##### Weekly Forecast, December 15, 2023: Longest Reversal Streak Approaching Now

As explained in Professor Robert Jarrow’s book cited below, forward rates contain a risk premium over and above market expectations for the 3-month forward rate. We document the magnitude of this risk premium in this graph, which shows the zero-coupon yield the curve implied by current Treasury bond prices compared to the annualized compound yield of 3-month Treasury bills that market participants would expect based on the daily movement of Treasury bond yields in 14 countries since 1962. The risk premium, the reward for a long-term investment, is large and extends over most of the 30-year maturity range. The chart also shows a sharp downward shift in yields in the early years, after which the decline continues at a slow but steady pace for nearly 30 years. We explain the details below.

More on this topic, see the analysis of government bond yields in 14 countries until November 30, 2023, given in the appendix.

**Inverted yields, negative rates and US Treasury probabilities 10 years ahead**

The negative 2-year/10-year Treasury spread has now persisted for 364 trading days. The spread is currently at negative 53 basis points compared to negative 48 last week. The chart below shows that the current inverted yield curve streak is the second longest in the US Treasury market since the 2-year Treasury yield was first released on June 1, 1976. The longest streak so far is 423 trading days starting on August 18, 1978, but that streak is in jeopardy .

This week’s forecast focuses on three elements of interest rate behavior: the future probability of a recession predicting an inverted yield curve, the probability of negative rates, and the probability distribution of US Treasury yields over the next decade.

We’ll start with the final US Treasury yield curve published daily by the US Treasury. Using the maximum forward rate fluidity approach, Friday’s implied forward rate curve shows a rapid rise in 1-month forward rates to an initial high of 5.50% from 5.25% last week. After an initial rise, there is a decline until rates peak again at 4.04%, compared to 4.42% last week. Rates eventually peaked again at 5.02%, compared to 5.38% last week, before falling to a lower plateau at the end of the 30-year horizon.

Using the methodology presented in the appendix, we simulate 100,000 future trajectories of the US Treasury yield curve up to thirty years. The next three sections summarize our conclusions from this simulation.

**Inverted Treasury yields: Inverted now, 76.5% probability by June 14, 2024**

A large number of economists have concluded that the US Treasury’s declining yield curve is an important indicator of future recessions. A recent example is this paper by Alex Domash and Lawrence H. Summers. We measure the probability that the 10-year Treasury coupon yield is lower than the 2-year Treasury coupon yield for each scenario in each of the first 80 quarterly periods in the simulation.(1) The following graph shows that the probability of the inverted yield has fallen and peaked at 76.5% compared with 76.1% a week earlier, in the 91-day quarterly period ending June 14, 2024.

**Negative T-bill yields: 13.4% probability by March 4, 2033**

The next chart plots the probability of negative 3-month Treasury bill rates for all but the first 3 months of the next 3 decades. The probability of negative rates starts near zero, but peaks at 13.4%, compared to 13.2% a week earlier, in the period ending March 4, 2033:

**Calculation of default risk due to interest rate maturity mismatch**

In light of Silicon Valley Bank’s March 10, 2023 rate-induced failure, we’ve added a table that applies equally to banks, institutional investors, and individual investors of the mismatch from buying long-term Treasuries with borrowed short-term funds. Assume that the only asset is a 10-year Treasury note purchased at time zero for a face value of $100. We analyze the default risk for four different ratios of initial market value of equity to market value of assets: 5%, 10%, 15% and 20%. For the banking example, we assume that the only class of liabilities are deposits that can be withdrawn at any time at face value. In the case of an institutional and retail investor, we assume that the commitment is essentially a margin loan/repurchase with the possibility of margin calls. For all investors, the call amount (95, 90, 85 or 80) represents the “strike price” of the put option held by the call holders. Default occurs through a supplementary payment call, bank run or regulatory takeover (in the banking case) when the value of assets falls below the value of liabilities.

The chart below shows the cumulative 10-year default probabilities for each of the 4 possible capital ratios when the asset has a 10-year maturity. At 5 percent, the probability of a default is 40.90%, compared to 40.95% last week.

This default probability analysis is updated weekly based on the US Treasury yield simulation described in the next section. The calculation process is the same for any portfolio of credit risk assets.

**US Treasury Probabilities 10 Years Ahead**

In this section, attention turns to the decade ahead. This week’s simulation shows that the most likely range for the ten-year 3-month US T-bill yield is 0% to 1%, unchanged from last week. There is a 28.16% probability that the 3-month yield will fall within this range, a change from 30.18% a week ago. The most likely range for the 10-year Treasury yield is between 2% and 3%, also unchanged from last week. The probability of being in this range is 23.00% compared to 21.53% a week ago.

IN a recent post on Seeking Alpha, we pointed out that predicting “heads” or “tails” in a coin toss misses critical information. What the sophisticated bettor needs to know is that on average for a fair coin the probability of heads is 50%. The prediction that the next coin toss will be “heads” has literally no value to investors because the outcome is purely random.

The same goes for interest rates.

In this section, we report detailed probability distributions for both the 3-month T-bill rate and the 10-year US T-bill yield 10 years ahead using semiannual time steps.(2) We report the probability that at each time step, rates will be 1 percent “ rate segments’. The 3-month government bond yield forecast is shown in this chart:

**US Treasury yield data for 3 months:**

The probability that the yield on a 3-month Treasury bill will be between 1% and 2% in 2 years is shown in column 4: 27.76%. The probability that the 3-month T-bill yield will be negative (as it often has been in Europe and Japan) in 2 years is 1.83% plus 0.05% plus 0.00% = 1.88% (difference due to rounding). Blue shaded cells represent a positive probability of occurrence, but the probability has been rounded to the nearest 0.01%. The shading scheme works like this:

- Dark blue: probability is greater than 0% but less than 1%
- Light blue: probability is greater than or equal to 1% and less than 5%
- Light yellow: 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: 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.

**10-year US Treasury yield data:**

**Appendix: Treasury simulation methodology**

Probabilities are derived using the same methodology that SAS Institute Inc. recommends KRIS® and Kamakura Risk Manager® to its clients. A slightly technical explanation is given later in the appendix, but first we’ll summarize it in plain English.

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

Step 2: We use the number of points on the yield curve that best explains historical yield curve shifts. Using daily data on government bond yields from 14 countries from 1962 to November 30, 2023, we conclude that 12 “factors” drive almost all movements in government bond yields. The countries on which the analysis is based are Australia, Canada, France, Germany, Italy, Japan and New Zealand. Russia, Singapore, Spain, Sweden, Thailand, United Kingdom and United States of America. After January 2022, no data from Russia is included.

Step 3: We measure the volatility of changes in these factors and how volatility has changed over the same period.

Step 4: Using these measured volatilities, we generate 100,000 random shocks at each time step and derive the resulting yield curve.

Step 5: We “validate” the model to make sure the simulation is EXACTLY pricing the initial Treasury curve and that it matches history as well as possible. The methodology for achieving this is described below.

Step 6: We take all 100,000 simulated yield curves and calculate the probabilities of the yield falling in each of the 1% “buckets” shown in the graph.

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

We showed you a recent post on Seeking Alpha that, on average, investors have almost always done better buying long-term bonds than rolling short-term Treasury bills. This means that market participants have generally (but not always) been accurate in predicting future inflation and adding a risk premium to that forecast.

The above breakdown helps investors estimate the probability of success over the long haul.

Finally, as noted each week in Friday’s Corporate Bond Investor roundup, the future expenses (both amount and timing) that all investors seek to cover with their investments are an important part of investment strategy. The author follows his own advice: cover short-term cash needs first, then step up to meet longer-term cash needs as savings and investment returns accumulate.

**Specifications**

Daily yields of government bonds from the 14 countries mentioned above form the basic historical data for adjusting the number of factors of the yield curve and their volatility. US historical data provided by US Treasury Department. The use of international bond data increases the number of observations to more than 105,000 and provides a more complete range of experience with both high and negative rates than the US data set alone.

The modeling process was published in very important paper by David Heath, Robert Jarrow and Andrew Morton in 1992:

For technically oriented readers, we recommend the book by Prof. Jarrow *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, 12 for the 14 country model, has been stable for some time.

**footnotes:**

(1) After the first 20 years in the simulation, the 10-year Treasury cannot be derived from the initial 30 years of Treasury yields.

(2) The full simulation uses 91-day time steps for 30 years ahead. This note summarizes only the first 10 years of the full simulation.