# Negative Convexity: Definition, Example, Simplified Formula

## What Is Negative Convexity?

Negative convexity exists when the shape of a bond's [yield curve](https://www.investopedia.com/terms/y/yieldcurve.asp) is concave. A bond's convexity is the [rate of change](https://www.investopedia.com/terms/r/rateofchange.asp) of its duration, and it is measured as the second [derivative](https://www.investopedia.com/terms/d/derivative.asp) of the bond's price with respect to its yield. Most mortgage bonds are negatively convex, and callable bonds usually exhibit negative convexity at lower yields.

### Key Takeaways

*   Negative convexity exists when the price of a bond falls as well as interest rates, resulting in a concave yield curve.
*   Assessing a bond's convexity is a great way to measure and manage a portfolio's exposure to market risk.

## Understanding Negative Convexity

A bond's [duration](https://www.investopedia.com/terms/d/duration.asp) refers to the degree to which a bond's price is impacted by the rise and fall of interest rates. Convexity demonstrates how the duration of a bond changes as the interest rate changes. Typically, when interest rates decrease, a bond's price increases. However, for bonds that have negative convexity, prices decrease as interest rates fall.

For example, with a callable bond, as interest rates fall, the incentive for the issuer to call the bond [at par](https://www.investopedia.com/terms/a/at-par.asp) increases; therefore, its price will not rise as quickly as the price of a non-callable bond. The price of a [callable bond](https://www.investopedia.com/terms/c/callablebond.asp) might actually drop as the likelihood that the bond will be called increases. This is why the shape of a callable bond's curve of price with respect to yield is concave or negatively convex.

## Convexity Calculation Example

Since duration is an imperfect price change estimator, investors, analysts, and traders [calculate a bond's convexity](https://www.investopedia.com/ask/answers/052615/how-can-i-calculate-convexity-excel.asp). Convexity is a useful risk-management tool and is used to measure and manage a portfolio's exposure to market risk. This helps to increase the accuracy of price-movement predictions.

While the exact formula for convexity is rather complicated, an approximation for convexity can be found using the following simplified formula:

Convexity approximation = (P(+) + P(-) - 2 x P(0)) / (2 x P(0) x dy ^2)

Where:

P(+) = bond price when interest rate is decreased

P(-) = bond price when interest rate is increased

P(0) = bond price

dy = change in interest rate in decimal form

For example, assume a bond is currently priced at $1,000. If interest rates are decreased by 1%, the bond's new price is $1,035. If interest rates are increased by 1%, the bond's new price is $970. The approximate convexity would be:

Convexity approximation = ($1,035 + $970 - 2 x $1,000) / (2 x $1,000 x 0.01^2) = $5 / $0.2 = 25

When applying this to estimate a bond's price using duration a [convexity adjustment](https://www.investopedia.com/terms/c/convexity-adjustment.asp) must be used. The formula for the convexity adjustment is:

Convexity adjustment = convexity x 100 x (dy)^2

In this example, the convexity adjustment would be:

Convexity adjustment = 25 x 100 x (0.01)^2 = 0.25

Finally, using duration and convexity to obtain an estimate of a bond's price for a given change in interest rates, an investor can use the following formula:

Bond price change = duration x yield change + convexity adjustment