In Part 1 of our series on the basic building blocks of yield curve smoothing, we listed 13 different approaches one could take to smoothing yields or forward rates. In this installment, we talk about how the definition of “best” yield curve or forward rate curve and the constraints one imposes on the resulting yield curve implies the mathematical function that is “best.” This is the right way to approach smoothing. The wrong way to approach smoothing is the exact opposite: choose a mathematical function from the infinite number of functions one could draw and argue qualitatively why your choice is the “right one.” In this post, we discuss the definition of “best” yield curve and the constraints commonly placed on the smoothing process.
In our prior post, we listed these approaches to yield curve smoothing that we’ve seen in the literature:
- 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
With the exception of the last two items on the menu above, the functional form listed is not arbitrary. Instead, the mathematical yield curve or forward rate smoothing formulation can be derived after going through the following steps:
Step 1: Should the smoothed curves fit the observable data exactly?
1a. Yes
1b. No
Except in the case where the data is known from the outset to be bad, there is no reason for “yes” not to be the standard choice here. If one does chose yes, the Nelson-Siegel approach to yield curve smoothing will be eliminated as a candidate because it is not flexible enough to fit 5 or more data points perfectly except by accident. Even if the number of data points is the same as the number of Nelson-Siegel parameters (4), the Nelson-Siegel function will fit the data exactly only by coincidence. For more on why the Nelson-Siegel approach to yield curve smoothing fails to meet best practice standards, see our five part blog series on www.kamakuraco.com and www.riskcenter.com:
Klein, Sean and Donald R. van Deventer, “Yield Curve Smoothing: Nelson-Siegel versus Spline Technologies, Part 1,” Kamakura blog, www.kamakuraco.com, July 21, 2009. Redistributed on www.riskcenter.com on July 23, 2009.
Klein, Sean and Donald R. van Deventer, “Yield Curve Smoothing: Nelson-Siegel versus Spline Technologies, Part 2, Kamakura blog, www.kamakuraco.com, August 14, 2009. Redistributed on www.riskcenter.com on August 17, 2009.
Klein, Sean and Donald R. van Deventer, “Yield Curve Smoothing: Nelson-Siegel versus Spline Technologies,” Part 3 Kamakura blog, www.kamakuraco.com, August 17, 2009. Redistributed on www.riskcenter.com on August 18, 2009.
Klein, Sean and Donald R. van Deventer, “Yield Curve Smoothing: Nelson-Siegel versus Spline Technologies, Part 4” Kamakura blog, www.kamakuraco.com, August 18, 2009. Redistributed on www.riskcenter.com on August 19, 2009.
van Deventer, Donald R. “Yield Curve Smoothing: Nelson-Siegel versus Spline Technologies, Part 5,” Kamakura blog, www.kamakuraco.com, September 8, 2009. Redistributed on www.riskcenter.com on September 9, 2009.
Step 2: Select the element of the yield curve and related curves for analysis
2a. Zero coupon yields
2b. Forward rates
2c. Continuous credit spreads
2d. Forward continuous credit spreads
Smoothing choices 2a and 2b, smoothing with respect to either yields or forward rates, should only be selected when smoothing a curve with no credit risk. If the issuer of the securities being analyzed has a non-zero default probability, smoothing should only be done with respect to credit spreads (the risky zero coupon yield minus the risk-free zero coupon yield of the same maturity) or forward credit spreads (the risky forward curve less the risk-free forward curve). See Chapters 8 and 18 of Advanced Financial Management (van Deventer, Imai and Mesler, John Wiley & Sons, 2004) on why this distinction is important.
Step 3: Define “best curve” in explicit mathematical terms
3a. Maximum smoothness
3b. Minimum length of curve
3c. Hybrid approach
The real objective of yield curve smoothing, broadly defined, is to estimate those points on the curve that are not visible in the marketplace with the maximum amount of realism. “Realism” in the original Adams and van Deventer (1994, summarized in Chapter 8 of Advanced Financial Risk Management) paper on maximum smoothness forward rates was defined as “maximum smoothness” for two reasons. First, it was a lack of smoothness in the forward rate curves produced by other techniques that was the criterion for their rejection by many analysts. Second, “maximum smoothness” was viewed as consistent with the feeling that economic conditions rarely change in a non-smooth way and, even when they do (i.e. if the Central Bank manages short term rates in such a way that they jump in 0.125% increments or multiples of that amount) the uncertainty about the timing of the jump makes future rate expectations smooth. Adams and van Deventer also used the “Shimko test” to see if this hypothesis about realism was in fact more accurate in predicting data points which were intentionally left out of the smoothing process, compared to other techniques. The maximum smoothness yield curve or forward rate curve g(s) (where s is the years to maturity for the continuous yield or forward) is the one that minimizes this function Z:
There is another criterion that can be used for “best” curve, the curve that is the shortest in length. This again can be argued to be most realistic in the sense that one would not expect big swings in the yield, even if the swings are smooth. The following article on www.wikipedia.com explains how to calculate the length of a curve given the mathematical function that produced the curve:
http://en.wikipedia.org/wiki/Arc_length
The length s of a yield curve or forward rate curve between maturities a and b is
where f’(x) is the first derivative of the yield curve or forward rate curve. Some analysts have suggested using “tension splines” that take a hybrid approach to these two criteria for best. Leif Andersen proposes such an approach in this paper in the Review of Derivatives Research in 2007:
Once the mathematical criterion for “best” is selected, we move on to specifications on curve fitting that will add to the realism of the fitted curves.
Step 4: Is the curve constrained to be continuous?
4a. Yes
4b. No
Why is this question being asked? It can be shown (as was done in an Appendix contributed by Oldrich Vasicek in the Adams and van Deventer paper, corrected in Chapter 8 of Advanced Financial Risk Management) that the smoothest way to connect N points on the yield curve is NOT to fit a single function, like the Nelson-Siegel function below, to the data. Note y(t) is the continuous yield and t is time to maturity:
It can be proven that the Nelson-Siegel curve will ALWAYS be inferior, regardless of which criterion for “best” is chosen, to N or N+1 line segments that are fit to the data. The nature of the functional form that is consistent with the criterion for best and the constraints imposed can be derived using the calculus of variations, as Vasicek does in his proof. Any other functional form will not be “best” by the method selected by the analyst.
Given this, we now need to decide whether or not we should constrain the yield curve or forward rate curve to “connect” in a continuous fashion where each of the line segments meet. It is not obvious that “yes” is the only answer to this question. As an example, Robert Jarrow and Yildiray Yildirim use a four-step piece-wise constant function for forward rates in this paper:
This forward rate curve would not be continuous but it is both perfectly smooth (except where the four segments join) and shorter in length than any non-constant function. Most analysts, however, would answer “yes” in this step.
Step 5: Is the curve differentiable?
5a. Yes
5b. No
If one answers “yes” to this question, the first derivative of the two line segments that join at “knot point” N have to be equal at that point in time. If one answers “no,” that allows the first derivatives to be different and the resulting “curve” will have kinks or elbows at each knot point, like you would get with linear yield curve smoothing—where each segment is linear and continuous over the full length of the curve but not differentiable at the points where the linear segments join. Most analysts would answer “yes” to this question.
Step 6: Is the curve twice differentiable?
6a. Yes
6b. No
This constraint means that the second derivatives of the line segments that join at maturity N have to be equal. If the answer to this question is “yes,” the segments will be neither linear nor quadratic.
Step 7: Is the curve thrice differentiable?
7a. Yes
7b. No
One can still further constrain the curve so that, even for the third derivative, the line segments paste together smoothly.
Step 8: At the spot date, time 0, is the curve constrained?
8a. Yes, the first derivative of the curve is set to zero or a non-zero value x.
8b. Yes, the second derivative of the curve is set to zero or a non-zero value y.
8c. No
This constraint is typically imposed in order to allow an explicit solution for the parameters of the curve for each line segment. See Chapter 8 of Advanced Financial Management for examples. The choice of non-zero values in 8a or 8b was suggested by Tibor Janosi and Robert Jarrow in a 2002 paper.
Step 9: At the longest maturity for which the curve is derived, time T, is the curve constrained?
9a. Yes, the first derivative of the curve is set to zero or a non-zero value j at time T.
9b. Yes, the second derivative of the curve is set to zero or a non-zero value k at time T.
9c. No
Step 9 is taken for two reasons. First, it is often necessary for an explicit, unique set of parameters to be derived for each segment as in step 8. Second and more importantly, it may be important for realistic curve fitting. If the longest maturity is 100 years, for instance, most analyst would expect the yield curve to be flat (choice 9a=0), rather than rising or falling rapidly, at the 100 year point. If the longest maturity is only 2 years, this constraint would not be appropriate and one would probably make choice 9b. Again, the suggestion of non-zero values in 9a and 9b is an insight of Janosi and Jarrow (2002).
Once all of these choices have been made, both the functional form of the line segments and the parameters that are consistent with the data can be explicitly derived. The resulting forward rate curve or yield curve that is produced by this method has these attributes:
- Given the constraints imposed on the curve and the raw data, the curve is the “best” than can be draw consistent with the analyst’s definition of “best”
- The data will be fit perfectly
- All constraints will be adhered to exactly
Contrast this happy set of results with what one would get if one used the Nelson-Siegel approach on the same data:
- The Nelson-Siegel curve would not be “best” according to the criterion selected by the analyst
- The Nelson-Siegel curve would not fit the data exactly if there are 5 or more data points
- The Nelson-Siegel curve may or may not fit the constraints imposed in Steps 8 or 9. If these constraints are imposed on Nelson-Siegel, the degree to which the curve fits the input data would be further degraded.
In this sense, the Nelson-Siegel approach is inferior to every single yield curve smoothing approach derived from a given combination of constraints and the criterion for “best.” We illustrate this in coming blogs where we go through these steps:
- We select the criterion for best
- We impose constraints we think are realistic given the problems at hand
- We derive the functional form and parameters for the curve
- We prove the curve is “better” than alternatives by our definition
To be continued!
Donald R. van Deventer
Kamakura Corporation
Honolulu, November 10, 2009