Bond Convexity Explained: When Duration Stops Being Enough

Bond Convexity Explained: When Duration Stops Being Enough

বাংলায় পড়ুন

Learning objective

By the end of this tutorial, you should be able to explain why duration is only a first estimate of a bond's price response, use convexity to improve that estimate for a larger yield move, and recognise when call or prepayment features can create negative convexity.

The short version

Bond prices and yields generally move in opposite directions. Duration estimates the slope of that relationship around today's yield. The full price-yield relationship is curved, however, so a straight-line duration estimate becomes less accurate as the yield change becomes larger. Convexity measures that curvature.

For an ordinary option-free fixed-rate bond with positive convexity, an equal fall in yield normally produces a slightly larger percentage gain than the percentage loss produced by an equal rise in yield. Convexity does not remove interest-rate risk, credit risk or liquidity risk. It refines one part of the estimate.

1. Why duration needs a second term

Modified duration gives a first-order approximation:

Estimated price change (%) ≈ −Modified duration × Yield change

The minus sign expresses the usual inverse relationship. If modified duration is 7 and yield rises by 0.50 percentage points, written as 0.005 in decimal form, the duration-only estimate is:

−7 × 0.005 = −0.035, or −3.50%.

That calculation treats the price-yield relationship like a straight line tangent to the curve at the starting yield. It works best for a small, parallel change in yield and a bond whose cash flows do not change. For a larger move, the curve pulls the actual estimate away from that tangent.

The convexity adjustment adds a second-order term:

Estimated price change (%) ≈ −Dmod × Δy + ½ × Convexity × (Δy)²

Here, Dmod is modified duration, Δy is the yield change in decimal form, and the convexity figure must use the same convention as the data source. Because the yield change is squared, this adjustment is positive for a conventional positive-convexity bond whether yields rise or fall.

2. Worked example

Assume a hypothetical option-free bond has:

• Current clean price: 100
• Modified duration: 7.0
• Convexity: 60
• Yield change: plus or minus 1.00 percentage point, or 0.01

If yield rises by 1.00 percentage point

Duration component:

−7.0 × 0.01 = −7.00%

Convexity component:

½ × 60 × (0.01)² = +0.30%

Combined estimate:

−7.00% + 0.30% = −6.70%

The estimated price becomes about 93.30 before allowing for accrued interest, spread changes, fees or other risks.

If yield falls by 1.00 percentage point

Duration component:

−7.0 × (−0.01) = +7.00%

The convexity component remains +0.30%, so the combined estimate is +7.30% and the estimated price is about 107.30.

The asymmetry is the point: in this simplified example the estimated gain from a 1-point yield fall is larger than the estimated loss from a 1-point rise. Exact repricing from the bond's cash flows is preferable when available. Duration plus convexity is still an approximation.

3. Positive and negative convexity

Most ordinary option-free bonds have positive convexity. Their cash flows are fixed, and the price-yield curve bows in the investor's favour in the limited sense shown above.

Embedded options can change the shape:

• Callable bond: when yields fall, the issuer may refinance and call the bond. The investor's upside can become capped because the expected cash flows shorten.
• Mortgage-backed security: falling rates can encourage borrowers to refinance, returning principal earlier; rising rates can slow prepayments and extend the security's effective duration.
• Puttable bond: the holder's put option can alter the downside behaviour when yields rise.

The Federal Reserve describes negative convexity in mortgage-backed securities as a situation where expected maturity can lengthen as rates rise, increasing extension risk. For securities with changing cash flows, effective duration and effective convexity based on a pricing model are more relevant than simple calculations from fixed cash flows.

4. Convexity is not automatically a free advantage

Two bonds with the same yield and duration may have different convexity, but the more positively convex instrument may already command a higher price or lower yield. Transaction costs, liquidity, credit spread and tax treatment can outweigh a modest convexity difference.

Convexity also says nothing by itself about whether rates will rise or fall. It describes the local shape of the price response under a set of assumptions. It is not a market-timing signal.

5. What can break the estimate

The duration-convexity formula often assumes one yield moves while everything else remains stable. Real portfolios can behave differently because:

1. The yield curve moves unevenly. Two-year and ten-year yields may change by different amounts. Read our steepener and flattener guide for the curve dimension.
2. Credit spreads change. A corporate bond can lose value even if the government benchmark yield falls. Our credit-spread guide separates these effects.
3. Cash-flow assumptions change. Calls, prepayments and defaults can alter timing and amount.
4. The quote is stale. Thinly traded bonds may not have a recent executable price.
5. The metric convention differs. Convexity can be reported with different scaling or compounding assumptions. Never copy a number into the formula without checking the provider's definition.

6. A practical comparison process

When comparing two bonds or funds:

1. Record yield, modified or effective duration, convexity, currency and valuation time.
2. Confirm whether the securities contain calls, puts or prepayment exposure.
3. Apply the same hypothetical yield shocks, such as ±25, ±50 and ±100 basis points.
4. Calculate the duration-only result and then add the convexity term.
5. If the security has embedded options, use the issuer or fund's effective measures and read the model assumptions.
6. Stress credit spread and curve-shape changes separately.
7. Compare the estimate with exact repricing when your platform provides it.
8. Include bid-ask spread, fees, tax and liquidity before making a portfolio decision.

Common mistakes

• Using 1 instead of 0.01 for a one-percentage-point yield change.
• Adding the convexity term without the one-half factor when the reported definition expects it.
• Treating maturity as duration.
• Assuming every bond has positive convexity.
• Comparing metrics from providers that use different conventions.
• Applying a single parallel-yield shock to a portfolio with substantial curve, currency or credit-spread exposure.
• Treating an approximation as a guaranteed future price.

Checklist before relying on convexity

• Is the yield change expressed in decimal form?
• Are duration and convexity measured at the same valuation time?
• Does the security contain a call, put or prepayment option?
• Are the cash-flow and volatility assumptions documented?
• Did you test both rising and falling yields?
• Did you test curve and credit-spread risk separately?
• Is the underlying price recent and executable?
• Have you checked costs and liquidity?

Sources

• Federal Reserve: The U.S. Treasury yield curve—duration and convexity
• FINRA: Duration and bond portfolio interest-rate risk
• Investor.gov: Fixed-income investments and interest-rate risk
• Federal Reserve: MBS negative convexity and extension risk

This tutorial was prepared with AI-assisted research and writing under The Money Man editorial workflow. It is educational information, not personalised investment advice or a bond recommendation. The worked figures are hypothetical and exclude credit-spread changes, costs, tax and liquidity effects.