Hamada's equation

In corporate finance, Hamada’s equation, named after Professor Robert Hamada, is used to separate the financial risk of a levered firm from its business risk. The equation combines the Modigliani-Miller theorem with the capital asset pricing model. It is used to help determine the levered beta and, through this, the optimal capital structure of firms.

Hamada’s equation relates the beta of a levered firm (a firm financed by both debt and equity) to that of its unlevered (i.e., a firm which has no debt) counterpart. It has proved useful in several areas of finance, including capital structuring, portfolio management and risk management, to name just a few. This formula is commonly taught in MBA Corporate Finance and Valuation classes. It is used to determine the cost of capital of a levered firm based on the cost of capital of comparable firms. Here, the comparable firms would be the ones having similar business risk and, thus, similar unlevered betas as the firm of interest.

The equation is^[1]

where β_L and β_U are the levered and unlevered betas, respectively, T the tax rate and φ the leverage, defined here as the ratio of debt, D, to equity, E, of the firm.

The importance of Hamada's equation is that it separates the risk of the business, reflected here by the beta of an unlevered firm, β_U, from that of its levered counterpart, β_L, which contains the financial risk of leverage. Apart from the effect of the tax rate, which is generally taken as constant, the discrepancy between the two betas can be attributed solely to how the business is financed.

The equation is often wrongly thought to hold in general. However, there are several key assumptions behind the Hamada equation:^[2]

1. The Hamada formula is based on Modigliani and Miller’s formulation of the tax shield values for constant debt, i.e. when the dollar amount of debt is constant over time. The formulas are not correct if the firm follows a constant leverage policy, i.e. the firm rebalances its capital structure so that debt capital remains at a constant percentage of equity capital, which is a more common and realistic assumption than a fixed dollar debt (Brealey, Myers, Allen, 2010). If the firm is assumed to rebalance its debt-to-equity ratio continuously, the Hamada equation is replaced with the Harris-Pringle equation; if the firm rebalances only periodically, such as once a year, the Miles-Ezzell equation is the one to be used.

2. The beta of debt β_D equals zero. This is the case if debt capital has negligible risk that interest and principal payments will not be made when owed. The timely interest payments imply that tax deductions on the interest expense will also be realized--in the period in which the interest is paid.

3. The discount rate used to calculate the tax shield is assumed to be equal to the cost of debt capital (thus, the tax shield has the same risk as debt). This and the constant debt assumption in (1) imply that the tax shield is proportionate to the market value of debt: Tax Shield = T×D.

References

• Brealey, R., Myers, S., and Allen, F. (2010) "Principles of Corporate Finance," McGraw-Hill, New York, NY, 10th edition, ch. 19, pp. 485-486.
• Cohen, R.D. (2007) "Incorporating Default Risk Into Hamada's Equation for Application to Capital Structure," Wilmott Magazine (download paper)
• Conine, T.E. and Tamarkin, M. (1985) "Divisional Cost of Capital Estimation: Adjusting for Leverage,” Financial Management 14, Spring issue, p.54.
• Harris, R. S. and Pringle, J. J. (1985) "Risk-Adjusted Discount Rates—Extensions from the Average Risk Case," Journal of Financial Research, (Fall 1985): 237–244.
• Miles, J. and Ezzell, J. (1980) "The Weighted Average Cost of Capital, Perfect Capital Markets and Project Life: a Clarification". Journal of Financial and Quantitative Analysis 15: 719–730.