Which formula correctly unleverages beta?

Which formula correctly unleverages beta?

• A. BetaUnlevered = BetaLevered / (1 + (1 - tax) * D/E)
• B. BetaUnlevered = BetaLevered * (1 + (1 - tax) * D/E)
• C. BetaUnlevered = BetaLevered + (1 - tax) * D/E
• D. BetaUnlevered = BetaLevered - (1 - tax) * D/E

Answer: (A)

The idea is to remove the financial leverage effect from the levered beta so you’re left with the firm’s pure business risk. In the Hamada framework, levered beta equals unlevered beta times a factor that captures the extra risk from debt: 1 plus the tax-adjusted debt-to-equity ratio, i.e., 1 + (1 − tax) × (D/E). To isolate the unlevered beta, you divide the levered beta by that same factor. Thus Beta_unlevered = Beta_levered / [1 + (1 − tax) × (D/E)]. This keeps Beta_unlevered smaller than Beta_levered when there is debt, reflecting only business risk. Multiplying, adding, or subtracting the debt term does not correctly remove leverage, hence those forms are not appropriate.

The idea is to remove the financial leverage effect from the levered beta so you’re left with the firm’s pure business risk. In the Hamada framework, levered beta equals unlevered beta times a factor that captures the extra risk from debt: 1 plus the tax-adjusted debt-to-equity ratio, i.e., 1 + (1 − tax) × (D/E). To isolate the unlevered beta, you divide the levered beta by that same factor.

Thus Beta_unlevered = Beta_levered / [1 + (1 − tax) × (D/E)]. This keeps Beta_unlevered smaller than Beta_levered when there is debt, reflecting only business risk. Multiplying, adding, or subtracting the debt term does not correctly remove leverage, hence those forms are not appropriate.