Durations of Non-Default-Free Securities

A bond subject to default risk trades at a higher interest rate than a comparable default-free bond in order to compensate investors for expected loss resulting from reduced or delayed promised payments; the difference between the yields to maturity on the two bonds is called the default yield premium. Bonds with the same default yield premium, however, may differ in regard to the timing of their reductions in payment. Each time pattern of cash payments translates into a different duration value for a given stochastic process of default-free interest rates. Failure to take this time pattern into account when computing durations of non-default-free bonds will result in misspecification of risk exposure. For instance, the duration of a bond that defaults on the first coupon will differ from the duration of a bond that defaults on the last payment, even if both bonds have the same default yield premium and initial maturity. Early defaulters will have durations that are consistently longer than their simple, unadjusted macaulay durations, while later defaulters will have consistently shorter durations. Furthermore, the error from not taking the stochastic process of default into account will be larger, the greater the default yield premium. More common than outright default is the situation where the issuer delays coupon and principal payments but eventually pays them in full some time after default. All default-adjusted durations for these bonds will exceed their corresponding unadjusted durations, with the size of the adjustment increasing with term to maturity and exceeding the length of the delay in payments.