TL;DR. Duration is the number that tells you, in one figure, how much a bond’s price should move when yields move. The rule of thumb: for a small change in yield, price moves by roughly −modified duration × Δy. A 7-year modified duration and a 1-percentage-point rise in yield implies about a −7% price move. Convexity is the correction term — a second derivative that captures the fact that the price–yield curve is bowed, not straight, so duration alone always underestimates price when yields fall and overestimates price when yields rise. Long-dated zero-coupon bonds have the highest duration; short T-bills the lowest. That is why long Treasuries lost roughly a third of their value in 2022 while T-bill investors barely noticed the same rate cycle.
Why duration exists at all
Bond prices move inversely to yields — that much every investor learns early. What is less obvious is how much a bond’s price should move for a given change in yield. The answer depends on the bond’s coupon, maturity, yield level, and payment frequency, all mashed together. Duration is the industry’s single-number summary of that sensitivity.
Frederick Macaulay introduced the concept in 1938 as the present-value-weighted average time to a bond’s cash flows. Modern practice uses a related figure, modified duration, that is directly interpretable as a percentage price move per unit yield move. Both are widely defined in fixed-income textbooks and in public reference materials such as the Wikipedia bond-duration entry.
The U.S. Securities and Exchange Commission’s Office of Investor Education has warned since at least 2013 that duration is the primary tool for gauging a bond portfolio’s interest-rate risk and that longer-duration portfolios can suffer “significant” losses when rates rise (SEC Investor Bulletin — Interest Rate Risk).
The three durations worth knowing
Practitioners use three related quantities. All three have units of years, and all three fall as coupons rise or as maturity shortens.
Macaulay duration
The original definition — the present-value-weighted average time to a bond’s cash flows:
DMac = ∑i ( ti · PV(Ci) ) ÷ Price
Each cash flow Ci arriving at time ti contributes its present value as a weight. A zero-coupon bond has the maximum possible Macaulay duration for its maturity: since the only cash flow is the face value at maturity, its duration equals its maturity.
Modified duration
Macaulay duration is a weighted average time; modified duration is what most desks actually quote. It is Macaulay duration divided by (1 + y/m), where y is the yield to maturity and m is the compounding frequency:
Dmod = DMac ÷ (1 + y/m)
The reason it matters is that modified duration is the first derivative of price with respect to yield, divided by price, with a sign flip. Which gives the first-order price-change rule of thumb everyone quotes:
ΔPrice/Price ≈ −Dmod × Δy
So a bond with modified duration 7.0 should lose about 7% of its price if yields rise by 100 basis points, and gain about 7% if yields fall by 100 basis points. Note that this is only a first-order approximation — and it gets worse as Δy gets bigger. That is where convexity comes in.
Effective duration
For bonds with embedded options — callable corporates, mortgage-backed securities, prepayable munis — Macaulay and modified duration mislead because the cash-flow schedule changes as yields change. Effective duration measures sensitivity numerically by shocking the yield curve up and down by a small amount and observing what a full valuation model does to the price:
Deff ≈ ( P− − P+ ) ÷ ( 2 · P0 · Δy )
where P− and P+ are model prices after small down- and up-shifts. For plain-vanilla non-callable bonds, effective duration collapses to modified duration; for a mortgage-backed pool it can be dramatically different (bond duration — effective duration).
Convexity: the correction term the straight-line rule misses
The price–yield relationship of a normal bond is a smooth downward-sloping curve, not a straight line. Modified duration draws the tangent to that curve at today’s yield. For very small yield moves the tangent is fine. For larger moves the curve bows away from the tangent — and always in the investor’s favour: prices fall by less than duration predicts when yields spike, and rise by more than duration predicts when yields collapse. This bowing is called convexity, and it is defined as the second derivative of price with respect to yield, normalised by price:
C = ( 1 ÷ Price ) · d2Price ÷ dy2
The full second-order approximation to a bond’s price change is:
ΔPrice/Price ≈ −Dmod · Δy + ½ · C · (Δy)2
Two things to notice. First, convexity is always positive for a plain-vanilla bond, so the correction term always adds to the price estimate. Second, it scales with (Δy)2, which is negligible for a 5-basis-point move and material for a 200-basis-point move. Long-dated bonds have far more convexity than short-dated bonds because their price–yield curves are far more bowed.
A worked example on a 10-year Treasury
The Federal Reserve H.15 release for the week ending August 11, 2026 shows the 10-year U.S. Treasury constant-maturity yield at 4.70% (Federal Reserve H.15). Take a stylised 10-year U.S. Treasury note with a 4.70% coupon, priced at par ($100) and paying semiannually, so y = 4.70% and m = 2.
Working through the standard bond-math for this par bond gives:
- Macaulay duration DMac ≈ 8.06 years
- Modified duration Dmod = 8.06 ÷ (1 + 0.047/2) ≈ 7.87 years
- Convexity C ≈ 77 (using the standard bond-math convention)
Now suppose yields rise by 100 basis points, from 4.70% to 5.70%. The duration-only estimate predicts a price change of:
−7.87 × 0.01 = −7.87%
The full second-order estimate adds the convexity correction:
−7.87 × 0.01 + ½ × 77 × (0.01)2 = −7.87% + 0.385% = −7.49%
The actual, exactly-repriced move on a 4.70%-coupon 10-year par bond going to a 5.70% yield is a price of about $92.44, a loss of −7.56%. The straight-duration rule over-predicts the loss by 31 basis points; the duration + convexity estimate under-predicts it by only 7. For a 200-basis-point shock the convexity term is roughly four times as large (it scales with the square) and matters a great deal more.
Duration rises with maturity, falls with coupon
Two intuitions the formula makes rigorous:
- Longer maturity ⇒ higher duration. Cash flows further in the future carry more weight and are discounted more heavily, so distant maturity dominates the calculation. A 30-year par bond has a duration close to 15 years; a 2-year par bond has a duration close to 2.
- Higher coupon ⇒ lower duration. A larger coupon returns more of your money sooner, shifting the weighted average time forward and shortening duration. A zero-coupon bond gets no coupons at all, so its Macaulay duration equals its maturity — the highest possible for its tenor.
The table below shows illustrative modified durations for the current Treasury yield curve, using each tenor’s constant-maturity yield from the latest H.15 release as both the coupon and the discount rate (a par-bond assumption).
| Tenor | Yield (%) | Modified duration (yrs) | Est. price move on +100 bps |
|---|---|---|---|
| 3-month T-bill | 3.89 | 0.25 | −0.25% |
| 2-year note | 4.22 | 1.90 | −1.90% |
| 5-year note | 4.39 | 4.42 | −4.42% |
| 7-year note | 4.54 | 5.94 | −5.94% |
| 10-year note | 4.70 | 7.87 | −7.87% |
| 20-year bond | 5.25 | 12.35 | −12.35% |
| 30-year bond | 5.24 | 15.42 | −15.42% |
The price–yield curve is bowed — that’s convexity in one picture
The two curves touch only at today’s yield (4.70% in this example). Move either way and the tangent line falls below the true curve — meaning the straight-line duration estimate under-predicts the price for any move in either direction. That gap is small for small moves and grows with the square of the yield change.
What history looks like when duration is very long
2022 is the cleanest recent object lesson in duration. The Bloomberg U.S. Aggregate Bond Index, which most balanced funds use as a fixed-income benchmark, has a duration of roughly six years. Its 2022 loss of about 13% is very close to what its duration times the year’s ~200-basis-point yield back-up would predict — with a small offset from convexity and the coupon return in the other direction.
Common mistakes retail investors make with duration
- Confusing duration with maturity. They are not the same. A 30-year bond with a 5% coupon has a duration around 15 years, not 30. A 30-year zero-coupon bond’s Macaulay duration is 30 years — and its price move for a 100-basis-point yield change is proportionally larger.
- Assuming the duration rule holds for big yield moves. For a 25-basis-point shock, ignoring convexity costs almost nothing. For a 200-basis-point shock in a long-duration bond, ignoring convexity can misprice the position by several percent.
- Using modified duration on callable or MBS securities. Cash flows change with rates in these securities. Use effective duration (or a full valuation model). Mortgage-backed securities can even have negative convexity at low yields because falling rates trigger refinancing.
- Forgetting duration changes as time passes. A bond’s duration shortens every day as its cash flows draw closer. Duration also changes with yield: as yields rise, duration falls modestly (the “convexity of duration”).
- Ignoring the coupon reinvestment side. Duration captures the price-risk half of interest-rate risk. Higher yields hurt price today but lift future coupon reinvestment income — the two effects offset over a holding period equal to the bond’s Macaulay duration. This is the classic duration immunization result.
Where duration sits in the bigger picture
Duration is the fixed-income analogue of beta on the equity side: a one-number sensitivity to the market’s dominant risk factor. Portfolio managers extend the idea in two natural directions. Key-rate durations (or partial durations) break out sensitivity to shifts at specific points on the yield curve — useful for portfolios that face non-parallel curve moves such as steepeners and flatteners. Spread duration captures sensitivity to credit-spread moves (as opposed to Treasury yields) and matters most for corporate and high-yield portfolios where credit spreads dominate the risk budget.
Related concepts and what to learn next
- Yields themselves — how bond quotes are converted into yield to maturity, and why yield to worst matters for callable bonds. See our companion piece: Yield to Maturity Explained: YTM, YTC, YTW.
- The shape of the Treasury curve and why it inverts before recessions: The Yield Curve Explained.
- What the underlying instruments are — bills, notes, bonds — and why their durations differ: T-Bills, T-Notes, and T-Bonds Explained.
Sources
- U.S. Securities and Exchange Commission — Investor Bulletin: Interest Rate Risk — When Interest Rates Go Up, Prices of Fixed-Rate Bonds Fall. Confirms duration as the primary measure of interest-rate risk and the price-decline-on-rising-rates relationship.
- Federal Reserve Board — H.15 Selected Interest Rates. Source for Treasury constant-maturity yields cited above (week ending August 11, 2026).
- Federal Reserve Board — FEDS Notes: How Do Bond Investors Measure Performance? Bonds-101 primer covering duration, convexity, and the drivers of the Bloomberg Aggregate’s 2022 drawdown.
- Wikipedia — Bond duration. Source for the standard Macaulay, modified, and effective-duration formulas and the second-order duration+convexity price approximation.
- Wikipedia — Bond convexity. Source for the second-derivative convexity definition and the sign-of-convexity discussion applied to callable and MBS securities.
Disclosure: This article is for informational purposes only and is not investment advice.