Question

Q5. Firm XYZ is currently financed entirely with equity that has a total market value of $100 million. Management is considering a debt-for-equity swap to add leverage to the firm's capital structure. Management recognizes two factors that would affect the value of the firm as leverage is added. First, the addition of permanent debt in the amount of ? would provide a tax shield that has a value of ??? where for firm XYZ, ??=0.34. The second, and offsetting, factor is the present value of expected costs of future financial distress, i.e., E[(CFFD)], associated with debt, PV[ E( ????????)], which increases at an accelerating rate with leverage. Management decides that the relationship of PV[ E( ????????)], to leverage can be approximated with the following equation: PV[ E( ????????)], = ??2, where ?= 0.005. Given these specifications, find the value of debt, ?⋆, that maximizes the value of the firm. What is the firm's debt ratio, and what is the market value of the firm if it has the optimal amount of debt?

Answer 1

The value of a levered firm = Value of an unlevered firm + PV of ITS - PV (Cost of financial distress)

Hence, V_L = V_U + ?_?? - ??² = 100 + 0.34D - 0.005D²

In order to maximize V_L; it's first derivative with respect to D should be zero and it's second derivative must be negative.

Recall the famous rule of differentiation: d(xⁿ) / dx = nx^n-1

If the value of debt, ?*maximizes the value of the firm then

0.34 - 2 x 0.005D* = 0

Hence, D* = 0.34 / (2 x 0.005) = 34

Hence, D* = $ 34 million is the level of debt that will maximize the value of the firm.

The market value of the firm if it has the optimal amount of debt, V_L = 100 + 0.34D* - 0.005D*² = $ 105.78

And the firm's debt ratio = D* / V_L = 34 / 105.78 = 32.14%