TL;DR. Duration is the single most useful number for thinking about how much a bond’s price will move when interest rates change. The rule of thumb: for a small yield change, a bond’s price moves by roughly minus duration times the change in yield. That is why a 30-year Treasury can lose 15% of its value when long yields rise one percentage point, while a 2-year note barely flinches. Convexity is the small adjustment that captures the curvature in that relationship.
What duration actually is
The intuition is older than the math. In 1938 the Canadian economist Frederick Macaulay asked a simple question: when you hold a coupon bond, how long do you really wait on average to get your money back? A bond that pays a coupon every six months returns cash to you steadily; a zero-coupon bond returns everything in one lump at maturity. Duration is the present-value-weighted average time to those cash flows. By construction, a zero-coupon bond’s Macaulay duration equals its maturity, and a coupon bond’s duration is always shorter than its maturity.
Why does that average matter for price? Because a bond’s price is just the present value of its future cash flows discounted at the prevailing yield. Push the yield up and every future dollar is worth a little less. The further out in time a cash flow sits, the more punishing the discount, so the bonds with the longest weighted-average wait fall the most. Duration is the unit-free way of summarising that sensitivity. It is measured in years, but it is really an elasticity in disguise.
The formula and the rule of thumb
Macaulay duration is the present-value-weighted average time to cash flow:
DMac = Σ ( ti × PVi ) / Price
Where ti is the time to each cash flow and PVi is its present value. Modified duration is the small adjustment that turns this time-weighted average into a clean price sensitivity:
Dmod = DMac / ( 1 + y / m )
Where y is the nominal yield and m is the number of compounding periods per year (2 for a standard U.S. Treasury). And then the rule that every fixed-income desk lives by:
ΔPrice / Price ≈ − Dmod × Δy
Translation: a bond with modified duration of 8 will lose about 8% of its price if its yield rises by 1 percentage point, and gain about 8% if its yield falls by 1 percentage point. The minus sign is the most important part of all of fixed income: price and yield move in opposite directions.
A worked example with current Treasury yields
The Federal Reserve publishes Treasury constant-maturity yields daily in the H.15 release. As of May 14, 2026, the curve looked like this:
| Treasury | Yield (May 14, 2026) | Approx. Modified Duration* | Price impact of +100 bps |
|---|---|---|---|
| 3-month T-bill | 3.69% | 0.25 | −0.25% |
| 2-year note | 4.00% | 1.90 | −1.9% |
| 5-year note | 4.13% | 4.47 | −4.5% |
| 10-year note | 4.47% | 7.96 | −8.0% |
| 30-year bond | 5.02% | 15.42 | −15.4% |
Notice how non-linear the table is. Tripling maturity from 10 to 30 years does not triple the price impact; it nearly doubles it, because the longer bond’s coupons stretch much further into the future. This is the single most important thing to internalise about duration: it grows with maturity, but it grows slowly, because each additional coupon payment pulls the present-value-weighted average back toward the present.
Picture the price-yield curve
The relationship between a bond’s price and its yield is not a straight line. It curves, and the curvature itself has a name: convexity. The chart below shows the price-yield curve for the 30-year Treasury in our example, with duration as the slope of the tangent line at today’s yield.
Two things to notice. First, the curve slopes down: higher yield, lower price. Duration is the slope. Second, the curve bends. The straight line is duration’s estimate of price at every yield; the real price is the curve. They agree only at the single point where the line is tangent to the curve – today’s yield.
Convexity: the small but important correction
Whenever the line and the curve disagree, the curve sits above the line. That is positive convexity. For an option-free bond it is mathematically guaranteed: if yields fall, your bond gains a little more than duration alone would predict; if yields rise, it loses a little less. The convexity term is the second-order Taylor adjustment to the duration approximation:
ΔPrice / Price ≈ − Dmod × Δy + ½ × C × (Δy)2
Where C is the bond’s convexity. For small yield moves the squared term is negligible. For big moves – the 1.5 percentage-point shifts that have actually happened in the long bond over the past two years – it can be worth more than a percentage point of bond price.
A few rules of thumb on convexity:
- Longer-maturity bonds have more convexity, because the price curve has more room to bend.
- Lower-coupon bonds have more convexity, because the cash flows are more back-loaded. A zero-coupon bond has the most convexity of any plain-vanilla bond of the same maturity.
- Callable bonds can have negative convexity, because the issuer’s option to refinance caps the upside when yields fall. Mortgage-backed securities behave this way – prepayments accelerate exactly when bondholders would otherwise be enjoying rate-fall gains.
Why long bonds are different
The chart below shows the estimated price hit for the same +100 basis-point move across the Treasury curve, using the durations from our table. The non-linearity in maturity is exactly what makes the 30-year bond the most rate-sensitive asset on the curve – and why long-duration funds tend to be the first place that bond-market stress shows up.
This is exactly why the long bond has been the most volatile part of the U.S. Treasury curve. When investors decide the term premium needs to be higher – or when foreign demand for long-dated paper softens – a 30 basis-point move in the 30-year drops about 5% off the price of a long-bond ETF. The same move in the 2-year barely registers.
Where the rule of thumb breaks down
Duration is a linear approximation, so it stops working well in three situations:
- Big yield moves. For a parallel shift of 200 basis points or more, the convexity term in the formula starts to matter. The duration estimate will overstate losses and understate gains.
- Non-parallel curve shifts. Real markets do not move every maturity by the same amount on the same day. If the long end sells off but the short end holds, a barbell portfolio with the same duration as a bullet portfolio can behave very differently.
- Embedded options. Callable bonds, putable bonds, and mortgage-backed securities have cash flows that depend on the path of rates. Use effective duration, which is calculated by repricing the bond under small up and down yield scenarios, not Macaulay or modified duration.
Common mistakes
- Confusing duration with maturity. A 10-year Treasury has a duration of about 8 years, not 10. The two only coincide for a zero-coupon bond held to maturity.
- Adding up durations naively. A portfolio’s duration is the dollar-weighted average of its holdings’ durations – not a simple average.
- Assuming duration captures all risk. Duration is a measure of interest rate risk only. Credit risk, liquidity risk, and convexity risk live in different statistics.
- Forgetting the sign. Duration is always reported as a positive number, but the price-yield relationship is always negative. When yields rise, prices fall.
What to learn next
Once duration and convexity click, the rest of fixed income gets much easier. Useful follow-ons: the difference between nominal and real yields (look at TIPS yields on FRED), how the shape of the yield curve compresses or expands duration risk across maturities, and the role of credit spreads in fixed-income returns. For the practitioner’s view of how callable mortgages get their negative convexity, the SIFMA primer on agency MBS is the standard reference.
Sources
- Federal Reserve, H.15 Selected Interest Rates – Treasury constant maturity yields as of May 14, 2026.
- Federal Reserve Bank of St. Louis (FRED), 30-Year Treasury Constant Maturity Rate (DGS30) and 10-Year Treasury Constant Maturity Rate (DGS10).
- CFA Institute, Introduction to Fixed Income Valuation – duration, convexity, and price-yield relationships.
- SEC Office of Investor Education and Advocacy, Investor Bulletins on bond market basics and interest rate risk.
- U.S. Department of the Treasury, Daily Treasury Yield Curve Rates.
Disclosure: This article was produced with AI assistance and reviewed before publication. It is for informational purposes only and is not investment advice.