Price-Yield Relationship
A conventional fixed-rate bond's price generally moves inversely to its yield. The relationship is curved rather than perfectly linear, which is why duration alone becomes less precise as the yield change becomes larger.
Advertisement
Bond Convexity Calculator
Enter the bond's characteristics to calculate its price, Macaulay duration, modified duration and convexity.
Bond Information
Enter the bond's face value, coupon, yield, maturity and payment frequency.
Example: 1,000
Enter the bond's stated annual coupon rate.
Enter the annualized yield to maturity.
Calculation Results
Results are calculated from the present value of the bond's projected cash flows.
Bond Convexity
—
Approximate convexity based on the bond's periodic
cash flows and yield.
Bond Price
—
Macaulay Duration
—
Modified Duration
—
Annual Coupon
—
Number of Periods
—
Bond Price Sensitivity
Compare a duration-only estimate with a duration-plus-convexity estimate for different yield changes.
—
Estimated price change when yield rises 1%
—
Estimated price change when yield falls 1%
—
Estimated price change when yield changes 0.5%
| Yield Change | Duration Only | Duration + Convexity |
|---|---|---|
| +1.00% | — | — |
| +0.50% | — | — |
| −0.50% | — | — |
| −1.00% | — | — |
These are approximations. Actual bond price changes can differ because the duration-and-convexity approximation does not capture every market factor.
Advertisement
What Is Bond Convexity?
Convexity describes the curvature of the relationship between bond prices and yields.
Duration
Modified duration provides a first-order approximation of the percentage price change resulting from a small change in yield.
Convexity
Convexity provides a second-order adjustment to the duration estimate and can improve the approximation when yields change by a larger amount.
Duration + Convexity Approximation:
ΔP / P ≈ −Dmod × Δy + ½ × C × (Δy)2
ΔP / P ≈ −Dmod × Δy + ½ × C × (Δy)2
Bond Convexity Formula
The calculator derives convexity from the present value of the bond's future cash flows.
Convexity:
C ≈ [1 / P] × Σ [ CFt × t(t + 1) / (1 + y)t + 2 ]
C ≈ [1 / P] × Σ [ CFt × t(t + 1) / (1 + y)t + 2 ]
In this calculation, the yield and time periods are expressed using the bond's coupon-payment frequency. The result is converted to an annualized convexity measure.
Macaulay Duration:
DMac = Σ[t × PV(CFt)] ÷ P
Modified Duration:
DMod = DMac ÷ (1 + y)
DMac = Σ[t × PV(CFt)] ÷ P
Modified Duration:
DMod = DMac ÷ (1 + y)
Why Convexity Matters for Bond Investors
Larger Yield Changes
Duration is most useful as a first-order approximation for relatively small changes in yield. Convexity can improve the estimate when the change is larger.
Portfolio Analysis
Fixed-income professionals can consider duration and convexity together when analyzing interest-rate sensitivity across bonds or portfolios.
Curvature
Convexity captures the curvature of the price-yield relationship that a simple duration estimate cannot fully represent.
Duration vs. Convexity
Both measures describe interest-rate sensitivity, but they serve different mathematical purposes.
| Measure | What It Measures | Order | Main Use |
|---|---|---|---|
| Macaulay Duration | Weighted average timing of cash flows | Time measure | Cash-flow timing |
| Modified Duration | Approximate price sensitivity to yield | First-order | Interest-rate sensitivity |
| Convexity | Curvature of the price-yield relationship | Second-order | Improve price-change estimates |
Bond Convexity Example
Understanding the duration-plus-convexity approximation.
Suppose a bond has a modified duration of 5 years and a convexity of 30.
If yield increases by 1%:
Duration effect: −5 × 0.01 = −5%
Convexity adjustment: ½ × 30 × (0.01)2 = 0.15%
Approximate total: −5% + 0.15% = −4.85%
Duration effect: −5 × 0.01 = −5%
Convexity adjustment: ½ × 30 × (0.01)2 = 0.15%
Approximate total: −5% + 0.15% = −4.85%
This is an illustration of the mathematical approximation, not a forecast of an actual bond's future market price.
Related Bond & Fixed-Income Calculators
Explore additional fixed-income tools on EZTradingHub.
Bond Price Calculator
Estimate bond prices from cash flows.
Bond Yield Calculator
Calculate current bond yield and related measures.
Bond YTM Calculator
Estimate yield to maturity.
Bond Duration Calculator
Calculate Macaulay and modified duration.
Bond Coupon Calculator
Calculate periodic coupon payments.
ETF Return Calculator
Estimate investment returns.
Stock Return Calculator
Calculate stock investment returns.
All Calculators
Browse the EZTradingHub calculator collection.
Bond Convexity Calculator FAQ
What is bond convexity?
Bond convexity measures the curvature of the relationship between a bond's price and its yield. It provides a second-order adjustment to a duration-based price estimate.
Why is convexity important?
Duration provides a first-order estimate of price sensitivity. Convexity can improve that estimate by accounting for the curvature of the price-yield relationship.
What is the difference between duration and convexity?
Modified duration estimates the first-order percentage price response to a yield change. Convexity describes the second-order curvature of that response.
Does higher convexity mean a bond is safer?
Convexity alone does not determine overall investment risk. Credit quality, duration, liquidity, maturity, market conditions and other factors can also affect risk.
Why does duration alone become less accurate?
Duration treats the price-yield relationship as approximately linear around the current yield. Because the actual relationship is curved, the approximation can become less precise as the yield change becomes larger.
Does this calculator account for convexity?
Yes. The calculator estimates convexity from the bond's projected cash flows and uses it with modified duration to provide an approximate price-change calculation.
Is this calculator financial advice?
No. EZTradingHub calculators are educational tools and do not provide personalized investment, financial, tax or legal advice.
Financial Disclaimer:
EZTradingHub provides calculators and educational information
for general informational purposes only. Results are estimates
based on the assumptions and information entered by the user.
Actual bond prices, yields and investment outcomes may differ
because of interest rates, credit conditions, liquidity,
transaction costs, taxes, market conditions, issuer actions
and other factors.
The duration-and-convexity price-change calculation is an
approximation and does not fully capture every factor that
can affect a bond's market value.
Nothing on this page constitutes personalized investment,
financial, tax or legal advice. Conduct your own research and
consider consulting a qualified financial professional before
making investment decisions.