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An investor is expected to analyze the market risk while investing in equity stocks. This is because the investor has to choose a portfolio which maximizes the return with a minimum risk. The mean-variance approach by Markowitz (1952) is a dominant method of portfolio optimization, which uses variance as a risk measure. The purpose of this paper is to replace this risk measure with modified expected shortfall, defined by Jadhav et al. (2013).

Modified expected shortfall introduced by Jadhav et al. (2013) is found to be a coherent risk measure under univariate and multivariate elliptical distributions. This paper presents an approach of portfolio optimization based on mean-modified expected shortfall for the elliptical family of distributions.

It is proved that the modified expected shortfall of a portfolio can be represented in the form of expected return and standard deviation of the portfolio return and modified expected shortfall of standard elliptical distribution. The authors also establish that the optimum portfolio through mean-modified expected shortfall approach exists and is located within the efficient frontier of the mean-variance portfolio. The results have been empirically illustrated using returns from stocks listed in National Stock Exchange of India, Shanghai Stock Exchange of China, London Stock Exchange of the UK and New York Stock Exchange of the USA for the period February 2005-June 2018. The results are found to be consistent across all the four stock markets.

The mean-modified expected shortfall portfolio approach presented in this paper is new and is a natural extension of the Markowitz’s mean-variance and mean-expected shortfall portfolio optimization discussed by Deng et al. (2009).

The purpose of this paper is to describe some optimization exercises which have proved to be very useful for introducing students to Markowitz‐style mean‐varience optimization.

This paper describes two exercises that walk students through the process of gathering security price and dividend data, estimating the parameters of the joint distribution of asset returns, and then using a portfolio optimizer to construct mean‐variance efficient portfolios. It describes the basic methodology, and the more complex formulations of the portfolio optimization problem that are used in practice.

Portfolio selection is typically taught in finance courses as an abstract solution to a system of equations, and does little to connect the portfolio construction process to Exchange Traded Funds, stocks, bonds and other assets that are traded in markets. This study offers a practical approach to teaching portfolio optimization, that starts with gathering market data and shows how a quadratic optimization system is used to construct mean‐variance optimal portfolios.

The exercises in this case study prepare students to construct mean‐variance efficient portfolios for asset allocation with Exchange Traded Funds, and for building stock and bond portfolios, using market data and a portfolio optimizer.

Aims to test to determine whether the selection of the historical return time interval (monthly, quarterly, semiannual, or annual) used for calculating real estate investment trust (REIT) returns has a significant effect on optimal portfolio allocations.

Using a mean‐variance utility function, optimal allocations to portfolios of stocks, bonds, bills, and REITs across different levels of assumed investor risk aversion are calculated. The average historical returns, standard deviations, and correlations (assuming different time intervals) of the various asset classes are used as mean‐variance inputs. Results are also compared using more recent data, since 1988, with, data from the full REIT history, which goes back to 1972.

Using the more recent REIT datarather than the full dataset results in optimal allocations to REITs that are considerably higher. Likewise, using monthly and quarterly returns tends to understate the variability of REITs and leads to higher portfolio allocations.

The results of this study are based on the limited historical return data that are currently available for REITs. The results of future time periods may not prove to be consistent with the findings.

Numerous research papers arbitrarily decide to employ monthly or quarterly returns in their analyses to increase the number of REIT observations they have available. These shorter interval returns are generally annualized. This paper addresses the consequences of those decisions.

It has been shown that the decision to use return estimation intervals shorter than a year does have dramatic consequences on the results obtained and, therefore, must be carefully considered and justified.

The purpose of this paper is to provide theory for some popular models and strategies used by practitioners in constructing optimal portfolios. King (2007), for example, advocated adding a diversification term to mean-variance problems to create better portfolios and provided clear empirical evidence that this is beneficial.

The authors provide an analytical framework to help us understand different portfolio construction practices that may incorporate diversification and conviction strategies; this allows us to connect our analysis to ideas in psychophysics and behavioural finance. The critical psychological ideas are cognitive dissonance and entropy; the economics are based on expected utility theory. The empirical section uses the theory outlined and provides the basis for constructing such portfolios.

The model presented allows the incorporation of different strategies within a mean-variance framework, ranging from diversification and conviction strategies to more ESG-oriented ones. The empirical analysis provides a practical application.

To the best of the authors’ knowledge, this model is the first to bridge the gap between portfolio optimisation and the psychological ideas mentioned in a coherent analytical framework.

This paper examines the effect of cryptocurrencies on the portfolio risk-adjusted returns of traditional and alternative investments within an emerging market economy.

The paper employs daily arithmetic returns from August 2015 to October 2018 of traditional assets (stocks, bonds, currencies), alternative assets (commodities, real estate) and cryptocurrencies. Using the mean-variance analysis, the Sharpe ratio, the conditional value-at-risk and the mean-variance spanning tests.

The paper documents evidence to support the diversification benefits of cryptocurrencies by utilising the mean-variance tests, improving the efficient frontier and the risk-adjusted returns of the emerging market economy portfolio of investments.

This paper firmly broadens the Modern Portfolio Theory by authenticating cryptocurrencies as assets with diversification benefits in an emerging market economy investment portfolio.

As far as the authors are concerned, this paper presents the first evidence of the effect of diversification benefits of cryptocurrencies on emerging market asset portfolios constructed using traditional and alternative assets.

Markowitz’s mean‐variance approach is used to identify the returns to vertical investment in the pork industry. In addition to previous efforts, this paper considers not only returns to stock ownership, but uses operating return on investment in pork slaughter and hog production to evaluate the impacts of vertical investment within the industry segment. Results suggest there are indeed diversification incentives for vertical investment in the pork industry. However, results do differ for vertical direct investment versus investment through stock ownership.

The existing literature on the Black-Litterman (BL) model does not offer adequate guidance on how to generate investors’ views in an objective manner. Therefore, the purpose of this paper is to establish a generalized multivariate Vector Error Correction Model (VECM)/Vector Auto-Regressive (VAR)-Dynamic Conditional Correlation (DCC)/Asymmetric DCC (ADCC) framework, and applies it to generate objective views to improve the practicality of the BL model.

This paper establishes a generalized VECM/VAR-DCC/ADCC framework that can be utilized to model multivariate financial time series in general, and produce objective views as inputs to the BL model in particular. To test the VECM/VAR-DCC/ADCC preconditioned BL model’s practical utility, it is applied to a six-asset China portfolio (including one risk-free asset).

With dynamically optimized view confidence parameters, the VECM/VAR-DCC/ADCC preconditioned BL model offers clear advantage over the standard mean-variance method, and provides an automated portfolio optimization alternative to the classic BL approach.

The VECM/VAR-DCC/ADCC framework and its application in the BL model proposed by this paper provide an alternative approach to the classic BL method. Since all the view parameters, including estimated mean return vectors, conditional covariance matrices and pick matrices, are generated in the VECM/VAR and DCC/ADCC preconditioning stage, the model improves the objectiveness of the inputs to the BL stage. In conclusion, the proposed model offers a practical choice for automated portfolio balancing and optimization in a China context.

This study investigates the diversification benefits of multiple cryptocurrencies and their usefulness as investment assets, individually or combined, in enhancing the performance of a well-diversified portfolio of traditional assets before and during the pandemic COVID-19.

This paper uses two optimization techniques, namely the mean-variance and the maximum Sharpe ratio. The naïve diversification rules are used for comparison. Besides, the Sharpe and the Sortino ratios are used as performance measures.

The results show that cryptocurrencies diversification benefits occur more during the COVID-19 pandemic rather than before it, with the maximum Sharpe ratio portfolio presenting its highest performance. Furthermore, the results suggest that, during COVID-19, the diversification benefits are slightly better when using a combination of cryptocurrencies to an already well-diversified portfolio of traditional assets rather than individual ones. This serves to improve the performance of the maximum Sharpe ratio portfolio, and to some extent, the naïve portfolio. Yet, cryptocurrencies, whether added individually or combined to a well-diversified portfolio of traditional assets, don't fit in the minimum variance portfolio. Besides, the efficient frontier during COVID-19 pandemic dominates the one before COVID-19 pandemic, giving the investor a better risk-return trade-off.

To the best of the author's knowledge, this is the first study that examines the diversification benefits of multiple cryptocurrencies both as individual investments and as additional asset classes, before and during COVID-19 pandemic. The paper covers all analyses performed separately in previous studies, which brings new evidence regarding the potential for cryptocurrencies in portfolio diversification under different portfolio strategies.

This paper aims to elaborate on the optimization of two particular cryptocurrency portfolios in a mean-variance framework. In general, cryptocurrencies can be classified to as coins and tokens where the first can be thought of as a medium of exchange and the latter accounts for security or utility tokens depending upon its design.

Against this backdrop, this empirical study distinguishes, in particular, between pure coin and token portfolios. Both portfolios are optimized by maximizing the Sharpe ratio and, subsequently, compared with alternative portfolio strategies.

The empirical findings demonstrate that the maximum utility portfolio of coins, with a risk aversion of λ = 10, outweighs alternative frameworks. The portfolios optimized by maximizing the Sharpe ratio for both coins and tokens indicate a rather poor performance. Testing the maximized utility for different levels of risk aversion confirms the findings of this empirical study and confers them more robustness.

Further investigation is strongly recommended as tokens represent a new phenomenon in the cryptocurrency universe, for which only a limited amount of data are available, which restricts the sampling. Furthermore, future study is to include more sophisticated optimization models using different constraints in portfolio creation.

In light of the persistently substantial volatility in cryptocurrency markets, the empirical findings assert that portfolio managers are advised to construct a global minimum variance portfolio. In the absence of sophisticated optimization models, private investors can invest according to the market values of cryptocurrencies. Despite minor differences in the risk and reward ratios of the portfolios tested, tokens tend to be more speculative, especially, if the Tether token is excluded, which may require enhanced supervision and investor protection by regulating authorities.

As the current literature investigates on diversification effects of blended cryptocurrency portfolios rather than making an explicit distinction, this paper reflects one of the first to explore the investability and role of diversifying coins and tokens using a classic Markowitz approach.

This chapter investigates the relative magnitude of the benefits of global diversification from the viewpoint of domestic investors in various countries by forming time-rolling efficient frontiers. To enhance feasibility of asset allocation strategies, the constraints of short-sales and over-weighting investments are taken into account. The empirical results suggest that local investors in less developed countries, particularly in Latin America, East Asia, and Southern Europe, comparatively benefit more from global diversification. Investors in the countries of civic-law origin tend to benefit more from global investment than the ones in the common-law states. Although the global market has become more integrated over the past decades, diversification benefits for domestic investors declined but did not vanish. The results of this chapter are useful for asset management professionals to determine target markets to promote the sales of international funds.