How would you define quasi-cubic in plain English?

Question

The context is the quasi-cubic hermite spline function used to calculate the yield curves of bonds. I know what hermite spline function is but am having trouble with quasi-cubic in that context.

Answer 1

I don't know what curves are used to calculate bond yields, but here is some info that might be relevant ...

You know what Hermite cubic splines are, you said, so I guess the mystery is the word "quasi". I can think of two possibilities:

(1) If the splines are not $C_2$ (continuous second derivatives), then some people would not regard them as "real" splines. Maybe the "quasi" qualifier is being used to indicate this lack of continuity. Hermite cubic splines computed from given values and derivatives are generally not $C_2$, so this explanation is plausible.

(2) In spline theory, there is a thing called a "quasi-interpolant" (references here). It's a "smoothing" function that approximates the given values but doesn't interpolate them. Maybe this is what's being referred to.

Answer 2

The US Treasury mentions the "quasi-cubic hermite spline function" they use for interpolating yields of US Treasury bonds on their web site. There is no precise description of what "quasi" refers to in this context, or any further description of the method. It is likely a modification of the cubic hermite spline method adapted to the specific requirements of bonds.

There are published papers on such methods, such as

James M. Hyman. Accurate monotonicity preserving cubic interpolation. SIAM Journal on Scientific and Statistical Computing, 4(4):645—654, 1983.

Patrick S. Hagan, Graeme West: Methods for Constructing a Yield Curve, May 2008, Wilmott Magazine. Specifically the "monotone convex (unameliorated)" method.

Most likely the method used by the USTR is one of the above two methods or a close relative thereof.