Application of factor models to risk premium estimation

Abstract

We provide a rigorous mathematical approach to the beta-pricing model, starting from the standard two-step cross-sectional regression, through Nonlinear Seemingly Unrelated Regression (NSUR) and Generalized Method of Moments (GMM), and finally compare the results with several linear approximation methods. The use of the linear approximation applied to a single-factor nonlinear system of equations is new in the literature and is one of the major contributions of this work. Our results show that, in the presence of heavy-tailed distributions, the L1-norm methods proposed in this study are more appropriate (exhibiting lower bias and variance) for risk price estimation than traditional L2-norm approaches. It is also the first time that the Capital Asset Pricing Model (CAPM) is applied systematically to compare the integration and segmentation between different markets and a given portfolio set. Our study, Penco and Lucas (2024), applies a two-factor integration model to the economies of Asia, Europe, Japan and North America, showing integration between the European and North American economies. We also extend the integration model to commodity markets. To capture more accurately the cross-sectional pricing of commodity risk we use the Cochrane factor mimicking approach and compare the results with alternative dependence-based integration measures using copulas. We show how the copula correlation between the Stochastic Discount Factor (SDF) and returns differentiates the contribution of joint dependence from the contribution of the risk prices. Finally, we introduce a penalised p-value Fama-MacBeth Generalized Least Squares (GLS) regularisation, which provides several advantages over other methods as it ensures that retained factors contribute not only to statistical fit but also to risk pricing. Unlike other approaches, this method regularises the pricing kernel directly. Factors that lack significance or explanatory power are penalised and removed, while priced and relevant sources of risk are preserved.