Shape Factors and Cross-Sectional Risk

Abstract

Galluccio and Roncoroni (2006) empirically demonstrate that cross-sectional data provide relevant information when assessing dynamic risk in fixed income markets. We propose a theoretical framework supporting that finding, which is based on a notion of “shape factors”. This notion represents cross-sectional risk in terms of stylized analytical deformations experienced by yield curves. We provide an econometric procedure to identify shape factors, and propose a continuous-time yield curve dynamic model driven by these factors. Our proposal consists of a function-valued dynamic term structure model driven by these factors. We also propose and develop the corresponding arbitrage pricing theory. We devise three applications of the proposed framework. First, we derive interest rate derivative pricing formulas. Second, we study the analytical properties exhibited by a finite factor restriction of term structure dynamics that are cross-sectionally consistent with a family of exponentially weighed polynomials. Finally, we conduct an empirical analysis of cross-sectional risk on US swap, Euro bond and oil price data sets. Results support our conclusion that shape factors outperform the classical yield/price factors (i.e., level, slope, and convexity) in explaining the underlying market risk. The methodology can in principle be used for understanding the intertemporal dynamics of any cross-sectional data, whether it be price curves, option implied volatility smiles, or cross-sections of equity returns.

Keywords: Risk measures, Infinite-dimensional stochastic calculus, Cross-sectional analysis

JEL Classification: C21, C22, C51, E43