For simplicity, assume that the BIPOs have a promised perpetual interest payment M per period, and that this payment is such that the firm can not default at t = 1.
bg~ |is less than or equal to~ Z |is less than or equal to~ T, the value of a firm that issues BIPOs is directly related to Z.
Consequently, the value of the firm which issues BIPOs with Z = |T.
Note that the amount for which the BIPOs are sold back to the company exceeds the refunding cost of repurchasable bonds.
Using Equations (B1) and (B2) of Appendix B, we calculate the values of the debt, the equity, and the expected refunding premium of a firm that issues BIPOs with M = 500 and Z = T.
Given the assumptions that lead to the results in Corollaries 1 and 2, and assuming that issuing BIPOs is equally as effective as issuing callable debt in mitigating agency costs, a formal proof is as follows.
We further demonstrate that, from a tax point of view, issuing optimally designed BIPOs dominates issuing repurchasable debt.
Note that although our results are stated in terms of a tax advantage of issuing BIPOs over issuing callable debt, an alternative implication is that new motivations for issuing callable debt should be explored.
However, if interest rates vary after t = 1, then we would expect that our model understates the benefits of issuing BIPOs for the same reasons as outlined in footnote 8.