What Is Bond Convexity? Definition and Examples

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Bond convexity is a measure of how much a bond’s duration changes as its yield changes. On a graph of price against yield, the line is not straight. It curves, and convexity quantifies that curve. Duration gives the slope of the price-yield relationship at the current yield. Convexity gives the rate at which that slope changes. Together they estimate how far a bond’s price will move for a given change in interest rates, and convexity becomes more important as the rate change gets larger.

Why Bond Prices and Yields Move in Opposite Directions

A fixed-rate bond pays a set coupon. When market rates rise, an existing bond paying less than new issues only attracts buyers at a lower price, which falls until its yield matches what the market demands. When rates fall, its higher coupon becomes more valuable and its price rises. Duration and convexity describe how large that price move is, and whether it is symmetric.

Duration: The Straight-Line Estimate

Duration measures how sensitive a bond’s price is to a change in yield. There are two versions you will see most often:

• Macaulay duration is the weighted average time, in years, until the bond’s cash flows are received, with each cash flow weighted by its present value.
• Modified duration converts that figure into price sensitivity: Macaulay duration divided by (1 + yield per period). It gives the approximate percentage price change for a 1 percentage point change in yield.

A bond with a modified duration of 5 is expected to lose about 5% of its value if yields rise by 1 percentage point, and gain about 5% if they fall by the same amount.

Duration is not the same as maturity. Maturity is simply the date the principal is repaid. Duration accounts for the timing and size of every coupon, so a coupon bond’s duration is shorter than its maturity. Only a zero-coupon bond has a Macaulay duration equal to its maturity.

The weakness of duration is that it predicts the same size move up and down. It draws a tangent line to the price-yield curve at today’s yield. For small rate changes, the tangent and the curve are close together. For larger changes, the curve bends away from the tangent, and duration’s estimate drifts off.

How Bond Convexity Works

The price-yield curve of an option-free bond bows toward the origin: it is steep at low yields and flattens at high yields. That shape has a direct consequence. As yields fall, the curve gets steeper, so the bond’s duration rises and each further drop in yield adds more to the price. As yields rise, the curve flattens, duration falls, and each further increase in yield takes less off the price.

Interpreting bond convexity

On a price-yield graph, the straight tangent line represents the duration estimate. The actual price curve sits above that line on both sides for a positively convex bond. The gap between them is the convexity effect. Two consequences follow:

1. When yields fall, the bond gains more than duration predicts.
2. When yields rise, the bond loses less than duration predicts.

What makes convexity higher or lower

• Maturity: longer-maturity bonds are more convex. Convexity grows faster than duration as maturity lengthens.
• Coupon: lower-coupon bonds are more convex than higher-coupon bonds of the same maturity, because more of their value sits in distant cash flows. Zero-coupon bonds have the highest convexity for a given maturity.
• Yield level: convexity is higher when yields are low, matching the steeper part of the curve.
• Cash flow dispersion: for a fixed duration, spreading cash flows out raises convexity. A barbell portfolio (short and long bonds) is more convex than a bullet portfolio concentrated near one maturity with the same duration.
• Embedded options: a call feature or prepayment option can reduce convexity and turn it negative over some range of yields.

Bond Convexity Formula

Mathematically, convexity is the second derivative of price with respect to yield, divided by price:

Convexity = (1 / P) × d²P / dy²

For a bond with annual cash flows, this works out to:

Convexity = [1 / (P × (1 + y)²)] × Σ [CFₜ × t × (t + 1) / (1 + y)ᵗ]

where P is the bond’s price, y is the yield to maturity, CFₜ is the cash flow at year t, and the sum runs over every cash flow.

The result is expressed in years squared. For bonds that pay semiannually, the same formula is applied per period and the result is divided by the number of periods per year squared (4 for semiannual) to annualize it.

Approximate (numerical) convexity

You can estimate convexity without calculus by repricing the bond at slightly higher and lower yields:

Convexity ≈ (P₋ + P₊ − 2P₀) / (P₀ × Δy²)

P₀ is the current price, P₋ is the price after yields fall by Δy, and P₊ is the price after yields rise by Δy, with Δy written as a decimal.

Using duration and convexity together

The combined estimate of the percentage price change is:

%ΔP ≈ −(Modified duration × Δy) + ½ × Convexity × (Δy)²

The duration term carries a negative sign and changes direction with Δy. The convexity term uses Δy squared, so for a positively convex bond it is always positive, whichever way yields move. That is the algebraic source of the asymmetry.

Example of Bond Convexity

Hypothetical example for illustration only. Take a 10-year zero-coupon bond with a face value of $100 and a yield of 5%, compounded annually.

• Price: 100 / 1.05¹⁰ = $61.39
• Modified duration: 10 / 1.05 = 9.52
• Convexity: for a zero-coupon bond, the formula simplifies to n(n + 1) / (1 + y)² = 110 / 1.1025 = 99.8

The numerical method gives nearly the same figure. At 4%, the price is $67.56. At 6%, it is $55.84. Applying the approximation: (67.56 + 55.84 − 2 × 61.39) / (61.39 × 0.0001) ≈ 99.9.

Now compare estimates with actual prices for a 1 percentage point move (Δy = 0.01):

The convexity term is ½ × 99.8 × 0.0001 ≈ 0.50%. Duration alone misses by about half a percentage point either way. With convexity included, the estimate is within a few hundredths of a point.

For a 2 percentage point move, the convexity term quadruples to about 2.0%, because Δy is squared. Duration alone predicts ±19.05%. The bond would actually gain about 21.2% if yields fell to 3% and lose about 17.2% if they rose to 7%. Duration plus convexity estimates +21.0% and −17.1%. The bigger the rate move, the more of the price change comes from convexity.

Positive vs. Negative Convexity

Positive convexity

A bond has positive convexity when its duration rises as yields fall and falls as yields rise, as in the example above. Every option-free fixed-rate bond has this shape.

Negative convexity

A bond has negative convexity when the price-yield curve bends the other way over part of its range. As yields fall, its price rises less than duration suggests. As yields rise, its duration can lengthen and the price can fall more than a comparable option-free bond.

The usual cause is an option held by the borrower:

• Callable bonds: when market rates drop well below the coupon, the issuer can redeem the bond and refinance more cheaply. Buyers will not pay much above the call price for a bond likely to be called, so the price flattens out as yields fall. This is often called price compression.
• Mortgage-backed securities: homeowners tend to refinance when mortgage rates fall, returning principal early just when reinvestment rates are lower. When rates rise, refinancing slows, principal comes back later than expected, and the security’s duration extends (extension risk). Investors lose on both sides of the curve compared with an option-free bond.

Which is better?

For the holder, positive convexity is the more favorable shape, and among bonds with the same duration and yield, higher convexity is better. Markets price this in. More convex bonds tend to trade at a higher price, and so a lower yield, while negatively convex securities usually offer extra yield to compensate investors for the options they have effectively sold to the borrower. The real choice is whether the extra convexity is worth the yield given up. If rates barely move, the higher-yielding, less convex bond earns more. If rates swing sharply, the more convex bond tends to come out ahead.

Effective Convexity

The formula above assumes the cash flows are fixed. For callable bonds, putable bonds and mortgage-backed securities, cash flows change with rates, so analysts use effective convexity instead. It applies the same numerical approximation, but the prices at the shifted yields come from a valuation model that reflects how the embedded option or prepayments respond. The yield curve is shifted up and down, not just the bond’s own yield. Effective convexity is only as good as the model and its prepayment or call assumptions, so two analysts can report different values for the same security.

How Convexity Is Used

• Comparing bonds with similar duration: two holdings can have the same rate sensitivity for small moves but behave quite differently in a large rally or sell-off. Convexity shows that difference.
• Portfolio construction: managers who expect volatile rates may favor barbell structures or noncallable bonds to add convexity. Those who expect stable rates may accept lower convexity in exchange for yield.
• Immunization and liability matching: pension funds and insurers typically match asset duration to liability duration first, then seek asset convexity at least equal to liability convexity, so the match holds up better when rates move more than a little.
• Hedging: a duration hedge has to be rebalanced as rates move, because duration itself drifts. Convexity tells the hedger how fast that drift happens.
• Bond funds: a portfolio’s convexity is approximately the market-value-weighted average of its holdings’ convexities. A fund heavy in mortgage-backed securities can have low or negative convexity even when its duration looks moderate.

Limitations of Bond Convexity

• It is still an approximation. Duration plus convexity captures the first two terms of the price-yield relationship. For very large rate changes, full repricing is more accurate.
• Parallel shift assumption. Standard duration and convexity assume every yield moves by the same amount. Real yield curves steepen, flatten and twist. Key-rate measures are used to capture those shapes.
• Rate risk only. Convexity says nothing about credit risk, changes in credit spreads, liquidity or inflation, all of which also move bond prices.
• Model dependence. Effective convexity for option-embedded securities relies on assumptions about borrower behavior and rate volatility.
• Reporting conventions differ. Some sources quote convexity in years squared (99.8 in the example), some divide by 100 (0.998), and some fold the ½ factor into the published number. Plugging a figure into the price-change formula without knowing its convention can double or distort the result.

Distinctions Readers Often Confuse

• Convexity vs. maturity: maturity is a date. Convexity depends on the full pattern of cash flows. Two bonds maturing in the same year can have very different convexity if their coupons differ.
• Negative convexity vs. negative returns: negative convexity describes the shape of the price response, not whether the bond loses money. A callable bond can still gain value when rates fall, just less than a comparable noncallable bond.
• Convexity in finance more broadly: the term also describes any payoff that curves upward, gaining more from favorable moves than it loses from adverse ones. Option traders call the equivalent second-order sensitivity gamma. In futures and interest rate derivatives, a “convexity adjustment” is a separate pricing correction, not a bond’s convexity figure.

The Bottom Line

Convexity matters most to bond investors, fund managers and liability-matching institutions facing large rate moves, where duration alone misprices the change. Among bonds with the same duration and yield, higher convexity is better, though markets charge for it with lower yields. Callable bonds and mortgage-backed securities pay extra yield for negative convexity, so the practical choice is whether added convexity is worth the yield given up given how much rates are expected to swing.