*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.*

As zero interest rate policies and negative interest rates ripple through world financial markets, many legacy interest rate risk systems and asset and liability management systems have been unable to keep pace. In this note, we use 100,000 scenarios from a modern 9 factor Heath, Jarrow and Morton interest rate simulation from Kamakura Corporation to illustrate the model validation issues that arise when one admits that negative interest rates have a probability that is not zero.

The model validation procedures we outline are used by Kamakura in both its Kamakura Risk Information Services macro factor scenario sets and in Kamakura Risk Manager (“KRM”). KRM has allowed users to simulate and analyze negative interest rates for more than 20 years.

**Negative Interest Rates: An International Perspective**

Even with negative interest rates making the headlines in European markets daily, one sometimes hears the phrase “It can’t happen in the United States.” The same phrase, of course, was used to deny the possibility that home prices could fall in the United States prior to the 2006-2010 financial crisis. Negative interest rates have already been observed in the secondary market for U.S. Treasury securities, as confirmed by this phrase quoted from the February 17 version of the U.S. Department of the Treasury yield reporting web page:

“Negative Yields and Nominal Constant Maturity Treasury Series Rates (CMTs). 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.”

What this means for an analyst of interest rate risk in the United States is important. The nominal U.S. Treasury has over-ridden or not used observations of negative interest rates in its reporting. That forces a careful analyst to be diligent in learning lessons about the behavior of negative interest rates in other markets. By way of contrast, yields for Treasury Inflation Protected Securities (TIPS) have often been negative and provide another modeling option.

**Japan:**

Japanese government bond yields have risen sharply in the last few weeks, but the Ministry of Finance web site is an excellent source of data showing negative yields in this important government bond market. During the period from January 1, 2010 through January 30, 2015, Ministry of Finance historical data reports negative rates for these maturities with these frequencies:

1 year 25 days

2 years 29 days

3 years 23 days

4 years 15 days

Most of the observations were in December 2014 and January 2015.

**Switzerland:**

The SARON (formerly overnight repo rate) reported by the Swiss National Bank was -0.72% at 12 noon on February 17. The Libor rate quoted for 3 months and reported by the Swiss National Bank was -0.90% on the same day.

**Germany:**

The Bundesrepublik Deutschland Finanzagentur GmbH reports in its auction results for 2014 that 19 government bond auctions settled at negative yields.

**Hong Kong:**

Negative interest rates have become a fairly frequent occurrence in Hong Kong. The website of the Hong Kong Monetary Authority reports daily interest rates. As just one example, at the 11 a.m. fixing on December 31, 2014 yields were negative at 1 week and 1 month maturities.

This section is just a small sampling of negative interest rates from around the world. When a careful analyst wants to allow the possibility of negative rates in a simulation, what are the key model validation issues? We turn to that question next.

**Key Model Validation Issues for Low Rates and Negative Rates**

In previous blogs, we have shown that there are a number of important steps that one should take in model validation regardless of the level of interest rates:

- Recognize that the common one factor term structure models are inadequate for realistic interest rate risk and asset and liability management.
- Establish the number of random factors needed to realistically simulate random interest rate movements. In the U.S. Treasury market, we show that between 9 and 11 risk factors are necessary for a level of accuracy that is consistent with the value at risk probabilities at common use by major banks.
- Confirm that a Monte Carlo simulation of interest rate movements correctly prices the full maturity spectrum of zero coupon bonds from the risk-free term structure.

One factor term structure models that cannot generate negative interest rates (in their continuous time limit) include Black-Derman-Toy (in which volatility is proportional to the level of the short rate) and Cox-Ingersoll-Ross (in which volatility is proportional to the square root of the level of the short rate). Other one factor models which assume an interest rate volatility that is constant over time, like Ho and Lee, Vasicek, and Extended Vasicek/Hull-White produce negative interest rates that are “too negative” when the starting yield curve is low. Only a multi-factor approach using the no arbitrage constraints of Heath, Jarrow and Morton can generate a realistic degree of negative rates. In doing so, an analyst needs to address a number of issues. In working with key risk management clients around the world, Kamakura Corporation, led by Prof. Robert Jarrow, has identified these issues relevant to zero or negative interest rate environments:

**If interest rate volatility is proportional to the level of a certain interest rate for interest rate factor k, what happens when that benchmark interest rate for factor k is negative? **

The Heath, Jarrow and Morton theory specifies that the shocks to the yield curve are normally distributed. If the interest rate volatility function is a positive constant times the level of rates, a positive “shock” leads to increases in interest rates when the benchmark rate is positive. A positive “shock” will lead to *decreases* in interest rates in the case where the benchmark rate is negative. A careful model validation procedure will judge this an invalid model because it blows up in both a theoretical and a practical sense. In a theoretical sense, positive rates can grow very high and negative rates can grow very negative. The results are nonsense, even when the model has been carefully adjusted to match observable zero coupon bond prices. From a practical interest rate hedging perspective, the model implies that a hedge against a positive shock in risk factor k goes in one direction when rates are positive and goes in the opposite direction when rates are negative. No such model should survive a careful model audit.

**If interest rate volatility changes its mathematical form for one or more of the n random factors driving the yield curve at different rate levels, we need to ask these questions as part of model validation: does each of these mathematical volatility functions (a) have a valid statistical basis, (b) a sensible projection into the future, and (c) a historical simulation of rates highly consistent with actual rate movements? **

Kamakura Managing Director Prof. Robert Jarrow is fond of saying “If one wants to model future interest rate movements accurately, it is essential to be able to model historical interest rate movements accurately.” In the example that follows, we use a 9 factor model in which 7 of the 9 factors have interest rate volatility functions that change over three interest rate ranges. Each of functions has been carefully estimated and validated over the relevant range of interest rates.

**Generally speaking, how is interest rate volatility different in a negative interest rate environment than in a positive interest rate environment? **

As noted in the section on the Japanese government bond market above, negative interest rates have been fairly common but the amount of data would lead the typical statistician to ask for more data. A careful model validation process would use a wide cross-section of international experience when data is sparse (say, in Hong Kong) or where negative interest rate data has not been reported by the government (as in the case of the United States because of policy decisions at the U.S. Department of the Treasury). Moreover, even with an international data base, good judgment will need to be applied in both model construction and model validation.

We now turn to a worked example using 100,000 scenarios over a 30 year time frame beginning with the U.S. Treasury yield curve on February 13, 2015.

**A Worked Example Using the U.S. Treasury Curve of February 13, 2015**

In this note, we use a 9 factor model that has been developed to be consistent with the no arbitrage constraints outlined by Heath, Jarrow and Morton in a series of papers written in the late 1980s and published five years later. Heath, Jarrow and Morton (1992) (“HJM”) posed this question: for a given set of assumptions about the number of risk factors and the associated impacts they have on interest rate volatility, what constraints are necessary to ensure “no arbitrage”? They defined “no arbitrage” as we define a valid model in this note: the assumptions of the interest rate simulation must be such that a Monte Carlo simulation of interest rates correctly prices the observable zero coupon bond prices at time zero. HJM concluded that, once the interest rate risk factors and their volatilities have been specified, the (risk-neutral) drift in interest rates is fully determined and cannot be independently specified.

We use those constraints on interest rate drift in our model for this worked example. The 9 factors driving the yield curve are assumed to be the idiosyncratic movements of forward rates at 9 points on the U.S. Treasury yield curve, ranging from 3 months to 30 years. The initial yield curve is the U.S. Treasury on-the-run yields reported by the U.S. Department of the Treasury for February 13, 2015:

The volatilities used are rich and complex. They are similar to the volatilities used in our note of February 10, “Essential Model Validation for Interest Rate Risk and Asset and Liability Management,” with some adjustments. We use volatility functions that allow both constant and rate-dependent interest rate volatility, with an “if/then” ability to change those functions at both high and low levels of interest rates. We have made these changes to the assumptions used in the February 10 analysis to achieve greater realism in the number of negative rate scenarios and to more accurately reflect interest rate volatility at both very high and very low levels:

- When the forward rate driving risk factor k gets very high, we dampen its influence on interest rate volatility as rates rise further.
- When the forward rate driving risk factor k turns negative, we ensure that the impact of a positive shock to risk factor k does not “change signs” of the resulting change in absolute rate levels.
- When the forward rate being modeled as a function of our 9 risk factors is already very low, we dampen the impact of all risk factors jointly because of the low level of interest rate volatility seen in periods of negative rates in Japan. We make the judgment in this regard that U.S. experience is likely to be more like Japan than Switzerland, for example.

We use this model because of its richness and consistency with the current level of low interest rates and the observable positive probability of negative interest rates. Again, what is important is not the question “is this the best term structure model one can construct?” because that is irrelevant to this note. What is important is the process by which we determine the validity and improve the accuracy of the model.

We now summarize our simulation results and perform the model validation.

**Simulation Results**

Using the term structure model above, Kamakura Risk Information Services produced 100,000 risk neutral scenarios for yield curve movements from the close of business U.S. Treasury curve on February 13, 2015. “Risk neutral” scenarios are scenarios used for the valuation of interest rate-sensitive securities. In general, “risk neutral” scenarios produce higher interest rates than expected (or “empirical”) future rates because risk neutral rates reflect the risk premium or term premium one earns from investing today at a long term fixed interest rate. Yield curve smoothing used to produce the initial zero coupon curve was the no arbitrage maximum smoothness forward rate approach of Adams and van Deventer (1994) as corrected in van Deventer and Imai (1996) and van Deventer, Imai and Mesler (2013) . The beginning U.S. Treasury forward rates and zero coupon yields are shown here:

The initial level for the 3 month U.S. Treasury yield was 0.01%, which has an impact on the probability of negative rates for a wide range of assumptions consistent with recent experience in Europe and Japan. We first review the percentile distribution of the 100,000 simulated paths of the 1 quarter, 1 year (4 quarterly periods) and 5 year (20 periods) zero coupon Treasury yields over the 30 year time horizon of the simulation.

The first graph shows the minimum, maximum, average, median and other percentiles of the simulated 3 month zero coupon Treasury yield:

The degree to which interest rates can be negative is controlled by the critical assumptions listed above. This set of projections produces negative rates with a much smaller probability and magnitude than our note of February 10. The second graph shows the same percentiles over 30 years for the 1 year zero coupon bond yield:

By the end of the 30 year simulation period, the lowest 1 year rate simulated was close to zero. The third graph shows the percentile distributions of the 100,000 simulated paths for the 5 year zero coupon Treasury yield.

The graph above shows that only one percent of the simulated 5 year zero coupon levels was below zero in 30 years, even starting from a 3 month U.S. Treasury rate of 0.01%. Next, we show the simulated distributions of the 1 quarter, 1 year and 5 year zero yields at these points in time:

- 1 quarter
- 1 year
- 5 years
- 10 years
- 20 years
- 30 years

We show the 5 year zero yield distributions in this section and the others in the Appendix. The distribution of the 5 year zero yield is nearly normally distributed after 1 quarter:

After 1 year, we get the following distribution:

The graph shows a small probability of negative yields for the 5 year zero coupon bond yield.

The distribution after 5 years takes on the shape below. There are more observable negative yields but there is also a clustering around the zero yield level like we have seen in the Japanese experience:

The graph below shows the distribution of the 5 year zero yield after 10 years.

Again, the graph shows a modest degree of negative yields and a more pronounced clustering of yields near zero. The next graph shows the same distribution at the 20 year point. The magnitude of negative yields is small and the clustering near the zero level is pronounced:

The final graph shows the simulated distribution of 100,000 paths of the 5 year zero yield after 30 years:

The final distribution of the 5 year zero coupon bond yield reflects two scenarios much discussed in the popular financial press: the “Japan” scenario where rates stay low for a very long time, and the “exploding inflation” scenario that many fear will be the result of quantitative easing.

**The Model Validation: Valuation of 120 Starting Zero Coupon Bond Prices**

How well did the simulated interest rates price the 120 starting values of U.S. Treasury zero coupon bond prices? The results were excellent. The average pricing error on a zero coupon bond with a principal value of $1 was -0.00000036. The mean absolute pricing error was 0.00000370.

**Conclusions**

We have shown the procedures for model validation associated with very low or negative interest rates. We adjusted our assumptions from February 10 to reduce the incidence of negative rates and to improve the realism and accuracy of rate movements as the 9 rate risk factors are impacted by very high and very low rates. Even with this level of complexity over 100,000 scenarios, we were able to demonstrate that the model is “valid,” consistent with the no arbitrage conditions set forth by Heath, Jarrow and Morton in 1992. All 120 quarterly zero coupon bond prices used as input to the simulation were correctly priced.

A disciplined approach to model validation is at the heart of development efforts for Kamakura Risk Information Services and Kamakura Corporation’s enterprise risk management software system Kamakura Risk Manager. For more information, please contact us at info@kamakuraco.com.

**References
**Adams, Kenneth J. and Donald R. van Deventer. “Fitting Yield Curves and Forward Rate Curves with Maximum Smoothness.”

*Journal of Fixed Income*, June 1994.

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.

Jarrow, Robert and Stuart Turnbull, *Derivative Securities*, 2^{nd} edition, South-Western College Publishing, Cincinnati, 2000.

van Deventer, Donald R. “Essential Model Validation for Interest Rate Risk and Asset and Liability Management,” Kamakura blog, www.kamakuraco.com, February 11, 2015.

van Deventer, Donald R. and Kenji Imai, *Financial Risk Analytics: A Term Structure Model Approach for Banking, Insurance, and Investment Management*, Chicago: McGraw Hill, 1996.

van Deventer, Donald R., Kenji Imai, and Mark Mesler, *Advanced Financial Risk Management*, 2^{nd} edition, Singapore, John Wiley & Sons, 2013.

**Related publications on interest rate risk and model validation**

As a convenience to the reader, we summarize our recent publications on interest rate risk management and related model validation here.

Adams, Kenneth J. and Donald R. van Deventer. “Fitting Yield Curves and Forward Rate Curves with Maximum Smoothness.” *Journal of Fixed Income*, June 1994.

van Deventer, Donald R. “Risk Management Model Validation: Checklist and Procedures,” Kamakura blog, www.kamakuraco.com, June 12, 2009. Redistributed on www.riskcenter.com on June 16, 2009.

van Deventer, Donald R. “Simulating the Term Structure of Interest Rates—How Many Factors are Necessary?” Kamakura blog, www.kamakuraco.com, July 7, 2009. Redistributed on www.riskcenter.com on July 8, 2009.

Miocinovic, Predrag and Donald R. van Deventer, “Common Pitfalls in Risk Management, Part 1: Confusing Pseudo Monte Carlo with the Real Thing,” Kamakura blog, www.kamakuraco.com, August 25, 2009. Redistributed on www.riskcenter.com on August 26, 2009.

van Deventer, Donald R. “Common Pitfalls in Risk Management, Part 2: Disco is Dead-Why Net Income Simulation and Saturday Night Fever are Necessary but Not Sufficient,” Kamakura blog, www.kamakuraco.com, August 28, 2009. Redistributed on www.riskcenter.com on August 31, 2009.

van Deventer, Donald R. “Common Pitfalls in Risk Management, Part 3: Comments from Bank Regulators and a JPMorgan Veteran on ‘Disco Risk Management,’” Kamakura blog, www.kamakuraco.com, September 1, 2009. Redistributed on www.riskcenter.com on September 2, 2009.

Miocinovic, Predrag, Alexandre Telnov, and Donald R. van Deventer, “Pitfalls in Asset and Liability Management: Interpolating Monte Carlo Results, Or How to Prove Augusta National is Not a Golf Course,” Kamakura blog, www.kamakuraco.com, October 7, 2009.

Dicker, Daniel T., Robert A. Jarrow and Donald R. van Deventer, “Inside the Kamakura Book of Yields: A Pictorial History of 50 Years of U.S. Treasury Forward Rates,” Kamakura Corporation memorandum, September 13, 2011.

Dickler, Daniel T. and Donald R. van Deventer, “Inside the Kamakura Book of Yields: An Analysis of 50 Years of Daily U.S. Treasury Forward Rates,” Kamakura blog, www.kamakuraco.com, September 14, 2011.

Dicker, Daniel T., Robert A. Jarrow and Donald R. van Deventer, “Inside the Kamakura Book of Yields: A Pictorial History of 50 Years of U.S. Treasury Zero Coupon Bond Yields,” Kamakura Corporation memorandum, September 26, 2011.

Dickler, Daniel T. and Donald R. van Deventer, “Inside the Kamakura Book of Yields: An Analysis of 50 Years of Daily U.S. Treasury Zero Coupon Bond Yields,” Kamakura blog, www.kamakuraco.com, September 26, 2011.

Dicker, Daniel T., Robert A. Jarrow and Donald R. van Deventer, “Inside the Kamakura Book of Yields: A Pictorial History of 50 Years of U.S. Treasury Par Coupon Bond Yields,” Kamakura Corporation memorandum, October 5, 2011.

Dickler, Daniel T. and Donald R. van Deventer, “Inside the Kamakura Book of Yields: An Analysis of 50 Years of Daily U.S. Par Coupon Bond Yields,” Kamakura blog, www.kamakuraco.com, October 6, 2011.

van Deventer, Donald R. “Pitfalls in Asset and Liability Management: One Factor Term Structure Models,” Kamakura blog, www.kamakuraco.com, November 7, 2011. Reprinted in Bank Asset and Liability Management Newsletter, January, 2012.

van Deventer, Donald R. “Pitfalls in Asset and Liability Management: One Factor Term Structure Models and the Libor-Swap Curve,” Kamakura blog, www.kamakuraco.com, November 23, 2011. Reprinted in Bank Asset and Liability Management Newsletter, February, 2012.

van Deventer, Donald R. “Stress Testing And Interest Rate Risk Models: How Many Risk Factors Are Necessary?” March 6, 2014, www.SeekingAlpha.com.

van Deventer, Donald R. “Stress Testing And Interest Rate Risk Models: A Multi-Factor Stress Testing Example,” March 18, 2014, www.SeekingAlpha.com.

van Deventer, Donald R. “Credit Spreads And Default Probabilities: A Simple Model Validation Example,” August 7, 2014, www.SeekingAlpha.com.

van Deventer, Donald R. “BAC: Best Practice Model Validation For Fed Stress-Testing, Value At Risk And Credit VAR,” October 21, 2014, www.SeekingAlpha.com.

van Deventer, Donald R. “Best Practice Model Validation for Stress-Testing Under the Fed’s CCAR 2015 Test Regime,” Kamakura Blog, www.kamakuraco.com, November 12, 2014.

van Deventer, Donald R. “Stress Testing: The Use and Abuse of “Intuitive Signs” on Credit Model Coefficients,” Kamakura Blog, www.kamakuraco.com, November 14, 2014. Forthcoming in Bank Asset and Liability Management Newsletter.

van Deventer, Donald R. “Stress Testing: The Use and Abuse of Lagged Default Probabilities in “Forbidden” Credit Models,” Kamakura Blog, www.kamakuraco.com, December 2, 2014.

**Appendix A: Simulated Values of the 1 Quarter Zero Coupon U.S. Treasury Yield**

**Appendix B: Simulated Values of the 1 Year Zero Coupon U.S. Treasury Yield**