13.16.2 Linear Interpolation

Cubic Splines have historically been the method preferred for yield curve smoothing. Despite the popularity of the cubic spline approach, market participants have often relied on linear yield curve smoothing as a technique that is especially easy to implement, but its limitations are well-known:

• Linear Yield Curves are continuous but not smooth; at each knot point, there is a kink in the yield curve.
• Forward rate curves associated with linear yield curves are linear and discontinuous at the knot points. This means that linear yield curve smoothing sometimes cannot be used with the Heath, Jarrow, and Morton term structure model because it usually assumes the existence of a continuous Forward Rate Curve.
• Estimates for the parameters associated with popular term structure models like the Extended Vasicek model are unreliable because the structure of the yield curve is unrealistic. The shape of the yield curve, because of its linearity, is fundamentally incompatible with an academically sound term structure model. Resulting parameter estimates are, therefore, often implausible.

Note:

As in the case of the Cubic Spline, we extrapolate for the maturities less than the first term yield and greater than the last term yield: in the former, the yield is set to be equal to the first term yield, and for the latter, it is set to be the last term yield.