The Shapley value decomposition of optimal portfolios
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The Shapley value decomposition of optimal portfolios Haim Shalit1 Received: 23 September 2020 / Accepted: 28 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Investors want the ability to evaluate the true and complete risk of the financial assets held in a portfolio. Yet, the current analytic methods provide only partial risk measures. I suggest that, by viewing a portfolio of securities as a cooperative game played by the assets that minimize portfolio risk, investors can calculate the exact value, each security contributes to the common payoff of the game, which is known as the Shapley value. It is determined by computing the contribution of each asset to the portfolio risk by looking at all the possible coalitions in which the asset would participate. I develop this concept in order to decompose the risk of mean-variance and mean-Gini efficient portfolios. This decomposition gives us a better rank of assets by their comprehensive contribution to the risk of optimal portfolios. Such a procedure allows investors to make unbiased decisions when they analyze the inherent risk of their holdings. The Shapley value is calculated for index classes and the empirical results based on asset allocation data are contrary to some of the findings of conventional wisdom and beta analysis. Keywords Mean-variance portfolios · Mean-Gini portfolios · Efficient frontier · Systematic risk · Asset allocation JEL Classification G11
1 Introduction It is well known that investment managers are concerned with the risk impact of adding securities to portfolios. Since the inception of modern portfolio theory, investors have measured how securities affect each other. The simplest risk as expressed by asset variance is not sufficient to establish sound investment decisions. For this reason, financial
I am grateful to anonymous referees for useful comments and to Ofer Zevin for directing me to the Shapley value in finance.
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Haim Shalit [email protected] Ben-Gurion University of the Negev, Beersheba, Israel
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H. Shalit
theoreticians and practitioners now also take into account correlations, covariances, and betas to establish the cross-effects among investments. My claim is that these risk measures although sufficient to build efficient portfolios are prone to error when measuring the true impact of a risky asset upon an optimal portfolio. Hence, when presumably rational and efficient efforts misjudge the true risk of assets in optimal portfolios, a totally new approach is required. To address this deficiency, I suggest applying the concept of Shapley value (Shapley 1953) to financial management theory and practice. Shapley value theory emerged from cooperative game theory so as to measure the exact contribution of agents playing the game. In a cooperative game, players interact in order to optimize a common objective whose utility is transferable. The Shapley value concept has been applied successfully in economic theory, politics, sports, and income inequality as evidenced by a new handbook edi
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