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This study aims to propose the Shapley value that originates from the game theory to quantify the relative risk of a security in an optimal portfolio.

Systematic risk as expressed by the relative covariance of stock returns to market returns is an essential measure in pricing risky securities. Although very much in use, the concept has become marginalized in recent years because of the difficulties that arise estimating beta. The idea is that portfolios can be viewed as cooperative games played by assets aiming at minimizing risk. With the Shapley value, investors can calculate the exact contribution of each risky asset to the joint payoff. For a portfolio of three stocks, this study exemplifies the Shapley value when risk is minimized regardless of portfolio return.

This study computes the Shapley value of stocks and indices for optimal mean-variance portfolios by using daily returns for the years 2016–2019. This results in the risk attributes allocated to securities in optimal portfolios. The Shapley values are analyzed and compared to the standard beta estimates to determine the ranking of assets with respect to pertinent risk and return.

An alternative approach to value risk and return in optimal portfolios is presented in this study. The logic and the mechanics of Shapley value theory in portfolio analysis have been explained, and its advantages relative to standard beta analysis are presented. Hence, financial analysts when adding or removing specific assets from present positions will have the true and exact impact of their actions by using the Shapley value instead of the beta.

When computing the Shapley value, portfolio risk is decomposed exactly among its assets because it considers all possible coalitions of portfolios. In that sense, financial analysts when adding or removing specific securities from present holdings will be able to predict the true and exact impact of their transactions by using the Shapley value instead of the beta. The main implication for investors is that risk is ultimately priced relative to their holdings. This prevents the subjective mispricing of securities, as standard beta is not used and might allow investors to gain from arbitrage conditions.

The logic and the methodology of Shapley value theory in portfolio analysis have been explained as an alternative to value risk and return in optimal portfolios by presenting its advantages relative to standard beta analysis. The conclusion is that the Shapley value theory contributes much more financial optimization than to standard systematic risk analysis because it enables looking at the contribution of each security to all possible coalitions of portfolios.

The purpose of this study is to estimate the convergence order of the Aumann–Serrano Riskiness Index.

This study uses the equivalent relation between the Aumann–Serrano Riskiness Index and the moment generating function and aggregately compares between each two statistical moments for statistical significance. Thus, this study enables to find the convergence order of the index to its stable value.

This study finds that the first-best estimation of the Aumann–Serrano Riskiness Index is reached in no less than its seventh statistical moment. However, this study also finds that its second-best approximation could be achieved with its second statistical moment.

The implications of this research support the standard deviation as a statistically sufficient approximation of Aumann–Serrano Riskiness Index, thus strengthening the CAPM methodology for asset pricing in the financial markets.

This research sheds a new light, both in theory and in practice, on understanding of the risk’s structure, as it may improve accuracy of asset pricing.

The purpose of this paper is to increase the accuracy of the efficient portfolios frontier and the capital market line using the Riskiness Index.

This paper will develop the mean-riskiness model for portfolio selection using the Riskiness Index.

This paper’s main result is establishing a mean-riskiness efficient set of portfolios. In addition, the paper presents two applications for the mean-riskiness portfolio management method: one that is based on the multi-normal distribution (which is identical to the MV model optimal portfolio) and one that is based on the multi-normal inverse Gaussian distribution (which increases the portfolio’s accuracy, as it includes the a-symmetry and tail-heaviness features in addition to the scale and diversification features of the MV model).

The Riskiness Index is not a coherent measurement of financial risk, and the mean-riskiness model application is based on a high-order approximation to the portfolio’s rate of return distribution.

The mean-riskiness model increases portfolio management accuracy using the Riskiness Index. As the approximation order increases, the portfolio’s accuracy increases as well. This result can lead to a more efficient asset allocation in the capital markets.