We show the Cox, Ingersoll, and Ross term structure framework can allow a variety of alternative equilibrium solutions for discount bond prices. This is important since it allows us additional flexibility in developing models that capture the properties of the term structure. As an example, we solve for the value of a discount bond when the short-term rate is absorbed at zero. We compare the yields implied by this model to those implied by the original Cox, Ingersoll, and Ross model. We also show that alternative equilibria can occur in other term structure models.