The market value of any cash flow stream, including bonds or bond portfolios, can be approximated by the sum of its cash flows, discounted at a given internal rate of return, or yield. In comparing cash flows, however, or in creating an asset-liability match, one must consider not only the market values of the cash flows, but also their price behavior over time, as yields change. Duration measures the percentage change in the market value of a cash flow for a given change in yield and comes close to true value when rate changes are small. With larger changes in rates, however, it is necessary to consider convexity—the curvature of the price-yield relationship. Positive convexity implies that price increases at a faster rate as yields drop than it decreases as rates rise.
The price-yield relationship measures bond return when the cash flow stream is known and certain. For a callable bond, where neither maturity nor cash flow is certain, the best way to estimate value is to consider the bond as a non-callable security with an imbedded short option—the option owned by the issuer to repurchase (call) the bond at a predetermined price and time. Using option-valuation techniques to value this option, one can derive an option–adjusted yield, maturity, duration and convexity for the callable bond.
The duration of the callable bond will be lower than the duration of the bond to maturity, but higher than the duration to call. The yield to the “synthetic” maturity date implied by this duration is a better indication of return of the callable bond than either the yield to call or the yield to maturity, because it takes into account the value of the call option and the bond’s market volatility. The convexity of the callable bond will never be greater than that of a comparable non-callable bond and may be negative, reflecting the slowing down of price appreciation as the price of the callable bond approaches the strike price of the option.