Size, book-to-market, and momentum during the business cycle

The Fama-French-Methodology (1993-1998) offers cross-sectional explanations of returns by taking the specially designed portfolios SMB and HML as additional factors. It is acknowledged that these factors are related to some forms of risk (they bear premia) which, by researchers is often proposed to be related to the uncertainty with respect to macroeconomic production and aggregate consumption. In more recent research a momentum factor is included in order to improve the explanatory power of the Fama-French-Model. We use data from business cycles 1926-2007 to show that SMB represents the risks related to the very early phase of an upswing while HML may be related to the uncertainty whether a business cycle will continue to gain depth and strength (or shifts back into recession). In contrast to SMB and HML, we do not find momentum to be related to risks associated with particular phases of the business cycl

Discrete Routh Reduction

This paper develops the theory of abelian Routh reduction for discrete mechanical systems and applies it to the variational integration of mechanical systems with abelian symmetry. The reduction of variational Runge-Kutta discretizations is considered, as well as the extent to which symmetry reduction and discretization commute. These reduced methods allow the direct simulation of dynamical features such as relative equilibria and relative periodic orbits that can be obscured or difficult to identify in the unreduced dynamics. The methods are demonstrated for the dynamics of an Earth orbiting satellite with a non-spherical $J_2$ correction, as well as the double spherical pendulum. The $J_2$ problem is interesting because in the unreduced picture, geometric phases inherent in the model and those due to numerical discretization can be hard to distinguish, but this issue does not appear in the reduced algorithm, where one can directly observe interesting dynamical structures in the reduced phase space (the cotangent bundle of shape space), in which the geometric phases have been removed. The main feature of the double spherical pendulum example is that it has a nontrivial magnetic term in its reduced symplectic form. Our method is still efficient as it can directly handle the essential non-canonical nature of the symplectic structure. In contrast, a traditional symplectic method for canonical systems could require repeated coordinate changes if one is evoking Darboux' theorem to transform the symplectic structure into canonical form, thereby incurring additional computational cost. Our method allows one to design reduced symplectic integrators in a natural way, despite the noncanonical nature of the symplectic structure.Comment: 24 pages, 7 figures, numerous minor improvements, references added, fixed typo