Yield curves and the term structure of interest rates
A government borrows for a month, for two years and for thirty, and each loan has its own yield. Drawn together they make the yield curve, the base of almost every other rate and one of the most watched signals in finance. This lesson reads the US Treasury curve for September 2026, explains why long and short yields differ, and shows how an inverted curve behaved from 2022 to 2024. A calculator sets the curve's shape and revalues two bonds. It ends with Wells Fargo and Bank of America, two banks that borrow short and lend long.
- Level
- Beginner
- Reading time
- About 28 minutes
- Know this first
- Central banks and interest rates
By the end you will be able to:
- read a yield curve and say whether it slopes upward, is flat or is inverted
- calculate term spreads such as the ten-year less two-year
- explain why long and short yields differ, and why long bonds move more in price
- describe what research says and does not say about inversions and recessions
- see how a bank's interest income responds to higher rates
One borrower, many interest rates
The last lesson ended with a central bank choosing one short-term interest rate. But the US government does not borrow at one rate. It borrows for a month, for two years, for ten, for thirty, and each loan has its own yield. Put those yields on a chart with the length of the loan along the bottom, and the line you get is the yield curve.
The curve matters because it is the base of almost every other rate. A company's bond, a mortgage and a bank's loan are each often priced by starting from a government yield of a similar length and adding something for risk. Its shape is also watched as a signal, since it has often changed before the economy has. This lesson reads the curve for September 2026, explains why its shape changes, and uses a calculator to show what a move in it does to the price of a bond. It ends with two large US banks, whose business is to borrow short and lend long.
Reading the curve
The US Treasury publishes the curve every trading day as a table of par yield rates. A par yield is the coupon rate a bond of that length would need to pay to be priced at exactly 100. Here is the table for the first and last trading days of September 2026, a month in which the Fed raised its policy rate.
| Maturity | 1 September 2026 | 30 September 2026 | Change, points |
|---|---|---|---|
| 1 month | 3.85% | 4.02% | +0.17 |
| 3 months | 3.92% | 4.20% | +0.28 |
| 6 months | 4.00% | 4.33% | +0.33 |
| 1 year | 4.18% | 4.54% | +0.36 |
| 2 years | 4.39% | 4.88% | +0.49 |
| 5 years | 4.55% | 5.09% | +0.54 |
| 10 years | 4.79% | 5.29% | +0.50 |
| 20 years | 5.27% | 5.68% | +0.41 |
| 30 years | 5.27% | 5.64% | +0.37 |
Two things stand out. First, the curve slopes upward on both dates: the longer the loan, the higher the yield. On 30 September the one-month yield was 4.02% and the ten-year yield was 5.29%. Second, the whole curve moved up during the month, by between 0.17 and 0.54 percentage points, with the largest rise at the five-year point. The ten-year yield rose from 4.79% to 5.29%. The very long end is not perfectly smooth: the 20-year yield on 30 September, 5.68%, is above the 30-year yield of 5.64%.
Why long and short rates differ
Three ideas explain most of the shape. The first is expectations. A two-year loan can be thought of as a series of short loans, so its yield reflects where short rates are expected to go. The Fed says the same thing about its own decisions: short-term rates move quickly with the federal funds rate, while longer-term rates respond to expectations of future policy. If people expect the policy rate to rise, the curve tends to slope up, and if they expect cuts it tends to slope down.
The second is compensation for risk. Lending for thirty years leaves more room for inflation to surprise, and the price of a long bond moves more when rates change, as the calculator below shows. Investors may ask for extra yield for taking on that risk, which economists call the term premium. The third is supply and demand for particular maturities, the idea from the first lesson applied to bonds. The sources used here do not separate the three, and the table cannot either.
Slopes, flats and inversions
Economists boil the shape down to a term spread: a long yield minus a short one. The two most quoted are the ten-year less the two-year and the ten-year less the three-month. When the spread is positive the curve slopes upward, which is the usual shape. When it is near zero the curve is flat. When it is negative the curve is inverted, meaning short-term yields are higher than long-term ones.
An inverted curve gets attention because of its record. Two Federal Reserve Bank of San Francisco researchers, writing in 2018, found that the curve has been a reliable predictor of US recessions, with the ten-year less three-month spread the best of the measures they tested. Since the 1970s, they report, such spreads have typically turned negative about one to two years before a recession began. They also give the cautions. The relationship is a regularity, not a certainty. It leaves open questions about cause and effect, since inversions might contribute to recessions, investors expecting a recession might push long yields down, or both. And they urge great caution in reading the evidence. A negative spread is a warning to look harder at other data like the indicators in the third lesson, and not a forecast.
The recent record shows the shape changing. The table uses the Federal Reserve's monthly averages of daily two-year and ten-year yields.
| Month (average) | 10-year yield | 2-year yield | 10-year less 2-year | Shape |
|---|---|---|---|---|
| December 2021 | 1.47% | 0.68% | 0.79 | Upward-sloping |
| June 2022 | 3.14% | 3.00% | 0.14 | Upward-sloping |
| December 2022 | 3.62% | 4.29% | −0.67 | Inverted |
| July 2023 | 3.90% | 4.83% | −0.93 | Inverted |
| December 2023 | 4.02% | 4.46% | −0.44 | Inverted |
| August 2024 | 3.87% | 3.97% | −0.10 | Inverted |
| January 2025 | 4.63% | 4.27% | 0.36 | Upward-sloping |
| December 2025 | 4.14% | 3.50% | 0.64 | Upward-sloping |
| June 2026 | 4.47% | 4.11% | 0.36 | Upward-sloping |
| September 2026 | 4.99% | 4.65% | 0.34 | Upward-sloping |
At the end of 2021 the ten-year yield was 1.47% and the two-year yield 0.68%, a gap of 0.79 points, after the Fed had held its rate near zero. As the Fed raised rates through 2022 the two-year yield rose faster than the ten-year yield. By December 2022 the curve was inverted, with a spread of −0.67 points, and the inversion was deepest in the months shown in July 2023, at −0.93 points. By August 2024 it had nearly closed at −0.10, and by January 2025 it sloped upward again. These are monthly averages at selected months, so they do not show the exact first or last day of the inversion. The same cautions apply to the other direction: the table shows what happened to the shape, not what it caused.
Move the curve
Set three points on the curve: the 3-month, 2-year and 10-year yields. The calculator works out the two spreads and names the shape. It then asks what a move has done to someone who bought on 1 September at par. A 10-year bond with a 4.79% coupon and a 2-year bond with a 4.39% coupon, the par yields of that day, are valued again at the yields you set. Prices are per 100 of face value, with semiannual coupons, and the move is treated as instant so the full term remains. It starts at the 30 September 2026 yields of 4.20%, 4.88% and 5.29%.
With a 3-month yield of 4.20%, a 2-year yield of 4.88% and a 10-year yield of 5.29%, the 10-year less 3-month spread is 1.09 points and the 10-year less 2-year spread is 0.41 points, so the curve slopes upward. A 10-year bond bought at par on 1 September with a 4.79% coupon is now worth 96.16 per 100 (−3.84%), and a 2-year bond bought at par with a 4.39% coupon is worth 99.08 (−0.92%).
Four experiments. First, leave the starting settings alone. The curve slopes upward, with a ten-year less three-month spread of 1.09 points. The 10-year bond has lost 3.84% of its value and the 2-year bond 0.92%, although the ten-year yield rose only 0.50 points and the two-year yield 0.49. Longer bonds fall further for each point of yield, which is one reason investors may ask for a term premium. Second, enter the monthly averages for July 2023: 3-month 5.25%, 2-year 4.83% and 10-year 3.90%. The calculator reports an inverted curve, with a ten-year less two-year spread of −0.93 points. The 10-year bond is now worth 107.31, a gain of 7.31%, because its yield fell, while the 2-year bond lost 0.83%. The 3-month figure there is the Treasury bill discount rate, a slightly different measure from the constant maturity yields used elsewhere on this page, so it is an illustration, not a like-for-like point. Third, set all three yields to 4.00%. The curve is flat, with both spreads at zero. Fourth, set the 3-month yield to 0% and the 2-year yield to 8%, with the 10-year at 0%. The calculator calls it mixed, because the ten-year yield is level with the three-month yield and below the two-year yield. Real curves rarely look like this, but the two spreads can disagree with each other, so it pays to say which one is quoted.
Two banks that borrow short and lend long
A bank takes in deposits, which are short-term and move with rates, and makes loans, which are often longer. The difference between what it earns and what it pays is its net interest income. The usual argument is that a steep curve helps this business, because loans are priced from longer yields and funding from shorter ones, and that an inverted curve squeezes it. The third quarter of 2023 is a test: it fell inside the inversion shown above, and both Wells Fargo and Bank of America report it against the third quarter of 2022.
| Third quarter 2023 against 2022 | Wells Fargo | Bank of America |
|---|---|---|
| Net interest income, $ millions | 13,105, against 12,098 | 14,379, against 13,765 |
| Change in net interest income | +8.3% | +4.5% |
| Average yield on loans | 6.23%, against 4.28% | 5.65%, against 3.94% |
| Cost of deposits | Rate on interest-bearing deposits: 1.92%, against 0.23% | Deposit interest expense: $7,340 million, against $1,235 million |
Both banks earned much more on loans: the average loan yield rose 1.95 points at Wells Fargo and 1.71 points at Bank of America. Both also paid far more to depositors. Wells Fargo's average rate on interest-bearing deposits went from 0.23% to 1.92%, and Bank of America's dollar interest expense on deposits rose from $1,235 million to $7,340 million, about 5.9 times as much. Even so, net interest income rose at both, by 8.3% at Wells Fargo and 4.5% at Bank of America. Wells Fargo's filing attributes the increase to "the impact of higher interest rates on earning assets," partly offset by higher expenses on deposits and long-term debt. So the inversion of the curve did not stop the two banks' interest income rising over that year, because the level of rates rose so much. It does not show that the slope never matters. The comparison is between two quarters a year apart, and the filings do not isolate the slope's effect from the level of rates, from loan growth or from the mix of deposits.
What this lesson leaves out
- One country's curve. The data are US Treasuries. Other governments have their own curves, which can look quite different.
- Par yields and averages. The September table is the Treasury's par yield curve for two days. The history uses the Fed's monthly averages of the constant maturity series, so the two do not match exactly.
- The bond sums are simplified. The calculator uses semiannual coupons, the full term remaining, and an instant move in yields. It ignores accrued interest, tax and trading costs, and it assumes the same yield change applies to the whole curve point.
- A loss on paper is not a loss held to maturity. A bond that is held to maturity pays its face value back. A fall in price matters to someone who must sell, and to anyone who values holdings at market prices.
- The curve is a signal, not a cause. The record of inversions before recessions is a regularity. The research cited here says it leaves open why.
- Two banks, one quarter. They show what happened in one year, not how every bank behaves.
Common mistakes
Calling any downward move an inversion
An inversion means short yields are above long yields. A curve can fall overall and still slope upward.
Mixing up which spread is quoted
The ten-year less two-year and the ten-year less three-month spreads can disagree, as the fourth experiment shows. Say which one.
Treating the curve as a forecast
Inversions have often come before recessions, but the research describes a regularity with open questions about cause. It does not describe a promise.
Forgetting that long bonds move more
The same one-point change in yield moves a ten-year bond's price much more than a two-year bond's.
Comparing yields from different series
Bill discount rates, par yields and monthly averages are different measures. Compare like with like.
Try it yourself
1. On 30 September 2026 the Treasury's par yields were 4.20% for 3 months, 4.88% for 2 years and 5.29% for 10 years. What are the two term spreads?
Show the working
Ten-year less two-year: 5.29 − 4.88 = 0.41. Ten-year less three-month: 5.29 − 4.20 = 1.09. Both are positive, so the curve slopes upward.
2. In December 2022 the ten-year yield averaged 3.62% and the two-year yield 4.29%. What was the spread, and what is the curve called?
Show the working
3.62 − 4.29 = −0.67. A negative spread means short yields are above long yields, so the curve was inverted.
3. The ten-year par yield rose from 4.79% on 1 September to 5.29% on 30 September 2026. By how many points, and what happened to a 10-year bond bought at par at 4.79%?
Show the working
5.29 − 4.79 = 0.50 points. Revalued at 5.29%, with semiannual coupons and ten years remaining, the bond is worth 96.16 per 100, a loss of 3.84%.
4. Bank of America's net interest income was $14,379 million in the third quarter of 2023 and $13,765 million a year earlier. What was the change?
Show the working
14,379 ÷ 13,765 − 1 = 4.5%.
5. Why did the 10-year bond lose more than the 2-year bond when yields rose?
Show the working
A longer bond has more payments far in the future, and these are worth less when the discount rate rises. In the calculator the 10-year bond fell 3.84% and the 2-year bond 0.92%, though their yields rose by about the same amount.
Where this is used
- A company issuing bonds looks at government yields of the same length as the starting point for what it will pay.
- A lender sets mortgage and loan rates by reference to the part of the curve that matches the loan.
- A bond investor judges how much price risk a maturity carries before buying it.
- An economist watches the term spreads as one of several signals about the economy.
- A bank manager studies the curve for its effect on the margin between loan income and funding cost.
Key terms
- Yield curve
- The yields of bonds of the same issuer and type across different lengths of loan.
- Par yield
- The coupon rate at which a bond of a given length would be priced at exactly its face value.
- Term spread
- A long-term yield minus a short-term yield.
- Inverted curve
- A curve on which short-term yields are higher than long-term yields.
- Flat curve
- A curve on which short and long yields are about equal.
- Term premium
- The extra yield investors may ask for lending for longer.
- Net interest income
- A bank's interest earned on loans and securities less interest paid on deposits and borrowings.
- Net interest margin
- Net interest income as a share of the assets that earn it.
- Coupon
- The fixed interest a bond pays, quoted as a percentage of face value.
Questions people ask
Why are long-term yields usually higher?
Because investors may ask for extra yield for lending longer, and because rates are often expected to rise. Neither source here separates the two.
Does an inverted yield curve mean a recession is coming?
The research cited here finds that inversions have typically come one to two years before US recessions, and that the relationship is not a certainty. A negative spread is a reason to look at other data.
Why does a bond lose value when rates rise?
New bonds pay more, so an old bond with a lower coupon must sell for less to give a buyer the same yield.
Which spread should I follow?
The San Francisco Fed researchers found the ten-year less three-month spread slightly better at predicting recessions than the others they tested. Always say which spread you mean.
Sources and review
- US Department of the Treasury, Daily Treasury par yield curve rates, September 2026, for the yields at each maturity on 1 and 30 September 2026.
- Board of Governors of the Federal Reserve System, Market yield on US Treasury securities at 10-year constant maturity (GS10), percent, monthly, as published by the Federal Reserve Bank of St. Louis (last updated 1 October 2026), for the ten-year yields in the history table.
- Board of Governors of the Federal Reserve System, Market yield on US Treasury securities at 2-year constant maturity (GS2), percent, monthly, as published by the Federal Reserve Bank of St. Louis (last updated 1 October 2026), for the two-year yields in the history table.
- Michael D. Bauer and Thomas M. Mertens, Federal Reserve Bank of San Francisco, Information in the yield curve about future recessions (Economic Letter 2018-20, 27 August 2018), for term spreads, their record before recessions and the stated caveats.
- Board of Governors of the Federal Reserve System, Monetary policy: what are its goals? How does it work? (page last updated 29 July 2021), for the Fed’s goals, the 2 percent PCE objective, its tools and the way a rate change spreads.
- Wells Fargo & Company, Form 10-Q for the quarter ended 30 September 2023, for net interest income, the average loan yield, the average rate on interest-bearing deposits and its explanation of the change.
- Bank of America Corporation, Form 8-K, Exhibit 99.3, third quarter 2023 financial tables (17 October 2023), for net interest income, deposit interest expense and the average yield on loans and leases.
How the numbers were checked: every spread, bond price and price change was recomputed in plain code, and the calculator was compared with that code across a grid of 3-month, 2-year and 10-year settings. The yield-curve table is copied from the Treasury's September 2026 table and the changes recomputed; the spreads in the history table were recomputed from the two Federal Reserve series; and the bank percentage changes and loan-yield differences were recomputed from the filings' figures. The bond prices are for hypothetical par bonds and use the formulas stated in the calculator section.