Dickler, Jarrow and van Deventer’s “Inside the Kamakura Book of Yields” is based on U.S. Treasury yields provided daily by the Board of Governors of the Federal Reserve System. Monthly forward rates1 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.
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 document the reasons for these conclusions and supplement them with other insights that stem from a quantitative analysis of 12,395 days of monthly U.S. Treasury forward rate data.
Dickler, Jarrow and van Deventer begin by noting 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. Typically, academic models of the term structure begin with an assumption about the stochastic processes (normally for one, two or three risk factors) and conclude by determining the shapes of the yield curves that are consistent with these assumptions and with no-arbitrage conditions. Recent examples of this approach include Duffie and Kan (1994), Duffie and Singleton (1997, 1999), and Dai and Singleton (2000, 2003). An alternative approach is that of Heath, Jarrow and Morton (1992), where the initial forward rate curve is taken as given and restrictions are derived on the stochastic processes driving the forward rate curve so that no arbitrage is possible. In this blog, we take an approach more consistent with Heath, Jarrow and Morton: we take the forward rate curves as facts and ask what type of yield curve modeling would produce curves and curve movements consistent with that history.
Forward Rate Curve Shapes
We first ask this question: from 1962 to 2011, over 12,395 days of data, what forward rate curve shapes have prevailed and how often have they prevailed? By number of occurrences, this chart summarizes the number of times the forward curve has been monotonically increasing or decreasing or humped:
Forward rate curves have only been monotonically downward sloping on 66 out of 12,395 business days since January 2, 1962. The forward curve has been upward sloping on 762 business days. No monotonically downward sloping curves have been observed since the 1977-1981 data regime. By probability of historical occurrence, we can summarize the data this way:
Monotonically downward sloping yield curves have occurred only 0.53% of business days since January 2, 1962. By contrast, the forward curve has been humped on 93.32% of the business days.
Complexity of Forward Yield Curve Shapes
A number of related questions arise. How complex are the shapes of the forward rate curves? 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 forward rate 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. A lack of data hides variations in the forward curve, as the follow charts reveal:
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:
By contrast, 2009 was a year in which complex forward rate curves frequently prevailed due to the collapse of Lehman Brothers, AIG, Wachovia, Washington Mutual and many others:
The dramatic decline of forward rates, 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.
Most Frequent Maturities at Which Optima Occur
We then pose a related question: At what maturities (in months) are optima most likely to occur? 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 5, 10, 22, 34, 58, 79, 113, 144, 202. These optima are close to knot points (data at which the Federal Reserve provides data) at 6, 12, 24, 36, 60, 84, 120, and 240 months.
Volatility of Forward Rates
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 it is the intermediate maturity forward rates which have moved over the narrowest range:
This is confirmed by the simple standard deviation of forward rates over the 1962-2011 period, which the standard deviation at intermediate maturities is considerably less than the shortest and longest maturities. This contradicts findings of some of the studies mentioned above, which relied on a shorter history of interest rates.
The complexity of forward rate movements is confirmed by looking at the standard deviation of daily changes in forward rates, both including and excluding the first day after a change in the Federal Reserve’s data regime:
The standard deviations of daily changes themselves show many humps that are associated with the maturities at which data is provided by the Federal Reserve. The standard deviation of daily changes including the first day after a regime change is larger because the addition or subtraction of yield information can cause big changes in derived forward rates, as we see here in the graph for 1977, the first year in which 30 year yields became available on the Federal Reserve’s H15 statistical index:
Smoothness of the Calculated Forward Rate 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 forward rates 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:
Substantially all of our 12,395 observations show a smoothness index (using forward rates expressed in percent) of less than 0.1. If we cap the number of observations, we see that very few of the 12, 395 observations have smoothness statistics over 0.5:
The “least smooth” observations are those with the highest smoothing index Z. We list those dates here:
The least smooth forward rate 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. We graph all 360 months of forward rates in this graph:
Given the disruptions occurring in financial markets at the time, the forward rate 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 forward rates, which is inevitable because of inputs from the Federal Reserve H15 statistical release on that day: 3 months=0.03%, 6 months=1.03%, and 1 year=1.50%.
The next graph plots the daily movements of the forward rate 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 its yield 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.
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 16 “least smooth” forward rate curves occurred between August 1, 1982 and September 30, 1982. We graph those forward curves here:
While the forward rate curves during this period are “less smooth” than other periods in history, the plotted curves are rational and reasonable. The volatility of forward rates 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 forward rates can be calculated for those days between January 2, 1962 and August 22, 2011 for which data exists. The graph below shows that the correlation between the first month’s forward (spot) rate and the 360 month forward is 65%:
Kamakura’s analysis of this data set is continuing. Upcoming blogs will focus on zero coupon bond yields and par coupon yield curves and other topics of interest. Suggestions are welcome at firstname.lastname@example.org.
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.
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
September 14, 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.