Yesterday’s blog post described an implementation of the popular Nelson-Siegel approach to yield curve smoothing. In today’s post, we do a similar worked example for the maximum smoothness forward rate technique of Adams and van Deventer (1993), which is described in more detail in Chapter 8 of Advanced Financial Risk Management (van Deventer, Imai and Mesler, John Wiley & Sons, 2004). The maximum smoothness forward rate technique is superior to the Nelson-Siegel approach in every dimension that matters: it can fit observable market data perfectly, and it produces smoother (and therefore more reasonable) forward rate curves given equivalent restrictions on the right hand side (longest maturity) of the yield curve being fitted. This post provides a worked example using the maximum smoothness forward rate approach.
Spline techniques dominate computer animation and other applications where one has to draw a line connecting two or more points. It is well known that the smoothest line connecting an arbitrary set of points is a cubic spline, a series of cubic functions constrained in such a way that they fit smoothly together at each ‘knot point” or data point. The calculus of variations is used to prove that no other functional form can produce a smoother line.
For financial applications, the fact that the yield curve is smooth is necessary but not sufficient for a high quality yield curve smoothing application. It is also very important that the forward rate curve be reasonable and plausible. Early critics of the use of cubic splines for yield curve smoothing in fact made the argument that the technique was not useful in finance because often it produced forward rate curves that were not smooth and therefore implausible. Van Deventer and Adams addressed this issue directly by deriving the functional form, using a proof by Oldrich Vasicek, that produces the smoothest forward rate curve that is consistent with observable yield data. It turns out that the smoothest forward rate curve that can be drawn is a series of quartic splines that are connected at each data point or knot point in a smooth way. The Vasicek proof that this functional form is the smoothest forward rate curve also uses the calculus of variations. The forward curve that is produced in this manner is the smoothest that can be drawn, conditional on what constraints the user imposes on the first or second derivative of the forward rate curve at the shortest and longest maturities on the yield curve being drawn.
Adams and van Deventer were motivated to research improved smoothing technologies because of complaints from market participants that existing techniques, including the Nelson-Siegel technique, were not useful in the Japanese government bond market where trading was common only in one bond issue, the cheapest to deliver in the JGB futures market.
In this post, we use the Russian Federation bond data from our August 14, 2009 post, “Yield Curve Smoothing: Nelson-Siegel versus Spline Technologies, Part 2.” That data includes 4 bond prices, so we could in theory use each of the 4 maturity dates, plus time 0 and a long maturity T to fit the maximum smoothness forward rate function using 5 quartic segments for the forward rate curve: the segment from time 0 to the first maturity, the segment from the first to the second maturity, and so on.
Although it is common to use N+1 segments for N data points, it is not necessary. In fact it is very common for fewer than N+1 line segments to be used if there is “noise” in the data from bid-asked spreads, lack of synchronicity in quotations, and so on. In this post, we will use two line segments to fit the maximum smoothness forward rate curve.
For both of the line segments that we fit, the maximum smoothness forward rate curve will have this form, but the values of a, b, c, d and e will be different for each line segment:
The values of the coefficients a_i,b_i,c_i,d_i, and e_i are determined by the value of t relative to the “knots” of the spline. That is, if the value of t is between the first and second knot, we will use the vector a_1,b_1,c_1,d_1,e_1 to describe our coefficients. As the number of observations increases, the number of knots will increase and the quartic spline can be used to approximate any functional form with arbitrary levels of accuracy. This is one reason why spline methods are so preferred: in the limit, the specification is fully flexible.
The algorithm for maximum smoothness is as follows:
Note that in Step 4d, the constraint on the right hand side of the yield curve is normally decided by the analyst depending on the nature of the yield curve. The common constraints are for the first or second derivative of the forward rate curve to be set to zero. As noted in our Part 2 blog post, Tibor Janosi of Cornell University has proposed that the constraint be such that the first derivative of the forward rate curve be set to a constant x, and that x be optimized to maximize the smoothness of the curve. In this example, we arbitrarily constrained the first derivative of the forward rate curve to be zero at 3.6802 years to maturity. The following spreadsheet contains example entries for the matrix of linear constraints and the associated inverse after the optimization is completed:
The forward rate curve, derived using the steps above, is shown below. The flattening of the forward rate curve on the right hand side of the forward rate curve is consistent with the constraint that we imposed.
After the final iteration, the zero coupon yields that were the best fitting inputs to the maximum smoothness forward rate process were as follows:
The sum of squared pricing errors on the three bonds was zero to 11 decimal places. The coefficients a, b, c, d and e for each of the two forward curve line segments are given in the previous chart.
This example has shown how to fit the maximum smoothness forward rate function using 2 knot points to bond price data. In a high volume risk management environment, this calculation is normally done within a high quality enterprise risk management system. In fact, the maximum smoothness forward rate approach has been used continuously in risk systems since 1992 and it has stood the test of time.
In Part 4 of this series, we compare the robustness of the Nelson-Siegel approach with the maximum smoothness approach in order to see where the differences between the two methods are greatest. As explained in the introduction, the maximum smoothness forward rate approach will always (a) produce pricing errors that are equal to or smaller than those produced by Nelson-Siegel and (b) produce a smoother forward rate curve given appropriate constraints on the left and right hand side of the forward rate curve. In Part 4, we explore these differences in detail.
Sean Klein and Donald R. van Deventer
Kamakura Corporation
Honolulu, August 17, 2009