Long Forward Probabilities, Recovery and the Term Structure of Bond Risk Premiums

Abstract

We show that the martingale component in the long-term factorization of the stochastic discount factor due to Alvarez and Jermann (2005) and Hansen and Scheinkman (2009) is highly volatile, produces a downward-sloping term structure of bond Sharpe ratios, and implies that the long bond is far from growth optimality. In contrast, the long forward probabilities forecast an upward sloping term structure of bond Sharpe ratios that starts from zero for short-term bonds and implies that the long bond is growth optimal. Thus, transition independence and degeneracy of the martingale component are implausible assumptions in the bond market.

Long-term factorization of affine pricing kernels

“…In this case, κ P = κ L so that the data-generating measure coincides with the long-forward measure. This is the condition of Ross' recovery theorem (see Qin and Linetsky (2016) for more details).…”

“…(3.10) holds. In this regard, we recall that Qin and Linetsky (2016) identified the unique recurrent eigenfunction π R of an affine pricing kernel with the minimal solution of the quadratic vector equation (see Appendix F in the on-line e-companion to Qin and Linetsky (2016)). We recall that, for a Markovian pricing kernel S (see Hansen and Scheinkman (2009) and Qin and Linetsky (2016)), we can associate a martingale…”

“…In general, positive eigenfunctions are not unique. Qin and Linetsky (2016) proved uniqueness of a recurrent eigenfunction π R defined as such a positive eigenfunction of the pricing kernel S, i.e.…”

“…If the exponential ergodicity assumption is not satisfied, they may differ, or one may exist, while the other does not exist. We refer the reader to Qin and Linetsky (2016) and Qin and Linetsky (2017) for the exponential ergodicity assumption. Analytical tractability of affine models allows us to provide fully explicit examples to illustrate these theoretical possibilities.…”

“…In addition to the references above, the growing literature on the long-term factorization and its applications includes Hansen and Scheinkman (2012), Hansen and Scheinkman (2017), , Borovička et al (2011), , Bakshi and Chabi-Yo (2012), Bakshi et al (2015), Christensen (2017), Christensen (2016), Qin and Linetsky (2016), Qin et al (2016), Backus et al (2015), Filipović et al (2017), Filipović et al (2016). Empirical investigations in this literature show that the martingale component in the long-term factorization is highly volatile and economically significant (see, in particular, Bakshi and Chabi-Yo (2012) for results based on pricing kernel bounds, Christensen (2017) for results based on structural asset pricing models connecting to the macro-economic fundamentals, and Qin et al (2016) for results based on explicit parameterizations of the pricing kernel, where, in particular, the relationship among the measures P, Q and L is empirically investigated).…”