Parameters for the following legacy one factor term structure models are also available as “challenger models” to subscribers for the Kamakura Risk Information Systems Term Structure Model Parameter Service:
Ho and Lee
Theoretical Vasicek
Extended Vasicek
Black, Derman and Toy
Black and Karasinski
Regime Change
A paper forthcoming from Kamakura Corporation as a companion to this note proves that these one factor models cannot compete with a best practice Heath, Jarrow and Morton multi-factor model if proper model validation procedures are followed.
Background on Key Aspects of the U.S. Treasury Yield History
The U.S. Department of the Treasury yield history is one of the longest government securities yield histories in the world. This long history is exceptionally important in benchmarking interest rate volatility functions that can both replicate historical movements in yields and forecast future movements in yields successfully. The U.S. Treasury yield history has a number of important attributes:
- The time series is an impressive 54 years in length.
- There is extensive experience in the history with high rates, that is rates of 10% and higher.
- There is extensive experience in the history with low rates during the recent credit crisis.
- Unlike central banks in Japan, Germany, Sweden, Spain, and Hong Kong, the U.S. Treasury overrides observed negative yields with zero values. As of the date of this note, the U.S. Treasury’s policy on negative rates is as follows:
- “Current financial market conditions, in conjunction with extraordinary low levels of interest rates, have resulted in negative yields for some Treasury securities trading in the secondary market. Negative yields for Treasury securities most often reflect highly technical factors in Treasury markets related to the cash and repurchase agreement markets, and are at times unrelated to the time value of money.
- “As such, Treasury will restrict the use of negative input yields for securities used in deriving interest rates for the Treasury nominal Constant Maturity Treasury series (CMTs). Any CMT input points with negative yields will be reset to zero percent prior to use as inputs in the CMT derivation. This decision is consistent with Treasury not accepting negative yields in Treasury nominal security auctions.
- “In addition, given that CMTs are used in many statutorily and regulatory determined loan and credit programs as well as for setting interest rates on non-marketable government securities, establishing a floor of zero more accurately reflects borrowing costs related to various programs.”
- While the impact of this override of negative yields can’t be measured precisely, Kamakura Corporation carefully compares parameters with other government markets where negative rate experience is extensive.
- The U.S. Treasury, on the positive side, does not restrict the shape of its yield curve data as happens with the yield curve smoothing methods of Svensson and Nelson-Siegel. The Treasury uses a form of cubic spline smoothing that is described on its website.
- In a perfect data environment, Kamakura Corporation would prefer to fit its term structure models using raw traded prices for government bonds. Such a price series is not made available from the U.S. Treasury itself, and, according to a recent conversation with the U.S. Treasury’s Office of Financial Research, such a historical U.S. Treasury traded bond price data base is not commercially available.
With this as background, we present the updated Heath, Jarrow and Morton term structure models.
No Arbitrage and Fitting the Current Yield Curve Perfectly
Market participants occasionally express a preference for one term structure model over others “because the model fits observable yields perfectly.” This comment, however, applies to all choices of term structure models if the no arbitrage constraints of Heath, Jarrow and Morton are followed correctly. “Fitting observable yields perfectly” is the most basic model validation condition imaginable, and all of the term structure models listed above conform to this requirement in accordance with best practice Heath, Jarrow and Morton procedures.
The Analysis of U.S. Treasury Yields
A multi-factor term structure model is the foundation for best practice asset and liability management, market risk, economic capital, interest rate risk in the banking book, stress-testing and the internal capital adequacy assessment process. Our objective in this note is to show the derivation of a multi-factor Heath, Jarrow, and Morton model of the U.S. Treasury yield curve. As a by-product, we are also able to apply standard tests of model validation to commonly used one factor term structure models in the U.S. Treasury market. Consistent with our prior studies of government securities markets in the United States, Canada, Japan, the United Kingdom, Australia, Sweden, Germany, and Singapore, we conclude that a rich multi-factor model is essential for accuracy and that common one factor models fail even the most basic model validation tests. For the detailed forthcoming paper in this regard, please contact us at info@kamakuraco.com.
Background for the Analysis
Kamakura Corporation provides regular updates of multi-factor term structure models in major bond markets around the world. The data is provided by Kamakura Corporation’s Kamakura Risk Information Services group, and the resulting parameters and documentation are available by subscription. These government securities markets have been reviewed in prior notes and are available at these links:
Australia Commonwealth Government Securities
Canada Government of Canada Securities
Germany German Bunds
Japan Japanese Government Bonds
Singapore Singapore Government Securities
Spain Spanish Government Bonds
Sweden Swedish Government Securities
United Kingdom United Kingdom Government Bonds
United States U.S. Treasury Bonds, previous update
In all nine of these studies, one factor models failed basic model validation tests and were judged unacceptable from an accuracy point of view. We now re-examine the same issues for government securities in the U.S. Treasury market.
We follow the same model validation process again in this note and show the reasons for these strong conclusions, using the experience in the U.S. Treasury market since 1962 through year-end, 2015. Readers who would like to see the difference between a best-practice Heath, Jarrow and Morton model and a common practice one factor model in a U.S. context are referred to this June 24 simulation analysis for the U.S. Treasury curve .
Defining “How Good is Good Enough?” for Interest Rate Risk Modeling
In our previous March 5, 2014 note “ Stress Testing and Interest Rate Risk Models: How Many Risk Factors are Necessary? ” we showed that nine interest rate risk factors were necessary for a best practice model of the U.S. Treasury curve. In a companion piece on March 18, 2014 titled “ Stress Testing and Interest Rate Risk Models: A Multi-Factor Stress Testing Example ,” we outlined the process for determining risk factors and the parameters used in a multi-factor interest rate model, again using U.S. Treasury data.
In the studies done so far, the number of statistically significant factors are summarized below:
Australia: Commonwealth Government Securities, 14 factors
Canada: Government of Canada Securities, 12 factors
Germany: Bunds, 14 factors
Japan: Japanese Government Bonds, 16 factors
Singapore: Singapore Government Securities, 9 factors
Spain: Spanish Government Securities, 11 factors
Sweden: Swedish Government Securities, 11 factors
United Kingdom: Government Securities, 14 factors
United States: Treasury Securities, prior version 9 factors
We now again address three critical questions relevant to modeling the U.S. Treasury yield curve:
- How do you measure the accuracy of an interest rate risk simulation technique?
- Given that measure of accuracy, how many risk factors are necessary?
- How does accuracy change as the number of factors increases?
In answering the question “how good is good enough” for interest rate risk modeling, we again follow the procedures that Bharath and Shumway (2008) used in testing the accuracy of the Merton model of risky debt versus the reduced form approach to credit risk modeling. We test these two hypotheses about one factor term structure models:
Strong form of hypothesis: One factor term structure models are so accurate that there are no other variables than the first factor that have statistically significant explanatory power.
Weaker form of hypothesis: There are other factors beyond the first factor that are statistically significant, but their impact is very modest and the benefits of using more than one factor are very minor.
Non-Parametric Tests of One Factor Term Structure Models
Jarrow, van Deventer and Wang (2003) (“JvDW”) provide another testing procedure that we address first. In examining the Merton model of risky debt, JvDW provide a very intuitive testing procedure that is independent of the parameters fitted to the model structure. They asked this question: “Are the implications of the model true or false?” Since no model is perfect, they answer this question with a probability.
We again address two classes of one factor term structure models in this section using data from the U.S. Treasury bond market:
One factor models with rate-dependent interest rate volatility;
Cox, Ingersoll and Ross (1985)
Black, Derman and Toy (1990)
Black and Karasinski (1991)
One factor models with constant interest rate volatility (affine models)
Vasicek (1977)
Ho and Lee (1986)
Extended Vasicek or Hull and White Model (1990, 1993)
Non-parametric test 1: The one factor models with rate-dependent interest rate volatility make it impossible for interest rates to be negative. Is this implication true or false? It is false, as Deutsche Bundesbank yield histories and Swedish Government Bond histories show frequent negative yields in in recent months. Statistics from Banco de España include 3,877 daily observations on Treasury bills with maturities between 34 and 94 days. As of September 3, 2015, yields had been negative on 62 days. Negative yields were also reported at Treasury bill maturities of 6 and 12 months. According to the Japan Ministry of Finance, there have been negative rates in Japanese government bill auctions at 2 months (once), 3 months (10 times), and 6 months (6 times) between April 7, 1999 and July 9, 2015. The Japan Ministry of Finance also reports on secondary market yields for maturities of 1 year or more on a daily basis. Negative yields have been reported for maturities of 1 year (49 days), 2 years (60 days), 3 years (32 days), and 4 years (15 days) through July 13, 2015. For this reason alone, we advise analysts to reject the one factor rate-dependent volatility models as inconsistent with historical facts.
Non-parametric test 2: The Vasicek, Ho and Lee, and Extended Vasicek/Hull and White models assume that interest rate volatility is a constant, independent of the level of interest rates. This assumption implies that both the level and the changes in interest rates are normally distributed over time. We use quarterly data on U.S. Treasury yields provided by the U.S. Department of the Treasury from January 2, 1962 through December 31, 2015. We extract quarterly zero coupon bond yields from this data using Kamakura Risk Manager version 8.1 and maximum smoothness forward rate smoothing . This graph shows the quarterly evolution of U.S. Treasury zero coupon yields over time:
The graph below shows the evolution of the first quarterly forward rate (the forward that applies from the 91^{st} day through the 182^{nd} day) over the same time period:
We use three statistical tests to determine whether or not the hypothesis of normality should be rejected at the 5% level for two sets of data:
- The absolute level of zero coupon bond yields over the 1962 to 2015 time period.
- The quarterly changes in 119 quarterly forward rates making up the 30 year U.S. Treasury yield curve.
The statistical tests we use include the Shapiro-Wilk test, theShapiro-Francia test, and the skew test, all of which are available in common statistical packages.
The chart above shows the p-values for these three statistical tests for the first 24 quarterly maturities. The null hypothesis of normality is rejected by all 3 tests for 87 of the 120 quarterly zero coupon yield maturities. For quarterly changes in forward rates, the null hypothesis of normality is rejected by all 3 tests for 115 of the 119 maturities. This is a strong rejection of the normality assumptions implicit in constant coefficient single factor term structure models like the Ho and Lee, Hull and White, and Vasicek/Extended Vasicek models, along with the hybrid regime change model. In most of the other countries studied, the hypothesis of normality has been rejected as well.
Non-parametric test 3: As commonly implemented, one factor term structure models imply that all yields will either (a) rise, (b) fall, or (c) remain unchanged. In Chapter 3 of Advanced Financial Management (second edition, 2013), van Deventer, Imai and Mesler show that this implication of one factor term structure models is rarely true in the U.S. Treasury market. We perform the same test using 13,487 days of zero coupon bond yields for the U.S. Treasury yield curve. We analyze the daily shifts in the 360 different monthly zero coupon bond yields on each day. The results are given here:
The results were not consistent with the implications of one factor term structure models. Yield curve shifts were all positive, all negative, or all zero 11.20%, 5.85%, and 0.01% of the time, a total of 17.05% of all business days. The predominant yield curve shift was a twist, with a mix of positive changes, negative changes, or zero changes. These figures are similar to those for the Japanese Government Bond, Government of Canada, German Bund and United Kingdom Government Bond yield curves. These twists, which happen 82.95% of the time in the U.S. Treasury market, cannot be modeled at all with one factor term structure models.
Non-parametric test 4 : A closely related test is discussed in Chapter 3 of van Deventer, Imai and Mesler. One factor term structure models cannot create a yield curve that has multiple humps in it. One simply has to count the humps in the U.S. Treasury yield curve to show that this is another serious problem with one factor term structure models:
The number of days with 0 or 1 humps (defined as the sum of local minima and maxima on that day’s yield curve) was 54.42% of the total observations in the data set. The remainder of the data set, 45.58% of the total, has yield curves with shapes that are inconsistent with a one factor term structure model.
Fitting a Multi-Factor Heath Jarrow and Morton Term Structure Model to U.S. Treasury Yields
We now fit a multi-factor Heath, Jarrow and Morton model to quarterly U.S. Treasury zero coupon yield data from January 2, 1962 to December 31, 2015. We use two data regimes. The first is for observations where no 20 year or 30 year yield was reported. The second is for those observations where the 20 year or 30 year yield was available.
The availability of data out to 30 years is fairly typical in government bond markets world-wide. The procedures used to derive the parameters of a Heath, Jarrow and Morton model are described in detail in these documents:
Jarrow, Robert A. and Donald R. van Deventer, “Parameter Estimation for Heath, Jarrow and Morton Term Structure Models,” Technical Guide, Version 2.0, Kamakura Corporation, June 30, 2015.
Jarrow, Robert A. and Donald R. van Deventer, Appendix A, Version 1.0: “U.S. Treasury Yields,” to “Parameter Estimation for Heath, Jarrow and Morton Term Structure Models,” Technical Guide, Kamakura Corporation, January 2016.
Jarrow, Robert A. and Donald R. van Deventer, “Monte Carlo Simulation in a Multi-Factor Heath, Jarrow and Morton Term Structure Model,” Technical Guide, Version 4.0, Kamakura Corporation, June 16, 2015.
We followed these steps to estimate the parameters of the model:
- We extract the zero coupon yields and zero coupon bond prices for all quarterly maturities out to 30 years for all daily observations for which the 30 year zero coupon yield is available. For other observations, we extended the analysis to the longest maturity available, 10 years. This is done using Kamakura Risk Manager, version 8.1, using the maximum smoothness forward rate approach to fill the quarterly maturity gaps in the zero coupon bond data.
- We drop the daily observations that are not the last observation of the quarter, to avoid overlapping quarterly observations and the resulting autocorrelated errors that would stem from that.
- We calculate the continuously compounded changes in forward returns as described in the parameter technical guide.
- We then begin the process of creating the orthogonalized risk factors that drive interest rates. These factors are assumed to be uncorrelated independent random variables that have a normal distribution with mean zero and standard deviation of 1.
- In the estimation process, we added factors to the model as long as each new factor provided incremental explanatory power.
We use the resulting parameters and accuracy tests to address the hypothesis that a one factor model is “good enough” for modeling U.S. Treasury yields.
Proof That One Factor Models Are Not Sufficient for Best Practice Risk Management
We now test the hypotheses about one factor term structure models using U.S. Treasury yield data.
Strong form of hypothesis: One factor term structure models are so accurate that there are no other variables than the first factor that have statistically significant explanatory power.
The following graph shows that a one factor term structure model omits a very large number of statistically significant risk factors driving U.S. Treasury yields:
Other than the first quarterly forward rate in the yield curve, there are as many as 17 explanatory variables that drive the 120 quarterly segments of the yield curve. The final U.S. Treasury term structure model from Kamakura Risk Information services has 9 independent risk factors that drive yields, and these factors also appear in combination with rate level variables. A total of 21 related candidate explanatory variables were used in the estimation process.
Conclusion: The strong form of the hypothesis is overwhelmingly rejected by the data on U.S. Treasury yields.
We now turn to the weaker hypothesis.
Weaker form of hypothesis: There are other factors beyond the first factor that are statistically significant, but their impact is very modest and the benefits of using more than one factor are very minor.
To address this hypothesis, we graph the adjusted r-squared of a one factor model which combines normally distributed and rate dependent one factor models with a best practice model which includes all statistically significant factors. The results are shown here:
The adjusted r-squared for the best practice model is plotted in blue and is near 100% for all 120 quarterly segments of the yield curve. The one factor model, by contrast, does a poor job of fitting quarterly movements in the quarterly forward rates. The adjusted r-squared is good, of course, for the first forward rate since the short rate is the standard risk factor in a one factor term structure model. Beyond the first quarter, however, explanatory power is extremely low. The adjusted r-squared of the one factor model never exceeds 30% after the first 20 quarterly forward rates and is far below that level at most maturities.
This result should not come as a surprise to a serious analyst, because it is very similar to the results of the best practice Heath, Jarrow and Morton term structure model for Government of Canada Bonds, United Kingdom Government Bonds, German Bunds, Australian Commonwealth Government Securities, Swedish Government Securities, Singapore Government Securities, and Japanese Government Bond yields.
We can confirm the low explanatory power of a one factor model with a one line principal components analysis in a common statistical package. The “PCA” analysis is not constrained to choose the short rate as the explanatory variable in a one factor model. In fact, there are many other factors that would be stronger candidates for a single factor model. The results of the principal components analysis on quarterly movements in U.S. Treasury forward rates are shown here:
The results show that at least 10 factors are needed to model the U.S. Treasury yield curve with cumulative accuracy comparable to the confidence levels most large financial institutions would use for value at risk analysis. The first factor explains only 48.9% of quarterly forward rate movements. In the other government bond markets that we have examined, the normal figure is typically in the 55% to 60% range.
Conclusion: The weak form of the hypothesis is also overwhelmingly rejected by the data on U.S. Treasury yields.
Can a One Factor Model be “Tweaked” with One or Two More Factors?
Many large financial institutions have been using one factor models for such a long time that hope springs eternal that they can be fixed with a small “tweak,” a second or third factor. In the next graph we show the adjusted r-squareds for 1, 2, 3, 6 and “all” factors in a model of the U.S. Treasury yield curve.
The results show that even a six factor model leaves a big chunk of long term yield curve movements unexplained. In the 21^{st} century, with modern big data technology, using all factors that matter, instead of just a few of them, is a simple step forward to best practice term structure modeling.
A similar plot of the root mean squared errors for 1 factor, 2 factor, 3 factor, 6 factor, and all factor term structure models shows the danger of half steps in improving interest rate risk technology:
We close with this plot of which maturities on the yield curve are statistically significant in predicting forward rate movements at each of the 120 quarterly segments on the 30 year U.S. Treasury yield curve. Statistical significance is represented by a dot at the combination of yield curve risk factor (by maturity, on the vertical axis) and quarterly forward rate number. The lack of a dot means that risk factor maturity is not statistically significant. An orange dot represents interest rate volatility that is constant or “affine.” A green dot represents interest rate volatility that is proportional to the level of interest rates. A blue dot represents interest rate volatility that is linear, combining both constant and proportional impacts on interest rate volatility.
At shorter term forward rates, the linear specification for interest rate volatility (in blue) is the most common specification. As maturities lengthen to maturities where low rate experience is more limited, the measured interest rate volatility is increasingly shown as orange, or constant. We caution readers, especially those in high interest rate environments, that the constant volatility result is very likely to be rejected as experience with low rates becomes more common. In most countries, a prudent risk manager would consider the merits of a “one world” term structure model that includes insights from countries with longer histories and a wider range of interest rate experience. This would be best practice in the U.S. Treasury market as well, given that there are no reported negative rates in the U.S. Department of the Treasury time series.
Note also that a “regime change” one factor model includes only those statistically significant variables on the bottom row of the chart. The explanatory power of such a model is very low because the variables on all of the other rows have been omitted. We reach the same conclusions as we did in the U.S. Treasury, Government of Canada, United Kingdom, German Bund, Swedish Government Securities, Australian Commonwealth Government Securities, Singapore Government Securities, and Japanese Government Bond cases: use of one factor models exposes the analyst and his or her employer to very significant model risk. A multi-factor Heath, Jarrow and Morton model is the best practice replacement for one factor models.
Appendix A: Moving Forward with Modern Interest Rate Risk Technology
Kamakura Corporation facilitates client progress in interest rate modeling in multiple ways via Kamakura Risk Information Services’ Macro Factor Sensitivity Products:
Research subscriptions to Heath, Jarrow and Morton term structure modeling
This is a good first step for regulatory agencies and financial institutions building their familiarity with modern interest rate risk technology. The subscription includes the Technical Guides describing the parameter estimation process, the underlying raw data, and the parameters themselves, updated annually. Models are available for all major government yield curves.
Production subscription to Heath, Jarrow and Morton term structure modeling
The production subscription includes formatting of parameters for use in Kamakura Risk Manager’s newest versions and immediate release of HJM parameter estimates as soon as the quality control process at Kamakura Corporation is completed.
Production subscription to HJM yield scenarios
Kamakura Risk Information Services also generates the scenarios in-house and provides the scenarios in standard Kamakura Risk Manager format for all major government yield curves at customized frequencies (daily, weekly, monthly, quarterly) for individual clients. Transfer of data is by file transfer protocol technology.
Heath, Jarrow and Morton Training
Kamakura Corporation, led by Managing Director Robert A. Jarrow (Cornell University) provides training in modern Heath, Jarrow and Morton interest rate risk technology for both clients and potential clients. Professor Jarrow usually participates by video link in these training sessions.
For inquiries about these and other products, please contact your Kamakura representative or e-mail Kamakura at info@kamakuraco.com.
Appendix B: Further Reading for the Technically Inclined Reader
References for random interest rate modeling are given here:
Heath, David, Robert A. Jarrow and Andrew Morton, "Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approach," Journal of Financial and Quantitative Analysis, 11190, pp. 419-440.
Heath, David, Robert A. Jarrow and Andrew Morton, "Contingent Claims Valuation with a Random Evolution of Interest Rates," The Review of Futures Markets, 9 (1), 11190, pp.54 -76.
Heath, David, Robert A. Jarrow and Andrew Morton, “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claim Valuation,” Econometrica, 60(1), 11192, pp. 77-105.
Heath, David, Robert A. Jarrow and Andrew Morton, "Easier Done than Said", RISK Magazine, October, 11192.
References for non-parametric methods of model testing are given here:
Bharath, Sreedhar and Tyler Shumway, "Forecasting Default with the Merton Distance to Default Model," Review of Financial Studies, May 2008, pp. 1339-1369.
Jarrow, Robert, Donald R. van Deventer and Xiaoming. Wang, “A Robust Test of Merton’s Structural Model for Credit Risk,” Journal of Risk, Fall 2003, pp. 39-58.
References for modeling traded securities (like bank stocks) in a random interest rate framework are given here:
Amin, Kaushik and Robert A. Jarrow, "Pricing American Options on Risky Assets in a Stochastic Interest Rate Economy," Mathematical Finance, October 11192, pp. 217-237.
Jarrow, Robert A. “Amin and Jarrow with Defaults,” Kamakura Corporation and Cornell University Working Paper, March 18, 2013.
The behavior of credit spreads when interest rates vary is discussed in these papers:
Campbell, John Y. & Glen B. Taksler, "Equity Volatility and Corporate Bond Yields," Journal of Finance, vol. 58(6), December 2003, pages 2321-2350.
Elton, Edwin J., Martin J. Gruber, Deepak Agrawal, and Christopher Mann, “Explaining the Rate Spread on Corporate Bonds,” Journal of Finance, February 2001, pp. 247-277.