Sparse Portfolio Selection via Bayesian Multiple Testing
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Sparse Portfolio Selection via Bayesian Multiple Testing Sourish Das Chennai Mathematical Institute, Chennai, India
Rituparna Sen Indian Statistical Institute, Bengaluru, India
Abstract We present Bayesian portfolio selection strategy, via the k factor asset pricing model. If the market is information efficient, the proposed strategy will mimic the market; otherwise, the strategy will outperform the market. The strategy depends on the selection of a portfolio via Bayesian multiple testing methodologies. We present the “discrete-mixture prior” model and the “hierarchical Bayes model with horseshoe prior.” We define the oracle set and prove that asymptotically the Bayes rule attains the risk of Bayes oracle up to O(1). Our proposed Bayes oracle test guarantees statistical power by providing the upper bound of the type-II error. Simulation study indicates that the proposed Bayes oracle test is suitable for the efficient market with few stocks inefficiently priced. The statistical power of the Bayes oracle portfolio is uniformly better for the k-factor model (k > 1) than the one factor CAPM. We present an empirical study, where we consider the 500 constituent stocks of S&P 500 from the New York Stock Exchange (NYSE), and S&P 500 index as the benchmark for thirteen years from the year 2006 to 2018. We show the out-sample risk and return performance of the four different portfolio selection strategies and compare with the S&P 500 index as the benchmark market index. Empirical results indicate that it is possible to propose a strategy which can outperform the market. All the R code and data are available in the following GitHub repository https://github.com/ sourish-cmi/sparse portfolio Bayes multiple test. AMS (2000) subject classification. Primary 62F15, 62P05; Secondary 91G10, 62J15. Keywords and phrases. CAPM, Discrete mixture prior, Hierarchical Bayes, Oracle, Factor model
Research supported by Bill, Melinda Gates Foundation grant, TATA Trust grant and Infosys Foundation grant to CMI.
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S. Das and R. Sen
1 Introduction Markowitz portfolio theory (Markowitz, 1952) in finance analytically formalizes the risk-return tradeoff in selecting optimal portfolios. An investor allocates the wealth among securities in such a way that the portfolio guarantees a certain level of expected returns and minimizes the ‘risk’ associated with it. The variance of the portfolio return is quantified as the risk. Markowitz portfolio optimization is very sensitive to errors in the estimates of the expected return vector and the covariance matrix, see, e.g., Fan et al. (2012). The problem is severe when the portfolio size is large. Several techniques have been suggested to reduce the sensitivity of the Markowitz optimal portfolios. One approach is to use a James-Stein estimator for means (i.e., expected return) (Chopra and Ziemba, 1993) and shrink the sample covariance matrix (Ledoit and Wolf, 2003; Das et al., 2017). Still, the curse of dimensionality kicks-in for a typically large portfolio (like mutual fund portfolio) and the procedure underestim
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