Bond Convexity: Why Duration Is Not Enough

For educational purposes only; not investment advice.

Direct answer

Bond convexity measures the curvature in the relationship between a bond’s price and its yield. Duration gives a first-order, straight-line estimate of how price changes when yields move. Convexity adds a second-order adjustment, which becomes more important when the yield move is large.

For many option-free fixed-rate bonds, convexity is positive: a yield decline tends to increase price by more than duration alone predicts, while a yield increase tends to reduce price by less than duration alone predicts. Callable bonds, mortgage-backed securities, and other bonds with changing cash flows can show negative convexity in some rate environments.

How the adjustment works

A common approximation is:

ΔP / P ≈ -D × Δy + 0.5 × C × (Δy)^2

D is modified duration, C is convexity, and Δy is the yield change written as a decimal. A 100 basis point move is 0.01, not 100.

The duration term is linear. If yields rise, it is negative; if yields fall, it is positive. The convexity term uses (Δy)^2, so positive convexity adds to the estimate in both directions. It softens the estimated loss when yields rise and increases the estimated gain when yields fall.

For bonds whose cash flows change when rates change, use effective convexity rather than a fixed-cash-flow calculation:

effective convexity ≈ (P- + P+ - 2P0) / (P0 × (Δy)^2)

Here P- is the modeled price after yields fall, P+ is the modeled price after yields rise, and P0 is the current modeled price.

Worked example

Suppose a bond is priced at 100, has modified duration 6, and convexity 50.

If yields rise by 100 basis points:

ΔP / P ≈ -6 × 0.01 + 0.5 × 50 × 0.01^2 = -5.75%

The estimated price becomes about 94.25. Duration alone would have estimated 94.00.

If yields fall by 100 basis points:

ΔP / P ≈ +6 × 0.01 + 0.5 × 50 × 0.01^2 = +6.25%

The estimated price becomes about 106.25.

For a 25 basis point move, the convexity adjustment is only about 0.0156%. For a 200 basis point move, it is about 1.0%. That is why convexity is often minor in small daily moves but meaningful in stress scenarios.

Practical checks

Use decimal yield changes: 50 bp = 0.005, 100 bp = 0.01.
Confirm whether the duration is modified, Macaulay, effective, or key-rate duration.
Confirm the convexity scale used by the data provider.
Do not use fixed-cash-flow convexity for callable bonds, mortgage-backed securities, or securities with material prepayment behavior.
Separate Treasury-rate risk from credit-spread risk; both can affect a corporate bond’s yield.
Do not assume the yield curve moves in parallel. Key-rate risk can matter more than one aggregate duration number.
Remember that higher positive convexity usually has a price, such as lower yield, higher premium, or different liquidity.

Common misconceptions

Positive convexity does not prevent losses. If rates rise enough, the duration effect can still dominate.

Convexity does not replace duration. Duration is the main first-order measure; convexity is the curvature adjustment.

Basis points cannot be entered directly into the formula. Using 100 instead of 0.01 creates a wildly wrong squared term.

Negative convexity does not mean a bond must lose money. It describes an unfavorable shape of price response, often linked to call or prepayment behavior.

Sources

FINRA, “Brush Up on Bonds: Interest Rate Changes and Duration.”
SEC, “Investor Bulletin: Interest Rate Risk — When Interest Rates Go Up, Prices of Fixed-Rate Bonds Fall.”
CFA Institute, “Yield-Based Bond Convexity and Portfolio Properties.”