Black’s Consol Rate Conjecture

This paper confirms a version of a conjecture by Fischer Black regarding consol rate models for the term structure of interest rates. A consol rate model is one in which the stochastic behavior of the short rate is influenced by the consol rate. Since the consol rate is itself determined, via the usual discounted present value formula, by the short rate, such models have an inherent fixed point aspect. Under an equivalent martingale measure, purely technical regularity conditions are given for the stochastic differential equation defining the short rate and the consol rate to be consistent with the definition of the consol rate as the yield on a perpetual annuity. The results are based on an extension of the theory for the forward-backward stochastic differential equations to infinite-horizon settings. Under additional compatibility conditions, we also show that the consol rate is uniquely determined and given as a function of the short rate.