KamakuraCorporation-USTreasury3Month.xlsx

Using the coin flip as an example, you can find two kinds of economic forecasts among commentators. The first kind of forecast has a projection like “heads.” This forecasting style is for readers who like to think that someone can actually forecast the future perfectly. The second kind of forecast has a projection like “there is a 50 percent chance of ‘heads’ and a 50 percent chance of ‘tails.’” This note is the second type of forecast for investors and interest rate risk analysts who want to know the best possible probability distribution for the three-month Treasury bill rate for (almost) 30 years forward (of course the full yield curve is available as well).

We start with the actual U.S. Treasury yield curve on Friday, July 7. We simulate it forward based on historical daily movements in the U.S. Treasury curve from January 2, 1962, through June 30, 2017. We use best practice interest rate risk technology like that required by the Bank for International Settlements in its regulations concerning “Interest Rate Risk in the Banking Book” and “The Fundamental Review of the Trading Book.”

The table below contains the full 30-year forecast for the three-month Treasury bill rate, with results stated quarterly. The table contains these elements:

The lowest rate simulated for that date among the 250,000 scenarios

The first percentile simulated rate, that is (when rates are ordered from lowest to highest), the 2500th lowest rate among the 250,000 scenarios.

The 10th percentile simulated rate (the 25,000th lowest rate),

the 25th percentile simulated rate (the 62,500th lowest rate),

the 50th percentile simulated rate (the 125,000th lowest rate),

the average simulated rate,

the 75th percentile simulated rate,

the 90th percentile simulated rate,

the 99th percentile simulated rate,

the highest rate simulated for that date,

the three-month forward rate maturing on that date as of the forecast date, July 7, 2017.

Let’s look at July 7, 2027, in the table, 10 years from now.

On July 7, 2027, the simulation shows a range of simulated three-month Treasury bills from a low of -1.68% (yes, rates can be negative and are negative in many countries) to a high of 22.54%. The average of all of the simulated rates on July 7, 2027, was 2.26%, and the median or 50th percentile was 1.52%. Note that the average and the median are well below the 2.71% forward rate that prevailed on July 7, 2017, for maturity on July 7, 2027. **That means that investors will on average earn a “term premium” above and beyond rolling over three-month Treasury bills if they are willing to buy a fixed rate Treasury bond on July 7, 2017.** Note that the average is much higher than the median because the distribution of interest rates has a “long tail” on the right hand side of the distribution (the high side) as shown below for July 7, 2027:

The word “period” in the graph refers to quarterly periods. The probability distribution of future interest rates affects the value of almost every security, which is why saying more than “heads” is critical when looking at future interest rates.

The spreadsheet is attached to this article for your convenience. The technical details are summarized in the Appendix for those who are quantitatively inclined. Good luck with your investments.

**Technical Details**

The interest rate model used was a 10 factor “stochastic volatility” model of the Heath, Jarrow and Morton type. The description of the model is given in the references below (see Donald R. van Deventer, June 28, 2017). The parameters of the model were updated to span the historical period from January 2, 1962, through June 30, 2017. The simulated interest rates price Treasury bonds perfectly on July 7, 2017, with no errors.

The rates displayed in the table above are “empirical” or “real world” rates. The full set of empirical and “risk neutral” simulated rates (the latter are used for valuation of securities) are available by subscription from Kamakura Risk Information Services.

**References**

Adams, Kenneth J. and Donald R. van Deventer, "Fitting Yield Curves and Forward Rate Curves with Maximum Smoothness," *Journal of Fixed Income*, 1994, pp. 52-61.

Adrian, Tobias, Richard K. Crump and Emanuel Moench, “Pricing the Term Structure with Linear Regressions,” Federal Reserve Bank of New York, Staff Report 340, August 2008, revised August 2013.

Campbell, John Y, Andrew W. Lo, and A. Craig McKinley, *The Econometrics of Financial Markets*, Princeton University Press, 1997.

Gelman, Andrew and John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin, *Bayesian Data Analysis*, third edition, CRC Press, 2013.

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 A. *Modeling Fixed Income Securities and Interest Rate Options*, second edition, Stanford University Press, Stanford, 2002.

Jarrow, Robert A. and Stuart Turnbull, *Derivative Securities*, second edition, South-Western College Publishing, Cincinnati, 2000.

Jarrow, Robert A. and Donald R. van Deventer, “Monte Carlo Simulation in a Multi-Factor Heath, Jarrow and Morton Term Structure Model,” Kamakura Corporation

Technical Guide, Version 4.0, June 16, 2015

Jarrow, Robert A. and Donald R. van Deventer, “Parameter Estimation for

Heath, Jarrow and Morton Term Structure Models,” Kamakura Corporation

Technical Guide, Version 4.0, May 5, 2017.

Kim, Don H. and Jonathan H. Wright, “An Arbitrage-Free Three Factor Term Structure Model and the Recent Behavior of Long-Term Yields and Distant-Horizon Forward Rates,” Finance and Economics Discussion Series, Federal Reserve Board, 2005-33.

van Deventer, Donald R. “ A 10 Factor Heath, Jarrow and Morton Model for the U.S. Treasury Yield Curve, January 1962 to March 2017: Bayesian Model Validation Given Negative Rates in Japan,” Kamakura Corporation working paper, www.kamakuraco.com, June 28, 2017.

**Supporting Documents**

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.