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

After a two-day reversal, the shapes and levels of US Treasury yields are returning to what was considered normal prior to the 2008-2010 financial crisis. In this week’s forecast, the focus is on three things the behavior of interest rates: the future probability of the inverted yield curve predicting recession, the probability of negative rates, and the probability distribution of US Treasury yields 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 3.14%, down 0.17% from to last week. After the initial rise, there is some volatility before rates peak again at 2.98%, up 0.09%. Rates eventually peak again at 3.74%, up 0.07%, then decline 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.

**Inverse Treasury yield: 34.9% probability by spring 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 yield curve has fallen, peaking at 34.9%, from 52.9% last week, during the 91-day quarterly period ending April 13, 2023.

**Negative treasury yields: 8.3% probability by spring 2028**

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 8.3%, up from 8.9% last week, in the period ending April 6, 2028:

**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 28.65% chance that the 3-month yield will fall within this range, a change from 29.94% last week. For the 10-year Treasury yield, the most likely range is 2% to 3%. The probability of being in this range is 26.53%, compared to 27.11% last week.

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:**

Click here for data.

The probability that the yield on 3-month Treasury bills will be between 1% and 2% in 2 years is shown in column 4: 35.04%. 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 1.82% plus 0.02% plus 1.82% plus 0, 02% = 1.82%, compared to 1.50% last week. 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:**

Clear here for data.

**Annex: Cash simulation methodology**

The probabilities are derived using the same methodology Kamakura recommends to its risk management 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. For today’s forecast, it’s April 14, 2022.

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 March 31, 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 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. We used daily data from January 2, 1962 to March 31, 2022 for today’s forecast. Historical data is provided by the US Department of Treasury.

An example of the modeling process using data up to December 31, 2021 is available on 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 US) has been stable for some time, but we update the volatilities for each factor monthly.