For any given bond, a graph of the relationship between price and yield is convex. This means that the graph forms a curve rather than a straightline (linear). The degree to which the graph is curved shows how much a bond's yield changes in response to a change in price. In this section we take a look at what affects convexity and how investors can use it to compare bonds.
Convexity and Duration
If we graph a tangent at a particular price of the bond (touching a point on the curved priceyield curve), the linear tangent is the bond's duration, which is shown in red on the graph below. The exact point where the two lines touch represents Macaulay duration. Modified duration, as we saw in the preceding section of this tutorial, must be used to measure how duration is affected by changes in interest rates. But modified duration does not account for large changes in price. If we were to use duration to estimate the price resulting from a significant change in yield, the estimation would be inaccurate. The yellow portions of the graph show the ranges in which using duration for estimating price would be inappropriate.
Furthermore, as yield moves further from Y*, the yellow space between the actual bond price and the prices estimated by duration (tangent line) increases.
The convexity calculation, therefore, accounts for the inaccuracies of the linear duration line. This calculation that plots the curved line uses a Taylor series, a very complicated calculus theory that we won't be describing here. The main thing for you to remember about convexity is that it shows how much a bond's yield changes in response to changes in price.
Properties of Convexity
Convexity is also useful for comparing bonds. If two bonds offer the same duration and yield but one exhibits greater convexity, changes in interest rates will affect each bond differently. A bond with greater convexity is less affected by interest rates than a bond with less convexity. Also, bonds with greater convexity will have a higher price than bonds with a lower convexity, regardless of whether interest rates rise or fall. This relationship is illustrated in the following diagram:
As you can see Bond A has greater convexity than Bond B, but they both have the same price and convexity when price equals *P and yield equals *Y. If interest rates change from this point by a very small amount, then both bonds would have approximately the same price, regardless of the convexity. When yield increases by a large amount, however, the prices of both Bond A and Bond B decrease, but Bond B's price decreases more than Bond A's. Notice how at **Y the price of Bond A remains higher, demonstrating that investors will have to pay more money (accept a lower yield to maturity) for a bond with greater convexity.
What Factors Affect Convexity?
Here is a summary of the different kinds of convexities produced by different types of bonds:
1) The graph of the priceyield relationship for a plain vanilla bond exhibits positive convexity. The priceyield curve will increase as yield decreases, and vice versa. Therefore, as market yields decrease, the duration increases (and vice versa).
2) In general, the higher the coupon rate, the lower the convexity of a bond. Zerocoupon bonds have the highest convexity.3) Callable bonds will exhibit negative convexity at certain priceyield combinations. Negative convexity means that as market yields decrease, duration decreases as well. See the chart below for an example of a convexity diagram of callable bonds.
Remember that for callable bonds, which we discuss in our section detailing types of bonds, modified duration can be used for an accurate estimate of bond price when there is no chance that the bond will be called. In the chart above, the callable bond will behave like an optionfree bond at any point to the right of *Y. This portion of the graph has positive convexity because, at yields greater than *Y, a company would not call its bond issue: doing so would mean the company would have to reissue new bonds at a higher interest rate. Remember that as bond yields increase, bond prices are decreasing and thus interest rates are increasing. A bond issuer would find it most optimal, or costeffective, to call the bond when prevailing interest rates have declined below the callable bond's interest (coupon) rate. For decreases in yields below *Y, the graph has negative convexity, as there is a higher risk that the bond issuer will call the bond. As such, at yields below *Y, the price of a callable bond won't rise as much as the price of a plain vanilla bond.
Convexity is the final major concept you need to know for gaining insight into the more technical aspects of the bond market. Understanding even the most basic characteristics of convexity allows the investor to better comprehend the way in which duration is best measured and how changes in interest rates affect the prices of both plain vanilla and callable bonds.

Investing
Explaining Convexity
Convexity is the measure of the curve in the relationship between a bondâ€™s price and its yield. 
Investing
Immunization Inoculates Against Interest Rate Risk
Bigmoney investors can hedge against bond portfolio losses caused by rate fluctuations. 
Investing
Use Duration And Convexity To Measure Bond Risk
Find out how this measure can help fixedincome investors manage their portfolios. 
Investing
Why You Should Avoid Fixating on Bond Duration
Financial advisors and their clients should then focus on a bond fundâ€™s portfolio rather than relying on any single metric like duration. 
Investing
Understanding Interest Rates, Inflation And Bonds
Get to know the relationships that determine a bond's price and its payout. 
Financial Advisor
The Effect of Fed Fund Rate Hikes on Your Bond Portfolio
Learn how an increase in the federal funds rate may impact a bond portfolio. Read about how investors can use the duration of their portfolio to reduce risk. 
Investing
Advanced Bond Concepts
Learn the complex concepts and calculations for trading bonds including bond pricing, yield, term structure of interest rates and duration. 
Financial Advisor
Why Bondholders Should Manage Duration Risk
Bonds and bond funds are fixedincome investments, but their duration, combined with changes to interest rates, can lead to price fluctuations. 
Investing
Investing in Bonds: 5 Mistakes to Avoid in Today's Market
Investors need to understand the five mistakes involving interest rate risk, credit risk, complex bonds, markups and inflation to avoid in the bond market. 
Investing
Corporate Bond Basics: Learn to Invest
Understand the basics of corporate bonds to increase your chances of positive returns.