Bond Duration and Convexity Calculator

Bond duration calculator inputs and unit conventions

This bond duration and convexity calculation treats the entered security as a plain fixed-coupon bond with regularly spaced payments.

• Face value is the principal repaid at maturity.
• Annual coupon (%) is the stated annual rate; the coupon per period is Face × coupon / payments per year.
• Yield to maturity (%) is treated as a nominal annual yield compounded at the coupon frequency (that is, yield per period = annual yield / payments per year).
• Years to maturity and payments per year determine the number of coupon periods n.
• Yield shock (%) is used only for the price-impact approximation. It is interpreted as an annual yield change. The calculator converts it to a decimal (for example, 1% becomes 0.01) and applies it consistently in the estimate.

Fixed-coupon bond duration and convexity formulas

The calculator discounts every scheduled coupon and the maturity payment using the periodic yield. Let:

• F = face value
• c = annual coupon rate (decimal)
• m = payments per year
• T = years to maturity
• n = total payments = T × m
• y = yield per period = (annual YTM as decimal) / m
• t = period index (1…n)
• CF_t = cash flow in period t (coupon each period; the final period also includes face value)

Bond price (present value) is:

Macaulay duration in periods is the present-value-weighted average of t:

D_Mac,periods = (∑ t × PV(CF_t)) / P

Converted to years:

D_Mac,years = D_Mac,periods / m

Modified duration, reported in years by this calculator, is:

D_Mod = D_Mac,years / (1 + y)

Convexity uses a discrete-compounding form and is converted from periods squared to years squared:

Cx = [∑ t(t+1) × PV(CF_t)] / [P × (1 + y)² × m²]

Estimating a bond price impact from a yield shock

For a bond yield change Δy in decimal terms (for example, 1% = 0.01), the calculator applies this Taylor approximation to percentage price change:

ΔP / P ≈ −D_Mod × Δy + ½ × Cx × (Δy)²

A positive bond yield shock generally reduces price; the convexity term offsets part of that decline and also increases the estimated gain when yields fall.

How to interpret bond duration and convexity results

Read the reported bond risk measures together: duration supplies the primary rate-sensitivity estimate, while convexity refines it.

• Macaulay duration (years): the average timing of discounted bond cash flows. A higher value means more value is received later and will usually indicate greater rate sensitivity.
• Modified duration: a first-order bond-price sensitivity. For example, D_Mod = 6 indicates an approximate 6% price decline for a 1% increase in yield, all else equal and under the small-move assumption.
• Convexity: the second-order adjustment to that estimate. Bonds can have similar modified duration but different convexity; higher convexity generally produces a more favorable estimated outcome when yields move materially in either direction.

Worked example: semiannual fixed-coupon bond duration

For this bond duration example, consider a $1,000 face-value bond that pays a 5% annual coupon and has eight years remaining to maturity.

• Face value F = $1,000
• Annual coupon = 5% (so $50 per year)
• Payments per year m = 2 (semiannual coupons of $25)
• Years to maturity T = 8 (so n = 16 periods)
• YTM = 4.2% (yield per period y = 0.042 / 2 = 0.021)

The calculator discounts each $25 coupon and the final $1,025 payment at (1 + 0.021)^t, sums the present values for price P, and uses those present-value weights for Macaulay duration and convexity.

If the entered yield shock is +1.00% (Δy = 0.01), the bond-price estimate uses:

ΔP / P ≈ −D_Mod·0.01 + ½·Cx·(0.01)²

This produces a quick percentage price estimate without repricing the bond at the shocked yield. For larger shocks, the duration-and-convexity estimate is generally more informative than duration alone, but it remains an approximation.

Bond duration and convexity assumptions and limitations

These results describe a simplified, option-free fixed-coupon bond and should be matched to the conventions used by the security or market data source.

• Fixed-coupon, plain-vanilla bond: no call, put, convertible features, sinking funds, step-up coupons, or other embedded options are modeled. Optionable bonds can have effective duration and convexity that differ materially.
• Equal spacing & integer periods: cash flows are assumed to occur exactly every 1/m years. Irregular first or final coupons are not modeled.
• No settlement date / accrued interest: results are based on a theoretical present value from time 0. Market quotes commonly depend on settlement conventions and accrued interest (clean versus dirty price).
• Simple compounding convention: YTM is treated as nominal annual yield compounded at the coupon frequency (y = YTM/m). Different compounding or continuous-compounding definitions will produce different values.
• No day-count conventions: ACT/ACT, 30/360, ACT/360, and similar conventions are not applied; timing is simplified to uniform periods.
• Approximation accuracy: the duration-plus-convexity price-impact formula is a Taylor approximation and is most reliable for small-to-moderate yield changes; large moves require full repricing at the new yield.
• Informational use: outputs are educational estimates, not investment advice. Use convention-matched analytics from your data source, broker, or terminal when making investment decisions.

Comparison table: bond duration versus convexity

This table separates the fixed-income role of each result reported by the calculator.

Bond duration and convexity references

The definitions used here follow standard fixed-income bond-math treatments found in professional finance curricula and widely used bond mathematics texts.