TL;DR. When yields rise, bond prices fall. Duration tells you approximately how much: a bond with a modified duration of 8 loses about 8% of its price for a one-percentage-point rise in yield. Convexity is the correction that keeps that estimate honest when yields move a lot — it makes actual losses smaller than duration predicts and actual gains larger. It is also why the 30-year Treasury bond has been the epicenter of every rates-driven selloff this cycle: its modified duration sits near 18.
Why bond prices and yields move opposite
Every bond promises a fixed schedule of coupon payments plus a principal at maturity. Once those cash flows are set, the only way for a bond to be more or less attractive is for its price to change. If new bonds are being issued at a 5% coupon, nobody will pay you par for your old 3% bond — you have to sell at a discount so the buyer’s effective yield matches the market. This is the fundamental inverse relationship the SEC describes for retail investors: rising rates make older, lower-coupon bonds worth less.
Duration and convexity are simply the tools that put a number on that intuition.
Duration, in one sentence
Modified duration is the approximate percentage price change of a bond for a 1-percentage-point change in yield. If a bond has modified duration of 6.2, and yields rise by 25 basis points (0.25%), the bond price falls by roughly 6.2 × 0.25 = 1.55%. That is the whole rule of thumb, and it accounts for most of what a bond does on any given day.
Under the hood, duration comes from Macaulay’s original 1938 idea: it is the present-value-weighted average time you wait to receive a bond’s cash flows. The formal Macaulay duration is:
DMac = Σi (ti · PVi) / Σi PVi
where ti is the time to each cash flow and PVi is its present value. Modified duration — the one you actually use for price sensitivity — is Macaulay duration divided by (1 + y/m), where y is yield and m is the number of coupon periods per year:
Dmod = DMac / (1 + y/m)
And the punchline formula every fixed-income analyst has memorised:
ΔP / P ≈ − Dmod · Δy
A worked example
Take a 10-year Treasury note trading at par with a 4.25% coupon paid semi-annually, and modified duration of roughly 8.1. Yields rise by 50 basis points overnight (say, on a hot CPI print). Duration predicts:
ΔP / P ≈ − 8.1 · 0.50% = − 4.05%
So a $100,000 face-value position drops by about $4,050. Not exactly — because of convexity, the actual loss will be slightly less — but that is a solid first-cut number for a trader who needs it in seconds.
Duration by tenor
The single most useful fact about duration: it grows almost linearly with maturity for coupon bonds, and it grows even faster for zero-coupon bonds (whose duration equals maturity). That is why the long end whipsaws harder than the short end on the same yield move.
| Treasury | Approx. yield | Modified duration | Price change per +100 bps |
|---|---|---|---|
| 2-year note | 4.0% | 1.9 | −1.9% |
| 5-year note | 4.2% | 4.5 | −4.5% |
| 10-year note | 4.4% | 8.1 | −8.1% |
| 20-year bond | 5.0% | 12.6 | −12.6% |
| 30-year bond | 5.2% | 17.5 | −17.5% |
Read the last column as: “if yields for that tenor rise by exactly 1 percentage point, price falls by roughly this much.” A 30-year bond loses roughly nine times what a 2-year loses on the same yield move. That single fact drives most of the pain when the long end sells off — as it did across major economies in August 2026, when the US 30-year yield touched 5.31% and Japan’s 30-year hit a record 4.13% inside a week.
Convexity: the second-order correction
Duration is a straight-line approximation. But the true price-yield relationship of a bond is a curve — specifically, a convex curve. As yields fall, price rises faster than duration predicts; as yields rise, price falls slower than duration predicts. That curvature is what convexity captures.
Mathematically, convexity is the second derivative of price with respect to yield, divided by price. The improved (second-order Taylor) formula is:
ΔP / P ≈ − Dmod · Δy + ½ · C · (Δy)2
Because the convexity term is squared, it is always positive for a plain-vanilla bond — whether yields rise or fall. That is why more convex bonds are said to have a free option: they gain more when rates fall than they lose when rates rise by the same amount.
The intuition: convexity is a portfolio’s protection against large moves. Two portfolios with identical duration will behave identically for a 1-basis-point yield move, but for a 100-basis-point move the higher-convexity portfolio ends up meaningfully better off — more gain in the rally, less loss in the selloff.
Convexity worked out
Same 10-year note as before: modified duration 8.1, convexity 82 (roughly typical for a 10-year around par). Yield rises 100 basis points.
- Duration-only estimate: − 8.1 × 1.00% = − 8.10%.
- Convexity adjustment: + 0.5 × 82 × (0.01)2 = + 0.41%.
- Total: − 7.69%.
The convexity term saved about 40 bps of loss. For a 200-bps move, the correction quadruples (because it scales with the square of yield change) — convexity really starts to matter when rates move a lot. That is exactly the regime the long end has been trading in.
Why the 30-year has been the story
Long bonds carry the most duration and the most convexity. In a rally they run first; in a selloff they blow up first. The chart below shows why traders talk about the 30-year the way they used to talk about tech mega-caps.
Common mistakes and where the math breaks
- Assuming duration is constant. Duration itself shifts as yields change — a phenomenon called duration drift. For a big move, recompute rather than extrapolating.
- Ignoring embedded options. Callable bonds and mortgage-backed securities have negative convexity at low yields, because their maturity effectively shortens when rates fall (the issuer or homeowner refinances). Duration and convexity for these have to come from an option-adjusted framework, not the plain formula above.
- Confusing Macaulay and modified duration. Macaulay duration is expressed in years and is closer to a waiting time; modified duration is the one you multiply against yield changes to get percent price change.
- Applying it to non-parallel shifts. Duration assumes the entire yield curve shifts by the same amount. When the short end and long end move differently — a curve steepening or flattening — you need key-rate durations, one per point on the curve, to be honest.
What to read next
If you found this useful, three natural follow-ups: convertible bonds, which mix duration with equity optionality; foreign-currency corporate bonds, where duration is only half the story and FX moves add another layer; and the mechanics of Treasury auctions themselves, which set the yields we have been multiplying against all article.
Sources
- SEC investor.gov — Bonds and the inverse price/yield relationship
- Bond duration: Macaulay and modified duration formulas
- Bond convexity and the second-order Taylor approximation
- US Treasury daily constant-maturity yield data
Disclosure: This article is for informational purposes only and is not investment advice.