Chapter 2. Duration and Convexity: Measuring the Sensitivity

If Chapter 1 taught us that bond prices move when interest rates change, Chapter 2 teaches us “By exactly how much?” To manage a bond portfolio, we need tools that quantify this risk. These tools are ==Duration== and ==Convexity==.

1. What is Duration? (The First Derivative)

Duration is not just time; it is a measure of a bond’s price sensitivity to interest rate changes.

Types of Duration

Term	Definition	Key Use Case
Macaulay Duration	The weighted average time until all cash flows are received.	Measuring the 'Effective Life' of a bond.
Modified Duration	Adjusted Macaulay duration that directly links % price change to % yield change.	Day-to-day risk management.
Effective Duration	Used for bonds with embedded options (like callable bonds) where cash flows can change.	Complex bond valuation.

2. The Linear Approximation: Using Duration

The most practical application of duration is the following formula: ==% Price Change ≈ -Modified Duration × Change in Yield==

Scenario

You hold a bond with a Modified Duration of 7.0

Shock

The market interest rate (YTM) rises by 1% (100 bps)

Calculation

Multiply Duration by the change: 7.0 * 0.01 = 0.07

Result

Your bond price will drop by approximately 7.0%

3. Beyond the Line: Convexity (The Second Derivative)

Duration assumes that the relationship between price and yield is a straight line. But in reality, it is a curve. ==Convexity== measures this curvature.

The Benefit of Convexity: For a “Convex” bond, the price increases more when rates fall than it decreases when rates rise.
The Cushion: Convexity provides a safety buffer during large interest rate swings that duration alone cannot capture.

Warning

Negative Convexity: Some bonds, like Mortgage-Backed Securities (MBS) or Callable Bonds, can have “Negative Convexity.” This means their price may not rise as much as expected when rates fall because the borrower might repay the debt early.

4. Conclusion: Mastering Risk

Duration tells you the direction and the magnitude; Convexity tells you the error in that estimate. By understanding these two, you can design a ==“Duration-Neutral”== portfolio or intentionally bet on the direction of interest rates with mathematical confidence.

📚 Prof. Sean’s Selected Library

[Fixed Income Mathematics] - Frank Fabozzi: The definitive technical guide on how to calculate these metrics.
[The Handbook of Fixed Income Securities]: A deep dive into how institutional managers use duration to hedge risk.
[Interest Rate Risk Management] - Various Authors: Understanding the trade-off between yield and sensitivity.

Next time, we will explore ‘Yield Curve Analysis’—learning how to read the economy’s future through the shape of interest rates across different maturities.