Using daily Treasury data from 1962, this week’s simulation shows an increase in market expectations gleaned from a rational examination of the history of US Treasury yields. The 1-month US Treasury forecast yield now peaks at 4.94%, up 26% points from last week. As explained in Professor Robert Jarrow’s book cited below, forward rates contain a risk premium above market expectations for the 1-month forward rate. We document the size of this risk premium in this chart, which shows the zero-coupon yield curve implied by current Treasury prices versus the annualized compound return on 3-month Treasury bills that market participants would expect in based on the daily movement of Treasury yields. since 1962. The risk premium, the reward for a long-term investment, is significant for the entire maturity range up to 30 years. The chart also shows that historical movements in Treasuries imply a steady decline in 3-month rates after an initial peak.

For more on this topic, see the analysis of US Treasury yields through March 31, 2022, presented in the appendix.

**Inverted Yields, Negative Rates and 10-Year US Treasury Probabilities**

The negative 2-year/10-year Treasury spread has now persisted for 63 trading days, narrowing this week to negative 39 basis points from negative 51 basis points last week. In this week’s forecast, the focus is on three elements of interest rate behavior: the future probability of the inverted yield curve predicting recession, the probability of negative rates, and the probability distribution of Treasury yields. American over the next decade.

We start from the closing US Treasury yield curve and interest rate swap quotes based on the guaranteed overnight funding rate published daily by the Federal Reserve Bank of New York. Using a maximum smoothing forward rate approach, Friday’s implied forward rate curve shows a rapid rise in 1-month rates to an initial peak of 4.53% from 4.50% last week. After the initial rise, there is some volatility before rates peak again at 4.59% from 4.24% a week ago. Rates eventually peak again at 4.94%, down from 4.69% last week, then ease back to a lower plateau at the end of the 30-year horizon.

Using the methodology described in the appendix, we simulate 500,000 future paths for the 30-year US Treasury yield curve. The following three sections summarize our conclusions from this simulation.

**Treasury Yield Inverted: Inverted now, 89.7% probability by March 31, 2023**

Many economists have concluded that a downward sloping US Treasury yield curve is an important indicator of future recessions. A recent example is this article by Alex Domash and Lawrence H. Summers. We measure the probability that the 10-year nominal coupon Treasury yield will be lower than the 2-year nominal coupon Treasury yield for each scenario in each of the first 80 quarterly periods of the simulation.^{[1]} The following chart shows that the probability of an inverted return remains elevated, peaking at 89.7%, from 94.2% a week earlier, during the 91-day quarterly period ending March 31, 2023.

**Negative treasury yields: 4.9% chance by September 20, 2030**

The following graph depicts the probability of negative 3-month Treasury bill rates for all the last 3 months of the next 3 decades. The probability of negative rates starts near zero but then increases steadily to peak at 4.9%, up from 7.6% a week earlier, in the period ending September 20, 2030:

**10-year US Treasury probabilities**

In this section, the focus is on the coming decade. This week’s simulation shows that the most likely range for the yield on 3-month US Treasuries ten years ahead is up 1 percentage point from 1% to 2%. There is a 24.89% chance that the 3-month yield will fall within this range, a change from 25.26% a week earlier. For the 10-year Treasury yield, the most likely range is also up 1 percentage point from last week, ranging from 3% to 4%. The probability of being in this range is 22.17%, compared to 21.83% a week before.

In a recent article on Seeking Alpha, we pointed out that a “heads” or “heads” prediction in a draw misses critical information. What a knowledgeable bettor should know is that on average, for a fair coin, the probability of heads is 50%. A prediction that the next coin toss will be heads is literally worthless to investors because the outcome is purely random.

The same is true for interest rates.

In this section, we present the detailed probability distribution for the 3-month Treasury bill rate and the 10-year US 10-year Treasury bill yield using semi-annual time steps. We present the probability of where rates will be at each time step in 1% “rate buckets”. The forecasts are presented in this graph:

**US 3-Month Treasury Yield Data:**

SAS3monthUST20220930.xlsx

The probability that the 3-month treasury bill yield will be between 1% and 2% in 2 years is shown in column 4: 16.80%. The probability that the 3-month Treasury yield will be negative (as has often been the case in Europe and Japan) in 2 years is 0.30% plus 0.00% plus 0.00% = 0, 30 %. Cells shaded blue represent positive probabilities of occurrence, but the probability has been rounded to the nearest 0.01%. The shading scheme works like this:

Dark blue: the probability is greater than 0% but less than 1%

Light blue: the probability is greater than or equal to 1% and less than 5%

Light yellow: the probability is greater than or equal to 5% and 10%

Medium yellow: the probability is greater than or equal to 10% and less than 20%

Orange: the probability is greater than or equal to 20% and less than 25%

Red: the probability is greater than 25%

The chart below shows the same probabilities for the 10-year US Treasury yield derived under the same simulation.

**US 10-Year Treasury Yield Data:**

** SAS10yearUST20220930.xlsx**

**Annex: Cash simulation methodology**

Probabilities are derived using the same methodology that SAS Institute Inc. recommends in its KRIS^{®} and Risk Manager of Kamakura^{®} clients, who currently have over $38 trillion in assets or assets under management. A moderately technical explanation is given later in the appendix, but we summarize it first in plain language.

Step 1: We take the closing of the US Treasury yield curve as a starting point.

Step 2: We use the number of points on the yield curve that best explain historical shifts in the yield curve. Using daily data from 1962 through September 30, 2022, we conclude that 10 “factors” determine almost all movements in US Treasury yields.

Step 3: We measure the volatility of changes in these factors and its evolution over the same period.

Step 4: Using these measured volatilities, we generate 500,000 random shocks at each time step and derive the resulting yield curve.

Step 5: We “validate” the model to ensure that the simulation EXACTLY assesses the starting Treasury curve and that it matches the story as closely as possible. The methodology for doing this is described below.

Step 6: We take the 500,000 simulated yield curves and calculate the probabilities of yields falling in each of the 1% “buckets” shown in the chart.

**Do Treasury yields accurately reflect expected future inflation?**

We showed in a recent article on Seeking Alpha that, on average, investors have almost always done better buying long-term bonds than rolling over short-term Treasuries. This means that market participants have generally (but not always) been accurate in their forecasts of future inflation and in adding a risk premium to those forecasts.

The above distribution helps investors estimate the probability of a successful long position.

Finally, as mentioned each week in The Corporate Bond Investor Friday preview, the future expenses (both amount and timing) that all investors try to cover with their investments are an important part of the investment strategy. . The author follows his own advice: cover short-term cash needs first, then cover more distant cash needs as savings and investment returns accumulate.

**Technical details**

Daily treasury bill returns are the basic historical data for adjusting the number of yield curve factors and their volatility. Historical data is provided by the US Department of Treasury.

An example of the modeling process using data up to March 31, 2022 is available at this link.

The modeling process has been published in a very important paper by David Heath Robert Jarrow and Andrew Morton in 1992:

For technically inclined readers, we recommend Professor Jarrow’s book *Modeling fixed income securities and interest rate options* for those who want to know exactly how the construction of the “HJM” model works.

The number of factors (10 for the United States) has been stable for some time.

**Footnotes :**

[1] After the first 20 years of the simulation, the 10-year Treasury bills cannot be derived from the yields of the first 30-year Treasury bills.