The Yield Curve: Shapes, Inversions and What They Signal
Read yield-curve shapes and spreads, calculate an implied forward rate and roll-down return, and judge inversion signals with care.
What you will learn
- How to describe a yield curve with its level, slope and curvature, and the standard slope measures.
- How to calculate the forward rate the curve implies.
- The four named curve moves: bull and bear steepening and flattening.
- What an inverted curve has historically signalled, and why it is not a timing tool.
Explanation
The macro course showed what a yield curve is. This lesson explains how professionals read and use it.
A yield curve plots the yields of bonds from the same issuer (usually a government) against their maturities. Only same-issuer bonds are compared, so that differences come from maturity, not from credit risk.
Level, slope and curvature
- Level: the general height of yields, driven mainly by current and expected central-bank policy rates and inflation.
- Slope: long-term minus short-term yield. The most quoted measures are the 2s10s spread (10-year yield minus 2-year yield) and the 3m10y spread (10-year minus 3-month). Spreads are quoted in basis points (bp; 1 bp = 0.01 percentage points).
- Curvature: whether the middle (around 5 years) is higher or lower than a straight line between the ends. A curve with a peak in the middle is called humped.
Shapes: normal (upward-sloping), flat, inverted (short yields above long yields) and humped.
Why long yields differ from short yields
Two forces explain most of the slope:
- Expectations: a long yield roughly reflects the average of expected future short-term rates. If markets expect rate cuts, long yields can sit below short yields.
- Term premium: extra yield investors demand for locking money up longer and bearing duration risk. It cannot be observed directly; it is estimated with models, and estimates differ.
Implied forward rates
The curve contains a "break-even" for future rates. With a 1-year yield y1 and a 2-year yield y2 (zero-coupon yields), the 1-year rate in one year that makes both strategies equal is:
f = (1 + y2)² ÷ (1 + y1) − 1
A forward rate is not a forecast you can rely on: it mixes expectations with the term premium.
Named curve moves
| Move | What happens | Typical background |
|---|---|---|
| Bull steepener | Short yields fall more than long yields | Markets expect rate cuts |
| Bear steepener | Long yields rise more than short yields | Inflation or debt worries, higher term premium |
| Bear flattener | Short yields rise more than long yields | Central bank raising rates |
| Bull flattener | Long yields fall more than short yields | Growth worries, demand for safe long bonds |
"Bull" means prices rise (yields fall); "bear" means prices fall (yields rise).
Inversions: historical observation, not a law
In the United States, inversions of the 3m10y and 2s10s spreads have often occurred before recessions in past decades. However, the lag between inversion and recession has varied widely, sometimes by more than a year, there have been periods of inversion without a following recession, and central-bank bond purchases may have changed how the curve behaves. Treat an inversion as one piece of evidence about expectations, not as a date for a crash.
Worked example
Hypothetical example: a government curve shows these zero-coupon yields: 3 months 3.2%, 1 year 3.0%, 2 years 2.6%, 3 years 3.0%, 10 years 2.9%. Fees and taxes are excluded.
1. Slope. 2s10s = 2.9% − 2.6% = +0.3 points = +30 bp. 3m10y = 2.9% − 3.2% = −30 bp. The two measures disagree: the very short end is inverted, the 2-to-10-year part is not.
2. Forward rate. f = 1.026² ÷ 1.030 − 1 = 1.052676 ÷ 1.030 − 1 = 2.20%. Buying a 2-year bond beats rolling 1-year bonds only if the 1-year rate in one year turns out below 2.20%.
3. Roll-down. You buy a 3-year bond with a 3% annual coupon at par (100, yield 3%). If the curve is unchanged after one year, the bond is now a 2-year bond and is priced at the 2-year yield of 2.6% (for simplicity, both remaining cash flows are discounted at 2.6%):
Price = 3/1.026 + 103/1.026² = 2.924 + 97.846 = 100.77.
One-year return = coupon 3.00 + price gain 0.77 = 3.77%, above the 3% yield you bought. This extra return from "rolling down" a curve segment that slopes upward is called roll-down. It disappears if yields rise.
Common mistake
Practice activity
On paper, using a hypothetical curve: 2-year yield 3.4%, 10-year yield 3.1%, 1-year yield 3.6%. (a) Calculate the 2s10s spread and name the shape of this segment. (b) Calculate the implied 1-year rate in one year. (c) Over the next month the 2-year yield falls to 2.9% and the 10-year to 3.0%. Name the move and give the new 2s10s.
Show solution
(a) 3.1% − 3.4% = −30 bp: this segment is inverted.
(b) f = 1.034² ÷ 1.036 − 1 = 1.069156 ÷ 1.036 − 1 ≈ 3.20%.
(c) Both yields fell (bull), and the short yield fell more (−50 bp against −10 bp), so the curve steepened: a bull steepener. New 2s10s = 3.0% − 2.9% = +10 bp.
Limitations: forward rates here use zero-coupon yields; coupon-bond yields give slightly different numbers.
Key takeaways
- Describe a curve by level, slope (2s10s, 3m10y) and curvature, always in basis points.
- Forward rates show the break-even future rate, mixing expectations and term premium.
- Bull/bear and steepener/flattener tell you which end of the curve is driving the move.
- Inversions have often preceded US recessions, but with long, variable lags and false signals.
Next step
In the next lesson you will apply these tools to German Bunds, US Treasuries and the yield spreads between euro-area governments.
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