Chapter 21, Problem 6P

Air Tampa has just been incorporated, and its board of directors is grappling with the question of optimal capital structure. The company plans to offer commuter air services between Tampa and smaller surrounding cities. Air Tampa believes it would have the same business risk as Jaxair, which is an airline that has been around for a few years and that has had zero growth. Jaxair’s market-determined beta is 2.1, and it has a current market value debt ratio (total debt to total assets) of 50% and a federal-plus-state tax rate of 25%. Air Tampa expects to have investment tax credits when it begins business, which reduces its federal-plus-state tax rate to 20%. Air Tampa’s owners expect that the total book and market value of the firm’s stock, if it uses zero debt, would be $10 million. Air Tampa’s CFO believes that the MM and Hamada formulas for the value of a levered firm and the levered firm’s cost of capital should be used because zero growth is expected.

1. a. Estimate the beta of an unlevered firm in the commuter airline business based on Jaxair’s market-determined beta.
2. b. Now assume that r_d = r_RF = 10% and that the market risk premium RP_M = 5%. Find the required rate of return on equity for an unlevered commuter airline.
3. c. Air Tampa is considering three capital structures: (1) $2 million debt, (2) $4 million debt, and (3) $6 million debt. Estimate Air Tampa’s r_s for these debt levels.
4. d. Suppose Air Tampa’s investment tax credits expire, causing it to have a 25% federal-plus-state tax rate. Calculate Air Tampa’s r_s at $6 million debt using the new tax rate. Compare this with your corresponding answer to Part c. (Hint: The increase in the tax rate causes V_U to drop to $9.375 million.)

Summary Introduction

To calculate: Beta of unlevered firm.

Introduction: Beta represents riskiness of a company’s shares in relation with market. It is used to determine the value of the company’s shares.

Explanation of Solution

Given information:

Beta of the company (Bl): 2.1

Tax rate: 25%

Expected tax rate: 20%

Debt to total asset ratio: 50%

Company’s book value (unlevered): $10 million

Formula to calculate beta of unlevered firm:

$\begin{array}{c}\text{Bl}=\text{Bu}\left(1+\left(1-\text{T}\right)\left(\frac{\text{D}}{\text{S}}\right)\right)\\ \text{Bu}=\frac{\text{Bl}}{\left(1+\left(1-\text{T}\right)\left(\frac{\text{D}}{\text{S}}\right)\right)}\end{array}$

Substitute 2.1 for Bl, 0.25 for tax rate, 0.5 for weight of debt, and 0.5 for weight of equity in the above formula.

$\begin{array}{c}\text{Bl}=\text{Bu}\left(1+\left(1-\text{T}\right)\left(\frac{\text{D}}{\text{S}}\right)\right)\\ \text{Bu}=\frac{\text{Bl}}{\left(1+\left(1-\text{T}\right)\left(\frac{\text{D}}{\text{S}}\right)\right)}\\ =\frac{2.1}{1+\left(1-0.25\right)\left(\frac{0.5}{0.5}\right)}\\ =1.2\end{array}$

Therefore, value of unlevered beta (Bu) is 1.2.

Summary Introduction

To calculate: Required return on equity for unlevered company.

Introduction: Rate at which investors expect return from the investments made in the shares of the company is referred as expected rate of return.

Explanation of Solution

Given information:

Beta of the unlevered company (Bu): 1.2

Risk free rate (Rrf): 10%

Market risk premium (Rpm): 5%

Formula to calculate required return on equity for unlevered company:

$\text{Rsu}=\text{Rrf}+\text{Bu}\times \left(\text{Rpm}\right)$

Substitute 10% for Rrf, 5% for Rpm, and 1.2 for Bu, in the above formula.

$\begin{array}{c}\text{Rsu}=10\%+1.2\left(5\%\right)\\ =16\%\end{array}$

Therefore, required return on equity for unlevered company is 16%.

Summary Introduction

To calculate: Cost of equity levered.

Introduction: Levered Company refers that the company has debt component in its capital structure in addition to equity. Cost of equity when a company has debt component is referred as levered cost of equity.

Explanation of Solution

Given information:

Debt (D): $2 million

Risk free rate (Rrf): 10%

Expected tax rate: 20%

Company’s book value (unlevered) (Vu): $10 million

Cost of equity unlevered (Rsu): 16%

Formula to calculate cost of equity levered:

$\begin{array}{c}\text{Rsl}=\text{Rsu}+\left(\text{Rsu}-\text{Rrf}\right)\left(1-\text{T}\right)\left(\frac{\text{D}}{\text{S}}\right)\\ =\text{Rsu}+\left(\text{Rsu}-\text{Rrf}\right)\left(1-\text{T}\right)\left(\frac{\text{D}}{\text{Vl}-\text{D}}\right)\end{array}$

Substitute 16% for Rsu, 10% for Rrf, 20% for tax rate, $2 million for debt, and $10.4 million for total value in the above formula.

$\begin{array}{c}\text{Rsl}=16\%+\left(16\%-10\%\right)\left(1-20\%\right)\left(\frac{\$2\text{ million}}{\left(\$10.4\text{ million1}-\$2\text{ million}\right)}\right)\\ =17.14\%\end{array}$

Working Notes:

1.

Calculate company’s book value levered:

$\begin{array}{c}\text{Vl}=\text{Vu}+\text{Debt Tax Shield}\\ =\text{Vu}+\left(\text{Tax Rate}\times \text{Debt}\right)\\ =\$10\text{ million}+\left(0.2\times \$2\text{ million}\right)\\ =\$10.4\text{ million}\end{array}$

Summary Introduction

To calculate: Cost of equity levered.

Introduction: Levered Company refers that the company has debt component in its capital structure in addition to equity. Cost of equity when a company has debt component is referred as levered cost of equity.

Explanation of Solution

Given information:

Debt (D): $4 million

Risk free rate (Rrf): 10%

Expected tax rate: 20%

Company’s book value (unlevered) (Vu): $10 million

Cost of equity unlevered (Rsu): 16%

Formula to calculate cost of equity levered:

$\begin{array}{c}\text{Rsl}=\text{Rsu}+\left(\text{Rsu}-\text{Rrf}\right)\left(1-\text{T}\right)\left(\frac{\text{D}}{\text{S}}\right)\\ =\text{Rsu}+\left(\text{Rsu}-\text{Rrf}\right)\left(1-\text{T}\right)\left(\frac{\text{D}}{\text{Vl}-\text{D}}\right)\end{array}$

Substitute 16% for Rsu, 10% for Rrf, 20% for tax rate, $4 million for debt, and $10.8 million for total value in the above formula.

$\begin{array}{c}\text{Rsl}=16\%+\left(16\%-10\%\right)\left(1-20\%\right)\left(\frac{\$4\text{ million}}{\left(\$10.8\text{ million1}-\$4\text{ million}\right)}\right)\\ =18.82\%\end{array}$

Working Notes:

1.

Calculate company’s book value levered:

$\begin{array}{c}\text{Vl}=\text{Vu}+\text{Debt Tax Shield}\\ =\text{Vu}+\left(\text{Tax Rate}\times \text{Debt}\right)\\ =\$10\text{ million}+\left(0.2\times \$4\text{ million}\right)\\ =\$10.8\text{ million}\end{array}$

Summary Introduction

To calculate: Cost of equity levered.

Introduction: Levered Company refers that the company has debt component in its capital structure in addition to equity. Cost of equity when a company has debt component is referred as levered cost of equity.

Explanation of Solution

Given information:

Debt (D): $6 million

Expected tax rate: 20%

Risk free rate (Rrf): 10%

Company’s book value (unlevered) (Vu): $10 million

Cost of equity unlevered (Rsu): 16%

Formula to calculate cost of equity levered:

$\begin{array}{c}\text{Rsl}=\text{Rsu}+\left(\text{Rsu}-\text{Rrf}\right)\left(1-\text{T}\right)\left(\frac{\text{D}}{\text{S}}\right)\\ =\text{Rsu}+\left(\text{Rsu}-\text{Rrf}\right)\left(1-\text{T}\right)\left(\frac{\text{D}}{\text{Vl}-\text{D}}\right)\end{array}$

Substitute 16% for Rsu, 10% for Rrf, 20% for tax rate, $6 million for debt, and $11.2 million for total value in the above formula.

$\begin{array}{c}\text{Rsl}=16\%+\left(16\%-10\%\right)\left(1-20\%\right)\left(\frac{\$6\text{ million}}{\left(\$11.2\text{ million1}-\$6\text{ million}\right)}\right)\\ =21.54\%\end{array}$

Working Notes:

1.

Calculate company’s book value levered:

$\begin{array}{c}\text{Vl}=\text{Vu}+\text{Debt Tax Shield}\\ =\text{Vu}+\left(\text{Tax Rate}\times \text{Debt}\right)\\ =\$10\text{ million}+\left(0.2\times \$6\text{ million}\right)\\ =\$11.2\text{ million}\end{array}$

Summary Introduction

To calculate: Cost of equity levered.

Introduction: Levered Company refers that the company has debt component in its capital structure in addition to equity. Cost of equity when a company has debt component is referred as levered cost of equity.

Explanation of Solution

Given information:

Debt (D): $6 million

Risk free rate (Rrf): 10%

Expected tax rate (T): 25%

Company’s book value (unlevered) (Vu): $9.375 million

Cost of equity unlevered (Rsu): 16%

Formula to calculate cost of equity levered:

$\begin{array}{c}\text{Rsl}=\text{Rsu}+\left(\text{Rsu}-\text{Rrf}\right)\left(1-\text{T}\right)\left(\frac{\text{D}}{\text{S}}\right)\\ =\text{Rsu}+\left(\text{Rsu}-\text{Rrf}\right)\left(1-\text{T}\right)\left(\frac{\text{D}}{\text{Vl}-\text{D}}\right)\end{array}$

Substitute 16% for Rsu, 10% for Rrf, 25% for tax rate, $6 million for debt, and $10.875 million for total value in the above formula.

$\begin{array}{c}\text{Rsl}=16\%+\left(16\%-10\%\right)\left(1-25\%\right)\left(\frac{\$6\text{ million}}{\left(\$10.875\text{ million1}-\$6\text{ million}\right)}\right)\\ =21.54\%\end{array}$

Working Notes:

1.

Calculate company’s book value levered:

$\begin{array}{c}\text{Vl}=\text{Vu}+\text{Debt Tax Shield}\\ =\text{Vu}+\left(\text{Tax Rate}\times \text{Debt}\right)\\ =\$9.375\text{ million}+\left(0.25\times \$6\text{ million}\right)\\ =\$10.875\text{ million}\end{array}$