A note on the long rate in factor models of the term structure

We show that, as a consequence of the Dybvig-Ingersoll-Ross theorem, the existence of a non-deterministic long rate in a factor model of the term structure implies that the model has an equivalent representation in which one of the state variables is nondecreasing. Moreover, for two-dimensional factor models, we prove that if the long rate is non-deterministic, the yield curve flattens out and the factor process is asymptotically non-deterministic, then the term structure is unbounded. Finally, following up on an open question in El Karoui et al. (1997), we provide an explicit example of a three-dimensional affine factor model with a non-deterministic yet finite long rate in which volatility of the factor process does not vanish over time.