What is duration and convexity?

Duration and convexity are key concepts in fixed income analysis, essential for understanding the sensitivity of bond prices to interest rate changes.

Duration

Definition: Duration measures the weighted average time it takes to receive all cash flows from a bond (coupon payments and principal repayment). It's a way to assess a bond's price sensitivity to interest rate changes.

Types:
- Macaulay Duration: The weighted average time to receive the bond's cash flows. It’s expressed in years.
- Modified Duration: Adjusts Macaulay duration to account for interest rate changes, providing a more direct measure of price sensitivity. Calculated by dividing Macaulay duration by 1 + (yield/n), where n is the number of compounding periods per year.

Importance:
- Interest Rate Risk: Duration indicates how much a bond's price will change with a 1% change in interest rates. For instance, a duration of 5 years suggests that a 1% increase in interest rates will result in a roughly 5% decrease in the bond's price.
- Portfolio Management: Helps investors match asset and liability durations to manage interest rate risk effectively.

Convexity

Definition: Convexity measures the curvature in the relationship between bond prices and interest rates. It accounts for changes in duration as interest rates change, providing a more accurate prediction of bond price movements for significant interest rate shifts.

Calculation: Convexity is calculated by taking the second derivative of the bond's price with respect to interest rates. The formula involves summing the present values of all cash flows weighted by the square of the time periods until receipt.

Importance:
- Precision: While duration provides a linear estimate of price sensitivity, convexity adjusts for the fact that the relationship between bond prices and yields is actually curved.
- Risk Assessment: Bonds with higher convexity experience less price volatility with interest rate changes compared to bonds with lower convexity. This is because the price-yield curve is steeper for lower convexity bonds.

Practical Implications

1. Interest Rate Changes: Both duration and convexity help investors understand and predict how bond prices will respond to interest rate fluctuations. Duration gives a first-order approximation, while convexity provides a second-order correction.
2. Portfolio Strategy: Bond portfolio managers use these metrics to structure portfolios that minimize interest rate risk. By balancing duration and convexity, they can achieve desired risk-return profiles.
3. Hedging: Investors can use these measures to hedge against interest rate risks, ensuring that portfolio values remain stable despite changes in market interest rates.

Examples

- Short-Term Bonds: Typically have lower duration and convexity, meaning they are less sensitive to interest rate changes.
- Long-Term Bonds: Higher duration and convexity, indicating greater sensitivity to interest rate shifts.

Conclusion

Understanding duration and convexity is crucial for fixed income investors. These metrics provide valuable insights into the potential price movements of bonds in response to interest rate changes, enabling more informed investment decisions and effective risk management.