Optimizing Sparse Mean-Reverting Portfolio

Abstract

Mean-reverting behavior of individuals assets is widely known in financial markets. In fact, we can construct a portfolio that has mean-reverting behavior and use it in trading strategies to extract profits. In this paper, we show that we are able to find the optimal weights of stocks to construct portfolio that has the fastest mean-reverting behavior. We further add minimum variance and sparsity constraints to the optimization problem and transform into Semidefinite Programming (SDP) problem to find the optimal weights. Using the optimal weights, we empirically compare the performance of contrarian strategies between non-sparse mean-reverting portfolio and sparse mean-reverting portfolio to argue that the latter provides higher returns when we take into account of transaction costs.

Figures (5)

• Figure 1: Example of mean-reverting portfolio using optimal weights derived from the closed-form solution (left) and equally-weighted portfolio (right)
• Figure 2: Example of non-sparse ($\rho=0$) mean-reverting portfolio using optimal weights derived from the optimization solution (left) and sparse ($\rho=0.2$) mean-reverting portfolio with minimum variance requirement ($\nu=0.1$) using optimal weights derived from the optimization solution (right)
• Figure 3: Individual weights of 12 stocks for sparse portfolio (blue) and non-sparse portfolio (red). We display log of weights multiplied by the sign of weights for visualization purpose. Here, we can see that the sparse portfolio is constructed of 5 stocks and non-sparse portfolio is constructed of 12 stocks.
• Figure 4: Change in wealth over time when there is no transaction costs
• Figure 5: Change in wealth over time for different transaction costs (0.04 cents, 0.08 cents, 0.12 cents, and 0.16 cents per contract)