Press Release

Kamakura Releases New Stochastic Volatility Model for U.S. Treasuries
Bayesian Out-of-Sample Model Validation
Predicts Modest Chance of Negative Rates

Prof. Robert Jarrow NEW YORK, April 24, 2019: Kamakura Corporation has released a new 10-factor term structure model for the U.S. Treasury yield curve for Kamakura Risk Manager and Kamakura Risk Information Services subscribers. Development of the model was led by Robert A. Jarrow, Kamakura’s Managing Director for Research and the Ronald P. and Susan E. Lynch Professor of Investment Management at Cornell University’s Johnson Graduate School of Management. Dr. Jarrow’s 1992 paper, written with David Heath and Andrew Morton, provides the underlying framework for the model’s estimations.

The menu of options includes 1-, 2-, 3-, 6-, and 10- factor models featuring both constant (“affine”) interest rate volatility and stochastic volatility. The best-fitting model is the stochastic volatility 10-factor, which is benchmarked on daily data from the U.S. Department of the Treasury from January 2, 1962 through December 31, 2018.

You can read a paper providing an overview of the model here:
A 10 Factor Heath, Jarrow and Morton Stochastic Volatility Model for the U.S. Treasury Yield Curve, Using Daily Data from January 1, 1962 through December 31, 2018

Martin Zorn, President and Chief Operating Officer for Kamakura Corporation Martin Zorn, President and Chief Operating Officer for Kamakura Corporation, said Wednesday, “Global traders and risk managers have spent most if not all of their careers in a declining rate environment. We have seen nominal rates go negative as central bankers have aimed for an expansionary effect. Given market conditions since the 2007-2009 financial crisis, traders need new models. Government yield curves are a critical input to the risk management calculations of central banks, bank regulators, major commercial banks, insurance firms, fund managers, pension funds, and endowments around the world. With the internationalization of fixed income investing, it is important to understand the dynamics of movements in yield curves worldwide.”

The updated Treasury model provides a more realistic roadmap for predicting the future than relying on personal experience, since it is based on 57 years of daily U.S. interest rate movements. The modeling project benefitted enormously from Kamakura’s broad first-hand knowledge across a diverse economic spectrum. Particularly helpful was the annually-released Japanese government bond model, which demonstrates interest rate movements in a negative rate economy and has been invaluable in maximizing the realism of the new U.S. Treasury model.

Figure 1: Distribution of 100,000 scenarios for 3-Month Treasury bill yields 20 years forward on both an empirical (red) and risk neutral (blue) basis.

The model documentation includes a significant section on Bayesian model validation, in which 100,000 out-of-sample scenarios were used to measure the consistency of the model’s predictions, taking into account both expert knowledge and history about interest rate movements worldwide.

The Bayesian process of model fitting, simulation, and revision is both analytically elegant and eminently practical. The model documentation for the U.S. Treasury HJM model shows realistic variation in both risk- neutral and empirical interest rates over time. The simulation is consistent with worldwide experience, showing that negative rates are possible, but much less likely than positive rates. Simulated rates at highs like those last seen in the 1970s and 1980s show a low probability of occurring.

The Kamakura standard Bayesian out-of-sample model validation provides assurance to risk managers, auditors, regulators and boards of directors that the interest rate simulation technology used in Kamakura Risk Information Services (KRIS) and Kamakura Risk Manager (KRM) sets a new standard for industry best practice and is exceptionally realistic.

Kamakura’s analytical team regularly updates term structure models from all major markets around the world. Model documentation and parameters are available by subscription to Kamakura Risk Information Services’ default probability and bond information services. All of the models mark to market observable securities prices. They provide the basis for very high scenario simulation of correlated risks and for interest rate factor-driven and other macro-factor-driven stress tests using Kamakura Risk Manager, both with and without default modeling.

To see Kamakura model and product announcements and other risk commentary by Kamakura on a daily basis, please follow:

Kamakura CEO Dr. Donald van Deventer (
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About Kamakura Corporation
Founded in 1990, Honolulu-based Kamakura Corporation is a leading provider of risk management information, processing, and software. Kamakura was recognized as a category leader in the Chartis Report,Technology Solutions for Credit Risk 2.0 2018. Kamakura was named to the World Finance 100 by the editor and readers of World Finance magazine in 2017, 2016 and 2012. In 2010, Kamakura was the only vendor to win two Credit Magazine innovation awards. ,Kamakura Risk Manager, first sold commercially in 1993 and now in version 10.0.5, is the first enterprise risk management system for 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 with assets ranging in size from $1.5 billion to $3.0 trillion. Its risk management products are currently used in 47 countries, including the United States, Canada, Germany, the Netherlands, France, Austria, Switzerland, the United Kingdom, Russia, Ukraine, South Africa, Australia, China, Hong Kong, India, Indonesia, Japan, Korea, Malaysia, Singapore, Sri Lanka, Taiwan, Thailand, Vietnam, and many other countries in Asia, Europe and the Middle East.

To follow risk commentary by Kamakura on a daily basis, please follow:

Kamakura CEO Dr. Donald van Deventer (
Kamakura President Martin Zorn ( and
Kamakura’s official twitter account (

For more information, please contact:
Kamakura Corporation
2222 Kalakaua Avenue, Suite 1400, Honolulu, Hawaii 96815
Telephone: 1-808-791-9888
Facsimile: 1-808-791-9898
Web site: