That step is easy to describe: a Monte Carlo simulation of the risk-free curve should produce values that exactly match the observable prices of securities traded in that risk-free market. In the U.S. context, a Monte Carlo simulation of the U.S. Treasury curve should produce valuations that exactly match the securities prices, either on coupon-bearing bonds or derived zero coupon bond prices, that are used as the benchmark yield curve at time zero. In this note, we describe this essential element of interest rate model validation that is in daily use at Kamakura Corporation.
A Worked Example Using the U.S. Treasury Curve of February 10, 2015
In previous notes, we have described the steps in model validation for interest rate risk management in detail. In both 2009 and 2014, we outlined how important it is for the interest rate risk manager to correctly measure and implement an approach to interest rate simulation that contains a realistic and accurate number of interest rate risk factors. In a recent note, we showed that at least 9 factors are needed to accurate model the movement of the forward rates along the 30 year length of the U.S. Treasury curve.
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 volatilities used are rich and complex. 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 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. 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 of the model, i.e. whether or not the model correctly prices the observable zero coupon bond prices from the U.S. Treasury market on February 10, 2015 as reported by the U.S. Department of the Treasury.
Two Invalid Model Validations
Before summarizing the results of the simulation, we need to point out two common approaches to valuation that are not valid tests of the accuracy of our Monte Carlo simulation parameters. The first is the common bond valuation formula
This formula says that a bond’s value is the sum of n terms, each of which is the amount of the cash flow at time t=t_{i }C_{i} and the present value factor for cash flow on that date. This formula is not a test of our Monte Carlo simulation parameters, obviously, because it doesn’t use those parameters at all. It only uses facts observable at time zero, the n cash flows on the bond and the zero coupon bond prices for each of those maturities. In fact, this formula is only valid if our Monte Carlo simulation is a valid model.
Another common but invalid approach to valuation asserts that the value of n different cash flows at a single time t_{i} is their expected value times the present value factor for time t_{i}.
This formula sounds reasonable at first hearing, but it ignores a critical aspect of interest rate risk analysis and valuation: the n simulated values for cash flow at time t_{i }may be correlated with the level of interest rates at time t_{i }(and earlier times). That correlation should and does change valuation, but the formula above ignores this fact. The formula above is correct only if the cash flow C_{i} is constant or uncorrelated with interest rates. For this reason, term structure model researchers beginning with Merton and Vasicek in the 1970s have all used a different discount factor for cash flow, which we turn to next.
The Necessary Criterion for a Valid Model
In this paragraph we follow Jarrow and Turnbull (2000), second edition, chapters 15 and 16, and Heath, Jarrow and Morton (1992). To properly discount our n simulated cash flows at time t_{i}, we need to know the cumulated value of a money market fund B(j,k) as of time t_{i}, which is j time steps from time 0 in scenario k. If the unannualized spot rate of interest at time zero is r(0,k), then the value of the money market fund B(0,k) is 1+r(0,k). In scenario k, as of time 1 with maturity at time 2, the spot rate may be r(1,k) and the money market fund is now worth B(1,k)=[1+r(0,k)][1+r(1,k)]. The general formula for the money market fund’s value as of time step j in scenario k is
The proper valuation of the n simulated values of cash flow C_{i }at time t_{i}, j time steps from time 0, is given by this formula:
Risk-neutral value, as used throughout the literature on the value of interest-sensitive securities, is the average of the n values of cash flow C _{m} divided by the value of the money market fund B in scenario m. Our model validation procedure is simple. If there is an observable zero coupon bond price with maturity at time t_{i}, j time steps from time 0, then the model is only valid if the simulated Monte Carlo value of C_{m} =1 (the payoff at maturity on the zero coupon bond) exactly equals the observable zero coupon bond price:
In our example, we have 120 observable zero coupon bond prices, one for each quarterly maturity of the U.S. Treasury curve out to 30 years. The model simulated is “valid” if these 120 zero coupon bonds are priced correctly. If they are not priced correctly, within the accuracy limits of computer science, then the model is invalid and technically wrong. We remind the reader that our test for “validity” does not certify a model as “best.” Instead, it certifies the model as “not wrong, and possibly but not necessarily best.” That distinction is important.
We now summarize our simulation results and perform the model validation.
Simulation Results
Using the term structure model above, Kamakura Risk Information Services produced 50,000 risk neutral scenarios for yield curve movements from the close of business U.S. Treasury curve on February 10, 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 starting yields on February 10, 2015 were reported as follows by the U.S. Treasury:
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 50,000 simulated paths of the 1 quarter, 1 year (4 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 a small number of parameters in the model. The second graph shows the same percentiles over 30 years for the 1 year zero coupon bond yield:
The third graph shows the percentile distributions of the 50,000 simulated paths for the 5 year zero coupon Treasury yield.
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
- 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 distribution after 5 years takes on this shape:
The graph below shows the distribution of the 5 year zero yield after 10 years.
The final graph shows the simulated distribution of 50,000 paths of the 5 year zero yield after 30 years:
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 -0.00000009. The mean absolute pricing error was 0.00000381. The results are summarized in these tables:
Conclusions
We have shown the procedures for essential model validation for an interest rate risk simulation driven by a rich set of assumptions: 9 risk factors, both constant and rate-dependent volatility, variation in the volatility formulas by rate level, and allowance of negative interest rates. Even with this level of complexity over 50,000 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. 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 Quarter Zero Coupon U.S. Treasury Yield