WACC is not the correct discount rate for general asset cash flows

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1 WACC is not the correct discount rate for general asset cash flows Jing Chen School of Business University of Northern British Columbia Prince George, BC Canada V2N 4Z9 Phone: Web: First draft: January 2017 This draft: November 2017 Abstract We show that the theoretical proofs of Modigliani and Miller propositions, which were formulated on specific assumptions of cash flows of investment projects, cannot be extended to more general patterns of cash flows. In general, total cash flows from an investment project discounted by WACC (Weighted Average of Cost of Capital) will not yield correct value of the project as the sum of debt and equity. As the Modigliani and Miller propositions and WACC have been applied to many corporate finance problems, fundamental issues in corporate finance need to be reexamined. This paper is revised from an earlier working paper, On the Non-equivalence of Two Definitions of Asset Values. I thank Leroy Carr, Michael Ehrhardt for helpful comments, which greatly improve the quality of this paper. 0

2 More than half century ago, Modigliani and Miller (1958) published some propositions on asset valuation and capital structures. These propositions have since become the foundation of the theory of corporate finance. Some important concepts, such as weighted average cost of capital (WACC), have sprouted from the Modigliani and Miller propositions. WACC has played a ubiquitous role in determining values and capital structures of assets with general cash flow patterns. Yet the Modigliani and Miller propositions were formulated on assets with constant expected cash flows into perpetuity. Will WACC provide correct discount rate for assets with general cash flow patterns? In this paper, we will discuss a specific type of assets that will provide constant debt payment and changing expected dividend payment over time. We prove that when the expected growth rate of dividend payout from an asset is positive (negative), discounting by WACC will overvalue (undervalue) the asset. In general, total cash flows from an investment project discounted by WACC will not yield correct value of the project as the sum of debt and equity. The theoretical propositions of Modigliani and Miller (1958) are no longer controversial and have been aepted into standard economic theory (Miller, 1988; Myers, 2001). Today, discussion on capital structure focus on empirical issues. Our finding shows that the theoretical propositions of Modigliani and Miller (1958), which were proved under very restrictive assumptions, cannot be extended to general cases. 1

3 A Brief Review of Modigliani and Miller Propositions We will concern ourselves with Propositions I and II in Modigliani and Miller s 1958 paper. We will preserve Modigliani and Miller s original words as much as possible in stating their propositions. Proposition I: Let XX stand for the expected return per year on the assets by the company. Denote by D the market value of the debts of the company; by S the market value of its common shares; by VV SS + DD the market value of all its securities or, as we shall say, the market value of the firm; and by ρρ the expected rate of return appropriate to its risk. Then our Proposition I asserts that we must have in equilibrium: VV SS + DD = XX ρρ (1) The market value of any firm is independent of its capital structure and is given by capitalizing its expected return at the rate ρρ appropriate to its risk. XX is the expected cash flow of the firm that is available for distribution to shareholders and debtholders. With zero growth (which means no need to a new assets) and zero taxes, then XX is sales minus costs minus depreciation. With no need to finance asset growth, XX is also equal to coupon payments plus dividend payments. 2

4 This proposition can be stated in an equivalent way in terms of the firm s average cost of capital, XX VV, which is the ratio of its expected return to the market value of all its securities. Our proposition then is XX XX = ρρ (2) SS+DD VV Proposition II. From Proposition I we can derive the following proposition concerning the rate of return on common stock in a company whose capital structure includes some debt: the expected rate of return or yield, rr ss, on the stock of the company is a linear function of leverage as follows: rr ss = ρρ + (ρρ rr DD ) DD SS (3) Where rr DD is the yield of the debt of the company. The above are Propositions I and II in MM s paper. We can rearrange equation (3) to obtain ρρ = DD DD + SS rr DD + SS DD + SS rr ss = DD VV rr DD + SS VV rr ss (4) This means that the company s average cost of capital is the weighted average of the costs of its debt and its equity. Propositions I and II are proved under the assumption that the expected return from the asset is constant over time and there are no taxes. However, formula (4) of WACC has since been used in literature and taught in textbooks as a general formula of cost of capital of firms. Some of the assumptions in the original MM 1958 paper had been relaxed (Stiglitz, 1969). Can formula (3) 3

5 and (4) be extended to value assets with general cash flows? We will examine this issue in the next section. Valuation of Asset Cash Flows with WACC The value of an asset is the sum of the values of its debt and equity. In corporate finance literature, asset value is also defined as cash flows discounted by weighted average cost of capital (WACC) (Ross, Westerfield, Jordan and Roberts, 2013). Modigliani and Miller (1958) proved that when the expected return of an asset is constant over time, two definitions give the same result. We shall prove, by providing a counter example, that in general, two definitions provide different valuations. Suppose an asset is financed by a perpetual bond and an equity issue. The bond pays coupon amount c per unit time. The equity is expected to pay dividend amount d next time period. The amount of dividend is expected to grow at a rate of g. The market value of the bond is D. Then the yield of the bond is rr DD = DD (5) The market value of the equity is S. The discount rate on the dividends is rr ss. Then the yield of equity is rr ss = + gg (6) SS 4

6 The asset value, V, is the sum of debt and equity. In corporate finance, the value of an asset is also defined as total cash flows discounted by WACC. Let VV represent the asset value calculated from this definition. VV = + + +(1+gg) + +(1+gg)2 + 1+ρρ (1+ρρ) 2 (1+ρρ) 3 = 1+ρρ + (1+ρρ) 2 + (1+ρρ) (1+gg) + (1+gg)2 + 1+ρρ (1+ρρ) 2 (1+ρρ) 3 = ρρ + ρρ gg = DD VV rr DD+ SS VV rr ss + DD VV rr DD+ SS VV rr ss gg = VV + DDrr DD +SSrr ss DDrr DD +SSrr ss VVVV So VV = VV + DDrr DD +SSrr ss DDrr DD +SSrr ss VVVV (7) The difference between VV aaaaaa VV would be VV VV = VV + DDrr DD +SSrr ss DDrr DD +SSrr ss VVVV 1 From (5) and (6), the above formula can be simplified into, = VV( + ++SSSS ++SSSS VVVV 1) = VV + ++SSSS + DDDD 1 5

7 = VV (+ DDDD)+(++SSSS) (++SSSS)(+ DDDD) (++SSSS)(+ DDDD) = VV ++SSSSgg2 (++SSSS)(+ DDDD) gggggg = VV DD + SS + gg ( + + SSSS)( + DDDD) = VV gggggg(rr ss rr DD ) (++SSSS)(+ DDDD) (8) When g = 0, term (8) is equal to zero. VV = VV In the special case when the expected growth rate of dividend payout is zero, WACC does provide correct valuation for asset cash flows, as proved by Modigliani and Miller (1958). From (8), when the growth rate is positive, g > 0, VV > VV discounting by WACC will overvalue the asset. When the growth rate is negative, g < 0, VV < VV discounting by WACC will undervalue the asset. In general, asset values calculated from cash flows discounted by WACC are not equal to the sum of values of debt and equity. Debt and equities can be traded in the markets and their values are observable quantities. Valuations by WACC, in contrast, are not observable quantities. Since valuation with WACC differs from the 6

8 sum of debt and equity values, the theoretical method of Modigliani and Miller cannot be extended to value assets with general cash flows. In practice, people usually choose a discount rate that equalizes two definitions of asset values and call it WACC (Brigham, Ehrhardt, Gessaroli and Nason, 2017, P. 655). This is to solve for r in the equation CCCCCChffffffffff ii (1 + rr) ii = DD + SS (9) ii=1 and call r WACC. However, this r in general is not equal to the weighted average of costs of debt and equity, DD DD + SS rr DD + SS DD + SS rr ss It should not be called WACC. The following numerical example will provide a more concrete illustration of the issue. Suppose a company is financed with a perpetual bond and common stock. The market value of bond is 100 million dollars and the market value of the equity is 100 million dollars as well. The asset value of the company, as the sum of debt and equity, is = 200 million dollars. 7

9 The company is expected to distribute coupon amount to 3 million dollar and dividend amount to 3 million dollars next year. So the yield of the perpetual bond is 3% and the dividend ratio is 3%. Assume the growth rate of the dividend is 4% per year. The cost of equity is Dividend yield + growth rate = 3%+4%=7% and WWWWWWWW = DD rr DD+SS DD + SS rr DD+SS ss = % + 7% = 5% The sum of cash flows discounted by WACC is 3+3(1+4%)ii 1 ii=1 (1+5%) ii 3 = ii=1 + 3(1+4%)ii 1 ii=1 (1+5%) ii = = = 360 mmmmmmmmmmmmmm 5% 5% 4% (1+5%) ii This is the asset value calculated from discounting by WACC. Compared with the asset value as the sum of debt and equity, which is 200 million, the asset value calculated from discounting by WACC is much higher. In the example above, the correct discount rate for total cash flows can be obtained by solving (9). As = = 200, 6% 6% 4% the discount rate is 6%. But this rate is not a weighted average of costs of debt and equity. 8

10 Empirical tests often find firms are underleveraged. This is often attributed to various types of market imperfection. However, further investigations often show that it is the designs or methodologies of these tests that are flawed (Molina, 2005; Chen, 2006). Our finding suggests some supposed underleveraging could also come from miscalculation caused by discounting with WACC. Concluding Remarks We prove that WACC is not the correct discount rate for general cash flows. The intuition behind the proof is very simple. Discount rate is not a linear factor. We shouldn t expect WACC, a linear combination of two discount rates, will provide correct measure of discounting for combined cash flows in general cases. Since discounting by WACC does not provide correct measurement of asset values in general, we recommend no more using WACC in valuing assets. Instead, we can calculate equity value and debt value separately with their corresponding discount rates. This will provide more aurate measurement of asset values as the sum of debt and equity values. The goal of financial management, as stated at the beginning of most textbooks, is to maximize shareholder value. After the introduction of the concept of WACC in the mile of the textbooks, the goal of financial management usually changes into the maximization of firm value, which is 9

11 the sum of equity value and debt value. The goal of maximizing shareholder value and maximizing frim value are not always consistent with each other. By calculating equity value and debt value separately, we can maintain a consistent approach of maximizing shareholder value throughout the whole teaching process. We can also determine the optimal capital structure from the process of maximizing shareholder value. The Modigliani and Miller propositions and WACC have been applied to many corporate finance problems. For example, the optimal capital structure, which is supposed to maximize firm value, is often defined as the debt equity ratio where WACC reaches its minimum. As the WACC discounting method does not provide correct measure of asset value, fundamental issues in corporate finance need to be reexamined. This finding will have deep implications to research, teaching and practice in corporate finance. 10

12 References Brigham, E. F., Ehrhardt, M. C., Gessaroli, J. and Nason, R.R. (2017). Financial management: Theory & practice. Third Canadian Edition, Nelson. Chen, J. (2006). Imperfect market or imperfect theory: a unified analytical theory of production and capital structure of firms. Miller, M. H. (1988). The Modigliani-Miller propositions after thirty years. The Journal of Economic Perspectives, 2(4), Modigliani, Franco, and Merton H. Miller. "The cost of capital, corporation finance and the theory of investment." The American economic review 48.3 (1958): Molina, C. A. (2005). Are firms underleveraged? An examination of the effect of leverage on default probabilities. The Journal of Finance, 60(3), Myers, S. C. (2001). Capital structure. The journal of economic perspectives, 15(2),

13 Ross SA, Westerfield R, Jordan BD. Roberts, G, Fundamentals of corporate finance. Eighth Canadian Edition, McGraw-Hill Education; Stiglitz, J. E. (1969). A re-examination of the Modigliani-Miller theorem. The American Economic Review, Villamil, A. P. (2008). The Modigliani-Miller Theorem. The New Palgrave Dictionary of Economics, Second Edition. Eds. Steven N. Durlauf and Lawrence E. Blume. Palgrave Macmillan, 6. 12