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Category Investments & Stocks

Face Value ($) Par / nominal value of the bond

Annual Coupon Rate (%) Annual coupon as a percentage of face value

Yield to Maturity (%) Market yield (annualized)

Years to Maturity Remaining life of the bond in years

Coupon Frequency Number of coupon payments per year

Yield Shift for Estimate (bps) Hypothetical yield change in basis points (1 bp = 0.01%)

Enter bond parameters and click Calculate to see convexity, duration, and price sensitivity analysis.

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About

Convexity measures the curvature in the relationship between a bond's price and its yield. Modified duration captures the first-order (linear) sensitivity: a 1% yield increase on a bond with modified duration 5.0 implies roughly a 5% price decline. But duration alone underestimates gains when yields fall and overestimates losses when yields rise. Convexity corrects this asymmetry. A bond with higher convexity outperforms a duration-matched bond in both rising and falling rate environments. For portfolios exceeding $10M in fixed-income exposure, ignoring convexity on a 100bps parallel shift can produce pricing errors of 0.5 - 2.0% of notional, translating directly to P&L misstatement.

This calculator enumerates every discrete cash flow from a plain-vanilla fixed-coupon bond, discounts each at the yield-to-maturity y, and computes Macaulay duration, modified duration, and annualized convexity C using the exact summation method. It then estimates the percentage and dollar price change for a user-specified yield shift using the second-order Taylor expansion. The model assumes a flat yield curve, no embedded options, and settlement on a coupon date (no accrued interest). For callable or putable bonds, effective convexity from an OAS model is required instead.

Formulas

The dirty price of a fixed-coupon bond paying k coupons per year over N total periods is the sum of discounted cash flows:

P = N∑t=1 CF_t(1 + yk)^t

where CF_t = c ⋅ Fk for t < N, and CF_N = c ⋅ Fk + F.

Macaulay Duration measures the weighted-average time to receive cash flows:

D_Mac = N∑t=1 t ⋅ CF_t(1 + yk)^tP

The result is in periods. Dividing by k converts to years. Modified Duration adjusts for compounding:

D_Mod = D_Mac1 + yk

Convexity captures the second derivative of price with respect to yield:

C = 1P ⋅ (1 + yk)² ⋅ N∑t=1 CF_t ⋅ t ⋅ (t + 1)(1 + yk)^t

The second-order price change estimate for a yield shift Δy is:

ΔPP ≈ −D_Mod ⋅ Δy + 12 ⋅ C ⋅ (Δy)²

Where P = bond price, F = face (par) value, c = annual coupon rate (decimal), y = yield to maturity (decimal), k = coupon payments per year, N = total number of coupon periods (k × years to maturity), t = period index, Δy = hypothetical yield change (decimal), D_Mac = Macaulay duration (years), D_Mod = modified duration (years), C = annualized convexity.

Reference Data

Bond Type	Typical Coupon	Maturity Range	Convexity Range	Modified Duration	Key Risk
US Treasury Note	2 - 5%	2 - 10yr	5 - 90	1.8 - 8.5	Interest rate
US Treasury Bond	3 - 6%	20 - 30yr	150 - 700	12 - 20	Interest rate
Investment-Grade Corporate	3 - 6%	5 - 30yr	20 - 500	4 - 15	Credit + Rate
High-Yield Corporate	5 - 10%	5 - 10yr	15 - 80	3 - 7	Credit spread
Municipal Bond	2 - 5%	10 - 30yr	50 - 400	5 - 14	Call risk + Rate
Zero-Coupon Bond (10yr)	0%	10yr	90 - 110	≈ Maturity	Maximum rate sensitivity
Zero-Coupon Bond (30yr)	0%	30yr	800 - 950	≈ Maturity	Maximum rate sensitivity
Floating Rate Note	SOFR + spread	1 - 5yr	≈ 0	0.1 - 0.5	Credit spread
Mortgage-Backed Security	3 - 6%	15 - 30yr	Negative possible	2 - 8	Prepayment (neg. convexity)
Callable Corporate	4 - 7%	10 - 30yr	Negative near call	3 - 10	Neg. convexity near call price
Inflation-Linked (TIPS)	0.5 - 2% real	5 - 30yr	20 - 600	4 - 18	Real rate + Inflation
Perpetual Bond	4 - 8%	∞	Very high	1/y	Extreme rate sensitivity
Sovereign EM Bond	5 - 12%	5 - 30yr	25 - 350	3 - 12	Credit + FX + Rate

Frequently Asked Questions

All cash flow is concentrated at maturity, so the weight in the convexity summation falls on the largest possible value of t · (t + 1). A coupon-bearing bond distributes cash flows earlier, reducing the weighted sum. For a 30-year zero-coupon at 5% yield, convexity exceeds 800, whereas a 30-year 6% coupon bond at the same yield typically shows convexity around 300-400.

Modified duration provides a linear (first-order) approximation of price sensitivity. For small yield changes (under 25 bps), duration alone is accurate to within a few basis points of actual price movement. For larger shifts (100+ bps), the convexity adjustment - ½ · C · (Δy)² - can account for 0.5-2.0% of price, depending on the bond's profile. The convexity term is always positive for option-free bonds, meaning duration-only estimates systematically overstate losses and understate gains.

Yes. Callable bonds and mortgage-backed securities exhibit negative convexity when yields drop near the call or prepayment threshold. As yields fall, the probability of early redemption rises, capping the bond's price upside. The price-yield curve bends downward (concave), producing negative convexity. This calculator assumes option-free bonds. For callable bonds, use an option-adjusted spread (OAS) model to compute effective convexity via finite differences.

The raw summation formula produces convexity in period-squared units. To annualize, divide by k² (where k is the number of coupon payments per year). This calculator reports annualized convexity directly. A semiannual bond with raw period convexity of 400 has annualized convexity of 400 / 4 = 100. Always verify which convention a source uses before comparing values across systems.

Higher coupon frequency (e.g., monthly vs. semiannual) slightly reduces both Macaulay duration and convexity because cash flows arrive sooner and more frequently, reducing the time-weighted present value. The effect is modest for typical bonds: switching from semiannual to quarterly on a 10-year 5% coupon bond at 5% yield changes convexity by roughly 1-3%. The dominant factors remain maturity and coupon rate.

Higher yields apply heavier discounting to distant cash flows, shifting the effective weight toward earlier periods. Since duration and convexity are weighted by PV-of-cash-flow / price, the distant cash flows contribute less to both measures. For a 20-year 4% coupon bond, raising yield from 3% to 7% can reduce modified duration from ~14 to ~11 years and convexity from ~280 to ~160.

For yield shifts up to approximately 200 bps, the duration + convexity (second-order Taylor) approximation is typically within 0.05-0.10% of the exact repriced value for option-free bonds. Beyond 300 bps, third-order effects become noticeable. Institutional desks handling tail-risk scenarios (e.g., stress tests with 500+ bps shocks) often reprice the full cash flow schedule rather than relying on the polynomial approximation.