A bond is a promise to pay a fixed set of amounts on fixed dates. Because those payments do not change, the only thing that can adjust when interest rates in the market move is the price someone will pay for them today. That is why bond prices and yields always move in opposite directions.
A bond is a fixed stream of payments
Take a 10-year government bond with a face value of $1,000 and a coupon of 4% a year, paid annually. Whoever holds it receives:
- $40 at the end of each year for ten years, and
- the $1,000 face value back at the end of year ten.
If the bond is issued when investors want a 4% return on 10-year government debt, it sells for its face value, $1,000. Its yield to maturity (the single annual return that makes the payments worth exactly the price) is 4%.
Worked example: market yields rise to 5%
Suppose that soon after, yields on new 10-year bonds rise to 5%. A new $1,000 bond now pays $50 a year. Nobody will pay $1,000 for the old bond's $40 a year when $50 is on offer, so its price has to fall until a buyer earns 5% on it too.
The price that gives a 5% yield is the value of each payment discounted at 5% a year:
- The ten $40 coupons are worth $40 × (1 − 1.05⁻¹⁰) ÷ 0.05 = $40 × 7.7217 = $308.87.
- The $1,000 repaid in year ten is worth $1,000 ÷ 1.05¹⁰ = $613.91.
Together that is $922.78, a fall of about 7.7%. A buyer at $922.78 collects the same $40 coupons plus $1,000 at the end, and that works out to exactly 5% a year: the coupons give 4.3% on the price paid ($40 ÷ $922.78), and the $77 gain as the price rises back to $1,000 by maturity makes up the rest.
And if yields fall to 3%
Now run it the other way. If yields on new bonds fall to 3%, the old bond's $40 coupon looks generous, and buyers bid its price up:
- Coupons: $40 × (1 − 1.03⁻¹⁰) ÷ 0.03 = $40 × 8.5302 = $341.21
- Face value: $1,000 ÷ 1.03¹⁰ = $744.09
The price rises to $1,085.30, a gain of about 8.5%. The bond still pays exactly what it always did; it is simply worth more when other investments pay less.
Notice that the gain for a 1 percentage point fall (+8.5%) is larger than the loss for a 1 point rise (−7.7%). That curvature is called convexity, and it is a feature of every ordinary bond.
Duration: why longer bonds move more
Repeat the example with a 2-year bond paying the same 4% coupon:
| Market yield | 2-year bond price | Change | 10-year bond price | Change |
|---|---|---|---|---|
| 3% | $1,019.13 | +1.9% | $1,085.30 | +8.5% |
| 4% | $1,000.00 | 0% | $1,000.00 | 0% |
| 5% | $981.41 | −1.9% | $922.78 | −7.7% |
The 10-year bond moves about four times as much. A higher yield only costs the 2-year holder two years of below-market coupons, but it costs the 10-year holder ten years of them, and the final repayment is also further away, so discounting hits it harder.
Duration puts a number on that sensitivity. Macaulay duration is the average time until you receive the bond's payments, weighted by how much each payment is worth today. For the 10-year bond at a 4% yield it is about 8.4 years; for the 2-year bond, about 2.0 years. Divide by (1 + yield) and you get modified duration: about 8.1 for the 10-year bond and 1.9 for the 2-year bond.
Modified duration is a rule of thumb for price changes:
% change in price ≈ − modified duration × change in yield (in percentage points)
For the 10-year bond, a 1 point rise in yields gives roughly −8.1 × 1 = −8.1%. The exact answer was −7.7%; the gap is the convexity described above, and it grows with bigger moves. For small moves, duration alone is a good guide.
Two things raise duration: a longer time to maturity, and a lower coupon (more of the value sits in the far-off repayment). A zero-coupon bond's duration equals its maturity.
A one-year version you can check in your head
A bond that pays $100 in one year and nothing else is worth $100 ÷ 1.05 = $95.24 when yields are 5%. If yields rise to 6%, it is worth $100 ÷ 1.06 = $94.34. The payment did not change; the discount rate did.
What this means when you read yield charts
- A rising yield means existing bonds have lost market value. When the 10-year yield on the economy dashboard climbs by a percentage point, a 10-year bond bought before the rise is worth roughly 8% less on the market.
- Holding to maturity is different from selling early. If the issuer pays in full, a bond held to maturity returns its face value whatever happened to prices along the way. The price change matters if you sell before then, or if you own a bond fund, whose unit price reflects market prices every day.
- Yields move because of expected policy rates, expected inflation and a term premium. Higher expected inflation or higher expected cash rates push yields up and bond prices down.
- Coupon and yield are different numbers. The coupon is fixed when the bond is issued. The yield changes every day with the price.
Key terms
- Face value: the amount repaid at maturity.
- Coupon: the fixed interest payment, as a percentage of face value.
- Yield to maturity: the annual return from buying at today's price and holding to maturity, assuming every payment is made.
- Duration: how sensitive a bond's price is to a change in yields, measured in years.
- Convexity: the curvature that makes price rises from falling yields larger than price falls from rising yields of the same size.
This article explains how bond pricing works. It is general information, not a recommendation about any bond or fund.