what is convexity 
convexity is apparent in the relationship between bond prices and bond yields convexity is the curvature in the relationship between bond prices and interest rates it reflects the rate at which the duration of a bond changes as interest rates change duration measures a bond s sensitivity to changes in interest rates it represents the expected percentage change in the price of a bond for a 1 change in interest rates investopedia joules garciaunderstanding convexityconvexity demonstrates how the duration of a bond changes as the interest rate changes portfolio managers will use convexity as a risk management tool to measure and manage the portfolio s exposure to interest rate risk in the example figure shown below bond a has a higher convexity than bond b which indicates that all else being equal bond a will always have a higher price than bond b as interest rates rise or fall as interest rates fall bond prices rise conversely rising market interest rates lead to falling bond prices the bond yield is the earnings or returns an investor can expect to make by buying and holding that particular security the bond price depends on several characteristics including the market interest rate and can change regularly if market rates rise new bond issues must also have higher rates to satisfy investor demand for lending money the price of bonds returning less than that rate will fall as there would be very little demand for them as bondholders will look to sell their existing bonds and opt for bonds with higher yields eventually the price of these bonds with the lower coupon rates will drop to a level where the rate of return is equal to the prevailing market interest rates bond durationbond duration measures the change in a bond s price when interest rates fluctuate if the duration of a bond is high it means the bond s price will move to a greater degree in the opposite direction of interest rates if rates rise by 1 a bond or bond fund with a 5 year average duration would likely lose approximately 5 of its value conversely when this figure is low the debt instrument will show less movement to the change in interest rates the higher a bond s duration the larger the change in its price when interest rates change and the greater its interest rate risk if an investor believes that interest rates are going to rise they should consider bonds with a lower duration bond duration should not be confused with its term to maturity though they both decline as the maturity date approaches the latter is simply a measure of the time during which the bondholder will receive coupon payments until the principal is paid if market rates rise by 1 a one year maturity bond price should decline by an equal 1 for bonds with long dated maturities the reaction increases as a general rule of thumb if rates rise by 1 bond prices fall by 1 for each year of maturity convexity and riskconvexity builds on the concept of duration by measuring the sensitivity of the duration of a bond as yields change convexity is a better measure of interest rate risk where duration assumes that interest rates and bond prices have a linear relationship convexity produces a slope duration can be a good measure of how bond prices may be affected due to small and sudden fluctuations in interest rates however the relationship between bond prices and yields is typically more sloped or convex therefore convexity is a better measure for assessing the impact on bond prices when there are large fluctuations in interest rates as convexity increases the systemic risk to which the portfolio is exposed increases for a fixed income portfolio as interest rates rise the existing fixed rate instruments are not as attractive as convexity decreases the exposure to market interest rates decreases and the bond portfolio can be considered hedged typically the higher the coupon rate or yield the lower the convexity or market risk of a bond example of convexitya bond issuer xyz corporation has two bonds on the market bond a and bond b both bonds have a face value of 100 000 and a coupon rate of 5 bond a however matures in 5 years while bond b matures in 10 years using the concept of duration we can calculate that bond a has a duration of 4 years while bond b has a duration of 5 5 years this means that for every 1 change in interest rates bond a s price will change by 4 while bond b s price will change by 5 5 if the interest rate increase by 2 the price of bond a should decrease by 8 while the price of bond b will decrease by 11 however using the concept of convexity we can predict that the price change for bond b will be less than expected based on its duration alone this is because bond b has a longer maturity which means it has a higher convexity the higher convexity of bond b acts as a buffer against changes in interest rates resulting in a relatively smaller price change than expected based on its duration alone negative and positive convexityif a bond s duration increases as yields increase the bond is said to have negative convexity the bond price will decline by a greater rate with a rise in yields than if yields had fallen therefore if a bond has negative convexity its duration would increase and the price would fall as interest rates rise the opposite is true if a bond s duration rises and yields fall the bond is said to have positive convexity as yields fall bond prices rise by a greater rate or duration than if yields rise positive convexity leads to increases in bond prices if a bond has positive convexity it would typically experience price increases as yields fall compared to price decreases when yields increase under normal market conditions the higher the coupon rate or yield the lower a bond s degree of convexity there s less risk to the investor when the bond has a high coupon or yield since market rates would have to increase significantly to surpass the bond s yield a portfolio of bonds with high yields would have low convexity and subsequently less risk of existing yields becoming unattractive as interest rates rise most mortgage backed securities mbs will have negative convexity because their yield is typically higher than traditional bonds as a result it would take a significant rise in yields to make an existing holder of an mbs have a lower yield or less attractive than the current market 