Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets

By Gary Chamberlain, M. Rothschild. 1983

We examine the implications of arbitrage in a market with many assets. The absence of arbitrage opportunities implies that the linear functionals that give the mean and cost of a portfolio are continuous; hence there exist unique portfolios that represent these functionals. These portfolios span the mean-variance efficient set. We resolve the question of when a market with many assets permits so much diversification that risk-free investment opportunities are available. Ross 112, 141 showed that if there is a factor structure, then the mean returns are approximately linear functions of factor loadings. We define an approximate factor structure and show that this weaker restriction is sufficient for Ross' result. If the covariance matrix of the asset returns has only K unbounded eigenvalues, then there is an approximate factor structure and it is unique. The corresponding K eigenvectors converge and play the role of factor loadings. Hence only a principal component analysis is needed in empirical work.

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

The Generalized Dynamic-Factor Model: Identification and Estimation (2000, methodology): “It encompasses as a special case the approximate-factor model of Chamberlain (1983) and Chamberlain and Rothschild (1983), which allows for correlated idiosyncratic components but is static.”
Generalized dynamic factor models and volatilities: recovering the market volatility shocks (2016, unknown): “(ii) The " low rank plus sparse " type decomposition of Γ corresponds to a factor model decomposition of {Y t } of the type considered by Chamberlain and Rothschild (1983), where q factors are loaded in a static way, that is, through a matrix of instantaneous loadings.”
Fitting dynamic factor models to non-stationary time series (2009, background): “Chamberlain and Rothschild (1983) defined the class of approximate factor models as a sequence of factor models where the covariance matrix ΣZN of the idiosyncratic components is a sequence of non-diagonal matrices with uniformly bounded eigenvalues.”
Do financial variables help forecasting inflation and real activity in the Euro area (2003, unknown): “This feature is shared with the static approximate factor model of Chamberlain and Rothschild (1983), for which our model provides a dynamic generalization.”
GLS Estimation of Dynamic Factor Models (2011, methodology): “Under the weak assumptions of an approximate factor model (Chamberlain and Rothschild 1983), the parameters can be consistently estimated by applying the traditional principal component (PC) estimator (Stock and Watson 2002b; Bai 2003) or—in the frequency domain—by using the dynamic principal…”
Eigenvalue Ratio Test for the Number of Factors (2013, methodology): “Preliminaries and Motivation We begin by defining the approximate factor model of Chamberlain and Rothschild (1983).”
Modeling Dependence in High Dimensions With Factor Copulas (2015, background): “Xi = h (Z; "i) , i = 1; 2; :::; N Z s Fz, "i s iid F", Z??"i 8 i (13)”
Panels with Nonstationary Multifactor Error Structures (2006, unknown): “See, for example, Chamberlain and Rothschild (1983), Connor and Korajzcyk (1986) and Connor and Korajzcyk (1988). But more recently the unobserved factor models have gained popularity for forecasting with a large number of variables as advocated by Stock and Watson (2002).”
Matrix Completion, Counterfactuals, and Factor Analysis of Missing Data (2019, methodology): “The econometrics literature on large dimensional factor analysis has largely adopted as framework the approximate factor model due to Chamberlain and Rothschild (1983) according to which the largest r population eigenvalues of X will increase with N and T while the remaining ones are bounded.”
A network analysis of the volatility of high dimensional financial series (2017, methodology): “In agreement with arbitrage pricing theory, the factor-driven, or common, component represents the non-diversifiable or systematic part of risk, whereas the idiosyncratic component becomes perfectly diversifiable as the dimension of the system grows (Chamberlain and Rothschild, 1983).”
STATISTICAL ANALYSIS OF FACTOR MODELS OF HIGH DIMENSION (2012, methodology): “Many applications or theoretical models involve data sets with the number of variables comparable with or even greater than the number of observations; see [11, 20, 22] and [23].”
Principal Component Analysis of High-Frequency Data (2016, unknown): “That being said, PCA is often employed to analyze (approximate) factor structures (Ross 1976; Chamberlain and Rothschild 1983), including estimating the number of approximate factors (see Connor and Korajczyk 1993; Bai and Ng 2002; Amengual andWatson 2007; Kapetanios 2010;Onatski 2010, and Trapani 2”
TESTING HYPOTHESES ABOUT THE NUMBER OF FACTORS IN LARGE FACTOR MODELS (2009, background): “In an important special case when Λ(L) does not depend on L but factors are not necessarily white noise, model (1) reduces to an approximate factor model of Chamberlain and Rothschild (1983). In such a case, we can test hypotheses about the number of approximate, as opposed to dynamic, factors using”
The Generalized Dynamic Factor Model (2005, unknown): “…in this article is an approximate factor model, in that the idiosyncratic component are allowed to be weakly correlated, as was done by Chamberlain (1983) and Chamberlain and Rothschild (1983) and in contrast to the approaches of Sargent and Sims (1977), Geweke (1977), and Quah and Sargent (1993).”