Bond Convexity Calculator
Calculate bond convexity to measure how bond prices respond to changes in interest rates. Convexity provides a more accurate measure of interest rate risk than duration alone.
Bond Parameters
Interest Rate Change Analysis
Convexity Results
Convexity:
0.0000
Modified Duration:
0.0000
Price Change (%):
0.00%
Risk Analysis
Interest Rate Risk:
N/A
Convexity Adjustment:
0.00%
Effective Duration:
0.0000
Price Impact
Duration Estimate:
$0.00
Convexity Adjustment:
$0.00
Total Price Change:
$0.00
Understanding Bond Convexity
Bond convexity measures the rate of change of bond duration as interest rates change. It provides a more accurate estimate of bond price changes than duration alone, especially for large interest rate movements.
Convexity Formula
Convexity Calculation
- Convexity = [S(PV × t × (t+1))] / [P × (1+y)²]
- Where PV = present value of cash flows
- t = time period
- P = current bond price
- y = yield to maturity
Price Change with Convexity
- ?P/P ˜ -Duration × ?y + 0.5 × Convexity × (?y)²
- Duration provides first-order approximation
- Convexity provides second-order adjustment
- More accurate for large rate changes
Duration vs Convexity
Interest Rate Risk Measures
Comparing duration and convexity
Duration
- First-order price sensitivity
- Linear approximation
- Accurate for small rate changes
- Macaulay, modified, or effective duration
Convexity
- Second-order price sensitivity
- Curvature of price-yield relationship
- Adjusts duration approximation
- Always positive for standard bonds
Factors Affecting Convexity
| Factor | Impact on Convexity | Reason |
|---|---|---|
| Coupon Rate | Higher coupons ? Lower convexity | More cash flows concentrated early |
| Time to Maturity | Longer maturity ? Higher convexity | More dispersed cash flows |
| Yield Level | Higher yields ? Lower convexity | Cash flows discounted more heavily |
Convexity Applications
Portfolio Management
- Hedging interest rate risk
- Immunization strategies
- Duration matching
- Risk-adjusted performance
Bond Valuation
- Accurate price change estimates
- Option-adjusted spreads
- Risk-neutral valuation
- Scenario analysis
Positive vs Negative Convexity
Positive Convexity
- Standard bonds
- Price increases more when rates fall
- Price decreases less when rates rise
- Desirable for investors
Negative Convexity
- Callable bonds
- Mortgage-backed securities
- Price increases less when rates fall
- Price decreases more when rates rise
Convexity and Bond Types
Zero-Coupon Bonds
- Highest convexity
- All cash flow at maturity
- Most sensitive to rate changes
- Duration equals maturity
Coupon Bonds
- Moderate convexity
- Cash flows throughout life
- Lower convexity than zeros
- Duration less than maturity
Practical Considerations
Measurement Issues
- Requires accurate yield curve
- Assumes parallel yield shifts
- Model risk in calculations
- Limited by market liquidity
Portfolio Applications
- Convexity matching
- Immunization strategies
- Risk budgeting
- Performance attribution
Key Takeaways for Bond Convexity Calculator
- Convexity measures the curvature in the relationship between bond prices and interest rates
- ?P/P ˜ -Duration × ?y + 0.5 × Convexity × (?y)² provides more accurate price change estimates
- Positive convexity means bond prices increase more when rates fall than they decrease when rates rise
- Zero-coupon bonds have the highest convexity, while high-coupon bonds have lower convexity
- Longer-maturity bonds have higher convexity than shorter-maturity bonds
- Convexity is always positive for standard bonds but can be negative for callable bonds
- Use convexity to improve duration-based hedging and portfolio immunization strategies
- Convexity adjustments become more important for large interest rate changes