# Bond Convexity: Definition, Formula & Examples

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• 0:04 Changing Bond Price
• 1:08 Convexity
• 2:01 Formula & Example
• 5:05 Lesson Summary
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Lesson Transcript
Instructor: Douglas Stockbridge

DJ Stockbridge is currently pursuing a Masters degree in Accounting.

In this lesson, you will learn about bond convexity. You'll learn the definition, formula and how to calculate convexity and the convexity adjustment, which is used to calculate the bond price changes.

## Changing Bond Price

Here's a quick question: What will the percentage change in a bond's price be if rates decrease by 1%? The duration of the bond is 10. What if rates increase by 2%? Did you get the answer? There is a rule of thumb in the bond world. In order to calculate the percentage change in a bond's price resulting from an interest rate change, all you need to do is multiply the duration by the change in yield. Yields and price are conversely related, so if the yield falls, the bond's price increases; and if the yield rises, then the bond's price decreases. In our example, if rates decrease by 1%, the bond price will increase by approximately 10%. And, if the yield increases by 2%, the bond's price decreases by approximately 20%. Notice that I said approximately. This is because the bond concept is called convexity. Convexity is the reason why the change in bond price does not exactly match our rule of thumb.

Let's define convexity, explore how to calculate it, and how to calculate the convexity adjustment, which allows analysts to understand with more accuracy the bond price changes with each change in yield.

## Convexity

Convexity is a measure of the curve in the relationship between a bond's price and a bond's yield. In our original example, the rule of thumb considered the relationship between yield and price to be linear (a straight line). For every X% change in interest rates, bond prices increased or decreased by the duration multiplied by X%. Convexity says that relationship is not linear. In fact, it's curved, with a steeper slope as interest rates decrease and a lower slope as interest rates increase.

The x-axis is interest rates (yield), and the y-axis is the bond's price. The curved line is the actual bond price at different levels of interest rates. The straight dotted line is the bond price if there was no convexity. In other words, it's the bond price using our simple rule of thumb mentioned in the introduction. Convexity, then, is the difference between the two lines. It measures and accounts for the curvature in actual bond prices.

## Formula & Example

Let's now calculate convexity and the convexity adjustment. The formula for convexity is:

P(i decrease) = price of the bond when interest rates decrease

P(i increase) = price of the bond when interest rates increase

FV = face value of the bonds

dY = change in interest rate in decimal form

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