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Bond Valuation Models

Bond Valuation Models & Formulas

Complete mathematical framework for bond valuation with detailed formulas, derivations, and practical applications

Mathematical ModelsFormula DerivationsPractical Examples

1. Zero-Coupon Bond Pricing Model

Bonds that pay no periodic interest but are sold at a discount to face value

Model Conditions & Assumptions

  • No periodic coupon payments during the life of the bond
  • Single payment of face value at maturity
  • Investors earn return through price appreciation
  • Common in U.S. Treasury STRIPS and some corporate bonds

Variables:

  • • M: Face value (par value) at maturity
  • • r: Yield to maturity (decimal)
  • • n: Years to maturity
  • • P: Current bond price

Formulas:

P = M / (1 + r)ⁿ

Example & Solution

Question: A 10-year Treasury STRIPS with $100,000 face value trades at 4.5% YTM. What is its price?

Given Conditions:

  • • Face value (M): $100,000
  • • Time to maturity (n): 10 years
  • • Yield to maturity (r): 4.5% or 0.045
  • • Zero-coupon bond (no intermediate payments)

Step-by-Step Calculation:

**Step-by-Step Calculation:** 1. **Identify variables**: - M = $100,000 - r = 0.045 - n = 10 2. **Apply the formula**: P = M / (1 + r)ⁿ 3. **Calculate (1 + r)ⁿ**: (1 + 0.045)¹⁰ = (1.045)¹⁰ = 1.552 4. **Compute present value**: P = 100,000 / 1.552 = $64,460 5. **Verify the discount**: Discount = M - P = $100,000 - $64,460 = $35,540 Discount percentage = ($35,540 / $100,000) × 100% = 35.54%

Answer: $64,460

Explanation: The bond is priced at a 35.54% discount to face value, reflecting the time value of money over the 10-year period at a 4.5% yield.

2. Coupon Bond Pricing Model

Traditional bonds with regular coupon payments at a fixed interest rate

Model Conditions & Assumptions

  • Regular periodic coupon payments throughout the bond's life
  • Face value payment at maturity
  • Coupon rate typically fixed at issuance
  • Most common type of bond in corporate and government markets

Variables:

  • • C: Annual coupon payment
  • • M: Face value at maturity
  • • r: Yield to maturity (decimal)
  • • n: Years to maturity
  • • P: Current bond price

Formulas:

P = Σ[C / (1 + r)ᵗ] + M / (1 + r)ⁿ

Example & Solution

Question: A 5-year corporate bond with 6% coupon, $1,000 face value trades at 5% YTM. What is its price?

Given Conditions:

  • • Face value (M): $1,000
  • • Coupon rate: 6% per year
  • • Time to maturity (n): 5 years
  • • Yield to maturity (r): 5% or 0.05
  • • Annual coupon payments: $60 per year

Step-by-Step Calculation:

**Step-by-Step Calculation:** 1. **Identify variables**: - M = $1,000 - C = $1,000 × 0.06 = $60 (annual coupon) - r = 0.05 - n = 5 2. **Apply the formula**: P = Σ[C / (1 + r)ᵗ] + M / (1 + r)ⁿ 3. **Calculate each term**: Year 1: $60 / (1.05)¹ = $60 / 1.05 = $57.14 Year 2: $60 / (1.05)² = $60 / 1.1025 = $54.42 Year 3: $60 / (1.05)³ = $60 / 1.1576 = $51.83 Year 4: $60 / (1.05)⁴ = $60 / 1.2155 = $49.36 Year 5: $60 / (1.05)⁵ = $60 / 1.2763 = $47.01 4. **Sum coupon payments**: $57.14 + $54.42 + $51.83 + $49.36 + $47.01 = $259.76 5. **Add face value**: $1,000 / (1.05)⁵ = $1,000 / 1.2763 = $783.53 6. **Total price**: P = $259.76 + $783.53 = $1,043.29 7. **Verify premium**: Premium = P - M = $1,043.29 - $1,000 = $43.29 Premium percentage = 4.33%

Answer: $1,043.29

Explanation: The bond trades at a premium to face value because its coupon rate (6%) exceeds the market yield (5%), making it attractive to investors.

3. Duration Models

Measures bond price sensitivity to interest rate changes

Model Conditions & Assumptions

  • Measures interest rate risk exposure
  • Weighted average time to receive cash flows
  • Essential for bond portfolio management
  • More accurate for small yield changes

Variables:

  • • Cᵗ: Cash flow at time t
  • • t: Time period
  • • r: Yield to maturity (decimal)
  • • P: Bond price
  • • D: Macaulay duration
  • • MD: Modified duration

Formulas:

D = Σ[t × Cᵗ / (1 + r)ᵗ] / P

MD = D / (1 + r)

ΔP/P ≈ -MD × Δr

Example & Solution

Question: A bond has 8-year modified duration. If yields increase by 50 basis points, what happens to price?

Given Conditions:

  • • Modified duration (MD): 8 years
  • • Yield change (Δr): +0.50% or +0.005
  • • Small yield change (duration approximation valid)
  • • No convexity effects considered in basic calculation

Step-by-Step Calculation:

**Step-by-Step Calculation:** 1. **Identify variables**: - MD = 8 years - Δr = +0.005 (50 basis points) 2. **Apply duration formula**: ΔP/P ≈ -MD × Δr 3. **Calculate price change**: ΔP/P ≈ -8 × 0.005 = -0.04 4. **Convert to percentage**: ΔP/P = -4% 5. **Interpretation**: The bond price will decline by approximately 4% for every 50 basis point increase in yield.

Answer: 4% price decline

Explanation: The bond's price will decrease by approximately 4% due to the 50 basis point increase in yield, demonstrating the inverse relationship between bond prices and interest rates.

4. Convexity Model

Second derivative of price-yield relationship, improves duration estimates

Model Conditions & Assumptions

  • Measures curvature of price-yield relationship
  • Improves duration-based price estimates
  • More important for large yield changes
  • Essential for long-term bonds and volatile markets

Variables:

  • • Cᵗ: Cash flow at time t
  • • t: Time period
  • • r: Yield to maturity (decimal)
  • • P: Bond price
  • • C: Convexity measure
  • • MD: Modified duration

Formulas:

C = Σ[t(t+1) × Cᵗ / (1 + r)ᵗ⁺²] / P

ΔP/P ≈ -MD × Δr + (1/2) × C × (Δr)²

Example & Solution

Question: A 30-year Treasury bond has 15.2-year duration and 280 convexity. If yields decline by 50 basis points, what's the price impact?

Given Conditions:

  • • Modified duration (MD): 15.2 years
  • • Convexity (C): 280
  • • Yield change (Δr): -0.50% or -0.005
  • • Large yield change (convexity effect significant)
  • • Long maturity bond (higher convexity)

Step-by-Step Calculation:

**Step-by-Step Calculation:** 1. **Identify variables**: - MD = 15.2 years - C = 280 - Δr = -0.005 (50 basis points decline) 2. **Duration effect**: Duration contribution = -MD × Δr = -15.2 × (-0.005) = +0.076 or +7.6% 3. **Convexity effect**: Convexity contribution = (1/2) × C × (Δr)² = 0.5 × 280 × (0.005)² = 140 × 0.000025 = 0.0035 or +0.35% 4. **Total price change**: ΔP/P ≈ +7.6% + 0.35% = +7.95% 5. **Comparison with duration-only estimate**: Duration-only estimate would be +7.6% (underestimating the price increase)

Answer: +7.95%

Explanation: The bond price increases by 7.95%, which is 0.35% more than the duration-only estimate of 7.6%. The convexity effect adds 0.35% to the price appreciation because bond prices rise more when yields fall than they decline when yields rise.

Bond Valuation Framework

Core Principles

Time Value of Money

All future cash flows must be discounted to present value

Risk-Return Relationship

Higher yields compensate for higher risk and longer maturities

Market Efficiency

Bond prices reflect all available information and risk factors

Applications

Investment Analysis

Compare intrinsic value with market price for investment decisions

Risk Management

Use duration and convexity for interest rate risk assessment

Portfolio Strategy

Optimize bond allocation based on objectives and market conditions

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