Bonds Price and Yield: Concepts, Formulas and Examples

A bond is a loan from investors to a government or company: the issuer pays regular coupons and returns the face value at maturity. The bond's price is simply the present value of those promised cash flows, discounted at the market's required yield.

This matters because bond prices move every day as market yields change, and the relationship runs in one fixed direction: when yields rise, prices fall, and vice versa. Understanding price and yield is the gateway to all fixed-income investing.

Key Concepts

• Face (par) value (F): the amount repaid at maturity, usually ₹1,000. Coupon (C): the periodic interest payment; the coupon rate is C / F.
• Yield to maturity (YTM, y): the single discount rate that equates the bond's price to the present value of its cash flows; the market's required return.
• Premium, discount, par: a bond trades at a premium when coupon rate > YTM, at a discount when coupon rate < YTM, and at par when they are equal.
• Current yield: annual coupon divided by current price, a quick but incomplete return measure that ignores capital gains or losses.

Formulas

Price of a coupon bond: P = Σ_{t=1}^{n} C / (1 + y)^t + F / (1 + y)^n. Zero-coupon bond: P = F / (1 + y)^n. Current yield: C / P. Perpetual bond (consol): P = C / y. These assume coupons are certain, a flat yield curve is used for discounting, and YTM as a realized return assumes coupons are reinvested at y.

Worked Example

Price a 3-year bond with face value ₹1,000, an 8% annual coupon (₹80 per year), when the market yield is 6%. Discount each cash flow: year 1: 80 / 1.06 = ₹75.47170; year 2: 80 / (1.06)² = 80 / 1.1236 = ₹71.19979; year 3: 1,080 / (1.06)³ = 1,080 / 1.191016 = ₹906.80633. Summing gives P = 75.47170 + 71.19979 + 906.80633 = ₹1,053.48. Because the 8% coupon exceeds the 6% yield, the bond trades at a premium of ₹53.48 above par. If the market yield were 10% instead: 80 / 1.1 = ₹72.72727, 80 / 1.21 = ₹66.11570, 1,080 / 1.331 = ₹811.42329, for a price of ₹950.27, a discount bond. The same bond falls ₹103.21 when the yield rises from 6% to 10%, demonstrating the inverse price-yield relationship.

Common Misconceptions

• Bond prices are fixed until maturity. Prices float daily with market yields; only the promised cash flows are fixed.
• The coupon rate is your return. Your true return is the yield to maturity, which blends coupons with any premium paid or discount received, and it materializes only if coupons are reinvested at that yield.
• Premium bonds are bad deals. The premium is exactly compensated by above-market coupons; at the market yield, premium, par, and discount bonds are fairly priced relative to each other.

Real-World Applications

Bond pricing values government securities, corporate debentures, and municipal bonds; it sets mortgage rates, anchors pension fund valuations, and gives central banks the transmission channel through which policy rates reach the economy.

A bond is worth the discounted value of its coupons plus principal, so its price always moves opposite to market yields, trading at a premium when its coupon beats the market and at a discount when it lags.