Yield Curve Analysis

When to Activate

• Analyzing term structure of interest rates for trading signals or macro assessment
• Designing curve trades (steepeners, flatteners, butterflies)
• Interpreting yield curve inversions and their economic implications
• Calculating key rate durations and understanding curve risk
• Evaluating central bank policy impact on different parts of the curve

Core Concepts

Yield Curve Shapes

Normal (Upward Sloping):

• Short rates < long rates
• Reflects term premium (compensation for holding longer duration) and expected rate path
• Typical in healthy economic expansion
• Spread (10Y - 2Y) historically averages 100-150 bps

Inverted:

• Short rates > long rates
• 2s10s inversion has preceded every US recession since 1955 (with variable lead time of 6-24 months)
• Reflects market expectation that rates will be cut due to economic slowdown
• Can be driven by aggressive Fed tightening (front end) or flight to safety (long end)
• False positives are rare but exist (1966, arguably 1998)

Flat:

• Minimal spread between short and long rates
• Transition state between normal and inverted, or vice versa
• Reduces bank profitability (banks borrow short, lend long)

Humped (Bear Flattener):

• Belly of the curve (5Y-7Y) yields higher than both ends
• Can occur during transitional monetary policy periods
• Less common, but creates butterfly trade opportunities

Term Structure Theories

Pure Expectations Theory:

• Forward rates equal expected future spot rates
• Long rates are geometric average of expected short rates
• Implication: inverted curve = market expects rate cuts
• Reality: ignores risk premium, but useful as baseline

Liquidity Premium Theory:

• Investors demand extra compensation for holding longer maturities
• Term premium should always be positive (not always true empirically)
• Term premium estimation: Adrian-Crump-Moench (ACM) model, Kim-Wright model
• Current term premium is observable via Fed models

Preferred Habitat Theory:

• Different investors have preferred maturities (pension funds want long, banks want short)
• Supply/demand imbalances at specific tenors drive relative pricing
• Explains humps and kinks in the curve
• QE disrupts natural supply/demand by removing duration from the market

Key Curve Metrics

Spreads:

2s10s = 10Y yield - 2Y yield (most watched recession indicator)
2s30s = 30Y yield - 2Y yield (steeper, more sensitive to term premium)
3m10Y = 10Y yield - 3M T-bill yield (Fed's preferred recession indicator)
5s30s = 30Y yield - 5Y yield (long-end steepness)
Fed funds - 2Y = front-end tightness

Term Premium:

10Y yield = Expected average of short rates over 10 years + Term premium

Term premium drivers:
- Inflation uncertainty (higher uncertainty = higher premium)
- Supply of duration (more Treasury issuance = higher premium)
- Central bank holdings (QE reduces premium, QT increases it)
- Foreign demand (strong foreign demand compresses premium)
- Historically: -0.5% to +2.5% range; negative term premium from 2015-2021 due to QE

Key Rate Durations:

• Sensitivity of a bond's price to a 1 bp change at a specific maturity point
• Used to decompose curve risk: how much do I lose if the 5Y rate rises 10 bps but 10Y stays flat?
• Essential for hedging: match key rate duration profile, not just overall duration

Curve Fitting Methods

Bootstrapping:

• Derive zero-coupon (spot) rates from observed coupon bond prices
• Sequential: use shortest maturity first, solve for each spot rate
• Produces exact fit to observed prices

Nelson-Siegel Model:

• Parameterizes the curve with 4 parameters: level, slope, curvature, decay
• y(tau) = beta0 + beta1 * ((1-exp(-tau/lambda))/(tau/lambda)) + beta2 * ((1-exp(-tau/lambda))/(tau/lambda) - exp(-tau/lambda))
• Smooth, parsimonious, widely used by central banks
• Svensson extension adds a second hump parameter

Spline Methods:

• Cubic splines fit piecewise polynomials through observed points
• More flexible but can overfit
• Used when precision at specific tenors matters

Methodology

Curve Trade Construction

Bull Steepener (expect rate cuts, long end stable):

View: Fed will cut rates; front end rallies more than long end
Trade: Long 2Y futures, Short 10Y futures (duration-weighted)
DV01 ratio: 2Y DV01 / 10Y DV01 to equalize interest rate sensitivity
Profit if 2s10s spread widens (steepens)
Risk: if long end sells off more than front end (bear steepener)

Bear Flattener (expect rate hikes, long end anchored):

View: Fed tightening; front end sells off, long end anchored by slowing growth expectations
Trade: Short 2Y futures, Long 10Y futures (duration-weighted)
Profit if 2s10s spread narrows (flattens)
Risk: if curve steepens instead (growth surprises to upside)

Butterfly:

View: belly of curve is mispriced relative to wings
Trade: 2s5s10s butterfly
- Long wings (2Y and 10Y), Short belly (5Y)
- Or vice versa

Weighting: match DV01 so total portfolio duration is zero
Example: Long 1x 2Y, Short 2x 5Y, Long 1x 10Y (approximate weights)
Profit: if 5Y yield moves relative to interpolated 2Y-10Y line

Barbell vs bullet: butterfly trades express view on curvature

Curve Signal Analysis

1. Plot current curve against historical percentiles for each tenor
2. Decompose moves into level, slope, and curvature using PCA (first 3 components explain ~99%)
3. Compare implied forwards with market expectations (Fed funds futures, OIS)
4. Assess term premium using ACM or Kim-Wright models
5. Monitor curve momentum — curve steepening/flattening trends tend to persist
6. Cross-reference with macro — leading indicators, PMI, inflation expectations

Recession Probability from Curve

Models:
- NY Fed model: uses 3m10Y spread to estimate recession probability
  Probit regression: P(recession in 12 months) = Phi(beta0 + beta1 * spread)
  When 3m10Y inverts: ~30-40% recession probability
  When 3m10Y at -100bps: ~60-80% recession probability

- 2s10s alternative: similar signal but historically less reliable than 3m10Y

Caveats:
- Lead time varies: 6-24 months from inversion to recession
- QE may distort term premium, making inversions less meaningful
- "This time is different" arguments emerge every cycle (and are usually wrong)

Examples

Example 1: Post-Inversion Steepener

Scenario: 2s10s has been inverted at -80bps for 6 months
3-month T-bill: 5.25%, 2Y: 4.80%, 10Y: 4.00%
Fed signaling end of tightening cycle

Trade: Bull steepener
- Long 2Y Treasury futures (2 contracts, DV01 = $40/bp each)
- Short 10Y Treasury futures (1 contract, DV01 = $80/bp)
- Net DV01: zero (duration neutral)

Thesis: 2Y will rally more than 10Y as Fed pivots to cuts
Target: 2s10s moves from -80bps to 0bps
P&L: 80bps * $80 DV01 = $6,400 per unit

Risk: if Fed delays cuts and front end stays elevated
Stop: if 2s10s widens to -120bps (loss = $3,200 per unit)
Risk/reward: 2:1

Example 2: Butterfly on Humped Curve

Current yields: 2Y = 4.50%, 5Y = 4.70%, 10Y = 4.40%
5Y looks rich relative to wings
Implied 5Y rate from 2Y-10Y interpolation: 4.45%
5Y is 25bps above theoretical (butterfly spread = +25bps)
Historical average butterfly: +5bps

Trade: Sell the butterfly (short belly, long wings)
- Long 1x 2Y futures
- Short 2.2x 5Y futures
- Long 1x 10Y futures
- DV01 neutral overall

Target: butterfly normalizes to +5bps (20bps profit)
Stop: butterfly widens to +40bps (15bps loss)
Catalyst: front-end rally on dovish Fed, or long-end selloff on supply concerns

Example 3: Curve and Equity Signal

Signal analysis:
- 2s10s inverted 14 months ago (currently at -20bps, steepening)
- Historical pattern: equities peak 3-6 months after inversion
- Steepening from inversion = recession approaching (Fed about to cut)
- S&P 500 has been flat for 3 months (topping pattern?)

Framework:
- Phase 1: Curve inverts -> equities continue rising (6-18 months)
- Phase 2: Curve begins steepening from inversion -> equities peak
- Phase 3: Recession begins -> curve steepens aggressively -> equities decline 20-40%
- Phase 4: Recovery -> curve normalizes -> equities bottom

Current assessment: Late Phase 2. Reduce equity exposure.
Position for further steepening. Build cash for Phase 3 opportunities.

Quality Gate

Before implementing curve trades, verify:

• Duration-weighted properly — curve trades should be DV01-neutral unless intentionally taking directional risk
• Carry and roll-down calculated — what does the trade earn if curve stays unchanged?
• Financing costs included — repo rates for funded positions, margin costs for futures
• Macro thesis is explicit — what economic scenario drives the trade? What would invalidate it?
• Historical context checked — where is the spread relative to historical range and percentiles?
• Term premium decomposition performed — is the spread driven by rate expectations or term premium?
• Central bank positioning assessed — QE/QT schedule, forward guidance, and how they affect specific tenors
• Liquidity considered — on-the-run vs off-the-run, auction cycle effects on pricing
• Roll dates accounted for — futures rolls create basis risk and transaction costs
• Cross-market confirmation — do credit spreads, equity vol, and FX carry confirm the macro view?