Ebook Dynamic Financial Index Models: Modeling Conditional Dependencies via Graphs
Since the seminal work of Sharpe (1964), Financial Index Models are in the core of asset pricing and portfolio allocation problems. These models assume that all systematic variation in the returns of financial securities can be explained by one, or a set of market indices (factors). The central empirical implication of this assumption is a highly structured covariance matrix for the distribution of returns as, after conditioning on the chosen set of market indices, the residual covariance matrix is diagonal. The attractiveness of this approach is immediate as it offers a very simple, economically justifiable and stable way to estimate potentially very large covariance matrices. A vast body of literature is dedicated to testing the validity of Index Models and the selection of indices we refer the reader to Cochrane (2001) for a detailed account of the area.
The covariance matrix of returns is a key input in building optimal portfolios and its estimation is often challenging as the number of parameters grows exponentially with the number of assets considered. It is necessary, therefore, to work with structured models that reduce the dimensionality of the problem and deliver more effective estimates and, in turn, better investment decisions. In this paper, we explore a generalization of Financial Index Models with more complex patterns of covariation between returns by allowing conditional dependencies via the introduction of graphical constraints. We work with the matrix-variate dynamic graphical model (DGM) framework of Carvalho & West (2007a,b) but, unlike their original work, graphs are used to increase complexity and not to reduce it.
We take the view that, given its popularity in empirical finance, Index Models such as the Capital Asset Pricing Model (CAPM) and the Fama-French (FF) are appropriate for the purpose of asset allocation. The central idea of our work it to show that it is possible to improve upon traditional estimates from Index Models and provide more flexible, efficient and still parsimonious strategies for estimating covariances. In addition, we provide two extensions to DGMs: (i) we consider the problem of sequential inference about the graphical structure and, (ii) define the sequential updating process in the presence of stochastic regressors.
The proposed forecasting model is tested on stock returns data in a portfolio selection exercise. Using 100 NYSE monthly stock returns from 1989 through 2008, we find that our strategy yields better out-of-sample forecast of realized covariance matrix and lower portfolio variance than the two traditional implementations of index models, the capital asset pricing model (CAPM) and the Fama-French (FF) model.
We start by describing Index Models in Section 2 along with its use in the dynamic linear model context. Section 3 and 4 present the necessary background of dynamic matrix-variate graphical models. In Section 5 we discuss issues of dealing with graph (model) uncertainty through time and a simulation study in presented in Section 6. Section 7 expands the DGM context to allow for random regressors. Finally, in Section 8 we explore the use of DGMs as a tool to improve the implementation of Financial Index Models.
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