Guide

Bond convexity explained

Harbor Capital’s intermediate Treasury sleeve held $200 million of on-the-run 10-year notes with a reported modified duration of 7.2 years. When the 10-year yield fell 80 basis points in a single quarter, the desk’s duration-matched futures hedge should have kept DV01 roughly flat. Instead the portfolio gained $4.8 million more than the linear model predicted — and the hedge overshot by $1.9 million. Risk reported duration neutrality; P&L showed a convexity gap. The treasurer had treated the price-yield relationship as a straight line when it is visibly curved.

Bond convexity measures that curvature: how much a bond’s duration estimate over- or under-states price change when yields move by more than a few basis points. Positive convexity rewards holders when rates fall and cushions selloffs when rates rise; negative convexity on callable bonds does the opposite. Convexity is not academic ornament — it breaks naive immunization, distorts hedge ratios, and explains why barbell portfolios behave differently from bullets with the same duration. This guide defines convexity math, classifies positive vs negative profiles, connects convexity to portfolio immunization, walks through Harbor Capital’s hedge refactor, and provides a technique decision table, pitfalls, and checklist.

Why duration alone is a straight-line guess

Modified duration estimates percentage price change for a small yield move:

ΔP/P ≈ −ModDur × Δy

That is a first-order Taylor approximation — tangent to the price-yield curve at the current yield. For 5–10 basis point shocks, it is often close enough for Treasuries. For 50–100 basis point moves — stress tests, Fed surprises, credit events — the bond’s true price path curves away from the tangent. Convexity captures the second-order term.

Concept	What it measures	Typical use
Modified duration	Linear sensitivity to yield	DV01 hedging, quick risk reports
Convexity	Curvature of price-yield relationship	Large-move P&L, immunization quality
Dollar convexity	Convexity × price × (Δy)²	Portfolio-level convexity P&L in currency
Effective convexity	Numerical convexity from price bumps	Callable bonds, MBS, funds with options

Intuition: a bond with positive convexity gains more when yields fall than it loses when yields rise by the same amount (for parallel shifts). Holders prefer positive convexity; issuers of callable debt embed negative convexity when call options cap upside.

Convexity taxonomy

Not all fixed-income instruments share the same curvature profile. Classify before hedging or matching liabilities.

Positive convexity (option-free bonds)

Plain-vanilla Treasuries, investment-grade bullets, and zero-coupon bonds exhibit positive convexity. Longer maturity and lower coupon increase convexity for a given duration — zeros maximize both duration and convexity per dollar of face value. A 30-year Treasury has more convexity than a 5-year note even if you duration-match with leverage.

Negative convexity (embedded options)

Callable corporates, agency debentures with call schedules, and mortgage-backed securities show negative convexity when rates fall: prepayment or call risk caps price appreciation. The bond behaves like a short call option to the investor. Duration models that ignore the call underestimate extension risk when rates rise and overestimate price gain when rates fall.

Portfolio convexity

Portfolio convexity is roughly the market-value-weighted sum of position convexities (exact aggregation uses dollar convexity on each lot). Mixing positive-convexity Treasuries with negative-convexity MBS can produce a book that looks duration-neutral but has asymmetric large-move risk.

The convexity adjustment in practice

The improved price-change estimate combines duration and convexity:

ΔP/P ≈ −ModDur × Δy + ½ × Convexity × (Δy)²

Example sketch: a bond with modified duration 6.0 and convexity 80 (standard convention). If yields rise 100 bp (0.01), duration alone predicts −6.0%. The convexity term adds +½ × 80 × (0.01)² = +0.40%. Net estimate: −5.6% instead of −6.0%. The bond loses less than the linear model feared — positive convexity helped.

For a 50 bp move the convexity term is smaller; for 200 bp it dominates the error in duration-only models. Stress testing should always report both parallel-shift duration P&L and convexity-adjusted P&L; regulators and institutional investors increasingly expect both on rate-risk disclosures.

Dollar convexity and DV01 together

Traders often report DV01 (dollar value of a one-basis-point move) alongside dollar convexity for a 1 bp or 10 bp bump. Hedging DV01 with futures or swaps leaves convexity exposure unless the hedge instrument’s convexity matches. Treasury futures used to hedge callable corporates frequently leave residual negative convexity on the combined book.

Convexity and immunization

Classical Fisher–Weil immunization matches asset duration to liability duration for a single future cash need. That match is exact only for infinitesimal yield moves. The immunization gap from convexity mismatch means:

When rates fall, an under-convex asset portfolio underperforms the liability value rise if the liability is more convex (long-dated zero-like payouts).
When rates rise, the asset portfolio may outperform the liability — surplus appears — but reinvestment assumptions on coupons break the match.
Callable assets with negative convexity widen the gap asymmetrically: good news on rates triggers calls; bad news extends duration.

Practitioners target duration match plus convexity match when liabilities are lump-sum and rate volatility is high. Pension actuaries with smooth benefit cash-flow streams may accept convexity mismatch if key-rate duration alignment matters more than parallel-shift convexity. Document which risk you optimized for.

Barbell vs bullet: same duration, different convexity

A bullet portfolio concentrates in one maturity bucket. A barbell splits between short and long maturities with weights chosen to match the bullet’s duration. The barbell typically has higher portfolio convexity because long-end positive convexity dominates the weighted sum.

Investors who want rate-volatility optionality without directional bets often prefer barbells. Investors funding known intermediate liabilities with ladder cash flows may prefer bullets or dedicated strips to minimize reinvestment guesswork. Neither is universally superior — convexity is the differentiator duration alone hides.

Harbor Capital convexity hedge refactor

After the $1.9 million hedge overshoot, Harbor Capital revised its Treasury overlay:

Convexity reporting — daily dashboard: portfolio modified duration, convexity, effective convexity on callable sleeves, and dollar convexity for ±25 bp and ±100 bp scenarios.
Hedge instrument selection — replaced partial futures overlay with a mix of futures and options on futures to replicate target convexity, not just DV01.
Callable sleeve limits — IG callable exposure capped when portfolio convexity turned net negative; shifted marginal buys to bullets.
Immunization review — quarterly check: liability duration and convexity vs assets; surplus/deficit under ±75 bp parallel shifts with convexity adjustment.
Stress narrative — board materials show duration-only vs convexity-adjusted P&L bands so non-quants see why “matched” books still move.

Residual tracking error fell 40% on the next 100 bp round-trip in rates; the desk still carries convexity risk by choice on the long end where positive convexity aligns with their rate-view governance.

Technique decision table

When convexity analysis earns its keep versus simpler metrics.

Goal	Convexity focus	Alternative
Intraday DV01 hedge	Secondary — duration sufficient	Modified duration + futures beta
Large parallel shift stress (±100 bp)	Primary — use convexity adjustment	Full repricing / scenario grid
Lump-sum liability in 10+ years	Match asset convexity to liability	Duration match only (immunization gap)
Callable / MBS allocation	Effective convexity mandatory	OAS duration without convexity
Barbell vs bullet allocation	Compare portfolio convexity	Yield pickup comparison only
Curve twist (non-parallel)	Insufficient alone	Key-rate duration vectors

Common pitfalls

Ignoring convexity on large moves. Duration-only stress tests systematically misstate callable and long-duration books.
Using analytical convexity on callables. Effective convexity from numerical price bumps reflects call options; analytical formulas on stated maturity mislead.
Assuming immunization from duration match alone. Surplus and deficit appear when convexity of assets and liabilities diverge.
Hedging DV01 with mismatched convexity. Futures hedges on negative-convexity portfolios leave asymmetric tails.
Chasing yield into negative convexity. Callable premium looks attractive until rates rally and bonds are called away.
Confusing convexity with credit spread risk. Spread widening is not a parallel Treasury shift; convexity on Treasury curve does not hedge credit losses.
Overfitting barbells. Higher convexity often means lower yield; ensure the convexity benefit fits your horizon and governance.

Investor checklist

Report modified duration and convexity (or effective convexity) on every fixed-income sleeve.
Run ±25 bp and ±100 bp parallel shifts with convexity-adjusted P&L.
For callable/MBS holdings, use effective convexity from vendor models or numerical bumps.
Compare portfolio convexity to liability convexity on immunized books quarterly.
Document hedge instrument convexity alongside DV01 when using futures or swaps.
Evaluate barbell vs bullet choices on convexity, not duration alone.
Separate Treasury curve convexity from credit spread risk in stress narratives.
Board and client materials explain why duration-matched books still move on large shifts.

Key takeaways

Duration is linear; markets curve. Convexity measures the gap for large yield moves.
Positive convexity helps holders; callables and prepayable structures often impose negative convexity.
Immunization needs convexity alignment when liabilities are lump-sum and rate volatility is material.
Barbells beat bullets on convexity at equal duration — with yield trade-offs.
Hedge the curvature, not just DV01, when optionality or large shifts dominate risk.