Bond Duration and Convexity Explained: Interest Rate Risk
Bond duration measures the sensitivity of a debt security’s price to changes in yields, while convexity describes the curvature of the bond’s price–yield relationship. Duration is fundamental to interest-rate risk management because two bonds with the same face value or maturity can respond very differently to the same movement in rates. Convexity improves the estimate when yield changes are too large for a purely linear approximation.
For traders and investors, the practical question is not simply “How many years until this bond matures?” but “How much money might this position gain or lose when a particular part of the yield curve moves?” This guide distinguishes Macaulay duration, modified duration, effective duration, DV01 and convexity with worked calculations and limitations.
Why maturity is not enough to measure interest-rate risk
A bond’s value is the present value of scheduled cash flows. A five-year zero-coupon security pays only at the end, while a five-year coupon bond distributes some value earlier. Their maturity dates are identical, but the timing of their cash flows differs and so does interest-rate sensitivity.
In general, cash paid later has greater sensitivity to changes in discount rates. Low-coupon and zero-coupon bonds tend to have greater duration than otherwise comparable high-coupon bonds with the same maturity. However, embedded options, floating coupons, credit risk and the shape of the discount curve can alter the relationship.
Macaulay duration explained
Macaulay duration measures the present-value-weighted average time until the bond’s cash flows are received. Its unit is time, commonly years. Each future coupon and principal payment is weighted by its share of the bond’s current present value; the result summarises the centre of gravity of the cash-flow schedule.
For a simplified bond with annual cash flows, Macaulay duration = Σ[t × PV(CFt)] ÷ bond price. A zero-coupon bond with a single maturity payment has Macaulay duration equal to its maturity under the standard fixed-cash-flow formulation. Coupon bonds usually have a lower Macaulay duration than their final maturity.
Macaulay duration is an important concept but is not itself the most direct percentage price-sensitivity coefficient. For that, modified duration or an appropriate effective duration measure is usually used.
Modified duration explained
Modified duration approximates the percentage change in bond price for a small change in yield, assuming the bond’s promised cash flows remain fixed. For yield expressed per annum with m compounding periods, modified duration is often calculated as Macaulay duration ÷ (1 + y/m), with consistent conventions.
The first-order relation is percentage price change ≈ −modified duration × yield change, using the yield change as a decimal. If modified duration is 6.0 and yield rises 0.25 percentage points, Δy = +0.0025, then approximate price change is −6 × 0.0025 = −1.5%.
The negative sign is essential: price and yield generally move inversely when the cash-flow structure is fixed. A 25-basis-point yield decline instead gives an approximate 1.5% price increase under the same first-order assumption.
Worked calculation: modified duration and price change
Imagine a bond trading at £102 per £100 face value with modified duration of 7.5. If the required yield increases by 50 basis points (0.005), the duration-only price estimate is −7.5 × 0.005 = −3.75%. Estimated new bond price is £102 × (1 − 0.0375) = £98.175, before convexity and all other changes.
If required yield falls 50 basis points, the same linear formula gives +3.75%, or approximately £105.825. The actual price response of an ordinary option-free bond is typically slightly curved rather than symmetric, meaning the two estimates are not exact. Convexity accounts for part of that discrepancy.
| Hypothetical yield change | Modified duration 7.5 | Approx. price from £102 |
|---|---|---|
| −100 bp | +7.50% | £109.65 |
| −50 bp | +3.75% | £105.83 |
| +50 bp | −3.75% | £98.18 |
| +100 bp | −7.50% | £94.35 |
Linear duration-only estimates; convexity and cash-flow/credit changes are excluded.
DV01 and PVBP: measuring money at risk
DV01, or dollar value of one basis point, measures the approximate change in a position’s market value for a one-basis-point change in yield, under a defined method. In sterling positions a similar concept may be called PVBP or the value of one basis point. It translates relative sensitivity into money terms.
For a bond valued at £10 million with modified duration 5, the approximate absolute value change for 1 bp is £10,000,000 × 5 × 0.0001 = £5,000 per bp. A ten-basis-point yield rise produces an approximate £50,000 loss under the linear approximation, before other market movements.
A trader comparing two bonds by DV01 can reduce the risk of confusing notional size with risk size. £5 million in a long-dated gilt can have more yield sensitivity than £10 million in short-dated debt. Yet a DV01 match at one yield point does not eliminate exposure to yield-curve twists or changes in credit spread.
What is convexity?
Convexity measures the curvature of a bond’s price as a function of yield, capturing the part of the movement not explained by the local straight-line duration approximation. For a typical option-free fixed-rate bond, the price–yield curve is positively convex: as yields fall, the price rises by slightly more than a linear approximation suggests; as yields rise, the price usually falls by slightly less.
The common second-order approximation is ΔP/P ≈ −Dmod × Δy + ½ × C × (Δy)2, where C is appropriately defined convexity and the yield change is in decimal form. The definition and units of convexity must match the compounding and sensitivity conventions used.
Because the convexity term includes the square of the yield change, its contribution becomes more important as yields move further from the starting point. It does not eliminate error from different cash-flow assumptions or large dislocations.
Worked duration-plus-convexity example
Suppose a fictional option-free bond has modified duration 8 and convexity 85 under compatible conventions. If yield rises by 100 basis points, Δy = +0.01. The duration contribution is −8 × 0.01 = −8%, while the convexity contribution is ½ × 85 × 0.01² = +0.425%. The combined approximation is −7.575%.
For a 100-basis-point yield decline, Δy = −0.01. The duration term is +8% and the convexity term remains +0.425%, giving an approximate +8.425%. The positive convexity term improves the symmetry-free estimate for an ordinary option-free bond. It is not a guarantee of those exact price changes.
Options embedded in debt can change the behaviour. A callable bond may exhibit limited upside when yields fall because the issuer becomes more likely to redeem the bond early. An instrument with material prepayment risk can have very different effective duration and convexity from a standard fixed-rate government bond.
Effective duration for bonds with options
Effective duration estimates price sensitivity by modelling how the instrument’s value and possibly its cash flows respond to a change in underlying interest rates. It is especially relevant for callable or mortgage-backed securities whose expected payments can change with rates.
The output depends on a valuation model, assumed volatility, borrower or issuer behaviour and other factors. A simple modified-duration calculation based on an unchanging payment schedule may substantially misstate risk when a bond is likely to be called or prepaid. Analysts should distinguish model assumptions from observed contract terms.
Positive and negative convexity
Ordinary option-free fixed-rate bonds typically have positive convexity. Callable bonds can exhibit reduced or negative effective convexity over some yield ranges, because lower rates increase the probability that the issuer redeems. In mortgage-backed securities, declining rates may accelerate prepayments and shorten expected cash flows, a form of negative-convexity exposure.
This explains why not all fixed income positions respond alike when bond yields fall. A plain gilt might benefit from lower yields, while a callable corporate bond’s upside is capped or reduced by expected redemption. A trader must inspect both yield risk and option structure.
Duration versus maturity: five differences
Maturity is the contractual endpoint; duration measures cash-flow timing or price sensitivity. Coupon payments shorten Macaulay duration relative to final maturity. Modified duration changes when required yields change. Callable securities may have effective duration that changes sharply as exercise becomes more likely. Finally, duration estimates can be reported under different pricing and compounding conventions.
For a zero-coupon bond, Macaulay duration equals maturity under standard conventions, but modified duration is slightly different because of the yield denominator. For a floating-rate note that resets frequently, effective rate sensitivity may be much lower than its final legal maturity would suggest, though credit-spread and basis risks remain.
Yield-curve risk: why a single duration number is incomplete
Modified duration generally assumes one yield move, often treated as parallel for a portfolio-level approximation. Real curves can steepen, flatten or twist. A portfolio concentrated in five-year bonds may react differently from one exposed to thirty-year bonds even when both report similar aggregate duration.
Key-rate duration measures sensitivity to changes at selected maturity points, holding other curve conditions under a defined methodology. These sensitivities help traders understand which part of the curve creates P&L under non-parallel shifts. The concept is developed further in Yield Curves Explained.
Credit-spread duration is different from government-rate duration
Corporate bond yields contain both a benchmark-rate component and compensation for credit, liquidity and other factors. A corporate bond can lose value because Treasury yields rise or because its credit spread widens. Spread duration measures sensitivity to changes in a specified credit spread, under relevant assumptions.
Two portfolios with the same government-rate duration can have very different credit exposures. A high-yield portfolio may experience substantial spread widening during a downturn even as government bond yields decline, leaving its overall returns negative. See Corporate Bond Credit Spreads and Default Risk.
Inflation-linked duration
Inflation-linked bonds are sensitive to real yield changes and inflation-indexation structures rather than being identical to nominal bonds of the same final maturity. A long-dated index-linked gilt can have high real-duration exposure and suffer a significant mark-to-market decline when real yields rise, even in a period of elevated reported inflation.
Traders should distinguish the nominal price, inflation-adjusted principal, quoted real clean price and valuation convention. Use the exact indexation terms instead of calculating an ordinary nominal duration based only on a fixed coupon.
Position risk example using DV01
A hypothetical portfolio has £3,000 DV01 in two-year exposure and £7,000 DV01 in ten-year exposure, both long duration. If two-year yields rise 20 basis points and ten-year yields rise 60 basis points, the simple first-order total loss is £3,000 × 20 + £7,000 × 60 = £480,000, ignoring convexity and other changes.
Using only the average or change in a single headline rate would obscure the larger contribution from ten-year exposure. The example highlights the value of maturity-bucket reporting and scenario testing rather than relying on a single total portfolio duration.
A practical duration and convexity checklist
- Verify the bond’s cash-flow schedule, settlement conventions and embedded options.
- Identify whether the reported metric is Macaulay, modified or effective duration.
- Calculate DV01 in portfolio currency using the relevant full market value.
- Check whether convexity is positive, low or potentially negative.
- Stress-test both yield rises and falls at realistic magnitudes.
- Analyse non-parallel maturity shifts using key-rate duration or scenarios.
- Measure credit-spread duration separately for risky issuers.
- Recalculate sensitivities after large rate moves, calls or portfolio changes.
DV01 hedging: why equal face amounts do not neutralise rate exposure
Suppose an investor holds £2 million of a corporate bond with modified duration 6, giving approximate government-rate DV01 of £2,000,000 × 6 × 0.0001 = £1,200 per basis point. To offset part of this rate exposure using a government bond with £800 DV01 per £1 million market value, a simple DV01-matched hedge might require £1.5 million market value on the opposite side under the stated approximations.
This is a sensitivity match, not a risk-free position. The corporate bond’s credit spread can still widen, while the hedge and corporate bond may respond to different yield-curve segments. Currency, contract multipliers, basis and market liquidity can also create residual exposure. A hedge ratio should be reviewed as yields and portfolio holdings change.
Measuring a non-parallel rate shock
Imagine a portfolio whose key-rate DV01 is £1,000 per bp at two years, £2,500 at five years and £3,000 at ten years. If the relevant yield points rise 10, 30 and 60 basis points respectively, a first-order scenario loss is about £10,000 + £75,000 + £180,000 = £265,000. Using a single assumed parallel 30-basis-point shift would give a different result and hide the larger long-end contribution.
Real key-rate calculations require an underlying curve model and consistent bump methodology. The simple sum can also miss interpolation, correlations and complex option-related behaviour. Nevertheless, it demonstrates why maturity location matters as well as aggregate duration.
Convexity is not protection against a credit event
The duration-and-convexity approximation assumes a reasonably continuous price response to a chosen yield factor. A sudden issuer default, a restructuring announcement or a loss of market liquidity can create discontinuous changes that are not well captured by a smooth yield curve. A bond may fall sharply because expected payments have changed, not merely because the rate used to discount unchanged payments increased.
For risky issuers, analyse default and recovery scenarios separately. The market value could fall dramatically even when benchmark government rates are stable or declining. Treat the smooth interest-rate model as one component of a broader risk system rather than as a maximum-loss calculation.
Checking the sensitivity after a price move
Duration is a local measure and can change as the bond’s yield, price or expected cash flows change. A callable bond can gain or lose effective duration when exercise becomes more or less likely. A mortgage-related security may alter expected cash-flow timing when borrowers change refinancing behaviour. Therefore a duration figure downloaded last month may no longer describe current risk accurately.
A professional risk note should record calculation date, price, model, yield convention, cash-flow assumptions and whether the measure is modified, effective or key-rate duration. Recalculation is especially important after large rate moves, corporate events or portfolio rebalancing.
Frequently asked questions
Is a bond with duration five guaranteed to mature in five years?
No. Duration is a cash-flow timing or price-sensitivity measure depending on its definition, and it can differ materially from final maturity.
What does duration seven mean for a 1% yield rise?
For modified duration seven, the first-order price estimate is approximately a 7% decline, before convexity and other changes. It is not a guaranteed or maximum loss.
Why is convexity useful?
It improves the estimate of bond-price changes when the relationship with yield is curved, especially for larger moves.
Does convexity always benefit the bondholder?
Typical option-free bonds have positive convexity, but callable or prepayable debt can display negative effective convexity in certain conditions.
What is DV01?
The approximate monetary change in a bond or portfolio’s value for a one-basis-point movement in a specified yield or rate factor.
Key takeaways
Duration translates yield movement into approximate price sensitivity, DV01 expresses that sensitivity in cash terms, and convexity accounts for curvature. For robust fixed income risk management, combine these measures with credit-spread risk, key-rate exposures, embedded options and actual market liquidity.
Related Fixed Income Education
Connect risk sensitivities to bond pricing and yields, bond fund duration exposure and the six-pillar fixed income framework.
Sources and further reading
- UK DMO — Duration, Convexity and Gilt Yield Glossary.
- SEC — Bond Risks and Interest Rates.
- US Treasury — Marketable Security Pricing.
Important notice: General market education, not trading advice. Sensitivities and illustrative figures are not live bond prices. Duration and convexity are approximations that can fail to capture large or nonlinear changes, losses or liquidity stress.