Sample Data for the Basic Building Blocks of Yield Curve Smoothing
In Part 5 of this series, we continue to use the same input data to the smoothing process that we used in Parts 3 and 4. We refer the reader to Part 3 for numerous examples of past U.S. Treasury yield curve data that have curves that are too complex for the Nelson-Siegel approach to fit the data exactly. As before, we continue to insist in this section of our series that any smoothing technique that does not fit the market exactly is unacceptable for practical use. In the meantime, we continue to fit this raw data with our derived “best” yield curve.
Example C: Linear Forward Rates and Related Yields
As always, unless otherwise noted, “yields” are always meant to be continuously compounded zero coupon bond yields and “forwards” are the continuous forward rates that are consistent with the yield curve. As in parts 3 and 4 of this series, the first step in exploring a yield curve smoothing technique is to define our criterion for best and to specify what constraints we impose on the “best” technique to fit our desired trade-off between simplicity and realism. We again answer the nine questions posed in Part 2 of this series. In this installment of the series, we make just one modification in our answers to those nine questions and DERIVE, not assert, the best yield curve consistent with the definition of “best” given the constraints we impose.
Step 1: Should the smoothed curves fit the observable data exactly?
1a. Yes. Our answer is unchanged. With only six data points at six different maturities, it would be a poor exercise in smoothing if we could not fit this data exactly.
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
2b. Forward rates is our choice, instead of zero coupon yields as in Parts 3 and 4 of the series. We focus on forward rates in Example C to address a specific problem that arose from linear yield curve smoothing in Example B. We found in that example that the yields were continuous but the forward rates were not. Instead, gaps opened and jumps in forward rates occurred at the knot points. We want to make that problem go away in our search for realism, although we always run the risk of taking one step backward from time to time. We note again that we would never choose 2a or 2b to smooth a curve where the underlying securities issuer is subject to default risk. In that case, we would make the choices in either 2c or 2d. We do that later in this series.
Step 3: Define “best curve” in explicit mathematical terms
3a. Maximum smoothness
3b. Minimum length of curve
3c. Hybrid approach
3b. Minimum length of curve. We continue with this criterion for best. As noted in earlier blogs, the following article on www.wikipedia.com explains how to calculate the length of a curve given the mathematical function that produced the curve:
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. We want to minimize s over the full length of the yield curve.
Step 4: Is the curve constrained to be continuous?
4b. Yes. This constraint is different in Example C in that it is imposed on forward rates instead of zero yields as in Example B. In Part 4, Example B, of this series, we found that requiring continuity in the “best” yield curve did indeed produce a very short yield curve, but the gaps in the forward rates at the knot points were unrealistic. We seek to remedy that in Part 5 by insisting on a continuous forward curve. It goes without saying that, when we impose more constraints on the “best” forward or yield curve, we will be intentionally selecting a yield curve that is not as “short” (under our current criterion for best) as the yield curve derived in Part 3 and Part 4. Again, we are willing to make that trade off in order to gain realism in our yield curve fitting.
The remaining five choices are the same as Parts 3 and 4 in this series. We will change our answers to questions 5-9 as we progress through this series on basic building blocks of yield curve smoothing.
Step 5: Is the curve differentiable?
5b. No. We know this may give us “kinks” where the five line segments we derive fit together, but at least for now we’re willing to tolerate this potential problem.
Step 6: Is the curve twice differentiable?
6b. No. As noted in Parts 3 and 4, the curve will not be twice differentiable at some points on the full length of the curve.
Step 7: Is the curve thrice differentiable?
7b. No. The reason is due to our choice of 5b.
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. For simplicity, we again answer No to this question.
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. Again, we choose No for simplicity and relax this assumption later in our series on the basic building blocks of yield curve smoothing.
Now that all of these choices have been made, both the functional form of the line segments for forward rates and the parameters that are consistent with the data can be explicitly derived from our sample data.
Deriving the Form of the Forward Curve Implied by Example C
The key questions in the list of 9 questions above are question 2 and question 4, where we again insist on the continuous nature of the curve, but we impose that continuity on forward rates instead of zero coupon yields. We note that by virtue of our choices in questions 5-9, these yield curve pieces are also not subject to any constraints except that the five forward “curves” join each other at the “knot points” in our data set, the maturities where two segments connect. Our data set has observable yield data at maturities of 0, 0.25 years, 1, 3, 5 and 10 years. This means we have 6 knot points in general and 4 knot points (0.25, 1, 3, and 5) where we require the adjacent forward rate curve segments to produce the same forward rate. All we have to do to get the “best” forward rate curve is to apply our criterion for “best”—the curve that produces the forward curve with shortest length—subject to the constraint of meeting at the four interior knot points
Since our objective is to join two dots on a piece of paper, as in Example B, we know from the old saying that “a straight line is the shortest distance between two points” that our “best” forward curve will consistent of five straight lines that, unlike Example A, will not be flat unless all of the observable yields are equal. In addition, unlike Example B, the forward rate curves must be drawn so they are continuous.
We can again measure the length of each line segment, thanks to Pythagoras, whom we invoked in Part 3 of this series:
As we noted in Parts 3 and 4 in this series, the functional form of the “best” forward curve given our definition and constraints can be derived more elegantly using the calculus of variations as Oldrich Vasicek did in the proof of the maximum smoothness forward rate approach in Adams and van Deventer (1994), reproduced in Chapter 8 in van Deventer, Imai and Mesler’s Advanced Financial Management (John Wiley & Sons, 2004). We omit that step here since the answer is so intuitive.
Each forward line segment has the linear form.
f(t)=a + bt
We call the first of the five functions f1(t) with intercept c1 and coefficient of t d11. The first “1” refers to line segment 1 and the second “1” refers to the power of t. Later in this series we will invoke higher powers of maturity t and we use this labeling now for consistency with later installments in the series. In example B, we had the good fortune to know the zero coupon yields at each knot point, so solving for the intercept and coefficient of time for each segment was independent of the other segments. In example C, we are not so lucky. We do know five constraints immediately however:
Equation 1: f1(0)=0.04 or c1+d11(0)=0.04
Equation 2: f1(0.25)=f2(0.25), or c1 + d11(0.25) – c2 – d21(0.25) = 0
Equation 3: f2(1)=f3(1), or c2 + d21(1) – c3 – d31(1)=0
Equation 4: f3(3)=f4(3), or c3 +d31(3) – c4 – d41(3)=0
Equation 5: f4(5)=f5(5), or c4 + d41(5) – c5 – d51(5)=0
Equations 2-5 require that the unobservable forward rates be equal where the line segments join at the knot points. Equation 1 requires that the forward rate at time 0 equal 0.04, since f(0) and y(0) are equal by definition.
In order to impose the constraints of observable market data, we make use of these relationships between the forward rate functions f, zero yields y, and zero prices p:
Consider the case of what must be true at two adjacent knot points, say tj and tk where tj<tk. We can link the observable zero coupon bond prices at those two knot points to the forward rate “curve” joining them using the second equation above:
This means that
Because we have imposed “maximum tension” or “shortest length” on our forward curve, we know that f(s) between tj and tk is linear and that this constraint becomes
The forward rate function f(s) between tk and tj is the jth segment in the forward curve. We use the zero coupon bond prices from our base data to derive this constraint on the first line segment f1 where:
We substitute 0.25 for t1, 0 for t0, 0.98820 for p(t1), and 1.00000 for p(t0). We have four more constraints just like it, which are based on the ratios of zero coupon bond prices at maturities of 1 year and 0.25 years, 3 years and 1 year, 5 years and 3 years, and 10 years and 5 years. This gives us 10 equations and 10 unknowns and we can solve for them directly using matrix inversion.
The system of equations we are trying to solve has this form in matrix notation:
The inverse of the coefficient matrix is shown here:
We then multiply this inverted matrix times the y vector above to solve for all 10 coefficients that are consistent with our 6 input yields:
We can also derive the coefficients for the yield curve segments consistent with these forward rate functions by using the first and second relationship of these links between continuously compounded yields, forwards and zero coupon bond prices:
For any time t between knot points tj and tk, where tj<=t<=tk, the zero coupon yield is related to the forward rate function by this relationship:
For any yield curve segment j, since tj and y(tj) are constants, the zero coupon yield that we have DERIVED, not assumed, from our specifications for the best forward rate curve, has the general form
How well does this combination of linear forward rates and related yields work from a realism point of view? Because of the nature of the observable data, we can see clearly that the implied forward rates are not realistic:
Like Example B, the good news is that the forward curve (in orange) and related yield curve (in red) both (a) fit the observable yield data, denoted by black dots, perfectly and (b) are continuous. The bad news is that the continuity restrictions on forwards were too simplistic to be realistic. The “best” criterion of shortest length forward curve subject to a continuity constraint and no other constraints produces a saw tooth movement in forward rates that’s too extreme to be realistic.
We gain some additional insights by overlaying the smoothing from Example C with the Nelson-Siegel yield curve and forward rates that we fitted in Part 3 of this series:
The Nelson-Siegel yield curve, plotted in green, is shorter (our criterion for best) than the Example C yield curve, but, again, for a very bad reason: The Nelson-Siegel yield curve doesn’t fit the data, so of course it’s shorter! We reject the Nelson-Siegel approach (again) for this reason. By examining what’s plotted above, however, we realize something else—the “shortness” of the yield curve is a logical virtue, but there is another one that we prize highly for its realism: the smoothness of the yield curve. We can see by inspection that constraining the line segments to fit together “smoothly’ is another virtue worth exploring.
The table below summarizes the criterion for “best” that we have used so far, along with the constraints we have imposed and the length of the yield curves derived. Since the curves are not everywhere differentiable, we’ve used a discrete approximation to length using the Pythagorean formula from Part 3 at time intervals of 1/12 of a year:
Examples A, B and C fit the data perfectly. Example A’s length, using 1/12 of a year intervals, measures longer when we count the discontinuities at the knot points. We can see that using linear forward rates produces a shorter (and therefore “better”) yield curve than the yield step function, but at great cost: we have wild swings up and down in the forward rate curve and its length balloons to 31.5921. For that reason, we will increasingly focus on smoothness of both yields and forward rates in this series. Because the Nelson-Siegel approach fails the most basic constraint, consistency with the observable market yields, we again reject it as unacceptable.
In Part 6, we impose more constraints on the “best yield curve” in order to maximize realism.
Donald R. van Deventer
November 30, 2009