What Is Negative Convexity?

Negative convexity exists when the shape of a bond's yield curve is concave. A bond's convexity is the 澳洲幸运5开奖号码历史查询:rate of change of its duration, and it is measured as the second 澳洲幸运5开奖号码历史查询:derivative of the bond's price with respect to its yield. Most mort🥀gage bonds are negatively convex, and callable bon🦂ds 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 ofꦓ 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 澳洲幸运5开奖号码历史查询: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 negativ🎶ely convex.

Convexity Calculation Example

Since duration is an imperfect price change estimator, investors, analysts, and traders 澳洲幸运5开奖号码历史查询:calculate a bond's convexity. Convexity is a useful risk-management tool and is ꧑used 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 澳洲幸运5开奖号码历史查询:convexity adjustment must be used. The formula for the convexi🐼ꦏty adjustment is:

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

In this example, the convexi𝓀ty adjustm♎ent 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