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Pricing Mechanics, Yields, and Duration

The market price of a bond is the sum of the present value of its remaining contractually obligated cash flows (periodic coupon payments and the principal repayment at maturity), discounted at the market’s required rate of return. This required rate of return is the bond’s yield to maturity (YTM).

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

Where:

— Market price of the bond
— Periodic coupon payment (annual coupon in dollars divided by the payment frequency per year)
— Discount rate per period (YTM divided by the payment frequency per year)
— Face value (par value, typically $1,000 for corporate and municipal debt)
— Total number of periods remaining until maturity

A bond trades at par when its market price equals its face value. If prevailing market interest rates rise above the bond’s coupon rate, the bond’s price falls below par, trading at a discount to provide a competitive yield to new buyers. If market rates fall below the coupon rate, the bond’s price rises above par, trading at a premium.

Work it once. A $1,000 bond paying a 5% coupon semiannually ( $C = $ 25$ , so $n = 20$ periods over ten years) in a market demanding a 6% YTM ( $r = 3 %$ per period):

P = 25 \times \frac{1 - 1.0 3^{- 20}}{0.03} + \frac{1, 000}{1.0 3^{20}} = $ 371.94 + $ 553.68 \approx $ 926

The price sits $74 below par because the coupon is a point short of what the market now pays — the discount is the present value of that shortfall. Rerun it at a 4% YTM and the same bond prices at about $1,082: same paper, same coupons, a 17% price swing purely from the discount rate. That swing, scaled by maturity, is what the duration machinery below measures.

Yield Metrics

To evaluate a bond’s return profile, you must distinguish between different yield measures:

Coupon Rate: The contractually fixed annual interest rate paid on the face value.
Current Yield: The annual coupon income divided by the current market price ( $C ∕ P$ ). This measures the immediate cash flow return but ignores the capital gain or loss realized as the bond converges to par at maturity.
Yield to Maturity (YTM): The internal rate of return (IRR) earned on the bond if held to maturity, assuming all coupon payments are reinvested at the same YTM rate.

Duration and Price Sensitivity

Interest rate risk is quantified using duration . Macaulay duration measures the weighted average time until all cash flows are received, expressed in years. To determine a bond’s price sensitivity to interest rate movements, you must calculate its modified duration ( $ModD$ ):

ModD = - \frac{1}{P} \frac{dP}{dy} = \frac{MacD}{1 + \frac{y}{m}}

Where $y$ is the annualized YTM and $m$ is the compounding frequency per year. The percentage change in a bond’s market price ( $Δ P ∕ P$ ) in response to a small change in yield ( $Δ y$ ) is approximated by:

\frac{Δ P}{P} \approx - ModD \times Δ y

For example, if a bond has a modified duration of 8.0 years and market interest rates increase by 100 basis points (1.00%), the bond’s market price will decline by approximately 8.0%. High-duration assets are highly volatile and sensitive to changes in the discount rate, whereas low-duration assets preserve capital stability.

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