Jamal Munshi, Sonoma State Univesity, 1993 | ||

ABSTRACT 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) +
[t*ROI-j]*P/E
Now substitute the CAPM relations
kL = Rf + bL * (Rm - Rf)
to obtain
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. REFERENCES Baxter Leverage, Risk of Ruin, and the Cost of Capital, Journal of Finance, Vol.22 No. 3, September 1967 Black, F. and Myron Scholes, The Pricing of Options and Corporate liability, Journal of Political Economy, May/June 1973. Boquist, John A., and William T. Moore, The Inter-Industry Leverage Differences and the DeAngelo-Masulis Tax Shield Hypothesis, Financial Management, Spring, 1984 Bowen, Robert with Lane Daley and Charles Huber, Evidence on the Existence and Determinants of Inter-Industry Differences in Leverage, Financial Management, Winter, 1982 Bowman, Robert, The Importance of a Market Value Assessment of Debt in Assessing Leverage, Journal of Accounting Research, Vol 18 No 1, Spring 1980 Bradley, Michael with Gregg Jarrell and Han Kim, On the Existence of an Optimal Capital Structure: Theory and Evidence, Journal of Finance, July 1984 Brennan, Michael with Eduardo Schwartz, Corporate Income Taxes, Valuation, and the Problem of Optimal Capital Structure, Journal of Business, January 1978 Brennan, Michael, Optimal Financial Policy and Firm Valuation, Journal of Finance, July 1984 Conine, Thomas and Maury Tamarkin, Divisional Cost of Capital Estimation: Adjusting for Leverage, Financial Management, Spring 1985, p54 DeAngelo, H. and R. Masulis, Optimal Capital Structure Under Corporate and Personal Taxes, Journal of Financial Economics, March 1980 Fischer, Edwin with Robert Heinkel and Josef Zechner, Dynamic Capital Structure Choice: Theory and Tests, Journal of Finance, Vol 44 No 1, March 1989 Hamada, Robert S., The Effect of the Firm's Capital Structure on the Systematic Risk of Common Stocks, The Journal of Finance, May 1972, p435 Jensen, M. C. and W. H. Meckling, Theory of the Firm: Managerial Behaviour, Agency Costs, and Ownership Structure, Journal of Financial Economics, October 1976 Kane, Alex with Alan Marcus and Robert McDonald, How Big is the Tax Advantage to Debt, The Journal of Finance, Vol 39 No 3, July 1984 Kim, Wi Saeng and Eric Sorenson, Evidence on the Impact of the Agency Cost of Debt on Corporate Finance, Journal of Financial and Quantitative Analysis, Vol 21 NO 2, June 1986 Mandelker, Gershon, and S. Ghon Rhee, The Impact of the Degrees of Operating and Financial Leverage on Systematic Risk of Common Stock, Journal of Financial and Quantitative Analysis, Vol 19 No 1, March 1984. Miller, Merton, Debt and Taxes, Journal of Finance, Vol 32 No 2, May 1977 Modigliani, Franco, and Merton Miller, The Cost of Capital, Corporation Finance, and the Theory of Investment, American Economic Review, Vol. 48 No. 3, June 1958 Myers, S. C., Determinants of Corporate Borrowing, Journal of Financial Economics, November 1977 |