A Firm Has A Levered Beta Of 1.60 And Its Debt To Equity Rat

A Firm Has A Levered Beta Of 160 And Its Debt To Equity Ratio Is

1. A firm has a levered beta of 1.60, and its debt to equity ratio is 2.0. What would the company be if it used no debt, i.e., what is its unlevered beta if the corporate tax rate is 20%?

2. There are two firms: Firm U and Firm L. Both firms have $100M total assets and $30M EBIT (earnings before interest and taxes). Firm U is an unleveraged firm without debt. Firm L is a leveraged firm with 50% of debt and 50% of common equity. The pre-tax cost of debt for Firm L is 10%. Both firms have 20% corporate tax rate. Calculate the return on equity (ROE) for the unleveraged firm U. Based on the information from Question 2, calculate the return on equity (ROE) for the leveraged firm L.

Paper For Above instruction

The financial leverage of a firm significantly influences its risk profile and, consequently, its beta. Understanding the relationship between leveraged beta and unlevered beta helps investors and managers assess the inherent business risk and the impact of capital structure decisions. This paper explores the calculation of unlevered beta from leveraged beta in the context of corporate tax considerations and examines how leveraging affects the return on equity (ROE) for different firm structures.

1. Calculating Unlevered Beta from Levered Beta

The levered beta (βL) reflects the risk of a firm considering its debt-equity structure, while the unlevered beta (βU) isolates the firm's core business risk, excluding financial leverage. The relationship between these two betas, considering corporate taxes, is given by the formula:

βU = βL / [1 + (1 - T) * D/E]

where T is the corporate tax rate, D/E is the debt-to-equity ratio, βL is the levered beta, and βU is the unlevered beta.

Given the values: βL = 1.60, D/E = 2.0, and T = 20%, substituting into the formula:

βU = 1.60 / [1 + (1 - 0.20)  2.0] = 1.60 / [1 + 0.80  2.0] = 1.60 / [1 + 1.6] = 1.60 / 2.6 ≈ 0.6154

Hence, the unlevered beta of the company is approximately 0.6154, indicating the firm's inherent business risk absent financial leverage.

2. Analyzing Return on Equity in Different Capital Structures

The two firms, U and L, offer a clear comparison between unleveraged and leveraged capital structures, with distinct impacts on their ROE. Both firms share identical assets and EBIT but differ in leverage, which amplifies the risk and return to equity holders.

Return on Equity for Firm U (Unleveraged)

Firm U's ROE can be computed directly from its EBIT, tax rate, and equity base since it has no debt. The calculation proceeds as follows:

Equity equals total assets for an unleveraged firm, so:

Equity = $100 million


EBIT = $30 million


Tax rate = 20%


Interest expense = $0 (no debt)

The net income after tax is:

Net income = EBIT  (1 - T) = $30 million  (1 - 0.20) = $24 million

Thus, the return on equity for Firm U is:

ROE_U = Net income / Equity = $24 million / $100 million = 0.24 or 24%

Return on Equity for Firm L (Leveraged)

Firm L's leverage introduces additional risk and amplifies ROE. The pre-tax cost of debt is 10%, and the debt equals 50% of assets, or $50 million:

Debt = $50 million


Interest expense = 10% of $50 million = $5 million


EBIT = $30 million


Earnings before taxes (EBT) = EBIT - interest = $30 million - $5 million = $25 million


Tax remains 20%, so:


Net income = $25 million * (1 - 0.20) = $20 million

The equity of Firm L is also 50% of assets, which is $50 million. The ROE for Firm L can be calculated as:

ROE_L = Net income / Equity = $20 million / $50 million = 0.40 or 40%

This higher ROE illustrates the impact of financial leverage, which increases the risk and return to equity shareholders. While ROE is higher in Firm L, it also bears higher financial risk owing to debt obligations, especially if EBIT declines or interest rates rise.

Conclusion

The analysis clearly demonstrates that leverage influences both the risk profile and the return metrics of firms. Calculating unlevered beta provides insights into intrinsic risk devoid of capital structure effects, facilitating better investment and managerial decisions. Furthermore, leveraging amplifies ROE, as seen in Firm L, but with increased financial risk. Strategic capital structure decisions must carefully balance these considerations to optimize shareholder value while managing risk exposure.

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