Hamada's Equation
Encyclopedia
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
where βL and βU 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, βU, from that of its levered counterpart, βL, 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:
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 βD 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.
Corporate finance
Corporate finance is the area of finance dealing with monetary decisions that business enterprises make and the tools and analysis used to make these decisions. The primary goal of corporate finance is to maximize shareholder value while managing the firm's financial risks...
, Hamada’s equation, named after Professor Robert Hamada
Robert Hamada (professor)
Robert Hamada is the former Edward Eagle Brown Distinguished Service Professor of Finance and former Dean of the University of Chicago Booth School of Business.-Early life:...
, is used to separate the financial risk of a levered
Leverage (finance)
In finance, leverage is a general term for any technique to multiply gains and losses. Common ways to attain leverage are borrowing money, buying fixed assets and using derivatives. Important examples are:* A public corporation may leverage its equity by borrowing money...
firm from its business risk. The equation combines the Modigliani-Miller theorem
Modigliani-Miller theorem
The Modigliani–Miller theorem forms the basis for modern thinking on capital structure. The basic theorem states that, under a certain market price process , in the absence of taxes, bankruptcy costs, agency costs, and asymmetric information, and in an efficient market, the value of a firm is...
with the capital asset pricing model
Capital asset pricing model
In finance, the capital asset pricing model is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be added to an already well-diversified portfolio, given that asset's non-diversifiable risk...
. It is used to help determine the levered beta and, through this, the optimal capital structure
Capital structure
In finance, capital structure refers to the way a corporation finances its assets through some combination of equity, debt, or hybrid securities. A firm's capital structure is then the composition or 'structure' of its liabilities. For example, a firm that sells $20 billion in equity and $80...
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
where βL and βU 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, βU, from that of its levered counterpart, βL, 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:
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 βD 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.