Bond Duration vs Convexity

First- and second-order rate sensitivity, demystified.

Bond duration vs convexity is the foundational risk framework every fixed-income practitioner uses to estimate how a bond's price responds to a shift in yields. Duration gives the first-order, linear approximation; convexity supplies the second-order curvature correction that duration alone ignores. Together they explain why two bonds with identical durations can behave very differently when rates move by 100 basis points or more — and why the price-yield relationship is not a straight line. If you want to go further with a live example, the bond pricing calculator lets you see exact price changes directly. The goal here is to show you how the math works and what it means in practice.

Duration: the first-order sensitivity

Modified duration measures the percentage price change of a bond for a one-unit (100 bp) change in its yield to maturity, holding everything else constant. The working formula is:

%ΔPrice ≈ −ModDuration × Δy

where Δy is expressed as a decimal (0.01 for 100 bp). A bond with a modified duration of 7.0 is expected to fall roughly 7% in price if its yield rises by 100 bp, and rise 7% if its yield falls by 100 bp. The sign is negative because price and yield move in opposite directions — the core mechanic behind how rates move bond prices.

Duration is a weighted average of when you get your cash flows back, with each flow discounted by the bond's yield. A zero-coupon bond's modified duration equals its maturity divided by (1 + y). A coupon bond's duration is shorter than its maturity because you receive cash before final redemption. Three properties follow directly:

• Longer maturity → higher duration (cash flows arrive later).
• Lower coupon → higher duration (less cash returned early).
• Higher yield → slightly lower duration (discounting compresses far cash flows more).

Duration is exact for infinitesimally small yield moves. For any move of practical size — 50 bp or more — it becomes an approximation, and the approximation has a consistent bias: it understates price gains when yields fall and overstates price losses when yields rise. That bias is convexity.

Convexity: the second-order curvature correction

The price-yield relationship is not linear; it is a convex curve. Duration corresponds to the slope of a tangent line at the current yield. Convexity measures how that slope itself changes as yield moves — the curvature of the curve. Adding the convexity term gives a second-order Taylor expansion:

%ΔPrice ≈ −D·Δy + ½·C·Δy²

where D is modified duration and C is the bond's convexity (expressed in years squared, though practitioners often drop the units and treat it as a dimensionless scalar). The ½·C·Δy² term is always positive for a standard bond — regardless of whether yields fall or rise, the actual price outcome is better than duration alone predicts. That is the benefit of positive convexity.

Convexity is computed as the second derivative of the price-yield function divided by price:

C = (1/P) × d²P/dy²

In practice it is estimated from a bond's cash flows: for each cash flow CF_t at time t, compute t·(t+1)·CF_t·(1+y)^−(t+2), sum across all flows, and divide by price. A 10-year Treasury with a 4% coupon priced near par carries convexity of roughly 80–90 in those units.

Worked example: duration estimate vs duration + convexity

Consider a 10-year, 4% annual coupon bond priced at par (yield = 4%). Suppose its modified duration is 8.11 and its convexity is 82.0. Now suppose yields jump by 150 bp (Δy = 0.015).

Duration-only estimate:

%ΔPrice ≈ −8.11 × 0.015 = −12.17%

Duration + convexity estimate:

%ΔPrice ≈ −8.11 × 0.015 + ½ × 82.0 × (0.015)² = −12.17% + 0.92% = −11.25%

The convexity correction recovers nearly a full percentage point of the price loss. For a $1,000,000 position that is roughly $9,200 the duration-only approach leaves on the table. Now run the same exercise in the other direction — yields fall 150 bp (Δy = −0.015):

Duration-only: +12.17%. Duration + convexity: +12.17% + 0.92% = +13.09%.

The asymmetry is the defining feature of positive convexity: price gains are larger than duration predicts, and price losses are smaller. The bond outperforms the linear approximation in both directions.

Which bonds have more or less convexity

Convexity varies substantially across bond types, and the differences matter when constructing a portfolio:

• Zero-coupon bonds have the highest convexity for a given duration because all cash flow is concentrated at maturity — the price-yield curve bows out sharply.
• Long-maturity coupon bonds carry more convexity than short-maturity bonds; the ½·C·Δy² term scales with time squared.
• Callable bonds can exhibit negative convexity. When yields fall far enough, the issuer's call option kicks in and caps the price appreciation — the curve bends backward, and the bond underperforms the linear estimate when rates drop. Duration-only estimates look misleadingly favorable for callables in a rally.
• Mortgage-backed securities behave similarly to callables: prepayment risk accelerates when rates fall, compressing price gains and creating negative convexity in low-rate environments.
• Short-maturity bonds have very low convexity — the correction term is negligible for moves less than 50 bp but can still matter in a sharp, sudden shift.

Duration vs convexity: a quick reference

Metric	What it measures	Formula	Units
Modified duration	First-order price sensitivity to yield	%ΔP ≈ −D·Δy	Years
Convexity	Second-order curvature of price-yield curve	%ΔP ≈ −D·Δy + ½·C·Δy²	Years²
Dollar duration (DV01)	Dollar price change per 1 bp move	DV01 = D × P × 0.0001	Dollars

Using both measures together

In practice, duration and convexity are complementary, not competing, tools. Duration handles the bulk of the estimation for small yield moves and sets your hedge ratio — how many futures contracts or interest rate swaps you need to neutralize rate exposure. Convexity tells you how that hedge degrades as yields move further from the point at which you set it, and it explains why long-convexity positions tend to outperform over time in volatile rate environments (you earn from the curvature in both directions).

For relative value work, comparing two bonds with the same duration but different convexities is a standard trade setup: in a volatile environment, the higher-convexity bond should outperform; in a stable, carry-dominated environment, the lower-convexity bond may be priced cheaply enough to compensate for the curvature disadvantage. Pricing that trade accurately requires going beyond the linear duration approximation and taking convexity seriously as a stand-alone risk measure.