Bond Pricing Examples
Step-by-step examples of pricing JGBs under different scenarios
Worked Example: Pricing a 5-Year JGB
Let’s price a typical 5-year JGB using hypothetical market conditions to illustrate the pricing formula:
ℹ️ Note: This example uses simplified numbers for teaching purposes. For a real-world example with current market data, see Example 2 below.
- Face value: ¥1,000,000
- Annual coupon rate: 0.50%
- Assumed market yield: 1.25% (annual)
Step 1: Calculate the semi-annual coupon payment:
\[\begin{align} C &= \frac{¥1,000,000 \times 0.005}{2} \\ &= ¥2,500 \end{align}\]Step 2: Convert annual yield to semi-annual:
\[\begin{align} r &= \frac{0.0125}{2} \\ &= 0.00625 \text{ per period} \end{align}\]Step 3: Calculate number of periods:
\[\begin{align} n &= 5 \text{ years} \times 2 \\ &= 10 \text{ periods} \end{align}\]Step 4: Apply the pricing formula:
\[P = \sum_{t=1}^{10} \frac{¥2,500}{(1.00625)^t} + \frac{¥1,000,000}{(1.00625)^{10}}\] \[\begin{align} P &\approx ¥24,162 + ¥939,596 \\ &= ¥963,757 \end{align}\]Result: This bond trades at a discount to par (below ¥1,000,000) because its coupon rate (0.50%) is lower than the market yield (1.25%). New buyers pay less upfront to compensate for receiving lower coupon payments.
Additional Pricing Examples: Different Bond Scenarios
Example 2: Current 10-Year Benchmark JGB
📊 Live Data: This example uses JGB #380 (1.7% coupon, maturity 2035-09-20) from the November 2025 auction.
Using the current benchmark 10-year JGB:
- Issue: JGB #380
- Face value: ¥1,000,000
- Coupon: 1.7% annual
- Maturity: 2035-09-20
- Assumed market yield: 1.66% annual (hypothetical scenario where bond trades at small premium)
- Periods: 20 semi-annual
Interpretation: In this scenario, if the market yield were slightly below the coupon (1.66% vs 1.7%), the bond would trade at a small premium above par because investors would pay more for the higher coupon payments.
Example 3: Super-Long 30-Year JGB
- Face value: ¥1,000,000
- Coupon: 2.50% annual (older bond)
- Assumed market yield: 3.07% annual
- Periods: 60 semi-annual
- Result: Price ≈ ¥888,769 (trades at steep discount because coupon « yield)
Example 4: Short-Term 2-Year JGB
- Face value: ¥1,000,000
- Coupon: 1.00% annual
- Assumed market yield: 0.95% annual
- Periods: 4 semi-annual
- Result: Price ≈ ¥1,000,988 (trades very close to par because coupon ≈ yield and short maturity)
Understanding Bonds as Portfolios of Zero-Coupon Bonds
Some textbooks describe a coupon-bearing bond as a "portfolio of zero-coupon bonds." This is a powerful conceptual insight. Let's see why using our 5-year JGB example with ¥2,500 semi-annual coupons:
Expanded Form (What You're Really Buying):
\[\begin{align} P &= \frac{¥2,500}{(1.00625)^1} + \frac{¥2,500}{(1.00625)^2} + \frac{¥2,500}{(1.00625)^3} + ... + \frac{¥2,500}{(1.00625)^{10}} + \frac{¥1,000,000}{(1.00625)^{10}} \\ &= \text{10 separate "mini-bonds" paying ¥2,500} + \text{1 "mini-bond" paying ¥1,000,000} \end{align}\]Why This Matters:
- Each term is essentially a zero-coupon bond (a bond that pays only at maturity, no coupons)
- The coupon bond is just a bundled portfolio of these zeros
- This insight allows traders to "strip" bonds—separating the coupons and principal to sell as individual zero-coupon securities
- It also explains why bond math uses summation notation (∑): you're literally adding up the present value of each individual payment
Key Insights:
- Longer maturities show greater price sensitivity to yield changes
- Bonds with coupons far from market yields trade at larger premiums/discounts
- Current JGB market (Oct 2025) shows many older low-coupon bonds trading at discounts due to rising yields
- A coupon bond is conceptually a portfolio of zero-coupon bonds
- Euler’s number represents the mathematical limit of continuous compounding