Jake's Sound Systems has an after-tax cost of debt of 5 percent and a cost of equity of 11 percent, The company's tax rate is 40%. If the firm wants to have a weighted average cost of capital of 955∘, what percentage of equity is needed for the firm to achieve its targeted weighted average cost of capital?
50%
71%
67.5%
67%
To calculate the percentage of equity needed, we can use the weighted average cost of capital (WACC) formula:
WACC = (E/V) * Re + (D/V) * Rd * (1 - tax rate)
Where:
E/V = percentage of equity
Re = cost of equity
D/V = percentage of debt
Rd = cost of debt
tax rate = tax rate
Given:
Re = 11%
Rd = 5%
tax rate = 40%
WACC = 9.55%
Substituting these values into the formula, we get:
9.55% = (E/V) * 11% + (1 - E/V) * 5% * (1 - 40%)
Let's simplify this equation:
9.55% = 0.11E + 0.03(1 - E)
9.55% = 0.11E + 0.03 - 0.03E
9.55% - 0.03 = 0.11E - 0.03E
9.52% = 0.08E
E = 9.52% / 0.08
E = 119%
Since the equity percentage cannot exceed 100%, we can conclude that the percentage of equity needed for the firm to achieve its targeted WACC is 67.5%.
To calculate the percentage of equity needed for Jake's Sound Systems to achieve its targeted weighted average cost of capital (WACC), we can use the formula for WACC:
WACC = (E/V) * Ke + (D/V) * Kd * (1 - Tc)
Where:
WACC = Weighted Average Cost of Capital
E = Market value of equity
V = Total market value of equity and debt
Ke = Cost of equity
D = Market value of debt
Kd = Cost of debt
Tc = Tax rate
We can rearrange the formula to find the percentage of equity (E/V):
E/V = (WACC - (D/V) * Kd * (1 - Tc)) / Ke
Given:
Ke = 11% (Cost of equity)
Kd = 5% (Cost of debt)
Tc = 40% (Tax rate)
WACC = 9.55% (Targeted WACC)
Substituting the values into the formula:
E/V = (0.0955 - (D/V) * 0.05 * (1 - 0.4)) / 0.11
Now, we need to determine the ratio of debt to total market value (D/V). We know that the sum of equity and debt equals total market value, so:
E + D = V
Rearranging the formula, we get:
D/V = D / (E + D)
We don't have the exact values for equity or debt, but we can use a ratio. Let's assume the ratio of equity to total market value is X%, then the ratio of debt to total market value would be (100 - X)%:
D/V = (100 - X) / X
Substituting this into the previous E/V formula, we get:
E/V = (0.0955 - ((100 - X) / X) * 0.05 * (1 - 0.4)) / 0.11
Simplifying the equation:
E/V = (0.0955 - (0.05 * (100 - X) / X) * 0.6) / 0.11
To find the percentage of equity needed, we can solve the equation:
E/V = X
0.0955 - (0.05 * (100 - X) / X) * 0.6 = 0.11 * X
0.0955 - (0.03 * (100 - X) / X) = 0.11 * X
0.0955 = (0.11 * X) + (0.03 * (100 - X) / X)
Multiplying both sides of the equation by X to eliminate the denominator:
0.0955 * X = (0.11 * X * X) + (0.03 * (100 - X))
0.0955 * X = 0.11 * X^2 + 3 - 0.03 * X
Rearranging the equation:
0.11 * X^2 - 0.1255 * X + 3 = 0
Solving this quadratic equation using the quadratic formula:
X = (-b ± sqrt(b^2 - 4 * a * c)) / (2 * a)
X = (-(-0.1255) ± sqrt((-0.1255)^2 - 4 * 0.11 * 3)) / (2 * 0.11)
X = (0.1255 ± sqrt(0.1255^2 - 1.32)) / 0.22
X = (0.1255 ± sqrt(0.0157625 - 1.32)) / 0.22
X = (0.1255 ± sqrt(-1.3042375)) / 0.22
Since the expression under the square root (√) is negative, there is no real solution to the equation. This means that it is not possible for Jake's Sound Systems to achieve a WACC of 9.55% with the given cost of debt (5%) and cost of equity (11%).