Eigenportfolios of US Equities for the Exponential Correlation Model

Abstract

In this paper, the eigendecomposition of a Toeplitz matrix populated by an exponential function in order to model empirical correlations of US equity returns is investigated. The closed-form expressions for eigenvalues and eigenvectors of such a matrix are available. These eigenvectors are used to design the eigenportfolios of the model, and we derive their performance for the two metrics. The Sharpe ratios and profit-and-loss curves (P&Ls) of eigenportfolios for twenty-eight of the thirty stocks in the Dow Jones Industrial Average index are calculated for the end-of-day returns from July 1, 1999 to November 1, 2018, several different subintervals and three other baskets in order to validate the model. The proposed method provides eigenportfolios that mimic those based on an empirical correlation matrix generated from market data. The model brings new insights into the design and evaluation of eigenportfolios for US equities and other asset classes. These eigenportfolios are used in the design of trading algorithms, including statistical arbitrage, and investment portfolios. Here, P&Ls and Sharpe ratios of minimum variance, market and eigenportfolios are compared along with the index and three sector exchange-traded funds (XLF, XLI and XLV) for the same time intervals. They show that the first eigenportfolio outperforms the others considered in the paper.

KEY MESSAGES

1) Empirical correlations of asset returns in a group of stocks are approximated by the exponential correlation model that populates a Toeplitz matrix with closed-form expressions for its eigenvalues and eigenvectors.

2) Exponential model based Toeplitz matrix and measurements based empirical correlation matrix are used to create their eigensubspaces and eigenportfolios for performance comparisons by using risk normalized annual returns.

3) It is demonstrated in the paper that the exponential approximation to empirical correlations provide a good model to design eigenportfolios and to evaluate their performance.

Keywords: Eigenportfolio, exponential correlation model, PNL, Sharpe Ratio

JEL Classification: G11