In part 10 of this series on yield curve smoothing, we included the maximum smoothness forward rate approach in our comparison of 23 different smoothing techniques, both in terms of smoothness and “tension” or length of the resulting forward and yield curves. In each of our worked examples, we showed how to derive unique forward rate curves and yield curves based on the same set of sample data. This sample data assumed that we had observable zero coupon yields or zero coupon bond prices to use as inputs. At most maturities, this will not be the case and the only observable inputs will be coupon-bearing bond prices. In this post, we show how to use coupon-bearing bond prices to derive maximum smoothness forward rates and yields. The same approach can be applied to the 22 other smoothing techniques summarized in Part 10 of this series.
Revised Inputs to the Smoothing Process
In the first 10 installments in this series, we used the following inputs to various yield curve smoothing approaches:
For the remaining parts of this series on basic building blocks of yield curve smoothing, we assume the following:
- The shortest maturity zero coupon yield is observable in the overnight market at 4%
- The 3 month zero coupon yield is observable at 4.75% in the market for short term instruments, like the U.S. Treasury bill market
- The only other observable instruments are 3 coupon bearing bonds with the following attributes:
Note that we are expressing value in terms of observable “net present value,” which is the sum of “price” and “accrued interest.” It is the total dollar amount that the bond buyer pays the bond seller. We ignore the arbitrary accounting division of this number into two pieces (“accrued interest” and “price”) because they are not relevant to the smoothing calculation. We assume all three bonds pay interest on a semiannual basis. In doing the smoothing calculation in practice, we would use exact day counts that recognize that “semiannual” could mean 179 or 183 days or some other number. For purposes of this example, we assume that the two halves of the year have equal length.
Note also that the yield to maturity on these bonds is irrelevant to the smoothing process. The historical “yield to maturity” calculation embeds a number of inconsistent assumptions about forward rates among the three bonds and other observable data. To use “yield to maturity” as an input to the smoothing process is a classic case of “garbage in/garbage out.”
Valuation Using Maximum Smoothness Forward Rates from Example H
The first question we ask ourselves is this: how far off the observable net present values is the net present value we could calculate using Example H Qf1a (where we constrained the forward rate curve to be flat at the 10 year point) from Part 10 of this blog series?
To make this present value calculation, we use the coefficients for the 5 forward rate curve segments from our Part 10 blog. Recall that we have 5 forward rate curve segments that are quartic functions of years to maturity:
The coefficients that we derived for the base case in Example H were these coefficients:
We can then use these coefficients in this equation to derive the relevant zero coupon bond yield for each payment date on our three bonds. The yield function in any yield curve segment j is
We note that y* denotes the observable value of y at the left hand side of the line segment where the maturity is t_{j}. Within the segment, y is a quintic function of t, divided by t. Using this formula, we lay out the cash flow timing and amounts for each of the bonds, identify which segment is relevant, calculate the zero coupon yield and the relevant discount factor using the formulas from Part 10 of our blog:
If we multiply each cash flow by the relevant discount factor, we can get the theoretical bond net present value from the coefficients derived in Example H of Part 10 in this blog series:
The table shows that the zero coupon yields that we used in the smoothing process produced incorrect net present values. As we shall see below, the problem is not the maximum smoothness forward rate technique itself. It is the inputs to the smoothing process at the 1, 3, 5 and 10 year maturity in our input table:
Given these inputs, our smoothing coefficients and NPVs follow directly as we have shown above and in part 10 of this series. We now improve our valuations by changing the zero yields used as inputs to the process.
Iterating on Zero Coupon Yields to the Smoothing Process
We know that the 4% yield for a maturity of zero and the 4.75% yield for 0.25 year maturity are consistent with observable market data. That is clearly not true for the 1, 3, 5 and 10 year zero yields because (a) there are no observable zero coupon bond yields at those maturities and (b) bond prices in the market are trading at net present values that are inconsistent with the yields we have been using at those maturities so far.
We now pose this question:
What values of zero coupon bond yields at maturities of 1, 3, 5, and 10 years will minimize the sum of squared pricing errors on our three observable bonds?
We can answer this question using a powerful enterprise-wide risk management like Kamakura Risk Manager (see www.kamakuraco.com) or even by using common spreadsheet software’s non-linear optimization routines. Using the latter approach, we find that this set of inputs eliminates the bond pricing errors:
The forward rate curve coefficients for maximum smoothness forward rate smoothing that are consistent with these inputs are given here:
Using these coefficients, we again lay out the maturities and amounts of all cash flows and compare the discount factors derived from these coefficients with those derived from the forward curve in Example H:
The table below confirms that the sum of the cash flows multiplied by the relevant discount factors produces bond net present values that exactly match those observable in the marketplace:
The yield curves that we derive from these observable bond net present values are of course different from those derived from the “guessed” zero coupon yield curve inputs of Example H:
This simple example shows that using bond prices as input to the smoothing process is only a small step forward from using zero coupon bond yields. It goes without saying that there is no need to use an arbitrary functional form, like the Nelson-Siegel approach, that is neither consistent with the observable bond prices nor optimum in terms of smoothness. It continues to mystify me why that technique is ever employed.
Extensions to the Analysis
For the remainder of this blog series, we will use bond prices as inputs. Two important questions come to mind:
- What if the bonds are issued by a firm with credit risk? How should the analysis differ from the case when the issuer of the bonds is assumed to be risk free?
- How do I know what maturities are appropriate and how many line segments are appropriate in the smoothing process?
We address the first set of questions in Part 13 of this blog series, smoothing credit spreads and credit spread forwards. We address the second question later in this series.
Donald R. van Deventer
Kamakura Corporation
Honolulu, January 20, 2010