We need to choose the functions carefully if we want the estimation to yield a reasonably estimated discount function. The typical discount function is nonlinear. It is a monotonically decreasing function and converges to zero asymptotically at infinity. Thus, fitting a straight line is not a good idea. One can try to fit a higher order polynomial to the discount function. This is not a satisfactory solution either. If we fit low-order polynomials, they are usually not flexible enough and don't fit well, especially at the short-term maturities. If we fit high-order polynomials, they may fit well but tend to produce wild swings at long-term maturities where relatively few bonds mature. These wild swings usually result in unrealistic term structure estimates.

Spline functions are functions that help solve this problem as their flexibility can be increased locally where needed, without raising the polynomial order of the estimated function. Estimating the term structure...