Three $l_1$ Based Nonconvex Methods in Constructing Sparse Mean Reverting Portfolios

Abstract

We study the problem of constructing sparse and fast mean reverting portfolios. The problem is motivated by convergence trading and formulated as a generalized eigenvalue problem with a cardinality constraint (d’Aspremont in Quant Finance 11(3):351–364, 2011). We use a proxy of mean reversion coefficient, the direct Ornstein–Uhlenbeck estimator, which can be applied to both stationary and nonstationary data. In addition, we introduce three different methods to enforce the sparsity of the solutions. One method uses the ratio of $l_1$ and $l_2$ norms and the other two use $l_1$ norm. We analyze various formulations of the resulting non-convex optimization problems and develop efficient algorithms to solve them for portfolio sizes as large as hundreds. By adopting a simple convergence trading strategy, we test the performance of our sparse mean reverting portfolios on both synthetic and historical real market data. In particular, the $l_1$ regularization method, in combination with quadratic program formulation as well as difference of convex functions and least angle regression treatment, gives fast and robust performance on large out-of-sample data set.

Appendices

Appendix A: Semi-definite Relaxation Method Derived From $l_1/l_2$

By setting $X = xx^T$, then the problem (12) is equivalent to

$$\begin{aligned} \begin{array}{clcr} \min _{X} &{} \mathrm {trace}(AX)/\mathrm {trace}(BX)\\ s.t. &{} \begin{array}[t]{ll} \frac{1^T |X| 1}{\mathrm {trace}(X)} \le m^2\\ rank(X) = 1\\ X \succeq 0 \end{array} \end{array} \end{aligned}$$

where |X| means we take the absolute value for each entry of X.

Then after a change of variables:

$$\begin{aligned} Y = \frac{X}{\mathrm {trace}(BX)},\quad z = \frac{1}{\mathrm {trace}(BX)} \end{aligned}$$

and dropping the rank constraint, the previous problem can be written as a semidefinite programming problem:

$$\begin{aligned} \begin{array}{clcr} \min _{Y} &{} \mathrm {trace}(AY)\\ s.t. &{} \begin{array}[t]{ll} 1^T|Y|1\le m^2 z\\ \mathrm {trace}(Y) - z = 0\\ \mathrm {trace}(BY) = 1\\ Y \succeq 0\\ \end{array} \end{array} \end{aligned}$$

(19)

If we set $\mathrm {Card}(x) = k = m^2$, this is exactly the semidefinite relaxation in [6].

Appendix B: Estimation of the Matrices A and B

For the estimations below, we assume that each column of the data matrix S represents an asset and its mean is 0. Its size is $l\times n$, so we have l observations and n assets.

We define $S_c$ and $S_f$ in the following way:

$$\begin{aligned} S_c= & {} \left( \begin{array}{c} S_1^T \\ \vdots \\ S_{l-1}^T \end{array} \right) =\left( \begin{array}{ccc} S_{11}&{} \ldots &{} S_{1n} \\ \vdots &{} \ddots &{}\vdots \\ S_{l-1,1}&{} \ldots &{} S_{l-1,n}\end{array} \right) \\ S_f= & {} \left( \begin{array}{c} S_2^T \\ \vdots \\ S_{l}^T \end{array} \right) =\left( \begin{array}{ccc} S_{21}&{} \ldots &{} S_{2n} \\ \vdots &{} \ddots &{}\vdots \\ S_{l1}&{} \ldots &{} S_{ln}\end{array} \right) \end{aligned}$$

where $S_{ti}$ is the value at time t of the ith asset.

1.1 Appendix B.1: Predictability

In [11], the authors discussed several methods in estimating $\beta$ and $\Gamma$ in VAR(1) model (4). In most cases, the number of the observations of assets values l is greater than the number of assets n. Under this case and previous assumptions, we could use the following estimates:

$$\begin{aligned} {\hat{\beta }} = (S_{c}^T S_{c})^{-1}(S^T_{c}S_{f})) \quad {\hat{\Gamma }}= \frac{1}{l-1}S^T_{c}S_{c} \end{aligned}$$

Therefore, the matrices in problem (7) can be estimated as:

$$\begin{aligned} {\hat{A}} = {\hat{\beta }}^T {\hat{\Gamma }}{\hat{\beta }}\quad {\hat{B}} = {\hat{\Gamma }} \end{aligned}$$

1.2 Appendix B.2: Direct OU Estimator

Using a similar method as B.1, we estimate the matrices A and B as follows:

$$\begin{aligned} {\hat{A}} = \frac{1}{l-1} \left( S_{c}^T S_{f}+S_{f}^T S_{c}\right) \quad {\hat{B}} = \frac{2}{l-1} S_{c}^T S_{c} \end{aligned}$$