The Shapley value of regression portfolios
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ORIGINAL ARTICLE
The Shapley value of regression portfolios Haim Shalit1 Revised: 28 June 2020 / Published online: 20 July 2020 © Springer Nature Limited 2020
Abstract 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. Keywords Efficient portfolios · Ordinary least-squares · Asset allocation
Introduction The purpose of this paper is to apply Shapley value imputation (Shapley 1953) to optimal portfolios being generated by ordinary least-squared (OLS) regressions on financial assets. Recently, Shalit (2017) used the Shapley value to decompose the risk of optimal mean-variance (MV) and mean-Gini (MG) portfolios. The Shapley value originated from cooperative game theory where it was derived for the purpose of measuring the exact contribution of players in a game. Since then, it has become a standard measure in economics, political science, sports, and income inequality. As evidenced by the recent handbook edited by Algaba et al. (2019), Shapley value has become the norm by which complex allocation problems are solved and priced. In finance, it has been shown to be most adequate in distributing costs of insurance companies, valuing corporate voting rights and attributing risk in the banking system. In investments and portfolio theory the use has been slower. Indeed, only recently have Ortmann (2016) and Colini-Baleschi et al.
* Haim Shalit [email protected] 1
Ben-Gurion University of the Negev, Beer‑Sheva, Israel
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(2018) implemented the Shapley value for portfolio analysis and for pricing the market risk of individual assets. The main idea behind the Shapley value is to look at all the feasible coalitions of participants in a cooperative game and calculate the benefits each player contributes to the common goal. As each contribution depends upon the order in which players join the coalition, the Shapley value is calculated by averaging the marginal contributions from the arrival of the various players to the specific coalitions. Since p
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