This presentation briefly describes the cash flow approach that fundamental value analysts use to estimate corporate value such as stock prices. It is important to understand how the analysts work, because informed investors are the only kind of investors that are able to give stock prices any informational value. This informational aspect is treated in ĎExhibition: Why the equilibrium stock price is fluctuatingí. The present text proceeds as follows: The first section presents the theoretical framework of the cash flow approach. The second section explains a few things about how the theoretical framework can be applied for actual pricing of corporate value. The final section discusses some illustrative calculations on the sensitivity of corporate value to the cost of capital.
2††††† The Theoretical Framework††
Fisherís [1907, 1930] books on interest rates made it clear that the value of an investment project is equal to the discounted cash flow that this investment generates to its owner(s). The most simple and intuitive formula illustrating this principle is the investment formula calculating the present value of a single investment project under certainty:
∑ NPVI† is the net present value of the investment project I.
∑ NCFIt† is the net cash flow in period t from the investment project I.
∑ rt† is the cost of capital in period t.
The framework underlying the cash flow approach for firm valuation is conceptually identical to this simple formula. The main difference is uncertainty. How does one determine the future values of NCFt and rt in a world of uncertainty? Complicating the task is the fact that firms are not single projects but rather bundles of interrelated investment projects. Furthermore, this bundle is not expected to expire at some date (n). Instead, it is expected to continue indefinitely either by replacements of old projects or by creating new but somehow related projects. This is, the so-called Ďgoing concerní assumption. Stock price analysts have tried to answer the uncertainty question for years and today, the cash flow approach has become the most applied method for sophisticated fundamental value analysis. The theoretical foundation of this framework is composed of a minimum of three theories: 1) A refined version of Fisherís cash flow formula. 2) Some version of the Modigliani and Millerís formula for the weighted average cost of capital that is part of their famous irrelevance of capital structure theorem. 3) Some version of the capital asset pricing model. This section presents the simplest versions of these three theories.
2.1†††††† The basic cash flow formula
Similar to Fisherís formula the basic formula calculating firm value is this:
∑ PVF is the estimated present value of the firm or the value of ownership. This is also called the equity value.
∑ E[FCFt] is the expected nominal free cash flow from the firms operations for period t. This is, the cash flow free to honor returns to debt and equity holders under the going concern assumption. The present value of the FCF stream is called the operating value.
∑ PVNOA is the present value of all non-operating assets. Among other things, it includes the value of 1) overfunded pension funds, 2) excess marketable securities. Together, the PVNOA and the operating value are called the entity value.
∑ PVNEL is the present value of all non-equity liabilities. Among other things, it includes the value of 1) long-term debt, 2) short-term debt 3) operating leases, 4) capital leases, 5) preferred stocks, 6) warrants, 7) convertible debt unlikely to be converted, and 8) stock options (this is stock value but when given as payment by the ďtrueĒ owners it should be considered as debt since it deludes the value of ownership). The PVNEL is also called the debt value.
∑ E[rt] is the expected average cost of capital in period t.
∑ E[r] is the expected average cost of capital. The removal of t is based on the assumption of constant cost of capital for all periods (E[rt] = E[r] for ).
∑ E[NNOPLATn+1] is the normalized net operating profit, less adjusted taxes in period n+1. It is normalized because this earnings measure should be free of any extraordinary influence.
It should be stressed that ĎModel: The General Cash Flow Modelí provides a detailed derivation of the cash flow formula (1), and its underlying assumptions. The essential accounting statistics that formula (1) is based upon are illustrated in ĎTable: The essential accounting statisticsí. This table shows exactly what is meant by FCF, NOPLAT, PVNOA, and PVNEL and how these concepts relate to the firmís accountings.
Formula (1) does not answer the question of how the discount rate (E[r]) is calculated. This is important because as demonstrated in ĎExhibition: Sensitivity analysis - Discounted cash flow frameworkí the discount factor is highly sensitive to the equity value determination. The model used to calculate the discount rate or time value of money is the weighted average cost of capital (WACC) formula. It expresses the opportunity cost that investors suffer by investing their funds in one particular business instead of others with equivalent risk. The idea of the formula is this: The FCF is generated by the economic activities of the firm. These activities are financed in different ways e.g. debt and equity. Each of these means-of-finance have a particular cost of capital. The cost of financing the firms FCF is then calculated as a weighted average of the different costs of capital, weighting each cost of capital (only) in proportion to the value that the financial instrument has relative to the total value of financial instruments. If debt and equity are the only financial instruments the WACC formula becomes:
∑ E[r] is the expected average cost of capital or the WACC.
∑ E[kb] is the pre-tax expected cost of capital on non-callable, non-convertible debt.
∑ T is the marginal tax rate of the entity being valued.
∑ B is the value of the non-callable, non-convertible debt. This is PVNEL in formula 1.
∑ E[ks] is the expected cost of capital on the equity capital of the firm.
∑ S is the value of equity. Actually this is the same as PVf in formula 1.
Equation (2) represents the simplest version of the WACC formula. It assumes two types of finance instruments: debt and equity. It also assumes no personal taxes, a corporate tax (T), and that debt is deductible. The WACC formula is not really a model but rather a definitory equation or a book-keeping relation. The generalized version of the WACC is:
∑ E[ki] is the expected cost of capital on finance segment Vi. E[ki] is calculated by APT or CAPM as shown below.
∑ Wi is the weight given to finance segment Vi according to how much value it represents of the total value: Wi = Vi/V.
∑ Vi is the value of capital segment i.
It should also be noted that the WACC is part of the irrelevance of capital structure theorem developed by Modigliani and Miller. For detailed derivations and proofs visit ĎModel: The irrelevance of capital structure in perfect capital marketsí.
2.3†††††† The capital asset pricing model or the CAPM
The equation does still not answer the essential question of how to calculate the different segment cost of capital, ks and kb. Fortunately, finance theory has developed behavioral models that provide answers to this question. E.g. the CAP-model (capital asset pricing model, sometimes called the pricing formula) and the APT-model (arbitrage pricing theory). The CAP-model is:
∑ E[k] is the expected cost of capital on the considered capital, e.g. a stock or a bond.
∑ kf is the expected return on a riskless portfolio. The latter is a portfolio that yields a constant return period after period with certainty.
∑ E[km] is the expected return on the market portfolio.
∑ bm is the systematic risk or the undiversifiable risk. bm measures the size of risk contrary to (E[km] - kf) that measures the price of risk.
∑ COV(k,km) is the expected covariance in returns between the capital being priced and the overall market portfolio.
∑ VAR(km) is the expected variance of the market portfolio.
Once more, this is the simplest model that finance theory has developed to price capital assets. This model assumes that E[km] is the only risk factor. The generalized version of the CAP-model is the arbitrage pricing theory (APT):
∑ E[ki] is the expected return on a portfolio with unit sensitivity to the iíth factor and zero sensitivity to all other factors -i. In other words, it is the expected rate of return on a portfolio that mimics the iíth factor and is independent of all others. These factors could for example be: market return, growth rates, short-term real rates, short-term inflation, long-term inflation and default risk.
∑ bi is the sensitivity on the expected return of the capital being priced to changes in the iíth factor. It has exactly the same meaning as the partial regression coefficients in a multiple linear regression of the ordinary least squares type.
∑ COV(k,ki) is the expected covariance in returns between the capital being priced and the portfolio with unit sensitivity to the iíth factor and zero sensitivity to all other factors ĖI.
∑ VAR(ki) is the expected variance in returns of the portfolio with unit sensitivity to the iíth factor and zero sensitivity to all other factors -i.
The mathematically interested should visit ĎModel: The CAP-model (CAPM)í where the CAP-model is derived and explained in detail. The above equations (1) to (3) are the theoretical foundation of the cash flow approach. The next section briefly discusses some of the practical problems that arise when this framework is applied for actual stock pricing.
3††††† From Theory to Practice
The fundamental value analysts need to know much more than the skeletal frame of the cash flow approach in order to make their value analyses predictive. This additional knowledge may come from the analystís familiarity with other theories such as accounting theory, transaction cost economics, corporate finance, organization theory, etc. However, perhaps the most important knowledge is the analystís tacit (personal) knowledge that has been accumulated through years of working experience pricing particular firms. The following concentrates on a few of the things that value analyst do in order to apply the cash flow approach in the real world.
††††††††††† The essential problem that the analysts face is how to manage the uncertainty of the real world. The formulas (1) to (3) are in principal incapable of handling this problem, because they are assuming risk, not uncertainty. E.g. the CAP-model assumes that asset returns are normally distributed. However, the real world is one of uncertainty. This is a situation where at least some of the relevant factors to a problem of decision is entirely unknown or cannot be determined. It is very important to understand the difference between certainty, risk and uncertainty. These important concepts are explained in ĎTable: Definitions - Certainty, risk, and uncertaintyí. Note that certainty applies to Fisherís investment formula, risk applies to formula (1) to (3), and uncertainty characterizes the world as it is.
The following mentions a few of the things that the analysts typically do in order to apply equation (1) to (3) in a world of uncertainty. Naturally, different analysts may organize their work in different ways. It is therefore impossible to present anything but a highly stylized picture of how the stock price analysts typically work.
3.1†††††† Estimating future cash flows
How do the analysts predict the future cash flows? One answer that simplifies the matters is to distinguish between historic cash flows, and cash flows in the short-, the medium-, and the long-term. This is illustrated in part 2 of ĎExhibition: Fundamental value analysis - The big picture.í Historical cash flows are of cause not used to calculate present stock value, but they are excellent for evaluating how good the analysts were with respect to previous predictions of cash flows. This is perhaps the best evidence available with regard to how much one can trust present estimations of future cash flows. The following considers the question of future cash flows.
THE SHORT-TERM. Cash flows that are created in the near future are more valuable in terms of present value than cash flows that are created in a distant future. Furthermore, predictions about the distant future are more uncertain than predictions about the near future. Considering these facts, the stock price analysts always devote more time and effort to produce predictions about the near future than they do with respect to the distant future. In particular, they make explicit estimates about the size of the free cash flows 3 to 5 years ahead. This is done by making detailed forecasts about how the different items on the earnings account, the balance sheet, and the cash flow account are expected to develop. E.g. the analysts may predict that sales will grow by 10%, 5%, and 20% the next three years and that investments will be $10, $50, and $30 millions. Such expectations are reasoned by analysis of the firmís plans for the future, the present state and trend of the firmís markets, coming changes in national regulation and so on. In other words, it is analyzed how expected changes and stresses in the internal- and the external fundamentals of the firm effect the value of the firm. This is illustrated in part 1 of ĎExhibition: Fundamental value analysis - The big picture.í The key instrument that the analysts use to do this job is a huge spreadsheet model over the firms accounting statistics. This model incorporates some version of formula (1) to (3) so that it is possible to run different forecasting scenarios and to see how they affect the firmís equity value (PVF). For instance, an optimistic scenario may assume high growth rates in sales and favorable market trends contrary to a pessimistic scenario that assumes small growth rates and increasing competition. The advantage of running different scenarios is that the analysts get a good feeling about the sensitivity of the different assumptions that they make about the development in the fundamentals. In general, the advantage of cash flow analyses is that they consider the timing of earnings and the investment required to generate e.g. new earnings and growth. This makes cash flow analysis more realistic and predictive than for example dividend models and equity-multiplier models.
The quality of the analytical work is enhanced further, by working in teams with specialized members. For example, some analysts follow the national regulation very closely and others are experts with respect to the firmís operational plans and follow product development and investments in new production facilities etc. The point is that a specialized team of analysts is able to possess an enormous amount of knowledge about the firmís fundamentals. This is illustrated in ĎExhibition: Deciding on fundamentals - The stylized case of credit ratingsí.
THE MEDIUM and THE LONG-TERM. When the analysts go beyond 3 to 5 years their analyses are less detailed and contain fewer but much more crude assumptions about the expected development of the particular accounting items. The culmination of such simplifications happens when the analysts consider the long run. This is the cash flows beyond about 12 to 18 years. After that time it does not make much sense to estimate the particulars of the accounting statistics, since nobody can really know, no matter how much they analyze. Instead the analysts focus on getting a single accounting statistic right: The normalized NOPLAT. With this figure plus a discount rate it is possible to calculate the present value of the long run cash flow. Essentially, the second term in formula (1) expresses how this is done. Note that this term is extremely simplifying. Essentially, by applying NOPLAT instead of FCF the formula becomes simpler. However, the cost of making such a simplification is that it assumes a couple of more or less unrealistic assumptions and more importantly that it is extremely difficult to produce an accurate estimate of such an aggregated statistic. For a detailed explanation of why this term uses NOPLAT instead of FCF visit ĎModel: The General Cash Flow Modelí.
††††††††† Finally, the medium-term should be mentioned. To use a metaphor, it is like a bridge connecting the extremely detailed short-term analysis with the extremely simplified long-term analyses by gradually trading off the degree of explicitness with the degree of simplification. For example, a typical medium term assumption is that sales are expected to grow by say 20% starting from year 5 and then gradually fall until it reaches the average nominal growth rate in the economy say 6% in year 18. The abnormally high growth rates could for example be explained by the fact that the firm operates in an emerging industry such as some IT industry.
3.2†††††† Estimating the future cost of capital
Apart from cash flows, the analysts also need to determine an appropriate discount rate in order to calculate the firmís equity value. The theory demands that a specific discount rate is calculated for each future period, but as a practical matter analysts often satisfy by calculating one discount rate and then apply that rate to all future periods. The WACC formula (2) makes it clear that the problem of discount rate determination can be separated into the problem of determining the financial weights and the problem of determining the segment cost of capital.
THE FINANCIAL WEIGHTS. In theory, the weights are calculated by dividing the true value of a particular source of finance with the total true value of all the financial instruments. The analystís problem is how to estimate the true values. The answer depends on the type of debt that is considered and the sources of information that are available about value. For example, debt will typically be valued by its market value if such information is available (e.g. publicly traded corporate bonds). If not, the value could be estimated as the present value of the promised payments discounted by the cost of capital from marketed debt with equal risk. If this is also impossible, the analyst may resort to the accounting value of the particular debt and possible correct its value by personal judgment. It is much more difficult to determine the weight of equity. The problem is that in order to determine the value of equity, we need to know the discount factor, but we cannot determine the discount rate without knowing the value of equity. This is known as the circularity problem. One way out of the problem is iteration. The analysts simply start to calculate PVF by using the present market value of the firmís stocks as an estimate of equity value. After that the PVF is recalculated by applying the former PVF value as a new estimate of equity value. The process continues until the difference between two subsequent PVF calculations is insignificant. Another way out of the problem is to assume that the firm has a target capital structure. For instance the management may target that 40% of the firmís financing comes from equity. This target is then applied as the equity weight.
THE COSTS OF CAPITAL. The CAPM formula (3) suggests how the different segment cost of capital in the WACC formula (2) could be calculated. The CAP-model is typically used for estimating the cost of equity financing, but it may equally well calculate the cost of debt financing. It should be stressed that, so far, there is no consensus among theorists or analysts about how the different elements in the CAPM formula ought to be estimated. For example, although they seem to agree that kf can be approximated by the return on a government bond, they do not agree on the bondís appropriate time to maturity (one month or 30 years?). Neither do they agree whether they should be satisfied with using the present bond return, or whether they should use the average rate of return on this bond through a period of time. Furthermore, those who agree on the latter may not agree on the length of time (10 or 50 years?) or whether one should use a geometric or an arithmetic average. The interested reader may refer to Copeland et al.  for more information on how to apply the cash flow framework.
4††††† Sensitivity Analysis
So far, the cash flow approach is the best technique available for estimating the equity value of large firms. In the hands of professional investors this technique is able to produce estimates of fundamental corporate values that are more accurate than market prices. An interesting question, however, is to ask how certain and how predictive this kind of analysis is? One way to find out is to make a sensitivity analysis. This is done in ĎExhibition: Sensitivity - Discounted cash flow frameworkí.
The numbers in this exhibition are calculated on a hypothetical firm that is 100% equity financed and produces $1 in cash flow each year in an indefinite future. Such a firm is worth $10 if its cost of capital is 10%, and $20 if it is 5%, see exhibition. The point is that a firmís equity value is extremely sensitive to the firmís cost of capital. The exhibition also calculates the percentage of the firmís equity value that can be attributed to a certain period. E.g. 38% of the hypothetical firmís value is produced within the first 5 years when the firmís cost of capital is 10%. The numbers in the exhibition are quite representative of many real world cases despite that they are calculated on an extremely simplified firm. For instance, the introduction of growing cash flows does not change the figures much, see exhibition. The reason is that after a certain time, the return to new investment generating this growth would have to be equal to its cost of capital (because of competition) and this is neutralizing the growthís influence on the present value of the firm.
5††††† Concluding Remarks
The conclusion is that firm value is highly sensitive with respect to the applied discount rate or weighted cost of capital, WACC. Furthermore, the accuracy of the cash flow framework is crucially dependent on the analystís experience and ability to make proper judgments. The latter is probably more a question of solid tacit knowledge than it is a question of the analystís knowledge of academic theories.
-†Copyright 1997-2018, H. Mathiesen. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.†Legal notice.
 For more information about the cash flow approach, cf. Copeland, Koller, and Murrin , and Copeland and Weston [1988, chapter 7, 12, 13, and chapter 20, pages 763-769], and Moskowitz .
 The mathematics behind this step is explained in detail in ĎModel: The General Cash Flow Modelí. It assumes 1) a constant growth rate in NNOPLAT, 2) that E[rt] = E[r] for, in words; the one period forward rate is constant over the entire life of the corporation, and 3) that the return on net new investment (i) is equal to the weighted average cost of capital (WACC = E[r]), that is E[r] = i.