Final Answer:
The value of the firm after borrowing $2,500,000 using the trade-off model would increase by the present value of interest tax shields (PVTS) derived from the tax benefit of debt. The exact calculation would require considering the tax shield effect on the firm's value.
Step-by-step explanation:
In the trade-off theory of capital structure, firms balance the benefits of debt interest tax shields against the costs of financial distress and agency conflicts. When Progressive borrows $2,500,000, it introduces tax shields by deducting interest payments from taxable income. This results in a tax benefit due to the shield provided by the interest expense.
The formula to determine the present value of interest tax shields (PVTS) involves multiplying the amount of debt borrowed by the corporate tax rate (Tc). The equation is PVTS = Debt * Tc.
For instance, if the corporate tax rate (Tc) is 30%, the present value of interest tax shields from borrowing $2,500,000 would be calculated as follows:
PVTS = $2,500,000 * 0.30 = $750,000.
The value of the firm would increase by the present value of the interest tax shields, which in this case, is $750,000. However, the trade-off model doesn't solely focus on the benefits of tax shields but also accounts for the costs associated with financial distress and agency problems.
Hence, while borrowing adds value through tax shields, it also introduces risks related to financial distress costs, potential agency conflicts, and changes in the firm's risk profile, which must be considered when evaluating the overall impact on the firm's value.