We will solve this step by step using the Weighted Average Cost of Capital (WACC) formula and Modigliani-Miller Proposition II (without taxes).

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(a) Find the company's cost of equity (Re) when there is no debt:

Since the company currently has no debt (D = 0), its WACC is simply equal to its cost of equity:

$WACC = Re$

$13\% = Re$

So, the cost of equity (Re) is 13%.

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(b) Find the cost of equity if the firm converts to 25% debt (D/V = 0.25)

Using Modigliani-Miller Proposition II (no taxes):

$Re = R_A + (R_A - R_D) \times \frac{D}{E}$

where:

• $R_A$ = Unlevered Cost of Equity (which is 13%)
• $R_D$ = Cost of debt (7.25%)
• $D/E$ = Debt-to-Equity ratio
  • Given $D/V = 0.25$, we find $E/V = 0.75$, so:
  • $D/E = \frac{0.25}{0.75} = 1/3$

Now, plug values into the equation:

$Re = 13\% + (13\% - 7.25\%) \times \frac{1}{3}$

$Re = 13\% + (5.75\%) \times \frac{1}{3}$

$Re = 13\% + 1.9167\%$

$Re = 14.92\%$

Cost of equity with 25% debt = 14.92%.

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(c) Find the cost of equity if the firm converts to 55% debt (D/V = 0.55)

• Given $D/V = 0.55$, we find $E/V = 0.45$, so:
  • $D/E = \frac{0.55}{0.45} = 11/9$

Using the same formula:

$Re = 13\% + (13\% - 7.25\%) \times \frac{11}{9}$

$Re = 13\% + (5.75\%) \times \frac{11}{9}$

$Re = 13\% + 7.0278\%$

$Re = 20.03\%$

Cost of equity with 55% debt = 20.03%.

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(d) Find WACC in parts (b) and (c)

WACC formula: $WACC = \left( \frac{E}{V} \times Re \right) + \left( \frac{D}{V} \times Rd \right)$

For (b) when $D/V = 0.25$:

$WACC = (0.75 \times 14.92\%) + (0.25 \times 7.25\%)$

$WACC = 11.19\% + 1.81\%$

$WACC = 13\%$

Since there are no corporate taxes, the WACC remains unchanged at 13%.

For (c) when $D/V = 0.55$:

$WACC = (0.45 \times 20.03\%) + (0.55 \times 7.25\%)$

$WACC = 9.01\% + 3.99\%$

$WACC = 13\%$

Again, the WACC remains unchanged at 13%.

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Final Answers

Scenario	Cost of Equity (Re)	WACC
(a) No Debt	13%	13%
(b) 25% Debt	14.92%	13%
(c) 55% Debt	20.03%	13%

Would you like a more detailed breakdown of any step?

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Related Questions

1. How does Modigliani-Miller Proposition II explain the impact of leverage on the cost of equity?
2. Why does WACC remain the same despite changes in debt levels?
3. How would corporate taxes change the results in this scenario?
4. What happens to the cost of equity if the firm moves to 75% debt?
5. How does financial risk increase with higher leverage?

Tip:

In a no-tax environment, increasing debt raises the cost of equity, but WACC remains constant. However, when corporate taxes are introduced, debt creates a tax shield, lowering WACC.