The Term Structure of Interest Rates: Spot, Par, and Forward Curves

Topic 9: The Term Structure of Interest Rates: Spot, Par, and Forward Curves

Learning Objectives Coverage

LO1: Define spot rates and the spot curve, and calculate the price of a bond using spot rates

Core Concept

• Definition: Spot rates are the yields on zero-coupon bonds for various maturities, representing the pure time value of money for each period. The spot curve plots these rates against maturity.
• Why it matters: Spot rates are the fundamental building blocks for valuing all fixed-income securities, providing arbitrage-free pricing and revealing market expectations about future interest rates.
• Key components:
  • Zero-coupon yields for each maturity
  • No-arbitrage pricing principle
  • Bootstrapping methodology
  • Term structure shapes (normal, flat, inverted)

Formulas & Calculations

• Bond Pricing with Spot Rates:

  PV = PMT/(1+z₁)¹ + PMT/(1+z₂)² + ... + (PMT+FV)/(1+zₙ)ᴺ
  

  Where: z₁, z₂, zₙ = spot rates for periods 1, 2, N

• Zero-Coupon Bond Price:

  PV = FV/(1+zₙ)ᴺ
  

• Bootstrapping Process:

  From 1-year bond: z₁ = YTM₁
  From 2-year bond: Solve for z₂ using known z₁
  Continue iteratively for longer maturities
  

• HP 12C Steps (Bond pricing with spot rates):

  Example: 3-year, 5% coupon, spots: 2%, 3%, 4%
  
  5 [ENTER] 1.02 [÷]        (Year 1: 4.902)
  5 [ENTER] 1.03 [x²] [÷]   (Year 2: 4.713)
  105 [ENTER] 1.04 [yˣ] 3 [÷] (Year 3: 93.345)
  [+] [+]                    (Total: 102.96)
  

Practical Examples

• Traditional Finance Example: Canadian government bond pricing

  5-year bond, 1% coupon
  Spot rates: 0.31%, 0.57%, 0.80%, 0.96%, 1.11%
  
  PV = 1/(1.0031) + 1/(1.0057)² + 1/(1.0080)³ + 1/(1.0096)⁴ + 101/(1.0111)⁵
  PV = 0.997 + 0.989 + 0.976 + 0.962 + 95.576 = 99.50
  

• Negative Rates Example: Swiss government bonds

  3-year bond, 0% coupon
  Spot rates: -0.79%, -0.71%, -0.64%
  PV = 100/(1-0.0064)³ = 101.94
  

DeFi Application

• Protocol example: Yield Protocol’s fixed-rate lending using fyTokens
• Implementation: fyTokens are zero-coupon tokens that mature to underlying asset, creating pure spot rates on-chain
• Advantages/Challenges:
  • Advantages: Transparent spot curve, composable with other protocols
  • Challenges: Limited maturity options, liquidity fragmentation
  • Example: fyDAI maturing in 6 months trading at 0.98 = 4.08% annualized spot rate

LO2: Define par and forward rates, and calculate par rates, forward rates from spot rates, spot rates from forward rates, and the price of a bond using forward rates

Core Concept

• Definition: Par rates are yields that price bonds at par value; forward rates are implied future spot rates between two periods derived from current spot rates.
• Why it matters: Par rates form the benchmark yield curves quoted in markets; forward rates reveal market expectations and enable hedging strategies.
• Key components:
  • Par-spot relationship
  • Forward rate extraction
  • No-arbitrage conditions
  • Expectations theory implications

Formulas & Calculations

• Par Rate Formula:

  100 = PMT/(1+z₁) + PMT/(1+z₂)² + ... + (PMT+100)/(1+zₙ)ᴺ
  Solve for PMT to get par rate
  

• Forward Rate Calculation:

  (1+z_A)^A × (1+IFR_{A,B-A})^{B-A} = (1+z_B)^B
  
  IFR_{A,B-A} = [(1+z_B)^B / (1+z_A)^A]^{1/(B-A)} - 1
  

• Bond Pricing with Forward Rates:

  PV = CF₁/(1+f₀,₁) + CF₂/[(1+f₀,₁)(1+f₁,₁)] + ... + CFₙ/[∏(1+fᵢ)]
  

• HP 12C Steps (Forward rate calculation):

  Example: Calculate 2y1y from z₂=3%, z₃=4%
  
  1.04 [ENTER] 3 [yˣ]       (1.1249)
  1.03 [ENTER] 2 [yˣ]       (1.0609)
  [÷]                        (1.0603)
  1 [-]                      (0.0603 = 6.03%)
  

Practical Examples

• Par Rate Calculation:

  Given spots: 5.26%, 5.62%, 6.36%, 7.01%
  Par rates: 5.26%, 5.61%, 6.31%, 6.90%
  Note: Par < Spot in upward sloping curve
  

• Forward Rate Example:

  3y1y forward with z₃=3.65%, z₄=4.18%
  (1.0365)³ × (1+f₃,₁) = (1.0418)⁴
  f₃,₁ = 5.79%
  

• Negative Forward Rates:

  Swiss 9y1y: z₉=-0.19%, z₁₀=-0.12%
  Forward rate = +0.54% (positive despite negative spots!)
  

DeFi Application

• Protocol example: Pendle Finance’s PT/YT decomposition
• Implementation: Principal Tokens (PT) trade at discount creating implied forward rates; Yield Tokens (YT) capture rate differential
• Advantages/Challenges:
  • Advantages: Market-determined forward rates, tradeable rate exposure
  • Challenges: Complex pricing, requires sophisticated users
  • Example: If stETH spot = 4% and PT implies 3% forward, YT captures the 1% differential

LO3: Compare the spot curve, par curve, and forward curve

Core Concept

• Definition: Three related representations of the term structure, each serving different analytical and practical purposes in fixed-income markets.
• Why it matters: Understanding relationships between curves enables proper security valuation, risk assessment, and trading strategy development.
• Key components:
  • Curve shapes and relationships
  • Information content differences
  • Market conventions
  • Arbitrage relationships

Formulas & Calculations

• Curve Relationships:

  Upward sloping: Forward > Spot > Par
  Flat: Forward = Spot = Par
  Inverted: Par > Spot > Forward
  

• Approximations:

  For small changes:
  Forward ≈ Spot + (Maturity × Slope of Spot Curve)
  Par ≈ Spot - (Duration × Convexity Adjustment)
  

Practical Examples

• Upward Sloping Curves (Normal):

  Maturity | Par    | Spot   | Forward
  1-year   | 2.00%  | 2.00%  | 2.00%
  2-year   | 2.98%  | 3.00%  | 4.01%
  3-year   | 3.93%  | 4.00%  | 6.03%
  

• Inverted Curves (Recession signal):

  Maturity | Par    | Spot   | Forward
  1-year   | 5.00%  | 5.00%  | 5.00%
  2-year   | 4.51%  | 4.50%  | 4.00%
  3-year   | 4.03%  | 4.00%  | 3.00%
  

DeFi Application

• Protocol example: Curve Finance’s gauge weights and veCRV voting
• Implementation: Different pools represent different “maturities” with varying yields, creating implicit term structure
• Advantages/Challenges:
  • Advantages: Market-driven rate discovery, governance participation
  • Challenges: Yields include token incentives, not pure interest rates
  • Observation: Longer lock periods (like veCRV) typically earn higher yields, mimicking upward sloping curve

Core Concepts Summary (80/20 Principle)

Essential Knowledge (20% that delivers 80% value)

1. Spot Rates Are Fundamental

   • All bond prices derive from spot rates
   • Zero-coupon yields for each maturity
   • No-arbitrage pricing principle

2. Forward Rates Are Expectations

   • Implied future spot rates
   • (1+z_long)^long = (1+z_short)^short × (1+forward)
   • Market’s best guess of future rates

3. Curve Shapes Tell Stories

   • Normal (upward): Growth expectations
   • Flat: Transition period
   • Inverted: Recession fears

4. Relationships Are Fixed

   • Upward: Forward > Spot > Par
   • Downward: Par > Spot > Forward
   • These relationships are mathematical certainties

Comprehensive Formula Sheet formula

Spot Rate Formulas

Zero-Coupon Bond: PV = FV/(1+z)ⁿ
Coupon Bond: PV = Σ CF_t/(1+z_t)^t
Spot from Price: z = (FV/PV)^(1/n) - 1
Bootstrapping: Use known spots to solve for unknown

Par Rate Formulas

Par Rate Definition: Price = 100 (par value)
100 = Σ PMT/(1+z_t)^t + 100/(1+z_n)^n
Par Rate = [100 - Σ PV(100)]/[Σ PV(1)]
For annual: Par ≈ (1 - 1/(1+z_n)^n) / Σ(1/(1+z_t)^t)

Forward Rate Formulas

Basic: (1+z_B)^B = (1+z_A)^A × (1+f_{A,B-A})^{B-A}
Forward Rate: f_{A,B-A} = [(1+z_B)^B/(1+z_A)^A]^{1/(B-A)} - 1
1-Period Forward: f_{n-1,1} = [(1+z_n)^n/(1+z_{n-1})^{n-1}] - 1
Pricing with Forwards: PV = CF₁/(1+f₀,₁) + CF₂/[(1+f₀,₁)(1+f₁,₁)] + ...

Curve Relationships

Upward Sloping: Forward > Spot > Par
Flat: Forward = Spot = Par
Inverted: Par > Spot > Forward
Steepness: Greater slope → Larger forward-spot spread

HP 12C Calculator Sequences hp12c

Calculate Bond Price Using Spot Rates

Example: 3-year, 4% coupon, spots: 2%, 2.5%, 3%

4 [ENTER] 1.02 [÷]           (Year 1: 3.922)
4 [ENTER] 1.025 [x²] [÷]     (Year 2: 3.810)
104 [ENTER] 1.03 [yˣ] 3 [÷]  (Year 3: 95.157)
[+] [+]                       (Total: 102.889)

Calculate Forward Rate

Example: 1y1y forward from z₁=2%, z₂=3%

1.03 [ENTER] [x²]    (1.0609)
1.02 [÷]             (1.0401)
1 [-]                (0.0401 = 4.01%)

Calculate Par Rate

Example: 2-year par rate from z₁=2%, z₂=3%

1 [ENTER] 1.02 [÷]        (0.9804)
1 [ENTER] 1.03 [x²] [÷]   (0.9426)
[+]                       (1.9230 = sum of PVs)
100 [ENTER] 1.03 [x²] [÷] (94.260 = PV of principal)
100 [x<>y] [-]            (5.740 = PV of coupons needed)
1.923 [÷]                 (2.985% = par rate)

Bootstrap Spot Rate

Example: Find z₂ given z₁=2%, 2-year 3% bond at par

100 [CHS] [PV]           (bond price)
3 [ENTER] 1.02 [÷] [+]   (add PV of first coupon = -97.06)
103 [FV]                 (final payment)
2 [n] 0 [PMT]            (setup for spot rate)
[i]                      (z₂ = 3.005%)

Practice Problems

Basic Level

1. Spot Rate Pricing

   • Q: Price a 2-year 4% annual bond with spots: z₁=3%, z₂=3.5%
   • A: PV = 4/1.03 + 104/1.035² = 3.883 + 97.017 = 100.90

2. Simple Forward Rate

   • Q: Calculate 1y1y forward from z₁=2%, z₂=2.5%
   • A: f₁,₁ = (1.025²/1.02) - 1 = 3.00%

3. Par vs Spot

   • Q: If 3-year spot = 4%, is 3-year par higher or lower?
   • A: Lower (par < spot in upward sloping curve)

Intermediate Level

1. Multi-Period Forward

   • Q: Find 2y2y forward from z₂=3%, z₄=4%
   • A: (1.04⁴/1.03²)^0.5 - 1 = 5.01%

2. Par Rate Calculation

   • Q: Calculate 3-year par from spots: 2%, 3%, 4%
   • A: 100 = C/1.02 + C/1.03² + (C+100)/1.04³ Solving: C = 3.94%

3. Negative Rates

   • Q: Price 2-year zero with spots: z₁=-0.5%, z₂=-0.3%
   • A: PV = 100/(1-0.003)² = 100.60

Advanced Level

1. Complete Curve Construction

   • Q: Given bonds: 1yr 2% at par, 2yr 3% at 101, find z₁, z₂, and f₁,₁
   • A: z₁ = 2%, z₂ = 2.48%, f₁,₁ = 2.96%

2. Arbitrage Opportunity

   • Q: 2yr zero at 96, z₁=2%, z₂=2.5%. Is there arbitrage?
   • A: Fair price = 100/1.025² = 95.18. Bond overpriced, sell bond and buy strips.

3. Forward Curve Inversion

   • Q: Given forwards: f₀,₁=3%, f₁,₁=2%, f₂,₁=1%. Find spot curve shape.
   • A: z₁=3%, z₂=2.50%, z₃=2.00%. Inverted spot curve.

DeFi Applications & Real-World Examples

Traditional Finance Examples

1. US Treasury Curve (Normal times)

   Maturity | Par Yield | Spot Rate | Forward
   1-year   | 2.50%     | 2.50%     | 2.50%
   5-year   | 3.00%     | 3.02%     | 3.52%
   10-year  | 3.50%     | 3.55%     | 4.55%
   30-year  | 4.00%     | 4.10%     | 5.30%
   

2. European Negative Rates

   Germany 10-year: -0.18% spot
   Switzerland 10-year: -0.12% spot
   Forward rates turning positive beyond 5 years
   

3. Emerging Market Curves

   Brazil: Steep upward slope (inflation expectations)
   Turkey: Very steep (currency risk premium)
   China: Relatively flat (controlled rates)
   

DeFi Protocol Comparisons

1. Notional Finance Term Structure

   3-month fDAI: 3.5% fixed rate
   6-month fDAI: 4.0% fixed rate
   1-year fDAI: 4.8% fixed rate
   Implied forward: 5.6% for 6m-1y period
   

2. Element Finance Principal Tokens

   3-month PT-yvUSDC: 0.985 (6.15% annualized)
   6-month PT-yvUSDC: 0.970 (6.12% annualized)
   Flat to slightly inverted curve
   

3. Curve Pool Implied Rates

   3pool base APY: 0.8%
   Tricrypto APY: 2.5%
   Term premium for volatility exposure: 170 bps
   

Innovative DeFi Structures

1. Pendle AMM for Yield Trading

   • Separate pools for each maturity
   • PT prices create spot curve
   • YT prices reveal forward rate expectations
   • Example: If 1yr PT at 0.95, spot = 5.26%

2. UMA Yield Dollar Synthetic

   • Creates synthetic fixed-rate exposure
   • Uses price feeds to determine payoffs
   • Enables custom term structures

3. 88mph Fixed-Rate Deposits

   • Floating depositors subsidize fixed depositors
   • Creates synthetic spot curve
   • Rates adjust based on pool utilization

Common Pitfalls & Exam Tips

Frequent Mistakes

1. Confusing Rate Types

   • Error: Using par rates for spot rate calculations
   • Fix: Par rates for quoted yields, spots for valuation
   • Remember: Published Treasury curves are par rates

2. Forward Rate Direction

   • Error: Thinking forwards predict future spots
   • Fix: Forwards are break-even rates, not predictions
   • Key: Include risk premiums in expectations

3. Compounding Confusion

   • Error: Not compounding forward rates correctly
   • Fix: Multiply (1+f) terms, don’t add rates
   • Example: Two 5% forwards ≠ 10% two-period rate

Exam Strategies

1. Quick Identifications

   • Upward curve: Use Forward > Spot > Par
   • See “zero-coupon”: Think spot rate
   • See “trades at par”: Think par rate

2. Calculator Efficiency

   • Program spots into memory registers
   • Use chain calculations for forward rates
   • Check: Forwards should amplify curve slope

3. Time Management

   • Start with curve shape identification
   • Use approximations for complex calculations
   • Verify relationships match curve shape

Conceptual Traps

1. Negative Rate Complications

   • Still use (1+r) format even if r < 0
   • Forward can be positive with negative spots
   • Price > Par for zero-coupon with negative yield

2. Liquidity Premium vs Expectations

   • Forward rates include risk premiums
   • Not pure expectations of future rates
   • Longer forwards have larger premiums

3. Curve Construction Issues

   • On-the-run vs off-the-run bonds
   • Tax effects distort municipal curves
   • Credit spreads affect corporate curves

Key Takeaways

Must-Know Concepts

1. Spot rates are fundamental zero-coupon yields
2. Forward rates are break-even future rates
3. Par rates are yields for bonds priced at par
4. Curve shape determines rate relationships
5. No-arbitrage links all three curves

Critical Formulas

• Spot pricing: PV = Σ CF/(1+z_t)
• Forward rate: (1+z_B)^B = (1+z_A)^A × (1+f)^(B-A)
• Curve relationships: Shape determines Forward vs Spot vs Par ordering

DeFi Applications

• Fixed-rate protocols create on-chain spot curves
• Yield tokenization enables forward rate trading
• AMM pools imply term structure through yields
• Governance tokens add complexity to pure rates

Cross-References & Additional Resources

Related Finance Topics

• Bond Valuation (uses spot rates)
• Yield Measures (par rates are yields)
• Duration (sensitivity to curve shifts) duration
• Curve-Based Risk Measures (key rate durations)

DeFi Resources

• Notional Docs - Fixed rate lending
• Pendle Docs - Yield tokenization
• Element Docs - Principal/Yield splitting
• Yield Protocol - fyToken system

Practice Platforms

• Bloomberg Terminal: GC functions for curve analysis
• Python: QuantLib for term structure modeling
• DeFi: Dune dashboards for protocol yields
• Excel: Data Analysis ToolPak for bootstrapping

Review Checklist

Conceptual Understanding

• Can explain difference between spot, par, and forward rates
• Understand no-arbitrage pricing principle
• Know curve shape implications
• Can interpret forward rates correctly

Calculation Proficiency

• Price bonds using spot rates
• Calculate forward rates from spots
• Derive par rates from spot curve
• Bootstrap spot rates from bond prices

Application Skills

• Compare bonds using appropriate rates
• Identify arbitrage opportunities
• Apply concepts to DeFi protocols
• Interpret real-world yield curves

DeFi Integration

• Understand fixed-rate protocol mechanics
• Can analyze yield tokenization
• Know how AMMs create implicit curves
• Can evaluate cross-protocol rate differences