# The Principle of Convexity between Bond Price and Yield

For investors, convexity is a tool used to help measure the amount of risk to which a bond portfolio is exposed. Specifically, the concept of convexity helps investors holding bonds determine the effect interest rates have on bonds over their lifetime. When graphing the relationship between price and yield (the return on investment) for any bond on an X-Y axis, an investor will get a convex (or curved) line, rather than one that is linear (or straight). The amount of curve or convexity on the graph indicates the change in yield, based on changes in price for the bond.

Using Convexity to Compare Bonds

In graph form, duration would be portrayed as a straight line (compared with convexity, which is curved). When comparing two bonds with the same duration, an investor can measure their differences by examining the degree of convexity (or curvature) for the two bonds. Changes in interest rates and how they affect the bonds can be seen in the convexity of the lines. For example, a bond with more curvature or convexity will be impacted less by interest rates. Similarly, those bonds with greater convexity will have a higher price, no matter whether interest rates fall or rise.

Factors Impacting Convexity

Different convexities result from different bond types. A simple bond, for example, has a positive convexity. The price-yield relationship (graphed out as a curve) will increase as yield lessens. Similarly, as market-yield lessens, the duration increases.

Callable bonds, on the other hand, will show negative convexity at certain price-yield points. That is, as market yields lessen, duration decreases too. It is important to consider that modified duration can be employed to correctly approximate a bond price when there is little chance that the bond will be called. It would be wise for investors to consider that, for those calling the bonds, the optimal time to call the bond is when interest rates have subsided, dipping below the coupon rate. Knowing that principle can help the investor choose the best time to move.

For coupon-bonds, zero-coupon bonds (those bonds that pay no interest, but instead are sold at a deep discount and have value only when reaching maturity) have the greatest convexity. In general, the higher the coupon rate, the more slight the convexity curve will be.

Applying Convexity’s Cues

Like other indicators, convexity is but one tool in the investor’s kit that can be used to make more informed decisions about bond holdings. Convexity offers more complex insight into the movement of bonds, offering some clues as to how duration affects bond capacity, as well as the consequences of interest rate changes on bond prices.