To find the firm's after-tax cost of equity, we first need to clarify that the concept of "after-tax cost of equity" is not standard as equity does not incur a tax in the same way debt does. Instead, we typically refer to the firm's overall cost of capital, which combines the cost of equity and the after-tax cost of debt.
However, based on your question, it seems you're looking for the impact of the firm's debt on its overall cost of capital.
To calculate the firm's overall weighted average cost of capital (WACC), we use the following formula:
\[ \text{WACC} = \left( E/V \times r_e \right) + \left( D/V \times r_d \times (1 - T) \right) \]
Where:
- \(E\) = market value of equity
- \(D\) = market value of debt
- \(V\) = total market value of financing (equity + debt)
- \(r_e\) = cost of equity
- \(r_d\) = cost of debt
- \(T\) = tax rate
Given data:
- \(E/V = 0.75\) (equity proportion)
- \(r_e = 0.13\) (cost of equity)
- \(D/V = 0.25\) (debt proportion, calculated as \(1 - 0.75\))
- \(r_d = 0.10\) (pre-tax cost of debt)
- \(T = 0.28\) (tax rate)
Now, let's calculate the after-tax cost of debt:
\[ \text{After-tax Cost of Debt} = r_d \times (1 - T) = 0.10 \times (1 - 0.28) = 0.10 \times 0.72 = 0.072 \]
Now we can substitute back into the WACC formula:
\[ \text{WACC} = \left(0.75 \times 0.13\right) + \left(0.25 \times 0.072\right) \] \[ \text{WACC} = 0.0975 + 0.018 \] \[ \text{WACC} = 0.1155 \text{ or } 11.55% \]
Thus, while we cannot specifically calculate an "after-tax cost of equity," the weighted average cost of capital, which includes both equity and after-tax debt, is approximately 11.55%.