were included as the research objects. According to different examination methods, the included patients were divided into the MSCT group, and the MSCT combined with the phlebography group, Then, patients in both groups received joint treatment of high ligation and varicectomy. The Gabor filtering-based algorithm and the Mean-Variance Efficient Frontier (MVEF) algorithm were used to smoothen the obtained images. Also, the operation rates and contradiction compositions of patients in both groups were analyzed. Patients who received surgical treatment were followed up for three months to one year; whether the deep vein thrombosis (DVT) occurred in them were observed, as well as the edema regression. Results: As for gender, grading, disease history, duration of disease, and onset age of both groups, no statistical significance was found (P > 0.05). Compared to group A, the positive detection rate of postoperative DVT was lower in group B, so did the operation rate. In group A, 32 cases of edema regressed significantly, and the disease condition was relieved. In group B, 35 cases of edema were regressed. Conclusion: Compared with MSCT, phlebography had a higher screening rate for indications of lower limb varicosis, which could effectively reduce the probability of DVT after operations and relieve the symptoms of edema in patients. In addition, although the sensitivity of MSCT was relatively low, it could be used for the primary screening diagnosis of lower limb varicosis.

The Shapley value of regression portfolios

By viewing portfolio optimization as a cooperative game played by the assets minimizing risk for a given return, investors can compute the exact value each security adds to the common payoff of the game. This is known the Shapley value that imputes the contribution of each asset, by looking at all the possible portfolios in which securities might participate. In this paper I use the Shapley value to decompose the risk and return of optimal portfolios that result from minimizing ordinary least squares. These regression portfolios are identical to tangency portfolios obtained by maximizing the Sharpe ratio of holdings on the mean-variance efficient frontiers. The Shapley value of individual assets is computed using the statistics resulting from the regressions. The value imputation prices assets by their comprehensive contribution to portfolio risk and return. This procedure allows investors to make unbiased decisions when analyzing the inherent risk of their holdings. By running OLS regressions, the Shapley value is calculated for asset allocation using Ibbotson’s aggregate financial data for the years 1926–2019.

A Machine Learning Efficient Frontier

We propose a simple approach to bridge between portfolio theory and machine learning. The outcome is an out-of-sample machine learning efficient frontier based on two assets, high risk and low risk. By rotating between the two assets, we show that the proposed frontier dominates the mean-variance efficient frontier out-of-sample. Our results, therefore, shed important light on the appeal of machine learning into portfolio selection under estimation risk.

Comparative Analysis of Mean Variance Efficient Frontier and Resampled Efficient Frontier For Optimal Stock Portfolio Formation

One way to form an optimal stock portfolio is to use the mean-variance efficient frontier (MVEF) method. However, the MVEF method is susceptible to changes in input. An optimal stock portfolio formation method has been developed to overcome this problem, known as the Resampled Efficient Frontier (REF) method. The REF method is a refinement of the MVEF method. REF is done by generating data repeatedly using the Montecarlo simulation. The parameters needed to form a stock portfolio using REF are the average input return μ and the covariance variance matrix Σ. The calculation results show that the weight of the REF portfolio is not negative (there is no short selling strategy). The Sharpe ratio of the REF portfolio is also higher than the MVEF. A larger Sharpe ratio indicates that the REF portfolio has a better performance compared to MVEF. However, empirically based on a case study simulation, the MVEF method experienced a smaller loss compared to the REF method.

Mean–Variance Portfolio Selection Under Volterra Heston Model

Motivated by empirical evidence for rough volatility models, this paper investigates continuous-time mean-variance (MV) portfolio selection under the Volterra Heston model. Due to the non-Markovian and non-semimartingale nature of the model, classic stochastic optimal control frameworks are not directly applicable to the associated optimization problem. By constructing an auxiliary stochastic process, we obtain the optimal investment strategy, which depends on the solution to a Riccati-Volterra equation. The MV efficient frontier is shown to maintain a quadratic curve. Numerical studies show that both roughness and volatility of volatility materially affect the optimal strategy.

Development of an efficient cluster-based portfolio optimization model under realistic market conditions

Modern portfolio theory introduced by Markowitz in 1952 is the most popular portfolio optimization framework established based on the trade-off between risk and return as an operation research model. The main shortcoming of applying Markowitz portfolio optimization in practice is that the obtained optimal weights are really sensitive to the embedded uncertainty in return series of stocks. In this paper, it is demonstrated how using a new methodology of time series clustering as a remedy can lead to a more robust and accurate portfolio in terms of the gap between mean variance efficient frontier obtained from the optimization model and the one observed in reality. In this regard, two similarity measures, the autocorrelation coefficients and the weighted dynamic time warping, are used in an innovative way to construct the desired portfolio optimization model. Moreover, the effectiveness of proposed approach is investigated in two different market conditions: semi-realistic and full-realistic. In the first one, it is assumed that the forecasted and realized stocks mean returns are the same; however, these returns are not necessarily equal in the second market conditions. Finally, a database of stock prices from the literature is utilized to show the robustness and accuracy of the proposed approach in empirical results in comparison with applied similarity measures in previous researches.

The Hansen Ratio in Mean-Variance Portfolio Theory

It is shown that the ratio between the mean and the L2-norm leads to a particularly parsimonious description of the mean-variance efficient frontier and the dual pricing kernel restrictions known as the Hansen-Jagannathan (HJ) bounds. Because this ratio has not appeared in economic theory previously, it seems appropriate to name it the Hansen ratio. The initial treatment of the mean-variance theory via the Hansen ratio is extended in two directions, to monotone mean-variance preferences and to arbitrary Hilbert space setting. A multi-period example with IID returns is also discussed.

John Bogle Had It Right: The Benefits of International Diversification Are Limited

Stock index research over long-term periods supports a U-shaped relationship of the total risk to the allocation to foreign stocks (adding foreign equities lowers risk at first then increases it). This study constructs mean–variance efficient frontiers using 28 country exchange-traded funds (ETFs) available to US investors over two periods: 1996–2018 (mostly developed countries) and 2010–2018 (many emerging markets) to determine the optimal global allocation. Over both periods, the US-only ETF had the highest Sharpe ratio. For 1996–2018, the empirical mean–variance “market” portfolio of 17 country ETFs had a 94.5% allocation to the United States, and the minimum-variance portfolio had 56.55% in the United States. Over the 2010–2018 period, the empirical tangent portfolio of 28 country ETFs had a 100% allocation to the United States, and the minimum-variance threshold for efficiency was 52.65%. For both periods, global portfolios based on gross domestic product or market capitalization weights were highly inefficient. Home-country bias may not be a bias at all. Perhaps investors just rationally maximize the return-to-risk trade-off. <b>TOPICS:</b>Wealth management, exchange-traded funds and applications, global markets <b>Key Findings</b> • The empirical “market portfolio” of 28 country ETFs over 2010–2018 shows a 100% allocation to the United States. • The empirical “market portfolio” of 17 country ETFs over 1996–2018 shows a 94.5% allocation to the United States. • Strong US home country bias mean-variance dominates cap- or GDP-weighting in building global index portfolios.