What sets a bond’s price
A bond pays a fixed coupon each period and returns its face value at maturity. Once it is issued, the coupon never changes, but market interest rates do. When rates rise, new bonds pay more, so an existing bond must sell for less to offer the same yield; when rates fall, it sells for more. The price is simply the present value of everything the bond will still pay, discounted at today’s market yield.
The formulas
With c the coupon per period, y the market yield per period and n the number of periods left:
- Price = c × (1 − (1 + y)^−n) ÷ y + face ÷ (1 + y)^n
- Current yield = annual coupon ÷ price
- Approximate yield to maturity at a quoted price P = (annual coupon + (face − P) ÷ years) ÷ ((face + P) ÷ 2)
- Macaulay duration: the weighted average time to receive the cash flows. Modified duration = Macaulay ÷ (1 + y), the approximate percentage price change for a 1-point move in yields.
The yield-to-maturity formula is a close shortcut; the exact YTM is the rate that makes the price formula equal the quoted price.
Worked example
A bond with a $1,000 face value pays a 4% coupon semiannually and matures in 10 years. Similar bonds now yield 4.5%.
- Coupon per period: $1,000 × 4% ÷ 2 = $20; y = 2.25%; n = 20.
- Price = 20 × (1 − 1.0225^−20) ÷ 0.0225 + 1,000 ÷ 1.0225^20 = $960.09, or 96% of face value: a discount, because the coupon is below the market yield.
- Current yield = $40 ÷ $960.09 = 4.17%.
- Duration is 8.3 years and modified duration 8.1: if yields rose one more point, the price would fall about 8%.
- If the bond is quoted at $950, its approximate yield to maturity is 4.62%.
Price at other yields
| Market yield | Price | Current yield |
|---|---|---|
| 3% | $1,085.84 | 3.68% |
| 4% | $1,000.00 | 4.00% |
| 4.5% | $960.09 | 4.17% |
| 5% | $922.05 | 4.34% |
| 6% | $851.23 | 4.70% |
At a 4% market yield the bond trades exactly at par, because the coupon equals the yield.
Practical notes
- Quotes are in percent of par. A Treasury quoted at 96 costs $960 per $1,000, plus accrued interest since the last coupon.
- Longer and lower-coupon bonds are more sensitive. A 30-year zero-coupon bond loses about 25% when yields rise by one point.
- Holding to maturity removes price risk: you get the face value back whatever happens to rates, unless the issuer defaults.
- Interest on Treasuries is exempt from state income tax; interest on most municipal bonds is exempt from federal tax.