In our blog of November 7, 2011, we showed that the one–factor term structure models in wide use in the financial services business for interest rate management analytics were consistent with actual daily movements of the U.S. Treasury curve less than 38% of 12,286 business days since 1962. In this blog, we repeat the analysis on the Libor-swap curve and reach an even more devastating conclusion: once the full interest rate swap curve came into view in 1988, daily yield curve shifts were consistent with one factor models less than 8% of the time. This blog explains how we arrived at those conclusions.

The Kamakura blog “Pitfalls in Asset and Liability Management: One Factor Term Structure Models” is available at this link:

In that blog, we followed the approach of Jarrow, van Deventer and Wang (2003) regarding model validation. In that paper, Jarrow, van Deventer and Wang tested the Merton model of risky debt by examining the validity of the Merton model of risky debt by determining whether the implications of the model were in fact true. Their Merton model test was a “non-parametric test,” i.e. a test of implications which would prevail for any parameter values. In our November 7 blog, we did a similar non-parametric test of one factor term structure models. We described the basis for the test on November 7 as follows:

“Among one factor term structure models, for any set of parameters, there is almost always one characteristic in common. The majority of one factor models make all interest rates positively correlated (which implies the time-dependent interest rate volatility sigma(T) is greater than 0). We impose this condition on a one factor model. The non-random time-dependent drift(T) in rates is small and, if risk premia are consistent and positive (for bonds) across all maturities T, then the drift will be positive as in the Heath Jarrow and Morton (1992) drift condition. With more than 200 business days per year, the daily drift is almost always between plus one and minus one basis point. Hence, one might want to exclude from consideration yield curve changes that are small, so the effect of the drift is removed. We ignore the impact of drift in what follows.”

We then examined the extent to which U.S. Treasury coupon bearing bond yields, zero coupon bond yields, and forward rates were consistent with this common characteristic of single factor term structure models. To what extent did all rates rise, fall, or remain constant together? The answer is summarized in this table for the U.S. Treasury market:

One factor term structure model yield movements occurred only 37.7% of the time with the par coupon bond yields quoted on the Federal Reserve’s H15 statistical release. Using smoothed yield curves and the associated monthly zero coupon bond yields and forward rates, the conclusions are even more dramatic. In the case of zero coupon bond yields, movements were consistent with one factor term structure models only 24.8% of the 12,386 business days examined. In the case of forward rates, movements were consistent with one factor models only 5.7% of the time.

We now use U.S. dollar Libor rates and interest rate swap rates to do a similar analysis on the heavily used Libor-swap curve.

**Data Regime for Libor-Swap Curve **

Data was loaded for Eurodollar deposit rates reported by the Federal Reserve on its H15 statistical release beginning on January 4, 1971. Note that these rates are slightly different from “official” Libor rates reported by the British Bankers Association on www.bbalibor.com. We used U.S. dollar interest rate swap yields reported by the Federal Reserve on the H15 release. The original source of the data is that collected by the broker ICAP on behalf of ISDA. This data series begins on July 3, 2000. We supplement the H15 data with interest rate swap data from Bloomberg, which for the most part extends back to November 1, 1988. We have four distinct data regimes since 1971:

Using only this raw data, with no yield curve smoothing analytics employed, we asked this question: On what percent of the business days were yield curve shifts all positive, all negative, or all zero? We report the results in the next section.

**Consistency with One Factor Term Structure Models in the Libor-Swap Market**

The results vary by the data regime studied. Of the 10,664 days on which some data was available, we eliminated all dates where the data was less than the full data regime maturity spectrum and those dates which represented the first day of a new data regime, which would distort the reported shift amounts. After deleting these data points, we are left with 10,150 business days from January 4, 1971 to November 17, 2011. From January 4, 1971 to October 31, 1988, there were only three observable points on the Libor-swap yield curve, at 1, 3 and 6 months. In large part because of the paucity of data points, it was more likely that all rates moved up together, down together, or remained unchanged today. This happened 51.6% of the time during this data regime, so one factor term structure models were consistent with actual rate movements a narrow majority of the time. For regimes with 9, 11, or 12 observable points on the yield curve, however, the percentage of business days that were consistent with one factor term structure models was only 7.2%. Over the full sample, including (perhaps inappropriately) the long period of time where there were only three observable points on the yield curve, the consistency ratio was 26.9%.

As in the case of the U.S. Treasury yield curve, we conclude that one factor term structure models are grossly inaccurate approximations to the true historical movements of the Libor-swap curve.

**Implications of Results**

As we noted on November 7, there are a number of very serious errors that can result from an interest rate risk and asset and liability management process that relies solely on the assumption that one factor term structure models are an accurate description of potential yield movements:

- Measured interest rate risk will be incorrect, and the degree of the error will not be known. Using the data above, 92.8% of the time actual yield curves will show a twist, but the modeled yield curve shifts will never show a twist.
- Hedging using the duration or one factor term structure model approach assumes that interest rate risk of one position (or portfolio) can be completely eliminated with the proper short position in one instrument with a different maturity. The duration/one factor term structure model approach assumes that if interest rates on the first position rise, interest rates will rise on the second position as well so “going short” is the right hedging direction. The data above shows that on 92.8% of the days from November 1, 1988 to November 17, 2011, this “same direction” assumption was potentially false (some maturities will show same direction changes and some will show opposite direction changes) and the hedge could actually ADD to risk, not reduce risk.
- All estimates of prepayments and interest rate-driven defaults will be measured inaccurately
- Economic capital will be measured inaccurately
- Liquidity risk will be measured inadequately
- Non-maturity deposit levels will be projected inaccurately

For both U.S. Treasury data and Libor-swap data, this is an extremely serious list of deficiencies. The only remedy is to move as soon as possible to a more general N-factor model of interest rate movements. This should be done using the best available econometric results like those from Kamakura Risk Information Services and a simulation system like Kamakura Risk Manager. Academic assumptions about the stochastic processes have been too simple to be realistic. Jarrow (2009) notes,

“Which forward rate curve evolutions (HJM volatility specifications) fit markets best? The literature, for analytic convenience, has favored the affine class but with disappointing results. More general evolutions, but with more complex computational demands, need to be studied. How many factors are needed in the term structure evolution? One or two factors are commonly used, but the evidence suggests three or four are needed to accurately price exotic interest rate derivatives.”

As noted in our November 7 blog, this view is shared by the Basel Committee on Banking Supervision. In its December 31, 2010 *Revisions to the Basel II Market Risk Framework*, the Committee states its requirements clearly on page 12:

“For material exposures to interest rate movements in the major currencies and markets, banks must model the yield curve using a minimum of six risk factors.”

We have just emerged (perhaps briefly) from a credit crisis which stemmed to a large degree from the use of credit models based on assumptions which were known at the time to be false. This blog and our November 7 blog present evidence that interest rate risk managers today are relying heavily on interest rate risk models and risk systems that are based on assumptions that are known to be false. Let us hope that history does not repeat itself before the industry moves to more realistic models of interest rate risk.

**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.

Black, Fischer, “Interest Rates as Options,” *Journal of Finance* (December), pp. 1371-1377, 1995.

Black, Fischer, E. Derman, W. Toy, “A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options,” *Financial Analysts Journal*, pp. 33-39, 1990.

Black, Fischer and Piotr Karasinski, “Bond and Option Pricing when Short Rates are Lognormal,” *Financial Analysts Journal*, pp. 52-59, 1991.

Cox, John C., Jonathan E. Ingersoll, Jr. and Stephen A. Ross, “An Analysis of Variable Rate Loan Contracts,” *Journal of Finance*, pp. 389-403, 1980.

Cox, John C., Jonathan E. Ingersoll, Jr., and Stephen A. Ross, “A Theory of the Term Structure of Interest Rates,” *Econometrica* 53, 385-407, 1985.

Dai, Qiang and Kenneth J. Singleton, “Specification Analysis of Affine Term Structure Models,” *The Journal of Finance*, Volume LV, Number 5, October 2000.

Dai, Qiang and Kenneth J. Singleton, “Term Structure Dynamics in Theory and Reality,” *The Review of Financial Studies*, Volume 16, Number 3, Fall 2003.

Dickler, 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 memorandum, September 13, 2011.

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

Dickler, Daniel T., Robert A. Jarrow and Donald R. van Deventer, “Inside the Kamakura Book of Yields, Volume III: A Pictorial History of 50 Years of U.S. Treasury Par Coupon Bond Yields,” Kamakura 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. Treasury Forward Rates,” Kamakura blog, www.kamakuraco.com, September 14, 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.

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.

Duffie, Darrell and Rui Kan, “Multi-factor Term Structure Models,” *Philosophical Transactions: Physical Sciences and Engineering*, Volume 347, Number 1684, Mathematical Models in Finance, June 1994.

Duffie, Darrell and Kenneth J. Singleton, “An Econometric Model of the Term Structure of Interest-Rate Swap Yields,” *The Journal of Finance*, Volume LII, Number 4, September 1997.

Duffie, Darrell and Kenneth J. Singleton, “Modeling Term Structures of Defaultable Bonds,” *The Review of Financial Studies*, Volume 12, Number 4, 1999.

Heath, David, Robert A. Jarrow and Andrew Morton, “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation,” *Econometrica*, 60(1), January 1992.

Ho, Thomas S. Y. and Sang-Bin Lee, “Term Structure Movements and Pricing Interest Rate Contingent Claims,” *Journal of Finance* 41, 1011-1029, 1986.

Hull, John and Alan White, “One-Factor Interest-Rate Models and the Valuation of Interest-Rate Derivative Securities,” *Journal of Financial and Quantitative Analysis* 28, 235-254, 1993.

Jamshidian, Farshid, “An Exact Bond Option Formula,” *Journal of Finance* 44, pp. 205-209, March 1989.

Jarrow, Robert A., “The Term Structure of Interest Rates,” *Annual Review of Financial Economics*, 1, 2009.

Jarrow, Robert A., Donald R. van Deventer, and Xiaoming Wang, “A Robust Test of Merton’s Structural Model for Credit Risk,” *Journal of Risk*, 6 (1), 2003.

Macaulay, Frederick R. *Some Theoretical Problems Suggested by Movements of Interest Rates, Bond Yields, and Stock Prices in the United States since 1856*. New York, Columbia University Press,1938.

Merton, Robert C. “A Dynamic General Equilibrium Model of the Asset Market and Its Application to the Pricing of the Capital Structure of the Firm,” Working Paper No. 497-70, A. P. Sloan School of Management, Massachusetts Institute of Technology, 1970. Reproduced as Chapter 11 in Robert C. Merton, Continuous Time Finance, Blackwell Publishers, Cambridge, Massachusetts, 1993.

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

van Deventer, Donald R. and Kenji Imai, *Financial Risk Analytics: A Term Structure Model Approach for Banking, Insurance, and Investment Management*, Irwin Professional Publishing, Chicago, 1997.

van Deventer, Donald R., Kenji Imai, and Mark Mesler, *Advanced Financial Risk Management*, John Wiley & Sons, 2004. Translated into modern Chinese and published by China Renmin University Press, Beijing, 2007.

Vasicek, Oldrich A., “An Equilibrium Characterization of the Term Structure,” *Journal of Financial Economics* 5, pp. 177-188, 1977.

Donald R. van Deventer

Kamakura Corporation

Honolulu, Hawaii

November 23, 2011

© Copyright 2011 by Donald R. van Deventer, All Rights Reserved.