Pitfalls in Smoothing Interest Rate Term Structure Data: Equilibrium Models and Spline Approximations

The interest rate term structure refers to the array of discount rates on a collection of pure discount bonds that differ one from another only by the timing of their redemption. The most common approximation to the term structure is, of course, the yield to maturity curve, which is usually depicted as a smooth curve that relates rates of return on such bonds held to maturity to their term to maturity. Other expressions of the term structure also could be constructed, but underlying them all is the discount or present value function that we may denote δ(t). δ is the discount applied to a unitary payment to be made t periods hence. Expressing the term structure in this way does not necessarily imply that the term structure is itself driven by t, payment timings. Most economists would generally agree, however, that it is possible to draw smooth discount curves over the time axis. It is necessary to assume only that yield curves are continuous and smooth. By resort to arbitrage arguments implicit in equilibrium theories of the term structure, many economists are willing to live with these assumptions.