One Thousand Scenarios For The U.S. Treasury Curve

by: Donald van Deventer


23,117 daily observations on 1 year government yields in the USA and Japan show that rates bunch near the zero yield level and can go negative.

We benchmark a 1,000 scenario simulation in today's yields so that our Monte Carlo simulation correctly prices the entire current Treasury curve.

We present 10 and 30 year comparisons of forward rates with expected rates and the "risk neutral" rates used for security valuation.

Many readers have asked us to do more than extract forward rates from the U.S. Treasury (NYSEARCA:TLT), (NYSEARCA:TBT), (NYSEARCA:SHY), (NYSEARCA:IEF) curve in our weekly interest rate outlook. Beginning this week, we are benchmarking our interest rate outlook in today's U.S. Treasury curve and simulating forward. The simulations are based on historical movements in the U.S. Treasury curve from 1962 to the present.

We simulate U.S. Treasury yields in two closely related ways. The first method is the so-called "risk neutral" yields that are used to value any securities that are tied to the U.S. Treasury curve. The 1,000 scenarios we generate price today's Treasuries at their exact market prices. Since these risk neutral scenarios contain a risk premium, over very long time horizons they are higher than the actual rates that market participants expect to come about.

We'll expand our commentary and go into more detail in coming weeks. In the meantime, we are pleased to introduce 1,000 views of the future with some historical data on past movements in interest rates.

Lessons from History

A necessary condition for the accurate simulation of future interest rates is the consistency of the simulation approach with actual historical data. A simulation approach that cannot replicate history successfully is highly unlikely to produce an accurate forward looking simulation. When the coefficients of the term structure model are assumed constant, zero coupon bond yields will have a normal distribution. The extended Vasicek or Hull and White model is perhaps the best known in its class. When we examine the actual distribution of 13,280 daily observations on the 1 year U.S. Treasury rate over the period from 1962 through March 10, 2015, it is obvious by inspection that historical interest rates in the United are not normally distributed.

A review of 9,837 days of Japanese government bond yields shows a similar pattern. We graph the 1 year Japanese government bond yield from 1974 through March 10, 2015 here:

The graph also shows that the 1 year Japanese government bond yield was negative for 25 of the 9,837 observations, 0.254% of the sample. The next graph combines the data from the United States and Japan for a total of 23,117 observations.

History makes it clear that (1) yields are not normally distributed, (2) yields often cluster near zero, and (3) they can go negative. We incorporate that in our simulations with U.S. Treasury yield curve of March 11, 2015 as a starting point.

Current U.S. Treasury Zero Coupon Bond Yields and Forward Rates

The current zero coupon bond yields and forward rates implied by today's U.S. Treasury curve are given in this graph:

A 1,000 Scenario Simulation

Today's analysis is based on a simulation of 1,000 scenarios using a 9 factor term structure model based on the analysis of Heath, Jarrow and Morton ("HJM"). The HJM model allow us to readily use multiple factors to drive the yield curve and allows us to use realistic assumptions about the volatility of interest rates along the 30 year length of the U.S. Treasury curve. We use 9 points on the yield curve as the risk factors driving the entire curve. We use a separate econometric relationship for each quarterly segment of the yield curve. The econometrics are done using quarterly data from the U.S. Department of the Treasury from 1962 through the end of 2014. We use a rate-dependent volatility function for the 9 risk factors. The simulation is done in such a way that all current Treasury bonds are priced correctly by our 1,000 scenario Monte Carlo simulation.

The Probability Distribution of U.S. Treasury Yields for the Next 10 Years

In this section, we report 40 quarters of simulated results for U.S. Treasury zero coupon bond yields at these maturities:

  1. 3 months
  2. 1 year
  3. 5 years

For each maturity, we report the following statistics for the simulated "risk neutral" yields in the 1,000 scenarios:

  1. The matched maturity forward rate at each maturity
  2. The lowest rate simulated
  3. The 1st percentile yield
  4. The 10th percentile yield
  5. The 25th percentile yield
  6. The 50th percentile yield
  7. The average yield
  8. The 75th percentile yield
  9. The 90th percentile yield
  10. The 99th percentile yield
  11. The highest yield simulated
  12. The average "empirical" or expected actual yield

The 10 Year Outlook for the 3 Month U.S. Treasury Yield

The simulated results for 3 month U.S. Treasury zero coupon yields are given here:

The 10 Year Outlook for the 1 Year U.S. Treasury Yield

The simulated results for 1 year U.S. Treasury zero coupon yields are given here:

The 10 Year Outlook for the 5 Year U.S. Treasury Yield

The simulated results for 5 year U.S. Treasury zero coupon yields are given here:

A 30 Year View of Rate Movements

In this section, we present the same percentiles graphically for the 3 month, 1 year, and 5 year U.S. Treasury zero coupon bond yields.

Background Information on Input Data and Smoothing

The Federal Reserve H15 statistical release is the source of most of the data used in this analysis. The Kamakura approach to forward rate derivation and the maximum smoothness forward rate approach to yield curve smoothing is detailed in Chapter 5 of van Deventer, Imai and Mesler (2013).

Younger readers may not be familiar with the dramatic movements in interest rates that have occurred in modern U.S. economic history. Older readers were once familiar with these rate movements, but they may have forgotten them. Kamakura Corporation has provided a video that shows the daily movements in forward rates from 1962 through August 2011. To view the video, follow this link.

Disclosure: The author has no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

The author wrote this article themselves, and it expresses their own opinions. The author is not receiving compensation for it (other than from Seeking Alpha). The author has no business relationship with any company whose stock is mentioned in this article.