## I Introduction

The discounted cash flow (DCF) model is a widely used and effective method for valuing enterprises. As such, it is suitable method for assessing damages associated with business valuation and enterprise lost profits in international arbitration. The strengths of the DCF model relate to its straightforward simplicity, the way in which it embraces the income and growth prospects for an enterprise, and the way it incorporates the market's assessment of investment risk in any part of the world. Particularly for cases involving infrastructure assets, such as regulated enterprises or those supporting competitive energy or transportation markets, the DCF model is popular among tribunals as an objective method that focuses on the essential underlying elements that spur such arbitrations – the interruption of stable, going-concern profitability.^{2}

Being so computationally straightforward, a paradox in the use of the DCF model is that it was invented only recently. Economist Myron Gordon sought to bring together economic theory and the long-standing rule-of-thumb practices of financial analysts of the late 1950s: what he called the 'imprecise' finance literature. He grounded those financial practices with a theoretical economic foundation. The resulting DCF model he described soon came to the attention of administrative regulatory agencies, courts and arbitral tribunals as a tractable tool for deriving enterprise value and the cost of capital when used with objective market, industry or enterprise data. Gordon's success made him famous. The 'Gordon growth model' is part of all modern finance courses and textbooks.

## II The DCF Among its Peers for Enterprise Valuation

In cases where courts or arbitral tribunals must value an enterprise in a dispute, it is widely accepted that there are three available methods of doing so. The first looks to the cost of the enterprise's assets. The second is based on the market value drawn from comparable companies or the arm's-length sale of similar enterprises. The third charts the expected capitalised income from the enterprise as a going concern brought back to the present – of which, the DCF model is the principal method.^{3} All three have useful attributes under the right circumstances.

### i The cost method

Valuation based on the cost of the assets involved – either recorded-book cost or replacement cost – has some direct applicability to regulated enterprises whose values may be built more or less directly on such costs. But the method is sharply limited when assessing the going-concern value of infrastructure assets that draw their value from unregulated markets. For infrastructure assets that provide services over time, the cost method has sharp practical limitations based on the way that capital costs are handled by accountants – principally, because of depreciation. There has always been an unresolved tension between economists' and accountants' views of such depreciation. To economists who must value enterprises, depreciation charges merely reflect an allocation of the costs for investment decisions already made. In other words, such depreciation charges:

refer to an expenditure that has already taken place, and are merely a special method of writing history. Depreciation accounting enables the business firm to make several ledger entries, instead of one, when a capital expenditure occurs.^{4}

As such, depreciation charges are only useful in determining the value of enterprises if future profitability is somehow a function of those depreciation charges – as is the case with regulated enterprises for which consumers' prices are derived from investment costs, including depreciation.^{5}

### ii The market value method

Valuations based on market value depend on comparable companies or comparability in arm's-length sales. As awards in court or before arbitral tribunals are designed to assess an objective value for an enterprise, a market price is generally only applicable to demonstratively similarly situated enterprises. This would be the case for a dispute related to land or housing where reasonable markets exist involving related transactions of sufficient numbers to create a usable sample of comparative transactions. But the nature of international arbitration often involves disputes with unique firms for which comparable companies or transactions can be rare or non-existent either because of the site-specific nature of the investments or the unique nature of the international market in question.

Economists have long studied whether the technologies inherent in particular businesses permit reasonable comparisons for regulation, damage assessment or the evaluation of organisational efficiency.^{6} Literature points to the problems inherent in making comparisons across enterprises in the private sector. Market comparisons between private enterprises are sharply limited in situations where input choice or 'environmental factors' (a broad category, including differing countries, institutional environments and industrial histories) cannot be controlled. Any direct comparison of sales prices among enterprises assumes that they all can attain the same level of production given their factor endowment – that is, that they belong to what economists would call the same 'production function'. Although this assumption may hold for some public sector enterprises (such as public schools) or for particular private sector applications (such as the ubiquitous Starbucks coffee shops), it does not generally hold for the types of assets involved in international arbitration.^{7}

### iii The income capitalisation method

The inherent difficulties associated with cost or market prices is a principal reason why the income approach represented by the DCF method is so evidently popular among tribunals. To be sure, the DCF method also draws the cost of capital specific to the type of enterprise in question from market benchmarks, and its location in the world from the competitive capital markets. Nevertheless, the principal innovation of the DCF method is how it uses that market information to tie finance and enterprise valuation to a reliable underlying economic theory.

## III Origin of the Discounted Cash Flow Model

In an era of intense interest and research into stock valuation models and methods, and highly sophisticated markets in financial instruments, it is useful to reflect briefly on the comparatively backward nature of the field of business valuation in the 1950s. When it came to valuing business enterprises at that time, there was no intersection between the practical rules of thumb developed by stock analysts or insurance actuaries and neoclassical theoretical economics.

Gordon wished to fill that gap. In 1962, he published, in book form, his theoretical work on the value of corporations, reflecting his published research in the late 1950s.^{8} His book constituted both a deconstruction of economists' various theories of investing and a straightforward presentation of a new method of valuing business enterprises. Prior to Gordon, economists had only highly abstract theoretical models of firm investment flowing from Keynesian theories of income determination. They had to rely on 'business practices' to investigate problems of investment and financing. Gordon offered something better: a theoretical foundation for explaining the value of a corporation that could, in turn, be used to find the cost of capital without resorting to ad hoc business methods.

The economic theories of the era embodied the assumptions that the future is certain and the firm can borrow freely at a given rate of interest.^{9} When trying to bring uncertainty into their theoretical analyses, economists became irretrievably bogged down in particular complicated cases that did little more than provide 'a very able statement of the generalization found in the unprecise literature of finance', as Gordon wrote.^{10}

Gordon sought a cogent tie between neoclassical economic theory and the generally accepted notions that a firm's value is a function of its investment. As he wrote:

under neoclassical theory, the objective of the firm is to maximize its value, the value is a function of its future income, and the future income is a function of its investment. The task of the theory is to provide information on the nature of these two functions.^{11}

As is often the case in economics, advances in theory are often less like Darwin's 'gradualism' and more like the late evolutionary biologist Stephen J Gould's 'punctuated equilibrium' – with episodic evolutionary leaps. Gordon's theoretical discovery was such a leap, taking one page of not-too-difficult mathematics to tie maximising value to future income and to future investment. The result was the first statement of the DCF model in economic literature:^{12}

Equation 1

Where:

*P*_{0}= price of a share of common stock;*b*= the fraction of income retained in future period (i.e., not paid out as dividends);*Y*_{0}= income per share;*k*= required return on investment; and*r*= expected return on investment.

Gordon recognised that for his theory to be of any practical use, it had to have some means for establishing *k*, an enterprise's cost of capital. Using his model to solve for the cost of capital using discrete time periods gives the following:

Equation 2

Where:

*g*= (*r*) × (*b*), expected dividend growth rate;*P*_{0}= price of stock;*D*_{0}= (1 –*b*)*Y*_{0}, previous dividend paid; and*k*_{e}= cost of equity.

Here, the rate of profit investors require on a share of stock is equal to the dividend yield at which it is selling (the first term on the right-hand side of the equation) plus the rate of growth in the dividend (the second term).

## IV Why the DCF Method Works

As outlined by Gordon when he first presented the theory, the key theoretical advance of the DCF model was that it did not presume that investors knew more than they could reasonably know. They had only broad notions of how investment conditions would change in future years. The DCF model worked because it simplified the future by not overstating the kind of information that investors use to make decisions in the market. Specifically, the values of *r* and *b* (the expected return on investment and the retention rate) are not dated – Gordon assumed them to be constant. Without such an assumption, any investment model would quickly become empirically unmanageable, which was the very problem that had vexed those earlier economists trying to link economic theory with finance. But, as Gordon correctly observed, investors rarely have any clear notion of how these variables change over time. And in any event, without specific information to the contrary it is reasonable to conclude that retention ratios and expected returns are stable for going concerns. Established corporations commonly follow a policy of paying a stable fraction of their earnings as dividends (*b*). Investors will also only, at best, have a broad idea as to any future change in the expected return on investments (*r*). With such an assumption, the value of a share of stock is merely a function of the current dividend and the rate of growth of those dividends (which is the product of *b* × *r*).

Equation 1 can be written out to include the whole future income stream discounted by a constant expected cost of capital, *k*e, as follows:

Equation 3

Alternatively:

Equation 4

This is the form of the DCF model (for which Equation 1 is merely a mathematically reduced form) that populates the values supporting damages estimates before arbitral tribunals. This DCF model values a share of stock according to a discounted stream of future dividends growing at a constant rate, *g*. Although the basic Gordon model uses dividends to compute a share value, dividend payments are not necessary for such an analysis – cash flow or earnings can be used instead as these are more reflective of the value that investors place in the future profitability of the enterprise. That assumption of a constant growth rate can be changed if data on near-term expected growth differs from long-term growth. For example, corporate business plans may specify near-term income growth, *g*, reflecting particular business conditions known by management – after which, a long-term growth rate deals with future years.

## V The Strength of the DCF Model in Industrial and Institutional Settings

The attraction of this theory for administrative agencies and arbitral tribunals soon revealed itself. Gordon himself describes how he was retained in 1966 to provide evidence before the US Federal Communications Commission (FCC) on the cost of capital of AT&T. The FCC appreciated the straightforward simplicity of such a theoretical model that rendered AT&T's cost of capital as the sum of a measurable dividend yield and a growth rate. The agency was highly complementary regarding Gordon's analysis, structuring its findings to be consistent with it, and encouraging further study of the DCF model.^{13} As a result of that first application to regulated enterprises, the DCF model has dominated tariff proceedings in the United States and Canada to find the cost of capital for regulated firms.^{14}

Used to derive the price of a share of stock, *P*0, as in Equation 4, the DCF model has become the principal 'income-based' method for valuation. In this form, the model has wide-ranging applicability in many contexts. It is useful to assess the loss of a business in its entirety and lost profits within an ongoing concern. It is also useful for both measuring direct damages (e.g., for the loss a productive asset) and indirect damages (e.g., flowing from changes in risk – affecting return – or prospective growth rates). Essentially, the DCF model is a framework within which to apply financial and operating information, from both the enterprise in question and from the surrounding markets (including capital markets). The accounting and operational data used to populate the model in any setting depends on what is available and, if none is readily available, the obvious alternative is to use the shortest and most objective path to developing useful proxies. This work is conceptually straightforward but, often enough, complex and subject to dispute in such settings as international arbitration.

The DCF model has specific applicability to industries characterised by the dedication of capital to certain sorts of going concerns relating to regulated or competitive markets – those industries where the assumptions of constant retention ratios and growth rates work well. Such businesses include regulated utilities, and oil and gas companies with capital facilities that support competitive markets (such as liquefied natural gas terminals, refineries, port facilities or airports) where long-lived infrastructure investments, in specific locations with specific supply and demand conditions and risks, require a valuation method that takes into account a long-term payoff for unique arrangements of assets.

## VI Modern Permutations of the DCF Method

Gordon gave an economic foundation to the 'imprecise' financial literature and the reasonably effective rules of thumb of financial analysts of the 1950s. But Gordon was not the only economist looking for a theory by which to value corporations in the 1950s. His contemporaries, Franco Modigliani and Merton Miller were looking for the same thing. Their model, called the Capital Asset Pricing Model (CAPM) first appeared in 1958.^{15} CAPM looks at the behaviour of stock prices in the market at large – not simply prospective earnings – to judge what investors require as the return on equity investments. It began to appear before administrative agencies (in North America and elsewhere) in the late 1970s and early 1980s.^{16}

Both methods propelled an explosion of scholarly literature on how practically to value businesses using each.^{17} This literature displays endless permutations, adjustments and increasingly thin slivers of time (with fractional exponents) to derive ever-finer special cases of the DCF model. To a certain extent, the complexity evident in this literature tends to obscure the essential nature of what Gordon provided – the method by which, with limited information in the capital market, capital cost and expected earnings from an enterprise relate to each other in an uncertain world. Mathematical models may be sliced in endless ways. But genuinely objective information, in the market, to drive the DCF model – reflecting what investors expect for the market's growth rate – is rare. More complicated modelling does not make this information any less rare.

## VII The Central Role of the Growth Rate

Gordon's DCF model uses enterprise value and future income as two sides of a coin: either side can face up. Equation 1 shows the enterprise value on top with the income capitalised at the growth rate (*r* × *b*) underneath. Equation 2 has the cost of capital on top, with the additive terms of dividend plus the growth rate underneath. Either way, the growth rate is on the bottom of the coin, operating unseen. The DCF model rests on that unseen growth rate – it either permits observed income to translate into value, or it permits an observed dividend yield to translate into the cost of capital (which is how Gordon himself used it in finding the regulated cost of equity for AT&T). Where is that growth rate from?

In the capital markets that tie the top and the bottom of the DCF coin together, the growth rate is in the mind of the investors: they price the shares in question based on the way the capital markets reflect their collective expectations. The growth rate in the DCF model exists in the world – the problem is to find it in a sufficiently objective way to be credible before judges or tribunals.

The long-standing depth of the US capital market provides many sources by which to gauge investors' growth expectations. The accuracy of these analyses, in the sense of whether they are predictive of the future, is not the issue. The crux of the matter is whether they reflect widely held expectations. In recent years, the availability of sources of financial analysis has expanded to include the following, widely considered objective and credible.

### i Annual Valuation Handbook – US Industry Cost of Capital, Duff & Phelps

Duff & Phelps, founded in 1932 in Chicago as an investment research firm, took over the widely respected publication of Roger G Ibbotson and Rex A Sinquefield, begun in 1977 (with stock, bond and inflation data reaching back to 1926), in 2014. Duff & Phelps uses data from Standard & Poor's and Barclay's to develop annual industry statistics, including return on equity, CAPM betas, enterprise valuation multiples and long-term earnings per share growth projections.^{18}

### ii Institutional Brokers Estimate System (I/B/E/S), Refinitiv

Refinitiv, a new company built from the Financial & Risk business of Thomson Reuters, publishes I/B/E/S, which compiles estimates from 18,000 analysts pertaining to companies in more than 90 countries. Established in 1976, available metrics include traditional financial indicators and I/B/E/S Key Performance Indicators that vary by industry.^{19}

### iii NYU Stern School of Business, Aswath Damodaran

Aswath Damodaran, professor and holder of the Kerschner Family Chair in Finance Education at the NYU Stern School of Business, publishes several useful current and historical data sets on his website. Professor Damodaran annually updates industry averages for US and global companies on corporate finance and valuation metrics, including return, capital structure, dividend yield, capital expenditures and depreciation, multiples and growth rates. He also provides data on country risk premiums and corporate tax rates. He uses various published sources of data.^{20}

### iv Value Line

Founded in 1931, Value Line is one of the oldest and largest independent, subscription-funded investment research services. Value Line employs independent analysts to assess various financial metrics such as individual company growth anticipated from new products, overall financial health and three-to-five-year projections.^{21}

### v Zacks Investment Research

Founded in 1978, Zacks uses mathematical models combined with industry analyst research to estimate earnings growth of particular companies and industries. Analysts provide commentary and quantitative research to provide expectations for stock prices, fund performance and other financial metrics.^{22}

### vi Final thoughts

The DCF model does not presume that investors are omniscient with perfect knowledge of the future. Similarly, those who employ the DCF model do not have perfect knowledge – but they do have the ability to draw objective, disinterested assessments, from sources such as those above.

## VIII Conclusion

There have been a number of recent assessments of the DCF model available to the audiences of legal practitioners in the field of international arbitration.^{23} Some look closely at the accounting inputs needed for the model, and others on the particular facts of the model's use in specific cases. Although such reviews are useful, they do not reflect what, to economists, is the ultimate underlying attraction of the model among the various methods of deriving value. That ultimate attraction is that the DCF model links together finance and economic theory in an uncertain world in a highly effective way. Far from being unduly abstract or restrictive as a model of investment behaviour, the DCF model has a demonstrated track record among administrative agencies, courts and arbitral tribunals. The method reasonably ties market expectations with measurable features of current markets, yet makes no aggressive claims about what is known about future business conditions. Its underlying economic assumptions strengthen the DCF model's use by international tribunals for whom objectivity is of paramount importance.

### Footnotes

^{1} Jeff D Makholm is managing director and Laura T W Olive is associate director at NERA Economic Consulting.

^{2} In 2019, 60 per cent of new cases registered under the World Bank's International Centre for Settlement of Investment Disputes (ICSID) Convention and Additional Facility Rules involved energy and transport concerns. ICSID, *The ICSID Caseload Statistics*, Issue 2020-1, Part II Chart 6, p. 25.

^{3} Financial literature mentions other income-based approaches, such as the adjusted present value and capitalised cash flow, but they are simply variants of the basic DCF model. See M Kantor, *Valuation for Arbitration: Compensation Standards, Valuation Methods and Expert Evidence*, Kluwer Law International, 2008.

^{4} CS Bell, 'Elementary Economics and Depreciation Accounting,' *The American Economic Review*, Vol. 50, no. 1. (March 1960), p. 154.

^{5} For a discussion of the issue of depreciation charges in regulated enterprises, where North America contrasts with the rest of the regulated word, see JD Makholm, *The Political Economy of Pipelines*, Chicago and London: University of Chicago Press, 2011, pp. 160–61, 240 (note 19)–241 (note 20).

^{6} A Charnes, WW Cooper and E Rhodes, 'Measuring the Efficiency of Decision Making Units,' *European Journal of Operational Research*, 1978, volume 2, pp. 429-444.

^{7} G Federico, 'Why are we all alive? The Growth of Agricultural Productivity and its Causes, 1800-2000', European University Institute, paper for the Sixth conference of the European Historical Economics Society, Instambul, 9-10 September 2005, pp. 4–5.

^{8} MJ Gordon, *The Investment, Financing and Valuation of the Corporation*, Homewood IL: Irwin, 1962.

^{9} F Lutz and V Lutz, *The Theory of Investment of the Firm*, Princeton, NJ: Princeton University Press, 1951.

^{10} Gordon, 1962, p. 24.

^{11} Gordon, 1962, p. 43.

^{12} Gordon cited M Gordon and E Shapiro, 'Capital Equipment Analysis: The Required Rate of Profit,' *Management Science*, 1956, Volume 3, Issue 1, as the first appearance of the continuous time theory. Gordon noted that other writers claimed the equation reflected a standard actuarial formula. He also noted that those formulae had no economic content and did not deal with the future values of b and, particularly, r.

^{13} MJ Gordon, *The Cost of Capital to a Public Utility*, 1974, East Lansing, Michigan: MSU Public Utilities Studies, pp. xvii–xviii.

^{14} Alfred Kahn gives the derivation of this model. See: Alfred E Kahn, *The Economics of Regulation; Principles and Institutions*, 1970, New York: John Wiley & Sons, Inc., Volume 1, Appendix A, pages 58–60. The persistent popularity of the DCF model for determining the cost of capital for US regulatory agencies contrasts with the way in which the capital asset pricing model, of economists Franco Modigliani and Merton Miller, dominates regulation in much of the rest of the world. See: JD Makholm, 'Mysterious Cost of Capital for Energy Utilities', *Natural Gas and Electricity*, June 2017, Vol. 34, No. 3, pp. 28–32.

^{15} F Modigliani and M Miller, 'The Cost of Capital, Corporation Finance and the Theory of Investment', *American Economic Review*, June 1958, Vol. 48, No 3.

^{16} Charles F Phillips, *The Regulation of Public Utilities*, Arlington, VA: Public Utilities Reports, 1993, pp. 396–397.

^{17} See: SP Pratt and AV Niculita, *Valuing a Business*, 5th edn, New York: McGraw Hill, 2008.

^{18} www.duffandphelps.com.

^{19} https://www.refinitiv.com/en/financial-data/company-data/institutional-brokers-estimate-system-ibes.

^{21} www.valueline.com.

^{22} www.zacks.com.

^{23} MA Abdala, 'Key Damage Compensation Issues in Oil and Gas International Arbitration Disputes', *American University International Law Review*, 2009, Vol 24, Issue 3, pp. 539–570; A Demuth, 'Income Approach and the Discounted Cash Flow Methodology', *The Guide to Damages in International Arbitration*, JA Trenor (Ed.), *Global Arbitration Review*, London: Law Business Research Ltd, 2016.