Negative Convexity: Definition, Example, Simplified Formula

What Is Negative Convexity?

Negative convexity exists when the shape of a bond's yield curve is concave. A bond's convexity is the rate of change of its duration, and it is measured as the second derivative 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.

Understanding Negative Convexity

A bond's duration 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 increases; therefore, its price will not rise as quickly as the price of a non-callable bond. The price of a callable bond 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. 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 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