Treasury yields were surprisingly little moved on the day of President Powell’s comments in Jackson Hole, but after reflection early last week, the middle sections of the curve have risen significantly. The forward 1-month US Treasury yield now peaks at 4.95%, up 27%. basis points.

### **Inverted yields, negative rates and** **10-year US Treasury probabilities**

The negative 2-year/10-year Treasury spread has now persisted for 43 days, narrowing this week to minus 20 basis points from minus 33 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 high of 3.70% from 3.64% last week. After the initial rise, there is some volatility before rates peak again at 3.56% from 3.06% a week ago. Rates eventually peak again at 4.95%, down from 4.68% 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 Yields Inverted: Inverted Now, 67.8% Probability by March 3** **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 67.8%, from 81.7% a week earlier, during the 91-day quarterly period ending March 3, 2023.

**Negative treasury yields: 9.1% probability by February 26, 2027**

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 9.1%, from 9.9% a week earlier, in the period ending February 26, 2027:

**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 from now is 0% to 1%. There is a 28.57% chance that the 3-month yield will fall within this range, a change from 28.91% a week earlier. For the 10-year Treasury yield, the most likely range is 2% to 3%. The probability of being in this range is 23.68%, compared to 24.59% a week earlier.

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:**

Kamakura3monthUST20220902.xlsx

The probability that the yield on 3-month Treasury bills will be between 1% and 2% in 2 years is shown in column 4: 32.12%. 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 1.15% plus 0.01% plus 0.00% = 1, 16%. 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:**

Kamakura10yearUST20220902.xlsx

**Annex: Cash simulation methodology**

Probabilities are derived using the same methodology that SAS Institute Inc. recommends to its KRIS and Kamakura Risk Manager 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 June 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 evaluates the starting Treasury curve and that it matches the history 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% “slices” 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 forecasting future inflation and adding a risk premium to that forecast.

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 :**

- 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.