This note addresses a simple but extremely important point: how big are the simulation errors from a one factor model for the term structure of interest rates, compared to a best practice multifactor model? The answer is also very simple: the errors are very large.
Background on Multiple Factor Term Structure Models
Heath, Jarrow and Morton (“HJM”) provided the framework for modern “no arbitrage” multifactor term structure models in the late 1980s. The main publication summarizing their work was published in Econometrica in 1992. Other related publications are given in the reference section. This framework provides the analytical framework for simulating the term structure of interest rates with whatever number of factors that are necessary for “accuracy.” What does accuracy mean in the context of interest rate risk in the banking book and asset and liability management (“ALM”)? It means a number of things:

A Monte Carlo simulation of interest rates, when used to value the traded securities underlying the yield curve at initial time t, will exactly match their observable prices. The restrictions on movements in future interest rates to ensure this “no arbitrage” restriction prevails is at the heart of the HJM analysis.

The distribution of interest rates for a long time horizon should produce a distribution that is (a) consistent with historical experience and (b) consistent with likely future experience.

The statement above, in particular, means that the negative rates that have prevailed in so many markets over the last few years are a real possibility and should not be ruled out. Jarrow and van Deventer discuss model validation of interest rate models in a world with negative interest rates in an article forthcoming in the Journal of Risk Management in Financial Institutions.
Model Error Measurement and Steps in Model Validation for Interest Rate Risk in the Banking Book
Step 1 in model validation is to be aware of the lessons of history. The graph below shows the distribution of daily 1 year government securities yields in the United States (since 1962), Japan (since 1974), and Canada (since 1949.
The implications of this graph are clear:
1A. Interest rates can be negative. This is inconsistent with these one factor term structure models which don’t allow for negative rates:
 Black, Derman and Toy
 Black and Karasinski
 Cox, Ingersoll and Ross
1B. Interest rates are not normally distributed. This is inconsistent with these one factor term structure models:
 Ho and Lee
 Vasicek
 Extended Vasicek/Hull and White
1C. Interest rates near zero are very common and represent the mode of a realistic interest rate simulation.
With one graph alone, we can see starkly that nearly the full list of commonly used one factor term structure models are inconsistent with history in a significant way.
Step 2 in model validation is to determine how many risk factors are necessary to model the most important yield curves with a high degree of accuracy. A recent paper from Kamakura Corporation (“ Stress testing and Interest Rate Risk Models: How Many Factors are Necessary ?” March 5, 2014) outlines the proper procedure to be followed and concludes that 9 factors are needed for accurate quarterly modeling of the U.S. Treasury yield curve.
Step 3 is a critical step and our primary focus in this note. What are the errors in the institution’s current modeling assumptions, when compared with a multifactor best practice HJM interest rate model?
Measuring Model Error
Although multifactor interest rate modeling has been available in enterprise risk management systems like Kamakura Risk Manager (“KRM”) for almost 20 years, it is still very common for major financial institutions to use one factor models. Let’s assume that the bank insists that negative rates be a possibility in the simulation, which rules out the 3 models listed above as candidates. Let us further assume that the bank selects the following modeling environment for the U.S. Treasury curve:
Number of factors: 1 factor, the short term rate of interest
Model: Extended Vasicek/Hull and White
Interest rate volatility: Varies by maturity and estimated from January 1, 1962 through December 31, 2014. There are 119 quarterly
forward rates underlying the 30 year yield curve, so there are 119 interest rate volatility parameters in the model.
Modeling horizon: 30 years
We seek to compare the “common practice” model with a best practice model with these characteristics:
Number of factors: 9 factors, spanning the key yield curve maturities
Model: Heath, Jarrow and Morton with ratedependent volatility
Interest rate volatility: Estimated from January 1, 1962 through December 31, 2014.
The volatilities are provided by Kamakura Risk Information Services.
Modeling horizon: 30 years
We chose a starting U.S. Treasury yield curve on June 12, 2015, as described on the website of the U.S. Department of the Treasury.
The initial zero coupon yield curves associated with the par coupon Treasury yields are the same for both models:
We simulate forward quarterly for 30 years, using 100,000 scenarios for each model. A detailed discussion of the Heath, Jarrow and Morton forecast is available here . How do the outcomes compare? We turn to that now.
The Distribution of the 1 Year Zero Coupon Yield
Appendix A gives the distribution of the 3 month zero coupon Treasury yield at various points over 30 years. Appendix B provides the same information for the 5 year zero coupon Treasury yield. The same analysis can be done for any U.S. Treasury security. In this section, we focus on the 1 year zero coupon Treasury yield for compatibility with our graph of historical experience in 3 countries.
In our simulation, the first quarter is not random, because the initial rates are observable. The first set of random rates will prevail in the three month period from month 4 to month 6, which we label Quarter 1. The distribution of the 1 year zero coupon yield for both models is shown here:
Even with just one period’s worth of simulation, the problems with the extended Vasicek model are apparent:
 The variability of simulated rates from the extended Vasicek model is too great
 The number of outcomes that are negative in the extended Vasicek model is too great
 We speculate that the historical levels of interest rate volatility (from 1962 to 2014) are the reason for this disturbing outcome, but we are doing a 30 year simulation. To restrict volatility to very recent experience is inconsistent with the modeling horizon.
The 9 factor HJM model, by comparison, features a compact outcome that is very reasonable. There is a small probability of negative rates, and there is a cluster of rates near zero.
Quarter 3, the End of the First Year
By the end of the first year, we reach quarter number 3. The simulations that result are shown here:
The extended Vasicek simulation is seriously off track, with interest rates that are unrealistically negative by any historical measure. The 9 factor HJM model shows a modest and realistic probability of negative rates.
Quarter 7, the End of the Second Year
The pattern continues at the end of the second year. We can see just the beginning of a left “shoulder” on the HJM 9 factor model near zero. The percentage of rates that are negative in the HJM model remains small. The extended Vasicek model’s problems continue.
IMPORTANT NOTE: Observe that the economic capital needs that would be simulated by the extended Vasicek model would be much higher than under the HJM model given the respective parameter assumptions.
Quarter 11, the End of Year Three
By the end of year 3, the extended Vasicek model continues to produce normally distributed rates that are very, very negative. The left shoulder on the HJM distribution, indicating a bunching of rates near zero, is becoming more obvious.
Quarter 19, the End of Year 5
By now, the realistic skew of the HJM model is becoming very reminiscent of the actual historical rates of the 3 countries in the introduction. Rates skew to the right and a bunching of rates near zero is apparent, along with a small percentage of negative rates. The problems with the extended Vasicek model are not going away.
Quarter 27, the End of Year 7
The bunching of rates near and just below zero is now clear in the HJM simulation, and the distribution is bimodal. The consistency with actual history is now quite obvious.
Quarter 39, the End of Year 10
The resemblance of the simulated HJM 1 year zero coupon yield to actual history is shaping up nicely. Meanwhile, the extended Vasicek outcomes now range across more than 25 percentage points, with a significant number of negative rates below minus 5%.
Quarter 79, the End of Year 20
The bunching of rates near zero, predominately above zero now, in the HJM model is very distinct. A significant number of scenarios fall below minus 5% and even minus 10% in the extended Vasicek model.
Quarter 119, the End of Year 30
By the end of year 30, the overwhelming majority of the HJM scenarios produce 1 year zero coupon yields between zero and 5%. There is a chance of negative rates, but it is a relatively small percentage of the 100,000 scenarios.
The extended Vasicek outcomes are much more extreme. The majority of outcomes are outside of the 0 to 5% range, with a significant minority of scenarios well below zero.
Can the 1 Factor Model Be Fixed? Conclusions
After excluding models that cannot produce negative rates, the extended Vasicek model is the sole remaining candidate. The assumption that interest rate volatility remains constant at its 19622014 levels produces an unrealistic set of outcomes. As a challenger model, the 9 factor Heath, Jarrow and Morton model is the obvious winner.
Three questions may arise in the reader’s mind. We give short answers here for now.
Can we “fix” the extended Vasicek model by dropping the negative rate scenarios?
One could do this, but then the bonds underlying the starting yield curve will no longer be correctly priced. This violates “no arbitrage” restrictions and should result in a model validation failure.
Can we “fix” the extended Vasicek model by using interest rate volatility observed more recently, say the last five years?
Again, one could do this, but what is the justification for using 5 years of history to set parameters for 30 years? One might as well just make up the numbers. This is another surefire highway to a model validation failure.
Can we “fix” a one factor model by combining models or switching from one model to another as rates move around?
From an analytical point of view, this is a bit like putting lipstick on a pig. The pig may be slightly more attractive, but it remains a pig. Hedging interest rate risk using a one factor model, no matter what the model is, will produce larger hedging errors by far than a model with a more realistic number of factors.
The right things to do for maximum accuracy and realism fall into two categories:
Use the number of factors indicated by proper statistical procedures. In most countries, depending on the periodicity of the simulation (monthly, quarterly, semiannual or annual), this will mean 9, 10 or more factors. This not a large number compared to other financial instruments. Most equity factor models in major fund management companies use 20 to 40 factors.
Derive the formulas by which interest rate volatility for each of the n factors varies over time. Clearly, it is wrong to assume that interest rate volatility is constant over time as we have seen in today’s simulation. Proper statistical procedures will make these formulas clear.
Further Reading for the Technically Inclined Reader
References for random interest rate modeling are given here:
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. 419440.
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. 77105.
Heath, David, Robert A. Jarrow and Andrew Morton, "Easier Done than Said", RISK Magazine, October, 1992.
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. 217237.
Jarrow, Robert A. “Amin and Jarrow with Defaults,” Kamakura Corporation and Cornell University Working Paper, March 18, 2013.
The impact of credit risk on securities returns is discussed in these papers:
Campbell, John Y., Jens Hilscher and Jan Szilagyi, "In Search of Distress Risk," Journal of Finance, December 2008, pp. 28992939.
Campbell, John Y., Jens Hilscher and Jan Szilagyi, "Predicting Financial Distress and the Performance of Distressed Stocks," Journal of Investment Management, 2011, pp. 121.
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 23212350.
Elton, Edwin J., Martin J. Gruber, Deepak Agrawal, and Christopher Mann, “Explaining the Rate Spread on Corporate Bonds,” Journal of Finance, February 2001, pp. 247277.
The valuation of bank deposits is explained in these papers:
Jarrow, Robert, Tibor Janosi and Ferdinando Zullo. “An Empirical Analysis of the Jarrowvan Deventer Model for Valuing NonMaturity Deposits,” The Journal of Derivatives, Fall 1999, pp. 831.
Jarrow, Robert and Donald R. van Deventer, "Power Swaps: Disease or Cure?" RISK magazine, February 1996.
Jarrow, Robert and Donald R. van Deventer, "The ArbitrageFree Valuation and Hedging of Demand Deposits and Credit Card Loans," Journal of Banking and Finance, March 1998, pp. 249272.
Appendix A: Simulation of 3 Month Zero Coupon Treasury Curve
For the curious reader, we present the simulation results in this appendix without the distraction of commentary that would be similar to the observations made above.
Appendix B: Simulation Results for the 5 Year Zero Coupon Yield
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