Bond Parameters
Quick Examples
Load a preset to explore different bond characteristics:
Assumptions
- Settlement at coupon date (no accrued interest)
- Yield compounded at payment frequency
- Equal period lengths assumed
- Negative yields not supported
Duration Metrics
Interest Rate Sensitivity
Estimated price change if yield moves:
Visualization
Enter bond parameters to see the visualization
Cash Flow Breakdown
-- periods| Period | Time (Years) | Cash Flow | Discount Factor | Present Value | Weight | Weighted Time |
|---|---|---|---|---|---|---|
| Total | -- | -- | -- | 100% | -- | |
Understanding Bond Duration
What is Bond Duration?
Bond duration measures the sensitivity of a bond's price to changes in interest rates. It represents the weighted average time until you receive the bond's cash flows, where each cash flow is weighted by its present value as a proportion of the bond's total price.
Duration is one of the most important concepts in fixed income investing because it helps investors understand and manage interest rate risk. A higher duration means greater price sensitivity to interest rate changes.
Macaulay Duration
Macaulay Duration is measured in years and represents the weighted average time to receive the bond's cash flows. It was developed by Frederick Macaulay in 1938 and answers the question: "On average, how long do I have to wait to receive my money back?"
For example, a 10-year bond paying a 5% coupon has a Macaulay Duration of about 8 years because you receive coupon payments throughout the life of the bond, not just at maturity.
Modified Duration
Modified Duration adjusts Macaulay Duration to directly measure price sensitivity. It tells you the approximate percentage price change for a 1% change in yield. This is the most commonly used duration measure for interest rate risk management.
Where y is the yield to maturity and m is the number of payments per year. For example, if a bond has a modified duration of 7, a 1% increase in interest rates would cause approximately a 7% decrease in the bond's price.
Effective Duration
Effective Duration measures price sensitivity empirically by actually calculating how the bond price changes when yields are shocked up and down. Unlike Modified Duration (which is derived analytically from Macaulay Duration), Effective Duration uses numerical methods.
Where P- is the price when yield decreases, P+ is the price when yield increases, P0 is the current price, and Δy is the yield shock (typically 25 basis points).
Why use Effective Duration? For plain vanilla bonds, Effective Duration equals Modified Duration. However, for bonds with embedded options (callable bonds, mortgage-backed securities), the cash flows change when rates change, making Modified Duration inaccurate. Effective Duration captures these effects.
Understanding Convexity
Convexity measures the curvature of the price-yield relationship. While duration provides a linear approximation of price changes, the actual relationship between bond prices and yields is curved. This curvature is convexity.
Why convexity matters:
- Duration underestimates gains: When yields fall significantly, the actual price increase is larger than duration alone predicts
- Duration overestimates losses: When yields rise significantly, the actual price decrease is smaller than duration alone predicts
- Positive convexity is good: It means you gain more when rates fall than you lose when rates rise by the same amount
The complete price change formula incorporating both duration and convexity is:
The Duration Sensitivity chart above illustrates this relationship: the curved blue line shows actual prices (including convexity), while the straight dashed line shows the duration-only approximation.
How to Use This Calculator
- Enter bond parameters - Input the face value, coupon rate, yield to maturity, years to maturity, and payment frequency
- Review key metrics - See the calculated Macaulay Duration, Modified Duration, Effective Duration, DV01, and Convexity
- Analyze sensitivity - The sensitivity table shows estimated price changes for various yield movements
- Explore the chart - The Duration Sensitivity chart shows the price-yield relationship and demonstrates convexity visually
- Explore cash flows - The breakdown table shows each payment's present value and weight
- Use presets - Try the example bonds to see how different characteristics affect duration and convexity
Why Duration and Convexity Matter
- Interest Rate Risk: Duration quantifies how much your bond portfolio will change in value when rates move; convexity refines this estimate for larger rate changes
- Portfolio Immunization: Match portfolio duration to your investment horizon to minimize interest rate risk
- Hedging: Use duration and convexity to calculate hedge ratios for interest rate derivatives
- Bond Selection: Given two bonds with equal duration, prefer the one with higher convexity (all else equal)
- Comparison: Compare bonds with different coupons and maturities on an equal basis
Duration and Convexity Limitations
While duration and convexity are powerful tools, they have important limitations:
- Parallel shift assumption: Duration assumes the entire yield curve shifts by the same amount, which rarely happens in practice
- Instantaneous change: The formulas assume yields change instantaneously; in reality, changes occur over time
- No credit risk: Duration doesn't account for the possibility of default
- Embedded options: For callable or putable bonds, use Effective Duration instead of Modified Duration
Frequently Asked Questions
Important Disclaimer
This calculator is for educational purposes only. It assumes settlement at coupon dates, equal period lengths, and yield compounded at payment frequency. Actual bond prices may differ due to accrued interest, day count conventions, call features, credit risk, and other factors. This is not financial advice.
