Jamal Munshi, Sonoma State Univesity, 1993
A careful algebraic derivation of the well-known Hamada equation for levered beta (Hamada 1972) reveals a refinement that implies that the Hamada equation under-estimates the effect of leverage on beta. When there is no projected growth of cash flows, the modified Hamada equation predicts that the cost of capital and the value of the firm are independent of capital structure. The additional tax savings due to debt are exactly offset by the increased riskiness of the equity holdings. This result is in agreement with the Irrelevance Theorem postulated by Miller (Miller 1977).
When the projected growth rate is positive, the modified equation shows a steady decrease in firm value when debt is added to the capital structure. This result is contrary to that predicted by the Hamada equation and supports the agency cost explanation of capital structure described by Jensen and Meckling (1976). Debt only reduces the accounting value of the firm and not necessarily the value of the firm net of agency costs. The results are consistent with the idea that with agency costs taken into consideration the net firm value may first rise to a maximum as savings in agency costs outweigh the loss of accounting value.
In deriving his equation, Hamada set the corporate cost of debt to the risk-free rate to simplify the algebra. This restriction was overcome in a later revision of the equation by Conine and Tamarkin (1985). However, since later works use Hamada's equation as the starting point, they suffer from the same under-estimation problem. In this paper, a generalized form of the equation is presented that makes no assumption about the cost of debt. The equation is simple and easy to implement into spreadsheet models for analyzing capital structure effects. A complete spreadsheet model that allows for risky debt and preferred shares is included in the paper.
DERIVATION OF THE MODIFIED HAMADA EQUATION
We Start with the equation for accounting returns to common shareholders of the levered firm (kL) as
kL = (NIAT - j*P)/E
where NIAT is net after-tax income, j*P represents dividends payable to preferred shareholders at a rate of j on a total outstanding value preferred shares of P. The variable E is the value of common equity. We further define I as the interest payable to debtholders as the product of i*D where i is the interest rate in debt and D is the debt outstanding. Imposing these accounting definitions we obtain,
kL = [t*(NOI-I)j*P]/E
Where t is 1 minus the corporate tax rate (or the `keep' rate), ROI is the return on assets, P is preferred equity outstanding, and j is the rate of return on preferred equity.
The result is, in somewhat more general form, what is known as the DuPont Model. In the absence of debt the middle term disappears and we have
kU = t*[ROI]+[t*ROI-j]*P/E
Substituting this expression for ROI in the equation for kL to get
kL = t*(kU - j*P/E)/(1+ t*P/E) + t*D/E)*((kU - +j*P/E)/(1+ t*P/E)-i) +
Now substitute the CAPM relations
kL = Rf + bL * (Rm - Rf)
Collecting terms we get
This equation simplifies considerably if we assume that i = Rf and that is what Hamada did. We thus obtain the correct Hamada equation as
bL = bU + (D/E)*bU +(D/E)*Rf/(Rm - Rf) -(D/E)*(1-t)*Rf/(Rm - Rf)
Denoting Rf/(Rm - Rf) as r and the capital structure D/E as d we have,
bL = bU + dbU + drt
bL = bU(1+ d) + drt
We will refer to the corrected Hamada equation as the Jamada equation. It differs from the Hamada equation normally stated as
bL = bU(1+ d(1-t))
A more interesting point to note is that the assumption of i=Rf is not really necessary since the equation is still quite manageable if we retain i as risky debt and use the second form of the Jamada equation.
bL = bU(1+ d) + dr[1 - z(1-t)]
Where z is the ratio of the interest rate on corporate debt to the risk free rate (i/Rf).
Under the assumptions of the Hamada equation, the Jamada equation predicts a larger effect of debt on beta and the additional effect exactly offsets the apparent advantage of debt financing predicted by the Hamada equation. This means that under the same set of assumptions Hamada predicts a decreasing cost of capital with debt while Jamada predicts that the cost of capital and therefore the value of the firm is invariant with capital structure. These relationships are developed below.
The weighted average cost of capital (WACC) of the firm with w fraction of its assets supported by debt and (1-w) supported by equity may be written as
WACC = wRf(1-t) + (1-w)[(bu+d(1-t))(Rm-Rf) + Rf]
expressing d as w/(1-w) we get
WACC = wRf(1-t) + (1-w)[(bu+w(1-t)/(1-w))(Rm-Rf) + Rf] = wRf(1-t) + (1-w)Rf + (1-w)(Rm-Rf)bu + w(1-t)(Rm-Rf) = (1-w)Rf + (1-w)Rmbu - (1-w)Rfbu + wRm(1-t) = Rf + Rmbu - Rfbu + w[Rm(1-t)-Rf-Rmbu+Rfbu]
The first derivative of WACC with respect to w is
d/dw[WACC] = (1-t)Rm - [Rf + (Rm-Rf)bu]
and the second derivative is zero. The Hamada relation thus predicts a linear relationship between the debt ratio and cost of capital
The returns from the unlevered firm ku may be expressed as ROI(1-t) where ROI is the operating income per dollar of total assets. The equation shows that as long as ku is greater than Rm(1-t) or ROI is greater than Rm debt reduces the cost of capital and the `optimal' capital structure (for maximum firm value) would be all debt. If ku=Rm(1-t), or Rm = ROI then firm value is invariant with capital structure. And if ku is less than Rm(1-t) or ROI is less than Rm, firm value falls with debt, i.e., the optimal capital structure is all equity.
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