Modified Duration: Bond Price Sensitivity Formula, Examples, and Limits

What is Modified Duration?

Modified duration estimates the percentage change in a conventional bond's price for a small change in the bond's yield, assuming the cash flows do not change.

It turns the timing concept behind Bond Duration into a practical first-order price-sensitivity measure.

A common approximation is:

1Percentage price change ≈ -Modified Duration × Change in Yield

If modified duration is 6.0 and yield rises by 0.50 percentage points, or 50 basis points:

1Δy = +0.0050
2
3Estimated price change
4≈ -6.0 × 0.0050
5≈ -3.0%

The negative sign captures the usual inverse relationship between bond prices and yields.

But modified duration is an approximation, not an exact pricing formula. The true bond price-yield relationship is curved, and the approximation becomes less accurate as the yield move gets larger.

Modified duration comes from Macaulay duration

For a conventional fixed-rate bond with periodic compounding, modified duration can be calculated from Macaulay duration:

1Modified Duration
2=
3Macaulay Duration / (1 + y / m)

where:

• y is the annual yield to maturity under the quoted convention; and
• m is the number of coupon or compounding periods per year.

Suppose a bond has:

1Macaulay duration: 6.30 years
2YTM:               5.00%
3Compounding:       semiannual
4m:                  2

Then:

1Modified duration
2= 6.30 / (1 + 0.05 / 2)
3= 6.30 / 1.025
4≈ 6.15

That 6.15 is a price-sensitivity coefficient, not a remaining maturity.

Why modified duration is not measured like maturity

Macaulay duration is expressed in time units because it is a present-value-weighted average time to cash flow.

Modified duration is derived from that measure but is used as a percentage price response per unit change in yield.

That distinction matters because the same word, "duration," is used for both.

A reader who sees:

1Duration: 6.2 years

on a bond or fund page should not automatically assume the provider is reporting Macaulay duration. Many investor-facing fixed-income products use effective or modified duration as their headline rate-risk measure.

Always check the methodology.

Convert basis points correctly

A common duration error comes from mixing percentage points, percentages, and basis points.

One basis point is:

11 bp = 0.01 percentage point = 0.0001 in decimal form

Therefore:

125 bps  = 0.25 percentage point = 0.0025
250 bps  = 0.50 percentage point = 0.0050
3100 bps = 1.00 percentage point = 0.0100

The duration formula uses the decimal change in yield.

If modified duration is 7.0 and yield rises 100 bps:

1Estimated change ≈ -7.0 × 0.0100 = -7.0%

Using 100 or 1 in the formula instead of 0.01 would produce a meaningless result.

A hypothetical 75-basis-point example

Assume an option-free fixed-rate bond has:

1Price:              $980
2Modified duration:   6.2
3Yield change:        +75 bps

Convert the yield move:

175 bps = 0.0075

Then:

1Estimated percentage price change
2≈ -6.2 × 0.0075
3≈ -4.65%

The approximate new price is:

1$980 × (1 - 0.0465)
2≈ $934.43

That is a useful first-order approximation.

It should not be interpreted as the exact repriced bond value because the calculation ignores curvature and other moving parts.

Modified duration is a slope, not a full price curve

The bond price-yield relationship is nonlinear.

Modified duration approximates the slope of that relationship around the current yield.

A straight tangent line can closely approximate a curve near the point of contact, but the error grows as the move gets larger.

That is why the duration-only approximation tends to be more useful for:

• relatively small yield changes;
• conventional option-free bonds; and
• quick sensitivity analysis.

It becomes less reliable for:

• large yield shocks;
• very long-duration bonds;
• low-yield bonds with strong curvature; and
• securities whose expected cash flows change with rates.

Convexity adds a second-order adjustment for the curvature.

Adding convexity improves the estimate

A common duration-plus-convexity approximation is:

1ΔP / P
2≈
3-D_mod × Δy
4+
50.5 × Convexity × (Δy)^2

The exact scale and annualization convention for convexity should match the source of the duration measure.

For an ordinary option-free fixed-rate bond, convexity is generally positive.

That means the convexity term can partially offset the duration-estimated loss when yields rise and add to the duration-estimated gain when yields fall.

This asymmetric behavior reflects the actual curved price-yield relationship.

The important lesson is not that convexity magically removes rate risk. It is that a first-order straight-line estimate is incomplete.

Modified duration assumes cash flows stay fixed

This assumption is critical.

For a standard option-free Treasury or corporate bond, contractual cash flows are generally fixed unless credit events intervene.

For a callable bond, however, falling rates can increase the chance that the issuer redeems the bond early.

For a mortgage-backed security, falling rates can increase refinancing and prepayment activity.

In those cases, the cash-flow path can change precisely because rates changed.

Yield-based modified duration cannot fully capture that behavior.

Effective duration is generally more appropriate for option-sensitive bonds because it revalues the security after shifting a benchmark yield curve while allowing modeled cash flows to change.

This distinction is central to Option-Adjusted Spread analysis.

Modified duration measures own-yield sensitivity

Another subtlety is the meaning of the yield change.

Modified duration is a yield-based measure. It describes sensitivity to a change in the bond's own yield to maturity under the assumed cash flows.

But a corporate bond's yield can change because of multiple forces:

1benchmark Treasury rates move
2+
3credit spread moves
4+
5liquidity premium moves
6+
7other pricing effects

A 50 bp change in the bond's own yield does not tell you which component changed.

This matters because investors often say "rates rose" when the actual move was mostly a widening Credit Spread.

For portfolio risk management, curve duration, key-rate duration, spread duration, or empirical measures may isolate those exposures more precisely.

Duration does not imply a symmetric realized outcome

The formula:

1ΔP / P ≈ -D_mod × Δy

looks symmetric.

If yields rise 50 bps, the linear estimate predicts one loss. If yields fall 50 bps, it predicts an equal-sized gain.

Actual option-free bond prices are generally convex, so the true gain from the yield decline is larger than the linear estimate and the true loss from the yield increase is smaller than the linear estimate, all else equal.

Callable bonds can behave differently and can exhibit negative convexity over relevant ranges.

Therefore, a duration number should not be treated as a complete two-sided payoff map.

Modified duration changes over time

Duration is not a permanent property attached to a bond at issuance.

As time passes and the bond approaches maturity, its cash-flow timing changes.

Duration also changes when:

• market yield changes;
• the bond's price changes;
• coupons are paid;
• embedded options move in or out of the money; or
• the expected cash-flow path changes.

A duration value is therefore a measurement at a point in time under a methodology.

Using an old duration number with a new yield shock can produce a stale sensitivity estimate.

Portfolio duration is a weighted exposure measure

For a portfolio of conventional bonds, duration can often be approximated as a market-value-weighted average of the individual bond durations.

That creates a useful portfolio-level interest-rate-risk summary.

Suppose a portfolio contains:

160% in bonds with duration 4
240% in bonds with duration 9

A simple weighted estimate is:

1Portfolio duration
2≈ 0.60 × 4 + 0.40 × 9
3≈ 6.0

That estimate still embeds assumptions, including how yields across different securities move together.

Real yield curves rarely shift by exactly the same amount at every maturity.

This is why professional managers may supplement total duration with key-rate durations across the curve.

Modified duration is not a holding-period return forecast

A bond with modified duration of 6 does not have an expected six-year return, nor does it imply a 6% return.

It also does not tell you the bond's yield.

Those concepts answer different questions:

1Yield to Maturity / Yield to Worst
2-> return measure under specified assumptions
3
4Modified Duration
5-> first-order price sensitivity to yield changes
6
7Macaulay Duration
8-> present-value-weighted cash-flow timing and horizon relationship
9
10Convexity
11-> curvature of the price-yield relationship

Combining the measures is more informative than substituting one for another.

High duration is not automatically bad

Duration is exposure, not a quality score.

High duration means greater sensitivity to yield changes under the measure's assumptions.

That can create larger losses when yields rise, but it can also create larger gains when yields fall.

An investor with a long-dated liability may deliberately seek duration to match the sensitivity of that liability.

A trader may seek duration because of a rate view.

A capital-preservation investor with a short horizon may prefer much less duration.

The same duration can therefore be appropriate or inappropriate depending on the investor's objective and horizon.

How investors should use modified duration

A disciplined use of modified duration looks like this:

1. Verify that the security is appropriate for a fixed-cash-flow yield-duration framework.
2. Confirm the exact duration methodology.
3. Express the yield shock in decimal form.
4. Use duration for a first-order estimate.
5. Add convexity for a better estimate when the move is material.
6. Separate benchmark-rate exposure from credit-spread exposure where relevant.
7. Use option-aware measures if cash flows can change with rates.
8. Remember that mark-to-market price sensitivity is not the same as realized holding-period return.

For current macro and market-risk context, the Macroeconomic Conditions Index and Cyclically Adjusted Risk Premium can provide separate research context without becoming live duration or bond-pricing inputs for this page.

Sources and further reading

• CFA Institute, 2026, Yield-Based Bond Duration Measures and Properties: https://www.cfainstitute.org/insights/professional-learning/refresher-readings/2026/yield-based-bond-duration-measures-and-properties
• CFA Institute, 2026, Yield-Based Bond Convexity and Portfolio Properties: https://www.cfainstitute.org/insights/professional-learning/refresher-readings/2026/yield-based-bond-convexity-and-portfolio-properties
• CFA Institute, 2026, Curve-Based and Empirical Fixed-Income Risk Measures: https://www.cfainstitute.org/insights/professional-learning/refresher-readings/2026/curve-based-and-empirical-fixed-income-risk-measures
• CFA Institute, 2026, Overview of Fixed-Income Portfolio Management: https://www.cfainstitute.org/insights/professional-learning/refresher-readings/2026/overview-fixed-income-portfolio-management

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