Donald R. Van Deventer, Ph.D.

Don founded Kamakura Corporation in April 1990 and currently serves as Co-Chair, Center for Applied Quantitative Finance, Risk Research and Quantitative Solutions at SAS. Don’s focus at SAS is quantitative finance, credit risk, asset and liability management, and portfolio management for the most sophisticated financial services firms in the world.

# Basic Building Blocks of Yield Curve Smoothing, Part 1

11/02/2009 12:53 PM

Our recent series on Nelson Siegel and other yield curve smoothing techniques has generated an enormous amount of interest. We’ve come to the conclusion that the art of constructing a yield curve deserves more attention than it’s gotten. This blog outlines
what we have planned and seeks your suggestions.

After our last series of posts on yield curve smoothing, we had a very productive exchange with Leif Andersen of Bank of America, who’s done a nice paper on tension splines.  We’ve also plunged into Bob Jarrow’s archives and are trying to unearth an unpublished gem of a paper that Prof. Jarrow and long-time research partner Tibor Janosi did in 2002.

In general, we want to illustrate how to build a yield curve which meets the criteria for “best” that are defined by the user. Given a specific set of criteria, what yield curve results? By the criteria defined, how do we measure “best” and compare the best yield curve we derive with alternatives?

There is a most important set of comparisons we need to make—among N different sets of criteria for “best yield curve,” which set of criteria is in fact “best,” and what does “best” mean in this context?

In starting out in this series of building blocks for yield curve smoothing, we have to realize how lucky we are in finance that smoothing is only a two dimensional, not a 3 dimension, problem. That makes our task much simpler than the same problem is for computer graphics experts and engineers.

Here’s brief set of yield curve smoothing techniques that we’ve seen in the literature. We plan on going through them one by one on the same set of data. If you have any suggested additions to the list, please e-mail us at info@kamakuraco.com.

• Linear yield curve smoothing with no “connectivity” between line segments
• Linear forward rate smoothing with no connectivity between line segments
• Linear yield curve smoothing that is continuous, but without requiring equal first or second derivatives at the knot points
• Linear forward rate smoothing that is continuous, but without requiring equal first or second derivatives at the knot points
• Yield curve smoothing that is continuous and with equal first derivatives (but not second derivatives) at the knot points
• Forward rate smoothing that is continuous and with equal first derivatives (but not second derivatives) at the knot points
• Yield curve smoothing that is continuous and with equal first and second derivatives at the knot points
• Forward rate smoothing that is continuous and with equal first and second derivatives at the knot points
• Maximum smoothness creation of yield curves and forward rate curves
• The Janosi-Jarrow extension of maximum smoothness
• Tension splines
• Smoothing to avoid negative interest rates
• Smoothing with data errors

For each of these techniques, we compare them with the well-known Nelson-Siegeltechnique to show the stark contrasts between the results given the definition of “best” and what is produced by Nelson Siegel.

Donald R. van Deventer
Kamakura Corporation
Honolulu, November 2, 2009