Bond Pricing and Yield — Fundamentals of Bond Investment

1. 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

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2. Bond Pricing

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

$$\begin{aligned} \text{Bond Price} &= \sum [\frac{C}{1 + r}^t] + \frac{FV}{1 + r}^n \\ C: \text{Annual coupon} &= \text{Face Value} \times \text{Coupon Rate} \\ &r: \text{Market discount rate} (YTM) \\ &n: \text{Years to maturity} \\ &FV: \text{Face value} \\ \text{Example}: FV &= \$1,000 | \text{Coupon rate} 5\% | \text{Maturity} 3 \text{years} | \text{Market rate} 6\% \\ C &= \$1,000 \times 5\% = \$50 \\ \text{Price} &= \frac{50}{1.06} + \frac{50}{1.06}^2 + \frac{50}{1.06}^3 + \frac{1,000}{1.06}^3 \\ &= 47.17 + 44.50 + 41.98 + 839.62 \\ &\approx \$973.27 \quad \text{(discount bond)} \end{aligned}$$

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3. 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

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4. 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

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5. 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

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6. 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

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7. 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

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8. 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%.