*The author wishes to thank his colleague, Managing Director for Research Prof. Robert A. Jarrow, for twenty years of guidance and helpful conversations on this critical topic.*

The German Bund yield curve and its history are critical components for the derivation and validation of interest rate risk models for asset and liability management, market risk, economic capital, interest rate risk in the banking book, stress-testing and the internal capital adequacy assessment process. There are some interesting contrasts between results for the German Bund yield curve analysis on a stand-alone basis and world-wide experience because the German Bund data set spans a fairly narrow range of rate levels and is a short data series by international standards.

Our objective in this note is to show the derivation of a multi-factor Heath Jarrow and Morton model of the German Bund yield curve. As a by-product, we are again able to apply standard tests of model validation to commonly used one factor term structure models in Germany. Consistent with our studies of government securities markets in the United States, Canada, Japan and the United Kingdom, 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.

**Background for the Analysis
**In February, we discussed “ Essential Model Validation for Interest Rate Risk and Asset and Liability Management ” using U.S. Treasury data for the model validation exercise. In another worked example, we focused on Government of Canada yields and addressed this question in the validation process: “Is there any version of a one factor term structure model which is sufficiently accurate for general risk management purposes and for asset and liability management?” We performed the same analysis on the fascinating history of Japanese Government Bond yields , with more than two decades of negative and very low rate experience. Our study of United Kingdom Government Bonds reached similar conclusions using a fine data set compiled by the Bank of England.

In all four of these studies, one factor models failed basic model validation tests and were judged unacceptable from an accuracy point of view.

We follow the same model validation process again in this note and shown the reasons for these strong conclusions, using the experience in the German Bund market since 1996. Readers who want 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 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 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 Canadian case, we found that 12 factors were statistically significant in explaining movements in the Government of Canada yield curve. In the Japanese Government Bond case, we found that 16 factors were statistically significant in model movements in the government bond yield curve. Modeling for the United Kingdom Government Bond yield curve showed that 14 factors were necessary for a best practice Heath, Jarrow and Morton model of the yield movements from 1979 through 2015.

In this note, we address the same three questions that we raised in the March 2015 piece, but this time the answers are derived for the German Bund 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 German Bund market:

One factor models with rate-dependent interest rate volatility;

Cox, Ingersoll and Ross (1985)

Black, Derman and Toy (11190)

Black and Karasinski (11191)

One factor models with constant interest rate volatility (affine models)

Vasicek (1977)

Ho and Lee (1986)

Extended Vasicek or Hull and White Model (11190, 11193)

**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 for German Bunds show frequent negative yields in the German Bund market in recent 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, 11199 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 German Bund yields provided by the Deutsche Bundesbank and Bloomberg from January 1, 1996 through June 30, 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 German Bund 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 1996 to 2015 time period
- The quarterly changes in 120 quarterly forward rates making up the 30 year German Bund 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 twelve quarterly maturities. The null hypothesis of normality is rejected by all 3 tests for the first 26 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 31 of the 120 maturities. Somewhat surprisingly, the hypothesis of normality is rejected less strongly in the German Bund market than in other markets studied. This is again due to the short history and narrow range of interest rates experienced in the German Bund market.

**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 5,077 days of zero coupon bond yields for the German Bund 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 rarely consistent with the implications of one factor term structure models. Yield curve shifts were all positive, all negative, or all zero 5.38%, 5.30%, and 0% of the time, a total of 10.68% 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 U.S. Treasury, Japanese Government Bond, Government of Canada, and United Kingdom Government Bond yield curves. These twists, which happen 89.32% of the time in Germany, 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 German Bund 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 46.36% of the total observations in the data set. The remainder of the data set, 53.64% 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
**

**German Bund Yields**

Given the poor performance of one factor models on the non-parametric tests above, it is no surprise that bank regulators are turning to multi-factor models around the world. The Federal Reserve’s Comprehensive Capital Analysis and Review stress testing regime has included 3 points on the U.S. Treasury yield curve since 2014. The Bank for International Settlements has required that at least six interest rate risk factors be used to model market risk since 2010. Adrian, Crump and Moench of the Federal Reserve Bank of New York use five factors in their U.S. Treasury term structure model.

We now fit a multi-factor Heath, Jarrow and Morton model to quarterly German Bund zero coupon yield data from January 1, 1996 to June 30, 2015. There were no changes in the data availability during this period, so we do not face the common issue of changes in the data regime.

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 D, Version 1.0: “German Bund Yields,” to “Parameter Estimation for Heath, Jarrow and Morton Term Structure Models,” Technical Guide, Kamakura Corporation, June 30, 2015.

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 (in this case, that is all observations). 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 German Bund 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 German Bund 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 German Bund yields:

Other than the first quarterly forward rate in the yield curve, there are as many as 21 explanatory variables that drive the 120 quarterly segments of the yield curve. The final German Bund term structure model from Kamakura Risk Information services has 14 independent risk factors that drive yields, and these factors also appear in combination with rate level variables. A total of 31 related candidate explanatory variables were used in the estimation process.

**Conclusion: The strong form of the hypothesis is overwhelmingly rejected by the data on German Bund 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 “regime change” 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 120% for nearly all 120 quarterly segments of the yield curve. The one factor model, by contrast, does a stunningly 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 40% after the first 15 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 U.S. Treasuries, Government of Canada Bonds, United Kingdom Government Bonds, 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 German Bund forward rates are shown here:

The results show that at least 14 factors are needed to model the German Bund 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 59% of quarterly forward rate movements. Readers should beware of the fact that principal components analysis uses only that part of the data set for which data exists at all maturities tested, 77 quarterly observations in the case of the German Bund market.

**Conclusion: The weak form of the hypothesis is also overwhelmingly rejected by the data on German Bund 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 German Bund 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 German Bund 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 is the most common specification. As maturities lengthen to maturities where low rate experience is very 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. The prevalence of the constant (orange) interest rate volatilities should be a concern to managers of interest rate risk in Europe, because the “stand alone-one country” interest rate modeling effort for Germany differs fairly strongly from results internationally. Outside of the German Bund market, interest rate volatility is most often linear in the level of interest rates rather than constant. The differing results for Germany are the result of the shortness of the rate history and narrow range of movements in the German Bund market. 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.

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, 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.

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