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So i have this question on leverage and cost of capital:

Leverage and the cost of capital Gamma Airlines has an asset beta of 1.5. The risk-free interest rate is 6%, and the market risk premium is 8%. Assume the capital asset pricing model is correct. Gamma pays taxes at a marginal rate of 35%. Draw a graph plotting Gam- ma’s cost of equity and after-tax WACC as a function of its debt-to-equity ratio D/E, from no debt to D/E = 1.0. Assume that Gamma’s debt is risk-free up to D/E = .25. Then the interest rate increases to 6.5% at D/E = .5, 7% at D/E = .8, and 8% at D/E = 1.0. As in Problem 21, you can assume that the firm’s overall beta (βA) is not affected by its capital structure or the taxes saved because debt interest is tax-deductible.

The answer sheet is this ( only a part of the table is shown)

..enter image description here

I understand how they got to the RA and how it doesn't change according to the capital structure.

I also know how to calculate D/V from D/E.

The problem is that i dont seem to understand which formula I should use to calculate RE.

If , for example, i use this formula

RE=RA+(RA-RD)D/E for D/E equal to 0.10 i get

RE=0.180+(0.180-0.06) · 0.10 = 0.192 which is not 0.1941.

Then which formula should i use ?

R=rf+B(rm-rf)?

Why do i need to calculate DV?

Somebody please explain how i can get accurate results of RE using which formula? Thanks i greatly appreciate it.

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Your formula is wrong. The derivation for the cost of equity $r_E$ is actually:

$r_A = r_D(1-t)\frac{D}{V}+r_E \frac{E}{V}\\ r_E \frac{E}{V} = r_A - r_D(1-t)\frac{D}{V}\\ r_E = r_A \frac{V}{E} - r_D(1-t)\frac{D}{V} \times\frac{V}{E}\\ \mathbf{r_E = r_A \frac{V}{E} - r_D(1-t)\frac{D}{E}}$

Since $\frac{D}{V}=\frac{D}{D+E}=0.09091$ when $r_A=0.1941$, then $E=10$ which means $D=0.10E=1$, making $V=D+E=11$ for use in the above.

$r_E = 0.180 \times\frac{11}{10}-0.06(1-0.35)\times 0.10 = 0.1941$

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