The price-yield relationship of a non-callable or a non-putable bond is convex because price and yield are inversely proportional. Figure 1 shows the price-yield relationship of a bond when it is callable and non-callable. The non-convex nature of a callable bond can be explained as follows:
Note: y* yield at which bond becomes callable.
From the definition of the callable bond, we know that a bond becomes callable only when the prevailing market yield is less than the coupon rate on comparable bonds. The price-yield curve of a bond is unaffected till the time it becomes callable. Investors may not be willing to pay the same price for a bond even when the coupon rate on the bond is just lower than the market yield as what would have been its price if it were a callable bond, for, there is a possibility of a further drop in the market yield and the issuer may call the bond. As yields decline, there is an increasing possibility of the issuer calling the bond. Though the exact yield level at which the bond would be called may not be known, the existence of such a level is certain. In Figure 1, for yield levels below y*, the price-yield relationship of a callable bond differs from that of a non-callable bond. For instance, the market yield is such that a bond would be selling for 107. But, if it is a callable bond then it might be called at 105 and hence the investors will not be willing to pay 107. Even if the investors purchase at 107 and if the bond is called, then they get only 105 and hence there is a loss of 2 units per bond. For a range of yields below y*, there is price compression. Here, by price compression we mean that as yields decline price appreciation is limited. This portion of the price-yield relationship of a callable bond below y* is termed negatively convex the reason being-increase in yields by a given number of basis points will result in a greater price decline compared to the price appreciation if yields decline by the same number of basis points.