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Don founded Kamakura Corporation in April 1990 and currently serves as its chairman and chief executive officer where he focuses on enterprise wide risk management and modern credit risk technology. His primary financial consulting and research interests involve the practical application of leading edge financial theory to solve critical financial risk management problems. Don was elected to the 50 member RISK Magazine Hall of Fame in 2002 for his work at Kamakura. Read More

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Kamakura Blog

Apr 7

Written by: Donald van Deventer
4/7/2010 1:47 AM 

In part 10 of this series on yield curve smoothing, we showed why the maximum smoothness forward rate approach is superior to the wide range of alternative smoothing methods we have introduced in this series, including the deeply flawed Nelson-Siegel approach.  In part 12 of this series, we showed that using bond prices as inputs to the smoothing process is a modest extension to the smoothing we did in part 10. In this blog, we show that it is incorrect, sometimes disastrously so, to apply smoothing to a yield curve for an issuer with credit risk, ignoring the risk free yield curve.  We show alternative methods for smoothing the zero coupon credit spread curve, relative to the risk free rate, to achieve far superior results.  Smoothing a curve for an issuer with credit risk without reference to the risk free curve is probably the biggest mistake one can make in yield curve smoothing and we show why below.

Setting the Scene: Smoothing Results for the Risk Free Curve

We start first by assuming that there is a risk-free bond issuer in the currency of interest.  Prior to the 2007-2010 credit crisis, this assumption might have been taken for granted, but given current concerns about sovereign risk we make the assumption here that the government yield curve is truly free of credit risk. We use exactly the approach in part 12 of this series, with slightly different inputs for expositional purposes.

We assume there are three semi-annual coupon paying bonds issued by the government with these observable net present values (i.e. price plus accrued interest):

We assume that a very short term rate is observable at 4.00% and a three month rate is observable that is consistent with a continuously compounded yield of 4.75%:

We are smoothing the risk free yield curve for two reasons: first, so we can derive pricing for other risk-free transactions than those with observable prices, and, second, so we can derive a high quality credit spread for ABC Company. The ABC yield curve will allow us to make a new financing proposal to ABC Company that is consistent with observable credit risky bonds of ABC Company.

Because of our second objective, we need a high quality zero coupon yield curve for the government out to 10 years.  We choose additional knot points, where the yield curve segments join, at 1, 3, 5 and 10 years.  This gives us a total of 5 segments: from 0 to 0.25 years, from 0.25 to 1 year, from 1 to 3, from 3 to 5 and from 5 to 10.  We use the maximum smoothness forward rate approach as in part 12 of this blog series.  We know the zero coupon yields at the 0 and 0.25 points because they are observable-the securities trading at those maturities have only one payment date so there is no mystery about the correct yields.  That is not true at the 1, 3, 5, and 10 year points because the only other securities trading are coupon-bearing bonds.  We guess the yields at 1, 3, 5 and 10 years and then iterate repeatedly until we can solve for the zero coupon yields that will result in a maximum smoothness forward rate curve that causes the coupon bearing bonds in the government market to have a theoretical value exactly equal to the observable market net present value (price plus accrued interest, not price alone) by minimizing the sum of squared pricing errors.

These are the zero coupon yields that produce that result:

We confirm that the theoretical bond prices and actual bond prices match exactly:

We can then use the coefficients of the maximum smoothness forward rate curve to produce this graph of zero coupon bond yields and forward rates for the risk-free government bond curve, exactly as we did in parts 10 and 12 of this blog series:

We now use this information to derive the best possible credit spreads for ABC Company.

A Naive Approach: Smoothing ABC Yields by Ignoring the Risk Free Curve

We start by first taking a naive approach that is extremely common in the yield curve literature: we smooth the yield curve of ABC Company on the assumption that we don’t need to know what the risk free curve is.  This mistake is made daily with respect to the U.S. dollar interest rate swap curve by nearly every major financial institution in the world.  They assume that the observable data points are so numerous that the results from this naive approach will be good, and then they blame the yield curve smoothing method when the results are obviously garbage. In truth, this is not one of Nassim Taleb’s black swans—it’s an error in finance made by the analyst, and the mistake is due to incorrect assumptions by the analyst rather than a flaw in the smoothing technique.

We show why in this example.  We assume that these zero coupon bonds are outstanding issues of ABC Company with observable net present values:

The risky zero yield is derived using continuous compounding from the observable zero coupon bond price.  We know what the risk-free zero coupon yield is from the smoothing of the government curve in the prior section.  The observable credit spread is simply the difference between the observable ABC Company zero coupon yield and the risk free yield that we derived above on exactly the same maturity date, to the exact day.  We now want to smooth the risky yield curve, naively assuming that the risk-free curve is irrelevant because we have six observable bonds and we think that’s enough to avoid problems.

We make the additional assumption that the credit spread for a very short maturity is also 45 basis points, so the assumed “zero maturity” yield for ABC Company is the risk free zero yield of 4.00% plus 0.45%, a total of 4.45%.  We apply the maximum smoothness forward rate technique to these zero coupon bonds, exactly as in part 10 of this series.  If instead ABC bonds were coupon bearing, we would have used the approach in part 12 of the series and that in the prior section.

We compare the zero coupon yield curve that we derive for ABC Company with the zero coupon yield curve for the government in this graph:

We can immediately see that we have a problem. ABC Company’s fitted yield curve, which ignored the government curve, causes negative credit spreads on the short end of the yield curve, because the ABC yield curve is less than the risk-free zero coupon yield curve.

When we compare the forward rates from the two curves, we can see this problem more clearly:

Taking one more view, we plot the zero coupon credit spread (which is the difference between the ABC Company zero yields and government yields) and the credit spread forwards (the difference between the ABC Company forward rates and the government forward rates) in this graph:

The blue line is the zero coupon credit spread.  It is volatile and goes negative in the maturity interval between 0.25 years and 1.25 years.  The problem with the credit spread forwards makes these issues even more obvious.  The forward credit spreads go negative by approximately 50 basis points.  This means that any transactions for ABC Company in this interval will be grossly mispriced.

What does the analyst do now?  Most analysts are like the football coach whose team loses because the coach had a bad game plan-the coach blames the quarterback.  In this case, most analysts would blame the smoothing method, but in fact the problem is caused by the analyst’s decision to ignore the risk free curve.  We show how to fix that in the next two sections.

Fitting Credit Spreads with Cubic Splines

A natural alternative to smoothing the yields of ABC Company is to smooth the zero coupon credit spreads.  One should never commit the unimaginable sin of smoothing the differences between the yield to maturity on an ABC bond with N years to maturity and the simple yield to maturity on a government bond that also has something close to N years to maturity.  The mistakes embedded in this simple calculation take up most of Chapter 18 in Advanced Financial Risk Management (van Deventer, Imai, and Mesler, John Wiley and Sons, 2004).  The fact that this mistake is made often doesn’t make the magnitude of the error any smaller.

We make two choices in this section.  First, we again assume that the credit spread at a zero maturity for ABC Company is the same 45 basis point spread that we see on the shortest maturity bond. Second, we choose to use a cubic spline to solve for the ABC credit spread curve and we require that the observable credit spreads and actual credit spreads be equal:

Each credit spread curve segment will have a cubic polynomial that looks like this:
c(t) = a+b1t+b2t2+b3t3
where t is the years to maturity of that instantaneous credit spread. In part 8 of this series, published on on December 10, 2009, we show how to fit cubic splines to any data. In that blog issue, it was zero coupon yields.  In this section, we’re fitting to zero coupon credit spreads.

We arbitrarily choose to use knot points of zero, 0.25 years, 1, 3, 5 and 10 years.  These are the same knot points we used for risk-free curve smoothing.  We could have just as easily used the observable yields and maturities in the chart above.  In that case, all that credit spread smoothing does is to connect the observable credit spreads: 45 basis points at a zero maturity, 45 basis points at 0.1 years, 49 basis points at 2.75 years, and so on.

We use the risk-free curve knot points and iterate until we have an (almost perfect) match to observable credit spreads:

The match can be made perfect by changing parameters in the optimization routine. The zero coupon credit spreads that produce this result are as follows:

The zero maturity is not shaded because it is not part of the iteration.  We have assumed that the zero maturity credit spread is the same 45 basis points as the observable data for 0.1 years to maturity.

Once we have the spline parameters as in blog part 8, we can plot zero coupon credit spreads and zero coupon forward credit spreads:

We see that the blue line, the derived credit spread, no longer goes negative, and the forward credit spreads are much less volatile than they were under the naive approach.  Note that we assumed that the first derivative of the credit spread curve was zero at the right hand side of the graph.  It is that assumption that causes the forward credit spreads to rise and then dive down on the right hand side of the curve.

One would think we can do better, and that’s correct.  We now turn to maximum smoothness forward credit spreads.

Maximum Smoothness Forward Credit Spreads

In this section, we apply the maximum smoothness forward rate approach of part 10 in our blog series to forward credit spreads, not forward rates.  The forward credit spreads are simply the difference between the ABC Company forward rates and the government yield curve forward rates. When we recognize that relationship, the approach in part 10 of this blog series can be used with extremely minor modifications.  We iterate on zero coupon credit spreads instead of zero coupon yields.

When we do that iteration, we get an almost perfect fit.  A perfect fit just requires tweaking the tolerance in the iteration routine:

The zero coupon credit spreads which produce this happy result are given here:

Again, the zero maturity spread of 45 basis points is not part of the iteration as above.

We can plot the zero coupon credit spreads and forward credit spreads that result:

The resulting credit spread curve is very well-behaved and the forward credit spread is smooth with only modest variation.  This confirms what we asserted in the naive example: there is no problem with maximum smoothness forward rate smoothing.  The early negative credit spreads resulted from the analyst’s error in ignoring risk free rates.

Comparing Results

We can compare our three examples by looking first at the credit spreads from risky yield smoothing, cubic credit spread smoothing, and maximum smoothness forward credit spread smoothing:

Most analysts would assert that the maximum smoothness forward credit approach is superior because the resulting zero coupon credit spread curve is smoother.  The same is true, by definition, for forward credit spreads:

These results make it obvious that standard yield curve smoothing techniques can be applied to the zero coupon credit spread curve.  The performance of alternative techniques is essentially identical to the performance of the same techniques applied to zero coupon yields themselves.

Practical Use of Credit Spread Smoothing

The automated application of advanced credit spread smoothing techniques is standard practice for most of the 200 clients that Kamakura Corporation serves in 32 countries.  Credit spread smoothing has been a standard feature of Kamakura Risk Manager, Kamakura’s enterprise wide risk management package, since 1995.

As today’s blog has shown, smoothing the swap curve without reference to the risk free curve can produce nonsensical results.  Alas, the nonsense that results is the fault of the analyst.  If one smoothes zero coupon credit spreads directly, the results will be far superior in almost every interest environment. At the very worst, say a flat yield curve environment, the results will be the same.

As always comments are welcome at  Daily updates of Kamakura’s yield curve smoothing technology are available at

Donald R. van Deventer
Kamakura Corporation
Honolulu, April 7, 2010