

1 Introduction 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 ‘Exhibition: Why the equilibrium stock price is fluctuating’. 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.[1] 2 The Theoretical Framework Fisher’s [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: _{} where, ·
NPV^{I} is the
net present value of the investment project I. ·
NCF^{I}_{t} is the
net cash flow in period t from the investment project I. ·
r_{t} 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 NCF_{t} and r_{t} 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 socalled ‘going concern’ 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 Fisher’s
cash flow formula. 2) Some version of the Modigliani
and Miller’s 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 Fisher’s formula the basic formula calculating
firm value is this: _{} =>[2] _{} (1) where, ·
PV^{F} is the estimated present value of the firm or
the value of ownership. This is also called the equity value. ·
E[FCF_{t}] 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 nonoperating 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 nonequity liabilities. Among other things, it
includes the value of 1) longterm debt, 2) shortterm 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 “true” owners it should be considered as debt
since it deludes the value of ownership). The PVNEL is also called the debt value. ·
E[r_{t}]
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[r_{t}] =
E[r] for ). ·
E[NNOPLAT_{n+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 ‘Model: The General Cash Flow Model’ 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 ‘Table: The essential
accounting statistics’. This table shows exactly what is meant by FCF,
NOPLAT, PVNOA, and PVNEL and how these concepts relate to the firm’s
accountings. 2.2 The weighted average cost of capital (WACC) formula Formula (1) does not answer the question of how the discount
rate (E[r]) is calculated. This is important because as demonstrated in ‘Exhibition: Sensitivity analysis 
Discounted cash flow framework’ 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
meansoffinance 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: _{} (2) where, ·
E[r] is
the expected average cost of capital or the WACC. ·
E[k_{b}]
is the pretax expected cost of capital on noncallable, nonconvertible
debt. ·
T is the
marginal tax rate of the entity being valued. ·
B is the
value of the noncallable, nonconvertible debt. This is PVNEL in formula 1. ·
E[k_{s}]
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 PV^{f} 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 bookkeeping relation. The
generalized version of the WACC is: _{} where, ·
E[k_{i}] is
the expected cost of capital on finance segment V_{i}. E[k_{i}]
is calculated by APT or CAPM as shown below. ·
W_{i} is
the weight given to finance segment V_{i} according to how much value
it represents of the total value: W_{i} = V_{i}/V. ·
V_{i}_{ }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 ‘Model:
The irrelevance of capital structure in perfect capital markets’. 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, k_{s} and
k_{b}. Fortunately, finance theory has developed behavioral models
that provide answers to this question. E.g. the CAPmodel (capital asset
pricing model, sometimes called the pricing formula) and the APTmodel
(arbitrage pricing theory). The CAPmodel is: _{} (3) where ·
E[k] is
the expected cost of capital on the considered capital, e.g. a stock or a
bond. ·
k_{f}
is the expected return on a riskless portfolio. The latter is a portfolio
that yields a constant return period after period with certainty. ·
E[k_{m}]
is the expected return on the market portfolio. ·
b_{m} is the systematic risk or the undiversifiable
risk. b_{m}
measures the size of risk contrary to (E[k_{m}]  k_{f}) that
measures the price of risk. ·
COV(k,k_{m})
is the expected covariance in returns between the capital being priced and
the overall market portfolio. ·
VAR(k_{m})
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[k_{m}] is the only risk factor. The
generalized version of the CAPmodel is the arbitrage pricing theory (APT): _{} where, ·
E[k_{i}] is
the expected return on a portfolio with unit sensitivity to the i’th 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 i’th factor and is
independent of all others. These factors could for example be: market return,
growth rates, shortterm real rates, shortterm inflation, longterm
inflation and default risk. ·
b_{i} is
the sensitivity on the expected return of the capital being priced to changes
in the i’th 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,k_{i}) is
the expected covariance in returns between the capital being priced and the
portfolio with unit sensitivity to the i’th factor and zero sensitivity to
all other factors –I. ·
VAR(k_{i}) is
the expected variance in returns of the portfolio with unit sensitivity to
the i’th factor and zero sensitivity to all other factors i. The mathematically interested should visit ‘Model: The CAPmodel (CAPM)’ where the CAPmodel 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 analyst’s familiarity with other
theories such as accounting theory, transaction cost economics, corporate
finance, organization theory, etc. However, perhaps the most important
knowledge is the analyst’s 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 CAPmodel 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 ‘Table:
Definitions  Certainty, risk, and uncertainty’. Note that certainty
applies to Fisher’s 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 longterm. This is
illustrated in part 2 of ‘Exhibition:
Fundamental value analysis  The big picture.’ 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 SHORTTERM. 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 firm’s plans for the future, the present state
and trend of the firm’s 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 ‘Exhibition: Fundamental value
analysis  The big picture.’ 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 firm’s equity value (PV^{F}).
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 equitymultiplier 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 firm’s
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 firm’s
fundamentals. This is illustrated in ‘Exhibition: Deciding on
fundamentals  The stylized case of credit ratings’. THE MEDIUM and THE LONGTERM. 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 ‘Model:
The
General Cash Flow Model’. Finally,
the mediumterm should be mentioned. To use a metaphor, it is like a bridge
connecting the extremely detailed shortterm analysis with the extremely
simplified longterm 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 firm’s 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 analyst’s 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 PV^{F} by using the present
market value of the firm’s stocks as an estimate of equity value. After that
the PV^{F} is recalculated by applying the former PV^{F}
value as a new estimate of equity value. The process continues until the
difference between two subsequent PV^{F} 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 firm’s 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 CAPmodel 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 k_{f} can be
approximated by the return on a government bond, they do not agree on the
bond’s 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.
[1990] 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 ‘Exhibition: Sensitivity 
Discounted cash flow framework’. 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
firm’s equity value is extremely sensitive to the firm’s cost of capital. The
exhibition also calculates the percentage of the firm’s equity value that can
be attributed to a certain period. E.g. 38% of the hypothetical firm’s value
is produced within the first 5 years when the firm’s 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 growth’s 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 analyst’s 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 analyst’s knowledge of academic
theories. 

 Copyright 19972018, H. Mathiesen. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Legal notice. 
[1] For more information about
the cash flow approach, cf. Copeland, Koller, and Murrin [1990], and Copeland
and Weston [1988, chapter 7, 12, 13, and chapter 20, pages 763769], and
Moskowitz [1988].
[2] The mathematics behind this
step is explained in detail in ‘Model: The
General Cash Flow Model’. It assumes 1) a constant growth rate in NNOPLAT,
2) that E[r_{t}] = 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.