Capital Structure: Basic Concepts

...

rB = the pre-tax cost of a firm’s debt

rS = the cost of a firm’s equity.

In this problem: B = $10,000,000

S = $20,000,000

rB = 14%

The Capital Asset Pricing Model (CAPM) must be used to calculate the cost of Acetate’s equity (rS)

According to the CAPM: rS = rf + βS{E(rm) – rf}

where rf = the risk-free rate of interest

E(rm) = the expected rate of return on the market portfolio

βS = the beta of a firm’s equity

In this problem: rf = 8%

E(rm) = 18%

βS = 0.9

Therefore, the cost of Acetate’s equity is:

rS = rf + βS{E(rm) – rf}

= 0.08 + 0.9( 0.18 – 0.08)

= 0.17

The cost of Acetate’s equity (rS) is 17%.

Acetate’s weighted average cost of capital equals:

rwacc = {B / (B+S)} rB + {S / (B+S)}rS

= ($10 million / $30 million)(0.14) + ($20 million / $30 million)(0.17)

= (1/3)(0.14) + (2/3)(0.17)

= 0.16

Therefore, Acetate’s weighted average cost of capital is 16%.

- According to Modigliani-Miller Proposition II (No Taxes):

rS = r0 + (B/S)(r0 – rB)

where r0 = the cost of capital for an all-equity firm

rS = the cost of equity for a levered firm

rB = the pre-tax cost of debt

In this problem: rS = 0.17

rB = 0.14

B = $10,000,000

S = $20,000,000

Thus: 0.17 = r0 + (1/2)(r0 – 0.14)

Solving for r0: r0 = 0.16

Therefore, the cost of capital for an otherwise identical all-equity firm is 16%.

This is consistent with Modigliani-Miller’s proposition that, in the absence of taxes, the cost of capital for an all-equity firm is equal to the weighted average cost of capital of an otherwise identical levered firm.

15.3 Since Unlevered is an all-equity firm, its value is equal to the market value of its outstanding

shares. Unlevered has 10 million shares of common stock outstanding, worth $80 per share.

Therefore, the value of Unlevered is $800 million (= 10 million shares * $80 per share).

Modigliani-Miller Proposition I states that, in the absence of taxes, the value of a levered firm equals the value of an otherwise identical unlevered firm. Since Levered is identical to Unlevered in every way except its capital structure and neither firm pays taxes, the value of the two firms should be equal.

Modigliani-Miller Proposition I (No Taxes): VL =VU

Therefore, the market value of Levered, Inc., should be $800 million also.

Since Levered has 4.5 million outstanding shares, worth $100 per share, the market value of Levered’s equity is $450 million. The market value of Levered’s debt is $275 million.

The value of a levered firm equals the market value of its debt plus the market value of its equity.

Therefore, the current market value of Levered, Inc. is:

VL = B + S

= $275 million + $450 million

= $725 million

The market value of Levered’s equity needs to be $525 million, $75 million higher than its current market value of $450 million, for MM Proposition I to hold.

Since Levered’s market value is less than Unlevered’s market value, Levered is relatively underpriced and an investor should buy shares of the firm’s stock.

15.4 a. Since the market value of Knight’s equity is $1,714,000, 5% of the firm’s equity costs $85,700 (= 0.05 * $1,714,000).

Since the market value of Veblen’s equity is $2,400,000, 5% of the firm’s equity costs $120,000 (= 0.05 * $2,400,000). In order to compare dollar returns, the initial net cost of both positions should be the same. Therefore, the investor will borrow $34,300 (= $120,000 - $87,500) at 6% per annum when purchasing $120,000 of Veblen’s equity for a net cost of $85,700 (= $120,000 - $34,300).

An investor who owns 5% of Knight’s equity will be entitled to 5% of the firm’s earnings available to common stock holders at the end of each year. While Knight’s expected operating income is $300,000, it must pay $60,000 to debt holders before distributing any of its earnings to stockholders. Knight’s expected earnings available to stockholders is $240,000 (= $300,000 -$60,000).

Therefore, an investor who owns 5% of Knight’s stock expects to receive a dollar return of $12,000 (= 0.05 * $240,000) at the end of each year based on an initial net cost of $85,700.

An investor who owns 5% of Veblen’s equity will be entitled to 5% of the firm’s earnings at the end of each year. Since Veblen is an all-equity firm, it owes none of its money to debt holders and can distribute all $300,000 of its earnings to stockholders. An investor who owns 5% of Veblen’s equity will expect to receive a dollar return of $15,000 at the end of each year. However, since this investor borrowed $34,300 at 6% per annum in order to fund his equity purchase, he owes $2,058 (= 0.06 * $34,300) in interest payments at the end of each year. This reduces his expected net dollar return to $12,942 (= $15,000 - $2,058).

Therefore, an investor who borrows $34,300