What is a bond? Key terms
A bond is a certificate saying the issuer has borrowed money from you and will pay it back with interest.
- Face value (F) or par value: the amount repaid at the end, for example ₹1,000.
- Coupon rate (c): the fixed yearly interest rate on the face value.
- Coupon (C): the interest paid each period: C = c × F. Many bonds pay half of it every 6 months.
- Maturity (n): the date (or number of years) when the face value is repaid.
- Market yield / required rate (r): the return investors want today on similar bonds. Also called yield to maturity (YTM) when worked out from the price.
- Current price (P): what the bond sells for today in the market.
- Current yield: C ÷ P, the coupon as a percentage of today's price.
Present value approach to valuing a bond
₹100 received in one year is worth less than ₹100 today, because today's money could earn interest. Its present value at rate r is 100 ÷ (1 + r).
A bond is just a list of future payments, so its value is the sum of their present values:
P = C/(1+r) + C/(1+r)² + … + C/(1+r)ⁿ + F/(1+r)ⁿ
The coupons form an annuity, so this simplifies to:
P = C × [1 − (1 + r)⁻ⁿ] ÷ r + F × (1 + r)⁻ⁿ
Example: F = ₹1,000, coupon 8% (C = 80), n = 5, r = 10%. Annuity factor = [1 − 1.1⁻⁵] ÷ 0.1 = 3.7908; 1.1⁻⁵ = 0.6209. P = 80 × 3.7908 + 1,000 × 0.6209 = 303.26 + 620.92 = ₹924.18.
Half-yearly coupons
Use C/2 per period, r/2 per period and 2n periods.
Zero-coupon bonds
No coupons, only F at maturity: P = F ÷ (1 + r)ⁿ.
Premium, par and discount: price vs yield
| Market yield r compared with coupon rate c | Price | Name |
|---|---|---|
| r < c | P > F | Premium bond |
| r = c | P = F | Par bond |
| r > c | P < F | Discount bond |
Why: if new bonds pay 10% but yours pays only 8%, nobody will pay full price for yours; its price drops until the buyer earns 10% overall. As maturity gets near, the price moves toward the face value.
Yield to maturity and current yield
Yield to maturity (YTM) is the rate r that makes the present value of all payments equal the current price. It is found by trial (or a calculator): try a rate, compute P, and adjust.
A quick estimate: YTM ≈ [C + (F − P) ÷ n] ÷ [(F + P) ÷ 2].
Current yield = C ÷ P × 100%. It ignores the gain or loss at maturity, so it differs from YTM unless the bond is at par.
Try it: beat the slider
Before moving the slider, write your guess for the price at 6%, 8% and 12%. Then check: ₹1,084.25 (premium), ₹1,000 (par), ₹855.81 (discount). Notice the price changes more when the yield drops than when it rises by the same amount.
Key formulas and definitions
- Coupon C = coupon rate × face value
- PV of an amount A after t years = A ÷ (1 + r)ᵗ
- Bond price P = C × [1 − (1 + r)⁻ⁿ] ÷ r + F ÷ (1 + r)ⁿ
- Half-yearly: use C/2, r/2 and 2n
- Zero-coupon bond: P = F ÷ (1 + r)ⁿ
- Current yield = C ÷ P; YTM ≈ [C + (F − P)/n] ÷ [(F + P)/2]
Worked examples
1. A bond has face value ₹1,000 and coupon rate 7%. Find the annual coupon.
C = 7% × 1,000 = ₹70.
2. Find the price of a 3-year zero-coupon bond of face value ₹1,000 if the market yield is 6%.
P = 1,000 ÷ 1.06³ = 1,000 ÷ 1.191016 = ₹839.62.
3. A ₹1,000 bond pays 8% annually for 5 years. Find its price at a 10% market yield.
Annuity factor = (1 − 1.1⁻⁵) ÷ 0.1 = 3.7908. PV of coupons = 80 × 3.7908 = 303.26. PV of face = 1,000 × 0.6209 = 620.92. P = ₹924.18 (discount, since 10% > 8%).
4. The same bond at a 6% market yield: find the price and name its type.
Annuity factor = (1 − 1.06⁻⁵) ÷ 0.06 = 4.2124. PV of coupons = 80 × 4.2124 = 336.99. PV of face = 1,000 × 0.7473 = 747.26. P = ₹1,084.25, a premium bond (6% < 8%).
5. A ₹1,000 bond pays a 10% coupon half-yearly for 3 years. The market yield is 8% a year. Find its price.
Per half-year: C = ₹50, i = 4%, N = 6. Annuity factor = (1 − 1.04⁻⁶) ÷ 0.04 = 5.2421. PV of coupons = 262.11. PV of face = 1,000 ÷ 1.04⁶ = 790.31. P = ₹1,052.42.
6. A ₹1,000, 8% bond with 5 years left trades at ₹924.18. Find its current yield and approximate YTM.
Current yield = 80 ÷ 924.18 = 8.66%. Approx. YTM = [80 + (1,000 − 924.18)/5] ÷ [(1,000 + 924.18)/2] = (80 + 15.16) ÷ 962.09 = 9.89% (exact YTM is 10%).
Common mistakes
- Using the market yield to find the coupon. The coupon always uses the fixed coupon rate on face value.
- Forgetting to add the present value of the face value at maturity.
- For half-yearly bonds, halving the coupon but not the rate, or not doubling the number of periods.
- Thinking a higher market yield raises the bond price. Price and yield move in opposite directions.