Ch6. Bond Pricing and Yield — Fundamentals of Bond Investment

What Is a Bond?

Bond: A debt instrument in which the issuer promises to pay periodic interest and repay the principal at maturity.

Bond Components:
Face Value (Par Value): Amount repaid at maturity
Coupon Rate:            Annual interest as a % of face value
Maturity:               Time until principal repayment
Coupon Frequency:       Annual, semi-annual, or quarterly

Bond Pricing

A bond’s price equals the present value of its future cash flows.

Bond Price = Σ [C / (1 + r)^t] + FV / (1 + r)^n

C:  Annual coupon = Face Value × Coupon Rate
r:  Market discount rate (YTM)
n:  Years to maturity
FV: Face value

Example: FV = $1,000 | Coupon rate 5% | Maturity 3 years | Market rate 6%
C = $1,000 × 5% = $50

Price = 50/(1.06) + 50/(1.06)² + 50/(1.06)³ + 1,000/(1.06)³
      = 47.17 + 44.50 + 41.98 + 839.62
      ≈ $973.27  (discount bond)

Interest Rates and Bond Prices

Market interest rate ↑  →  Bond price ↓
Market interest rate ↓  →  Bond price ↑

Bond Types by Price vs Par:
Discount bond:  Coupon rate < Market rate  →  Price < Par
Premium bond:   Coupon rate > Market rate  →  Price > Par
Par bond:       Coupon rate = Market rate  →  Price = Par

YTM (Yield to Maturity)

YTM: The effective annual return earned by holding a bond from the current price to maturity.

Bond Price = Σ [C / (1 + YTM)^t] + FV / (1 + YTM)^n

→ Bond price and YTM move inversely
→ YTM is solved by trial-and-error or a financial calculator

YTM Approximation Formula:

YTM ≈ [C + (FV − P) / n] / [(FV + P) / 2]

P: Current bond price
n: Years to maturity

Duration

Duration: The weighted-average time to receive a bond’s cash flows. Measures interest-rate risk.

Macaulay Duration:
D = Σ [t × PV(CF_t)] / Bond Price

Modified Duration:
D* = D / (1 + YTM)

Price sensitivity to interest-rate changes:
ΔP / P ≈ −D* × Δr

Example: D* = 3,  Δr = 1% (0.01)
ΔP / P ≈ −3 × 0.01 = −3%
→ A 1% rise in rates → bond price falls ~3%

Duration Properties:

Longer maturity → higher duration
Lower coupon rate → higher duration (less cash received early)
Zero-coupon bond: Duration = Maturity

Yield Curve

Normal (Upward-Sloping):  Longer maturities offer higher yields (typical)
Inverted:                  Short-term yields > long-term yields (recession signal)
Flat:                      Similar yields across all maturities

Key Concept Cards

Inverse Relationship: Rates and Bond Prices ★★★★★ : Rising interest rates → falling bond prices. Existing lower-coupon bonds become less attractive. Memory tip: Rate ↑ = Bond Price ↓ (inverse)

Modified Duration ★★★★★ : ΔP/P ≈ −D* × Δr. Measures bond price sensitivity to interest-rate changes. Memory tip: % Price change = −Modified Duration × Δ rate

YTM vs Coupon Rate ★★★★☆ : YTM > coupon rate → discount bond (price < par). YTM < coupon rate → premium bond. Memory tip: YTM > coupon = discount bond

Practice Quiz

Q. A bond has a face value of $1,000, a 4% coupon, and 2-year maturity. The market rate rises to 6%. What is the bond price?

Price = 40/(1.06) + 40/(1.06)² + 1,000/(1.06)² = $37.74 +$ 35.60 + $890.00 = **$ 963.34** (discount bond)

Q. A bond with a modified duration of 5 sees interest rates fall by 0.5%. How does its price change?

ΔP/P ≈ −5 × (−0.005) = +2.5%. The bond price rises approximately 2.5%.

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