Week 31: Yield Curves — Shape, Level, Slope, Curvature

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1. Why This Is Important

The yield curve is the single chart that, more than any other, tells you what the bond market thinks. It is the price of money for every maturity from one day to thirty years, drawn as a line. Its level prices every mortgage and every corporate bond. Its slope prices every recession probability the market is willing to bet on. Its curvature prices the path of Fed policy. About 95% of all the variance in how the curve moves can be summarised in three numbers.

You need to read this chart for four reasons.

The curve is the foundation of every other rate. Mortgages

sit a couple of points above the 10-year. Investment-grade corporate bonds sit one to two points above the matched-maturity Treasury. High-yield credit sits four to six points above. If you understand the Treasury curve, you understand the price of credit risk in the entire economy because every other yield is a spread over it.

Inversions have predicted every US recession since 1970. Not

most of them — every single one. The 2y/10y spread inverted in July 2022, stayed inverted for over two years, and uninverted in late 2024. By the textbook playbook a recession should have followed within 6 to 24 months. Whether or not it did, an investor who could read this chart in 2022 had cleaner hands than one who could not.

Three factors explain almost all curve movement. Principal

component analysis on Treasury yields decomposes the curve into level (parallel shifts, ~70% of variance), slope (steepening or flattening, ~20%), and curvature (the belly versus the wings, ~5%). Every interest-rate trade you will ever see — duration bets, steepeners, butterflies — maps cleanly onto these three factors.

Forwards are the market's expected future spot rates. The

forward rate two years out is what the curve is pricing for the short rate two years from today. If you have a different view, you have a tradeable view. If you have the same view, you do not. Reading forwards turns the curve into a probability distribution over future Fed policy.

This lesson covers construction (par/spot/forward), the three PCA factors, the regime taxonomy (bull-flattener/bear-steepener and the two "wrong" cousins), butterfly trades on curvature, and the April 2026 curve in context.

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2. What You Need to Know

2.1 What the Curve Is, Exactly

A yield curve is a graph of yield versus maturity for a set of bonds of equivalent credit quality. The Treasury curve is the canonical example because the credit risk is constant (zero, by US convention) and the only thing varying is time-to-maturity.

The five tenors that get watched are 3-month, 2-year, 5-year, 10-year, and 30-year. Their FRED ticker family is DGS3MO, DGS2, DGS5,

DGS10, DGS30. The 3-month sits closest to the Fed funds rate. The

2-year reflects expected Fed policy over the next two years. The 10- and 30-year embed everything from inflation expectations to term premium to global savings flow.

Read the chart left to right. The 2007 curve is nearly flat — that flatness was the warning bell before the GFC. The 2020 curve is pinned near zero across the front because the Fed hit the lower bound. The 2023 curve is inverted: the 3-month is above the 10-year by more than 150 basis points, the deepest inversion since Volcker. The April 2026 curve is back to a normal upward shape — a regime change that itself is the story this week.

2.2 Par, Spot, and Forward — Three Rates from the Same Bond

When a financial paper says "the 10-year yield is 4.0%," it almost always means the par yield — the coupon a 10-year Treasury would need to pay to be priced exactly at 100. Par is the quoted rate.

The spot rate (or zero-coupon rate) is the yield on a bond that pays nothing until maturity, then pays back principal. Spots are the building blocks: every coupon bond is a portfolio of zero-coupon strips at different maturities.

The forward rate is the implied future spot. If today's 1-year spot is $r_1$ and today's 2-year spot is $r_2$, then the 1-year forward 1-year-out, $f_{1,2}$, is the rate that satisfies:

$$ (1 + r_2)^2 = (1 + r_1) \cdot (1 + f_{1,2}) $$

Solving:

$$ f_{1,2} = \frac{(1+r_2)^2}{1+r_1} - 1 $$

This is not a forecast — it is a no-arbitrage identity. If the forward rate were any different from this, you could borrow and lend across maturities to extract a riskless profit. Markets price this identity to within a basis point in every developed-market sovereign bond market on earth.

The interpretation, however, is economic. The forward rate embodies the rate the market expects (plus a term premium). A flat curve says the market expects roughly today's short rate to persist. An upward-sloping curve says the market expects the short rate to rise. An inverted curve says the market expects the short rate to fall. That last case is, mechanically, what an inversion is: a collective bet on Fed cuts.

2.3 The Three PCA Factors — Level, Slope, Curvature

In 1991 Robert Litterman and José Scheinkman published a working paper at Goldman Sachs that decomposed Treasury curve movements via principal component analysis. They found three factors:

Factor	Loosely	% of curve variance	What moves it
Level	All yields up or down together	~70%	Inflation expectations, term premium
Slope	Long end versus short end	~20%	Fed policy expectations, growth
Curvature	Belly versus wings	~5%	Convexity demand, hedging flow

Together the three factors explain about 95% of all daily Treasury yield variance. A duration trade is a level bet. A steepener or flattener is a slope bet. A butterfly trade is a curvature bet.

The level factor is the boring one and also the largest. When the Fed signals tighter, the whole curve shifts up by a roughly parallel amount. When inflation prints cooler, the whole curve drops.

The slope factor is the interesting one because it is what predicts recessions. When the curve flattens or inverts, it is the slope factor going negative — the market saying the short rate has further to fall than the long rate.

The curvature factor is the trader's one. Pension funds and insurance companies buy long-dated bonds for convexity matching; butterfly trades exploit the relative cheapness or richness of the belly versus the wings. We come back to this in §2.6.

2.4 The Four Curve Regimes — Bull/Bear × Steepener/Flattener

Slope movements decompose into four regimes by *which end is moving more and in which direction*:

Regime	What happens	Macro driver	Asset playbook
Bull steepener	Short rates fall faster than long rates → curve steepens, both ends drop	Fed cutting into a slowdown	Long bonds rally, equities mixed; lengthen duration
Bull flattener	Long rates fall faster than short → curve flattens, both ends drop	Recession fear, flight to quality	Long bonds rally most; flight-to-quality
Bear steepener	Long rates rise faster than short → curve steepens, both ends climb	Reflation, rising inflation expectations	Short duration, value/cyclical stocks, commodities
Bear flattener	Short rates rise faster than long → curve flattens or inverts	Fed hiking faster than the long end believes	Cash beats bonds, growth stocks penalised

The 2022-23 cycle was a textbook bear flattener turning into an inversion. The Fed hiked from 0.25% to 5.50% while the 10-year went from 1.5% to 4.0%. Short rates moved 5 points; long rates moved 2.5 points; the curve inverted by 100 basis points. Then in 2024-25 the Fed cut, the short end fell faster, and the curve bull-steepened back to normal.

The honest reading of this chart: every single dip below zero has been followed, within 6 to 24 months, by a recession dated by the NBER Business Cycle Dating Committee. The 2022 inversion is the longest on the chart. As of April 2026 the spread has uninverted but only just — the curve is still flatter than its long-run average.

2.5 Reading Today's Curve — April 2026

Pull up the chart at the top of §2.1. As of April 2026:

• 3-month: ~3.5%, well below the cycle peak of 5.5%.
• 2-year: ~3.6%, pricing roughly one more Fed cut over the next

two years.

• 5-year: ~3.8%.
• 10-year: ~4.0%.
• 30-year: ~4.3%.

The 10y-2y spread is back to roughly +40 basis points after spending two years deeply negative. The 10y-3m spread is back to slightly positive. The curve has fully uninverted but it is not yet steep in the historical sense — the long-run median 10y-2y spread is roughly +95 basis points.

The forward curve embedded in these spots is gently rising — the market is pricing the short rate to drift roughly back to today's 10-year over the next decade, suggesting the market believes neutral real rates have repriced higher than the 2010-2020 norm. This matters for every long-duration asset, including equities. The 40-year bond bull regime ended in 2022 — this chart gives that story a quiet confirmation.

**Horace's view — the curve is a regime dial, and the regime sets the allocation.** Most textbooks treat the un-inversion as a recession- timing signal and stop there. In my own book I read it as something broader: a regime print that re-prices every other sleeve in the portfolio. The shape I run is a barbell with a small persistent long-volatility and tail-hedge sleeve sitting alongside the equity end — inspired by Chris Cole's Dragon work but tilted by current regime rather than held statically. The curve is one of the inputs that decides how that tilt sits.

When the curve is deeply inverted with the Fed visibly fighting inflation, the long-equity sleeve is the trade that has been paying, and the tail-hedge sleeve runs at a low single-digit weight — cheap insurance, structured to expire worthless in the regime that's actually in force. The day the curve un-inverts and starts to bull- steepen — short rates falling faster than long, the Fed cutting into a slowdown — the historical base rate says the recession lands in that window, not before. That is the moment the tail-hedge weight goes up and the equity sleeve gets re-examined leg by leg, not because the model says "sell," but because the conditional payoff on the tail hedge has just improved materially. The palette stays the same; the weights move. Reading the curve is one of the few inputs that changes the weights honestly rather than emotionally.

2.6 Butterfly Trades — Trading Curvature

A butterfly is a market-neutral trade against the curvature of the curve. The classic structure is long the belly, short the wings, weighted to be duration-neutral. For example: long $50M of 5-year Treasuries; short $25M of 2-year and $25M of 30-year, weighted by DV01 so the position has zero net duration.

If the belly is cheap relative to the wings, the trade earns when the curvature mean-reverts. If the belly is rich, the inverse butterfly profits.

Pension funds and insurers create persistent richness at the long end (they need 30-year duration) and at the very short end (they park cash there). The belly tends to be relatively cheap in normal times. Trader convention quotes the butterfly as $2 \cdot y_{\text{belly}} - y_{\text{short}} - y_{\text{long}}$, which equals zero when the curve is perfectly straight, positive when the belly is cheap (high yield), and negative when the belly is rich.

This is a relative-value fixed-income trade — one of the structural alpha sources institutions actually run. It is not retail-friendly because the gross notional involved is large and the carry is small. But it lives on the same chart you have been staring at all week, and the interactive panel below lets you compute the butterfly statistic from any curve you draw.

2.7 The Practical Toolkit — Which Number Matters Most

For a retail investor reading the curve once a month, the priority order is:

Level of the 10-year. Sets the discount rate for every long-

duration asset on earth. Through Week 5 and Week 21 we already know stocks compete with bonds — the 10-year is the bond.

10y-2y or 10y-3m spread. Recession indicator and slope

factor in one. Inverted = late-cycle warning.

Forward 1-year, 1-year-out. Where the market thinks the

short rate goes next year. Compare against the Fed dot plot for a quick consensus check.

Curvature. Optional. Useful if you trade fixed income

actively; ignorable for buy-and-hold.

The interactive at the bottom of this lesson lets you slide all five tenors and watch the slope, curvature, and 1y1y forward update live. Hit the preset for "1981" and see what 16% rates with a deeply inverted curve look like. Hit "2020" and see the zero-bound. Hit "today" and see April 2026.

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3. Common Misconceptions

1. "An inversion causes a recession." It does not. The inversion is the market's prediction of a recession, expressed through forwards. The causation runs the other way: late-cycle conditions cause the Fed to be tight, which pushes the short end up while long-rate inflation expectations anchor or fall. Inversion is a symptom, not the disease.

2. "The yield curve is always upward-sloping in normal times." Often, but not always. About 80% of post-1976 monthly observations show 10y > 2y. The other 20% include genuine normal flat periods as well as inversions. Calling the curve "broken" because it is not steep is reading the median as the law.

3. "Long bonds are safer than short bonds." Lower default risk on US Treasuries is the same at every maturity. Long bonds have more interest-rate risk because their duration is higher. A 30-year zero loses ~30% on a one-percentage-point yield rise; a 2-year loses ~2%. Long bonds are more volatile, not safer. The vol tail wags the dog.

4. "Forwards are forecasts." Forwards are no-arbitrage identities derived from spots. They embody the market's expected path plus a term premium. They are not anyone's forecast in particular, and the market's expectation is wrong much of the time — see Eugene Fama's 1984 work showing forward rates are biased estimators of future spots.

5. "If I think rates are going down, buy long bonds." Often correct, but the magnitude depends on whether the *whole curve* shifts down (level) or just the long end (bull flattener). A bull steepener — short end falls faster — does not benefit long-duration positions much. Decompose the trade you want into level, slope, and curvature before sizing.

6. "The Fed controls the yield curve." The Fed controls the very front (overnight rate, and via quantitative easing the long end somewhat). The middle and long end are set by global savings flow, term premium, and inflation expectations. In 2022-23 the Fed hiked 5 percentage points; the 10-year moved only 2.5. The market does not always follow.

7. "All inversions are alike." They are not. The 1980 inversion under Volcker was caused by the short rate being driven to 19%. The 2006-07 inversion was a gentler 50bp inversion driven by housing optimism on the long end. The 2022-23 inversion was a 150bp post-COVID bear flattener. Same shape; very different macro story each time.

8. "Butterfly trades are for retail investors." They are not. Carry is small, gross notional is large, and the position has to be DV01-balanced and hedged. This is an institutional or hedge-fund activity. Retail can express the direction of the curvature view by overweighting or underweighting the 5-year sleeve of a bond ladder.

9. "Real yields don't matter for the curve." They are most of the long-end story. Decompose nominal = real + breakeven inflation. The 10-year real yield (TIPS, FRED DFII10) moved from -1.0% in 2021 to +2.0% in 2024. Two thirds of the 2022-23 long-rate rise was a real-rate move, not an inflation- expectation move. The bond regime really did change.

10. "Once I read the curve, I can time the market." You can read the curve. You can position duration. You can probability-weight an equity drawdown. You cannot time the market with the curve as your only signal. The 2022 inversion was followed by an 18-month equity rally before any meaningful correction. Signals are probabilistic; sizing matters more than conviction.

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4. Q&A

Q1: How do I actually read a yield curve plot for the first time?

A: Three glances. (a) Look at the level — where is the 10-year? Above 4% is meaningfully restrictive territory in modern terms; below 2% is extraordinarily loose. (b) Look at the slope — is the right end higher than the left? Upward = normal. Flat or down = late-cycle. (c) Look at the belly — is the 5-year sitting on a straight line between the 2-year and the 30-year, or is it bowed? Bowed up = market pricing rate cuts through the medium term; bowed down = sudden tightening expected.

Q2: What is the difference between the 10y-2y and the 10y-3m spreads?

A: Both are slope measures. The 10y-2y is more popular among traders because the 2-year integrates Fed expectations over a longer window. The 10y-3m is the academic favourite (Estrella and Mishkin, 1996, showed it is the strongest single recession predictor). In practice they invert and uninvert at slightly different times; the 10y-3m typically inverts later but is regarded as a more reliable signal.

Q3: Why does the curve invert before recessions?

A: Late-cycle Fed tightening drives the short end up. Long-end inflation expectations are forward-looking — if traders believe the hiking will eventually slow growth and inflation, the long end stays anchored or falls. The result: short above long. The market is saying "the Fed is too tight; cuts are coming." When those cuts arrive, they typically arrive into a slowing economy. So the inversion is the market pricing the rate cuts that the recession will bring.

Q4: Can I trade the yield curve as a retail investor?

A: Yes, with limits. The cleanest tools: SHV (1-3 month T-bills, ~0 duration), IEF (7-10 year, ~8 duration), TLT (20+ year, ~17 duration). A long IEF / short SHV trade is a long-duration / curve level bet. A long TLT / short IEF is a long-end steepening bet against the belly. Margin and futures (UB and TY contracts) give cleaner expressions but require more capital and more risk management. Butterfly trades are not really doable retail.

Q5: How does the curve relate to mortgage rates?

A: 30-year mortgage rates track the 10-year Treasury plus a 1.5–2.5% spread (the "MBS basis"). When the 10-year rises 100bp, mortgages typically rise about 100bp. When the curve inverts, banks' net interest margin compresses, lending tightens, and mortgage spreads sometimes widen further. The full chain — Fed → 10-year → mortgage → housing — is the topic of Week 18.

Q6: What is the "term premium" and why does it matter?

A: The term premium is the extra yield long bonds pay above the expected average of future short rates, to compensate investors for locking up money. Estimates vary, but the New York Fed's ACM model puts the 10-year term premium near 0% in 2020, rising to ~80bp by

A higher term premium steepens the curve without any change

in expected Fed policy. Some of the post-2022 long-end rise is term premium repricing, not inflation expectations.

Q7: Why was the 2022-24 inversion so unusually long?

A: Two reasons. (1) Inflation was the most persistent in 40 years, so the Fed had to hold restrictive policy longer than any prior post-1990 cycle. (2) The economy did not slow sharply enough to force cuts — fiscal stimulus and onshoring kept demand robust. The market kept pricing cuts in the forwards; the Fed kept not cutting; and the inversion persisted. By the time uninversion arrived in late 2024, the inversion had lasted ~26 months — the longest since records began.

Q8: What does a steep curve mean for stocks?

A: A bull steepener (Fed cutting, short end falling) is typically good for equities because the discount rate is dropping. A *bear steepener* (long end rising on growth or inflation) is mixed: cyclical and value sectors benefit; long-duration growth stocks suffer because their cash flows are far in the future and a higher discount rate hits them hardest. Reading the type of steepening matters more than the steepening itself.

Q9: Are forward rates the same as Fed dot-plot expectations?

A: Usually similar, sometimes very different. The Fed's own forecast is published quarterly in the Summary of Economic Projections (the "dot plot"). Market forwards are continuous and unbiased of any individual member's forecast. Through 2022-23 the market consistently priced cuts that the Fed dot-plot had not yet endorsed; the market turned out to be early. The gap between forwards and the dot-plot is one of the more reliable contrarian signals — pay attention when they diverge.

Q10: How does this connect to the rest of the course?

A: Week 5 introduced bonds; Week 18 traced the rate cascade through mortgages and credit; Week 10 used the inversion as a regime indicator; Week 23 discussed factor decay including the term factor; Week 32 covers duration and convexity (the math of how curve movements turn into bond returns). The yield curve is the spinal column the entire fixed-income half of this course is built on. After this week you should be able to read a Bloomberg yield curve panel as fluently as you read a price chart of SPY.

The interactive below lets you draw your own curve. Drag the five sliders. Watch the slope, the curvature, and the implied 1y1y forward update. Hit the presets to see history. Decide which world you think we are heading back to.