We consider a convergence model of interest rates, in which the behaviour of the domestic instantaneous interest rate (so called short rate) depends on the short rate in a monetary union that the country is going to join. The short rate in the monetary union is modelled by a two-factor model, which leads to a three-factor model for the domestic rate. In this setting, term structures of interest rates are computed from bond prices, which are obtained as solutions to a parabolic partial differential equation. A closed-form solution is known only in special cases. An analytical approximation formula for the domestic bond prices has been proposed, with the error estimate only for certain parameter values, when the solution has a separable form. In this paper, we derive the order of accuracy in the general case. We also study a special case, which makes it possible to model the phenomenon of negative interest rates that were observed in the previous years. It turns out that it leads to a higher accuracy than the one achieved in the general case without restriction on parameters.