Hamada Equation

What Is the Hamada Equation?

The Hamada equation is a fundamental analysis method of analyzing a firm's cost of capital as it uses additional financial leverage, and how that relates to the overall riskiness of the firm. The measure is used to summarize the effects this type of leverage has on a firm's cost of capital — over and above the cost of capital as if the firm had no debt.

How the Hamada Equation Works

Robert Hamada is a former professor of finance at the University of Chicago Booth School of Business. Hamada started teaching at the university in 1966 and served as the dean of the business school from 1993 to 2001. His equation appeared in his paper, "The Effect of the Firm's Capital Structure on the Systemic Risk of Common Stocks" in the Journal of Finance in May 1972.

The formula for the Hamada equation is:

β L = β U [ 1 + ( 1 − T ) ( D E ) ] where: β L = Levered beta β U = Unlevered beta* T = Tax rate D / E = Debt to equity ratio* \begin{aligned} &\beta_L = \beta_U \left [ 1 + ( 1 - T) \left ( \frac{ D }{ E } \right ) \right ]\\ &\textbf{where:} \\ &\beta_L = \text{Levered beta} \\ &\beta_U = \text{Unlevered beta*} \\ &T = \text{Tax rate} \\ &D/E = \text{Debt to equity ratio*} \\ \end{aligned} βL=βU[1+(1−T)(ED)]where:βL=Levered betaβU=Unlevered beta*T=Tax rateD/E=Debt to equity ratio*

* Unlevered beta is the market risk of a company without the impact of debt.

* Debt-to-equity ratio is a measure of a company's financial leverage.

How to Calculate the Hamada Equation

The Hamada equation is calculated by:

1. Dividing the company’s debt by its equity.
2. Finding one less the tax rate.
3. Multiplying the result from no. 1 and no. 2 and adding one.
4. Taking the unlevered beta and multiplying it by the result from no. 3.

What Does the Hamada Equation Tell You?

The equation draws upon the Modigliani-Miller theorem on capital structure and extends an analysis to quantify the effect of financial leverage on a firm. Beta is a measure of volatility or systemic risk relative to the overall market. The Hamada equation, then, shows how the beta of a firm changes with leverage. The higher the beta coefficient, the higher the risk associated with the firm.

Example of the Hamada Equation

A firm has a debt-to-equity ratio of 0.60, a tax rate of 33%, and an unlevered beta of 0.75. The Hamada coefficient would be 0.75 [1 + (1 - 0.33)(0.60)], or 1.05. This means that financial leverage for this firm increases the overall risk by a beta amount of 0.30, which is 1.05 less 0.75 or 40% (0.3 / 0.75).

Or consider retailer Target (NYSE: TGT), which has a current unlevered beta of 0.82. Its debt-to-equity ratio is 1.05 and the effective annual tax rate is 20%. Thus, the Hamada coefficient is 0.99, or 0.82 [1 + (1 - 0.2) (0.26)]. Thus, leverage for a firm increases the beta amount by 0.17, or 21%.

The Difference Between Hamada Equation and Weighted Average Cost of Capital (WACC)

The Hamada equation is part of the weighted average cost of capital (WACC). The WACC involves unlevering the beta to relever it to find an ideal capital structure. The act of relevering the beta is the Hamada equation.

Limitations of Using the Hamada Equation

The Hamada equation is used in finding optimal capital structures, yet the equation doesn’t include default risk. While there have been modifications to account for such a risk, they still lack a robust way to incorporate credit spreads and the risk of default. To gain a better understanding of how to use the Hamada equation, it’s useful to understand what the beta is and how to calculate it.