On September 13, 2011 Kamakura Corporation released the study “Inside the Kamakura Book of Yields: A Pictorial History of 50 Years of Daily U.S. Treasury Forward Rates,” which was followed by the release on September 26, 2011 of “Inside the Kamakura Book of Yields, Volume II: A Pictorial History of 50 Years of Daily U.S. Treasury Zero Coupon Bond Yields.” This blog entry quantifies some of the key conclusions of this 50 year study of zero coupon bond yields, which includes every business day of U.S. Treasury yields from January 2, 1962 through August 22, 2011.

Dickler, Jarrow and van Deventer’s “Inside the Kamakura Book of Yields” series is based on U.S. Treasury yields provided daily by the Board of Governors of the Federal Reserve System. Monthly forward rates^{1} are extracted from these yields using the maximum smoothness forward rate smoothing approach developed by Adams and van Deventer (*Journal of Fixed Income*, 1994) and corrected in van Deventer and Imai, *Financial Risk Analytics* (1996). Calculations were done using Kamakura Risk Manager version 7.3. The same data produces monthly forward rates, monthly zero coupon bond yields, and a semi-annual par coupon bond yield curve on a fully consistent basis.

Dickler, Jarrow and van Deventer summarized their conclusions from looking at 12,395 days of U.S. Treasury forward rate curves as follows:

- Forward rate curves display a richness of shapes that are not accurately captured by the academic literature on this topic.
- The evolution of the term structure of interest rates is driven by a larger number of risk factors than the 1, 2 or 3 factor models commonly employed in the academic literature.
- The maximum smoothness forward rate smoothing procedure is robust, generating smooth and stable forward rates even during periods of severe disruption in the financial markets, like the credit crisis of 2007-2008, the Russian debt crisis in 1998, and in the 1980-1982 period when interest rates were at historic highs.
- Large movements in forward rates appear to be almost exclusively triggered by one of two events: a change in the maturities reported by the Federal Reserve, which adds or subtracts key information used in the smoothing process, or specific financial market events like the September 14, 2008 bankruptcy announcement by Lehman Brothers.
- In response to a “flight to quality” in the U.S. Treasury market, the 30 year bond yield declines more than yields at intermediate maturities, causing a hump in the forward rate curve that can persist for extensive lengths of time. This phenomenon is not readily captured by existing academic models of the term structure.

In the remainder of this blog, we ask whether the same conclusions apply equally well to the 12,395 days of monthly U.S. Treasury zero coupon bond yields.

The Dickler, Jarrow and van Deventer paper “Inside the Kamakura Book of Yields: A Pictorial History of 50 Years of U.S. Treasury Forward Rates” is available at this link:

http://www.kamakuraco.com/Portals/0/PDF/KamakuraYieldBook-ForwardRates-FinalVersion-20110913.pdf

The Dickler, Jarrow and van Deventer paper “Inside the Kamakura Book of Yields, Volume II: A Pictorial History of 50 Years of U.S. Treasury Zero Coupon Bond Yields” is available at this link:

http://www.kamakuraco.com/Portals/0/PDF/KamakuraYieldBook-ZeroCouponBondYields-Final-20110926.pdf

The blog entry by Dickler and van Deventer “Inside the Kamakura Book of Yields: An Analysis of 50 Years of Daily U.S. Treasury Forward Rates” is available at this link:

Dickler, Jarrow and van Deventer noted that the maturities at which the Federal Reserve provided yield data in its H15 statistical release changed fairly frequently over the 1962-2011 period:

The first issue considered by Dickler, Jarrow and van Deventer was the variability of shapes of the forward rate curve actually observed over this period of history. In this blog, we take an approach consistent with Heath, Jarrow and Morton: we take the forward rate curves and related zero coupon bond yield curves as facts and ask what type of yield curve modeling would produce curves and curve movements consistent with that history.

**Zero Coupon Bond Yield Curve Shapes**

Over 12,395 days of data, what zero coupon bond yield curve shapes have prevailed and how often have they prevailed? By number of occurrences, this chart summarizes the number of times the zero coupon bond yield curve has been monotonically increasing or decreasing or humped:

Surprisingly, the zero coupon bond yield curve has never been monotonically downward sloping during the 50 year period studied, compared with 66 days in which the monthly forward rate curve was monotonically downward sloping. The zero coupon bond yield curve was monotonically upward sloping on 3,601 of the 12,395 business days studied.

By probability of historical occurrence, we can summarize the data this way:

The zero coupon bond yield curve has been humped on 70.95% of business days, compared to a figure of 93.32% for the monthly forward rate curve.

**Complexity of Zero Coupon Bond Yield Curve Shapes**

A number of related questions arise. How complex are the shapes of the zero coupon bond yield curve? When they show a humped shape, is there just one hump or many? For the most part, commonly used academic term structure models implicitly assume that there are zero or very few humps in the zero coupon bond yield curve. The analysis of 12,395 days of data show that there have been as many as 9 optima (local maximums or minimums) in the forward rate curve and that this is in part a result of the number of input maturities to the smoothing process. The chart below describes the results for the zero coupon bond yield curve:

While there were fewer optima (local minima and maxima) for the zero coupon bond yield curve than for the associated forward curve, a yield curve model would have to be capable of producing a zero coupon yield curve with as many as 4 optima to capture 90 percent of the shapes that have occurred between January 2, 1962 and August 22, 2011. To accurately model 100% of the actual yield curve shapes, a functional form that can produce up to 8 optima would be necessary. The year 2010 was a year in which the Federal Reserve’s near zero interest rate policy created a forward rate curve with very few optima. The related zero coupon bond yield curve for 2010 is shown here:

By contrast, 2009 was a year in which complex forward rate curves frequently prevailed due to the late 2008 collapse of Lehman Brothers, AIG, Wachovia, Washington Mutual and many others. The shapes of the associated zero coupon bond yield curves are somewhat less complex. The zero yields for 2009 are shown in this graph:

The dramatic decline of long term forward rates in 2009, as explained by Dickler, Jarrow and van Deventer, comes about because a “flight to quality” tends to drive down 30 year U.S. Treasury yields much more than intermediate yields, forcing forward rates in the long maturities to drop by more. This is what happened in the fourth quarter of 2008, persisting through most of 2009. The impact of these forward rate moves are reflected in a more subtle way in the associated zero coupon bond yield curves.

**Most Frequent Maturities at Which Optima Occur**

We then pose a related question: At what maturities (in months) are optima most likely to occur in the zero coupon bond yield curve? The results are summarized in this graph:

As one might imagine, the optima are most likely to occur near key “knot points” in the data provided by the Federal Reserve. When one looks at the 3 month moving average of the probability of an optima, the peak probabilities come at the following months:

The most frequent optima are at months 3, 7, 15, 30, 44, 69, 74, 90, 98, 130, 143, 180, 187, and 244. These optima are close to knot points (data at which the Federal Reserve provides data) at months 3, 6, 12, 48, 60, 84, 120, and 240.

**Volatility of Zero Coupon Bond Yields**

The popular single factor term structure model of Vasicek (see van Deventer, Imai and Mesler, 2004) implies that zero coupon bond yields and forward rates will be more volatile at the shortest maturities, with volatility declining monotonically as the maturity lengthens. A simple graph of the high, low and average monthly forward rates by month of maturity from 1962 to 2011 shows, however, that zero coupon bond yields at the long end of the curve may in fact cover a wider range than zero coupon bond yields at intermediate maturities:

The simple standard deviation of zero coupon bond yields over the 1962-2011 period, however, generally declines as maturity lengthens, a much different volatility profile than the one for monthly forward rates.

The complexity of forward rate movements and their impact on zero coupon bond yields, however, is much more complex than the simple standard deviation of zero coupon bond yields reveals. The graph below shows the standard deviation of daily changes in zero coupon bond yields, both including and excluding the first day after a change in the Federal Reserve’s data regime. In the case of zero yields, the difference between standard deviations “with and without” the data after a regime change is not visible given the scale of the graph:

The standard deviations of daily changes are largest at the short end of the curve, the long end of the curve, and about the 15 year point where there is a 10 year gap between the 10 year and 20 year yields provided by the Federal Reserve.

**Smoothness of the Calculated Zero Coupon Bond Yield Curves**

Adams and van Deventer (1994) adopt a common definition of smoothness used in engineering and computer graphics, the integral of the squared second derivative of the forward rate over the length of the forward rate curve. We can approximate this by taking the sum of the squared second differences of the (up to) 360 monthly zero coupon bond yields on each of the 12,395 business days in the data set.

If we calculate Z[t] for t=1,12395 over our entire data set, we get a distribution of smoothness indices that show an extremely high degree of smoothness, even compared with the associated forward rate curves, which were very smooth:

Substantially all of our 12,395 observations show a smoothness index (using zero coupon bond yields expressed in percent) of less than 0.03. If we cap the number of observations, we see that none of the 12,395 observations has a smoothness statistic over 0.13:

The “least smooth” observations are those with the highest smoothing index Z. We list those dates here:

The least smooth zero coupon bond yield curve, not surprisingly, comes 2 business days after the September 15, 2008 bankruptcy filing by Lehman Brothers and 3 calendar days after the announcement on Sunday September 14, 2008 of Lehman’s intent to file for bankruptcy. This is the same finding as in the forward rate curve case, but the smoothing index is only 1/8th as large in the zero coupon bond yield case. We graph all 360 months of forward rates in this graph:

Given the disruptions occurring in financial markets at the time, the zero coupon bond yield curve is extraordinarily smooth, even though by quantitative standards it has the highest (i.e. least smooth) smoothing index during the period from January 2, 1962 to August 22, 2011. The lack of smoothness is primarily due to volatility of the short maturity U.S. Treasury rates, which is inevitable because of inputs from the Federal Reserve H15 statistical release on that day: 1 month=0.07%, 3 months=0.03%, 6 months=1.03%, and 1 year=1.50%. The implied rate for a 2 month maturity 1 month forward is 0.01%.

The next graph plots the daily movements of the zero coupon bond yield curves for the period September 1, 2008 through December 31, 2008. It shows clearly the flight to quality, via huge investments in the 30 year Treasury bond, causing long term zero coupon bond yields to drop relative to intermediate maturities. This in turn drives down long term forward rates and causes the “crisis hump” often observed by Dicker, Jarrow and van Deventer in “Inside the Kamakura Book of Yields: A Pictorial History of 50 Years of U.S. Treasury Forward Rates.”

Next, we turn to September 1982, a period of extremely high and volatile interest rates in the United States. In terms of the smoothness index, 14 of the 25 “least smooth” zero coupon bond yield curves occurred between August 1, 1982 and September 30, 1982. We graph those zero coupon bond yield curves here:

While the zero coupon bond yield curves during this period are “less smooth” than other periods in history, the plotted curves are rational and reasonable. The volatility of zero coupon bond yields on the long end of the curve are simply a reflection of the high volatility of Federal Reserve quotations on 30 year coupon bearing U.S. Treasury yields during this period, relative to intermediate term yields. 30 year yields, as quoted by the Federal Reserve, varied from 11.75% to 13.41% during this two month period.

**Correlation among Forward Rates **

The correlation among the 360 zero coupon bond yields can be calculated for those days between January 2, 1962 and August 22, 2011 for which the 30 year yield data exists, a total of 7,631 business days. The graph below shows that the correlation between the first month’s zero coupon bond yield and the 360 month zero coupon bond yield is 81%, which is very nearly the average correlation for the 360 forward rate maturities reported on in a previous blog. The month 360 forward rate correlation with the 1 month forward (spot) rate was 65%:

**Next Steps**

Kamakura’s analysis of this data set is continuing. Upcoming blogs will focus on par coupon yield curves and other topics of interest. Suggestions are welcome at info@kamakuraco.com.

**References**

Adams, Kenneth J. and Donald R. van Deventer. “Fitting Yield Curves and Forward Rate Curves with Maximum Smoothness.” *Journal of Fixed Income*, June 1994.

Dai, Qiang and Kenneth J. Singleton, “Specification Analysis of Affine Term Structure Models,” *The Journal of Finance*, Volume LV, Number 5, October 2000.

Dai, Qiang and Kenneth J. Singleton, “Term Structure Dynamics in Theory and Reality,” *The Review of Financial Studies*, Volume 16, Number 3, Fall 2003.

Dickler, Daniel T., Robert A. Jarrow and Donald R. van Deventer, “Inside the Kamakura Book of Yields: A Pictorial History of 50 Years of U.S. Treasury Forward Rates,” Kamakura Corporation memorandum, September 13, 2011.

Dickler, Daniel T., Robert A. Jarrow and Donald R. van Deventer, “Inside the Kamakura Book of Yields, Volume II: A Pictorial History of 50 Years of U.S. Treasury Zero Coupon Bond Yields,” Kamakura Corporation memorandum, September 26, 2011.

Dickler, Daniel T., Robert A. Jarrow and Donald R. van Deventer, “Inside the Kamakura Book of Yields, Volume III: A Pictorial History of 50 Years of U.S. Treasury Par Coupon Bond Yields, Kamakura Corporation memorandum, October 5, 2011.

Duffie, Darrell and Rui Kan, “Multi-factor Term Structure Models,” *Philosophical Transactions: Physical Sciences and Engineering*, Volume 347, Number 1684, Mathematical Models in Finance, June 1994.

Duffie, Darrell and Kenneth J. Singleton, “An Econometric Model of the Term Structure of Interest-Rate Swap Yields,” *The Journal of Finance*, Volume LII, Number 4, September 1997.

Duffie, Darrell and Kenneth J. Singleton, “Modeling Term Structures of Defaultable Bonds,” *The Review of Financial Studies*, Volume 12, Number 4, 1999.

Heath, David, Robert A. Jarrow and Andrew Morton, “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation,” *Econometrica*, 60(1), January 1992.

van Deventer, Donald R. and Kenji Imai, *Financial Risk Analytics: A Term Structure Model Approach for Banking, Insurance, and Investment Management*, Irwin Professional Publishing, Chicago, 1997.

van Deventer, Donald R., Kenji Imai, and Mark Mesler, *Advanced Financial Risk Management*, John Wiley & Sons, 2004. Translated into modern Chinese and published by China Renmin University Press, Beijing, 2007.

Donald R. van Deventer and Daniel T. Dickler

Kamakura Corporation

Honolulu, Hawaii

September 26, 2011

© Copyright 2011 by Donald R. van Deventer, All Rights Reserved.

^{1} For computational convenience, it was assumed that each month is the same length, 1/12 of one year, rather than using exact day count of 28, 29, 30 or 31 days for the length of the month.