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Kamakura Releases New 10 Factor HJM Stochastic Volatility Model for U.S. Treasuries
Parameters, Documentation and Training Provided to KRIS Subscribers
NEW YORK, January 26, 2017: Kamakura Corporation reported Thursday that the December 31, 2016 version of the firm’s Heath, Jarrow and Morton 10-factor model for the U.S. Treasury yield curve has been released to subscribers of the Kamakura Risk Information Services Macro Factor Service. The newest version of the model was developed under the direction of Prof. Robert A. Jarrow, Kamakura Corporation’s Managing Director for research since 1995. The December 2016 version of the HJM model was developed using stochastic volatility assumptions for 10 factors driving movements in the U.S. Treasury curve based on daily U.S. Treasury yields from January 1962 to December 2016. The U.S. Treasury 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. 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.
Background on Heath, Jarrow and Morton Yield Curve Models
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. New market risk guidelines released by the Bank for International Settlements in early 2016 require interest rate models with at least 10 factors, for example. Today’s announcement from Kamakura Corporation 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
A new model for the Thai government bond market will be released in coming weeks.
In all of these studies, commonly used 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. The graph below shows the complexity of U.S. Treasury yield curve movements over the 1962-2016 period:
Kamakura Corporation’s founder and Chief Executive Officer Dr. Donald R. van Deventer commented on the December 2016 version of the U.S. Treasury model on Thursday, “We are extremely pleased at the world-wide interest in Kamakura’s Macro Factor Service given dramatic movements in interest rates, commodity prices, and default risk around the world. We applaud the research that has been done by the Federal Reserve Board and the Federal Reserve Bank of New York in providing a firm foundation for the use of multi-factor yield curve models both for setting strategy in the conduct of monetary policy and for monitoring the safety and soundness of the country’s largest financial institutions. Our newest model carries on in the tradition of Fed models by Kim and Wright (2005) and Adrian, Crump and Moench (2008-2013). We have made a number of extensions to the analysis to maximize both in-sample accuracy and the realism of very long term simulations that are essential in the management of the Social Security System, pension funds, endowments, life insurance firms, and commercial banks. We relax the ‘affine’ or constant interest rate volatility assumptions of the early multi-factor models and allow U.S. history to reveal the degree to which interest rate volatility changes with the level of interest rates.”
Martin Zorn, Kamakura Corporation’s President and Chief Operating Officer, elaborated on the risk management implications of the December 2016 release: “Many large banks around the world rely on legacy risk management systems that assume only one factor drives the movement of interest rates. This assumption implies that all yields will either rise together, fall together, or remain unchanged. The reason that the BIS requires 10 factors for market risk is that this implication of one factor models is true only 17% of the time in the U.S. Treasury market. On 83% of business days, the yield curve twists instead of moving in a 100% correlated way.”
Mr. Zorn added, “Many of the legacy interest rate risk systems rely heavily on models like the Vasicek, Extended Vasicek, Hull and White or Ho and Lee models that assume that interest rate volatility is not random. The truth of the matter is, however, that the volatility of interest rates in the United States is a random function of the level of rates, as this graph shows clearly.”
Mr. Zorn continued, “By recognizing the stochastic nature of interest rate volatility and extracting the 10 key factors that drive U.S. Treasuries, the new December 2016 HJM version achieves amazing in-sample accuracy with a root mean squared error in forecasting 91-day yield curve shifts of less than 0.02% for all 120 quarterly segments of the U.S. Treasury 30 year yield curve. The average root mean squared error across the full yield curve is closer to 0.005%.”
Fitting a Multi-Factor Heath Jarrow and Morton Term Structure Model to U.S. Treasury Yields
The KRIS Macro Factor Service HJM models apply the no-arbitrage constraints of Heath, Jarrow and Morton (Econometrica, 1992) in structuring the econometric approach to parameter fitting. Non-linear least squares is used to fit the discrete change in forward returns over 91 day overlapping intervals for the full period from 1962 to 2016. Because the overlapping periods induce autocorrelation, HAC (“heteroskedasticity and autocorrelation consistent”) standard errors are used in the model fitting process. Interest rate volatility is assumed to be a polynomial function of the beginning of period annualized forward rates for each of the 10 risk factors. More information is available to KRIS subscribers in the following 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 A, Version 1.0: “U.S. Treasury Yields,” to “Parameter Estimation for Heath, Jarrow and Morton Term Structure Models,” Technical Guide, Kamakura Corporation, November 1, 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.
For more information about Kamakura Risk Manager or the Kamakura Risk Information Services Macro Factor Service, please contact firstname.lastname@example.org.
Related Kamakura Corporation References
Adrian, Tobias, Richard K. Crump and Emanuel Moench, “Pricing the Term Structure with Linear Regressions,” Federal Reserve Bank of New York Staff Report No. 340, August 2008, Revised April 2013.
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.
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, "Easier Done than Said", RISK Magazine, October, 1992.
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.
Jarrow, Robert A. “Amin and Jarrow with Defaults,” Kamakura Corporation and Cornell University Working Paper, March 18, 2013.
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.
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. “A Multi-Factor Heath Jarrow and Morton Model of the United Kingdom Government Bond Yield Curve,” Kamakura Corporation working paper, www.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, www.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, www.kamakuraco.com, August 27, 2015.
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, www.kamakuraco.com, February 11, 2015.
van Deventer, Donald R. “Interest Rate Risk: Lessons from 2 Decades of Low Interest Rates in Japan,” Kamakura Corporation working paper, www.kamakuraco.com, August 11, 2016.
van Deventer, Donald R. “A Multi-Factor Heath Jarrow and Morton Model of the Swedish Government Bond Yield Curve,” Kamakura Corporation working paper, www.kamakuraco.com, September 3, 2015.
van Deventer, Donald R. “Model Validation for Asset and Liability Management: A Worked Example” using Canadian Government Securities, Kamakura Corporation working paper, www.kamakuraco.com, July 20, 2016.
van Deventer, Donald R. “Singapore Government Securities Yields: A Multi-Factor Heath Jarrow and Morton Model,” Kamakura Corporation working paper, www.kamakuraco.com, September 22, 2015.
van Deventer, Donald R. “Spanish Government Bond Yields: A Multi-Factor Heath Jarrow and Morton Model,” Kamakura Corporation working paper, www.kamakuraco.com, September 10, 2015.
van Deventer, Donald R. “The Regime Change Term Structure Model: A Simple Model Validation Approach,” Kamakura Corporation working paper, www.kamakuraco.com, January 26, 2016.
About Kamakura Corporation
Founded in 1990, Honolulu-based Kamakura Corporation is a leading provider of risk management information, processing and software. Kamakura was named to the World Finance 100 by the Editor and readers of World Finance magazine in 2012 and 2016. In 2010, Kamakura was the only vendor to win 2 Credit Magazine innovation awards. Kamakura Risk Manager, first sold commercially in 1993 and now in version 8.1, is the first enterprise risk management system with users focused on credit risk, asset and liability management, market risk, stress testing, liquidity risk, counterparty credit risk, and capital allocation from a single software solution. The KRIS public firm default service was launched in 2002. The KRIS sovereign default service, the world’s first, was launched in 2008, and the KRIS non-public firm default service was offered beginning in 2011. Kamakura added its U.S. Bank default probability service in 2014. Kamakura has served more than 330 clients ranging in size from $1.5 billion to $1.6 trillion in assets. Kamakura’s risk management products are currently used in 43 countries, including the United States, Canada, Germany, the Netherlands, France, Austria, Switzerland, the United Kingdom, Russia, the Ukraine, Eastern Europe, the Middle East, Africa, South America, Australia, Japan, China, Korea, India and many other countries in Asia.
Kamakura has world-wide alliances with Fiserv (www.fiserv.com) and SCSK Corporation (http://www.scsk.jp/index_en.html) making Kamakura products available in almost every major city around the globe.
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Web site: www.kamakuraco.com