Yield and Yield Spread Measures for Fixed-Rate Bonds

Topic 7: Yield and Yield Spread Measures for Fixed-Rate Bonds

Learning Objectives Coverage

LO1: Calculate annual yield on a bond for varying compounding periods in a year

Core Concept

• Definition: Bond yields can be expressed with different compounding frequencies (periodicity), requiring conversion between conventions for accurate comparison. The periodicity indicates how many times per year interest compounds.
• Why it matters: Different markets and securities use different yield conventions - comparing yields without proper conversion leads to incorrect investment decisions
• Key components:
  • Effective annual rate (periodicity = 1)
  • Semiannual bond basis yield (periodicity = 2, most common)
  • Quarterly and monthly compounding yields
  • Zero-coupon bond yield calculations

Formulas & Calculations

• Periodicity Conversion Formula:

  (1 + APRm/m)^m = (1 + APRn/n)^n
  

  Where: APRm = annual rate for m periods, APRn = annual rate for n periods

• Zero-Coupon Bond Yield:

  PV = FV/(1 + r)^N
  r = (FV/PV)^(1/N) - 1
  

• Effective Annual Rate from Periodic Rate:

  EAR = (1 + r/m)^m - 1
  

  Where: r = stated annual rate, m = compounding periods per year

• HP 12C Steps (Zero-coupon bond yield):

  80 [CHS] [PV]    (bond price negative)
  100 [FV]         (face value)
  5 [n]            (years to maturity)
  [i]              (compute annual yield = 4.564%)
  

Practical Examples

• Traditional Finance Example: Five-year zero-coupon bond priced at 80 per 100 of par
  • Annual compounding yield: 4.564%
  • Quarterly compounding: 4.488% (80 = 100/(1+r)^20; r = 1.122% × 4)
  • Monthly compounding: 4.476%
• Negative Yield Example: German 5-year zero at 103.72
  • Effective annual rate: -0.7278%
  • Semiannual equivalent: -0.7291%
  • Monthly equivalent: -0.7303%

DeFi Application

• Protocol example: Yearn Finance yield aggregator reporting
• Implementation: Yearn vaults display APY (annual percentage yield) with daily compounding, while lending protocols like Aave show APR (annual percentage rate) without compounding
• Advantages/Challenges:
  • Advantages: Transparent yield comparison across protocols with different compounding frequencies
  • Challenges: Users often confuse APR and APY, leading to suboptimal allocation decisions
  • Key insight: A 10% APR with daily compounding = 10.52% APY

LO2: Compare, calculate, and interpret yield and yield spread measures for fixed-rate bonds

Core Concept

• Definition: Multiple yield measures exist to evaluate bond returns and relative value, including current yield, YTM, yield-to-call, and various spread measures (G-spread, Z-spread, OAS)
• Why it matters: Different yield measures serve different analytical purposes - choosing the wrong measure can lead to mispricing and poor investment decisions
• Key components:
  • Current yield (income return only)
  • Yield-to-maturity (total return if held to maturity)
  • Yield-to-call and yield-to-worst (for callable bonds)
  • Spread measures for credit and liquidity risk assessment

Formulas & Calculations

• Current Yield:

  CY = Annual Coupon / Bond Price
  

• G-Spread (Government Spread):

  G-Spread = Bond YTM - Benchmark Government YTM
  

• Z-Spread Formula:

  PV = PMT/(1+z₁+Z)¹ + PMT/(1+z₂+Z)² + ... + (PMT+FV)/(1+zN+Z)^N
  

  Where: z = spot rates, Z = constant spread

• Option-Adjusted Spread:

  OAS = Z-spread - Option value (in basis points)
  

• Government Equivalent Yield:

  YieldACT/ACT = (365/360) × Yield30/360
  

Practical Examples

• Antelas AG vs BRWA Comparison:

  Antelas AG: 3.20% quarterly coupon, Price 94
  - Current yield: 3.40%
  - YTM: 4.548%
  
  BRWA: 2.50% semiannual coupon, Price 98.70
  - Current yield: 2.53%
  - YTM: 2.780%
  
  Spread = 176.8 basis points
  

• VIVU Callable Bond Analysis:

  6.5% Seven-Year Callable Notes:
  - Yield-to-first call: 5.149%
  - Yield-to-second call: 5.247%
  - Yield-to-maturity: 5.374%
  - Yield-to-worst: 5.149% (lowest scenario)
  

• Apple Bond Spread Analysis (Sept 2047):

  3.75% coupon:
  - Treasury spread: 72 bps
  - G-spread: 76 bps
  - I-spread: 106 bps
  - Z-spread: 109 bps
  

DeFi Application

• Protocol example: Maple Finance institutional lending spreads
• Implementation: Maple pools show spreads over USDC base rates, similar to traditional credit spreads over risk-free rates
• Advantages/Challenges:
  • Advantages: Transparent risk pricing, real-time spread updates based on pool utilization
  • Challenges: No true “risk-free” rate in DeFi, spreads can be volatile during market stress
  • Example: Corporate lending pool at 12% vs USDC base rate of 4% = 800 bps spread

Core Concepts Summary (80/20 Principle)

Essential Knowledge (20% that delivers 80% value)

1. Yield Periodicity Conversions

   • Always convert to same periodicity before comparing yields
   • Higher compounding frequency = lower stated rate for same effective yield
   • Formula: (1 + r/m)^m = (1 + r/n)

2. Yield Measure Hierarchy

   • Current Yield: Quick income measure (Coupon/Price)
   • YTM: Total return if held to maturity
   • YTC: Return if called early
   • YTW: Conservative measure for callable bonds

3. Spread Measures

   • G-spread: Simple spread over government bond
   • Z-spread: Constant spread over entire spot curve
   • OAS: Z-spread adjusted for embedded options
   • Rule: OAS < Z-spread for callable bonds

4. Key Relationships

   • Premium bonds: Coupon > YTM, Price > Par
   • Discount bonds: Coupon < YTM, Price < Par
   • Callable bonds: YTW ≤ YTM

Comprehensive Formula Sheet formula

Yield Conversions

Periodicity Conversion: (1 + APRm/m)^m = (1 + APRn/n)^n
Effective Annual Rate: EAR = (1 + r/m)^m - 1
Semiannual to Annual: (1 + r/2)^2 = 1 + EAR
Zero-Coupon Yield: r = (FV/PV)^(1/N) - 1

Yield Measures

Current Yield: CY = Annual Coupon / Bond Price
Simple Yield: (Coupon + (Par-Price)/Years) / Price
YTM: Solve for r in: PV = Σ CF/(1+r)^t
YTC: Same as YTM but using call price and call date

Spread Measures

G-Spread = Bond YTM - Government Bond YTM
I-Spread = Bond YTM - Swap Rate
Z-Spread: Constant spread Z where PV = Σ CF/(1+spot+Z)^t
OAS = Z-Spread - Option Value

Day Count Conversions

Act/Act to 30/360: Yield(30/360) = Yield(Act/Act) × (360/365)
30/360 to Act/Act: Yield(Act/Act) = Yield(30/360) × (365/360)

HP 12C Calculator Sequences hp12c

Calculate YTM (Semiannual Bond)

Example: 5% coupon, 5 years, price 95
95 [CHS] [PV]     (bond price, negative)
100 [FV]          (face value)
2.5 [PMT]         (semiannual coupon)
10 [n]            (periods = years × 2)
[i]               (compute semiannual yield)
2 [×]             (annualize)
Result: 6.18% annual

Calculate Current Yield

Example: 5% coupon, price 95
5 [ENTER]         (annual coupon)
95 [÷]            (divide by price)
100 [×]           (convert to percentage)
Result: 5.26%

Calculate Price from YTM

Example: 4% coupon, 3 years, 5% YTM
100 [FV]          (face value)
2 [PMT]           (semiannual coupon)
6 [n]             (periods)
2.5 [i]           (semiannual yield)
[PV]              (compute price)
Result: -97.33 (ignore negative sign)

Zero-Coupon Bond Yield

Example: Price 75, 10 years to maturity
75 [CHS] [PV]     (current price)
100 [FV]          (maturity value)
10 [n]            (years)
[i]               (compute yield)
Result: 2.92%

Practice Problems

Basic Level

1. Periodicity Conversion

   • Q: Convert 6% semiannual bond basis yield to quarterly compounding
   • A: (1 + 0.06/2)^2 = (1 + r/4)^4; r = 5.96% quarterly

2. Current Yield

   • Q: 4% coupon bond trading at 92. Calculate current yield.
   • A: CY = 4/92 = 4.35%

3. G-Spread

   • Q: Corporate bond YTM 5.5%, Treasury YTM 3.0%. Calculate G-spread.
   • A: G-spread = 5.5% - 3.0% = 250 basis points

Intermediate Level

1. YTM Calculation

   • Q: 3-year bond, 5% annual coupon, price 102. Calculate YTM.
   • A: 102 = 5/(1+r) + 5/(1+r)² + 105/(1+r)³; r = 4.26%

2. Yield-to-Call

   • Q: 7% bond callable at 102 in 2 years, currently at 105. Calculate YTC.
   • A: 105 = 7/(1+r) + 109/(1+r)²; r = 4.31%

3. Z-Spread Application

   • Q: Bond trading with 150 bps Z-spread, embedded call worth 40 bps. Find OAS.
   • A: OAS = 150 - 40 = 110 basis points

Advanced Level

1. Complex Spread Analysis

   • Q: Russian 30-year bond at 3.756%, interpolated Treasury at 2.10%. If Z-spread is 180 bps and option value is 25 bps, calculate G-spread and OAS.
   • A: G-spread = 375.6 - 210 = 165.6 bps; OAS = 180 - 25 = 155 bps

2. Negative Yield Scenario

   • Q: European bond priced at 105 for 3-year maturity, no coupon. Calculate annual and monthly yields.
   • A: Annual: (100/105)^(1/3) - 1 = -1.61%; Monthly: -1.62%

3. Callable Bond Valuation

   • Q: 6% bond, 5 years to maturity, callable at 101 after year 2. Market yield 4%. Calculate price, YTM, YTC, and identify YTW.
   • A: Price = 108.98, YTM = 3.86%, YTC = 2.95%, YTW = 2.95% (YTC is lowest)

DeFi Applications & Real-World Examples

Traditional Finance Examples

1. Investment Grade Spread Analysis

   • Apple 30-year bond: G-spread 76 bps, Z-spread 109 bps
   • Microsoft 10-year: G-spread 45 bps, Z-spread 48 bps
   • Interpretation: Longer maturity shows larger G-Z spread difference

2. High Yield Analysis

   • Tesla 5.3% notes: G-spread 350 bps
   • Ford 7.45% notes: G-spread 475 bps
   • Shows credit quality differentiation

3. Callable Bond Trading

   • AT&T callable bonds consistently trade at YTW
   • Verizon non-callable bonds trade at YTM
   • Premium for call protection: 20-30 bps

DeFi Protocol Comparisons

1. Compound vs Aave Yield Comparison

   Compound USDC: 4.2% APR (continuous compounding)
   Aave USDC: 3.9% APY (annual compounding)
   Adjusted comparison: Compound = 4.29% APY
   Spread: 39 basis points favoring Compound
   

2. Fixed vs Variable Rate Protocols

   Notional Fixed: 5.5% for 6-month maturity
   Compound Variable: Currently 4.8%, 30-day avg 5.2%
   Fixed premium: 30 bps over average variable
   

3. Cross-Chain Yield Spreads

   Ethereum Aave USDC: 3.5%
   Polygon Aave USDC: 4.8%
   Arbitrum Aave USDC: 4.2%
   Chain risk premium: 130 bps for Polygon, 70 bps for Arbitrum
   

Innovative DeFi Yield Structures

1. Pendle Yield Tokenization

   • Splits yield-bearing assets into PT (Principal) and YT (Yield)
   • PT trades like zero-coupon bond
   • YT captures variable yield upside
   • Example: stETH PT with 5% implied yield vs 4% current staking

2. Curve Finance Gauge Yields

   • Base lending yield: 2%
   • CRV rewards: 3% (at current prices)
   • Total yield: 5%
   • Spread decomposition mirrors traditional bond spreads

3. GMX GLP Token Yields

   • Trading fee share: 15-30% APR (variable)
   • Comparable to high-yield bonds with profit participation
   • Risk factors: Protocol revenue, trader PnL exposure

Common Pitfalls & Exam Tips

Frequent Mistakes

1. Periodicity Confusion

   • Error: Comparing yields without converting to same periodicity
   • Fix: Always check and align compounding frequencies
   • Example: 6% semiannual ≠ 6% annual (actually 6.09% annual)

2. Spread Misinterpretation

   • Error: Using G-spread when Z-spread more appropriate
   • Fix: Use Z-spread for accurate valuation, G-spread for quick comparison
   • Remember: Z-spread > G-spread for normal yield curves

3. YTW Miscalculation

   • Error: Assuming YTM is always YTW
   • Fix: Calculate all possible yields (YTM, all YTCs)
   • Rule: YTW = minimum(YTM, all YTCs)

Exam Strategies

1. Quick Checks

   • Premium bond: YTC < YTM (always)
   • Discount bond: YTC > YTM (always)
   • Current yield between coupon rate and YTM

2. Calculator Efficiency

   • Store common values (100 FV, standard n values)
   • Use memory functions for multi-step calculations
   • Double-check sign conventions (PV negative, FV positive)

3. Time Management

   • Start with current yield (quick calculation)
   • Estimate YTM before calculating
   • Skip complex Z-spread calculations if time-constrained

Conceptual Traps

1. Negative Yields

   • Still follow same mathematical relationships
   • Higher negative yield = lower (more negative) return
   • Common in European government bonds

2. Street vs True Convention

   • Street convention ignores holidays/weekends
   • True yield accounts for actual payment delays
   • Difference usually 1-3 basis points

3. Simple vs Compound Yields

   • Money market uses simple interest (< 1 year)
   • Bonds use compound interest
   • Conversion needed for comparison

Key Takeaways

Must-Know Concepts

1. Yield periodicity conversion is essential for comparing bonds
2. YTM is the primary measure but has limitations for callable bonds
3. Spread measures quantify credit and liquidity risk premiums
4. Z-spread is superior to G-spread for accurate valuation
5. OAS is the only appropriate spread measure for bonds with embedded options

Critical Formulas

• Periodicity: (1 + r/m)^m = (1 + r/n)
• Current Yield: Annual Coupon / Price
• G-Spread: Bond YTM - Treasury YTM
• OAS: Z-spread - Option Value

DeFi Applications

• APR vs APY confusion is the DeFi equivalent of periodicity issues
• Protocol spreads reflect smart contract and liquidity risks
• Yield tokenization creates synthetic fixed-income instruments
• Cross-chain spreads capture bridge and ecosystem risks

Cross-References & Additional Resources

Related Finance Topics

• Bond Valuation (foundation for yield calculations)
• Term Structure (spot rates for Z-spread) yield-curve
• Duration (yield sensitivity measures) duration
• Credit Risk (spread drivers) credit-analysis

DeFi Resources

• Notional Finance Docs - Fixed rate lending
• Pendle Finance Docs - Yield tokenization
• DeFi Rate - Current lending rates across protocols
• Yearn Watch - Vault yield analytics

Practice Platforms

• Bloomberg Terminal: FIRV function for spread analysis
• TradingView: Bond yield charts and spreads
• DeFi Pulse: Protocol TVL and yield tracking
• Dune Analytics: On-chain yield dashboards

Review Checklist

Conceptual Understanding

• Can explain difference between yield periodicity conventions
• Understand when to use each yield measure (current, YTM, YTC, YTW)
• Know relationship between G-spread, Z-spread, and OAS
• Can identify factors affecting yield spreads

Calculation Proficiency

• Convert yields between different compounding frequencies
• Calculate current yield and simple yield
• Compute YTM using calculator or spreadsheet
• Determine yield-to-worst for callable bonds
• Calculate basic spread measures

Application Skills

• Compare bonds with different payment frequencies
• Interpret spread changes in market context
• Apply concepts to DeFi lending protocols
• Recognize common pitfalls in yield analysis

DeFi Integration

• Understand APR vs APY in DeFi context
• Can analyze protocol yield spreads
• Know how yield tokenization works
• Can evaluate cross-chain yield opportunities