Current Yield Curve and Recent Yield Movements

[Pages:20]The 3 Month T-Bill Rate: Average of 100,000 Scenarios Up 0.23% to 3.46% in 2025 Donald R. van Deventer June 29, 2015 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 100,000 scenarios used in this study. We use 100,000 scenarios for the U.S. Treasury (TLT) yield curve to examine the outlook for Treasuries and the magnitude of the term premium (or risk premium) above and beyond expected rate levels that is embedded in the yield curve. The U.S. Treasury yield curve from which simulations are initiated is the June 26, 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.

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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:

o The biggest yield change of the week was 0.23%, which occurred for the ten year Treasury yield.

o The smallest yield change of the week was 0.00%, which took place at the 1 month and 3 month maturities.

o The 1 month Treasury yield finished the week at 0%, the tenth day in a row at the zero point. 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 June 26, 2015.

o The 1 year Treasury yield finished the week at 0.29%, a change of 0.04% since last week. This compares to the recent high and low values of 0.3% 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.

o The 10 year Treasury yield finished the week at 2.49%, a change of 0.23% since last week. This compares to the recent high and low values of 2.5% and 1.68%. The all-time high in the 10 year yield was 15.84%

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recorded on September 30, 1981. The all-time low was 1.43% on July 25, 2012. o The 30 year Treasury yield completed the week at 3.25%, a change of 0.20% since last week and a new high for 2015. This compares to the recent low value of 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:

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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 0.613% in one year, compared to the 0.613% quarterly forward rate implied by current yields.

The mean risk neutral 3 month U.S. Treasury bill rate in ten years rises to 3.459%, compared to the forward rate at that time of 3.139%. This is an increase of 0.23% from our simulation using data from June 12, 2015.

The simulated mean empirical expected 3 month T-bill rate at the 1 and 10 year horizons was 0.628% and 4.009%. A change in assumptions (and therefore parameters) would change these estimates.

The probability of a negative risk neutral 3 month Treasury bill rate is 0.815% in one year and 17.500% 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 4.627%.

At ten years, the empirical expected zero coupon bond yield is 2.939%. The actual U.S. Treasury zero yield is 2.540%.

The term premium at ten years is -0.399%. By the 30 year maturity, the empirical expected zero yield is 4.550% compared

to the actual 3.456% 30 year U.S. Treasury zero yield. At 30 years, the term premium on a zero coupon bond basis is -1.093% given

the parameter assumptions we have made.

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The next graph shows the difference in simulations for the current yield curve versus the most recent prior simulation results. We show the distribution of the risk neutral values of the three month Treasury bill rate in quarter 39, the ending quarter at a 10 year horizon. The current simulation (in black) has many fewer scenarios than the prior simulation (in brown) at low and negative rates, and slightly more scenarios for the higher rate levels. This results in the 0.23% rise in the average simulated 3 month Treasury bill rate:

The Analysis We generate 100,000 scenarios for U.S. Treasury yields using Kamakura Risk Manager ("KRM") version 8.1 and the U.S. Treasury yield curve for June 26, 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 socalled "risk neutral" yields that are used to value any securities that are tied to the U.S. Treasury curve. The 100,000 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.

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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 June 26, 2015:

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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.

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