Variance Bounds on the Permanent and Transitory Components of Stochastic Discount Factors

When the transitory component of the stochastic discount factors (SDFs) prices the long-term bond, and the permanent component prices other assets, we develop lower bounds on the variance of the permanent component and the transitory component, and on the variance of the ratio of the permanent to the transitory components of SDFs. A salient feature of our bounds is that they incorporate information from average returns and the variance-covariance matrix of returns corresponding to a generic set of assets. Relevant to economic modeling, we examine the tightness of our bounds relative to Alvarez and Jermann (2005, Econometrica). Exactly solved eigenfunction problems are then used to study the empirical attributes of asset pricing models that incorporate long-run risk, external habit persistence, and rare disasters. Specific quantitative implications are developed for the variance of the permanent and the transitory components, the return behavior of the long-term (infinite-maturity) bond, and the comovement between the transitory and the permanent components of SDFs.