This paper studies the optimal replacement policy of an item that experiences stochastic geometric growth in maintenance costs. The model integrates corporate taxes, tax credits, depreciation, and salvage value. We extend this traditional application to cover the cost of replacement with the payout from two bonds. The two-bond portfolio is passively immunized. The intersections between the continuation and replacement boundaries are computed using the Interval-Newton Generalized-Bisection (IN/GB) method. We allow small fluctuations of the replacement boundary. With these fluctuations, multiple intersections of the two boundaries are determined. The IN/GB method finds all these intersections without the need for initial guesses of the problem variables. This is a major computational improvement over traditional single-root finding implementations that require multiple initial guesses and provide no guarantees of existence or uniqueness. We demonstrate that without fluctuations one would expect to find a single optimal replacement time. However with fluctuations, there are several intersections of the continuation and replacement boundaries and the bond weight fractions may change by more than 200% between intersection points. These large changes in portfolio wealth allocation highlight the fragility of the idealized solution in the realm of fluctuations in replacement costs.
The risk free rate on bonds is a very important quantity that allows calculation of premium values on bonds. This quantity of stochastic nature has been modeled with different degrees of sophistication. This paper reviews the major models utilized in the estimation of the risk free rate and gives an example of the behavior generated by one of these models.Keywords: Interest Rates, Term Structure, Stock Dynamics.
ResumenLa tasa de interés de un Banco Estatal para sus bonos comerciales es una cantidad muy importante porque permite el cálculo del interés adicional proveído por un bono comercial. Esa tasa de interés estatal, que no tiene riesgo para el inversionista, ha sido modelada con diferentes niveles de sofistificación. Este artículo compila y compara los principales modelos utilizados en la estimación de esa tasa libre de riesgo y da un ejemplo del comportamiento generado por uno de esos modelos.
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