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Don founded Kamakura Corporation in April 1990 and currently serves as its chairman and chief executive officer where he focuses on enterprise wide risk management and modern credit risk technology. His primary financial consulting and research interests involve the practical application of leading edge financial theory to solve critical financial risk management problems. Don was elected to the 50 member RISK Magazine Hall of Fame in 2002 for his work at Kamakura. Read More

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Kamakura Blog

Jun 1

Written by: Donald van Deventer
6/1/2015 1:08 AM 

The author wishes to thank Prof. Robert Jarrow for many years of helpful conversations on this important topic. We thank the staff of Kamakura Risk Information Services for the 50,000 scenarios used in this study.

We use 50,000 scenarios for the U.S. Treasury (TLT) yield curve to examine the outlook for Treasuries and the magnitude of the term premium (or riskpremium) above and beyond expected rate levels that is embedded in the yield curve. The U.S. Treasury yield curve from which simulations are initiated isthe May 29, 2015 yield curve as reported by the U.S. Department of the Treasury.

Current Yield Curve and Recent Yield Movements
The chart below lists current U.S. Treasury yields, recent changes, and recent highs and lows since January 1, 2015. We also list the all-time highs and lows and the most recent dates on which those highs and lows have occurred.


Movements in 1 month and 6 month U.S. Treasury yields since January 1, 2015 are shown here:


The movements in the curve can be summarized this way, where “recent” refers to changes since January 1, 2015:

  • The biggest yield change of the week was 0.03%, which occurred for the one year Treasury yield.
  • The smallest yield change of the week was -0.11%, which took place at the 30 year maturity.
  • The 1 month Treasury yield finished the week at 0.01%, a change of 0.00% since last week. This compares to the recent high and low values of 0.05% and 0%. The all-time high in the 1 month yield was 5.27% recorded on February 21, 2007. The all-time low was 0% on May 14, 2015.
  • The 1 year Treasury yield finished the week at 0.26%, a change of 0.03% since last week. This compares to the recent high and low values of 0.28% and 0.16%. The all-time high in the 1 year yield was 17.31% recorded on September 3, 1981. The all-time low was 0.08% on September 19, 2011.
  • The 5 year Treasury yield closed at 1.49%, a change of -0.08% since last week. This compares to the recent high and low values of 1.7% and 1.18%. The all-time high in the 5 year yield was 16.27% recorded on September 30, 1981. The all-time low was 0.56% on July 25, 2012.
  • The 10 year Treasury yield finished the week at 2.12%, a change of -0.09% since last week. This compares to the recent high and low values of 2.28% and 1.68%. The all-time high in the 10 year yield was 15.84% recorded on September 30, 1981. The all-time low was 1.43% on July 25, 2012.
  • The 30 year Treasury yield completed the week at 2.88%, a change of -0.11% since last week. This compares to the recent high and low values of 3.07% and 2.25%. The all-time high in the 30 year yield was 15.21% recorded on October 26, 1981. The all-time low was 2.25% on February 2, 2015.

Recent movements in 1 year, 10 year, and 30 year Treasury yields are shown in this graph:

The long-term history of rate movements from 1962 shows a large rise in rates followed by a large fall in rates, a caution to all who are supremely confident about their forecast for rates: 

History is interesting, but what lies ahead? We turn to that now. We reach these conclusions from our simulation:

  • This week’s simulation shows the risk-neutral 3 month U.S. Treasury bill rate rising to a mean risk neutral level of 0.571% in one year, compared to the 0.571% quarterly forward rate implied by current yields.
  • The mean risk neutral 3 month U.S. Treasury bill rate in ten years rises to 3.019%, compared to the forward rate at that time of 2.735%.
  • The simulated mean empirical expected 3 month T-bill rate at the 1 and 10 year horizons was 0.587% and 3.526%. A change in assumptions (and therefore parameters) would change these estimates.
  • The probability of a negative risk neutral 3 month Treasury bill rate is 1.924% in one year and 21.720% in ten years.
  • The simulations first show the risk neutral 3 month Treasury bill rate hitting 10% in a few scenarios in quarter 9. By the ten year point, the 39th quarter, the probability that the 3 month Treasury bill rate is over 10% rises to 3.456%.
  • At ten years, the empirical expected zero coupon bond yield is 2.503%. The actual U.S. Treasury zero yield is 2.156%.
  • The term premium at ten years is -0.347%.
  • By the 30 year maturity, the empirical expected zero yield is 4.052% compared to the actual 3.049% 30 year U.S. Treasury zero yield.
  • At 30 years, the term premium on a zero coupon bond basis is -1.003% given the parameter assumptions we have made.

The Analysis
We generate 50,000 scenarios for U.S. Treasury yields using Kamakura Risk Manager (“KRM”) version 8.1 and the U.S. Treasury yield curve for May 29, 2015 as a starting point. The simulations are based on historical movements in the U.S. Treasury curve from January 1, 1962 to the present. We used this statistical approach rather than analyzing the credit risk of the U.S. government (via a dynamic stochastic general equilibrium model) and the thought process of the current Federal Open Market Committee members and their successors over the next 30 years. Brad DeLong did a nice post on the latter approach April 6, 2015 . Instead, we simulate U.S. Treasury yields in two closely related ways. The first method is to simulate the so-called “risk neutral” yields that are used to value any securities that are tied to the U.S. Treasury curve. The 50,000 scenarios we generate price today’s Treasuries at their exact market prices, accurate to 8 decimal places. Since these risk neutral scenarios in theory contain a risk premium (the “term premium”), over very long time horizons they have historically been higher than the actual rates that market participants expect to come about, the so-called “empirical rates.” We examine the possibility that Dr. Bernanke and Adrian, Crump and Moench discuss: that the term premium may well be negative today.

Our analysis differs from Adrian, Crump, and Moench (2013) in several ways.

  • First, we employ more risk factors, a total of 9 risk factors representing the idiosyncratic movements of 9 points on the yield curve. We explained why 9 factors are necessary in a recent note .
  • Second, we do not make the assumption that the coefficients of our Heath, Jarrow and Morton (1992) term structure model are constant. When the coefficients are constant, the term structure model is labeled “affine” and implies normally distributed future yields. We do not make this assumption because of the very powerful empirical evidence that interest rate volatilities are a complex function of the level of rates and because the drift in both empirical and risk neutral rates varies as a result. For a historical perspective and summary of the actual distribution of one year rates in the U.S. and Japan, see our interest rate analysis of March 11, 2015. When interest rate volatilities and the non-random drift in rates are not constant, the term premium must be calculated via a simulation rather than via a closed form solution. We illustrate the process in this note.
  • Third, we do not assume that the empirical drift in interest rates, which determines the term premium, is constant. Instead, we allow the empirical drift to be a non-linear function of the level of interest rates. When simulated rates are negative, we assume that the empirical drift in that case is zero.

We convert the simulated three month empirical expected rates to an “expected” zero coupon yield curve using the following steps:

  • We calculated the expected empirical 3 month zero coupon bond price explicitly, since by Jensen’s inequality the expected 3 month zero coupon bond price does not equal 1/(the expected empirical 3 month yield).
  • We construct a term structure of empirical expected zero coupon bond prices with maturities out to thirty years.
  • We then convert these zero coupon bond prices to continuously compounded zero coupon bond yields. We plot these zero coupon yields versus the actual zero coupon yields that prevailed in the U.S. Treasury market on May 29, 2015:


The empirical expected zero coupon bond yields are very close to the actual zero coupon bond yields for maturities out to 4 years. Beyond that, the term premium becomes more strongly negative.

This table compares the first ten years of zero coupon bond maturities with the zero coupon bond yields derived from empirical expected yields. 

Is there a rationale for a negative premium? Dr. Bernanke discusses this in his April 13 paper. It is possible that investors fear a continuing decline in rates that will make it increasingly difficult, if not impossible, for a retiree to live on income from fixed income investments. If that fear is widespread, then it is conceivable that current bond prices have been bid up, above and beyond the mean path of interest rates, as a way of avoiding this worst-case scenario. This is an important behavioral economics question that is likely to fascinate scholars for many years.

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: 


Forward Rate Comparison with Last Week
This week’s 1 month forward rates for the first ten years are compared to forward rates in last week’s analysis in this graph: 


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:

  • 3 months
  • 1 year
  • 5 years

For each maturity, we report the following statistics for the simulated “risk neutral” yields in the 50,000 scenarios:

  • The matched maturity forward rate at each maturity
  • The lowest rate simulated
  • The 1st percentile yield
  • The 10th percentile yield
  • The 25th percentile yield
  • The 50th percentile yield
  • The average yield
  • The 75th percentile yield
  • The 90th percentile yield
  • The 99th percentile yield
  • The highest yield simulated
  • The average “empirical” or expected actual yield

We start with this 30 year graphical summary of risk neutral yields for the 3 month Treasury bill yield, which was 0.02% on May 29. We emphasize the first 10 years of the forecast, since the current Treasury yield curve extends only 30 years (compared to 40 years in Japan and 100 years in Mexico). Any simulation of a yield of x years in maturity beyond 30 – x years is completely determined by the analyst’s assumptions for current rates beyond the 30 year point.

The risk neutral evolution of the 3 month Treasury yield is shown here: 

The graph shows a small probability of negative risk-neutral rates, consistent with recent experience in Europe. The mean risk-neutral 3 month T-bill rate is essentially indistinguishable from the corresponding 3 month forward rate prevailing as of May 29. The next graph shows the same 30 year outlook for empirical rates, in which the non-random drift in interest rates is assumed to be dependent on the current level of rates. 

The mean empirical 3 month Treasury bill rate is actually slightly higher than the forward rate curve, given our assumptions and coefficients, implying a term premium that starts near zero and then becomes negative, a possibility that Chairman Bernanke discussed.

We can plot the mean risk neutral and empirical Treasury bill rates jointly, as shown here: 

The light blue lines represent the highest and lowest 3 month empirical Treasury bill rates simulated among our 50,000 scenarios. The risk neutral mean (in red) and the empirical mean (in blue) are very close to each other and to the forward rate curve.

We can compare their distributions at various forward points in time. The first graph shows their distribution in quarter 19, which represents the 5 year time horizon. 

The distributions diverge more significantly at quarter 39, which represents the 10 year point in the simulation: 

Both distributions are clearly non-normal and cluster near zero, consistent with historical data from the United States and Japan discussed in the link above.

At quarter 79, the 20 year point, the distributions’ differences become more obvious: 


The 10 Year Outlook for the 3 Month U.S. Treasury Yield
We conclude that the current term premium is negative using the parameter set and assumptions described above. In subsequent posts, we will examine how sensitive estimates of the term premium are to the parameters and related assumptions employed. We close with a tabular summary of the simulation results.

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: 


Frequently Asked Questions
Over the last few years of publishing U.S. Treasury forward rates and interest rate simulations, a few questions are asked regularly by both retail and institutions investors. We answer a selected subset of them in this appendix.

Why is an interest rate forecast necessary?
For large institutional investors, a forward looking simulation of interest rates and macro factors is the standard first step in optimizing their equity and fixed income portfolios. This is standard best practice in both theory and practice. For many retail investors, this may be a new concept to some, even though it’s obvious that one should refinance a floating rate mortgage if one expects rates to rise. Another clear conclusion from a forward looking simulation is this: one shouldn’t own the stock of one’s employer because the stock price is likely to be low in the scenario where the investor gets laid off from work. These two examples are just common sense, but careful investors know that there are no $20 bills to be found on Wall Street and that understanding the full range of scenarios that might come about leads to better investment strategy. If one believes that bank stocks rise with high interest rates, what is the probability of that happening? We answer that question in this note. These quantitative simulations are essential for realistic risk management of assets and liabilities ("ALM").

What is the term premium and why is it important?
Former Fed Chairman Ben Bernanke had an insightful post April 13, 2015 on the term premium embedded in the U.S. Treasury curve and how that premium has changed over time. In the post, he very appropriately cites some careful analysis done at the New York Fed by Adrian, Crump and Moench. How big is the potential model risk in estimating the term premium in the U.S. Treasury curve? Prof. John Cochrane at the University of Chicago recently posted a thoughtful piece on the difficulties of extracting the market’s expected future interest rates from the current yield curve. Modeling assumptions are very important in the measured levels of the term premium.

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 original source of the U.S. Treasury yield information on the H15 statistical release is the U.S. Department of the Treasury. 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). The original publication, Adams and van Deventer (1994), was modified in van Deventer and Imai (1996).

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.