Convexity, in the context of finance and investments, refers to the curvature or nonlinear relationship between bond prices and interest rates. It is a measure of how the duration of a bond changes as interest rates fluctuate. Convexity provides additional insight beyond duration by accounting for the curvature in the price-yield curve.

Here are key points related to convexity:

1. **Bond Price and Interest Rates:** Bond prices and interest rates have an inverse relationship. As interest rates rise, bond prices generally fall, and as interest rates fall, bond prices typically rise. This relationship is described by the concept of duration.

2. **Duration:** Duration is a measure of the sensitivity of a bond’s price to changes in interest rates. It helps estimate the percentage change in a bond’s price for a 1% change in interest rates.

3. **Convexity’s Role:** While duration provides a linear approximation of the bond price-yield relationship, convexity accounts for the curvature or the nonlinear aspects. It measures how the duration itself changes as interest rates change.

4. **Mathematical Definition:** Convexity is the second derivative of the bond price with respect to the interest rate. Mathematically, it is expressed as the percentage change in bond price for a 1% change in yield, divided by the square of the change in yield.

\[ \text{Convexity} = \frac{1}{2} \times \left( \frac{P^- + P^+ – 2P^0}{(P^0) \times (\Delta y)^2} \right) \]

Where:

– \( P^- \) is the bond price when yields decrease by \(\Delta y\).

– \( P^+ \) is the bond price when yields increase by \(\Delta y\).

– \( P^0 \) is the initial bond price.

– \( \Delta y \) is the change in yield.

5. **Impact on Price Changes:** Convexity provides information on how much a bond’s price is expected to change for a given change in interest rates. It helps refine the estimate provided by duration, especially when interest rate changes are significant.

6. **Higher Convexity:** Bonds with higher convexity are more responsive to interest rate changes and are less affected by the direction of the change. This can be an advantage for investors, especially in a scenario where interest rates are expected to exhibit significant volatility.

Understanding convexity is particularly relevant for bond investors and portfolio managers. It provides a more accurate estimate of the potential price change of a bond in response to interest rate movements and allows for better risk management and portfolio optimization. Investors often consider both duration and convexity when assessing interest rate risk and making investment decisions in fixed-income markets.