Bond Duration and Convexity: A Full Worked Example (Price Change Estimate)

Duration tells you how much a bond's price should move for a 1-percentage-point change in yield. Convexity tells you how wrong that estimate is — and how much worse the error gets as the yield move grows larger. Most explanations stop at the formulas and one plugged-in number. Below, one bond is carried through the entire calculation: price, Macaulay duration, modified duration, convexity, a duration-only estimate, a duration-plus-convexity estimate, and the bond's actual repriced value, so you can see exactly how close each estimate lands.

The formulas

For a bond paying cash flow CF_t at each future annual period t, priced at yield y per period, with the present value of each payment PV_t = CF_t ÷ (1+y)^t and total price P = ΣPV_t:

Macaulay duration (D_Mac) = Σ(t × PV_t) ÷ P

Modified duration (D_Mod) = D_Mac ÷ (1 + y)

Convexity (C) = Σ[t × (t+1) × PV_t] ÷ [P × (1+y)²]

Estimated % price change ≈ −D_Mod × Δy + ½ × C × (Δy)²

Modified duration alone gives the first-order, straight-line estimate. The convexity term is the second-order correction: a bond's actual price-yield relationship curves rather than following a straight line, so a duration-only estimate increasingly understates the bond's true price as the yield move gets larger, whether yields rise or fall. FINRA's own investor guidance states the underlying rule of thumb directly: for every 1 percentage-point change in interest rates, a bond's price moves in the opposite direction by roughly its duration. What follows is the layer of precision that sits on top of that rule of thumb.

The bond

The bond pays annually, on purpose, to keep every number in whole percentage points. Real corporate and Treasury bonds usually pay semiannually, which halves the period length used in the formulas above and the modified-duration divisor, but changes nothing about the method.

Step 1: price and Macaulay duration

Discounting each cash flow at the initial 6% yield:

The price comes out to exactly $1,000.00 — unsurprising, since a bond whose coupon rate equals its yield always prices at face value. Macaulay duration is the present-value-weighted average time to receive the bond's cash flows, using the "t × PV" column above: D_Mac = 4,465.10 ÷ 1,000.00 = 4.4651 years.

Step 2: modified duration and the duration-only estimate

D_Mod = 4.4651 ÷ 1.06 = 4.2124

For a 1-percentage-point rise in yield (Δy = +0.01), the duration-only estimate is −4.2124 × 0.01 = −4.2124%, implying a new price of $1,000 × (1 − 0.042124) = $957.88.

Step 3: convexity and the combined estimate

Using the "t(t+1) × PV" column's sum: C = 25,751.46 ÷ ($1,000.00 × 1.06²) = 25,751.46 ÷ 1,123.60 = 22.92.

Adding the convexity correction to the duration-only estimate: −4.2124% + ½ × 22.92 × 0.01² = −4.2124% + 0.1146% = −4.0978%, implying a price of $1,000 × (1 − 0.040978) = $959.02.

Checking both estimates against the actual price

Repricing the same five cash flows directly at a 7% yield (no shortcuts) gives an actual price of $958.998, which rounds to $959.00 — a true change of −4.1002%.

The duration-only estimate overstates the price drop by about a dollar, because it assumes the price-yield relationship is a straight line when it actually curves upward (toward higher prices) relative to that line. Adding the convexity term pulls the estimate back to within two cents of the actual repriced value.

Why the gap grows for bigger moves

Run the same bond through a 3-percentage-point yield increase (6% to 9%, Δy = +0.03) instead of 1 point, using the identical D_Mod = 4.2124 and C = 22.92:

Tripling the yield move roughly triples the duration-only estimate's percentage error, but the dollar error from ignoring convexity entirely grows far faster — from $1.12 at a 1-point move to $9.68 at a 3-point move, about 8.6 times larger for a move only three times as big. That's the practical reason duration alone is treated as reliable for small, routine yield shifts, while convexity becomes essential for stress-testing a bond or portfolio against a large rate shock: the linear approximation's blind spot compounds faster than the move itself.

Quick reference

Both of this bond's existing risk questions — comparing a short- and long-duration bond, and recognizing convexity as the correction to a linear duration estimate — are drawn from the same mechanics worked through above. To practice applying them, try PassDrill's fixed-income practice questions.

This page is educational material to help you understand and practice bond-pricing math; it is not investment, financial, or professional advice, and no specific security is recommended.

Source: FINRA Investor Insights, "Brush Up on Bonds: Interest Rate Changes and Duration" (finra.org/investors/insights/bonds-interest-rate-changes-duration), for the core duration rule of thumb; CFA Institute Level I curriculum, Fixed Income topic area, for the standard Macaulay duration, modified duration, and convexity formulas, cross-checked against AnalystPrep's CFA-aligned worked treatment of the same formulas since the CFA Institute's own curriculum text sits behind a paywall. All figures above are independently calculated for this example, not reproduced from either source.

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