Callable bonds duration

For instance, consider a hypothetical case in which a callable bond has a price equal to 103.78. Suppose that the benchmark yield curve changes by 1 basis point. For a downward parallel shift, the price obtained is 103.83, while for an upward shift the price is 103.73. Therefore, applying Formula (11.1) the bond s effective duration is around 4.93. 

The convexity measure increases with the square of maturity it decreases as both coupon and yield rise. It is a function of modified duration, so index-linked bonds, which have greater duration than conventional bonds of similar maturities, also have greater convexity. For a conventional vanilla bond, convexity is almost always positive. Negative convexity occurs most frequently with callable bonds. 

Effective duration recognizes that yield changes may effect the future cash flow of a bond and so its price. For bonds with embedded options the difference between traditional duration and effective duration can be significant. The effective duration of a callable bond, for example, is sometimes half its traditional duration. As noted in chapter l4, for mortgage-backed securities, the difference is sometimes greater still. 

The modified duration and convexity methods we have described are only suitable for use in the analysis of conventional fixed-income instruments with known fixed cash flows and maturity dates. They are not satisfactory for use with bonds that contain embedded options such as callable bonds or instruments with unknown final redemption dates such as mortgage-backed bonds. For these and other bonds that exhibit uncertainties in their cash flow pattern and redemption date, so-called option-adjusted measures are used. The most common of these is option-adjusted spread (OAS) and option-adjusted duration (OAD). The techniques were developed to allow for the uncertain cash flow structure of non-vanilla fixed-income instruments, and model the effect of the option element of such bonds. 

To calculate the modified duration of a bond with an embedded option, the bondholder must assume a fixed maturity date based on the bond s current price. When it is unclear what redemption date to use, modified duration may be calculated to both the first call date and the final maturity date. This is an unsatisfactory compromise, however, since neither date, and so neither measure, may be appropriate. The problem is more acute for bonds that are continuously callable or putable from the first call or put date until maturity. 

Option-adjusted spread analysis uses simulated interest rate paths as part of its calculation of bond yield and convexity. Therefore an OAS model is a stochastic model. The OAS refers to the yield spread between a callable or mortgage-backed bond and a government benchmark bond. The government bond chosen ideally will have similar coupon and duration values. 

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