Sparse Portfolio Selection via Bayesian Multiple Testing

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.

Appendices

Appendix A

Markowitz portfolio optimization can be expressed as the following quadratic programming problem:

$$ \begin{array}{@{}rcl@{}} \min ~~ \mathbf{w}'{\Sigma} \mathbf{w}\quad \text{subject to}\quad\mathbf{w}'\mathbf{1}_{P}=1 ~~\mathrm{ and}\quad \mathbf{w}'{\boldsymbol{\mu}}= \mu_{k}. \end{array} $$

(A.1)

Here 1_P is a P-dimensional vector with one in every entry and μ_k is the desired level of return.

The portfolio covariance can be decomposed into two parts as,

$$ \begin{array}{@{}rcl@{}} \mathbf{w}'{\Sigma} \mathbf{w} &= &\mathbf{w}'[\boldsymbol{B}^{T}\mathbf{\Sigma}_{m}\boldsymbol{B}+\mathbf{\Sigma}_{\epsilon}] \mathbf{w}\\ & = & \mathbf{w}'\boldsymbol{B}^{T}\mathbf{\Sigma}_{m}\boldsymbol{B}\mathbf{w} + \mathbf{w}' \mathbf{\Sigma}_{\epsilon} \mathbf{w}, \end{array} $$

where first part explains the portfolio volatility due to market volatility and the second part explains portfolio volatility due to idiosyncratic behaviour of the stock. We assume σ_i’s are bounded ∀i. Then

$$ \begin{array}{@{}rcl@{}} \mathbf{w}' \mathbf{\Sigma}_{\epsilon} \mathbf{w} &=& \sum\limits_{i=1}^{P}{\omega_{i}^{2}}{\sigma_{i}^{2}}\\ &\leq& \sigma_{max}^{2}\sum\limits_{i=1}^{P}{\omega_{i}^{2}},~~\sigma_{max}^{2}=\max\{\mathbf{\Sigma}_{\epsilon}\}<\infty,\\ &\leq& \sigma_{max}^{2}M_{\omega:P}\sum\limits_{i=1}^{P}\omega_{i},~~M_{\omega:P}=\max\{\mathbf{w}\},\\ &=& \sigma_{max}^{2}M_{\omega:P}. \end{array} $$

Clearly, if $P\rightarrow \infty$ and $M_{\omega :P} \rightarrow 0 \implies \mathbf {w}' \mathbf {\Sigma }_{\epsilon } \mathbf {w} \rightarrow 0$. Hence we have the following result.

Result 1.

Under the CAPM model (2.1), covariance matrix (2.4) and assumption (A.1), if $P\rightarrow \infty$ and $M_{\omega :P} \rightarrow 0$ and $\sigma _{max}^{2}=\max \limits \{\mathbf {\Sigma }_{\epsilon }\}<\infty$, then

$$ \lim_{P\rightarrow \infty} \mathbf{w}' \mathbf{\Sigma}_{\epsilon} \mathbf{w} =0. $$

Remark 6.

Under the standard asset pricing theory, as the size of the portfolio increases, and the maximum weight of any asset is bounded, the idiosyncratic risk of the portfolio will be washed out. The portfolio’s risk and return will be a function of the systematic risk explained by major market indices.

Appendix B

Table 2 Out-sample annual return of equal weight portfolio

Table 3 Annualized Volatility of out-sample return of equal weight portfolio

Table 4 Value at Risk (VaR) of out-sample return of equal weight portfolio

Table 5 Risk adjusted out-sample return of equal weight portfolio

Figure 8

The line-plot of out of sample annualised return, volatility, Value at Risk and annualised risk-adjusted return for S&P 500, CAPM, Fan’s model, Factor Model with horseshoe prior, Factor model with ABOS for Markowitz’s weight portfolio in Empirical study

Table 6 Out-sample annual return with Markowitz’s weight

Table 7 Annualized volatility of out-sample return of Markowitz’s portfolio

Table 8 Value at Risk (VaR) of out-sample return of Markowitz’s portfolio

Table 9 Risk adjusted out-sample return of Markowitz’s portfolio