Treasury yields are doing things again. The 10-year sits at 4.51%, the 30-year at 4.95%, and the 2-year at 4.24% as of the Federal Reserve H.15 release dated June 23, 2026. Every time those numbers move, bond portfolios reprice. The single most useful concept for understanding by how much is duration. The second most useful is convexity — the correction that explains why duration alone always lies a little bit.
TL;DR: A bond’s price falls when yields rise because the new, higher coupons available on freshly issued bonds make the old bond’s lower coupons less attractive. Modified duration tells you the approximate percentage price change for a 1% change in yields. A bond with a duration of 8 will lose roughly 8% if yields rise 1%, and gain roughly 8% if they fall 1%. Convexity says that approximation understates gains and overstates losses — the price-yield relationship curves, it does not bend on a straight line. Get duration wrong and a portfolio that looked “defensive” against rate moves can blow up.
Why bond prices and yields move in opposite directions
A bond is a promise to pay fixed coupons and return par at maturity. Its cash flows do not change. What changes is the market’s required yield on a bond of that risk and maturity. If a new 10-year Treasury is issued at a 5% coupon and you own an old one paying 3%, your bond has to fall in price until its yield to a new buyer also equals 5%. Price falls; yield rises. Mathematically, the bond’s price is the present value of its cash flows discounted at the yield, and present value falls as the discount rate rises:
Price = Σ [ Coupont / (1 + y)t ] + Par / (1 + y)T
That formula does all the work. Duration and convexity are simply convenient summaries of how sensitive that present value is to a change in y.
What duration actually measures
There are three durations worth knowing. They are related, and they get confused often.
- Macaulay duration is the weighted-average time until the bond’s cash flows arrive, with each cash flow’s weight equal to its share of the bond’s present value. It is measured in years. For a zero-coupon bond, Macaulay duration equals maturity. For any coupon bond, it is shorter than maturity because some of the value is paid back earlier.
- Modified duration is Macaulay duration adjusted for the yield's compounding period. It is what people usually mean when they say “duration is 8.” A modified duration of 8 means a 1% change in yield produces an approximate 8% change in price in the opposite direction.
- Effective duration measures price sensitivity for bonds whose cash flows themselves depend on yields — callables, mortgage-backed securities, and other path-dependent instruments. It is computed numerically by shocking the yield curve up and down and observing the price.
The formulas are clean. The widely cited reference set for them is the CFA Institute fixed-income refresher and Aswath Damodaran’s corporate finance and valuation materials at NYU Stern.
Modified Duration = Macaulay Duration / (1 + y / k)
Where y is the yield and k is the number of coupon payments per year (2 for US Treasuries and corporates that pay semiannually). The price-change rule of thumb is:
ΔPrice / Price ≈ − Modified Duration × Δy
A worked example: the on-the-run 10-year Treasury
Take a stylised on-the-run 10-year US Treasury with a 4.50% coupon, $1,000 par, paid semiannually, and a yield to maturity of 4.51% — almost exactly the current H.15 close. The math gives a Macaulay duration of 8.16 years and a modified duration of 7.98. So a 100 basis-point parallel rise in yields produces a price move of approximately:
ΔPrice / Price ≈ −7.98 × 0.01 = −7.98%
Almost an 8% loss for a single percentage-point move in long-term rates. That is a serious capital loss on what most investors think of as the safest asset on Earth. The 30-year bond at a similar coupon has a modified duration closer to 16 — a 100bp rise wipes out 16% of its market value. The 2-year barely moves: its modified duration is under 2, so the same 100bp move costs less than 2% in price.
| US Treasury | Yield (H.15, 6/22/26) | Approx. modified duration | Approx. price impact, +100bp |
|---|---|---|---|
| 2-year note | 4.24% | 1.9 | −1.9% |
| 5-year note | 4.29% | 4.5 | −4.5% |
| 10-year note | 4.51% | 8.0 | −8.0% |
| 20-year bond | 4.88% | 12.7 | −12.7% |
| 30-year bond | 4.95% | 15.8 | −15.8% |
Convexity: where duration starts lying
Duration assumes a linear relationship between yield and price. The actual relationship is a curve. As yields rise, the price falls — but at a decreasing rate. As yields fall, the price rises at an increasing rate. The shape is bowed toward the origin and convex from below. Convexity is the second-derivative term that captures this curvature.
ΔPrice / Price ≈ − Modified Duration × Δy + ½ × Convexity × (Δy)2
The convexity term is always positive for a vanilla, option-free bond. It adds to your gains when yields fall and subtracts from your losses when yields rise. Two bonds with identical duration but different convexity will perform differently in a big rate move — the higher-convexity bond wins both ways. Long bonds and zeros have the highest convexity. That positive convexity is why duration alone tends to overstate the losses you actually take in a sharp selloff.
Convexity can flip negative when a bond has embedded options. Callable corporates and mortgage-backed securities show negative convexity around the strike: as yields fall and prepayments accelerate, the upside is capped. That is why a portfolio manager who buys MBS for yield has to manage convexity hedges — the duration-only model misses the worst part of the risk.
The five mistakes that show up in real portfolios
1. Treating duration as a constant
Duration itself drifts as yields move. A bond’s duration shortens as yields rise and lengthens as yields fall — the opposite of what a panicked seller wants. A duration computed at 3% yields can understate the rate sensitivity of the same bond if rates are now at 5%.
2. Confusing maturity with duration
A 30-year Treasury has a duration near 16, not 30. A 30-year zero-coupon Treasury, on the other hand, has a duration of essentially 30. Same maturity, very different rate risk. Anyone sizing a position by maturity rather than duration is taking a different bet than they think.
3. Forgetting credit and option features
Modified duration assumes cash flows are fixed. They are not, for callable corporates, putable bonds, mortgage-backed securities, or convertibles. Effective duration — computed numerically — is the right measure for anything with an embedded option. Treating a callable bond like a non-callable one routinely understates downside.
4. Ignoring convexity in big moves
For a 25bp move, duration is fine. For a 200bp move, the convexity correction matters: it can be the difference between a 16% loss and an 18% loss on a 30-year bond. In 2022 the Bloomberg US Aggregate Bond Index lost 13.0%, the worst calendar year in its history — a move large enough that convexity adjustments were not optional, per Bloomberg market data.
5. Adding durations without weighting
A portfolio’s duration is the dollar-weighted average of its constituents’ durations, not the simple average. A 90/10 mix of 2-year and 30-year bonds has a portfolio duration around 3.4, not 16. The difference matters for hedging — matching the duration of assets and liabilities is the entire business of a pension fund or insurer.
How this fits with the rest of the curve
Duration and convexity are the bricks of fixed-income analysis. The shape of the yield curve determines which durations are most exposed in a given move. The Treasury basis trade uses small mispricings between cash bond duration and futures-implied duration. Pension and insurance balance-sheet management is essentially a duration-matching exercise. Anyone touching bonds, MBS, leveraged loans, or rate derivatives is, knowingly or not, taking views on duration and convexity.
The standard references remain the CFA Institute fixed-income curriculum, Frank Fabozzi’s bond-market handbooks, and the Federal Reserve’s own FEDS Notes on rate-risk and duration.
Sources
- Federal Reserve – H.15 Selected Interest Rates (Treasury constant maturity yields)
- CFA Institute – Fixed-Income Securities: Defining Elements
- Aswath Damodaran – Corporate Finance and Valuation (NYU Stern)
- Federal Reserve – FEDS Notes (rates and duration research)
- SEC Office of Investor Education – Investor.gov
- Bloomberg – US bond market suffers worst year on record (2022 context)
Disclosure: This article was produced with AI assistance and reviewed before publication. It is for informational purposes only and is not investment advice.