A dynamic conditional approach to forecasting portfolio weights

Abstract

From the autoregressive representation of the portfolio-variance optimization problem, we derive a novel model for conditional portfolio weights as a linear function of past conditional and realized (and, hence, observable) terms. This dynamic conditional weights ($\mathsf{DCW}$) approach is benchmarked against popular model-based and model-free specifications in terms of weights forecasts and portfolio allocations. Next to portfolio turnover and variance, we introduce the break-even transaction cost as an additional measure that identifies the range of transaction costs for which one allocation is preferred to another. By comparing minimum-variance portfolios built on the components of the Dow Jones 30 Index, the proposed $\mathsf{DCW}$ attains the best allocations overall with respect to the measures considered, for any degree of risk aversion, transaction costs, and exposure.

Introduction

A key aspect of active portfolio management revolves around the projection of the weights that optimize portfolio holdings with respect to some representative measure of the investor’s preferences. Since Markowitz (1952), such optimal portfolio weights forecasts are generally derived from the forecasts of conditional moments of asset returns. More recently, the availability of realized measures from high-frequency data has allowed for model-based and model-free forecasting of conditional variance–covariance (var-cov) matrices and, by successive manipulation thereof, of optimal portfolio weights: see, e.g., Aït-Sahalia et al. (2010), Christensen et al. (2010), Barndorff-Nielsen et al. (2011), Zhang (2011), and Bibinger et al. (2014), among others.

Model-based approaches are inspired by the logic behind multivariate GARCH (MGARCH) models.¹ Examples of these approaches are the fractionally integrated processes of Chiriac and Voev (2011), the vector autoregressions of Callot et al. (2017), and the specifications based on the Wishart distribution of Gourieroux et al. (2009), Golosnoy et al. (2012), Noureldin et al. (2012), and Jin and Maheu (2010), among others. Within this framework it is not uncommon to separately model conditional variances² and correlation matrices to achieve a good balance between parameter parsimony and richness in the description of the second-order dynamics.

On the other hand, model-free approaches, also referred to as nonparametric, impose driftless random-walk dynamics on the conditional second moments, and thus eliminate the parameter estimation problem altogether. However, for large cross-sectional dimensions, the lag-1 realized var-cov matrices may result in extreme portfolio weights, poor portfolio performance out-of-sample (OOS), and even positive-semidefiniteness (psd). To mitigate this problem, various shrinkage approaches are available: the most direct is to impose constraints on the portfolio weights³ as in Jagannathan and Ma (2003), El Karoui (2010), Fan et al. (2012), and Gandy and Veraart (2013). Shrinkage of the realized var-cov matrix has been proposed by Fan et al. (2008), Fan et al. (2011), Tao et al. (2011), Ledoit and Wolf (2012), Tao et al. (2013), Fan et al. (2016), and Aït-Sahalia and Xiu (2017), to name a few. The ideas behind these approaches may also be traced back to the MGARCH literature and consist of imposing a factor structure on the returns and a sparse error var-cov matrix with blocks defined by some characteristics of the assets, such as sector, industry, etc.

In this paper, we introduce dynamic conditional weights ($\mathsf{DCW}$), an approach inspired by the autoregressive representation of the portfolio-variance optimization problem, originally expressed with time-varying coefficients and simplified to a vector ARMA process for the realized weights with time-independent parameters. The resulting specification provides conditional portfolio weights forecasts from a linear function of past conditional and realized (and, hence, observable) terms. With respect to model-based approaches, which model the $M(M+1)∕2$ elements of the conditional var-cov matrix, $\mathsf{DCW}$ requires modeling of only $M$ portfolio weights. Furthermore, while the dynamic specifications of model-based approaches must enforce positive-definiteness on the ensuing predictions and forecasts, the constraint associated to the $\mathsf{DCW}$ is that the resulting portfolio weights add to unity (a condition imposed with a simple rescaling when needed). With respect to model-free approaches, $\mathsf{DCW}$ does not require the imposition of particular var-cov matrix structures, nor discretionary choices about the level of shrinkage.

Modeling weights directly has a twofold motivation. First, as mentioned, optimal weights are a centerpiece in portfolio management, so for any utility functional representing investors’ preferences, the $\mathsf{DCW}$ framework may be applied to the corresponding time series of realized optimal portfolio weights, bypassing the need to specify var-cov dynamics.⁴ In this respect, the estimation of $\mathsf{DCW}$ (especially for the equation-by-equation estimation with least squares, in the diagonal case) is as simple as the standard least-squares estimation of empirical asset pricing models in the financial literature. Second, within a more financial econometrics perspective, $\mathsf{DCW}$ provides a parsimonious and challenging benchmark when forecasting conditional var-cov matrices is involved, given that the performance of different specifications is customarily evaluated in terms of the portfolio allocations via the ensuing optimal weights.

Focusing on the minimum-variance allocation, empirical results on a panel of 28 DJ-30 assets show that $\mathsf{DCW}$ outperforms model-based and model-free approaches in terms of out-of-sample portfolio variance and turnover (De Miguel et al., 2009). Since transaction costs can significantly alter the outlook in the performance of the approaches, we introduce the break-even transaction cost as a more comprehensive measure of forecasting performance, to confirm the goodness of the $\mathsf{DCW}$ allocation in terms of minimal portfolio variance and turnover. Since research on model-based and model-free approaches has proceeded on somewhat parallel tracks,⁵ a further contribution of this paper is a comparison across approaches, assessing the quality of the respective forecasts and portfolio allocations.

The paper is organized as follows. Section 2 introduces the optimal portfolio allocation problem. Section 3 introduces direct modeling of the portfolio weights. Model-based and model-free approaches are discussed in Section 4. Data, measures of performance, and results are presented in Section 5. Section 6 is devoted to the evolution of empirical $\mathsf{DCW}$ forecasts, when aggregated by sector and averaged by month. Section 7 concludes.