Identifier: C2T4L3 | Revision: 2026-10-26

Decision and objectives

A duration estimate predicts equal-sized gains and losses for equal opposite yield moves. Full repricing usually does not. Convexity is the curvature adjustment for a fixed cash-flow price-yield relationship. It improves a local estimate, but cannot make uncertain callable cash flows fixed.

Objectives: Calculate a convexity-adjusted estimate; Compare an estimate with full repricing; Identify changing cash-flow limitations.

Prerequisites: C2T4L2.

Mechanics

For a conventional option-free bond with positive payments, the price-yield curve bends upward. When yields fall, price gains tend to exceed the linear estimate; when yields rise, losses tend to be smaller. A callable bond can behave differently as an issuer's redemption right limits upside. A mortgage-related structure can change principal timing when borrowers prepay. Those effects require scenarios with different cash schedules.

Price and return drivers

For annual effective yield, a practical fixed-schedule convexity measure C = sum[t × (t+1) × CFt / (1+y)^(t+2)] / price. This is a weighted arithmetic formula; use it without deriving it. Approximate percentage change = −Dmod × Δy + 0.5 × C × Δy². Rates are decimals, so a two-percentage-point shock is 0.02; the squared term becomes 0.0004.

Alternatively infer effective sensitivity from supplied up/down scenario prices: effective duration = (Pdown − Pup)/(2 × P0 × shock). Effective convexity = (Pdown + Pup − 2P0)/(P0 × shock²). For embedded options, the scenario pricing must allow the relevant exercise behavior to change; otherwise the adjective effective adds no useful protection.

Worked case

Hypothetical two-year zero: face 100; annual effective yield 10%; P0 = 100/1.10² = 82.644628. Dmod = 2/1.10 = 1.818182. C = 2 × 3 / 1.10² = 4.958678. For a rise to 12%, linear change = −1.818182 × 0.02 = −3.636364%. Convexity contribution = 0.5 × 4.958678 × 0.02² = +0.099174%. Combined estimate = −3.537190%; estimated price = 79.721331. Full price = 100/1.12² = 79.719388; actual change = −3.539541%. Rounding is to six decimals, and no fees or tax are included.

Adverse case

For a fall to 8%, full price = 100/1.08² = 85.733882; gain = 3.737997%, larger than the loss percentage for the equal upward shock. Now consider a separate hypothetical callable issue: current price 100, down-shock scenario price 101, up-shock price 96 for ±1 percentage point. Effective duration = (101−96)/(2×100×0.01) = 2.5. Effective convexity = (101+96−200)/(100×0.01²) = −300. Limited upside and larger downside produce negative effective convexity. These are supplied scenario valuations, not contractual expiry payoffs or executable quotes.

Practical use

Read whether a reported convexity is modified, effective, scaled by 100 or quoted in another convention. Do not compare numbers without units and scenario size. Call eligibility, coupon resets and redemption assumptions belong with the valuation. Ask for full scenario repricing when an instrument contains a barrier, conversion, loss absorption or unusual redemption. Model outputs remain subject to actual dealer liquidity and cash costs.

Decision boundaries

Convexity improves a small-shock estimate under a fixed schedule; it is not a universal correction factor. Large moves can change duration and option exercise incentives. Credit default is not just a wider yield with unchanged promised cash. Decline a risk claim that applies an option-free convexity number to a perpetual callable issue without a call/extension model.

Practice and references

Exercise: supplied hypothetical model value TRY 200,000, Dmod 4, convexity 22; yield rise 1 percentage point; no time or cash changes. Answer: fraction = −4×0.01 + 0.5×22×0.01² = −0.0389, or −3.89%; estimated money change = TRY −7,780; new value TRY 192,220. Full repricing remains necessary for a precise valuation.

Takeaway: curvature explains asymmetric price moves; changing contractual cash needs additional scenarios.

Reference guide: callable-debt, bond-convexity, convexity-price-change, Convexity, Effective duration.

Next: C2T4L4. Topic test: C2T4.

Primary sources (checked 2026-10-26; changing terms require a publication recheck): SEC Investor Bulletin: What Are Corporate Bonds? — Interest rate risk, credit risk, call risk and seniority.

Shared reference guide