An Updated 10-Factor Heath, Jarrow And Morton Model For The U.S. Treasury Curve

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Summary

Many large financial institutions around the world use legacy interest rate risk systems for enterprise risk management that assume only one random factor drives yield curve movements.

This technical article shows that such an assumption is grossly inaccurate. Banks using such legacy systems are unprepared for the risks in a rising interest rate environment.

Both regulators and shareholders would be well-served to intensely scrutinize financial firms' total balance sheet risk management practices and systems to avoid unpleasant surprises when rates rise.

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

The U.S. Treasury (NYSEARCA:TLT) yield curve is a critical input to the risk management calculations of major banks, insurance firms, fund managers, pension funds, and endowments around the world. This note presents an updated 10 factor no arbitrage model of the U.S. Treasury yield curve using the Heath, Jarrow and Morton [1992] framework for the period from January 1962 through September 2016.

The U.S. Treasury history made available by the U.S. Department of the Treasury begins on January 2, 1962. While the U.S. Treasury does not report any negative rates during this period, the wide variation in U.S. Treasury yields over this period provides an important benchmark for government yield curve modeling in other countries where historical experience is not as varied, particularly in Germany and Australia.

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 illustrate 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 legacy asset and liability management systems still in use in many large banks around the world. Consistent with our studies of government securities markets in Canada, Japan, the United Kingdom, Australia, Sweden, Spain, Singapore and Germany, 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

This note is part of a series on 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. Previous reviews have covered the following bond market sectors:

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 Securities

* Prior version

In all of these studies, one-factor models failed basic model validation tests and were judged unacceptable from an accuracy point of view. A recent study prepared for a major U.S. bank regulator also confirmed that a one-factor "regime shift" term structure model made essentially no incremental contribution toward resolving the persistent lack of accuracy in one-factor term structure models.

We now examine the same issues for government securities in the United States.

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. 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, 2015 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 at least 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. This note updates those earlier studies.

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

SpainSpanish Government Securities, 11 factors

Sweden:Swedish Government Securities, 11 factors

United Kingdom: Government Securities, 14 factors

United States: Treasury Securities, 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 provides 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, all of which are special cases of the Heath, Jarrow and Morton framework, in this section using data from the U.S. Treasury 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, Swedish Government Bond histories, Japanese Government Bond histories, and yields in many other countries show frequent negative yields in recent years.

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 daily data on U.S. Treasury yields provided by the U.S. Department of the Treasury from January 2, 1962 through September 30, 2016. 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 daily evolution of U.S. Treasury zero coupon yields over time:

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The graph below shows the evolution of the first quarterly forward rate (the forward that applies from the 91st day through the 182nd day) over the same time period:

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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 2016 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, the Shapiro-Francia test, and the skew test, all of which are available in common statistical packages.

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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 all 120 of the 120 quarterly zero coupon yield maturities. For quarterly changes in forward rates, the null hypothesis of normality is again rejected by all 3 tests for all 119 of the 119 maturities for changes in forward rates. This is a powerful 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. In most of the other countries studied, the hypothesis of normality has been rejected strongly 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, and United Kingdom Government Bond yield curves. These twists, which happen 82.95% of the time in the United States, 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 U.S. Treasury zero coupon yield data from January 2, 1962 to September 30, 2016. 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 yield or the 30-year yield (or both) was available.

The availability of data out to 30 years is fairly typical in government bond markets worldwide. 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, November 1, 2016.

Jarrow, Robert A. and Donald R. van Deventer, Appendix I, Version 2.0: "U.S. Treasury Yields," to "Parameter Estimation for Heath, Jarrow and Morton Term Structure Models," Technical Guide, Kamakura Corporation, September 30, 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 use overlapping 91-day intervals to measure changes in forward rates, avoiding the use of quarterly data because of the unequal lengths of calendar quarters. Because overlapping observations trigger auto-correlation, "HAC" (heteroscedasticity and autocorrelation adjusted) standard errors are used. The methodology is that of Newey-West with 91 day lags. We consider ten potential explanatory factors: the idiosyncratic portion of the movements in quarterly forward rates that mature in 6 months, 1 year, 1.5 years, 2, 3, 5, 7, 10, 20 and 30 years. This is the same number of factors required by the Bank for International Settlements market risk guidelines published in January 2016. We calculate the discrete changes in forward returns as described in the parameter technical guide. Because the discrete changes are non-linear in the no-arbitrage framework of Heath, Jarrow and Morton, we use non-linear least squares to fit interest rate volatility. We then begin the process of creating the orthogonalized risk factors that drive interest rates using the Gram-Schmidt procedure. These factors are assumed to be uncorrelated independent random variables that have a normal distribution with mean zero and standard deviation of 1. Because interest volatility is assumed to be stochastic, simulated out of sample forward rates will not in general be normally distributed. In the estimation process, we added factors to the model as long as each new factor provided incremental explanatory power.

We postulate that interest rate volatility for each forward rate maturity is a cubic function of the annualized forward rate that prevails for the relevant risk factor at the beginning of each 91-day period:

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:

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Because of the volatility structure above, there are potentially 4 x 10 factors = 40 parameters that are statistically significant in fitting forward rates over the 13,420 observations that we have for maturities of 10 years or less. The final U.S. Treasury term structure model from Kamakura Risk Information services has 40 potential risk factors (the ten basic risk factors plus the product of each risk factor with the annualized forward rate for that risk factor maturity, its squared value, and its cubed value) that drive yields. All 40 factors are significant at one point on the yield curve and at least 8 factors are significant at the shortest maturity forward rate.

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 with the adjusted r-squared of a best practice model which includes all statistically significant factors. The results are shown here:

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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 91-day 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 22% after the first 22 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, Singapore Government Securities, Spanish Government Securities, Swedish 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 91-day movements in U.S. Treasury quarterly 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 that most large financial institutions would use for value at risk analysis. The first factor explains only 52.9% of 91-day changes in quarterly forward rates. In the other government bond markets that we have examined, the normal figure is typically in the 50% 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.

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The results show that even a six factor model leaves a large amount of long-term yield curve movements unexplained. In the 21st 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:

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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 for that forward rate segment. 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 (i.e. the constant term in the cubic function above is zero). A blue dot represents interest rate volatility that combines both a non-zero constant term and cubic proportional impacts on interest rate volatility.

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At shorter-term forward rates, the most common specification for interest rate volatility (in blue) is the cubic function with non-zero constant term, which allows for negative rates. As maturities lengthen to maturities where low rate experience is more limited, the measured interest rate volatility is occasionally 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.

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 Government of Canada, United Kingdom, German Bund, Spanish Government Securities, Swedish Government Securities, Australian Commonwealth Government Securities, Singapore Government Securities, and Japanese Government Bond cases: the 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

General references on econometrics:

Angrist, Joshua D. and Jorn-Steffen Pischke, Mostly Harmless Econometrics: An Empiricist's Companion, Princeton University Press, Princeton, 2009.

Berger, James O. Statistical Decision Theory and Bayesian Analysis, second edition, Springer-Verlag, 1985.

Berry, Donald A. Statistics: A Bayesian Perspective, Wadsworth Publishing Company, 1996.

Campbell, John Y, Andrew W. Lo, and A. Craig McKinley, The Econometrics of Financial Markets, Princeton University Press, 1997.

Gelman, Andrew and John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin, Bayesian Data Analysis, third edition, CRC Press, 2013.

Goldberger, Arthur S. A Course in Econometrics, Harvard University Press, 1991.

Hamilton, James D. Times Series Analysis, Princeton University Press, 1994.

Hansen, Bruce E. Econometrics, University of Wisconsin, January 15, 2015.

Hastie, Trevor, Robert Tibshirani and Jerome Friedman, Elements of Statistical Learning: Data Mining, Inference and Prediction, Springer, second edition, tenth printing, 2013.

Johnston, J. Econometric Methods, McGraw-Hill, 1972

Maddala, G. S. Introduction to Econometrics, third edition, John Wiley & Sons, 2005.

Papke, Leslie E. and Jeffrey M. Wooldridge, "Econometric Methods for Fractional Response Variables with an Application to 401(NYSE:K) Plan Participation Rates," Journal of Applied Econometrics, Volume 11, 619-632, 1996.

Robert, Christian P. The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation, second edition, Springer Science+Business Media LLC, 2007.

Stock, James H. and Mark W. Watson, Introduction to Econometrics, third edition, Pearson/Addison Wesley, 2015.

Studenmund, A. H. Using Econometrics: A Practical Guide, Addison-Wesley Educational Publishers, 1997.

Theil, Henri. Principles of Econometrics, John Wiley & Sons, 1971.

Woolridge, Jeffrey M. Econometric Analysis of Cross Section and Panel Data, The MIT Press, 2002.

References for random interest rate modeling are given here:

Adrian, Tobias, Richard K. Crump and Emanuel Moench, "Pricing the Term Structure with Linear Regressions," Federal Reserve Bank of New York, Staff Report 340, August 2008, revised August 2013.

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, 1990, 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), 1990, 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), 1992, pp. 77-105.

Heath, David, Robert A. Jarrow and Andrew Morton, "Easier Done than Said", RISK Magazine, October, 1992.

Kim, Don H. and Jonathan H. Wright, "An Arbitrage-Free Three Factor Term Structure Model and the Recent Behavior of Long-Term Yields and Distant-Horizon Forward Rates," Finance and Economics Discussion Series, Federal Reserve Board, 2005-33.

van Deventer, Donald R. "Essential Model Validation for Interest Rate Risk and Asset and Liability Management," an HJM model for the U.S. Treasury curve, Kamakura Corporation working paper, kamakuraco.com, February 11, 2015.

van Deventer, Donald R. "Model Validation for Asset and Liability Management: A Worked Example" using Canadian Government Securities, Kamakura Corporation working paper, kamakuraco.com, July 20, 2016.

van Deventer, Donald R. "Interest Rate Risk: Lessons from 2 Decades of Low Interest Rates in Japan," Kamakura Corporation working paper, kamakuraco.com, August 11, 2016.

van Deventer, Donald R. "A Multi-Factor Heath Jarrow and Morton Model of the United Kingdom Government Bond Yield Curve," Kamakura Corporation working paper, kamakuraco.com, August 17, 2015.

van Deventer, Donald R. "A Multi-Factor Heath Jarrow and Morton Model of the German Bund Yield Curve," Kamakura Corporation working paper, kamakuraco.com, August 21, 2015.

van Deventer, Donald R. "A Multi-Factor Heath Jarrow and Morton Model of the Australia Commonwealth Government Securities Yield Curve," Kamakura Corporation working paper, kamakuraco.com, August 27, 2015.

van Deventer, Donald R. "A Multi-Factor Heath Jarrow and Morton Model of the Swedish Government Bond Yield Curve," Kamakura Corporation working paper, kamakuraco.com, September 3, 2015.

van Deventer, Donald R. "Spanish Government Bond Yields: A Multi-Factor Heath Jarrow and Morton Model," Kamakura Corporation working paper, kamakuraco.com, September 10, 2015.

van Deventer, Donald R. "Singapore Government Securities Yields: A Multi-Factor Heath Jarrow and Morton Model," Kamakura Corporation working paper, kamakuraco.com, September 22, 2015.

van Deventer, Donald R. "The Regime Change Term Structure Model: A Simple Model Validation Approach," Kamakura Corporation working paper, kamakuraco.com, January 26, 2016.

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 1992, 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.

Disclosure: I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it. I have no business relationship with any company whose stock is mentioned in this article.