# Bond Duration & Interest Rate Risk Explained (Macaulay & Modified Duration)

Two bonds can share the same yield to maturity and still react completely differently when rates move. Duration is the number that tells you which one — with verified worked examples for Macaulay duration, Modified duration, and price sensitivity.

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- **Canonical URL:** https://dothecalculation.com/blog/finance/bond-duration-interest-rate-risk-guide
- **Category:** Finance
- **Author:** Do The Calculation Team
- **Published:** 2026-08-03
- **Last updated:** 2026-09-27
- **Reading time:** 16 min read
- **Publisher:** Do The Calculation (https://dothecalculation.com)
- **Methodology:** https://dothecalculation.com/methodology

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## Bond Duration & Interest Rate Risk Explained

Bond duration tells you how much a bond's price moves when yields change: a 10-year, 5% coupon bond priced at par has a modified duration of 7.72, so a 1-percentage-point rise in yields cuts its price by about 7.72% (7.36% when fully repriced), and a 1-point fall lifts it by 8.11%.

Two bonds can have the exact same yield to maturity and still behave completely differently the moment interest rates move. A 2-year Treasury and a 30-year Treasury yielding the same 5% will not lose the same amount of value when rates rise by 1%. Duration is the single number that predicts which bond moves more — and by roughly how much.

- What Macaulay duration actually measures (it is not just "years to maturity")
- Modified duration and how to use it to estimate a price move before it happens
- Why the actual price change never quite matches the duration estimate — and what convexity has to do with it
- Why higher coupons and shorter maturities both mean lower duration
- How to compare interest rate risk across bonds before you buy

Tool: [Try the Bond Yield Calculator](https://dothecalculation.com/calculators/bond-yield-calculator) — Enter face value, coupon rate, price, and maturity to get YTM, current yield, Macaulay duration, and Modified duration in one result.

## What Is Bond Duration?

Macaulay duration is the weighted-average time it takes to receive all of a bond's cash flows, where each cash flow is weighted by its present value. It is measured in years, and it is always less than or equal to the bond's time to maturity — because coupon payments arrive before the final maturity date and pull the average forward.

**Macaulay Duration**

```
Macaulay Duration = Σ [t × PV(Cash Flow at t)] ÷ Bond Price
```
- t = time (in years) until each cash flow
- PV(Cash Flow at t) = that cash flow discounted at the bond's YTM
- For a zero-coupon bond, Macaulay Duration always equals the years to maturity exactly

**Worked Example: $1,000 Face, 5% Annual Coupon, 10 Years to Maturity, $950 Price**
| Metric | Value |
| --- | --- |
| Yield to Maturity | 5.67% |
| Macaulay Duration | 8.05 years |
| Time to Maturity | 10 years |

> **Why 8.05 Years, Not 10?** — Every coupon payment between now and maturity returns some of your money early. Those earlier cash flows pull the weighted-average time down from 10 years to 8.05 years — the higher the coupon, the more weight sits in the early years, and the shorter the duration.

> **Annual vs Semi-Annual Coupons** — Every example in this guide assumes one coupon payment a year, so you can reproduce it with a short spreadsheet. The Bond Yield Calculator defaults to semi-annual coupons, as most US Treasury and corporate bonds pay. That gives slightly different figures: for the 10-year, 5% bond priced at par, Macaulay duration is 7.99 years and modified duration 7.79 with semi-annual payments, against 8.11 years and 7.72 with annual payments. Set the calculator's frequency to annual to match the tables here.

## Modified Duration: Turning Duration Into a Price-Move Estimate

Modified duration converts Macaulay duration into a direct estimate of how much a bond's price will move for a 1-percentage-point change in yield. This is the number that actually matters for risk management.

**Modified Duration**

```
Modified Duration = Macaulay Duration ÷ (1 + YTM ÷ frequency)
```
- For the example above (Macaulay 8.05 years, YTM 5.67%, annual coupons): Modified Duration = 8.05 ÷ 1.0567 = 7.62

**Estimating a Price Change**

```
% Change in Price ≈ −Modified Duration × Change in Yield
```
- A negative sign: yields up means price down, and vice versa — the core inverse relationship of fixed income.

**Modified Duration in Action: $1,000 Face, 5% Coupon, 10-Year Bond Priced at Par (YTM 5%, Modified Duration 7.72)**
| Yield Change | Duration Estimate | Actual Repriced Value | Actual % Change |
| --- | --- | --- | --- |
| −2.0% (yield falls to 3%) | +15.44% | $1,170.60 | +17.06% |
| −1.0% (yield falls to 4%) | +7.72% | $1,081.11 | +8.11% |
| +1.0% (yield rises to 6%) | −7.72% | $926.40 | −7.36% |
| +2.0% (yield rises to 7%) | −15.44% | $859.53 | −14.05% |

_[Figure: Duration Estimate vs Actual Price Change — Same 10-year, 5% coupon bond priced at par — the duration estimate is a straight line; the actual price path curves.]_

## Why the Estimate Is Always a Little Off: Convexity

Modified duration is a straight-line (linear) approximation of a relationship that is actually curved. That curve is called convexity, and it is a small but predictable effect: it makes gains a little bigger than duration predicts when yields fall, and makes losses a little smaller than duration predicts when yields rise. For a 1-percentage-point move, the gap is usually small (in the table above, 0.36 percentage points when yields rise and 0.39 points when they fall). For a 2-point move, the gap grows to 1.39 and 1.62 points, roughly four times as large, because the convexity term grows with the square of the yield change. That is why duration alone is unreliable for large rate swings, and why bond desks add a convexity adjustment on top of it.

## Worked Example: A Convexity-Adjusted Price Estimate

Adding a second term to the duration estimate closes most of the gap. The example below uses the same $1,000-face, 5% annual coupon, 10-year bond priced at par (yield 5%) and asks what happens if yields jump 2 percentage points to 7%. Every figure can be reproduced in a spreadsheet with ten rows.

**Price Change With a Convexity Adjustment**

```
% Change in Price ≈ (−Modified Duration × Δy) + (½ × Convexity × Δy²)
Convexity (annual coupons) = Σ [t × (t + 1) × PV(Cash Flow at t)] ÷ [Price × (1 + y)²]
```
- Δy = change in yield as a decimal (2 percentage points = 0.02)
- y = the current yield to maturity per period (0.05 here)
- The convexity term is always positive for an ordinary option-free bond, so it adds to gains and subtracts from losses

**Step 1: Discount Each Cash Flow at 5% ($1,000 Face, 5% Annual Coupon, 10 Years)**
| Year (t) | Cash Flow | PV at 5% | t × PV | t × (t + 1) × PV |
| --- | --- | --- | --- | --- |
| 1 | $50 | $47.62 | 47.62 | 95.24 |
| 2 | $50 | $45.35 | 90.70 | 272.11 |
| 3 | $50 | $43.19 | 129.58 | 518.30 |
| 4 | $50 | $41.14 | 164.54 | 822.70 |
| 5 | $50 | $39.18 | 195.88 | 1,175.29 |
| 6 | $50 | $37.31 | 223.86 | 1,567.05 |
| 7 | $50 | $35.53 | 248.74 | 1,989.91 |
| 8 | $50 | $33.84 | 270.74 | 2,436.62 |
| 9 | $50 | $32.23 | 290.07 | 2,900.74 |
| 10 | $1,050 | $644.61 | 6,446.09 | 70,906.98 |
| Total |  | $1,000.00 | 8,107.82 | 82,684.94 |

- Step 2, Macaulay duration: 8,107.82 ÷ 1,000.00 = 8.108 years.
- Step 3, modified duration: 8.108 ÷ 1.05 = 7.722.
- Step 4, convexity: 82,684.94 ÷ (1,000.00 × 1.05²) = 82,684.94 ÷ 1,102.50 = 75.00.
- Step 5, duration term for a 2-point rise: −7.722 × 0.02 = −15.44%.
- Step 6, convexity term: ½ × 75.00 × 0.02² = 0.5 × 75.00 × 0.0004 = +1.50%.
- Step 7, combined estimate: −15.44% + 1.50% = −13.94%, a price of about $860.56.

**How Close Each Estimate Gets (Same Bond, Yield Moves 2 Points)**
| Yield Change | Duration Only | Duration + Convexity | Actual Repriced Value | Error of Duration + Convexity |
| --- | --- | --- | --- | --- |
| +2.0% (to 7%) | $845.57 (−15.44%) | $860.56 (−13.94%) | $859.53 (−14.05%) | About $1 too high |
| −2.0% (to 3%) | $1,154.43 (+15.44%) | $1,169.43 (+16.94%) | $1,170.60 (+17.06%) | About $1 too low |

> **What the Example Shows** — Duration alone missed the true price by about $14 on the way down and $16 on the way up. Adding one convexity term cut the error to about $1 in both directions. The small remaining error comes from higher-order terms, which matter only for very large yield moves or very long bonds.

## What Actually Drives a Bond's Duration

Two variables move duration the most: time to maturity, and coupon rate.

_[Figure: Duration Rises With Maturity (5% Coupon Bond, Priced at Par) — Macaulay duration for the same coupon rate at five different maturities.]_

_[Figure: Duration Falls as Coupon Rises (10-Year Bond, Priced at Par) — A higher coupon returns more cash to you earlier, which shortens the weighted-average time to receive your money back.]_

**What Raises or Lowers Duration**
| Factor | Effect on Duration | Why |
| --- | --- | --- |
| Longer time to maturity | Increases duration | More total time before the largest cash flow (face value) arrives |
| Higher coupon rate | Decreases duration | More cash returned early pulls the weighted average forward |
| Lower coupon rate | Increases duration | Less early cash flow — more of the bond's value sits in the final face-value payment |
| Zero-coupon structure | Maximizes duration | All value arrives on a single date — duration equals maturity exactly |
| Higher market yield | Slightly decreases duration | Future cash flows are discounted more heavily, reducing their weight in the average |

## Comparing Interest Rate Risk Before You Buy

_[Figure: Short-Duration vs Long-Duration Bonds — Same 1% Rate Move — Modified duration 1.86 (2-year, 5% annual coupon bond) versus 15.37 (30-year, 5% annual coupon bond), both priced at par with a 5% yield.]_

## Managing Interest Rate Risk in Practice

- Match duration to your time horizon — a bond you plan to hold to maturity carries less practical rate risk than one you might need to sell early.
- Shorten average portfolio duration when you expect rates to rise; lengthen it when you expect rates to fall.
- Diversify across maturities rather than concentrating duration in one bucket — see our guide on [bond funds vs individual bonds](/blog/finance/bond-funds-vs-individual-bonds) for how funds handle this automatically, or [how to build a bond ladder](/blog/finance/how-to-build-a-bond-ladder) to do it yourself with individual bonds. In that guide's example (yields of 4.0% to 4.4%), a five-rung, one-to-five-year ladder has a modified duration of about 2.72, against 4.40 for a single 5-year bond.
- Watch the shape of the curve, not just the level of rates. When short yields are above long yields, extending duration pays you less, not more; the [yield curve explained](/blog/finance/yield-curve-explained) guide shows what normal, flat and inverted curves mean for that trade-off.
- Remember duration measures interest rate risk only — it says nothing about credit/default risk. Compare that separately; see [Treasury vs Corporate vs Municipal Bonds](/blog/finance/treasury-vs-corporate-vs-municipal-bonds).
- For a bond you already hold, check its Modified Duration on our [Bond Yield Calculator](/calculators/bond-yield-calculator) before assuming a rate move will not matter.

**What is bond duration in simple terms?**

Duration measures how sensitive a bond's price is to changes in interest rates. Macaulay duration expresses this as the weighted-average time to receive all cash flows; Modified duration converts that into a direct price-sensitivity percentage.

**What is the difference between Macaulay duration and Modified duration?**

Macaulay duration is measured in years — the weighted-average time to cash flows. Modified duration is derived from it (Macaulay ÷ (1 + YTM/frequency)) and estimates the percentage price change for a 1-point yield change.

**Is duration the same as maturity?**

No. Duration is always less than or equal to maturity for a coupon-paying bond, because coupon payments return some value before the maturity date. Only a zero-coupon bond has duration exactly equal to its maturity.

**Why does a higher coupon rate lower duration?**

A higher coupon returns more cash to the investor earlier in the bond's life, which pulls the weighted-average time to receive cash flows forward — shortening duration even though maturity is unchanged.

**How do I estimate a bond price change from duration?**

Use % Change in Price ≈ −Modified Duration × Change in Yield. It is an approximation — accurate for small yield moves, and increasingly imprecise for larger ones due to convexity.

**What is convexity and why does it matter?**

Convexity is the curvature in the actual price-yield relationship that duration (a straight-line estimate) misses. It benefits bondholders: real price gains from falling yields exceed the duration estimate, and real losses from rising yields fall short of it.

**How do I add convexity to a duration estimate?**

Add half the convexity times the squared yield change: % change ≈ (−Modified Duration × Δy) + (½ × Convexity × Δy²). For a 10-year, 5% annual coupon bond at par (modified duration 7.72, convexity 75.0) and a 2-point rise, that is −15.44% + 1.50% = −13.94%, within about $1 of the actual repriced value of $859.53.

**What does a modified duration of 8 mean in practice?**

It means the bond's price changes by roughly 8% for every 1-percentage-point change in its yield, in the opposite direction. A $10,000 position would move by about $800 for a 1-point change and about $80 for a 0.1-point (10 basis point) change. The estimate is best for small moves; for large ones, add the convexity term.

**Why does my calculator show a different duration from this guide?**

Usually because of coupon frequency. This guide uses annual coupons. With semi-annual coupons, the default on the Bond Yield Calculator, the same 10-year, 5% par bond has a Macaulay duration of 7.99 years and a modified duration of 7.79, instead of 8.11 and 7.72.

**Which bonds have the highest interest rate risk?**

Long-maturity, low-coupon bonds have the highest duration and therefore the highest interest rate risk. Zero-coupon long-term bonds carry the most risk of all for a given maturity.

**Does duration measure credit risk?**

No. Duration measures interest rate risk only — how much a bond's price moves when yields change. It says nothing about the issuer's ability to pay, which is credit risk and must be assessed separately.

**How can I lower my portfolio's interest rate risk?**

Shorten average duration by holding shorter-maturity or higher-coupon bonds, or diversify across a range of maturities (a bond ladder or a diversified bond fund) rather than concentrating in long-duration bonds.

**Where can I calculate a bond's duration?**

Our Bond Yield Calculator returns Macaulay and Modified duration alongside YTM and current yield — just enter face value, coupon rate, price, and years to maturity.

## Related Bond Guides and Calculators

- [How to Calculate Bond Yield](/blog/finance/how-to-calculate-bond-yield) — current yield, YTM and YTC with worked examples, the inputs that duration depends on.
- [Bond Yield and YTM Complete Guide](/blog/finance/bond-yield-ytm-complete-guide) — how the Bond Yield Calculator solves for yield to maturity.
- [How to Build a Bond Ladder](/blog/finance/how-to-build-a-bond-ladder) — spread duration across maturities with individual bonds.
- [The Yield Curve Explained](/blog/finance/yield-curve-explained) — why short and long bonds can pay very different yields at the same time.
- [Bond Funds vs Individual Bonds](/blog/finance/bond-funds-vs-individual-bonds) — why a fund keeps its duration while a held bond's duration shrinks to zero at maturity.
- [Bond Yield Calculator](/calculators/bond-yield-calculator) — Macaulay and modified duration for any bond you enter.

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_Source: [Do The Calculation](https://dothecalculation.com/blog/finance/bond-duration-interest-rate-risk-guide). Quote freely with attribution and a link to this page._