Markowitz Efficient Frontier Research Articles

Comparative Analysis of Portfolio Management Models: Markowitz, CAPM, and Multi-Factor Models

This paper presents a comparative analysis of three foundational portfolio management models: the Markowitz Efficient Frontier, the Capital Asset Pricing Model (CAPM), and Multi-Factor Models. It aims to explore their theoretical foundations, empirical applicability, and practical usage to assess their strengths and limitations in financial portfolio management. The Markowitz model is appreciated for its detailed risk-return optimization framework but criticized for its reliance on historical data and computational demands. The CAPM simplifies risk assessment through market beta, yet its assumptions of market efficiency and a risk-free rate are viewed as impractical under current market conditions. Multi-Factor Models address some limitations of the CAPM by incorporating various risk factors, offering enhanced explanatory power but also adding complexity in factor selection.The study concludes that while no single model is universally superior, the choice of model should depend on the specific needs of the portfolio manager, including risk-return objectives and market environment. The paper suggests a blended or adaptive approach to model selection, highlighting the need for flexibility in response to evolving market dynamics and advances in computational finance.

Investable Real Estate Allocations in a Mixed Asset Portfolio; Both Long Term and During Different Cycles

The NCREIF Property Index (NPI) data starts in 1978 and has been used as a benchmark index for over 45 years. In 2006 NCREIF created the NCREIF Open-End Diversified Core Equity Index (ODCE) and used historic fund level data to create performance data back to 1978. ODCE was the first “investable index ” in direct real estate for institutional investors, family offices, and high net worth investors who can meet these direct funds’ minimum investment levels (typically $5 million). Individual investors with less money can also access ODCE returns through Interval Funds or “Fund-of-Funds” that invest in and are designed to track or beat the ODCE Index. Since 2009 general partners of funds are required to update their net asset value (NAV) to “fair market value” on a quarterly basis as a result of Topic 820 of the Financial Accounting Standards Board. Both ODCE funds and interval funds have improved their liquidity with either monthly or quarterly redemption options, making them more competitive (from a liquidity aspect) with publicly traded securities. We analyze portfolio allocations over 45 years – the longest time frame ever studied and over 6 NEBR economic cycles and 4 ODCE real estate return cycles using Markowitz efficient frontier analysis. We inspect up-cycle and down-cycle periods separately and add to the literature by analyzing the ¼, ½, and ¾ points (low, medium, and high risk/return points) along the Markowitz efficient frontier. Our findings support many of the 45 studies published, but conflict with some depending upon their study methodology and time frame (9 to 25 years) studied. We conclude that real estate would have improved historic risk adjusted returns in many cycle periods.

Developing and Evaluating Sector Rotation Strategy for Indian Equity Markets

This study examines the effectiveness of rotating in and out of a sector based on the stage of a business cycle. Here, we assume that business cycles exist, are part of nature, and neither repeat at a regular intervals nor their intensity or duration are constant. However, we assume that with a perfect set of high-frequency indicators, an investor can perfectly time the business cycle phases. This study divides India's real GDP growth into High, Moderate, Low, and Minuscule growth. GDP Growth is used as an indication of a start/end of a business cycle. In an investor were to create four different portfolios, optimized for each of the business cycle phases, will he/she be able to outperform the SENSEX returns of the buy-and-hold strategy? Here, an attempt to create such four models has been done using Python based on Markowitz Efficient Frontier and Modern Portfolio Theory. Once developed using historical data from September 2005 to March 2020, the study tries to estimate its performance in out-of-sample data, i.e., June 2021-April 2023. Can an investor make higher absolute (not alpha) returns with perfect foresight? We try to evaluate such in this study. Some obvious drawbacks of this model are inapplicability for large investors, mutual funds, etc. However, an investor with small capital and access to data can have an accurate foresight, thus, applying the model in practice can earn higher absolute returns. Even with a tax consideration of moving in and out of sectors, an investor can make a higher absolute return compared to a buy-and-hold strategy.

Tail risk adjusted clean energy portfolios in P2P transactive markets using Rachev ratio

AbstractRenewable energy sources (RES) and electric vehicles (EVs) pose ‘energy‐risk’ in peer energy commitments due to their temporal and spatial uncertainties. Thus, optimistic commitments in the peer‐to‐peer transactive energy market (P2P TEM) are improbable. This paper proposes a two‐stage master–slave portfolio optimization approach for combining energy‐risk of RES and EVs, and welfare‐risk of peers, in building clean energy portfolios. The master portfolio (MP) refers to the shares of renewable and EVs in P2P market settlement, whereas the slave portfolio (SP) gives the wind‐solar mix within renewables. Here, Rachev Ratio (RR), an index used in financial portfolio selection for tail‐risk management, is adapted and combined with Markowitz Efficient Frontier (EF) to find the optimal slave portfolio. Both the extreme tails are optimized, encouraging energy outputs far above forecast and discouraging those far below forecast. The master portfolio is obtained by maximizing the sum of the average welfare of the peers at the best (right) and worst (left) tails of the welfare distribution curve using Stochastic Weight Trade‐off Particle Swarm Optimization (SWT‐PSO). The proposed portfolio selection approach is better in terms of increased expected energy output, improved utilization of RES and EVs, and better collective peer welfare.

Portfolio Management for Multi-industry

In the financial field, portfolio management is an important measure in the direction of investment or hedging. This paper mainly focuses on the optimization for the portfolio composed of assets from five industries, which is education, banking, automobile manufacturing, parts industry and e-commerce, and considers the allocation of assets to optimize the returns. In this paper, five representative assets from these five industries are selected. The Markowitz efficient frontier is plotted by Monte-Carlo method, using the return data of assets. Then the portfolio is optimized by mean-variance analysis and the maximum Sharpe ratio portfolio as well as minimum variance portfolio can be calculated. Finally, this paper analyzes the performance of the two portfolios, considering the influence of individual assets on the portfolio weight, and uses the Fama-French three factor model to analyze the performance of the portfolio. The results show that PTAIY and PSO from parts manufacturing and education occupy a large proportion in the maximum Sharpe ratio portfolio as well as the minimum variance portfolio. The findings could help investors interested in these five areas.

Bridging the Gap Between Markowitz Planning and Deep Reinforcement Learning

While researchers in the asset management industry have mostly focused on techniques based on financial and risk planning techniques like Markowitz efficient frontier, minimum variance, maximum diversification or equal risk parity, in parallel, another community in machine learning has started working on reinforcement learning and more particularly deep reinforcement learning to solve other decision making problems for challenging task like autonomous driving, robot learning, and on a more conceptual side games solving like Go. This paper aims to bridge the gap between these two approaches by showing Deep Reinforcement Learning (DRL) techniques can shed new lights on portfolio allocation thanks to a more general optimization setting that casts portfolio allocation as an optimal control problem that is not just a one-step optimization, but rather a continuous control optimization with a delayed reward. The advantages are numerous: (i) DRL maps directly market conditions to actions by design and hence should adapt to changing environment, (ii) DRL does not rely on any traditional financial risk assumptions like that risk is represented by variance, (iii) DRL can incorporate additional data and be a multi inputs method as opposed to more traditional optimization methods. We present on an experiment some encouraging results using convolution networks.

The SOCIMIs in Spain. An investment opportunity in the real estate sector

The purpose of the present work is to analyze the SOCIMIs (Real estate listed investment trusts in Spain), its performance and its potential as a diversification asset between June 2014 and December 2017. The tax advantages they offer, the main incentive for the creation of these types of companies, are bringing about an increase in development of real estate investment in Spain, which is behind other developed markets where they are generally called REITs (Real Estate Investment Trusts). First, the characteristics, operation, evolution in the market, portfolios and shareholders’ composition are studied. Next, the analysis shows high performance levels measured by the modified Sharpe Ratio and the Jensen’s Alpha for the main SOCIMIs on the market as well as a high portfolio diversification potential demonstrated by the Markowitz Efficient Frontier Model. SOCIMIs are characterized by being less risky investments than the market in general. Compared to traditional Real estate listed companies on the continuous market, SOCIMIs show better performance and lower levels of risk and, in general, all the companies studied –SOCIMIs and traditional Real estate listed companies- can be defined as defensive investments, less risky than the market.

The Probability Frontier or, Covariance Crunch: A New Paradigm for Mean-Variance Optimization

Three innovative concepts are combined here to create a new and unique framework for optimizing a portfolio of investments or bets. These inventions are: 1) The Probability Frontier, a generalization of the Markowitz Efficient Frontier; 2) The Positive Probability Estimate, which estimates the chance of a market to move up in the coming period, regardless of the magnitude of the move; and 3) The Directional Covariance, which measures the tendency of markets to move together, regardless of the size of the moves. Looking at both market moves and their covariances in terms of pure directionality, treating their magnitudes as “noise” on the premise (demonstrated decisively in this paper) that direction is more robustly predictable, dramatically improves the optimized system performance. Indeed, if our estimates can’t even get market directions right, what’s the point of quibbling over tenths of a percentage point? Don’t shrink that covariance matrix; crush it!

Growth‐instability frontier and industrial diversification: Evidence from European gross value added

AbstractThe paper applies the modern portfolio theory and stochastic frontier analysis to gross value added growth rates in Europe and presents evidence of convex growth‐instability frontiers. Using their national accounts by economic activity, numerical simulations of gross value added shares show that in 2011, industry portfolios were below Markowitz's efficient frontier and that rearranging the share of economic activities lead to efficiency gains in terms of lower output instability and higher growth rates. Gross value added in EU countries in general, and core countries in particular, should be contributed by the “service provider” and “financial activities” sectors, while optimal industry portfolios of periphery countries are more diversified. Stochastic frontier analysis confirms the convex growth‐instability frontiers and finds that technical inefficiencies are explained by differences in the growth rate of labour and the degree of industrial diversification. Getting more efficient is a combination of country characteristics and sector specifics, where efficiency gains are inversely related to industrial diversification during economic development. Reported results are robust to model specifications and data restrictions.

The Impact Made on Project Portfolio Optimisation by the Selection of Various Risk Measures

In this study, the effect of selecting an appropriate risk measure and the impact of this choice on the efficient frontier of the organization’s project portfolio, has been studied. Proper choice of firm’s project portfolio has a great impact on organizational success. Each portfolio manager selects the best projects with different criteria, and consistent with firm’s strategic objectives. We have selected the best projects of the organization, using Markowitz efficient frontier method. The choice of proper measures is influential in making this decision and can change the organization’s portfolio. For doing this, the standard deviation is applied and the relevant optimization have been done. Then, in order to differentiate favorable and unfavorable opportunities, semi-standard deviation has also been applied. Afterwards, Value at Risk and Expected Shortfall are applied as the appropriate risk measures, to have a better estimate of the tail risks. With all of these risk measures the best possible projects were selected. Managers should select the appropriate risk measures according to their objectives, estimation of their projects’ distribution, and characteristic of the projects. In this research, the best measures consistent with construction projects and the effect of changes in these measures have been studied.DOI: http://dx.doi.org/10.5755/j01.ee.29.2.17405

Building the Markowitz Efficient Frontier

In this paper we combine some algorithms in the literature in order to effectively build the Markowitz efficient frontier, as a solution of the optimal asset allocation problem. The proposed method works for any size of the investment portfolio, and for any reasonable equality and inequality constraints.

Valuación de una nota estructurada que vincula el rendimiento de un bono cupón cero con una opción en un portafolio de inversión

This paper poses the creation of a structured note consisting of assets from three different markets: fixed income, equity and derivatives. To value the zero-coupon bond it is assumed that the interest rate is stochastic and driven by a Cox, Ingersoll and Ross (1985) process. Stocks from the IPC (Mexican Stock Exchange Index) were used to create investment portfolios with aggressive and conservative risk profiles. The optimal portfolio tangent to the Markowitz (1952) efficient frontier was determined using the Sharpe´s (1970) model. Subsequently, a call option on the investment portfolio is valued by using Monte Carlo simulation. The results show that an economic agent with an aggressive risk profile investment porfolio will be better off investing in the suggested structured note.

Emergence of Commodity Derivatives as Defensive Instrument in Portfolio Risk Hedging: A Case of Indian Commodity Markets

Abstract This paper empirically examines whether commodity derivatives can be used as an alternative investment asset in India where commodity markets are at emerging state and provides the same diversification benefit as they provide in developed commodity markets. In India only commodity futures are prevalent so various commodity indices representing various sectors has been used in the study. Diversification aspect of commodity derivatives has been tested initially by using correlation analysis. Compounded Daily Growth rate and Relative Standard deviation has been used as a measure of calculating risk and return of daily data of SENSEX, BOND and four Commodity Indices (MCX Comdex, MCX AGRI, MCX Metal, MCX Energy). Markowitz Efficient Frontier theory has been used to calculate portfolio risk return and Sharpe risk adjusted ratio has been used to evaluate the various portfolios. Optimal portfolio has been obtained for the combination of equity, bond and commodity and overall results of the study indicate that an investor who is risk averse will prefer to invest in combination of SENSEX, BOND & MCX Energy whereas an investor who gets utility by taking more risk for more returns will prefer to invest in combination of SENSEX, BOND & MCX Metal. Investor having inclination towards moderate risk return would tend to invest in MCX AGRI along with SENSEX and BOND.

Leverage and Uncertainty

How risk and uncertainty are perceived depends on experience, luck, skills, and modelling, and, unsurprisingly, they are hard to disentangle. Yet asset allocation – leverage – should depend on accurately assessing quantifiable return and risk and unquantifiable uncertainty. Looking beyond risk and evaluating how uncertainty constrains optimal leverage is explored by extending the Kelly Criterion to a simple probabilistic model with an additional tail risk outcome associated with uncertainty. For a single asset, the fractional Kelly Criterion is justified by the negative skew and positive kurtosis of the probability distribution. The Portfolio Theory based on the Kelly Criterion is shown to be a general theory, which includes the Markowitz Portfolio Theory and Risk Parity as limiting cases. The fat tail distributions we see in markets, which stem from the transient nature and uncertainty of markets, underscore loss aversion and the use of leverage. Markowitz efficient risk-return frontier becomes concave due to the loss aversion and fat tails. Decisions about uncertainty and leverage are among the most important aspects of financial markets and global macro investing.

Volatility and Correlations for Stock Markets in the Emerging Economies of Central and Eastern Europe: Implications for European Investors

This paper examines the implications for European investors of the recent EU expansion to encompass former Eastern block economies. What were the risk and return characteristics of these markets pre- and post-EU? What are the implications for investors within the Euro zone? Should investors diversify outside the CEE? The former Eastern block economies constitute emerging markets which typically offer attractive risk adjusted returns for international investors. In this paper, we explore a number of aspects of this important issue and their implications for CEE based investors culminating in a Markowitz efficient frontier analysis of these markets pre- and post-EU expansion. The results suggest the ‘advanced emerging’ markets of Poland, Hungary, and Czech Republic are less attractive post-EU to a portfolio based investor seeking diversification benefits than the ‘other emerging markets.’

Tax Advantaged Roth Conversions

The recharacterization option for conversion of a traditional IRA to a Roth IRA has a non-negative value, just like most other options. The purpose of this paper is to establish that value under a variety of scenarios. A Markowitz Efficient Frontier is established relating the Rate of Return to the standard deviation of the Rate of Return.

High Dimensionality Effects on the Efficient Frontier: A Tri-Nation Study

Markowitz Portfolio theory under-estimates the risk associated with the return of a portfolio in case of high dimensional data. El Karoui mathematically proved this in [1] and suggested improved estimators for unbiased estimation of this risk under specific model assumptions. Norm constrained portfolios have recently been studied to keep the effective dimension low. In this paper we consider three sets of high dimensional data, the stock market prices for three countries, namely US, UK and India. We compare the Markowitz efficient frontier to those obtained by unbiasedness corrections and imposing norm-constraints in these real data scenarios. We also study the out-of-sample performance of the different procedures. We find that the 2-norm constrained portfolio has best overall performance.

Investor sentiment and portfolio selection

We provide a theoretical framework to examine how investor sentiment impacts the mean-variance tradeoff. We derive a sentiment-adjusted Markowitz efficient frontier in which investor sentiment alters the first two moments of asset returns, the minimum-variance frontier as well as the Capital Market Line. Our theoretical results are consistent with empirical findings that heightened sentiment-related noise trading activity drives perceived prices away from fundamental and increases market volatility. A rational investor neglecting the effect of investor sentiment may end up selecting a sub-optimal portfolio.

Comment on 'Markowitz versus Michaud: Portfolio Optimization Strategies Reconsidered,' Becker, GGrtler and Hibbeln, European Journal of Finance, 21(4): 2015

We are pleased that Becker, Gurtler and Hibbeln (BGH), authors of “Markowitz versus Michaud: Portfolio Optimization Strategies Reconsidered,” are interested in assessing the investment value of Michaud optimization (Michaud 1990, Michaud and Michaud 2008), relative to Markowitz (1952, 1959). As they note, Markowitz and Usmen (MU) (2003) compared Markowitz vs. Michaud efficient frontier optimization in a simulation study. Their results showed that Michaud optimization was superior to Markowitz on average and in every one of the 30 simulation tests performed in spite of the fact that the Markowitz player used a Bayesian estimation procedure to add investment value while the Michaud player did not. BGH also attempt to compare Markowitz to Michaud and, in contrast to MU, finds little evidence of superiority of the Michaud procedure.

A Comparison of Rebalanced and Buy and Hold Portfolios: Does Monetary Policy Matter?

We examine buy and hold and a number of rebalancing strategies on a portfolio of indices that are tracked by ETFs. The indices include Barkley's treasuries and MSCI indices on emerging markets, Pacific and European markets, value funds, and growth funds. Portfolios are rebalanced using threshold schemes and compared with portfolios rebalanced using weights from different points on the Markowitz efficient frontier. We also examine portfolios that are rebalanced across monetary policy regimes. We find that portfolios rebalanced using Markowitz weights that allow for shifts in monetary policy regimes easily outperform both threshold rebalancing and buy and hold strategies.