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Bond convexity

Convexity measures how a bond's price-yield sensitivity changes. It supplements duration for larger shocks but depends on the curve, model, cash flows and embedded optionality.

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Who this page is for — Anyone using duration who needs to measure curvature in the price-yield relationship, especially when a shock is not infinitesimal or cash flows contain optionality.

Bond convexity is a second-order sensitivity. While duration describes the local slope of the price-yield relationship, convexity describes how that slope changes with yield:

C = (1 / P) × ∂²P / ∂y²

P is full price under the calculation convention and y is the specified yield. If yield is a decimal annual rate, normalised convexity often has units interpretable as years squared. Other implementations use different scales or finite shocks, so the label alone does not establish the unit.

Duration and convexity approximate the price–yield curve DV01 expresses a conventional small yield move in money terms Duration and convexity approximate the price–yield curve DV01 expresses a conventional small yield move in money terms 1 bpyield →priceprice–yield curveduration approximation Modified duration Local relative slope: firstapproximation of the pricechange. Convexity Curvature term that improvesthe approximation fornon-infinitesimal moves. DV01 Money amount associated with1 bp for a defined curve,yield and position. Limitations Options, non-parallel curvesand spreads need extrameasures. Cyclepedia · source-checked conceptual map
The duration tangent loses precision away from the starting point; convexity adds the quadratic term.

Second-order correction

For a fixed-cash-flow security and a coherent yield shock:

ΔP / P ≈ −D_mod × Δy + ½ × C × (Δy)²

The first term is the linear modified-duration estimate; the second corrects for curvature. Δy must be decimal: 100 basis points is 0.01. The ½ factor comes from the Taylor approximation and must not be dropped.

Illustrative example: D_mod = 5, C = 40, shock +0.01.

ΔP/P ≈ −5 × 0.01 + ½ × 40 × 0.01² = −4.8%

Duration alone would estimate −5%. The example demonstrates the mechanism, not a forecast: yield curve, spread, cash flows and model can move together.

Positive and negative convexity

A plain non-callable bond normally has a convex price-yield relationship. For equal yield shocks, the price gain from a yield decline exceeds the duration- only estimate by more than the equivalent loss from a yield rise. This is called positive convexity.

Not every instrument keeps that shape. A callable bond's appreciation may be limited when rates fall because the likelihood of early redemption increases. In mortgage-backed securities, prepayment can shorten cash flows when rates fall and extend them when rates rise. Parts of the profile may show negative convexity.

Sign belongs to both instrument and position. Selling positive convexity reverses its contribution to the portfolio. Swaptions, spreads and multi-leg structures need signed aggregation and cannot be classified from a strategy name.

Effective convexity

When cash flows change with rates, convexity is often estimated using model-revalued prices:

C_eff = (P₋ + P₊ − 2P₀) / (P₀ × (Δy)²)

P₋ is price after the lower-yield scenario, P₊ after the higher-yield scenario and P₀ the initial price. The result depends on bump size and how the model rebuilds cash flows, exercise, prepayment, curves and spreads. Different shocks can generate different effective-convexity estimates.

A measure calculated by moving the entire risk-free curve is not the same as convexity to credit spread. The shocked factor and held-fixed inputs must be named.

Relationship to DV01 and curve risk

DV01 is local monetary sensitivity to one basis point. When convexity is non-zero, DV01 changes as yield moves: the tangent to the price-yield curve becomes steeper or flatter. A DV01-neutral hedge built today can therefore drift after a shock.

Total convexity does not describe its distribution along the curve. Two portfolios may have similar total duration and convexity but different exposures at two-, five- and thirty-year nodes. Key-rate duration, DV01 by node and steepening, flattening and twist scenarios complete the view.

Economic comparison needs coherent:

  • currency and full position value;
  • shocked curve and node;
  • yield compounding and day count;
  • clean or dirty/full price;
  • sign, quantity and netting perimeter;
  • model for optional cash flows.

Convexity is not free protection

More positive convexity can be desirable under larger shocks, but it has a price embedded in quotations or carry. It does not prove higher expected return and does not remove credit, spread, liquidity, inflation or currency risk. Negative convexity likewise does not mean loss in every scenario: it describes local geometry to the selected factor.

For very large shocks, the second-order approximation may remain inadequate. Barriers, discrete calls and behavioural changes can create kinks or discontinuities. Full scenario revaluation is then primary; duration and convexity explain and control the local result.

Portfolio convexity should be retained gross and net by currency and curve. Opposing totals may not offset if factors move differently or optional cash flows react asymmetrically.

Common mistake — Applying convexity calculated to yield to maturity to a different curve or spread shock without stating the mapping. A sensitivity belongs to the factor actually perturbed.

Checklist

  1. Is convexity analytical, effective or finite-difference based?
  2. Which yield, curve or spread is perturbed?
  3. Which units and bump size are used?
  4. Are price, quantity, currency and sign coherent?
  5. Are cash flows fixed or dependent on calls and prepayment?
  6. Does curve-node risk remain visible?
  7. Has full revaluation been run for material shocks?

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