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For an ordinary fixed-rate bond, convexity is the upward bend in the price–yield relationship. It adds a positive correction to the price change estimated from duration, whether yields rise or fall. In a holding-period return, that correction sits alongside carry and the duration effect; it does not make extra return free.
Why duration needs a second term
Modified duration gives the slope of the bond’s price–yield curve at the starting yield. That straight-line estimate is useful for a small yield move, but the slope changes as the yield moves. Convexity measures the curvature that duration leaves out. The CFA Institute explanation of yield-based convexity confirms that the correction raises the estimated price relative to a duration-only estimate for an option-free fixed-rate bond.
The duration term changes sign with the yield move. The convexity term does not: squaring the yield change makes the correction positive when the bond has positive convexity. For a rise in yield, it reduces the estimated price loss; for a fall, it increases the estimated price gain. It matters more as the size of the yield move grows, because the move is squared.
An illustrative bond return
All figures in this example are invented for explanation, not market observations. Suppose a bond starts with modified duration of 4 and convexity of 20. Its yield rises by one percentage point over the holding period, written as 0.01 in the calculation. Assume carry and roll-down would contribute 3% of starting price over that period if the yield did not change.
- Duration alone estimates a price change of minus 4 times 0.01, or minus 4%.
- The convexity correction is one half times 20 times 0.01 squared, or plus 0.1%.
- The combined price-change estimate is minus 3.9%. Add the assumed 3% carry and roll-down contribution to obtain an approximate total return of minus 0.9%, before any other effects.
A duration-only return estimate under these illustrative assumptions is minus 1%. The 0.1 percentage point difference comes from curvature. The calculation is an approximation around the starting yield, not a promise about the bond’s realised price. Credit-spread moves, changing cash flows and non-parallel yield-curve moves can make the realised return differ.
Why the exam cares about the starting rate
Convexity is also a pricing idea. When uncertain future rates produce different bond prices, averaging those prices is not generally the same as pricing at the average rate. The positive curvature means the average of scenario prices is higher for an otherwise comparable option-free bond. A market price can reflect that expected benefit in its starting forward rate, so a higher convexity estimate alone is not evidence of a higher expected return.
GARP identifies term-structure readings and their learning objectives in its FRM study materials. For Part II practice, separate the question being asked: a price-change estimate needs duration and convexity; a holding-period return also needs the starting carry assumption; a comparison of expected returns must consider what is already priced into the starting rate.
Mistakes to catch before moving on
- Percentage points are not decimals. Enter a one-point yield rise as 0.01 in the formula, not as 1.
- The squared term keeps its sign. Positive convexity adds to the duration estimate even when yields rise; it does not reverse the basic inverse price–yield relationship.
- Price change is not total return. Add the relevant carry and roll-down assumption only when the question asks for a holding-period return.
- Convexity is not a free expected-return bonus. Compare starting prices or forward rates as well as the payoff from a rate move.
The FRM programme overview locates this within the wider risk curriculum. The Omni study method is a useful place to plan a short revisit of the sign and unit checks after practice.
Sources
- FRM Study Materials | GARP · garp.org · Retrieved
- Yield-Based Bond Convexity and Portfolio Properties | CFA Institute · cfainstitute.org · Retrieved
- Appendix. Interest Rate Concepts | CFA Institute Research Foundation · rpc.cfainstitute.org · Retrieved
