Bond Convexity Calculator
Estimate bond price, duration, convexity, and approximate price impact from an interest-rate move.
What is bond convexity?
Bond convexity measures how the duration of a bond changes as yields change. Duration gives a first-order (linear) estimate of price sensitivity, while convexity provides the second-order adjustment that improves accuracy when rate changes are larger. In plain English: duration gives you the straight-line estimate; convexity bends that line into a curve that better matches reality.
Most plain-vanilla bonds have positive convexity. That means when rates fall, bond prices tend to rise by a bit more than duration alone predicts; when rates rise, prices tend to fall by a bit less than duration alone predicts.
Why investors care about convexity
- Risk control: Helps portfolio managers estimate losses under yield shocks more realistically.
- Relative value: Two bonds with similar duration can behave differently if convexity differs.
- Hedging: Duration hedges can drift; convexity helps explain hedge slippage when rates move materially.
- Scenario analysis: Improves stress-test quality for pension funds, insurers, and fixed-income traders.
Inputs used in this calculator
1) Face value
The principal repaid at maturity. Commonly $1,000 for corporate or municipal issues.
2) Coupon rate
Annual coupon as a percentage of face value. A 5% coupon on a $1,000 bond pays $50/year, split according to payment frequency.
3) Yield to maturity (YTM)
The discount rate that sets present value of all cash flows equal to current price. In this tool, YTM is entered as an annual nominal rate.
4) Years to maturity and payment frequency
These determine total periods. A 10-year semi-annual bond has 20 coupon periods.
5) Yield shock in basis points
This optional scenario lets you estimate percentage and dollar price change from a rate move using duration+convexity approximation, then compare it with exact repricing.
Core formulas
Let:
- P = bond price
- y = yield per period
- CFt = cash flow at period t
- N = total periods
- m = payments per year
Price: \( P = \sum_{t=1}^{N}\frac{CF_t}{(1+y)^t} \)
Macaulay duration (years): \( D_{Mac} = \frac{1}{m}\cdot \frac{\sum t\cdot \frac{CF_t}{(1+y)^t}}{P} \)
Modified duration (years): \( D_{Mod} = \frac{D_{Mac}}{1+y} \)
Convexity (annualized years²): \( C = \frac{1}{m^2}\cdot\frac{\sum CF_t\cdot t(t+1)/(1+y)^{t+2}}{P} \)
Price change approximation: \( \Delta P/P \approx -D_{Mod}\Delta r + \frac{1}{2}C(\Delta r)^2 \)
Duration vs convexity: quick intuition
If yields move by only a few basis points, duration usually does a decent job. But as moves get larger (50–200 bps or more), linear estimates can become noticeably off. Convexity is the correction term that captures curvature.
For option-free bonds, positive convexity is typically beneficial because upside from falling yields is amplified relative to downside from rising yields of equal size. For callable bonds and MBS, effective convexity can be lower or even negative in certain regions due to embedded options.
How to use this in practice
- Compute baseline price, duration, and convexity.
- Set a plausible rate shock (e.g., ±50 bps, ±100 bps).
- Compare approximation to exact repricing.
- Repeat across holdings to build a portfolio-level sensitivity map.
Frequently asked questions
Is higher convexity always better?
Not always. Higher convexity usually comes with lower yield or higher price. Whether it is “better” depends on your objectives, risk budget, and market view.
Can I use this for zero-coupon bonds?
Yes. Set coupon rate to 0. The tool will still compute price, duration, and convexity correctly for fixed-rate zeroes.
Does this handle callable bonds?
No. Callable, putable, and mortgage-backed bonds require option-adjusted methods and path-dependent assumptions. This calculator is for plain, fixed-rate, option-free structures.
Bottom line
Convexity is a key second-order risk metric for fixed-income analysis. If you only use duration, you may under- or over-estimate bond price moves when yields shift meaningfully. Use both together for more reliable scenario planning and clearer portfolio decisions.