How to Parameterize a Bond Yield Curve?

Question

Suppose I have a Bond Yield Curve (assume Semi-Annual Compounding), at term 1M, 3M... 1Y, 2Y... 10Y, 15Y ...30Y (x-axis is maturity / term).

How should I parameterize this yield curve? Any recommendations? Is there any known formula of Yield as a function of Maturity (or any approximations)?

And I also have some questions about the property of this yield curve:

First of all, is yield curve (strictly) monotone increase? Does the first order derivative have any meaning?

Secondly, does yield curve has an asymptote, as x -> Inf, y -> constant? Is the y-values bounded by a lower bound when x=0?

Thirdly, what can we say about the second order derivative f''? Does f'' has an upper-bound? Should the f'' be strictly non-negative? Or should we expect f'' change sign? If f'' did change sign, what does it tell us?

Finally, does the Area-Under-Curve has any meaning? (like those ROC curve has a meaningful AUC)

Answer 1

Nelson-Siegel and models in its succession (e.g. Diebold-Li) attempt to fit the yield curve as you describe. The reason for the development and research of these models answers your first additional question. The yield curve can take different shapes and the models attempt to model rising, inverted, flat, and humped yield curves.

For your second question, when maturity approaches 0, the bound is the risk free rate which is usually some type of government note (e.g. in the US, Fed Funds). When maturity approaches infinity, the rate is described in affine term structure models as mean reverting and should approach a constant.

For your questions relating to the calculus of the yield curve, perhaps you would get better answers by starting with the components of NS model which incidentally match the principal components of the curve. Since the curve isn't exactly a differentiable, continuous function (smoothing can create arbitrage conditions in the forward curve), it is more descriptive to use models of level, slope, and curvature. You should be able to quickly see the parallel to what you are looking for.

The integral under the zero curve will be the price of a zero coupon bond when compounded continuously.

I recommend learning about the Diebold-Li model, it will answer much of what you are searching.