Bond Duration and Convexity Explained: How Prices Move

TL;DR. Duration is a single number that tells you how much a bond’s price
will move if yields change by 1%. A bond with a modified duration of 7 will lose roughly 7%
of its price if yields rise 100 basis points and gain roughly 7% if yields fall 100 bps.
Convexity is the second-order correction — it says the true price–yield relationship
is curved, so duration alone understates gains when yields fall and overstates
losses when yields rise. The bigger the yield move and the longer the bond, the more convexity
matters.

The core idea: bond prices and yields move opposite each other

A bond’s coupon is fixed at issue. If market yields rise after you buy, newly issued bonds
of similar credit and maturity offer more income, so the price of your older, lower-coupon bond
has to fall until its yield-to-maturity matches the market. If yields fall, the reverse: your
bond’s above-market coupon is suddenly attractive, so its price rises.

The SEC’s Office of Investor Education uses a simple example: a 10-year Treasury bond with a
3% coupon and $1,000 face value. If market rates fall from 3% to 2% one year in, the bond’s
price rises to roughly $1,082. If rates instead rise from 3% to 4%, the price
falls to roughly $925 (SEC Investor Bulletin: Interest Rate Risk, 2013).

Duration is the number that lets you skip the discounted-cash-flow math and estimate this
move directly.

Duration: three flavors, one useful number

Three definitions get thrown around interchangeably, but they mean slightly different things.

• Macaulay duration is the weighted average time (in years) until you
  receive the bond’s cash flows, where each cash flow is weighted by its present value. It’s the
  original definition, invented by Frederick Macaulay in 1938.
• Modified duration is Macaulay duration divided by (1 + y/n),
  where y is the yield and n is the number of coupon periods per year. This
  is the number that actually estimates price sensitivity: %ΔPrice ≈
  −ModDuration × Δy.
• Effective duration is used for bonds with embedded options
  (callables, MBS, putable bonds). It estimates price sensitivity by repricing the bond after
  small parallel shifts in the yield curve, capturing the fact that the cash flows themselves
  change with rates.

For plain vanilla Treasuries and corporates, modified duration is what fund fact sheets and
Bloomberg terminals report. For MBS, callable munis, or preferreds with call features, use
effective duration.

The rule of thumb

For small yield moves, this approximation is close enough for most work:

So a bond with a modified duration of 5 loses about 5% if yields rise 100 bps and gains
about 5% if yields fall 100 bps. A 20-year zero-coupon Treasury has a duration close to 20;
if the 20-year yield jumps 50 bps, expect a roughly 10% price drop.

Worked example: a 10-year 3% coupon Treasury

Take a 10-year Treasury bond with a 3% coupon, semiannual payments, and a yield to maturity
of 3%. Running the present-value math on the 20 semiannual cash flows gives a Macaulay duration
of roughly 8.79 years and a modified duration of about
8.66 years. If the 10-year yield rises 100 bps to 4%, duration predicts a
price drop of about 8.66%. The SEC bulletin’s discounted-cash-flow calculation gives a drop from
$1,000 to $925 — a 7.5% fall. Duration overshoots because it’s a straight-line estimate
of a curved relationship. That gap between the linear estimate and the actual price is where
convexity comes in.

Convexity: the curvature correction

The true price–yield relationship is convex. Plot price on the vertical axis and yield
on the horizontal, and the curve is bowed toward the origin. Duration is the slope of the
tangent line at a given yield. Two consequences:

• Duration is only accurate for infinitesimally small yield moves.
• Because the true curve sits above the tangent line on both sides, duration
  understates gains when yields fall and overstates losses when
  yields rise. Convexity is always the investor’s friend when it’s positive (which it is for
  option-free bonds).

The second-order Taylor expansion gives a cleaner estimate:

For the 10-year 3% coupon Treasury above, convexity is roughly 87. Plugging in a
+100 bp yield shock: −8.66% + 0.5 × 87 × (0.01)² = −8.66% +
0.44% = −8.22% — much closer to the actual 7.5% drop, though still
a touch high because bond math isn’t perfectly quadratic either.

Duration snapshot: how sensitive is what you own?

Modified durations vary enormously by structure. Here’s a snapshot of common instruments
with typical durations at yields near the September 2026 Treasury curve
(Fed
H.15 as of 2026-09-02: 2Y 4.39%, 10Y 4.79%, 30Y 5.27%).

Instrument	Modified duration (yrs)	% price change if yields +100 bps
2-year Treasury note	~1.9	−1.9%
10-year Treasury note (4.79% coupon)	~7.9	−7.9%
30-year Treasury bond (5.27% coupon)	~15.5	−15.5%
30-year zero-coupon Treasury	~30.0	−30.0%
Investment-grade corporate index (~10-yr avg maturity)	~7.0	−7.0%
High-yield corporate index (~5-yr avg maturity)	~3.5	−3.5%
Floating-rate note (coupon resets quarterly)	~0.25	−0.25%

Two things jump out. First, the 30-year zero has almost double the duration of the 30-year
coupon bond, because you’re waiting the full 30 years for the single payment — there are
no intermediate coupons pulling the weighted-average time forward. Second, floaters have near-
zero interest-rate duration because the coupon itself resets to market. That’s why bank loan
funds held up in 2022 while long-duration Treasury funds got crushed.

What duration missed in 2022

2022 was the worst year for the Bloomberg U.S. Aggregate Bond Index in its history because
duration collided with a historic yield shock. The Fed took the funds rate from near zero to
4.5% over the year and the 10-year yield roughly doubled. With the Agg’s duration around 6.5
years, a ~230 bp move in intermediate yields translated to roughly a 15% drawdown —
almost exactly what duration would predict. Investors who owned long-dated Treasuries thinking
they were “safe” learned the hard way that credit risk and interest-rate risk are separate
animals.

Common mistakes

• Confusing maturity with duration. A 30-year bond with a fat coupon can
  have a duration well below 20; a 30-year zero has a duration of 30. Coupon matters.
• Applying duration to giant yield moves. The linear approximation breaks
  down for shocks of 200 bps or more. Use duration plus convexity or reprice from the
  cash-flow model.
• Ignoring credit spread duration. A high-yield bond has both interest-rate
  duration and spread duration. In a recession, spreads widen even as rates fall — the
  two can move against each other.
• Assuming bond funds behave like individual bonds. A held-to-maturity
  bond returns par at the end. A bond fund never matures — its NAV reflects marked-to-
  market duration risk indefinitely (see our related piece,
  Bond ETFs vs Individual Bonds).
• Forgetting negative convexity in callables and MBS. A callable bond gets
  called away when rates fall, capping your gains. Mortgages prepay faster in rallies, shortening
  duration exactly when you’d want more of it.

What to learn next

Duration and convexity are the workhorses of fixed-income risk management. Once they click,
the next stops are DV01 (dollar duration for one basis point), key-rate durations (how a bond
reacts to twists in the curve rather than parallel shifts), and how the Bloomberg U.S. Aggregate
gets its 6-to-7-year duration from a blend of Treasuries, agency MBS, and corporates. For a
refresher on how yields set the shape of the curve you’re durating against, see our companion
piece on the yield curve and what inversion has (and hasn’t) predicted.

Sources

• SEC Office of Investor Education, Investor Bulletin: Interest Rate Risk — When Interest Rates
  Go Up, Prices of Fixed-Rate Bonds Fall (SEC Pub. No. 151, 2013).
• Federal Reserve, H.15 Selected Interest Rates (release of 2026-09-02).
• Federal Reserve Bank of St. Louis, 10-Year Treasury Constant Maturity Rate (DGS10).
• Federal Reserve Bank of St. Louis, 2-Year Treasury Constant Maturity Rate (DGS2).
• Federal Reserve Bank of St. Louis, 30-Year Treasury Constant Maturity Rate (DGS30).
• TreasuryDirect, Treasury Bonds — product overview.
• Federal Reserve Bank of St. Louis, ICE BofA US Corporate Index Total Return (BAMLCC0A0CMTRIV).

Disclosure: This article is for informational purposes only and is not investment
advice.