Bond duration and convexity: measuring interest-rate risk

Every bond loses value when interest rates rise, but not by the same amount. A 2-year Treasury barely notices a one-point rise in yields; a 30-year Treasury can lose 15% of its price. Duration is the number that captures that difference. It is the most useful single measure of a bond's interest-rate risk, and it is the figure bond fund providers publish so investors can compare funds.

Duration has a limit. It is a straight-line estimate of a curved relationship, accurate for small moves and increasingly wrong for large ones. Convexity measures that curve. This page works through both with a 10-year bond calculated line by line, then uses them to explain what happened to Treasury ETFs when yields rose four to five percentage points between 2020 and 2023. Bond basics, such as coupons, yields and the price formula, are in Bonds and fixed income: how bonds work, pay and lose value.

What duration measures

The word means two related things, and both are used daily.

Macaulay duration is a length of time. Take each of a bond's payments, find its present value, and ask: on average, weighted by those present values, how many years do I wait to be paid? A zero-coupon bond pays everything at maturity, so its Macaulay duration equals its maturity. A coupon bond returns some money early, so its duration is shorter than its maturity.

Modified duration is a sensitivity. It answers: by roughly what percentage does the price change for a one-percentage-point change in yield? FINRA describes duration in exactly these terms, as a measure in years of a bond's price sensitivity to a 1% change in rates.

Δy is the change in yield written as a decimal (one percentage point = 0.01). The second line adds the convexity correction, covered below.

A worked example: a 10-year, 4% bond

Take a bond with $100 of face value, a 4% coupon paid once a year, 10 years to maturity, and a market yield of 4%, so it is priced at exactly $100. Each payment is discounted at 4%, and its weight is its present value divided by the $100 price:

The final payment, coupon plus principal, carries 70% of the value, so the weighted average time sits close to the end: 8.44 years. That is the Macaulay duration.

Modified duration is 8.44 ÷ 1.04 = 8.11. A one-point rise in yield, from 4% to 5%, should cut the price by about 8.11%. Re-pricing the bond at 5% gives $92.28, an actual fall of 7.72%. A one-point fall, to 3%, should raise the price by about 8.11%; re-pricing gives $108.53, an actual gain of 8.53%.

Convexity: why duration misses on big moves

The estimates above were off in a consistent direction: duration predicted a loss of 8.11% where the real loss was 7.72%, and a gain of 8.11% where the real gain was 8.53%. In both cases the bond did better than duration predicted. The reason is that the price-yield relationship is a curve, bowed toward the origin, while duration is the slope of a straight line touching it at today's yield.

Convexity measures how strongly the curve bends. For the 10-year bond it is about 80.75 (in years squared). Adding the correction term ½ × convexity × (Δy)² closes most of the gap:

For a one-point move, duration alone is within half a point of the truth. For a three-point move it overstates the loss by more than three points, and even the convexity-corrected estimate is off by a third of a point. The lesson: duration is a local tool. For big moves, re-price the bond.

Positive convexity is a good property to own: gains are larger than losses for the same size of move. Bonds that the issuer can call early, and mortgage-backed securities, whose borrowers refinance when rates fall, can have negative convexity. Their price rises less when yields fall, because the bond is likely to be repaid early, and for them analysts use effective duration, estimated by re-pricing the bond under small yield shifts rather than from a fixed payment schedule.

What makes duration longer or shorter

Three features drive duration:

Maturity. The chart re-prices 4% semiannual-coupon bonds of four maturities when yields rise from 4% to 5%. The losses are 1.9%, 4.4%, 7.8% and 15.5%. Their modified durations are about 1.90, 4.49, 8.18 and 17.38. Duration rises with maturity, but more slowly: the 30-year bond has 15 times the maturity of the 2-year and about 9 times its duration, and about 8 times its price loss, because convexity softens the larger move.

Coupon. Higher coupons return more money earlier, which shortens duration. At a 4% yield, 10-year bonds with coupons of 0%, 2%, 4% and 8% have Macaulay durations of 10.0, 9.07, 8.44 and 7.64 years.

Yield level. Lower yields give distant payments more weight, which lengthens duration. A 10-year 4% semiannual bond has a Macaulay duration of 8.57 years at a 1% yield, 8.34 at 4% and 7.99 at 8%. This is the effect that made 2020 so dangerous for bond holders: yields were near record lows, so durations were near their longest.

Duration as a break-even horizon

Duration has a second, useful reading. Suppose you buy the 10-year 4% bond at par and yields immediately jump to 5%. The price falls to $92.28, but from then on every coupon is reinvested at 5% instead of 4%. Over time the higher reinvestment income makes up for the price loss.

The approximation holds for a single parallel shift in yields, and it is the logic behind a common rule of thumb: an investor whose horizon matches a bond's or a fund's duration is roughly protected from a one-time rate change, because price risk and reinvestment risk offset each other.

What duration did to Treasury ETFs, 2020–2024

Between 2020 and 2024, Treasury ETFs ran a real-world version of the maturity chart. The deepest monthly-close falls were 6.7% for SHY (1–3 year Treasuries), 22.8% for AGG (the broad investment-grade market, an intermediate mix of Treasuries, mortgage bonds and corporates), 27.0% for IEF (7–10 year Treasuries) and 51.1% for TLT (20+ year Treasuries). The order is the order of their maturities, and therefore their durations.

To see why TLT's fall was so large, take a hypothetical 20-year Treasury bought at par in July 2020, when the 20-year yield averaged 1.09%. With a coupon that low, its modified duration would be about 17.9. By October 2023 the 20-year yield averaged 5.13%, a rise of 4.04 points. For a move that size:

• duration alone predicts a loss of 17.9 × 4.04 ≈ 72%;
• duration plus convexity predicts about 44%;
• actually re-pricing the bond gives 50%, before counting the three years of coupons and the shorter maturity by then.

Neither shortcut is close for a move that large. The convexity term (about 353 for this bond) is huge because the yield started so low, and the second-order estimate overshoots. The fund itself is a moving portfolio that keeps rolling into new long bonds, so its path differs from any single bond's, but the scale of its 51% fall is what this arithmetic implies. On monthly closes TLT fell further still, to about 55% below its July 2020 level in September 2026, when the 20-year yield averaged 5.40%.

Pitfalls in using duration

It assumes a parallel shift. Duration treats all yields as moving by the same amount. In reality short and long rates often move differently; from July 2020 to October 2023 the 2-year yield rose about five points and the 10-year about four. Analysts handle this with key-rate durations, sensitivities to each point on the curve.

It is local. As the TLT example shows, for moves of several points the straight-line estimate can be wildly wrong. Re-price instead.

It changes over time. A bond's duration shortens as it ages and lengthens when yields fall. A fund's stays roughly constant because the manager keeps buying new bonds.

It ignores credit. Corporate bond prices also move with credit spreads; duration only measures sensitivity to the yield, not why the yield changed.

Callable bonds need effective duration. For bonds with embedded options, a standard Macaulay calculation based on the stated maturity overstates how long the bond is likely to last.

Duration measures compared

The short definition is in Duration; how maturity choices affect funds versus single bonds is in Bond ETFs vs individual bonds: maturity, risk and cost compared; how credit adds a second kind of yield risk is in Credit ratings and credit spreads: how bond risk is priced.

How Finabase does it

Finabase does not calculate bond durations or convexity and has no bond-level data. What it shows are the yields that duration translates into prices and the funds where the effect is visible:

• The Treasury page lists yields from the 1-month bill to the 30-year bond, each with its own chart, such as the 10-year. Comparing the 1-week to 2-year change columns shows when long and short yields move by different amounts, the case where a single duration number is least reliable.
• Treasury ETFs of different maturities, such as SHY, IEF, AGG and TLT, have their own symbol pages. Charting them over the same range shows the duration ladder directly.

The worked example and both charts on this page are ours: the bond math is calculated, the ETF falls use Finabase's monthly prices, and the yields come from Federal Reserve data.

Frequently asked questions

What does a duration of 7 mean?

It means the bond's price will change by roughly 7% for each one-percentage-point change in its yield, falling if yields rise and rising if they fall. For moves larger than about one point, convexity makes the real change smaller on the way down and larger on the way up.

What is the difference between Macaulay and modified duration?

Macaulay duration is a time, the weighted average years until a bond's payments arrive. Modified duration is a sensitivity, the percentage price change per point of yield, and equals Macaulay duration divided by one plus the periodic yield. For the 10-year 4% bond they are 8.44 years and 8.11.

Why do long-term bonds have more interest-rate risk?

Because more of their value comes from payments far in the future, and distant payments are the most affected by a change in the discount rate. A 1-point rise cuts a 4% 30-year bond's price by about 15.5%, against 1.9% for a 2-year.

Is high convexity good?

For an option-free bond, yes, other things equal: it means larger gains when yields fall and smaller losses when they rise. The trade-off is that bonds with more convexity usually offer a slightly lower yield.

Summary

Duration is the bridge between a change in yields and a change in bond prices. Macaulay duration measures when, on average, a bond pays you back; modified duration converts that into a percentage price change per point of yield. For a 10-year 4% bond they are 8.44 years and 8.11, so a one-point rise costs about 8%. Convexity corrects for the curve in the price-yield relationship and matters more the bigger the move. Longer maturities, lower coupons and lower yields all raise duration, which is why long Treasury funds bought at 2020's record-low yields fell by about half when yields climbed four points, while short Treasury funds lost under 7%.