{ "id": "749808b6-51a0-5ef3-9fc4-a28d258dcf13", "slug": "negative-convexity", "term": "Negative Convexity", "aliases": [], "category": "Fixed Income", "category_slug": "fixed-income", "difficulty": "advanced", "definition": "Negative convexity describes a bond's price-yield relationship where the bond's duration decreases as yields fall (and increases as yields rise), causing the bond to underperform a standard bullet bond in both bull and bear rate scenarios. It is the defining characteristic of callable bonds and mortgage-backed securities.", "key_takeaways": [ "Negatively convex bonds exhibit price appreciation that slows as yields decline — the opposite of positive convexity.", "Mortgage-backed securities are the most common negatively convex instruments due to homeowner prepayment optionality.", "The embedded call option that borrowers or issuers hold creates negative convexity for bond investors.", "Option-Adjusted Spread (OAS) analysis is used to strip out the option component and assess the pure credit spread.", "Investors in negatively convex bonds are implicitly short a call option and require yield compensation (positive OAS) for bearing this risk." ], "detailed_explanation": "Convexity measures the curvature of the price-yield relationship for a fixed income security. For standard bullet bonds, convexity is positive: as yields decline, prices rise at an accelerating rate, and as yields rise, prices fall at a decelerating rate. Negative convexity reverses this dynamic — the bond's price appreciation is capped on the upside and its losses are not cushioned on the downside, making it a structurally disadvantaged position relative to a comparable-duration bullet bond.\n\nThe source of negative convexity is the embedded option held by the borrower or issuer. For callable corporate bonds, the issuer retains the right to call the bond at par when interest rates fall sufficiently — precisely when the investor would benefit most from continued coupon payments. For mortgage-backed securities (MBS), homeowners effectively hold prepayment options: they refinance when rates drop, returning principal to investors at the worst possible time and forcing reinvestment at lower rates. This optionality is valuable to the option holder and costly to the investor.\n\nThe price of a negatively convex bond approaches a ceiling — its call price or par value — as yields decline. This 'price compression' creates the kinked price-yield curve that characterizes callable bonds and MBS. Mathematically, the convexity term in the bond price approximation (Price ≈ Duration × Δy + ½ × Convexity × Δy²) is negative, meaning the second-order adjustment works against the investor rather than for them.\n\nFed tightening cycles create particularly acute pain for holders of negatively convex securities. As rates rise, prepayment speeds slow on MBS (homeowners are locked into low-rate mortgages), causing the security's effective duration to lengthen — exposing investors to greater rate sensitivity precisely when prices are already falling. This dynamic, known as 'extension risk,' compounded losses in agency MBS portfolios during the 2022 rate hiking cycle.\n\nThe NOB spread (Notes Over Bonds) and the implied repo rate are both relevant because they inform relative value comparisons between instruments with differing convexity profiles. Sophisticated fixed income portfolios manage convexity explicitly through swaption overlays, Treasury futures, or interest rate caps and floors, seeking to maintain a target convexity profile without sacrificing too much yield.", "example": "Consider a 30-year agency MBS pool with a 3.5% coupon trading at $95 per $100 face value when 30-year mortgage rates are at 7%. The bond's effective duration is 8.2 years and its convexity is −1.8. If interest rates fall 100 basis points, a standard bond with the same duration would be expected to appreciate by approximately 8.2% + ½ × positive convexity benefit. But the MBS pool experiences accelerating prepayments as homeowners refinance — shortening the effective duration to 5.5 years as the rally progresses. Instead of the expected ~8.2% gain, the MBS appreciates only ~5.5%, underperforming by approximately 2.7%. This underperformance is the cost of negative convexity, and investors demand a higher OAS (in this case, perhaps 45–60 bps above Treasuries) to compensate.", "formula": "Price Change ≈ −Duration × Δy + ½ × Convexity × (Δy)²; for negative convexity, the Convexity term is negative", "formula_latex": null, "interactive_type": "chart", "calculator_id": null, "related_terms": [ "basis", "bond", "bullet-bond", "convexity", "credit-rating", "duration", "effective-duration", "extension-risk", "face-value", "green-bond", "implied-repo-rate", "interest-rate", "nob-spread", "option", "par-value" ], "backlinks": [ "amortizing-bond", "callable-bond", "cheapest-to-deliver", "collateralized-mortgage-obligation", "convexity", "credit-rating", "dirty-price", "effective-duration", "extension-risk", "key-rate-duration", "prepayment-risk", "reinvestment-risk", "yield-to-worst" ], "cross_references": [ "basis", "bond", "bullet-bond", "convexity", "duration", "effective-duration", "extension-risk", "face-value", "implied-repo-rate", "interest-rate", "nob-spread", "option", "par-value", "rally", "relative-value", "repo", "swaption", "yield", "yield-curve" ], "tags": [ "level:advanced", "cat:fixed-income" ], "asset_classes": [ "fixed-income" ], "regulators": [], "see_also": [], "sources": [], "wordcount": 623, "checksum": "08f9f88a18a8cdbe", "version": "2026.05.03", "license": "CC-BY-4.0", "updated_at": "2026-09-07T02:15:24+00:00", "_links": { "self": "https://hedgefund.wiki/api/v1/terms/negative-convexity", "jsonld": "https://hedgefund.wiki/api/v1/terms/negative-convexity?format=jsonld", "markdown": "https://hedgefund.wiki/api/v1/terms/negative-convexity?format=md", "graph": "https://hedgefund.wiki/api/v1/graph/negative-convexity", "category": "https://hedgefund.wiki/api/v1/categories/fixed-income", "schema": "https://hedgefund.wiki/schema/term.schema.json", "html": "https://hedgefund.wiki/#/terms/negative-convexity" } }