Duration is the approximate percentage a bond's price moves for every 1 percentage point change in its yield, in the opposite direction. Take a hypothetical newly issued 10-year Treasury with a coupon equal to the 5.09% par yield on the October 16, 2026 Treasury curve: its modified duration works out to 7.76. If its yield jumps a full point to 6.09%, the duration estimate says the price drops about 7.76%, from $100 to roughly $92.24. Exact repricing puts it at $92.59, a 7.41% loss. That gap is convexity, and it gets bigger as maturities get longer.
That is the whole idea in one example. The rest of this piece shows the formulas, runs the same math on the 2-year and 30-year, and puts 2026's yield swings into dollars. For the news side of the bond market, the What's On Finance homepage tracks it daily.
The formulas, without the fog
Every bond is a stream of cash flows: coupons, then the face value at maturity. Price is those cash flows discounted at the yield. For a bond with semiannual coupons and annual yield y, each period's discount factor is (1 + y/2) raised to the period number.
- Macaulay duration is the weighted-average time until you get paid, measured in years, where each cash flow's weight is its present value divided by the bond's price.
- Modified duration = Macaulay duration / (1 + y/2). This is the number that converts a yield move into a price move: the % price change is approximately minus modified duration times the change in yield.
- Convexity measures how much that relationship bends. The better estimate is: % change = (minus modified duration times the yield change) + (one half times convexity times the yield change squared), with the yield change written as a decimal.
- DV01 (dollar value of a basis point) = modified duration times price times 0.0001. It is the dollar move per $100 of face value for a 0.01 point yield change.
We computed every number below in Python by discounting each semiannual cash flow directly, then repricing the bond at the new yield to get the exact answer. No shortcuts, no bond calculator.
Worked example: the 2-, 10- and 30-year on October 16, 2026
The US Treasury's daily par yield curve for October 16, 2026 showed the 2-year at 4.74%, the 10-year at 5.09% and the 30-year at 5.43%. A par yield is, by construction, the coupon at which a bond prices at exactly $100. So each example below is a hypothetical newly issued bond with a coupon equal to that day's par yield, priced at $100.
| Par bond (Oct 16, 2026 par yield) | Macaulay duration | Modified duration | Convexity | DV01 per $100 |
|---|---|---|---|---|
| 2-year at 4.74% | 1.93 years | 1.89 | 4.55 | $0.019 |
| 10-year at 5.09% | 7.96 years | 7.76 | 73.18 | $0.078 |
| 30-year at 5.43% | 15.13 years | 14.73 | 326.74 | $0.147 |
Look at the jump from 10 to 30 years. Maturity triples, duration roughly doubles, and convexity more than quadruples. Now shock each bond's yield and compare the duration-only estimate with exact repricing.
| Yield change | 2-yr: duration est. | 2-yr: exact | 10-yr: duration est. | 10-yr: exact | 30-yr: duration est. | 30-yr: exact |
|---|---|---|---|---|---|---|
| +1.00 pt | -1.89% | -1.86% | -7.76% | -7.41% | -14.73% | -13.22% |
| +0.50 pt | -0.94% | -0.94% | -3.88% | -3.79% | -7.36% | -6.97% |
| -0.50 pt | +0.94% | +0.95% | +3.88% | +3.97% | +7.36% | +7.79% |
| -1.00 pt | +1.89% | +1.91% | +7.76% | +8.14% | +14.73% | +16.51% |
Three things jump out. First, the 2-year barely moves: a full point of yield costs it under 2%. Second, the duration estimate is symmetric, but real prices are not. The 30-year loses 13.22% on a one-point rise and gains 16.51% on a one-point fall. Third, for the 30-year the duration-only estimate overstates the one-point loss by about 1.5 points. Adding the convexity term fixes most of that: it gives -13.09% against the exact -13.22%, and -7.40% against -7.41% for the 10-year.
What 2026's yield moves meant in price terms
Now the real tape. On the Treasury par curve, the 10-year yield went from 4.18% on December 31, 2025 to 5.09% on October 16, 2026, a rise of 0.91 points. Along the way it hit a 2026 low of 3.97% on February 27 and a high of 5.31% on October 5. The 30-year went from 4.84% at the end of 2025 to a 2026 high of 5.67% on October 7.
Apply the math to a hypothetical 10-year par bond issued at 4.18% on December 31, 2025. Its modified duration was 8.11. Times 0.91 points, that is a duration estimate of about a 7.4% price drop. Exact repricing, treating it as an instant shift with 10 years still to run, gives $92.94, a 7.06% loss. More on what drove the move is in our piece on why record-high Treasury yields matter for markets.
| Hypothetical par bond and 2026 move (Treasury par curve) | Yield change | Modified duration at start | Exact price change |
|---|---|---|---|
| 2-year: 3.47% (Dec 31, 2025) to 4.74% (Oct 16, 2026) | +1.27 pt | 1.92 | -2.40% |
| 10-year: 4.18% (Dec 31, 2025) to 5.09% (Oct 16, 2026) | +0.91 pt | 8.11 | -7.06% |
| 10-year: 3.97% (Feb 27, 2026 low) to 5.31% (Oct 5, 2026 high) | +1.34 pt | 8.19 | -10.29% |
| 30-year: 4.84% (Dec 31, 2025) to 5.67% (Oct 7, 2026 high) | +0.83 pt | 15.74 | -11.90% |
Notice the 2-year. Its yield rose more than the 10-year's, by 1.27 points against 0.91, yet the price hit was about a third of the size. That is duration doing its job. The table shows price change only, assuming the remaining term stayed fixed. Coupon income, paid at the bond's starting coupon rate, offsets part of the loss in a total-return calculation, and a real bond also rolls a little closer to maturity as the months pass.
Macaulay, modified, effective: which one to use
Macaulay duration is a time measure in years. It is useful for matching a bond portfolio to a future liability, but it does not directly give a price sensitivity.
Modified duration is the price-sensitivity number for plain bonds whose cash flows do not change when rates move, like the Treasuries above. When most people say "duration," this is what they mean.
Effective duration is estimated by repricing a bond a little up and a little down in yield, using a model that lets the cash flows change. It is the measure for callable corporates, mortgage-backed securities and the funds that hold them, because a borrower that can refinance or call early changes the bond's cash flows exactly when rates fall.
Convexity sits on top of all three. Plain bonds have positive convexity, which is why the 30-year's gain on a one-point drop beat its loss on a one-point rise. Callable bonds and mortgage securities can have negative convexity: price gains get capped as rates fall because the bonds get called or prepaid.
Why long bonds and long-duration stocks take the hit
The logic that makes a 30-year Treasury twitchy applies to anything whose value sits far in the future. A company priced on cash flows expected a decade or more out behaves like a long-duration asset: raise the discount rate and the present value of those distant dollars shrinks the most. That is why fast-growing tech names with thin current profits often trade like long bonds when yields climb. We covered that tension in rising Treasury yields putting equity valuations on watch.
The flip side matters too. When yields are high, the same duration math means a drop in rates produces bigger gains, which is the argument explored in whether the bond market is becoming attractive again. Duration cuts both ways; it does not tell you which way rates go.
Common mistakes with duration
- Treating duration as maturity. The 10-year par bond above has a Macaulay duration of 7.96 years, not 10, because coupons pay you back early. Only a zero-coupon bond has a Macaulay duration equal to its maturity: a hypothetical 10-year zero at 5.09% has a Macaulay duration of 10 years and a modified duration of 9.75.
- Ignoring convexity on big moves. For a half-point move in the 10-year, the duration estimate is off by less than 0.1 point. For a one-point move in the 30-year it is off by 1.5 points on the downside and almost 1.8 points on the upside.
- Confusing holding to maturity with mark-to-market. A Treasury held to maturity still pays its face value; the price drop is a mark-to-market loss, plus an opportunity cost versus newer bonds paying higher coupons. Sell before maturity and the loss is real.
- Assuming a bond fund behaves like one bond. A fund has its own average duration, published by the issuer, and it never matures because the manager keeps rolling holdings. Check the fund's stated effective duration rather than guessing from its name.
- Forgetting the curve does not move in parallel. These examples shift one yield at a time. In 2026, the 2-year, 10-year and 30-year all moved by different amounts, so a portfolio's real loss depends on where along the curve its holdings sit.
Taxes on bond gains, losses and interest vary by account and jurisdiction; this is not tax advice. Nothing here is investment advice. More macro explainers live in our Economy section.
Bull/Bear Verdict
Bull Case: Duration turns a vague fear of "rising rates" into a number. With a modified duration of 7.76 on a 10-year par bond at the October 16, 2026 par yield, an investor can size the hit from a half-point or full-point move in seconds, and compare a 2-year's sensitivity of about 1.9 against a 30-year's of about 14.7.
Bear Case: It is a straight-line approximation of a curved relationship. On big moves and long maturities it overstates losses and understates gains, it assumes the cash flows never change (wrong for callables and mortgage bonds), and it says nothing about which direction yields head next.
FAQ
What is a simple bond duration example?
A hypothetical 10-year Treasury with a 5.09% coupon, equal to the October 16, 2026 par yield, has a modified duration of 7.76. A 1-point rise in yield cuts its price by about 7.76% on the duration estimate, and by 7.41%, to $92.59, on exact repricing.
How do you calculate modified duration?
Compute Macaulay duration as the present-value-weighted average time of each cash flow, then divide by (1 + yield per period). For semiannual bonds that is Macaulay duration divided by (1 + y/2).
Is a higher duration good or bad?
Neither. Higher duration means bigger price swings in both directions. It hurts when yields rise and helps when they fall.
Why is duration shorter than maturity?
Coupons return part of your money before the final payment, which pulls the weighted-average timing forward. The higher the coupon, the shorter the duration relative to maturity. Only zero-coupon bonds have a Macaulay duration equal to their maturity.