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Statistical Proxy based Mean-Reverting Portfolios with Sparsity and Volatility Constraints

Ahmad Mousavi, George Michailidis

TL;DR

This paper develops an effective unifying algorithm that is enabled to obtain a Karush-Kuhn-Tucker point under mild regularity conditions and establishes that the convergence analysis can be extended to a nonconvex objective function case if the starting penalty parameter is larger than a finite bound and the objective function has a bounded level set.

Abstract

Mean-reverting portfolios with volatility and sparsity constraints are of prime interest to practitioners in finance since they are both profitable and well-diversified, while also managing risk and minimizing transaction costs. Three main measures that serve as statistical proxies to capture the mean-reversion property are predictability, portmanteau criterion, and crossing statistics. If in addition, reasonable volatility and sparsity for the portfolio are desired, a convex quadratic or quartic objective function, subject to nonconvex quadratic and cardinality constraints needs to be minimized. In this paper, we introduce and investigate a comprehensive modeling framework that incorporates all the previous proxies proposed in the literature and develop an effective unifying algorithm that is enabled to obtain a Karush-Kuhn-Tucker (KKT) point under mild regularity conditions. Specifically, we present a tailored penalty decomposition method that approximately solves a sequence of penalized subproblems by a block coordinate descent algorithm. To the best of our knowledge, our proposed algorithm is the first for finding volatile, sparse, and mean-reverting portfolios based on the portmanteau criterion and crossing statistics proxies. Further, we establish that the convergence analysis can be extended to a nonconvex objective function case if the starting penalty parameter is larger than a finite bound and the objective function has a bounded level set. Numerical experiments on the S&P 500 data set demonstrate the efficiency of the proposed algorithm in comparison to a semidefinite relaxation-based approach and suggest that the crossing statistics proxy yields more desirable portfolios.

Statistical Proxy based Mean-Reverting Portfolios with Sparsity and Volatility Constraints

TL;DR

This paper develops an effective unifying algorithm that is enabled to obtain a Karush-Kuhn-Tucker point under mild regularity conditions and establishes that the convergence analysis can be extended to a nonconvex objective function case if the starting penalty parameter is larger than a finite bound and the objective function has a bounded level set.

Abstract

Mean-reverting portfolios with volatility and sparsity constraints are of prime interest to practitioners in finance since they are both profitable and well-diversified, while also managing risk and minimizing transaction costs. Three main measures that serve as statistical proxies to capture the mean-reversion property are predictability, portmanteau criterion, and crossing statistics. If in addition, reasonable volatility and sparsity for the portfolio are desired, a convex quadratic or quartic objective function, subject to nonconvex quadratic and cardinality constraints needs to be minimized. In this paper, we introduce and investigate a comprehensive modeling framework that incorporates all the previous proxies proposed in the literature and develop an effective unifying algorithm that is enabled to obtain a Karush-Kuhn-Tucker (KKT) point under mild regularity conditions. Specifically, we present a tailored penalty decomposition method that approximately solves a sequence of penalized subproblems by a block coordinate descent algorithm. To the best of our knowledge, our proposed algorithm is the first for finding volatile, sparse, and mean-reverting portfolios based on the portmanteau criterion and crossing statistics proxies. Further, we establish that the convergence analysis can be extended to a nonconvex objective function case if the starting penalty parameter is larger than a finite bound and the objective function has a bounded level set. Numerical experiments on the S&P 500 data set demonstrate the efficiency of the proposed algorithm in comparison to a semidefinite relaxation-based approach and suggest that the crossing statistics proxy yields more desirable portfolios.
Paper Structure (11 sections, 4 theorems, 45 equations, 12 figures, 2 algorithms)

This paper contains 11 sections, 4 theorems, 45 equations, 12 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. A Penalty Decomposition Method for Statistical Proxy-based Portfolios with Sparsity and Volatility Constraints
  3. Details of PPC and BCD Algorithms
  4. Convergence Analysis
  5. Numerical Experiments
  6. Conclusion
  7. Appendix
  8. Proof of Lemma \ref{['lem: BCD boundedness']}.
  9. Proof of Theorem \ref{['thm: BCD convergence']}.
  10. Proof of Theorem \ref{['thm: PPC convergence']}.
  11. Proof of Theorem \ref{['thm: nonconvex convergence']}.

Key Result

Lemma 2.1

Let $\alpha\in \{0,1\},\gamma\ge 0, \rho>0, A_0\succ 0,$ and $A_i\succeq 0$ for $i\in [q]$. Consider the iterates of Algorithm algo: BCD, that is, $x_{l}\in \hbox{Argmin}_{x\in \mathcal{X}} \ q_{\rho}(x,y_{l-1},z_{l-1})$, $y_{l}\in \hbox{Argmin}_{y\in \mathcal{Y}} \ q_{\rho}(x_{l},y,z_{l-1})$ and fu

Figures (12)

  • Figure 1: Statistical proxy- and SDP-based portfolios for $k=5$.
  • Figure 2: Statistical proxy- and SDP-based portfolios for $k=10$.
  • Figure 3: Statistical proxy- and SDP-based portfolios for $k=17$.
  • Figure 4: Statistical proxy-based portfolios for $k=5,10,17$.
  • Figure :
  • ...and 7 more figures

Theorems & Definitions (8)

  • Lemma 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • proof
  • proof
  • proof