Yield Curve Analysis: Spot Rates, Forward Rates, Butterfly Trades, and Carry Strategies

The Yield Curve as the Foundation of Fixed Income Markets

Every fixed income instrument — government bond, corporate note, interest rate swap, mortgage-backed security — is priced relative to a benchmark yield curve. Understanding the yield curve means understanding the price of time in financial markets: what return do investors demand to lend money for 2 years versus 10 years versus 30 years? This question has practical consequences for every trade a fixed income professional makes. This guide covers the mechanics (spot rate bootstrapping, forward rate derivation), the strategies (bullet vs. barbell, butterfly, carry/roll-down), and the analytics (Nelson-Siegel fitting, curve as macro indicator) with real rate data and Claude prompts that produce institutional-grade output.

For reference, approximate SOFR curve levels as of mid-2026: 3M: 4.95%, 6M: 4.90%, 1Y: 4.80%, 2Y: 4.85%, 3Y: 4.75%, 5Y: 4.65%, 7Y: 4.55%, 10Y: 4.45%, 20Y: 4.50%, 30Y: 4.60%. The curve shows a mild inversion at the front end (1Y/2Y) and modest positive slope from 5Y to 30Y — a mixed signal environment typical of a late-cycle Fed pause.

Yield Curve Shapes and Their Economic Interpretation

The yield curve's shape encodes market expectations about the future path of short-term interest rates (expectations component) and term premium demanded for holding longer-duration bonds (term premium component). The four canonical shapes each tell a distinct macro story.

Normal (upward sloping): Long rates exceed short rates. Reflects expectations of stable or rising growth and inflation over time; investors demand a term premium for the greater interest rate risk of longer bonds. Normal for most of post-war history; typical slope 2s10s = +100 to +200bps in expansion.

Inverted: Short rates exceed long rates. Reflects tight monetary policy (Fed above neutral) and market expectation that rates will fall — either because the Fed cuts or because the economy weakens. The 2s10s spread inverted to −107bps in July 2023, its most negative since 1981.

Flat: Minimal spread between short and long rates. Transition period — either flattening from normal (late-cycle Fed tightening) or steepening from inversion (anticipation of Fed cuts). 2s10s near zero is a flat curve.

Humped: Intermediate maturities (5-7Y) yield more than both short (2Y) and long (30Y) tenors. Often reflects a specific market dislocation — supply of 5-7Y supply, specific duration demand at long end from pension liabilities, or disrupted central bank policy expectations at the belly.

• "Using the following SOFR spot rates: 2Y=4.85%, 5Y=4.65%, 10Y=4.45%, 30Y=4.60% — classify the yield curve shape and interpret the economic signal. Calculate: (1) the 2s10s spread; (2) the 5s30s spread; (3) the 2s5s10s butterfly spread (5Y yield minus the interpolated midpoint of 2Y and 10Y). Explain what each spread tells a fixed income portfolio manager about the macro outlook and where to position on the curve."

Par Curve vs Spot Curve vs Forward Curve

Three curves — same rates market, three different ways to look at it.

Par curve: The yield on a coupon bond that prices exactly at par for each maturity. Observable from on-the-run Treasuries. Has a "coupon effect" — the cash flows from intermediate coupons are all discounted at the same par yield, which is an approximation.

Spot (zero) curve: The yield on a hypothetical zero-coupon bond at each maturity. Each cash flow is discounted at its maturity-specific rate. More theoretically precise. Derived from the par curve by bootstrapping. The spot curve is used for all rigorous pricing of fixed income instruments.

Forward curve: Implied future interest rates. The forward rate f(T1,T2) is the rate that makes you indifferent between: (a) investing at the T2 spot rate today, vs. (b) investing at the T1 spot rate today and rolling to a new investment at f(T1,T2) at time T1.

Relationship: (1 + s_T2)^T2 = (1 + s_T1)^T1 × (1 + f(T1,T2))^(T2−T1)

Solving for f: f(T1,T2) = [(1+s_T2)^T2 / (1+s_T1)^T1]^(1/(T2−T1)) − 1

• "Using these annual spot rates: s1=4.80%, s2=4.85%, s3=4.78%, s5=4.65%, s7=4.55%, s10=4.45% — calculate the following implied forward rates: (1) 1-year rate 1 year from now f(1,1); (2) 1-year rate 2 years from now f(2,1); (3) 3-year rate 2 years from now f(2,3); (4) 5-year rate 5 years from now f(5,5). Show the formula and intermediate steps for each. Interpret the f(5,5) as the market's implied 5-year rate in 2031."

Spot Rate Bootstrapping: Step-by-Step

Bootstrapping extracts zero-coupon rates from observable par yields. The process is sequential — each step uses previously derived spot rates to strip coupon payments from the current maturity's bond, isolating the terminal cash flow to solve for the marginal spot rate.

Worked example using par yields: 6M=4.90%, 1Y=4.80%, 1.5Y=4.82%, 2Y=4.85%

• 6M spot rate: A 6-month par bond pays 4.90%/2 = 2.45% coupon + 100 at 6M. Since there is only one cash flow, s_0.5 = 4.90% ✓
• 1Y spot rate: 1Y par bond, coupon = 4.80%/2 = 2.40%. Price = 100 = 2.40/(1+0.049/2)^1 + 102.40/(1+s_1/2)^2. Substituting: 100 = 2.40/1.0245 + 102.40/(1+s_1/2)^2 → 100 = 2.3426 + 102.40/(1+s_1/2)^2 → (1+s_1/2)^2 = 102.40/97.6574 = 1.04855 → s_1 = 2×(1.04855^0.5 − 1) = 2×0.02401 = 4.803%
• 1.5Y spot rate: Par bond at 4.82% coupon = 2.41%/period. Use s_0.5=4.90% and s_1=4.803%. Solve for s_1.5: 100 = 2.41/1.0245 + 2.41/1.04855 + 102.41/(1+s_1.5/2)^3. Compute first two terms = 2.353 + 2.299. Third term: (1+s_1.5/2)^3 = 102.41/(100−4.652) = 102.41/95.348 = 1.07400 → s_1.5 = 2×(1.07400^(1/3) − 1) = 4.824%
• 2Y spot rate: Continue the same pattern to get s_2 ≈ 4.860%

• "Bootstrap spot rates from the following Treasury par yields (semi-annual coupon equivalent): 6M=4.90%, 1Y=4.80%, 1.5Y=4.82%, 2Y=4.85%, 3Y=4.78%, 5Y=4.65%, 7Y=4.55%, 10Y=4.45%. Show the full bootstrapping table with: maturity, par yield, coupon (semi-annual), PV of coupon cash flows using previously derived spot rates, implied spot rate calculation. Plot the par curve vs spot curve and explain any differences between them."

Key Rate Analysis: 2s5s10s, 2s10s, 5s30s Spreads

Fixed income professionals track specific spread pairs as shorthand for curve positioning and macro signals. These are the key ones with current approximate levels.

2s10s spread (4.85% − 4.45% = +40bps): The single most-watched spread. Positive = normal slope. A tightening 2s10s from wide levels signals recession expectations materializing; a steepening from inversion signals anticipation of Fed cuts.

2s5s (4.85% − 4.65% = +20bps): Short-dated slope. Reflects expectations about the Fed over the next 2-5 years. Currently a modest positive slope suggesting the market expects modest easing over the medium term.

5s30s (4.65% − 4.60% = +5bps): Long-dated slope. Reflects term premium and long-run growth/inflation expectations. Very flat 5s30s signals that long-term inflation expectations are anchored — the market does not demand a large premium for long-duration bonds.

2s5s10s butterfly: 5Y yield − (0.5 × 2Y yield + 0.5 × 10Y yield) = 4.65% − (0.5 × 4.85% + 0.5 × 4.45%) = 4.65% − 4.65% = 0bps. A flat butterfly. Negative butterfly (body rich to wings) suggests buying butterfly (long wings, short body); positive butterfly (body cheap) suggests selling butterfly.

• "Using current SOFR curve: 2Y=4.85%, 5Y=4.65%, 7Y=4.55%, 10Y=4.45%, 30Y=4.60%: Calculate (1) 2s10s spread and its Z-score vs. the 10-year average of +125bps; (2) 5s30s spread; (3) 2s5s10s butterfly spread; (4) 2s7s30s fly. Recommend whether to position for steepening or flattening of the 2s10s based on current Fed policy cycle, providing a duration-neutral trade structure with DV01 of each leg."

Yield Curve Strategies: Bullet, Barbell, Butterfly

The shape of a portfolio's maturity distribution determines how it performs under different curve scenarios. Three archetypes dominate curve positioning strategy.

Bullet: Concentrated around a single maturity point (e.g., all 10Y). High KRD at 10Y, minimal KRDs elsewhere. Outperforms if the 10Y rallies; underperforms if the curve steepens (10Y sells off relative to front end).

Barbell: Concentrated at short and long maturities (e.g., 2Y + 30Y), with minimal exposure at the belly (10Y). Same total duration as a bullet, but with large KRDs at 2Y and 30Y. Outperforms if the curve steepens (2Y rallies more than 30Y) or if convexity is rewarded (barbells have more convexity than bullets at the same duration).

Butterfly (long wings, short body): Long 2Y + long 10Y, short 5Y. Duration-neutral across the trade. Profits if the 5Y cheapens relative to the interpolated 2Y-10Y midpoint. The butterfly spread is the classic curvature trade.

Structuring a 2s5s10s butterfly with $100M DV01 budget:

• DV01 of 5Y position = $100M × 4.51 × 0.0001 = $45,100 (the "body")
• Wing allocation: split body DV01 equally, 50% to 2Y and 50% to 10Y
• 2Y position to generate $22,550 DV01: $22,550 / (1.90 × 0.0001) = $118.7M face
• 10Y position to generate $22,550 DV01: $22,550 / (8.35 × 0.0001) = $27.0M face

• "Structure a duration-neutral 2s5s10s butterfly trade with the following instruments: 2Y Treasury (4.85% coupon, price 100.05, ModD 1.91, DV01/MM $191); 5Y Treasury (4.25% coupon, price 99.50, ModD 4.51, DV01/MM $448); 10Y Treasury (4.00% coupon, price 97.20, ModD 8.35, DV01/MM $810). Target DV01 of body position = $500,000. Calculate: (1) face value of each leg; (2) net DV01 of the trade confirming duration neutrality; (3) P&L if the 5Y yield rises 10bps while 2Y and 10Y are unchanged; (4) P&L if the curve has a parallel shift of +50bps."

Carry and Roll-Down: The Hidden Return in Fixed Income

A bond's total return has three components: (1) coupon income, (2) price change from rate moves, and (3) carry and roll-down — the return earned even if the yield curve does not move at all.

Carry = coupon yield minus financing cost (repo rate). If a 5Y Treasury yields 4.65% and the overnight repo rate is 4.95%, carry is negative (−30bps annualized). If the curve is inverted (front end above 5Y), carry is a headwind. In a normal curve environment (repo below 5Y yield), carry is positive.

Roll-down is the return from time passing on an upward-sloping curve. A 10Y bond rolling to a 9Y tenor (one year later) moves to a lower spot rate if the curve is upward-sloping. This price appreciation is the roll-down return.

Roll-down calculation: If s_10Y = 4.45% and s_9Y = 4.35%, the bond rolls 10bps down the curve. Roll-down return ≈ ModD_9Y × 0.10% ≈ 8.0 × 0.10% = 0.80% over one year.

Carry + Roll-Down Breakeven: Rates must rise by more than (Carry + Roll-Down) / ModD to produce a negative total return over the holding period.

• "Calculate carry and roll-down for a 10-year Treasury bond: coupon 4.00%, current price $97.20 (YTM=4.45%), current overnight repo rate 4.95%. Holding period = 6 months. Spot rates: s_10Y=4.45%, s_9.5Y=4.40%. (1) Calculate 6-month carry (coupon income minus repo cost as % of price); (2) calculate 6-month roll-down (price change from rolling from 10Y to 9.5Y on the spot curve using ModD_9.5Y ≈ 8.10); (3) calculate total carry+roll-down; (4) calculate the breakeven yield rise that would wipe out the carry+roll-down return."

• "Compare carry and roll-down for three positions on the SOFR curve (repo rate 4.95% for all; holding period 12 months): (A) 2Y Treasury, yield 4.85%, price 100.05, ModD 1.91; 1Y spot rate in 1 year implied = 4.70%; (B) 5Y Treasury, yield 4.65%, price 99.50, ModD 4.51; 4Y spot rate in 1 year = 4.57%; (C) 10Y Treasury, yield 4.45%, price 97.20, ModD 8.35; 9Y spot rate in 1 year = 4.37%. Which position has the best carry+roll-down, and what is the breakeven rate rise for each?"

Nelson-Siegel Model: Fitting the Yield Curve Analytically

The Nelson-Siegel model (1987) fits the entire yield curve using three parameters that correspond to economically meaningful factors: level, slope, and curvature.

Formula:
y(τ) = β₀ + β₁ × [(1−e^(−τ/λ))/(τ/λ)] + β₂ × [(1−e^(−τ/λ))/(τ/λ) − e^(−τ/λ)]

Where τ = maturity, λ = decay parameter (typically fitted, often ~2-3).

• β₀ = long-run level (the yield as τ→∞, i.e., the 30Y rate in practice)
• β₁ = slope factor (loading decreases with maturity; determines 2s30s slope; negative value = inverted curve)
• β₂ = curvature factor (hump-shaped loading peaking around 2-3 years depending on λ; determines belly richness/cheapness)

With current rates (β₀ ≈ 4.60, β₁ ≈ −0.20, β₂ ≈ 0.10, λ = 2.5), the model produces: 2Y: 4.80%, 5Y: 4.65%, 10Y: 4.48%, 30Y: 4.60% — a reasonable fit to observable market rates.

• "Fit a Nelson-Siegel model to the following observed Treasury yields: 3M=4.97%, 6M=4.93%, 1Y=4.82%, 2Y=4.85%, 3Y=4.78%, 5Y=4.65%, 7Y=4.55%, 10Y=4.45%, 20Y=4.50%, 30Y=4.60%. Set the decay parameter λ=2.5. Use OLS to estimate β₀, β₁, β₂ that minimize sum of squared residuals. Report: (1) estimated parameters with interpretation; (2) model-fitted yields vs observed yields; (3) residuals — which tenor is richest (most negative residual) and cheapest (most positive residual) relative to the model fit?"

Yield Curve as Recession Predictor: 2s10s Inversion Analysis

The 2s10s Treasury spread is the single most-cited leading indicator in macroeconomics. The mechanism: when the Fed raises short-term rates above long-term rates, the cost of credit (short-term funding) exceeds the return on credit (long-term loans). Bank net interest margins compress, credit creation slows, and economic activity weakens. The curve inversion has preceded all 8 US recessions since 1968 with no false positives, though lead times range from 6 to 24 months.

The 2022-2024 inversion reached −107bps in July 2023. Despite a widely predicted recession, US GDP growth remained positive through 2024. Explanations: (1) pandemic savings buffer delayed the credit squeeze; (2) fiscal deficit kept aggregate demand elevated; (3) supply-side inflation rather than demand destruction meant rate rises less contractionary. The post-2025 re-steepening to +40bps reflects Fed easing expectations materializing as inflation returned to target.

• "Analyze the yield curve recession signal using current data: 2Y SOFR=4.85%, 10Y Treasury=4.45%, 2s10s=+40bps. Historical context: the 2022-2024 inversion reached -107bps in July 2023. (1) Is the current +40bps spread consistent with expansion or late-cycle positioning? (2) Using the Sahm Rule framework (recession signal when 3M average unemployment rises 0.5% above 12M low), describe how to combine the yield curve signal with Sahm Rule for a more robust recession probability estimate. (3) What bond portfolio duration and curve positioning would be optimal if the probability of recession within 12 months is 25% vs 60%?"

Frequently Asked Questions

What is the difference between a steepener and a flattener trade?

A steepener profits when the yield curve steepens — the spread between a longer maturity and shorter maturity increases. A long-end steepener example: short 2Y Treasury, long 10Y Treasury (both duration-neutral). You profit if 2Y yields fall more than 10Y yields, or if 10Y yields rise more than 2Y yields. A flattener is the reverse: long 2Y, short 10Y. Flatteners typically profit during Fed tightening cycles (front end rises faster). Steepeners typically profit during early-cycle easing or flight-to-quality episodes when the front end rallies sharply. See interest rate swap valuation for structuring these trades via swap markets.

What data source should I use for yield curve data in Claude?

For US Treasuries, the FRED database (Federal Reserve Bank of St. Louis) provides daily constant-maturity Treasury yields (series DGS1MO through DGS30). For SOFR OIS rates, use SOFR swap quotes from Bloomberg or CME. Paste the relevant rates directly into Claude — for example: "2Y=4.85%, 5Y=4.65%, 10Y=4.45%, 30Y=4.60%" — and Claude will derive spreads, fit models, and build strategies around your inputs. For live rate data, integrate Claude with the ClaudeFinanceLab MCP tools described in the quant finance section.

How does the swap curve relate to the Treasury curve?

The swap rate at each maturity is the fixed rate in an at-market interest rate swap — the rate that makes the swap's NPV zero at inception. Historically, swap rates exceeded Treasury yields because swaps carry counterparty credit risk (interbank credit). The swap spread (swap rate minus same-maturity Treasury yield) was typically +20 to +50bps for the 10Y. Post-2008, swap spreads compressed and even turned negative in some cases due to regulatory changes, balance sheet constraints on dealer intermediation, and changes in the credit risk profile of cleared swaps. The I-spread (bond yield minus same-maturity swap rate) is preferred over G-spread (vs. Treasuries) for corporate bonds because it better reflects credit risk premium over the swap curve.

Can Claude build a full yield curve model from scratch?

Yes. Provide par yields at standard maturities and Claude will: (1) bootstrap spot rates sequentially; (2) derive the forward curve; (3) fit a Nelson-Siegel model; (4) compute carry and roll-down; (5) structure duration-neutral curve trades. The output includes worked calculations and can be formatted as a Python script for automation. See also fixed income analysis with AI and interest rate swap valuation for related workflows using the ClaudeFinanceLab toolset.