Bond Convexity: Price, Yield & Interest-Rate Risk

Bond convexity measures the curve in the relationship between a bond’s price and yield. Duration estimates how much a bond’s price may change for a small change in interest rates, but bond convexity refines that estimate because price-yield behavior is not a straight line. When yields move by more than a small amount, or when a bond has a long maturity or low coupon, convexity can meaningfully affect the estimated price change. In practical terms, convexity helps explain why a duration-only estimate may understate or overstate the actual price movement.

What Bond Convexity Means

A bond is a debt security that generally pays interest and returns principal according to stated terms, unless the issuer defaults or the bond has special features that change those cash flows. Because most bonds promise future cash flows, their prices are sensitive to changes in market yields.

The basic price-yield rule is:

• When yields rise, existing bond prices generally fall.
• When yields fall, existing bond prices generally rise.

That relationship is inverse, but it is also curved. Bond convexity describes that curvature.

Duration is usually the first sensitivity measure investors learn. It estimates the approximate percentage price change for a given change in yield. For example, a bond with modified duration of 5 years might be expected to lose about 5% if yields rise by 1 percentage point, before considering convexity and other factors.

Convexity adds a second layer: it estimates how much the duration estimate itself changes as yields move. The CFA Institute describes duration as a linear approximation and explains that the true price-yield relationship is a curved, convex line, making convexity a complementary risk measure for improving duration-based estimates when yields move meaningfully (CFA Institute).

This article is for educational purposes only and does not constitute financial or investment advice. Finelo does not recommend any security, strategy, or transaction. Investing involves risk, including possible loss of principal.

How Convexity Works With Duration

Duration is useful because it gives a quick first approximation. But it treats the price-yield relationship as if it were a straight line. Convexity matters because the line bends.

A common approximation is:

Estimated % price change
≈ (-Modified duration × Change in yield)
+ (0.5 × Convexity × Change in yield²)

Where:

• Modified duration is measured in years.
• Change in yield is expressed as a decimal, so 1 percentage point is 0.01.
• Convexity is often expressed in years squared, though data providers may scale or label it differently.
• The result is an estimated percentage price change.

The first term, -Modified duration × Change in yield, is the duration estimate. The second term, 0.5 × Convexity × Change in yield², is the convexity adjustment.

Because the yield change is squared, the convexity adjustment is usually small for tiny rate moves and more noticeable for larger moves. That is why convexity often matters more in stress tests, scenario analysis, and comparisons among longer-maturity bonds.

Fidelity similarly notes that duration may be a good estimate for small and sudden rate changes, but may be less effective for larger changes; the difference between the linear duration estimate and the actual price change reflects convexity (Fidelity).

Worked Example: Estimating Price Change With Convexity

Assume a plain-vanilla annual-pay bond with these simplified terms:

These figures are for education only. In real markets, duration and convexity depend on the exact bond terms, pricing date, yield convention, optionality, settlement, and data provider methodology.

Scenario A: Yields Rise by 1 Percentage Point

The yield rises from 4% to 5%.

Express the change in yield as a decimal:

Δy = 0.01

Use the duration-only estimate:

-Duration × Δy
= -4.45 × 0.01
= -0.0445
= -4.45%

Now add the convexity adjustment:

0.5 × Convexity × Δy²
= 0.5 × 25.0 × (0.01)²
= 12.5 × 0.0001
= 0.00125
= 0.125%

Combine the two:

Estimated % price change
= -4.45% + 0.125%
= -4.325%

Apply that to the $1,000 price:

Estimated dollar price change
= $1,000 × -4.325%
= -$43.25

Estimated new price:

$1,000 - $43.25 = $956.75

If you price the same bond directly at a 5% yield, the approximate present value is:

PV of coupons = $40 × [1 - (1.05)^-5] ÷ 0.05
              ≈ $40 × 4.3295
              ≈ $173.18

PV of principal = $1,000 ÷ (1.05)^5
                ≈ $783.53

Total price ≈ $173.18 + $783.53
            ≈ $956.71

The convexity-adjusted estimate of $956.75 is close to the direct present-value estimate of about $956.71.

Scenario B: Yields Fall by 1 Percentage Point

The yield falls from 4% to 3%.

Δy = -0.01

Duration-only estimate:

-Duration × Δy
= -4.45 × -0.01
= +0.0445
= +4.45%

Convexity adjustment:

0.5 × 25.0 × (-0.01)²
= 0.5 × 25.0 × 0.0001
= 0.00125
= 0.125%

Combined estimate:

Estimated % price change
= 4.45% + 0.125%
= 4.575%

Estimated price:

$1,000 × 1.04575 = $1,045.75

Direct present-value estimate at a 3% yield:

PV of coupons = $40 × [1 - (1.03)^-5] ÷ 0.03
              ≈ $40 × 4.5797
              ≈ $183.19

PV of principal = $1,000 ÷ (1.03)^5
                ≈ $862.61

Total price ≈ $183.19 + $862.61
            ≈ $1,045.80

Again, the convexity-adjusted estimate is close.

What This Example Shows

The duration-only estimate would have predicted:

• About $955.50 after a 1 percentage point yield increase.
• About $1,044.50 after a 1 percentage point yield decrease.

Adding convexity improved the estimate in both directions. For this conventional bond, the convexity term is positive whether yields rise or fall because the yield change is squared. That is why positive convexity can soften estimated losses when yields rise and add slightly to estimated gains when yields fall, compared with duration alone.

Positive Convexity, Negative Convexity, and Optionality

Many traditional fixed-rate bonds have positive convexity. This means their price-yield curve bends in a way that can be favorable relative to a straight-line duration estimate:

• If yields rise, the price may fall slightly less than duration alone predicts.
• If yields fall, the price may rise slightly more than duration alone predicts.

Fidelity notes that the impact of convexity is more pronounced in long-duration bonds with small coupons and describes this as positive convexity that can magnify the price volatility measure indicated by duration (Fidelity).

However, not all fixed-income securities behave like simple noncallable bonds. Some have embedded options. For example:

• A callable bond may allow the issuer to redeem the bond early.
• A mortgage-backed security may be affected by borrower prepayment behavior.
• A putable bond may give the investor the right to sell the bond back under specified terms.

These features can change convexity. In some cases, a bond or bond-like security can have negative convexity over certain yield ranges. Negative convexity means the price behavior may become less favorable than a plain duration-and-positive-convexity assumption would suggest.

A simplified callable bond example illustrates the issue. If rates fall, the price of a noncallable bond may rise substantially. But a callable bond’s price may rise less because investors may expect the issuer to call the bond and refinance at lower rates. The upside can become limited. In that range, the bond may not show the same positive convexity as a comparable noncallable bond.

This is one reason convexity should be read together with the bond’s structure, not as an isolated number.