Fixed-Income Bond Valuation: Prices and Yields

Topic 6: Fixed-Income Bond Valuation: Prices and Yields

Learning Objectives Coverage

LO1: Calculate a bond’s price given a yield-to-maturity on or between coupon dates

Core Concept

Bond pricing is the application of discounted cash flow analysis — grounded in the time value of money concepts from Quantitative Methods — where the bond’s price equals the present value of all future cash flows (coupons and principal) discounted at the market discount rate (yield-to-maturity). This is fundamental to fixed-income investing, enabling investors to determine fair value, compare investment opportunities, and assess market conditions. The price-yield inverse relationship is the single most important concept in fixed income and underpins all of the duration and convexity analysis that follows. exam-focus

• Key components:
  • Market discount rate (required yield)
  • Coupon payments (periodic cash flows)
  • Principal repayment (face value at maturity)
  • Time to maturity and payment frequency

Formulas & Calculations

• Bond Price Formula:

  PV = PMT₁/(1+r)¹ + PMT₂/(1+r)² + ... + (PMTₙ + FV)/(1+r)ᴺ
  

  Where: PMT = coupon payment, r = discount rate per period, FV = face value, N = number of periods

• Price Between Coupon Dates:

  PVᶠᵘˡˡ = PV × (1 + r)^(t/T)
  

  Where: t = days since last coupon, T = days in coupon period

• HP 12C steps (Annual coupon bond):

  1000 [FV]    (face value)
  50 [PMT]     (annual coupon)
  5 [n]        (years to maturity)
  6 [i]        (YTM %)
  [PV]         (compute price)
  

Practical Examples

• Traditional Finance Example: 5-year BRWA bond, 3.2% annual coupon, $100 face value
  • At 2.4% YTM: Price = $103.75 (premium)
  • At 3.2% YTM: Price = $100.00 (par)
  • At 4.0% YTM: Price = $96.41 (discount)
• Calculation walkthrough:

  For 3.2% coupon, 4% YTM, 5 years:
  PV = 3.2/(1.04) + 3.2/(1.04)² + 3.2/(1.04)³ + 3.2/(1.04)⁴ + 103.2/(1.04)⁵
  PV = 3.08 + 2.96 + 2.85 + 2.74 + 84.78 = $96.41
  

• Interpretation: Bond trades at discount because coupon rate < required yield

DeFi Application

• Protocol example: Notional Finance fixed-rate lending
• Implementation: Users can lock in fixed yields by purchasing fCash tokens at a discount, similar to zero-coupon bonds
• Advantages/Challenges:
  • Advantages: Transparent pricing, no counterparty risk, instant settlement
  • Challenges: Limited maturity options, liquidity constraints, smart contract risk

LO2: Identify the relationships among a bond’s price, coupon rate, maturity, and yield-to-maturity

Core Concept

• Definition: Bond prices exhibit specific relationships with their characteristics - inverse relationship with yields, direct relationship with coupon rates, and complex maturity effects
• Why it matters: These relationships drive trading strategies, risk management decisions, and portfolio construction
• Key components:
  • Price-yield inverse relationship (fundamental law)
  • Coupon effect on price sensitivity
  • Maturity effect on price volatility
  • Pull-to-par phenomenon

Formulas & Calculations

• Price-Yield Relationship: As yield ↑, price ↓ (inverse)
• Premium/Discount Rules:
  • If Coupon Rate > YTM → Premium (Price > Par)
  • If Coupon Rate = YTM → Par (Price = Par)
  • If Coupon Rate < YTM → Discount (Price < Par)
• HP 12C Demonstration:

  Compare two bonds with same maturity:
  Bond A: 5% coupon, Bond B: 3% coupon
  At 4% YTM:
  Bond A: 1000 FV, 50 PMT, 5 n, 4 i → PV = 1044.52 (premium)
  Bond B: 1000 FV, 30 PMT, 5 n, 4 i → PV = 955.48 (discount)
  

Practical Examples

• Traditional Finance Example: Comparing price sensitivity
  • 2% coupon 30-year bond: 1% yield increase → 20% price decrease
  • 5% coupon 30-year bond: 1% yield increase → 17% price decrease
  • 2% coupon 5-year bond: 1% yield increase → 4.5% price decrease
• Interpretation: Lower coupon and longer maturity = higher price sensitivity

DeFi Application

• Protocol example: Pendle Finance yield tokenization
• Implementation: Separates yield-bearing assets into Principal Tokens (PT) and Yield Tokens (YT), creating fixed-income instruments
• Advantages/Challenges:
  • Advantages: Enables fixed-rate strategies in variable yield environment
  • Challenges: Complex pricing due to underlying asset volatility

LO3: Describe matrix pricing

Core Concept

• Definition: Matrix pricing estimates the value of illiquid or new bonds by using yields from comparable bonds with similar characteristics (maturity, coupon, credit quality)
• Why it matters: Most bonds trade infrequently, making real-time pricing difficult; matrix pricing provides fair value estimates for portfolio valuation and trading
• Key components:
  • Comparable bond selection
  • Yield interpolation/extrapolation
  • Credit spread adjustment
  • Liquidity premium consideration

Formulas & Calculations

• Linear Interpolation Formula:

  YTM_target = YTM₁ + [(YTM₂ - YTM₁) × (Maturity_target - Maturity₁)/(Maturity₂ - Maturity₁)]
  

• Matrix Pricing Steps:
  1. Find comparable bonds (similar credit, structure)
  2. Calculate their YTMs
  3. Interpolate to estimate target bond YTM
  4. Price target bond using estimated YTM

Practical Examples

• Traditional Finance Example: Pricing a new 7-year A-rated corporate bond
  • 5-year A-rated benchmark: 4.5% YTM
  • 10-year A-rated benchmark: 5.2% YTM
  • Interpolated 7-year YTM: 4.5% + (5.2%-4.5%) × (7-5)/(10-5) = 4.78%
  • Price 7-year bond using 4.78% discount rate
• Interpretation: Fair value estimate enables trading despite lack of market prices

DeFi Application

• Protocol example: Euler Finance risk-based pricing
• Implementation: Uses oracle-based pricing matrices to value collateral and determine lending rates
• Advantages/Challenges:
  • Advantages: Automated pricing, transparent methodology
  • Challenges: Oracle manipulation risk, limited historical data

Core Concepts Summary (80/20 Principle)

Must-Know Concepts

1. Price = PV of cash flows: Bond price equals present value of all future payments
2. Inverse price-yield relationship: Prices fall when yields rise, and vice versa
3. YTM definition: Single discount rate that equates PV of cash flows to price
4. Premium/par/discount: Determined by coupon rate vs YTM relationship
5. Full vs flat price: Full price = Flat price + Accrued interest
6. Convexity effect: Price increases exceed price decreases for equal yield changes
7. Pull-to-par: Bond prices converge to face value as maturity approaches
8. Matrix pricing: Estimates illiquid bond prices using comparable securities

Quick Reference Table

Relationship	Direction	Impact	DeFi Equivalent
Yield ↑	Price ↓	Inverse	APY ↑, Token price ↓
Coupon ↑	Price ↑	Direct	Higher rewards = Higher value
Maturity ↑	Volatility ↑	Amplified	Longer locks = More risk
Credit risk ↑	Yield ↑	Risk premium	Protocol risk = Higher APY
Time passage	Price → Par	Convergence	Expiry tokens → Settlement

Comprehensive Formula Sheet formula

Essential Formulas

Bond Price (Annual Coupons):
PV = C/(1+r) + C/(1+r)² + ... + (C+FV)/(1+r)ⁿ
Where: C = annual coupon, r = YTM, FV = face value, n = years

Bond Price (Semiannual Coupons):
PV = C/2/(1+r/2) + C/2/(1+r/2)² + ... + (C/2+FV)/(1+r/2)^(2n)
Where: C = annual coupon, r = annual YTM

Full Price Between Coupons:
PVᶠᵘˡˡ = PVᶠˡᵃᵗ + Accrued Interest
Accrued Interest = (t/T) × Coupon Payment
Where: t = days since last coupon, T = days in period

Current Yield:
Current Yield = Annual Coupon / Clean Price

Zero-Coupon Bond Price:
PV = FV/(1+r)ⁿ

YTM Approximation (for quick estimates):
YTM ≈ [C + (FV-PV)/n] / [(FV+PV)/2]

HP 12C Calculator Sequences

Standard Bond Pricing:
1000 [FV]      (face value)
60 [PMT]       (annual coupon)
10 [n]         (years)
5 [i]          (YTM)
[PV]           → -1077.95 (negative = cash outflow to buy)

Finding YTM:
1000 [FV]      (face value)
70 [PMT]       (coupon)
5 [n]          (years)
950 [CHS][PV]  (current price, negative)
[i]            → 8.53% (YTM)

Semiannual Bond:
1000 [FV]      (face value)
25 [PMT]       (semiannual coupon = 50/2)
20 [n]         (periods = 10 years × 2)
3 [i]          (semiannual yield = 6%/2)
[PV]           → -925.61

Zero-Coupon Bond:
1000 [FV]      (maturity value)
0 [PMT]        (no coupons)
5 [n]          (years)
6 [i]          (YTM)
[PV]           → -747.26

Practice Problems

Basic Level (Understanding)

1. Problem: Calculate price of 3-year, 4% annual coupon bond, $1000 par, 5% YTM

   • Given: FV = $1000,C=$40, n = 3, r = 5%
   • Find: Bond price
   • Solution:

     PV = 40/(1.05) + 40/(1.05)² + 1040/(1.05)³
     PV = 38.10 + 36.28 + 898.38 = $972.76
     

   • Answer: Bond trades at $972.76 (discount to par)

2. Problem: Bond has 6% coupon, 6% YTM. What is its price relative to par?

   • Given: Coupon rate = YTM = 6%
   • Find: Price relationship
   • Solution: When coupon rate = YTM, bond trades at par
   • Answer: Price = $1000 (par value)

Intermediate Level (Application)

1. Problem: 10-year bond, 5% semiannual coupon, $1000par,tradesat$950. Find YTM

   • Given: n = 20 periods, PMT = $25,FV=$1000, PV = -$950
   • Find: YTM
   • Solution:

     HP 12C: 1000 FV, 25 PMT, 20 n, 950 CHS PV, i
     Result: 2.82% semiannual
     Annual YTM = 2.82% × 2 = 5.64%
     

   • Answer: YTM = 5.64% annually

2. Problem: Using matrix pricing, estimate YTM for 6-year bond given:

   • 5-year similar bond: 4.2% YTM
   • 7-year similar bond: 4.6% YTM
   • Solution:

     YTM₆ = 4.2% + (4.6% - 4.2%) × (6-5)/(7-5)
     YTM₆ = 4.2% + 0.4% × 0.5 = 4.4%
     

   • Answer: Estimated YTM = 4.4%

Advanced Level (Analysis)

1. Problem: Compare two bonds when yields rise from 4% to 5%:
   • Bond A: 2% coupon, 30-year maturity
   • Bond B: 6% coupon, 5-year maturity
   • Given: Initial YTM = 4%, New YTM = 5%
   • Find: Price changes and analysis
   • Solution:

     Bond A initial: PV @ 4% = $691.79
     Bond A new: PV @ 5% = $537.68
     Change: -22.3%
     
     Bond B initial: PV @ 4% = $1089.04
     Bond B new: PV @ 5% = $1043.29
     Change: -4.2%
     

   • Answer: Bond A (low coupon, long maturity) has 5× greater price sensitivity

DeFi Applications & Real-World Examples

Traditional Finance Context

• Institution Example: BlackRock uses matrix pricing for $10 trillion in assets, especially for municipal bonds
• Market Application: Bloomberg BVAL provides matrix pricing for 2.5 million+ securities daily
• Historical Case: 2013 “Taper Tantrum” - 10-year Treasury yields rose from 1.6% to 3%, causing bond fund losses of 15%+

DeFi Protocol Implementations

1. Element Finance Fixed Rates:

   • Creates fixed-rate positions by splitting yield-bearing assets
   • Principal tokens trade at discount like zero-coupon bonds
   • Example: 1 year ETH principal token at 0.95 ETH = 5.26% fixed yield

2. 88mph Fixed Yield Protocol:

   • Offers fixed interest rates on variable yield assets
   • Uses floating-rate bonds to hedge yield variations
   • Depositors receive fixed yields while protocol manages rate risk

3. BarnBridge SMART Yield:

   • Tranches variable yields into fixed and leveraged positions
   • Senior tranches receive fixed yields (bond-like)
   • Junior tranches absorb yield volatility (equity-like)

Comparative Analysis

Aspect	TradFi Bonds	DeFi Fixed Income
Price Discovery	Matrix pricing for illiquid	AMM/Oracle based
Settlement	T+1 or T+2	Instant
Minimum Investment	Often $1,000+	No minimum
Yield Source	Issuer payments	Protocol revenues
Default Risk	Credit risk	Smart contract risk
Regulatory Protection	Yes	No

Common Pitfalls & Exam Tips

Frequent Mistakes

1. Forgetting payment frequency - Semiannual bonds require halving coupon and doubling periods
2. Sign convention confusion - PV is negative (outflow) when calculating, positive when stating price
3. Mixing YTM with coupon rate - YTM is market-determined; coupon rate is fixed at issuance
4. Ignoring accrued interest - Full price includes accrued interest; flat price doesn’t
5. Assuming linear price-yield relationship - Relationship is convex, not linear

Exam Strategies

• Quick check: Coupon > YTM = Premium, Coupon < YTM = Discount
• Calculator tip: Always clear TVM registers before new calculation
• Time saver: For YTM questions, try the coupon rate first
• Matrix pricing: Linear interpolation usually sufficient for exam
• Remember convexity: Price increases > price decreases for equal yield changes

Memory Aids

• PPDY: Price Premium when Discount Yield (below coupon)
• Five Cs: Coupon, Current price, Compounding, Cash flows, Calculate
• IMP: Inverse (yield-price), Maturity (affects sensitivity), Premium/discount (coupon vs yield)

Key Takeaways

1. Bond pricing is pure time value of money - PV of all future cash flows at market discount rate
2. Price-yield inverse relationship is fundamental - Never moves in same direction
3. YTM assumes reinvestment at same rate - Realized return may differ
4. Matrix pricing enables illiquid bond valuation - Critical for portfolio management
5. DeFi recreates fixed income through tokenization - Similar economics, different implementation

Cross-References & Additional Resources

Related Topics

• Duration (price sensitivity measurement) duration
• Convexity (second-order price effects)
• Term Structure (multiple discount rates) yield-curve
• Quantitative Methods: Time Value of Money — see 03-Quantitative-Methods

External Resources

• FINRA TRACE - Real-time bond price data
• Bloomberg Terminal BVAL - Professional matrix pricing
• Dune Analytics - DeFi protocol yields and TVL
• Element Finance Docs - Fixed rate implementation

Further Reading

• “Fixed Income Analysis” by Fabozzi - Comprehensive valuation techniques
• “The Bond Book” by Thau - Practical pricing examples
• “DeFi and the Future of Finance” by Harvey - Tokenized fixed income

Review Checklist

✅ Conceptual Understanding

• Can explain why bond prices and yields move inversely
• Understand premium/par/discount relationships
• Know three conditions for earning stated YTM
• Grasp matrix pricing methodology

✅ Calculations

• Calculate bond price given YTM
• Find YTM given bond price
• Compute accrued interest and full price
• Apply matrix pricing interpolation

✅ Applications

• Compare price sensitivity across bonds
• Identify arbitrage opportunities
• Value DeFi yield-bearing tokens
• Estimate illiquid bond prices

✅ Exam Readiness

• Memorized key formulas
• Practiced HP 12C sequences
• Completed all practice problems
• Reviewed common mistakes