Mean Reversion with a Variance Threshold

International audienceStarting from a multivariate data set, we study several techniques to isolate affine combinations of the variables with a maximum amount of mean reversion, while constraining the variance to be larger than a given threshold. We show that many of the optimization problems arising in this context can be solved exactly using semidefinite programming and some variant of the S-lemma. In finance, these methods are used to isolate statistical arbitrage opportunities, i.e. mean reverting portfolios with enough variance to overcome market friction. In a more general setting, mean reversion and its generalizations are also used as a proxy for stationarity, while variance simply measures signal strength

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

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

Maximally Machine-Learnable Portfolios

When it comes to stock returns, any form of predictability can bolster risk-adjusted profitability. We develop a collaborative machine learning algorithm that optimizes portfolio weights so that the resulting synthetic security is maximally predictable. Precisely, we introduce MACE, a multivariate extension of Alternating Conditional Expectations that achieves the aforementioned goal by wielding a Random Forest on one side of the equation, and a constrained Ridge Regression on the other. There are two key improvements with respect to Lo and MacKinlay's original maximally predictable portfolio approach. First, it accommodates for any (nonlinear) forecasting algorithm and predictor set. Second, it handles large portfolios. We conduct exercises at the daily and monthly frequency and report significant increases in predictability and profitability using very little conditioning information. Interestingly, predictability is found in bad as well as good times, and MACE successfully navigates the debacle of 2022