# The AI (ChatGPT) Bond Investing Handbook: Bond Convexity

Here is the response from ChatGPT for the prompt bond convexity entered on February 19, 2023.

Bond convexity is a measure of how sensitive the price of a bond is to changes in interest rates. It is a second-order approximation that takes into account the non-linear relationship between bond prices and yields. Specifically, convexity measures the curvature of the price-yield relationship of a bond.

When interest rates change, the price of a bond will typically move in the opposite direction. However, the degree of this price change is not constant, and bond convexity helps to explain why. Bonds with high convexity have a more pronounced price response to changes in interest rates than bonds with low convexity.

Convexity is an important concept for bond investors because it can affect the performance of a bond portfolio. By understanding the convexity of their bond holdings, investors can better estimate the potential impact of interest rate changes on their investments. In general, high convexity bonds may provide greater protection against rising interest rates, while low convexity bonds may be more vulnerable to interest rate risk. However, it's important to note that convexity is just one factor to consider when evaluating bond investments, and should be considered in conjunction with other factors, such as credit risk and liquidity.

## Bond Convexity Example

Here is the response from ChatGPT for the prompt bond convexity example entered on February 19, 2023.

Here's an example to illustrate bond convexity:

Let's say you own a 5-year bond with a face value of \$1,000 and a coupon rate of 5%. The bond has a modified duration of 4 years and a convexity of 10.

Suppose interest rates rise by 1%. Using modified duration, we can estimate that the bond's price would fall by about 4% (the modified duration multiplied by the change in interest rates). So, the bond's new price would be:

\$1,000 x (1 - 0.04) = \$960

However, this estimate does not take into account the effect of convexity. With a convexity of 10, the bond's price should actually increase slightly due to the convexity effect. The magnitude of this effect can be calculated using the following formula:

Convexity effect = (1/2) x convexity x (change in interest rate)^2 x face value

In this case, the convexity effect would be:

(1/2) x 10 x (0.01)^2 x \$1,000 = \$0.50

So, the bond's actual price after the 1% interest rate increase would be:

\$960 + \$0.50 = \$960.50

This example shows how bond convexity can impact the price of a bond in response to changes in interest rates. While modified duration provides a good estimate of the price change, convexity can help to refine that estimate and provide a more accurate picture of how the bond's price will react to interest rate changes.