Expected Inflation and the Constant-Growth Valuation Model* by Michael Bradley, Duke University, and Gregg A

Home , Market value, Present value, The Constant

VOLUME 20 | NUMBER 2 | spring 2008

Journal of APPLIED CORPORATE FINANCE A MORGAN STANLEY PUBLICATION

In This Issue: Valuation and Corporate Portfolio Management

Corporate Portfolio Management Roundtable 8 Panelists: Robert Bruner, University of Virginia; Robert Pozen, Presented by Ernst & Young MFS Investment Management; Anne Madden, Honeywell International; Aileen Stockburger, Johnson & Johnson; Forbes Alexander, Jabil Circuit; Steve Munger and Don Chew, Morgan Stanley. Moderated by Jeff Greene, Ernst & Young

Liquidity, the Value of the Firm, and Corporate Finance 32 Yakov Amihud, New York University, and Haim Mendelson, Stanford University

Real Asset Valuation: A Back-to-Basics Approach 46 David Laughton, University of Alberta; Raul Guerrero, Asymmetric Strategy LLC; and Donald Lessard, MIT Sloan School of Management

Expected Inflation and the Constant-Growth Valuation Model 66 Michael Bradley, Duke University, and Gregg Jarrell, University of Rochester

Single vs. Multiple Discount Rates: How to Limit “Influence Costs” 79 John Martin, Baylor University, and Sheridan Titman, in the Capital Allocation Process University of Texas at Austin

The Era of Cross-Border M&A: How Current Market Dynamics are 84 Marc Zenner, Matt Matthews, Jeff Marks, and Changing the M&A Landscape Nishant Mago, J.P. Morgan Chase & Co.

Transfer Pricing for Corporate Treasury in the Multinational Enterprise 97 Stephen L. Curtis, Ernst & Young

The Equity Market Risk Premium and Valuation of Overseas Investments 113 Luc Soenen,Universidad Catolica del Peru, and Robert Johnson, University of San Diego

Stock Option Expensing: The Role of Corporate Governance 122 Sanjay Deshmukh, Keith M. Howe, and Carl Luft, DePaul University

Real Options Valuation: A Case Study of an E-commerce Company 129 Rocío Sáenz-Diez, Universidad Pontificia Comillas de Madrid, Ricardo Gimeno, Banco de España, and Carlos de Abajo, Morgan Stanley Expected Inflation and the Constant-Growth Valuation Model* by Michael Bradley, Duke University, and Gregg A. Jarrell, University of Rochester

he Constant-Growth model is a discounted cash the incorrect transformations of this formula found throughout flow method of valuing companies and their the valuation literature for the value of a company that makes T projects that is taught in all top-tier business no new investments or invests only in zero NPV projects. schools and used widely throughout the finan- Perhaps the most common application of the Constant- cial community. It is found in virtually all graduate-level Growth model is its use in estimating what is referred to in corporate finance textbooks and valuation manuals. But, as the valuation literature as a company’s “continuation value” or, we found during an extensive review of this literature, there alternatively, its “terminal value.” When valuing a company, has no been careful, systematic analysis of the effects of infla- it is standard practice to estimate a company’s free cash flows tion on this model.1 As we show in the pages that follow, over a finite (say, five-year) forecast period, and then assume the failure to account properly for the effects of inflation that the firm will simply earn its cost of capital thereafter. The has led to what academics call a “misspecification,” and thus justification for this practice is the standard assumption of an incorrect use, of the model in a particular set of cases— financial economists that the arrival of competitors, along with those where a company is assumed either to make no net technological innovation and obsolescence, cause above-normal new investments or to invest only in zero net present value returns to become normal returns over time, and that, after the (NPV) projects. We show that the error produced by this forecast period, the firm will earn a normal rate of return on its misapplication of the model is significant even at moderate investments into perpetuity. The continuation value, as given levels of expected inflation. by Equation (1), is the present value of the expected free cash In addition to its reliance on operating cash flows rather flows beyond the forecast period into perpetuity.4 than earnings or P/E multiples, the main appeal of the The assumption that the company will earn only normal Constant-Growth valuation model is its simplicity. As shown rates of return during the post-forecast period is equiva- in Equation (1), lent to assuming that either the firm will make no new net investments—that is, capital expenditures will equal depreci- FCF (1)ation—or that any investments that are made will have zero V = 1 0 WG− NPVs. Using this “zero-rent” argument, financial economists often assert that the terminal value of the firm can be estimated the market value of the firm (V0) is a function of just three vari- by a simple perpetuity of next period’s free cash flows, with the ables: the expected free cash flows in the next (or first future) capitalization rate being the firm’s nominal cost of capital. This period (FCF1); the firm’s cost of capital (W); and the projected formulation is equivalent to setting G, the nominal growth growth rate of the firm’s future free cash flows (G).2 This model, rate in Equation (1), equal to zero: which can be found in virtually all finance textbooks, is always written in nominal terms as in Equation (1).3 FCF1 (2) V0 We have no quarrel with this equation. It is simply the W formula for a growing perpetuity. Rather, our quarrel is with

* We thank Michael Barclay, John Coleman, Magnus Dahlquist, Doug Foster, Jennifer 2. According to Brealey, Myers and Allen, p. 65, this formula was first developed in Francis, John Graham, Campbell Harvey, David Hsieh, Albert “Pete” Kyle, Richmond 1938 by J.B. Williams and rediscovered in 1956 by M.J. Gordon and E. Shapiro. It is Mathews, Michael Moore, Stephen Penman, Michael Roberts, Frank Torchio, S. “Vish” often called the Gordon Growth Model. Viswanathan and Ross Watts for helpful comments. We have benefited from many dis- 3. Throughout this paper we adopt the convention of stating nominal variables in up- cussions over the years regarding these and related issues with Robert Dammon, Tim per-case letters and real variables in lower-case letters. Also, since the models developed Eynon, and, especially, Al Rappaport. herein are forward-looking, all rates should be thought of as expectations. 1. The popular valuation texts, Rappaport (1998), Copeland, Koller and Murin (1994) 4. Rappaport (1998) refers to the continuing value as the firm’s residual value, pp. and Cornell (1993) all discuss various aspects of the effects of inflation on the valuation 40-47. Also see Copeland et al. (1994), Chapter 9, “Estimating Continuing Value,” and process. However, none develops the effects of inflation on the Constant-Growth model Cornell (1993), Chapter 6, “Estimating the Continuing Value at the Terminal Date.” It from first principles, as we do in this paper. This is also the case for the leading textbook should be noted that the continuing value as given by Equation (1) is the value of the in the field, Brealey, Myers and Allen (2006). Our analysis most closely resembles that firm at the end of the forecast period. Thus, in order to find the present value as of to- of Rappaport. On page 47 he presents a formula for a “perpetuity with inflation” that is day, the terminal value has to be discounted by (1+W)T, where T is the end of the last equivalent to an important relation that we develop in this paper. forecast period.

66 Journal of Applied Corporate Finance • Volume 20 Number 2 A Morgan Stanley Publication • Spring 2008 We accordingly refer to this version of Equation (1) as the overvaluation of the firm. We develop a correction factor that, Zero-Nominal-Growth (“ZNG”) model. when added to the (nominal) M&M WACC formula, yields Use of the ZNG model, or the simple perpetuity formula, a company’s true nominal WACC in the face of inflation.6 is typically justified by the following reasoning: (1) with zero Finally, to complete our analysis, we show that the nominal net new investments, there will be no growth; and (2) growth WACC model developed by two finance academics—James through the acceptance of zero NPV projects does not affect Miles and John Ezzell—is compatible with any level of growth, the (present) value of the firm. Therefore, under either condi- whether attributable to inflation or real investments.7 tion, it is appropriate to set G = 0 in Equation (1) and rely on Equation (2). Although this logic might appear to be The Constant-Growth Model with Inflation sound, we show that this ZNG model is based on a mistaken Our analysis of the effects of inflation on the Constant- specification of the nominal growth in free cash flows—G in Growth model begins with a derivation of an expression for

Equation (1)—in the presence of inflation. the firm’s nominal free cash flows (FCFt). We then derive Specifically, the generally accepted expression for the value an expression for the growth in nominal cash flows (G). We of a “zero-investment” or a “zero-NPV” firm—as presented defer our discussion of the appropriate discount rate (W) throughout the finance literature and typically applied in until our later discussion of the firm’s WACC. For present practice—ignores the effect of inflation on the company’s total purposes, suffice it to say that in the subsequent analysis, we (accumulated) invested capital. In the traditional Constant- assume that the firm’s nominal cost of capital (W) is consis- Growth model without inflation, if there is no new investment, tent with the Fisher Equation: there is no growth. However, in the presence of inflation, the value of the initial invested capital will grow at the rate of infla- W w0 + w=+ 0 (3) tion. And, assuming a constant real return on invested capital and the replenishment of depreciated assets, the company’s where w is the firm’s real cost of capital andΠ is the expected (nominal) free cash flows stemming from those investments rate of inflation.8 will grow at the same rate. In other words, the proper application of the Constant-Growth model is based on the market or The Real Return on Investment, Net Cash Flows, replacement value of assets, which of course is affected by infla- and Free Cash Flows tion. By ignoring the effects of expected inflation, the generally We begin our derivation of free cash flows with the following accepted expression—Equation (2)— understates the true definition of the real return on investment (r): value of the firm. And the higher the rate of inflation relative NCF to the real cost of capital, the greater the understatement. r = t (4) We develop the appropriate expression for the nominal (1+ 0)K t− 1 growth in cash flows in the presence of inflation and show that the correct model assuming either zero investments or zero net where NCFt is the firm’s net cash flow at the end of period t present value investments is not the ZNG model, but rather and Kt-1 is its total capital stock at the beginning of the period. the traditional Constant-Growth model, with the nominal Rearranging Equation (4) yields: growth rate set equal to the rate of inflation. We also show that this expression for the nominal growth term yields a valuation NCFt K t 1 r(1 0 ) (5) model that is independent of expected inflation.5 A second contribution of this paper to the valuation Equation (5) states that the net cash flow to the firm in period literature is an analysis of the effects of expected inflation t is given by the amount of capital at the beginning of the on a company’s weighted-average cost of capital (WACC), period (Kt-1), times the constant real return on investment the discount factor most often associated with the Constant- (r), times 1 plus the rate of inflation.9 Free cash flow in any Growth model. We find that the WACC methodology (at period equals the firm’s net cash flow for the period less any least in its classic form, as developed by Miller and Modigli- net new investment: ani) is incorrect if expected inflation is positive. Substituting nominal values into the M&M WACC equation results in an FCFt NCFt NNIt (6)

5. If by “inflation” we mean a proportionate increase in all nominal interest rates and to W in Equation 1, where tX is the firm’s tax rate and L is its debt to value ratio. Since the prices of all goods and services, then expected inflation should have no effect on both tX and L are less than 1.0, the M&M adjustment is much smaller than the adjust- present values, i.e., present values should be inflation neutral. ment to the ZNG model. 6. One might be tempted to speculate that these two errors are offsetting, since the 7. Miles and Ezzell (1980) error in the ZNG model leads to an underestimate of the value of the firm and the er- 8. Irving Fisher (1930). Also see Brealey, Myers and Allen, pages 116-118. ror in the M&M WACC leads to an overestimate. However, as developed subsequently, 9. Recall that we have adopted the convention of expressing nominal variables in the adjustment to the ZNG model is to subtract Π, the rate of inflation, from W in the upper-case letters and real variables in lower-case letters and that all rates and future denominator of Equation (2), whereas the adjustment to the M&M model is to add txΠL values should be interpreted as expectations.

Journal of Applied Corporate Finance • Volume 20 Number 2 A Morgan Stanley Publication • Spring 2008 67 Define k as the firm’s plowback rate, which is the fraction the inflationary growth of the company’s invested capital will of NCFt that is retained by the firm to finance net new be augmented by the nominal growth from new investments. investments:10 Finally, if k = 1, the firm is plowing back its entire net cash flows. Consequently, the firm will generate no free cash flows

NNI t (7) into perpetuity and therefore would be theoretically worthless, k NCFt since by assumption there would never be a distribution to security holders.12 However, when k = 1, the value of the firm Substituting Equation (7) into Equation (6) yields: will grow with the value of its capital stock, which, as given by Equation (10), will equal its nominal return on investment R.

FCFt NCFt (1 k) (8) The value of the firm will continue to grow at this rate until the firm is liquidated. The Growth in Free Cash Flows In addition to Equation (10), the analysis in Appendix A Clearly, a critical parameter in the Constant-Growth model is develops three additional relations that will be helpful in our G, the projected constant growth in the firm’s free cash flows. subsequent analysis. Consistent with the Fisher Equation we In Appendix A we derive the expression for the growth in show that: free cash flows based on two assumptions: (1) the (expected) real return on investment, as defined in Equation (4) above, G g 0 0 g (11) remains constant; and (2) the Fisher Equation holds such that the nominal return on investment (R) can be written as: Gand gG 0that:g g 0k 0 gr 0 g

R r 0 0 r (9) g k gr k r (12) where r is the expected real return on investment and Π is where k is the firm’s plowback rate, r is the real return on the expected rate of inflation. investment and g is the real growth rate. Substituting Equa- Making these assumptions, we develop the following tionG (12) kr into 0 Equation 0k r (11) yields: expression for the growth in the firm’s free cash flows: G krG 0 kr 0 0k r 0k r (13) G = k R + ( 1 - k ) 0 (10) Transformations of the Constant-Growth Model The first term in Equation (10) represents the growth in We now derive the appropriate expressions for the two most the firm’s free cash flows from new investments. The second frequent transformations of the Constant-Growth model term represents the increase in cash flows attributable to the found in the finance literature: the value of a company that increase in the nominal value of the firm’s fixed and working either (1) makes no net new investments or (2) accepts only capital that results solely from inflation. Thus, there are two zero NPV projects. As we demonstrate below, these two forces that account for the growth in nominal cash flows. The conditions yield the same valuation expression. first is the increase in cash flows due to the nominal return on new investments. The second is the increase in nominal Zero Investments cash flows due to the fact that the nominal value of the firm’s The value of the firm, according to the Constant-Growth capital stock is higher by the rate of inflation. As we demon- model, assuming zero net new investment, is found by substi- strate later, it is this second term that is ignored in the existing tuting Equation (8) into the numerator of Equation (1), finance literature—in other words, according to the literature, substituting G from Equation (10) into the denominator of this G = k R. equation, and setting k, the reinvestment rate, equal to zero: As indicated by Equation (10), the effect of inflation on NCF1 (1 k) a company’s stock of capital will depend on the percentage of V0 NCF1 (1 k) )41( WV0 (kR (1 k) 0 ) )41( cash flows that are reinvested in the firm (k). If the plowback W (kR (1 k) 0 ) (14) rate is zero (k = 0), then G = Π and all growth is attributable to NCF the nominal growth in the company’s initial fixed and working 1 (15) V0 NCF 1 )51( 11 WV0 0 )51( capital stemming from inflation. Under these conditions, W 0 the firm’s total invested capital invested will remain constant in real terms into perpetuity. If the plowback rate k > 0, then

10. It follows that (1-k) is the firm’s payout ratio. 11. See Appendix A for a demonstration of this proposition. 12. This is equivalent to the value of a zero coupon bond that never matures.

68 Journal of Applied Corporate Finance • Volume 20 Number 2 A Morgan Stanley Publication • Spring 2008 The intuition behind Equation (15) is that even though which, again, is the ZRG or Inflationary-Growth model. Note the firm makes no net new investments (k = 0), it still makes that real growth g is not necessarily zero under the zero NPV replacement investments sufficient to offset its economic assumption. Indeed, since g = kr, if r = w > 0, then real growth depreciation, so that the real net cash flows must be constant g will be positive if k is positive. But any positive real growth over time.13 Thus, the future nominal cash flows (revenues, that may be projected will not increase the value of the firm costs, and profits) emanating from a stream of constant real because all future investments have zero NPV (R = W and r NCFs logically must be projected to grow at the rate of infla- = w). Hence, g is absent from Equation (17). tion, as is reflected in Equation (15). The preceding analysis demonstrates that the expression We refer to this valuation equation as the Zero-Real- for the value of a zero-investment firm is the same as the Growth (“ZRG”) model, since the formula is equivalent to expression for a zero-NPV firm—that is, Equations (15) and assuming that k = 0 and therefore g = 0 in Equation (12), (17) are identical. This result—that the same expression is which implies that the real growth in cash flows is equal to relevant for firms that either invest only in zero-NPV projects zero. This in turn implies that all observed growth is attribut- or invest nothing at all—will be useful later when we develop able to inflation, which justifies our designation of this model the appropriate discount factor for the Constant-Growth as the ZRG model. (Of course, we could have just as easily model under either of these conditions. labeled this expression the Inflationary-Growth model.) The ZRG model is inflation neutral in that it generates a Review of the Literature value for the firm that is independent of the level of expected In this section we retrace the literature’s development of the inflation. To see this, note that the formula can be written expressions for the value of the firm assuming (1) zero net new exclusively in real terms. Given that W = w + Π + wΠ, it investments and (2) investments only in zero NPV projects. follows that W – Π = w (1+Π). Substituting this expression The purpose is to demonstrate that the formulas found in the in the denominator and the expression for nominal cash flows literature are incorrect when expected inflation is positive. into Equation (15) yields the following: Zero Investments ncf (1 0 ) ncf V =1 = 1 (16) )61( The expression for the value of a firm that makes no net new 0 w(1 0 ) w investments found in the literature is based on an erroneous expression for the growth in cash flows. Although incorrect, Zero Net Present Value Investments it is nevertheless common to define the nominal growth in The Constant-Growth model, under the assumption of zero net cash flows as: net present value investments, is found by making the same substitutions as above and setting R = W: G kR (19)

NCF(1 k) NCF(1 k) where R is the firm’s nominal return on investment.14 The V 1 1 0 W (kR (1 k) 0 ) W(1 k) (1 k) 0 correct expression for the growth in net cash flows, as we saw earlier in Equation (10), is: NCF1 (17) V0 W 0 G = k R + (1 - k) 0 (20)

Thus, the expression for the value of the firm when r = w Again, it is the second term in this equation that is ignored (or equivalently R = W) is simply: in the literature. To derive what the literature describes as the zero-investment formula, simply substitute G from Equation NCF ncf (1 0 ) ncf V 1 1 1 (18) (18)(19) into Equation (1), 0 W 0 w(1 0 ) w

13. See Appendix A. per share will also increase by 5.3%: 14. Applying the Gordon Growth Model for the valuation of common stock, Brealey, Dividend growth rate = g = plowback ratio x ROE.” Myers and Allen write on page 65: DIV1 (Emphasis added.) Again, on page 69 they write: “Dividend growth rate = plowback P0 = r - g ratio x ROE.” where P0 is the price of the stock at time 0, DIV1 is the (expected) dividend in period 1, Other references that assert that G=kR include: Ross, Westerfield and Jaffee, p. r is the firm’s capitalization rate and g is the growth in dividends. It is important to note 128, Grinblatt and Titman, p. 832, Rao, p. 403, Bodie and Merton, p. 123, Emery and that BMA do not indicate whether r or g is stated in real or nominal terms. However, Finnerty, p. 149, Shapiro and Balbirer, p. 159, Benninga and Sarig, p. 9, Van Horn, since DIV1 is presumably a nominal number, stated in period 1 dollars, we must presume pp. 30-31, Martin, Petty, Keown and Scott, p.123. It is interesting to note that none of that both r and g are nominal values. Instead of deriving an expression for g, the nominal these authorities feels compelled to prove this relation. They all rely on it as though it is growth in earnings, they simply assert on page 67: self-evident. Not only is this expression for the nominal growth rate not self-evident, it is “If Cascade earns 12 percent of book equity and reinvests 44 percent of income, then incorrect as we demonstrate by the derivation of Equation 10 in the text. book equity will increase by .44 x .12 = .053, or 5.3 percent. Earnings and dividends

Journal of Applied Corporate Finance • Volume 20 Number 2 A Morgan Stanley Publication • Spring 2008 69 NCF (1 k) (21) the parameters of this model. Failure to do so has led to V 1 )12( 0 W kR confusion in the valuation literature, particularly regarding the value of G and the values of companies that make and set k, the reinvestment rate, equal to zero. no new investments or invest only in zero net present value projects. We earlier labeled Equation (22) the Zero-Nominal- NCF1 (22) Growth model, since it is equal to Equation (1) with G, V0 )22( W the nominal growth rate, set equal to zero. We also use the term to emphasize the internal inconsistency inherent in Equation (22) is what the finance literature defines as the this expression. A Zero-Nominal-Growth model in a world “no-growth” value of the firm. For example, the most recent of inflation is an economic oxymoron. Since inflation is the edition of a bestselling finance text describes Equation (22) only distinction between real and nominal values, what does as the value of a company “that does not grow at all. It does it mean to have inflation and zero nominal growth? Under not plow back any earnings and simply produces a constant reasonable assumptions, you can’t have both at the same stream of dividends.”15 Another popular text states that the time. Based on Equation (11) only under the highly unlikely above expression “is the value of the firm if it rested on its and therefore mostly irrelevant case of a negative real growth laurels, that is, if it simply distributed all earnings to the rate—one that just offsets a positive rate of inflation—can stockholders.”16 you have zero nominal growth and positive inflation.18 While it may seem perfectly logical that with no net new Since expected inflation is almost always positive, the investments a firm would generate a constant stream of cash implication of negative real growth into perpetuity makes flows into perpetuity, this reasoning ignores the effects of the Zero-Nominal-Growth model useless for most real- inflation on the firm’s initial invested capital. If a company’s world valuation applications. In addition, the theoretical investments simply keep up with obsolescence and economic conditions under which real growth can be negative into depreciation, its total invested capital stock will increase at perpetuity are unlikely to occur in practice. As Equation the rate of inflation; and, if we assume a constant real return (13) indicates, a negative real growth rate implies that either on investment, its cash flows will also increase at the rate of the real return on investment (r) or the plowback ratio (k) is inflation. Put simply, one cannot express the value of the firm negative into perpetuity. Since the Constant-Growth model as a perpetuity in nominal terms—which is what Equation is forward looking, companies cannot be expected to accept (22) does—when inflation is positive. As we demonstrated a project with an ex ante negative return. Thus, a negative earlier in Equation (15), the presence of inflation requires real growth rate must be due to a negative k, which repre- that the value of a zero-investment firm be expressed as a sents a steady liquidation of the firm over time. Obviously, growing perpetuity, with the growth rate set equal to the this condition is not appropriate for most corporate valua- rate of inflation. tion applications.19 To be clear, we are not implying that either the authors of the texts cited above or the rest of the finance profession Zero Net Present Value Investments are unaware of the effects of expected inflation on the value The derivation of the expression for the value of a company of capital assets. Indeed, in the most influential text in the that invests only in zero NPV projects found throughout profession, after the authors develop the Constant-Growth the finance literature relies on the same mistaken expres- model, they present the Fisher Equation and admonish sion for the growth in nominal cash flows given in Equation the reader to “Discount nominal cash flows at a nominal (19). Zero NPV implies that R = W. Substituting Equations discount rate. Discount real cash flows at a real rate. Never (8) and (19) into Equation (1) and imposing this zero-NPV mix real cash flows with nominal discount rates or nominal condition (R = W) also yields the Zero-Nominal-Growth flows with real rates.”17 But the authors never revisit the formula: Constant-Growth model and discuss how inflation impacts

15. Brealey, Myers and Allen (2006), p. 73. 16. Ross, Westerfield and Jaffee (1993), pp. 130-131. We have examined a number of other texts and have been unable to find the expressions developed in this paper – expressions that are necessary for an inflation-neutral valuation model. 17. Brealey, Myers and Allen, p.118. 0 18. Specifically, if g , then G = 0 even though Π > 0. 1 0 19. Of course k can be negative or, with an influx of capital, even be greater than 1 over a finite period. However, only in very rare circumstances can these extreme values of k persist into perpetuity.

70 Journal of Applied Corporate Finance • Volume 20 Number 2 A Morgan Stanley Publication • Spring 2008 Table 1 Percentage Underestimate According to the Zero-Nominal-Growth Model

Expected Rate of Inflation 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15%

1% 0 50 66 74 79 83 85 87 88 89 90 91 91 92 92 93 2% 0 33 50 59 66 70 74 77 79 81 82 83 84 85 86 87 3% 0 25 40 49 56 61 65 69 71 73 75 77 78 79 80 81 4% 0 20 33 42 49 54 59 62 65 67 69 71 73 74 75 77 5% 0 17 28 37 43 49 53 57 60 62 65 66 68 70 71 72 6% 0 14 25 33 39 44 49 52 55 58 60 62 64 66 67 68 7% 0 12 22 29 35 40 45 48 51 54 56 59 60 62 64 65 8% 0 11 20 27 32 37 41 45 48 51 53 55 57 59 61 62 9% 0 10 18 24 30 35 39 42 45 48 50 52 54 56 58 59

Capital of Cost Real 10% 0 9 16 23 28 32 36 40 43 45 48 50 52 53 55 57 11% 0 8 15 21 26 30 34 37 40 43 45 47 49 51 53 54 12% 0 8 14 20 24 28 32 35 38 41 43 45 47 49 51 52 13% 0 7 13 18 23 27 30 33 36 39 41 43 45 47 49 50 14% 0 7 12 17 22 25 29 32 35 37 39 41 43 45 47 48 15% 0 6 12 16 20 24 27 30 33 36 38 40 42 43 45 47

NCF(1 k) NCF(1 k) NCF(1 k inflation—what we have labeled the Zero-Real-Growth or V 1 1 1 0 W kR W kW W(1 k) Inflationary-Growth model.

NCF1 (23) The General Model V0 W Thus far, we have focused our analysis on two specific cases: A number of financial economists have argued that (1) the value of a firm that makes no new investments; and (2) Equation (23) is the expression for the value of a firm that the value of a firm that invests only in zero NPV projects. But only invests in zero NPV projects.20 Although it may seem our criticisms of the treatment of inflation in the Constant- logical that since investments in zero NPV projects do not Growth model also apply to the estimation of G in general. create value, G should not appear in Equation (23). Intuitively, As developed above, the expression for the growth in nominal the discounted value of cash flows from additional investments cash flows that is found throughout the literature is: just equals the present value of the new investments assuming the return on investment equals the cost of capital. Thus, value G kR (24) is not created by these additional investments. And of course G kR this logic is correct. whereas the correct expression for the nominal growth But, with positive inflation, the value of invested capital term is will grow at the rate of inflation and, assuming a constant real G kR (1 k) 0 rate of return, the net cash flows from these investments (the G kR (1 k) 0 (25) numerator in Equation (23)) will also grow at the rate of inflation. The denominator in Equation (23) ignores this growth To the extent that researchers and practitioners use the and therefore understates the value of the firm. former expression to calculate G, they are understating the true In sum, the correct expression for the value of the value of the firm. Thus, our criticism of the Constant-Growth firm that invests in only zero NPV projects also must be a model, as it is presented in the literature, goes beyond the two growing perpetuity with the growth rate equal to the rate of special cases of zero NPV and zero investments. Our criticisms

20. Copeland, Koller and Murrin (1994), pp. 282-283. Rappaport (1998) p.42 also do not specify whether their variables are nominal or real. The clear implication though, asserts that Equation (23) is the value of the firm if all of its projects have zero NPV. Cope- is that they are all nominal values. Cornell (1993, p. 156) makes the same error. He land et. al. pp. 513-515, derive the relation (stated in our notation) FCFt = NPVt ( 1 - k explicitly states that G is the nominal rate of growth in free cash flows and distinguishes

). They then substitute G/R and write FCFt = NPVt ( 1 - G/R ). From this equation they it from g, which he earlier defines on page 148 as the “long-run growth in real returns.” follow the derivation as outlined in the text. Thus, these authors erroneously assume, like Nevertheless, Cornell assumes that k=G/R. Weston et. al. (1998, p. 186) make the Brealey, Myers and Allen (2006), that k = G/R, when in fact the correct relation is k = g/r. same, erroneous substitution. Rappaport (1998, pp. 40-44) rationalizes Equation (23) This substitution leads them to Equation (23). It is important to note that Copeland et. al. verbally, making the same arguments that are embodied in the algebra above.

Journal of Applied Corporate Finance • Volume 20 Number 2 A Morgan Stanley Publication • Spring 2008 71 also apply to the general model, provided the growth term is tion neutral when stated in nominal terms. We show that calculated according to Equation (24) instead of Equation the equation systematically understates the true WACC and (25). therefore overstates the value of the firm if expected inflation is positive. We provide an expression for this underestimate The Error Rate of the Zero-Nominal-Growth Model of the firm’s cost of capital that, when added to the M&M under Inflation WACC formula, generates the correct discount factor for the As noted above, the Zero-Real-Growth (ZRG) model is infla- Zero-Real-Growth (Inflationary-Growth) model. We show tion-neutral, since the model can be written exclusively in real that using this correction factor generates a value of the firm terms, as we saw in Equation (18). But, somewhat ironically, that is independent of expected inflation. the Zero-Nominal-Growth (ZNG) model, as presented in the There are two basic approaches to valuing the interest literature, is not inflation-neutral. According to the model, the tax shields of a levered firm into perpetuity: the Adjusted value of the firm is inversely related to expected inflation— Present Value (APV) method and the WACC method. The that is, the higher expected inflation, the higher the discount APV method recognizes that the value of a levered firm (VL) factor and hence the lower the value of the firm. is equal to the value of the firm if it were un-levered (VU) plus The percentage by which the Zero-Nominal-Growth (ZNG) the present value of the interest tax shields (PVITS): model understates the value of a firm with zero net new investments or zero NPV investments can be readily calculated by VLU = V + PVITS (27) subtracting the ratio of Equation (22) to Equation (15) from 1: In general, the interest tax shield per period can be written V ZNG W– 0 Percentage Underestimate = 1 = 1 as: V ZRG W w(1 0) Percentage Underestimate = 1 ITSt = t X W D D (28) w(1 0) 0 (26)

where tX is the corporate tax rate, WD is the nominal cost of The entries in Table 1 illustrate the percentage under- debt and D is the amount of debt outstanding at the begin- valuation generated by the Zero-Nominal-Growth model for ning of the period.22 various levels of the real cost of capital (w) and the expected The main insight behind the alternative WACC method- rate of inflation Π( ). For example, if expected inflation is 2% ology is that under certain circumstances, the value of a levered and the real rate of return is 3%, then the Zero-Nominal- firm can be found directly by discounting its future free cash Growth model will underestimate the true value of the firm flows by its tax-adjusted, weighted-average cost of capital. In by 40%! In other words if the true value of the firm was $100, other words, the value of a levered firm can be found by substi- the Zero-Nominal-Growth formula will value the firm at $60. tuting the following definition of WACC for W in Equations Note also that the undervaluation is greater when the expected (15) and (17): rate of inflation is higher than the expected real interest rate. D E L In other words, the values in the north-east triangle of the WACC ( 1 tXD ) W W E (29) table are greater than those in the south-west triangle. VL VL

Expected Inflation and the Zero-Real-Growth where E is the market value of equity, V L is the market value L Discount Factor of the levered firm (VL=D+E) and W E is the firm’s nominal We now develop the appropriate discount factor for the Zero- cost of (or required rate of return on) its levered equity.23 Real-Growth model, W in Equations (15) and (17). Most of Because of the tax subsidy to debt financing, the value of the valuation literature advocates the use of the weighted- the firm increases with greater leverage. Intuitively, as the firm average cost of capital (WACC) methodology developed by substitutes debt for equity, a greater weight is placed on the L Miller and Modigliani (M&M), which is designed to account first term of Equation (29). Since tX > 0 and WD < W E, the for the increase in the firm’s net cash flows stemming from greater the debt-to-value ratio, the lower the WACC—and the tax deductibility of interest payments.21 As we show, the lower the WACC, the higher the value of the firm.24 As however, the M&M WACC model (equation) is not infla- already noted, under certain circumstances, the increase in the

L 21. “The appropriate rate for discounting the company’s cash flow stream is the 24. Of course, according to M&M, if markets are perfect and taxes are zero, then W E weighted average of the costs of debt and equity capital,” Rappaport, page 37. Also see will adjust to offset exactly any change in weights, leaving WACC unchanged. Brealey, Myers and Allen (2006), Chapter 19, Copeland et al., Chapter 8 and Cornell, Chapter 7. 22. If the bond is issued at par, the cost of debt would be the bond’s coupon rate. 23. The analysis assumes that the firm can take full advantage of the tax deduction of interest payments.

72 Journal of Applied Corporate Finance • Volume 20 Number 2 A Morgan Stanley Publication • Spring 2008 value of the firm caused by discounting its net cash flows by Real WACC Under the Zero-Real-Growth Assumption this lower rate—namely its WACC—is exactly equivalent to Stating the M&M model in real terms, the cost of levered the present value of the debt tax shields created by the increase equity is equal to: in debt financing, provided the cost of debt is unaffected by 25 L D the increase in leverage. wE w U (w U w D )(1 t x ) (32) (32) E 26 28 The Present Value of the M&M Tax Shields where wU is the firm’s un-levered cost of capital. Substitut- Under the assumptions of the M&M model, the firm’s cash ing Equation (32) into Equation (29) and solving for the real flows and the amount of debt outstanding are constant weighted average cost of capital yields: into perpetuity. The interest tax shield in Equation (28) is M&M therefore a simple perpetuity. If we assume that the interest wacc wUX (1 t L) (33) payments are as risky as the debt itself, the present value of this perpetuity is: where L is the firm’s debt-to-value ratio, D . Now, according to the M&M model: VL PVITS = ( t W D ) / W = t D. (30) XD DX fcf PVITS = ( tXD W D ) / WDX = t D. 1 (34) VL M&M VU tX D Thus, substituting Equation (30) into (27), under wacc M&M, V = V + t D. To see that Equation (34) holds, substitute Equation (33) LUX M&M VLUX = V + t D. (31) for wacc and show that the resulting expression is equal

to VL: The M&M WACC Model fcf The M&M WACC model posits a relation between a firm’s 1 (35) VL = cost of equity capital and its leverage. Specifically, the cost wU ( 1 tX L ) of equity to a levered firm is equal to the cost of capital if the firm was un-levered plus the difference between the un-levered cost of capital and the (constant) cost of debt Noting that fcf 1 V times 1 minus the firm’s tax rate times the firm’s debt-to- U wU equity ratio. Although this formula can be found in most corporate finance textbooks, like the Constant-Growth VU (36) VL = model in general, the authors never state whether this rela- ( 1 tX L ) tion holds for nominal or real variables.27 Unfortunately, this silence and the context in which the M&M model is typically presented in the literature clearly imply that the relation holds in nominal terms. But, as we show below, this is not correct. The fact that the M&M model assumes fixed cash flows (37) and a fixed amount of debt outstanding immediately raises doubts that the model holds in nominal terms. As we will Combining the above results, the value of a levered firm, see, this intuition is correct. Since the M&M model is based assuming zero real growth, can be written as either Equation on the assumption of constant cash flows, it follows that the (34) or Equation (35), which implies the following: M&M WACC provides the correct valuation when stated fcf 1 in real terms. But, when stated in nominal terms, the M&M VLUX = V + t D = M&M . (38) model systematically understates the firm’s true nominal wacc WACC when expected inflation is positive. We provide an In other words, the M&M model holds in real terms. adjustment factor that, when added to the nominal M&M formula, makes it compatible with our Zero-Real-Growth (Inflationary-Growth) model.

25. Note that the WACC methodology ignores bankruptcy costs and any other lever- 26. This analysis follows the presentation of the M&M WACC model in Brealey, Mey-

age-related costs. In fact the M&M model assumes that WD is equal to the risk-free rate ers and Allen (2006) pp. 517-520. regardless of the degree of leverage. This is an obvious limitation to the M&M WACC 27. See references in Footnote 3. model since there is ample empirical evidence that leverage and the cost of debt are 28. Note that we continue the convention of stating real variables in lower-case letters positively related. and nominal values in upper-case letters.

Journal of Applied Corporate Finance • Volume 20 Number 2 A Morgan Stanley Publication • Spring 2008 73 Table 2 Percentage Overvaluation from Using M&M Nominal Parameters

Expected Rate of Inflation 0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15%

0% 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 10% 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 3.9 4.3 4.7 5.0 5.4 5.7 20% 0.0 0.9 1.7 2.6 3.5 4.3 5.2 6.0 6.9 7.7 8.6 9.4 10.3 11.1 12.0 12.8 30% 0.0 1.4 2.7 4.1 5.5 6.9 8.4 9.8 11.2 12.7 14.2 15.6 17.1 18.6 20.1 21.6 40% 0.0 1.9 3.9 5.9 7.9 10.0 12.1 14.2 16.4 18.7 20.9 23.3 25.6 28.1 30.5 33.1 50% 0.0 2.5 5.2 7.9 10.6 13.5 16.5 19.6 22.7 26.0 29.4 32.9 36.6 40.4 44.3 48.4

Value to Debt 60% 0.0 3.2 6.6 10.1 13.8 17.7 21.8 26.0 30.5 35.3 40.3 45.5 51.1 57.1 63.3 70.0 70% 0.0 4.0 8.3 12.8 17.6 22.7 28.2 34.1 40.5 47.3 54.7 62.7 71.4 81.0 91.4 102.9 80% 0.0 4.9 10.2 15.9 22.1 28.9 36.3 44.5 53.5 63.5 74.8 87.4 101.7 118.0 136.9 158.9

Nominal WACC under the Zero-Real-Growth Re-writing Equation (40) Assumption The assumptions of the basic M&M model imply that it (44) is inflation neutral in that a change in expected inflation will not have any effect on the value of the firm. However the M&M equation, stated in nominal terms, is not inflation So that, neutral. Inflation neutrality requires that rates of return adhere (45) to the Fisher Equation. Thus, for the nominal version of the M&M WACC to be inflation neutral, it must be the case that: True M&M WACC = WACC 0 tX L (46) WACCTrue = wacc + 0 + 0 wacc (39) (39) Equation (46) shows that the “true” nominal WACC is

Substituting the definition of wacc from Equation (33) equal to the M&M WACC plus the term ∏tX L. In other into (39) yields: words the M&M WACC under-states the appropriate value

by the factor ∏tX L. Intuitively, the nominal WACC model True WACC wU (1 t x L) 0 0 wUX (1 t L) ( 40) assumes(40) that an increase in inflation will increase the firm’s tax shield from debt financing. However, this violates a basic Stating the M&M WACC formula (Equation (33)) in nomi- assumption of the M&M model that the amount of debt is nal terms: fixed and independent of its cost of capital as illustrated in Equation (31). M&M WACC WUX (1 tL) (41) Equation (46) also shows that it is easy to correct the nominal WACC formula so that it is consistent with the Fisher

Invoking the Fisher Equation and substituting for WU, Equation and yields the correct value. One can simply add the amount ∏tX L to the computed WACC using the standard M&M M&M nominal WACC formula. Adding the quantity WACC (wU 0 wU 0 ) (1tL) X (42) ∏tX L to the standard M&M WACC yields the correct nominal WACC, which is consistent with the Fisher Equation.29 Expanding Equation (42) In Table 2 above we illustrate that the error generated by the nominal version of the M&M WACC equation is positively WACC M&M (43) related to the expected rate of inflation and that the error is greater, the greater the firm’s leverage ratio. Intuitively, the wUX (1 t L) 0 wUX (1 t L) 0 (1 tX L) nominal M&M model assumes correctly that an increase in

29. Although not shown, it should be noted that the nominal version of the M&M model is also vitiated if the real growth term is positive. Intuitively, the M&M model is incompatible with growth of any sort, be it from inflation or new investment.

74 Journal of Applied Corporate Finance • Volume 20 Number 2 A Morgan Stanley Publication • Spring 2008 inflation will increase the firm’s WACC. However, the model Ezzell (M&E) to determine the appropriate discount factor for incorrectly assumes that the government will absorb the amount a firm in which nominal growth is not zero. Note that since

∏tX L of the increase. In fact, as discussed above, an increase in the M&E analysis generates the appropriate WACC assum- the expected rate of inflation—at the time of valuation—will ing a positive growth in nominal net cash flows, it does not cause a proportionate increase in both the firm’s coupon rate matter whether the growth is generated by real investment and the cost of debt, leaving the (present) value of the debt tax or inflation. shields, and hence the value of the firm, unaffected. Before moving on, it is important to clarify the distinction Table 2 illustrates the economic magnitude of the error between the analysis in the previous section and the analysis generated by the nominal M&M model. Each entry in the in this section. In the previous section, the objective was to table gives the percentage overvaluation caused by using the determine the appropriate expression for the discount factor nominal M&M WACC model instead of the correct model, under the assumption of zero real growth and positive expected assuming zero real growth. Specifically we calculate the follow- inflation—the W in Equations (15) and (17). In this section, ing expression: our objective is to determine the appropriate expression for the M&M discount factor when either real growth or expected inflation V Percentage Overvaluation = True 1 (47) is(47) positive—the W in Equation (1). V The intuition behind the M&E analysis is that if debt is assuming that the real cost of capital and the real return on a constant proportion of the value of the firm, then so is the investment are both equal to 10%, the tax rate is 40%, and value of its interest tax shields. Moreover, since the amount of the combination of the expected rate of inflation and the firm’s debt outstanding is a constant fraction of the value of the firm, leverage ratio are those given in the table. Casual inspection of the risk of these tax shields is equivalent to the risk of the firm’s the table reveals that the error becomes material at combina- cash flows, which means that both should be discounted at the tions of 40% debt and 5% inflation. A 40% leverage ratio and same nominal rate. The M&E analysis relies on this logic. a 5% rate of inflation results in a 10% overvaluation of the In Appendix B, we demonstrate the validity of the M&E firm applying the M&M model in nominal terms.30 WACC in nominal terms. We follow the same procedure as Note that with zero inflation Π( ), zero leverage (L) or we do in the previous section. We assume that the value of M&M TRUE a zero tax rate (tX), V = V from Equation (46) and, a levered firm is equal to the value if it were un-levered plus according to Equation (47), the percentage overestimate would the present value of its interest tax shields.31 We define the be zero. Note also that the percentage error is independent of present value of the interest tax shields assuming a constant the level of free cash flows. Intuitively, since the free cash flows debt-to-value ratio and add this to the present value of the in the first period are in the numerator of both valuation un-levered firm calculated according to Equation (1). We then formulas, they cancel out when calculating the ratio of the demonstrate that the M&E WACC, stated in nominal terms, two. Finally, recall that the “fix” for the nominal M&M model yields this same present value of the firm. As developed in is to add ∏tX L to the M&M WACC. Thus the percentage Appendix B, the M&E WACC formula is: error increases with increases in t , L or Π. X t W L ( 1 W ) M&E XD U (48) WACC WU (48) Expected Inflation and theNominal-Growth 1 WD Discount Factor Conclusion In the previous section we developed the discount factor for We have shown that the generally accepted expression for the ZRG model. There we argued that if the analysis were the value of a firm that either makes no new investments or done using real variables, the M&M WACC, stated in real invests only in zero net present value projects is misspecified if terms, was the appropriate discount rate. We also showed that inflation is positive. We identified the source of the misspeci- if the analysis were done using nominal variables, an adjust- fication and traced its origins to the misspecification of the ment to the nominal M&M WACC was necessary in order growth term in the Constant-Growth valuation formula. We to be consistent with Zero-Real-Growth. derived the correct expression for the growth term, and in Recall that the rationale behind the ZRG model is the turn the correct expression for the value of the firm under assumption that either the firm makes no new net invest- either of these conditions. ments or invests only in zero NPV projects. Thus, this model We have also shown that the most widely used method is inappropriate if investment is not zero or if certain projects of estimating a firm’s WACC is inappropriate when inflation have positive net present values. However, if one is willing to is positive, unless the analysis is performed in real terms. The assume that a firm’s ratio of debt to value is constant through M&M equation for the WACC, stated in nominal terms, time, then one can rely on the analysis provided by Miles and underestimates the firm’s true WACC and thus overestimates

30. Although not shown, the overestimate is higher the lower the real cost of capital 31. We reiterate that this literature, and hence the models in this paper, assume zero (the real return on investment). bankruptcy and other leverage-related costs.

Journal of Applied Corporate Finance • Volume 20 Number 2 A Morgan Stanley Publication • Spring 2008 75 its value. The M&M model can be expressed in nominal terms Appendix A provided the real WACC is calculated first according to the M&M formula, and then the nominal WACC is calculated Derivation of the Growth Rate in the from the Fisher Equation. Alternatively, the nominal version Constant-Growth Model of the M&M model can be found by first calculating the We begin our derivation of the growth in free cash flows WACC in nominal terms, and then adding the correction under the assumptions of the Constant-Growth model by factor ∏tX L. Finally, we show that if the ratio of the firm’s debt establishing the relation between the firm’s capital stock and to value remains constant over time, then the nominal WACC its net cash flows. The total capital stock at the beginning of according to Miles & Ezzell is appropriate and requires no each period (Kt) is equal to the level of the capital stock at the adjustment. beginning of the previous period (Kt-1), less deterioration due to wear, tear and obsolescence over the period (DETt-1), times michael bradley is the F.M. Kirby Professor of Investment Bank- one plus the rate of inflation (Π), plus capital expenditures ing, Fuqua School of Business and Professor of Law, Duke University Ktmade (K tat 1 time DET) t t(CAPX 1 (1 0t): ) CA PX t (1) Kt (K t 1 DET) t 1 (1 0 ) CA PX t (1) ([email protected]). K (KK t DET) (K t 1 (1 DET) 0) t 1 CA(1 PX 0 ) CA PX t (1) (1) gregg jarrell is Professor of Finance and Economics, William E. t K t t 1 (K t 1 t 1 DET) t 1 (1 0 ) tCA PX t (1) (1) Simon Graduate School of Business Administration, University of Roch- ester ([email protected]). We divide CAPX in any period into two components:

replacement expenditures (REPt) and net new investments (NNIt ): CAPXttt REP NNI )2( )2( References CAPXttt REP NNI Brealey, R.A., S. C. Myers and F. Allen, Principles of Corpo- CAPX REPCAPX ttt NNI REP NNI (2) )2( )2( tttCAPXttt REP NNI )2( rate Finance, 8th ed. (New York: McGraw-Hill, 2006). Benninga, Simon and Oded Sarig, Corporate Finance: A Substituting Equation (2) into Equation (1) yields: Kt (K t 1 DETt 1 )(1 0 ) REPt NNI t (3) Valuation Approach (McGraw-Hill, 1997). Kt (K t 1 DE Tt 1 )(1 0 ) REPt NNI t (3) Bodie, Z. and R. Merton, Finance (New Jersey: Prentice K (K DET )(1 0 ) REP NNI (3) (3) Kt (KK t 1 t DE(KTt t 1 1 )(1DE T 0 ) )(1t 1 REP 0 )t REPNNI t t NNI t (3) (3) Hall, 2000). t t 1 t 1 t t Copeland, T., T. Koller and J. Murin, Valuation, 2nd ed. Replacement expenditures are equal to the economic (New York: John Wiley & Sons,1994). depreciation in the firm’s fixed and working capital over any Cornell, B., Corporate Valuation (New York: Irwin, arbitrary period, and are the costs of keeping the firm’s real 1993). capital stock at a constant level throughout the perpetuity Fisher, I., The Theory of Interest (New York: Augustus M. period. Note that we are replacing deteriorating assets at their Kelley, Publishers, 1965). Reprinted from the 1930 edition. current, nominal costs. Martin, J., J.W. Petty, A. Keown, D. Scott, Basic Financial DETt 1(1 0 ) DETt REPt . )4( Management, 5th ed. (New Jersey: Prentice Hall, 1991). DET t 1(1 0 ) DETt REPt . (4) )4( Miles, J. and J. Ezzell, The Weighted Cost of Capital, DET (1 0 ) DET REP . )4( DETt 1(1DET 0 )(1 t 1 DET )t DET REPt . t REP . t )4( )4( Perfect Capital Markets, and Project Life: A Clarification, Given Equationt 1 (4) 0, we can rewritet Equationt (3) as: Journal of Financial and Quantitative Analysis 15 (1980), pp.

719-730. K tK K t K 1 (1 (1 0 ) 0 NNI) NNIt (5) )5( )5( Modigliani, F. and M. H. Miller, “Corporate Income Taxes t t 1 t K K (1K t 0 K) t NNI 1 (1 0 ) NNIt )5( )5( and the Cost of Capital: A Correction,” American Economic Givent that theK t 1t real K t return 1 (1 0 on )t investment NNIt can be written as )5( Review, Vol. 53: 433-443 (June 1963). NCF Myers, S., “Determinants of Corporate Borrowing,” r t (6) Journal of Financial Economics 5 (1977) 147-175. (1 0 )K t 1 NCF Rao, R., Financial Management, (Ohio: Southwestern r t NCFt College Publishing, 1995). ItNCF follows r that K(1 0(1r )K t 1 ) t t 1 0(1 0 )K Rappaport, A., Creating Shareholder Value, 2nd ed. (New t 1 York: The Free Press, 1998). NCF r K (1 0 ) (7) tNCF t 1 r K (1 0 ) Ross, S.A., R. Westerfield and J. Jaffe,Corporate Finance,NCF NCF (K Kt ) r (1 t 1 0) (8) 3rd Edition, (Illinois: Irwin, 1993. t 1and t t t 1 Shapiro, A. and S. Balbirer, Modern Corporate Finance NCFt 1 NCF t (K t K t 1 ) r (1 0) (8) (New Jersey: Prentice-Hall, 2000). NCFt 1 NCF t (K t K t 1 ) r (1 0) (8) (8) Stewart III, G.B., The Quest for Value (USA: Harper Business, 1991).

76 Journal of Applied Corporate Finance • Volume 20 Number 2 A Morgan Stanley Publication • Spring 2008 Substituting Kt from Equation (5) into Equation (8) yields: Appendix B

(9) WACC According to Miles and Ezzell The WACC methodology developed by Miles and Ezzell Note that the first and last variables in the bracketed term (M&E) does hold in nominal terms. This model assumes

on the RHS of Equation (9) are the same – Kt-1 . Expanding that the ratio of debt to value remains constant through time. Equation (9) and canceling out the Kt-1 terms yields: Thus, if the nominal value of the firm grows with inflation, so does the amount of debt outstanding and, as a result, the

NCFt 1 NCF t [K t 1 0 NNIt ] r (1 0 ) (10) per period(10) interest tax shield. All else equal, the value of a levered firm (the market value of its outstanding securities)

Dividing Equation (10) by NCFt defines the growth in NCF is positively related to the rate of inflation. The value of the

and, since FCFt = NCFt (1–k), the growth in FCF as well: firm is not inflation neutral under the M&E model. Since the value of the firm’s tax shields increases with inflation, NCF NCF FCF FCF G t 1 t t 1 t (11) the M&E model is not correct if the parameters are stated NCFt FCFt in real terms. According to the M&E assumptions, the present value of 0 K r (1 0 ) NNI r (1 0 ) (12) the firm’s interest tax shield is equal to: G t 1 t K r (1 0 ) K r (1 0 ) t t-1 t 1 t 1 t W L V ( 1 G ) PVITS XDL (19) ¤ t-1 Recall that t 1 ( 1 WD ) ( 1 WU )

NNI NNI (13) The numerator of Equation (19) is the interest tax shield k t t in period t – the tax rate times the interest rate times the debt NCFt K t 1 r(1 0 ) to value ratio times the value of the firm at the beginning Substituting (13) into (12) yields: of the first period times the appropriate growth factor. Note that the interest tax shield is based on the beginning value of G 0 + k r (1+ 0 ) = 0 + kr + k r 0 (14) the firm. The denominator of Equation (19) reflects the fact that under the M&E assumptions, the amount of debt in the Define g as real growth in free cash flows. Since G = g when first period is known and therefore the interest tax shield is

Π = 0, it follows from Equation (14) that: discounted at the rate on debt, WD. Thereafter, the amount of debt outstanding depends on the value of the firm. Since g = kr (15) L is constant, the firm issues debt when the value of the firm rises and retires debt when the market value falls. Thus, the Equation (15) states that the real growth in FCF is equal size of the interest tax shield rises and falls with the value of to the plowback ratio times the real return on investment. the firm. Consequently, the interest tax shield is as risky as To obtain the equation for nominal growth (G) in terms of the firm itself, which accounts for discounting the per period

nominal returns (R) and inflation (Π) add the quantity (+ tax shield at the un-levered cost of capital, WU, from period kΠ – kΠ) to Equation (14) t = 2 onward. For purposes that will become clear shortly, we isolate the G = 0 G + = kr 0 + + kr kr 0 + + kr k 0 0 +- k k 0 0 - k 0 (16(16)) first(16) year’s tax shield G = k ( r + 0 +r 0 ) + ( 1 - k ) 0 (17) t W L V collectGG = terms, =0 k (+ r kr+ 0 + +r kr 0 0 ) ++ (k 1 0 - k- k) 0 (16)(17) PVITS XDL 1 WD t t-1 G = k ( r + 0 +r 0 ) + ( 1 - k ) 0 (17) (17) tXDL W L V ( 1 G ) (20) ¤ t-1 t 2 ( 1 WD ) ( 1 WU ) and substitute R = r + Π + r Π from the Fisher Equation for the nominal return to investment:

G = k R + ( 1 - k ) 0 )81( G = k R + ( 1 - k ) 0 (18) where,)81( V L is the value of the levered firm in period 0. Adjust- ing the time subscripts in the second term: This isG the = kgeneral R + ( 1expression - k ) 0 for the nominal growth in )81( free cash flows.

Journal of Applied Corporate Finance • Volume 20 Number 2 A Morgan Stanley Publication • Spring 2008 77 t W L V (21) (27) x D L PVITS 1 WD t t W L V ( 1 G ) t Substituting Q yields: x D L ¤ t t 1 ( 1 WD ) ( 1 WU ) (28)

Define Q as: Substituting WACCM&E into the denominator of the basic (22) valuation formula yields:

(29) Substituting (22) into (21) and applying the formula for the value of a growing perpetuity:

(23) Multiplying through by the denominator yields:

(30) Recall that the value of a levered firm is equal to the value of an un-levered firm plus the present value of interest tax (31) shields (PVITS): (32) Q (1 W ) V V U (24) L U W - G U Dividing through by ( WU – G ) yields:

We now demonstrate that the WACC model developed by (33) M&E yields this same expression. Under the M&E assumptions, the firm’s nominal cost of equity is: (34) (25)

Thus, Equation (29) equals Equation (24), which implies Substituting Equation 25 into the weighted into the that the M&E model is correct in nominal terms. weighted average cost of capital formula: Note that since the M&E model assumes a constant ratio of debt to equity, the firm’s debt, and hence its tax shields, will (26) increase at the rate of inflation, and the M&E model, stated in real terms, cannot account for this increase in value. yields:

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