Skip to content
Financial Context Wiki

Bond Convexity: Why Duration Is Not Enough

Use bond convexity with duration to estimate price sensitivity, distinguish yield-based from effective measures, handle embedded options, and avoid unit, scaling, and curve-shock errors.

Updated

For educational purposes only; not investment advice. Investing may result in loss.

Direct answer

Bond convexity measures curvature in a specified relationship between a bond’s price and yield or benchmark curve. Duration gives the first-order slope; convexity adds a second-order adjustment. Together they provide a local sensitivity estimate, not a forecast or a complete measure of return.

For a standard option-free fixed-rate bond with positive promised cash flows, yield-based convexity is positive: an equal yield decline tends to increase price by more than duration alone predicts, while an equal yield increase tends to reduce price by less. Callable bonds, mortgage-backed securities, and other instruments whose expected cash flows change with rates can have negative effective convexity in some rate environments.

7-year, 5% annual coupon, 5% base yield

Exact repricingDuration estimateDuration + convexity
12110181
Interactive duration and convexity comparison
-300 bp0 bp+300 bp
Exact price
$94.42
Modified duration
5.79
Convexity
41.9
Adjusted error
$0.01

How the adjustment works

A common approximation for a small change in a consistently defined yield is:

ΔP / P ≈ -D × Δy + 0.5 × C × (Δy)^2

D is modified duration, C is annual convexity under the convention shown, and Δy is the yield change written as a decimal. A 100-basis-point move is 0.01, not 100. Some providers report money convexity, scaled convexity, or a convexity adjustment that already incorporates a factor; confirm the definition before inserting a quoted number into this formula.

The duration term is linear. If yields rise, it is negative; if yields fall, it is positive. The convexity term uses (Δy)^2, so positive convexity adds to the estimate in both directions. It softens the estimated loss when yields rise and increases the estimated gain when yields fall.

For bonds whose expected cash flows change when rates change, use model-based effective convexity rather than a fixed-cash-flow calculation:

effective convexity ≈ (P- + P+ - 2P0) / (P0 × (Δy)^2)

Here P- is the modeled full price after the benchmark curve falls by the chosen shock, P+ is the modeled full price after it rises by the same shock, and P0 is the current modeled full price. The three valuations should use consistent option-adjusted-spread and volatility assumptions unless the test deliberately changes them. Effective convexity can vary materially with the shock size and rate environment.

Keep the risk factor consistent. Yield-based duration and convexity usually describe a parallel change in one yield to maturity for fixed cash flows. Effective measures generally shock a benchmark curve and revalue cash flows. Neither automatically captures a change in credit spread, curve slope or curvature, liquidity, default expectations, or option volatility.

Worked example

Suppose a bond has a full price of 100, modified duration 6, and annual convexity 50 under the formula above.

If yields rise by 100 basis points:

ΔP / P ≈ -6 × 0.01 + 0.5 × 50 × 0.01^2 = -5.75%

The estimated percentage change is -5.75%, so the full price becomes about 94.25. Duration alone would have estimated -6.00%, or 94.00.

If yields fall by 100 basis points:

ΔP / P ≈ +6 × 0.01 + 0.5 × 50 × 0.01^2 = +6.25%

The estimated percentage change is +6.25%, so the full price becomes about 106.25.

For a 25-basis-point move, 0.5 × 50 × 0.0025^2 = 0.015625%, so the convexity adjustment is about 0.0156%. For a 200-basis-point move, 0.5 × 50 × 0.02^2 = 1.00%. The squared term grows quickly, although the second-order approximation itself can become less accurate for large shocks; reprice the instrument when the decision is material.

Practical checks

  • Convert the shock correctly: 1 bp = 0.0001, 50 bp = 0.005, and 100 bp = 0.01.
  • Confirm whether duration is Macaulay, modified, effective, spread, or key-rate duration and whether convexity is yield-based, effective, money, or scaled.
  • Use the same yield definition, compounding frequency, settlement date, accrued-interest treatment, and full or clean price convention in every input and shocked valuation.
  • Do not use fixed-cash-flow convexity for callable bonds, mortgage-backed securities, or instruments with material prepayment, extension, put, conversion, or other path-dependent behavior.
  • For effective measures, document the benchmark curve shock, shock size, pricing model, option-adjusted spread, volatility, and cash-flow assumptions.
  • Separate benchmark-rate risk from credit-spread, default, liquidity, currency, and inflation risk; several can move simultaneously.
  • Test nonparallel curve moves with key-rate measures or direct scenario repricing instead of relying only on one aggregate duration and convexity pair.
  • For a portfolio, verify that market-value weighting, currencies, hedges, derivatives, and measure definitions are compatible before aggregating security-level values.
  • Compare the approximation with direct repricing for large shocks, concentrated positions, leveraged portfolios, or securities near an option exercise boundary.
  • Remember that greater positive convexity usually has an economic price, such as a lower yield, higher market price, or different liquidity and optionality.

Common misconceptions

“Positive convexity prevents losses when rates rise.” It only makes the local price response more favorable than the duration-only tangent. The negative duration effect, spread widening, default risk, or forced selling can still produce a loss.

“Convexity replaces duration.” Duration remains the first-order sensitivity; convexity adjusts for curvature. Both depend on the selected risk factor and assumptions.

“Basis points can be entered directly into the formula.” Using 100 instead of 0.01 creates a wildly wrong linear term and an even more extreme squared term.

“Negative convexity means the bond must lose money.” It describes an unfavorable local shape of price response, often linked to call or prepayment behavior; realized return also depends on income, carry, spread, default, trading price, and the actual rate path.

Authoritative sources

Navigation

Search the wiki...