In Part 4 of our series on the basic building blocks of yield curve
smoothing, we tweak our constraints on the “best” yield curve and find that our
criterion for best implies linear segments for both yields and forwards.
We compare the results to the popular but flawed Nelson-Siegel approach and gain
insights on how to further improve the realism of our smoothing techniques, a
step forward we will make in part 5 of this series.
Sample Data for the Basic Building Blocks of Yield Curve Smoothing
In Part 4 of this series, we continue to use the same input data to the
smoothing process that we used in Part 3. 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. 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. We
deal with the issue of “bad data” in a later post in this series. In the
meantime, we continue to fit this raw data with our derived “best” yield
curve.
Example B: Linear Yields and Related Forward Rates
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
part 3 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 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
1b. No
1a. Yes. As we noted in Part 3, 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
2a. Zero coupon yields is our choice as in Part 3 of the 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
for a few more installments in this series. 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. We want to minimize s over the full length of the yield
curve.
Step 4: Is the curve constrained to be continuous?
4a.
Yes
4b. No
4b. Yes. This is the big change for Example B. In Part 3
of this series, we found that allowing discontinuities in the yield curve did
indeed produce a very short yield curve, but the gaps in the step-wise constant
yields and forward rates were unrealistic. We seek to remedy that in Part
4 by insisting on a continuous yield curve. It goes without saying that,
when we impose more constraints on the “best” 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. 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 Part 3 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?
5a.
Yes
5b. No
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?
6a.
Yes
6b. No
6b. No. As noted in Part 3, the curve will not be twice differentiable
at some points on the full length of the curve.
Step 7: Is the curve thrice differentiable?
7a.
Yes
7b. No
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-ze | 
