Hamada Equation

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Hamada Equation

Definition

The Hamada equation quantifies how financial leverage (debt) affects a firm’s market risk as measured by beta. It adjusts an unlevered beta (the beta of an otherwise identical all-equity firm) to produce a levered beta that reflects the additional systematic risk introduced by debt.

Formula

β_L = β_U [1 + (1 − T) (D/E)]

where:
* β_L = levered beta (equity beta with debt)
* β_U = unlevered beta (asset beta, without debt)
* T = corporate tax rate
* D/E = debt-to-equity ratio (use market values)

You can also solve for unlevered beta:
β_U = β_L / [1 + (1 − T) (D/E)]

How to calculate (step-by-step)

  1. Determine the firm’s market-value debt (D) and equity (E) and compute D/E.
  2. Pick the appropriate corporate tax rate (T).
  3. Compute the leverage multiplier: 1 + (1 − T) × (D/E).
  4. Multiply the unlevered beta (β_U) by that multiplier to get the levered beta (β_L).

Examples

  1. Simple example
  2. D/E = 0.60, T = 33% (0.33), β_U = 0.75
  3. β_L = 0.75 × [1 + (1 − 0.33) × 0.60] = 0.75 × [1 + 0.402] = 0.75 × 1.402 = 1.05

  4. Leverage increases beta by 0.30 (40% increase relative to β_U).

  5. Corrected illustrative example

  6. β_U = 0.82, D/E = 1.05, T = 20% (0.20)
  7. β_L = 0.82 × [1 + (1 − 0.20) × 1.05] = 0.82 × 1.84 ≈ 1.51
  8. Leverage increases beta by ≈0.69 (≈84% increase relative to β_U).

Relation to WACC and valuation

The Hamada equation is commonly used when unlevering and relevering betas in valuation and WACC calculations:
* Unlever beta to remove a firm’s current capital structure effects (use β_U in comparisons).
* Relever beta to apply a peer or target capital structure when estimating the cost of equity for WACC.

Assumptions and limitations

  • Assumes corporate taxes but ignores default risk and credit spreads; debt is treated as less risky than equity without modeling bankruptcy costs.
  • Assumes a constant capital structure and that market-value D and E are used.
  • Relies on the Modigliani–Miller framework; real-world frictions (bankruptcy costs, agency problems, non-debt tax shields) can make the adjustments imperfect.
  • Sensitive to measurement choices (e.g., using book vs. market values, choice of tax rate, and time window for beta estimation).

Practical tips

  • Use market values for debt and equity when calculating D/E.
  • Ensure the unlevered beta is appropriate for the asset base and operating risk of the firm being analyzed.
  • When comparing firms, unlever betas before averaging or using peer betas, then relever to the target capital structure.
  • Consider alternative adjustments or models if default risk or credit spreads are material.

Key takeaways

  • The Hamada equation links financial leverage to equity beta, showing how debt increases a firm’s systematic risk.
  • It is a standard tool for unlevering/relevering betas in valuation and WACC calculations.
  • Use with caution: it omits default risk and other real-world capital structure frictions and depends on careful measurement of inputs.