Chapter 1 1.8

Bond Pricing Fundamentals

How JGB prices are calculated and their relationship to yields

The Price-Yield Relationship

The most fundamental concept in bond investing: bond prices and yields move inversely.

  • When yields rise → Bond prices fall
  • When yields fall → Bond prices rise

Think about it this way: If you want a higher yield (more return on your investment), do you need to pay more or less for the bond? The answer is less—you pay a lower price to get the same fixed coupon payments, which makes your effective return (yield) higher. So, the way I personally think about it is: is the current coupon rate enough for you, assuming you hold the bond to maturity and you didn’t have a view on interest rates? Conversely, if you accept a lower yield, you’re willing to pay more for the bond and vice versa.

This inverse relationship exists because a bond’s future cash flows are discounted (converted to present value) at the current market yield. (see below)

Present Value and Bond Pricing

The theoretical price of a JGB equals the present value of all its future cash flows, discounted at the current market yield. This is based on the Discounted Cash Flow (DCF) method, the fundamental algorithm for valuing any stream of future payments.

Connecting YTM to the Pricing Formula

Recall from earlier that Yield to Maturity (YTM) is the single discount rate that makes the present value of all future cash flows equal to the current market price. When we use YTM in the pricing formula:

  • If you know the price → you can solve for YTM (what the market requires)
  • If you know the YTM → you can solve for the fair price

The formula works both ways. In practice, bond prices are quoted in the market, and we calculate the implied YTM. But to understand pricing, we start with a known yield and calculate the theoretical price.

The Bond Pricing Formula (Discrete Compounding)

Formula Name: Standard Bond Pricing Equation (Semi-Annual Coupon)

This formula is essential—memorize it:

\[P = \sum_{t=1}^{n} \frac{C}{(1+r)^t} + \frac{F}{(1+r)^n}\]

Where:

  • $P$ = Current bond price (what you pay today)
  • $C$ = Coupon payment per period (semi-annual for JGBs)
  • $r$ = Market yield per period = YTM ÷ 2 (because JGBs pay semi-annually)
  • $F$ = Face value / par amount (typically ¥100 or ¥1,000,000)
  • $n$ = Total number of periods until maturity = years × 2 (semi-annual periods)

What This Means: The bond price is the sum of two components:

  1. Present value of all coupon payments (the summation term)
  2. Present value of the face value (the final term)

Each future payment is “discounted” back to today’s value using the market yield ($r$), which represents the required return for taking on the bond’s risk and locking up your money for that time period.

Interactive Bond Price Calculator

Bond Price Calculator

Note: Maturity input uses years for simplicity. Actual JGB pricing uses day counts (Actual/365 convention).

¥1,000,000
¥100.00
Par
¥2,500
20

Alternative Notations You May Encounter

The bond pricing formula can be expressed in several mathematically equivalent forms. You may encounter these in textbooks, research papers, or financial systems:

**1. Expanded Form** (showing each cash flow explicitly): $$P = \frac{C}{(1+r)^1} + \frac{C}{(1+r)^2} + \frac{C}{(1+r)^3} + ... + \frac{C + F}{(1+r)^n}$$ This notation makes it clear that the final payment includes both the last coupon ($C$) and the face value ($F$).
**2. Using Present Value Factors**: $$P = C \times \text{PVA}(r,n) + F \times \text{PVF}(r,n)$$ Where: - **PVA** = Present Value of Annuity factor - **PVF** = Present Value Factor (discount factor for the principal) This form is common in Excel and financial calculators.
**3. Continuous Compounding** (quantitative models only): $$P = \int_{0}^{T} C \cdot e^{-rt} \, dt + F \cdot e^{-rT}$$ Where $e$ is Euler's number (~2.71828), $T$ is time to maturity in years, and $r$ is the continuously compounded yield. ⚠️ **Note**: JGBs use **discrete semi-annual compounding** in practice. This continuous form is primarily used in academic models and derivatives pricing.

FYI: Premium, Par, and Discount

Quick reference—this follows logically from the pricing formula:

  • Premium: Bond price > Face value (when coupon rate > market yield)
  • Par: Bond price = Face value (when coupon rate = market yield)
  • Discount: Bond price < Face value (when coupon rate < market yield)

This is just a naming convention for describing where a bond trades relative to its face value.