OPTIMAL INVESTMENT STRATEGY VIA INTERVAL ARITHMETIC
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.
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