Finance Notes by Dr. J. Kashefi Finance Notes by Dr. J. Kashefi
BUSINESS VALUATION MODELS
BUSINESS VALUATION MODELS
This paper deals with the basic theory underlying valuation models. It begins by discussing This paper deals with the basic theory underlying valuation models. It begins by discussing such concepts as free cash flow, cost of capital, and epected growth rates, as they are basic to an such concepts as free cash flow, cost of capital, and epected growth rates, as they are basic to an understanding of valuation theory.
understanding of valuation theory. The ultim
The ultimate goal ate goal of a of a corporatcorporate e manmanager is to ager is to mamaimimi!i!e the e the wealwealth of th of shashareholreholdersders.. "hareholders weal
"hareholders wealth is the value of the firth is the value of the firm #s colm #s collectivlective assets. e assets. To maimiTo maimi!e the sharehold!e the shareholdersers wealth, in general, is e$uivalent to maimi!ing the value of the firm. Therefore, it is imperative that wealth, in general, is e$uivalent to maimi!ing the value of the firm. Therefore, it is imperative that the corpora
the corporate officers %now how to te officers %now how to determine the value of the firmdetermine the value of the firm..
Determining the value of a company is a difficult tas%, since there are various definitions of Determining the value of a company is a difficult tas%, since there are various definitions of &value&, depending on the needs and usage of different users.
&value&, depending on the needs and usage of different users. &'alue& can be
&'alue& can be considered from two different perspectives. considered from two different perspectives. The first The first is is mar%et value, whichmar%et value, which reflects what investors are willing to pay for a firm, which can be determined simply by multiplying reflects what investors are willing to pay for a firm, which can be determined simply by multiplying the price per share by the number of s
the price per share by the number of shares outstanding. hares outstanding. FinancFinancial ial economisteconomists often argue that s often argue that thethe stoc% mar%et is efficient, and therefore, the mar%et value of the firm should be the price the firm stoc% mar%et is efficient, and therefore, the mar%et value of the firm should be the price the firm would bring if sold in the mar%et today.
would bring if sold in the mar%et today.
(lthough the stoc% mar%et as a whole is considered to be efficient, many financial analysts (lthough the stoc% mar%et as a whole is considered to be efficient, many financial analysts feel
feel that individual that individual stoc%stoc%s and firms and firms are seldom s are seldom fairly fairly valued. valued. ThereforeTherefore, a second method, a second method, which is, which is based on the present value of epected cash f
based on the present value of epected cash flows, or what we call ilows, or what we call intrinntrinsic sic finafinancial ncial value, has bevalue, has beenen suggested.
suggested.
)nder effi
)nder efficicient mar%et ent mar%et hyphypothesothesis, we is, we epecepect t the the marmar%et %et valvalue ue to to refrefleclect t the the intintrinrinsisicc financial
financial value. value. *oweve*owever, in recent years, valuation of most fir, in recent years, valuation of most firms rms in min mergers and ac$uisitions, haveergers and ac$uisitions, have resulted in
resulted in errors, and undervaluation of errors, and undervaluation of the companies. the companies. (s a result, these two values (s a result, these two values have not beenhave not been the same, and the focus of this paper will be on the techni$ues of determining the intrinsic financial the same, and the focus of this paper will be on the techni$ues of determining the intrinsic financial value.
value.
THE BASI
THE BASICS OF INTRICS OF INTRINSIC FINANNSIC FINANCIAL VALUECIAL VALUE++
In finance, the intrinsic value of a firm is the present value of a firm#s epected free cash In finance, the intrinsic value of a firm is the present value of a firm#s epected free cash flows over a very long period in theory, to
flows over a very long period in theory, to infinfiniinity. ty. "ince this is "ince this is obviously obviously impossimpossiblible, there is ae, there is a need to
need to develdevelop op modimodifificatications -or ons -or sisimplmplifificicatiations of ons of thithis s modemodel, l, in order in order to to imimpleplemement nt it it forfor practical purpose
practical purposes of valuatios of valuation.n.
This paper will discuss three simplifications of the free cash flow method of valuations that This paper will discuss three simplifications of the free cash flow method of valuations that are, in fact, suitable for practical application. This simplification is based on the growth patterns of are, in fact, suitable for practical application. This simplification is based on the growth patterns of the
the companiecompanies. In s. In general, companies# growth pattegeneral, companies# growth patterns, which can rns, which can be identifibe identified from their industrialed from their industrial life cycle, may be classified as+ -/ supernormal or generally nonconstant growth, -0 normal or life cycle, may be classified as+ -/ supernormal or generally nonconstant growth, -0 normal or constant growth, and -1 !ero growth.
constant growth, and -1 !ero growth.
The industrial life cycle is a representation of an industry#s life as it goes through four stages The industrial life cycle is a representation of an industry#s life as it goes through four stages of growth, shown in (ppendi (, Figure /. The initial pioneering stage or supernormal growth of growth, shown in (ppendi (, Figure /. The initial pioneering stage or supernormal growth stage, the superdecline, maturity or normal growth stage, and the last stage is a decline phase, stage, the superdecline, maturity or normal growth stage, and the last stage is a decline phase, where usually the companies face etinction.
where usually the companies face etinction.
From Figure /, it is possible to determine the period of superior growth by ascertaining From Figure /, it is possible to determine the period of superior growth by ascertaining where the company of interest stands with respect to the cycle. *owever, a word of caution in where the company of interest stands with respect to the cycle. *owever, a word of caution in respect to
respect to the life the life cycle theory+ cycle theory+ it provides only it provides only a framewor%, and has some defia framewor%, and has some deficiencies. ciencies. Firstly, theFirstly, the theory is a highly ideali!ed, and more or less a sub2ective point of view, of how companies or theory is a highly ideali!ed, and more or less a sub2ective point of view, of how companies or industries grow and mature. "econdly, there is no guarantee that a specific industry or a group of industries grow and mature. "econdly, there is no guarantee that a specific industry or a group of companies will systematically pass through these life cycles. Finally, there is nothing inherent in companies will systematically pass through these life cycles. Finally, there is nothing inherent in theory, that provides a way of identifying in advance when a company or an industry will pass from theory, that provides a way of identifying in advance when a company or an industry will pass from
one stage to another. one stage to another.
*owever, it can be stated that the normal growth companies can be characteri!ed as those *owever, it can be stated that the normal growth companies can be characteri!ed as those epected to grow at rates in line with the economy, while supernormal growth companies are epected to grow at rates in line with the economy, while supernormal growth companies are characteri!ed as those that grow at superior rates.
characteri!ed as those that grow at superior rates.
The distinction factors between the models are as follows. First, we epect super growth The distinction factors between the models are as follows. First, we epect super growth companies to have high retention rates and high profitability rates, as measured by rate of return on companies to have high retention rates and high profitability rates, as measured by rate of return on investment or some combination of the two. "econd, we epect normal growth companies to have investment or some combination of the two. "econd, we epect normal growth companies to have low retention rates of profit and3or relatively lower profitability rates. 4orrespondingly, we epect low retention rates of profit and3or relatively lower profitability rates. 4orrespondingly, we epect comp
companianies sustaies sustaininning g highigh h growth growth rates to rates to pay !ero pay !ero or or relarelatitivelvely lower y lower divdivideidends, and nds, and normnormalal growth companies to pay higher dividends.
growth companies to pay higher dividends.
VALUATION MODELS VALUATION MODELS
In theory, there are several valuation models. These models are Discounted 4ash Flow, In theory, there are several valuation models. These models are Discounted 4ash Flow, 5ar%et36oo%
5ar%et36oo% 'alue, and 738 ratio, 'alue, and 738 ratio, which could be used to which could be used to estimestimate the value of a ate the value of a firfirm.m.
The most commonly used approach to valuing a firm is the epected free cash flow -F4F The most commonly used approach to valuing a firm is the epected free cash flow -F4F method of valuation. The princi
method of valuation. The principles of vaples of valuation of a company re$uirluation of a company re$uires the followies the following. ng. First, theFirst, the epec
epected ted frefree e cash flocash flows to ws to all clall classeasses of s of the the capitcapital provideal providers be rs be calcalculculated accordinated according g to to thethe economic setting of the company, its industry, and the economy as a whole, which determines the economic setting of the company, its industry, and the economy as a whole, which determines the si!e and duration of the epected free cash flows. The second principle re$uires that the discount si!e and duration of the epected free cash flows. The second principle re$uires that the discount rate be consistent with the ris% of the epected free cash flows. The consistency between the rate be consistent with the ris% of the epected free cash flows. The consistency between the valuation models, discount rates, and appropriate epected free cash flows is shown in Figure 0. valuation models, discount rates, and appropriate epected free cash flows is shown in Figure 0.
VALUATION TECHNIQUES FOR A FIRM VALUATION TECHNIQUES FOR A FIRM
Discounted Cash Flo Discounted Cash Flo I! "ei#hted A$e%a#e Cost o& Ca'ital ("ACC) A''%oach* I! "ei#hted A$e%a#e Cost o& Ca'ital ("ACC) A''%oach*
The
The 9ei9eighted (veragghted (verage e 4ost 4ost of of 4api4apital estimtal estimates a ates a compcompanyany:s value by :s value by disdiscounticounting ng itsits
unle$e%ed
unle$e%ed &%ee cash &los&%ee cash &los using a using aconstantconstantweighted average cost of weighted average cost of capital. This mcapital. This method re$uiresethod re$uires the calculation of four variables shown in e$uation -/+
the calculation of four variables shown in e$uation -/+
+
+ )nlevered free cash flows -)F4F )nlevered free cash flows -)F4F
,
, )nlevered terminal free cash flow -)TF4F )nlevered terminal free cash flow -)TF4F
-- 8pected future growth rate -g 8pected future growth rate -g
.
. 9eighted (verage 4ost of 4apital, ; 9eighted (verage 4ost of 4apital, ; 9(449(44
(1)
(1)
Equation
Equation
) )T
T
R
R WACC
WACC
+
+
(1
(1
UTFCF
UTFCF
+
+
) )t t
R
R WACC
WACC
+
+
(1
(1
UFCF
UFCF t t
T
T
11
=
=
t t
=
=
V
V
∑∑
8$uation / can also be stated 8$uation / can also be stated as+as+
) ) R R + + (1 (1 UTFCF UTFCF + + ) ) R R + + (1 (1 UFCF UFCF + + + + ) ) R R + + (1 (1 UFCF UFCF + + ) ) R R + + (1 (1 UFCF UFCF = = V V T T WACC WACC T T WACC WACC N N 2 2 WACC WACC 2 2 WACC WACC 1 1 (2) (2) Equation Equation g) g) + + (1 (1 UFCF UFCF = = UFCF UFCF Which Which g g -- R R UFCF UFCF = = UTFCF UTFCF Where Where (T (T ++1)1) T T WACC WACC 1) 1) + + (T (T
and the cost of capital is e$ual to+
r )
Equation
(3)
V
( +
r )
V
!
( t)-(1=
R
WACC # " +! Unle$e%ed F%ee Cash Flos (UFCF)*The unlevered free cash flow which is defined as net operating profit after ta -N<7(T plus various ad2ustments and addbac%s such as depreciation, deferred taes, as well as certain necessary ependiture, -net wor%ing capitalN94 or capital ependiture48 deducted, as a more dependable measure of the firm#s valuation than profits or earnings after ta. 4hanges in profits or earnings can occur without any corresponding changes in cash flows. 5ichael Jensen- /=>> defines the concept of the free cash flow as, &cash flow in ecess of that re$uired to fund all pro2ects that have positive net present values when discounted at the relevant cost of capital&. The basic measurement of unlevered free cash flow -)F4F is as follows+
86IT
?ess+ 4ash Taes on 86IT
7lus+ Depreciation and (morti!ation
O'e%atin# Cash Flo (OCF)
?ess+ Increase in Net 9or%ing 4apital -a
?ess+ Increase in 4apital 8penditures -a
?ess+ Increase in other operating (ssets -b
reduced by increase in other noninterest bearing liabilities
Unle$e%ed F%ee Cash Flo (UFCF)
The )F4F reflects the firm:s true operating cash flow after all re$uired investments. The )F4F is not affected by the company:s capital structure. The capital structure will affect the discount rate, hence value.
-a Increase in net wor%ing capital and capital ependiture+ Investment in net wor%ing capital is often estimated by finding the relationship of net wor%ing capital to sales. Increase in wor%ing capital e$uals net wor%ing capital at the end of the year less net wor%ing capital at the beginning of the year. @enerally, ca'ital e/'enditu%e could be the summation of capital ependiture re$uired, net of replacement of eiting capacity. The formula for these two items is+
Ca'ital In$est0ents 1 CE 2 N"C
where 48 is capital ependiture in fied assets.
9ith identified components of free cash flows, we can epress the )nlevered free cash flow as follows/+
UFCF 1 OCF 3 ( N"C)
(4) Increase in other operating assets reduced by increase in other noninterest bearing liabilities sweeps up any other assets or noninterest bearing liabilities that are related to the operations of the business. These would eclude any current assets or ad2ustment or special investments0.
,! Te%0inal F%ee Cash Flos*
)nless there are specific plans or reasons for terminating the business in the nearterm, the
assumption of &ongoing concern& will re$uire estimating the value of the unlevered free cash flows for an indefinite period. In practice, one ma%es a detailed pro2ection of free cash flow for a period of years, usually A/B years, and then estimates the present value of the free cash flow beyond that period by calculating the terminal free cash flow. The calculation of the )F4F and )TF4F, in
general, depends on the assumption of the epected future growth rate.
The valuation of the business is usually sensitive to the estimate of the terminal free cash flows -TF4F. There are several approaches for arriving at that value.
/ 6oo% 'alue+ ( simple approach for estimating terminal value would be the summation of the net fied asset and net wor%ing capital at the end of terminal year, T.
0 5ar%et 'alue to 6oo% 'alue ;atio+ This method re$uires calculation of the mar%et value3boo% value ratio -5'36', which is multiplied by the Cend of the period cash flow -F4FT+
TF4FE -5'36'-F4FT
1 Discounted Terminal Free 4ash Flows+ This method assumes that the pro2ected cash flow continues beyond the forecast period -T forever. 8$uation 0 shows the estimation of the terminal free cash flow beyond the forecast period, as long as the growth rate is less than the discount rate.
(ssuming that free cash flow is increasing at a constant growth rate -gT after T period, and
so long as the growth rate is less than the cost of capital, the terminal free cash flow is estimated by e$uation -0.
FORECASTIN5 CASH FLO"
The pro2ection of Free 4ash Flows begins with a pro2ection of sales. This pro2ection should ta%e into consideration+
• (n assessment of the company:s recent historical finance performance.
• (n ad2ustment of historical financial performance for nonrecurring and non
operating income and epense items.
• 4ertain assumptions regarding the company:s prospects for the future.
In case of a ta%e over, the forecast of future cash flows should reflect the epected results of the target company assuming that it is managed by the ac$uirer. Therefore, we must ta%e into account+
• 8pected overhead reductions • "everance payments
• "hut downs costs
• 8conomies of scale or synergies
• 4hanges in business strategy -different pricing policy, different level of service, and
mar%eting strategy
• Net proceeds from divestitures
It would be important to note that a great deal of guesswor% would be involved in estimating epected F4F for the above mentioned methods. 5any factors might enter the
estimation of the cash flows, nontheless the F4F model represents a fairly accurate position of the company#s future financial prospects.
Techni$ues used to forecast free cash flows and terminal free cash flow vary widely in degree of sophistication. The simplest method is to loo% at the historical rate of growth and etrapolate this growth into the future. For eample, if sales have been growing at a certain rate in recent years, then the future sales figure could be pro2ected by that rate. If we assume that the various items of the balance sheet and income statement, such as direct costs, general and administrative costs will continue to maintain their historical relationship to sales, then based on pro forma financial statements, free cash flows can be pro2ected annually.
It is important that we understand clearly what is meant by the term growth rate when used in the contet of valuing a company. To value a company, we need to %now how the F4F and TF4F are going to be estimated.
In general, the growth of free cash flows and terminal free cash flow are based on the profitability of the business operation and epected growth of the free cash flows. ( company#s free cash flows can grow in a variety of ways. It can grow by borrowing money, issuing common stoc%s, retention of epected profits or merging with another company. In all of these cases the firm#s free cash flows is growing and this &growth& potential has important implication in valuing the company. "ince growth epectations are unobservable, one needs to resort to proies to assess the sort of growth, that an analyst epects a company or a group of companies to have. There are several growth models that will be discussed in subse$uent sections1.
Conclusion* The techni$ue of using a constant weighted average cost of capital for valuation is widely applied in practice. The usual capital budgeting approach re$uires that prospective investments offer a return in ecess of some Churdle rate.
9hat is often overloo%ed is that the 9(44 will be constant onl6 if the firm maintains a constant debt to capital - or debt3e$uity ratio in mar%et value terms, and the ris% of the investments remains constant. There are a number of situations in which a firm does not intend to maintain a fied debttovalue structure. In these cases, the 9(44 changes over time. 8ach year:s cash flow could have a different 9(44G For eample, in a levered buyout, a firm:s owners begin with a heavy debt load, but are epected to pay off outstanding principal according to a specific timetable, and often very rapidly. The firm:s debt3e$uity ratio declines each year. )nder these circumstances,
Ad7usted 8%esent Value is the most practical method to value cash flows.
II! AD9USTED 8RESENT VALUE+
This is another techni$ue of valuing a company. The (7' and 9(44 use the same form of cash flows. 9ith this method the value of the company is simply the summation of the present value of epected unlevered free cash flows to e$uity holders, and the present value of the ta benefit due to the amount of debt in the capital structure of the company. The value of the
company using (d2usted 7resent 'alue -(7' can be epressed asH+
(7' E7resent value of all 7resent value of the side effects e$uity finance Investment associated with the financing.
($)
Equation
)T
R#
+
(1
R#
)
T C
%(
& T
)t
R#
+
(1
)
T C
%(
& t
T
1
=
t
+
)T
Ru"
+
(1
UTFCF T
)t
Ru"
+
(1
UFCF t
T
1
=
t
=
V
∑
+
∑
+
e$uity
(d2usted present value is very fleible and accommodates any particular pattern of cash flows
and debt financing with its associated ta shields. 9ith this method, we can specify the structure of debt financing and calculate the present value of the interest ta shields, and add it to the present value of epected unlevered cash flows to the e$uity holders. *owever, this process can be tedious if a large number of periods are involved. If we assume that debt is epected to be a constant proportion of the value of the firm, and that the income ta rate is constant, we can simplify the
calculation to a certain degree.
There are two different approaches to (7' if the debt ratio is constant over time. 9ith the first approach, the present value of interest ta shields is calculated using the cost of debtA. 9ith the
second approach, because the debt ratio is epected to be constant, the interest ta shields are tied to the value of the firm and therefore are uncertain. 4onse$uently, the unlevered cost of e$uity should be used to calculate the present value of the ta shields .-this model is not presented here
III! VALUATION OF EQUIT:*
In addition to valuing an entire company, the e$uity portion of a firm can also be valued by setting the second term in e$uation A to !ero. That is+
(')
Equation
)T
Ru"
+
(1
FCF +1
T
+
)t
Ru"
+
(1
FCF t
T
1
=
t
=
V
∑
This model is good only for a total e$uity financed company. If the company has debt in its capital structure, the $alue o& e;uit6 is 4ased on le$e%ed &%ee cash &los and le$e%ed cost o& e;uit6 .
That is+
Equation
()
)T
R
+
(1
T
FCF
+
)t
R
+
(1
FCF t
T
1
=
t
=
V
∑
+
/The difference between e$uation - and -> is due to the amount of debt in the capital structure of the company. The free cash flows to the e$uity investors must account for not only operating cash flow but also for any net return of principal -new debt issues principal repayments to debt holders. The ?F4F is a measure of what a firm can afford to pay out as dividends. 6elow is the estimation of ?F4F.
86IT
?ess+ Interest payment
?ess+ 4ash Taes on 86IT
7lus+ Depreciation and (morti!ation
O'e%atin# Cash Flo (OCF)
?ess+ Increase in Net 9or%ing 4apital ?ess+ Increase in 4apital 8penditures
?ess+ new debt issues minus debt repayments -ND8 ?ess+ Increase in other operating (ssets
reduced by increase in other noninterest bearing liabilities
?evered Free 4ash Flow -?F4F
LFCF 1 OCF 3 ( N"C+ND8)
-! Esti0ation o& 5%oth Rates and Valuation Models<*
The value of a firm is ultimately determined not by current cash flows but by epected future cash flows. The cash flows earned by the company in each period might change, depending on the stability of earnings, it:s future prospects for increasing or decreasing, or other factors. <bviously, there are many factors to consider when attempting to forecast future growth rates of the cash flows.
This section provides an eplanation of how the growth rate is estimated.
A! The Use o& Histo%ical 5%oth Rates
There is a connection between past growth rates and epected future growth rates, but the reliability of this connection is open to $uestion. This section eplores various ways of using historical growth rates to predict future growth.
Usin# A$e%a#e 5%oth Rates &%o0 the 8ast
This approach uses the average growth rate from the past as the predicted growth rate for the future. There are several estimation issues related to coming up with an average growth rate. "ome of these are discussed herein.
+! A%ith0etic A$e%a#e $e%sus 5eo0et%ic A$e%a#e
The average growth rate can be very different depending upon whether it is an arithmetic average or a geometric average. The arithmetic average is the mean of past growth rates, while the geometric average ta%es into account the compounding effect. The latter is clearly a much more accurate measure of true growth in past earnings, especially when yeartoyear growth has been erratic. This can be illustrated with a simple eample.
The following are the earnings per share at LLL 4ompany, starting in /=>= and ending in /==H+
Illust%ation +* Usin# a%ith0etic a$e%a#e $e%sus #eo0et%ic a$e%a#e
:ea% E8S 5%oth Rate ('e%cent)
/=>= MB. /==B B.=B 1.1 /==/ B.=/ /.// /==0 /.0 1=.A /==1 /./1 //.B0 /==H /.0 /0.1=
(rithmetic average E -1.1 /.// 1=.A //.B0 /0.1=3AE /A.> @eometric average E -M/.03M./3A / E /1.==
The geometric average will be lower than the arithmetic average, and the difference will increase with the variability in earnings.
(n alternative to the standard calculation of the arithmetic average is a weighted average,
with growth rates in more recent years being weighted more heavily than growth rates in earlier years. This would lead to a much lower estimate of the average for LLL.
,! Esti0ation 8e%iod
The average growth rate is sensitive to the starting and ending periods for the estimation. Thus, the growth rate in earnings over the last five years may be very different from the estimated growth rate over the last si years. The length of the estimation period is sub2ect to the analyst#s 2udgment, but the sensitivity of historical growth estimates to the length of the period should be a
factor considered in deciding how much to weigh these past growth rates in predictions.
The following table provides earnings per share at LLL, starting in /=>> instead of /=>= and uses si years of growth rather than five to estimate the arithmetic and geometric averages.
Illust%ation ,* Sensiti$it6 o& histo%ical #%oth %ates to the len#th o& the esti0ation 'e%iod* ===
Inc!
Ti0e (t) :ea% E8S 5%oth Rate ('e%cent)
/ /=>> MB.A 0 /=>= B. /.AH 1 /==B B.=B 1.1 H /==/ B.=/ /.// A /==0 /.0 1=.A /==1 /./1 //.B0 /==H /.0 /0.1= (rithmetic average E /1.10
@eometric average E -M/.03M.A/3 / E //.>/
The growth rate drops significantly if earnings per share starting in /=>> are used rather than earnings per share starting in /=>=, with the arithmetic average dropping from /A.> to /1.10.
-! Linea% and La#3Linea% Re#%ession Models
The arithmetic average weighs percentage changes in earnings in each period e$ually, and ignores compounding effects in earnings. The geometric average considers compounding effects, but focuses on the first and the last earnings observations in the seriesit ignores the information in the intermediate observations and any trend in growth rates that may have developed over the period. These problems are at least partially overcome by using <rdinary ?east "$uared regressions
of earnings per share against time. The linear version of this model is+ 87"t E a bT
where 87"t E earnings per share in period t
T E time period
The slope coefficient on the time variable is a measure of earnings change per time period. The problem, however, with the linear model is that it specifies growth in terms of dollar 87" and is not appropriate for pro2ecting future growth, given compounding.
In-87"t E a bT
where In-87" t E natural logarithm of earnings per share in period t
The coefficient b on the time variable becomes a measure of the percentage change in earnings per unit time. The earnings per share from /=>> to /==H are provided for LLL, and the linear and loglinear regressions are done below.
Illust%ation -* Linea% and lo#3linea% 0odels o& #%oth* === Co0'an6
+
Ti0e(t) :ea% E8S In(E8S)
/ /=>> MB.A B.H1 0 /=>= B. B.H0 1 /==B B.=B B.// H /==/ B.=/ B.B= A /==0 /.0 B.0H /==1 /./1 B./0 /==H /.0 B.0H
?inear regression+ 87" E B.A// B.//10T ?oglinear regression+ In-87" E B.AAA1 B./00AT
The slope from the loglinear regression -B./00A provides an estimate of growth rate of /0.0A in earnings. The slope from the linear regression is in dollar terms. The prediction for /==A from each regression is as follows+
8pected 87" -/==A+ linear regression E B.A// B.//10-> E M/.H0
8pected 87" -/==A+ loglinear regression E e-B.AAA1 B./00A ->E M/.A1
.! Dealin# ith Ne#ati$e Ea%nin#s
5easures of historical growth are distorted by the presence of negative earnings numbers. The percentage change in earnings on a yearbyyear basis is defined as+
8e%centa#e chan#es in E8S in 'e%iod t 1 (E8St 3 E8St3+ )>E8St3+
If 87"t/ is negative, this calculation yields a meaningless number. This etends into the
calculation of the geometric mean. If the 87" in the initial time period is negative or B, the geometric mean is not meaningful.
"imilar problems arise in loglinear regressions, since the 87" has to be greater than B for the log transformation to eit. There are at least two ways of trying to get meaningful estimates of earnings growth for firms with negative earnings. <ne is to run the linear regression of 87" against time specified in the previous regression, then the growth rate can be approimated as follows+
@rowth ;ate in 87" E b 3 average 87" over the time period of the regression
This assumes that the average 87" over the time period is positive. (nother approach to estimating growth for these firms, suggested by (rnott -/=>A, is as follows+
8e%centa#e chan#e in E8S 1 ( E8St 3 E8St3+ )>Ma/( E8St ? E8St3+ )
Note that these approaches to estimating historical growth do not provide information on whether these growth rates are useful in predicting future growth. It is not incorrect and, in fact, may be appropriate to conclude that historical growth rate is ¬ meaningful& when earnings are negative, and to ignore it in predicting future growth.
Illust%ation . Dealin# ith ne#ati$e ea%nin#s
Ti0e(t) :ea% E8S Io#(E8S)
5%oth Rate Modi&ied 5%oth Rate / /=>> M1.A /.0 0 /=>= /. B.A AB.0> AB.0> 1 /==B /.B B.B 1=.AA 1=.AA H /==/ B. B.HB 1.1> 1.1> A /==0 B.B> 0.A1 >>.B >>.B /==1 -B./B N5F 00A.BB 00A.BB /==H B.1H /.B> HHB.BB /0=.H/ (pproach /+ )sing the slope coefficient from the linear regression
87" E 1.///H B.A/1=T
(verage 87" -/=>>/==H E M/.B @rowth rate E B.A/1=3/.B E H>.H>
(pproach 0+ )sing the minimum or maimum of earnings as the denominator (rithmetic average, using modified growth rates E A/.>/
.! 8e% Sha%e versus Total Ea%nin#s
The growth rate in net income can be misleading for firms that have issued substantial amounts of new e$uity during the estimation time period. The funds raised from these e$uity issues will generate income that, in turn, will create growth in total net income. *ence, it ma%es sense to ad2ust income for the number of shares issued and to loo% at growth in earnings per share, rather than net income.
The number of shares to be used in calculating earnings per share is also an issue since accountants measure earnings per share relative to both the actual number of shares outstanding -primary 87" as well as in terms of the potential number of shares that could be outstanding, assuming conversion of warrants and convertible bonds -diluted 87". The primary earnings per share, using the actual number of shares outstanding, is the appropriate number to use in calculating earnings growth. The potential dilution effects of convertible bonds and warrants on value can be better assessed using option pricing models to value these securities.
@!Ti0e Se%ies Models to 8%edict Ea%nin#s 'e% Sha%e
Time series models use the same historical information, as the simpler models described in the previous section. They attempt to etract better predictions from this data, however, through the use of sophisticated statistical techni$ues.
The Value o& 8ast 5%oth in 8%edictin# Futu%e 5%oth
7ast growth rates are useful in forecasting future growth, but can seldom be considered sufficient information. In a study of the relationship between past growth rates and future growth rates, ?ittle -/=B coined the term &higgledy piggledy growth,& because he found little evidence that firms that grew fast in one period, continued to grow fast in the net period. In the process of running a series of correlation between growth rates in consecutive periods of different length, he fre$uently found negative correlation between growth rates in the two periods, and the average correlation across the two periods was close to B -B.B0. (n updated study using growth rates in two more recent fiveyear time periods /=>/ /=>A and /=>/==B find that the correlation coefficient in earnings growth, while positive, is still not significantly different from B.
The value of past growth in predicting future growth is determined by a number of factors, including the following.
+! Va%ia4ilit6 in #%oth %ates+ The usefulness of past growth rates in predicting future growth is inversely related to the variability in these growth rates. This variability can be measured in a number of ways. ( simple measure is the standard deviation in growth rates in past 87"+
1
-n
)
g
- g
(
=
2 t n =1 t g∑
σwhere σg E standard deviation in growth rate in earnings per share
gt E growth rate in earnings per share in year t
g E average growth rate in earnings per share over n periods n E number of periods of historical data
the variability can be measured by the standard error of the coefficient estimates as well as the ; s$uared of the regression. In general, analysts should be cautious about using past growth rates in forecasting future growth rates if there is significant volatility in these rates.
,! Sie o& the &i%0* "ince the growth rate is stated in percentage terms, the role of si!e has to be weighed in the analysis. It is easier for a firm with M/B million in earnings to generate a AB growth rate than it is for a firm with MABB million in earnings. "ince it becomes harder for firms to sustain high growth rates as they become larger, past growth rates for firms that have grown dramatically in si!e and profits, may find it difficult to sustain it in the future.
(mgen increased its net income from M/=./ million in /=>= to MH1B million in /==H. The following table shows the growth in net income for (mgen from /=>= to /==H, in both percentage and dollar terms.
Illust%ation <!@* The e&&ect o& sie on #%oth* A0#en
:ea% Net Inco0e 5%oth Rate Net Inco0e
/=>= M /=./B /==B >.1B 1A/.1/ M ./B /==/ />.1B //./1 /BB./B /==0 1B.B H.1 /0B.HB /==1 1AH.=B /A.0 H>.0B /==H H1B.BB 0/./ A./B
@eometric (verage @rowth ;ate E >.H0
(ssuming that this growth rate continues for the net five years+
:ea% Net Inco0e 5%oth Rate Net Inco0e
/==A M >B/.B >.H0 M1/.B /== /,H=H.10 >.H0 =0.0 /== 0,>A. >.H0 /,0=/.1A /==> A,/=0.=> >.H0 0,HB.1/ /=== =,>B.1 >.H0 H,H>.A
The dollar increase in net income needed each year to sustain an >.H0 growth rate becomes larger and larger and rises to MH.H> billion by /===. 8ven if (mgen remains a wellrun and successful firm, it will become progressively more difficult over time to deliver theses high growth rates.
-! C6clicalit6 in Econo06+ *istorical growth rates for cyclical firms are strongly influenced by where in the business cycle the economy is at the time of the estimation. If historical growth rates for cyclical firms are estimated in the middle of a recession, the growth rates are li%ely to be very negative. The reverse is generally true if the estimation of historical growth is done at an economic
pea%. These growth rates are of little value. *owever, in predicting future growth, it would be more useful to estimate growth across two or more economic cycles for these firms.
.! Chan#es in &unda0entals* The observed growth rate is the result of fundamental decisions made by the firm on business mi, pro2ect choice, capital structure, and dividend policy. If a firm changes in any or all of these dimensions, the historical growth rate may not be a reliable indicator for future growth. For instance, the restructuring of a firm often changes both its asset and its liability mi, and ma%es past growth rates fairly meaningless in predicting future growth.
The other problem with using past growth rates arises when the business in which the firm is operating changes, either as a result of mar%et forces or government regulation. These changes in fundamentals may cause a shift upward or downward in growth for all companies in that business and have to be factored into predictions. For instance, pharmaceutical companies at the end of /==0 had en2oyed a decade of high growth as medical technology advanced and health care costs surged. ?oo%ing into the future, however, mar%et forces and the potential for health care reform ma%e it unli%ely that these growth rates would continue.
@! Qualit6 o& ea%nin#s* (ll earnings growth is not e$ual. 8arnings growth created by changes in accounting policy or ac$uisitions is inherently less reliable than growth created by increasing units sold, and should be weighted less in forecasting future growth.
B! The Dete%0inants o& Ea%nin#s 5%oth
9hile growth in a firm may be measured using history or analyst forecast, it is determined by fundamental decision that a firm ma%es on product lines, profit margin, financial leverage and
dividend policy.
Sustaina4le 5%oth? Retention Rate and Retu%n On E;uit6
It is a fairly direct and ob2ective measure of the growth prospect of a company, although it suffers from the usual deficiencies associated with accounting data. *owever, it reveals the interdependence among financial policies when firms pursue a policy of a constant debt ratio and does not sell new stoc%. )nder the assumption of no eternal e$uity financing, the assets of the firm can grow only as fast as it retains earnings -*iggins /=.
The simplest relationship is+
# 1 RR / ROE
In this relationship, growth in earnings is an increasing function of both the retention rate and the return on e$uity. This relationship is also referred to as sustainable growth rate.
(lternative specification of sustainable growth that is consistent with the above e$uation is+
. -. - R*E RR% -1 R*E RR% = g u"taina, e
Inte%nal 5%oth Rate It is the maimum possible growth rate for a firm that relies only on internal financing. . -. - R*A RR% -1 R*A RR% = g &nterna,
;<( is the return on asset.
The return on e$uity and, by implication, the growth rate is affected by the leverage decisions of the firm. In the broadest terms, increasing leverage will lead to a higher return of e$uity if the preinterest, afterta return on pro2ects -assets eceeds the afterta interest rate paid on debt. This is captured in the following formulation of return on e$uity+
ROE 1 ROA 2 D>EROA 3 R d(+ 3 TC)
where ;<( E 86IT- / t36' of Total (ssets D38 E 6' of Debt36' of 8$uity
rd E Interest 8pense on Debt36' of Debt
TC E ta rate on ordinary income
6' E 6oo% 'alue
Note that 6' of (ssets E 6' of Debt 6' of 8$uity.
)sing this epanded version of ;<8, the growth rate can be written as+
# 1 RRROA 2 (ROA 3 R
d( + 3
TC)D>S
The advantage of this formulation is that it allows eplicitly for changes in leverage and the conse$uent effects on growth. It is a useful way of analy!ing the effects of restructuring on growth and value. There are generally three dimensions to restructuring+
1. Restructuring assets/projects- (sset restructuring generally ta%es the form of eliminating unprofitable pro2ects and divisions and3or ac$uiring new assets. The ob2ective in asset restructuring is to increase the firm#s return on assets, which in turn increases growth. The changing of a firm#s asset mi also leads to a change in the ris%iness of the firm, which causes discount rates to shift. The net effect of these changes in growth and ris% will determine the change in the firm:s value. The effect of changing ;<( on growth can be obtained fairly simply using the growth formulation
above+ ) ! + RR(1 = R*A g δ δ
Thus, the effect of a change in the ;<( on growth will depend upon the firm#s retention rate -;; and debt3e$uity ratio -D3".
2. Changing capital structure+ (nother component in financial restructuring is a change in leveragean increase or decrease in the debt financing in the firm. The effect on future growth can be captured by changing the debt3e$uity ratio and interest rate in the growth rate formulation, and recalculating the gro.th rate% The change in leverage will also lead to a change in ris%increa"e" in ,e/erage .i,, increa"e ri"0 .hi,e decreases in leverage .i,, reduce ris% and in discount rates. (gain, the net effect can be either positive or negative.
) C T - )(1 ! (NEW ! r # - ) C T -(1 R # - R*A = ) ! ( g δ δ δ δ
Ne. !1E= #e+t1equit3 ratio a2ter the change in ,e/erage
3. Change dividend policy* The final aspect of financial restructuring is changing dividend policy. ( decrease -increase in dividends will lead to an increase-decrease in the retention rate and an increase in the epected growth rate. This has to be offset, however, by the effects of changing payout ratio on the epected dividends. The tradeoff usually ta%es the form of higher growth for
lower dividends, and the net effect on value can again be either positive or negative.
))4 C T -(1 R # -(R*A ! + -15R*A = 6a3out g δ δ
In general terms, this formulation of epected growth rates ties value to corporate financial policycapital budgeting, capital structure, and dividend policy decisions.
Retu%n on Assets? 8%o&it Ma%#in? and Asset Tu%no$e%
The analysis of return on assets can be carried forward one step if it is related to profit margins and sales.
;<( E 86IT -/ TC3Total (ssets
E86IT -/ TC3"ales L -"ales3Total (ssets
E@ross 7rofit 5argin L (sset Turnover
The return on assets is an increasing function of both the gross profit margin and the asset turnover. The relationship is made more interesting, however, by the tradeoff between the two variablesincreasing profit margins will generally reduce asset turnover, and reducing profit margins will increase asset turnover. The net effect will depend upon the elasticity of demand for the product.
8%oduct Line Anal6sis
<ne criticism that is often leveled at analysts is that by focusing on aggregates at the firm level, they might be missing significant trends in the profitability of individual product lines. Thus, a firm with an aging product line mi may loo% healthy in terms of historical growth and current profitability, but it is not li%ely to sustain this growth into the future. The analysis of growth for a firm can be made more complete by loo%ing at its individual product lines and eamining where they stand in terms of the product life.
The preceding section indicated that the relevant variables of interest are, the epected free cash flows -F4F, growth rate, cost of capital and the valuation model as epressed as in e$uation -/. (s noted, this model can be simplified based on the life cycle theory into three growth models+
VALUATION MODELS and 5RO"TH RATES
The basic model for valuing a business is based on the present value of epected future free cash flows. This section eplores the general model for different assumptions about future growth rate.
ONE3STA5E 5RO"TH MODELS
The most basic simplification of the model applies to the case of !ero growth. (ssuming that sales and net income are epected not to grow at all, the value of the firm, based on e$uation -/, is+
R FCF =
V
This model states that the value ' or present value of the perpetuity is simply the fied free cash flow F4F, discounted by the cost of capital -r of the company.
,! Constant o% No%0al 5%oth Model
This model developed for stoc% valuation by several authors, is probably the most simple and yet practical approach to valuation of companies. "pecifically, this model applies to those companies which have reached the third stage of the industrial life cycles. (t this stage, it is assumed that the companies are growing at a constant -normal growth rate for an indefinite period of time.
9ith this assumption, the value of the firm, by simplifying e$uation -/ is+
g - R g) + FCF(1 = V
Few companies can reasonably meet the assumption of constant growth model. Ooung, developing firms are li%ely to eperience different growth rates for several years initially and then settle down to a constant growth rate, perhaps after a period of five to ten years, thereafter. The formula for finding the value of the company under this condition is stated in e$uation -0.
T"O3STA5E 5RO"TH MODEL*
The general formula applied to twostage growth model is+
()
Equation
)T
R
+
(1
TFCF
+
)t
R
+
(1
)t
g "
+
(1
FCF 7
T
1
=
t
=
V
∑
o%
)T R + (1 TFCF + ) N R + (1 )T g + (1 FCF 7 + + )2 R + (1 )2 g + (1 FCF 7 + R) + (1 ) g + (1 FCF 7 = V g T - R ) g T + (1 )T g + (1 FCF 7 = TFCFWhere and ; is the cost of capital
+ ! = W " + ! ! = W # .here R " W " + R# W # t) -(1 = R WACC
In this e$uation,
g
" represent the super growth rate from tE/ through period T, and gT is the longrun constant growth rate after year T.
It is important to note that the value of the company is a function of the present value of the two cash flows+ -/ the present value of free cash flows from period / to T, which we will call '/, and
the present value of free cash flows from period T / to infinity, referred to as '0.
' E '/ '0
T*;88"T(@8 @;<9T* 5<D8?
The twostage and threestage models differ essentially in the way that they allow for a shift in supernormal growth to growth in line with corporate maturity. The difference between the two
models is that, unli%e the twostage model that assumes a sudden decline of super growth rate to normal growth rate, the threestage model offers a more realistic portrayal of the gradual decline of supergrowth rate to a normal growth rate.
5olodovs%y, 5ay, and 4hottiner -/=Adeveloped the threestage dividend growth model based on the company#s three stage growth life cycle. The threestage model allows for a gradual tapering of the super growth rate to a normal growth rate by provision of the middle stage growth rate. This middle stage growth rate offers a generally more realistic way of portraying the real world pattern of growth and decline than the twostage model. <f course, by setting the middle stage
growth e$ual to !ero we will have the usual twostage growth model.
If we modify this model based on free cash flows, a threestage valuation model based on Figure / shown as e$uation -= can be epressed as+
(8)
Equation
)
T "
+
T #
r
+
)(1
g T
- R "
(
)
g T
+
(1
CF
)
+T #
(
T "
+
)t
R
+
(1
)
g 9
+
(1
CF T "
T #
+
T "
9
+
T "
=
t
+
)t
R
+
(1
)t
g "
+
(1
CF 7
T "
1
=
t
=
V
∑
∑
(17) Equation ) N 9 )( g - g ( - g = g # n " " 9 u9ect g " g # g T 9here t E /,..., T" and 2 E /,..., Td8ssentially the model assumes that free cash flow per share, FCF? will follow the life cycle pattern and go through the three stages of the growth rate. The cash flow will, in the first stage, grow at a super growth rate of #s forTs periods, after which the growth rate declines linearly at # 7,
over Td periods -second stage, as it is calculated by e$uation , until it reaches a normal growth
rate of #n for an indefinite period of time.
.! COST OF CA8ITAL*
'aluation of any asset re$uires the determination of the proper discount rate. There are different approaches to defining the discount rate that are conceptually e$uivalent. This concept is closely lin%ed to the financial concept of re$uired rate of return. The re$uired rate return is defined as the minimum rate of return necessary to induce an investor to buy and hold an asset.
The cost of capital is the minimum re$uired rate of return that investors, bondholders and stoc%holders, will demand as compensation for the ris% they bear if they are not to employ their savings elsewhere, in alternative, identically ris%y securities. That is, management must epect to earn on any new investments at least as much for the shareholders, as the shareholders can anticipate earning elsewhere.
The cost of capital for valuation models depend on the company and its capital structure. If a company has debt in its capital structure, then the appropriate cost of capital is+
R
"ACC1 "
d(+3t)R
d2 "
'! R
'2 "
s! R
swhere r d E cost of debt
r p E cost of preferred stoc%
r s E cost of e$uity
*owever, for a firm with no debt -e$uity financed in its capital structure the cost of capital is the same as the cost of e$uity.
ESTIMATION OF THE COST OF EQUIT:*
There are several methods to estimate the cost of e$uity capital. These methods are+ 4omparable 4ompanies, Discounted 4ash Flow, 4(75, and ;is% 7remium 7ositioning.
+! COM8ARABLE COM8ANIES METHOD
The comparable method typically starts by selecting a sample of firms believed to be of comparable si!e, earnings, capital structure, and ris%. The procedure used to select the comparable method varies widely, depending on the financial analysts# 2udgement of what factors indicates si!e, earnings, and ris%. "ales, total assets, the line of business or other factors could be used. 6ut there is no generally accepted way of defining comparable method.
<nce the sample of comparable companies is determined, the financial analyst calculates the return on e$uity, ;<8, for companies in the sample. ;<8 is the boo% rate of return to stoc%holders. The cost of capital for the company is inferred from this rate either as a simple average, or after some ad2ustments.
,! DISCOUNTED CASH FLO"
The discounted cash flow -D4F method of calculating the cost of e$uity has been based on the @ordon model. It was the first widely used alternative to the comparable method, and remains the most widely used alternative today. ( simple form of D4F is based on the sum of the epected dividend yield D/37B , and the epected growth rate of dividends in the future.
That is+ + g
: 7 !1 = R "
To estimate the cost of e$uity one needs the current stoc% price, an estimate of epected dividends over the net period, and the estimated longterm growth rate of dividends. Dividends and the epected growth rate can be estimated in several ways.
The dividends for the net period can be estimated by multiplying this year#s dividends by the estimated rate of growth. *owever, estimation of an epected growth rate, g, is a more difficult tas%. Two approaches are common.
A! *istorical growth rates of dividends over some period. "ometimes post growth in earnings or boo% value per share is used as a proy.
B! Sustaina4le #%oth %ate, g, is estimated by multiplying ;<8 by the retention rate -;;. That is
# 1 ROE ! RR
In the 4(75, the rate of return on e$uity is based on the ris%free rate and ris% premium which is measured by the 6eta times the mar%ris% premium. That is+
%S 1 Ris3&%ee %ate 2 Beta(Histo%ical Ma%et Ris 8%e0iu0)
R
s1 R
F2
8$uation Awhere is the coefficient of systematic ris% for a levered firm.
%S is the estimated cost of e$uity capital,%F, the ris%free rate, which is estimated as the average or
epected rate of return on Treasury bills in the future, and %0is the rate of return on the mar%et
portfolio.
If a firm has no debt in its capital structure, the unlevered beta is the business ris% inherent in the cost of e$uity. That is+
R
s1 R
F2
)
e$uation )nder the assumptions of the 4(75 the relationship between levered and unlevered betas can be stated based on the *amada#s relationship+
C) D>S
If the ta rate is e$ual to !ero, tEo, then
This relationship is based on the asset beta of a firm, which is a weighted average of its debt and e$uity beta. That is
( firm:s asset beta reflects its business ris%. The difference between its e$uity and asset beta reflects financial ris%. 5ore debt means more financial ris%. Now, if a company decides to use more debt and less e$uity, this would not affect the firm:s business ris%. There would be no change in the firm:s asset beta, and no change in the beta of a portfolio of all the firm:s debt e$uity security. The e$uity beta would change+ ' E D
From this e$uation, we can write+ E
P D>S
if
EB
L( R
M3 R
F)
L U( R
M3 R
F L E U + 2 (+3 T L E U + 2 D>Sasset 1 'o%t&olio
1
de4t(D>V) 2
e;uit6(S>V)
asset de4t e;uit6
asset asset
asset de4tQ
de4t
ESTIMATION OF THE COST OF EQUIT:
The return on a share of common stoc% is from two sources+
%
s1 ca'ital #ain 6ields 2 di$idend 6ield
The one period rate of return, then, is e$ual to+
/
−
t : : 1t - : t + : 1t - !t = R "The beta coefficient can be estimated by an ordinary leasts$uares regression of r " on r m.
.! RISG 8REMIUM
;is% 7remium method is less used as a standalone method. This method is based on adding an eplicit premium for ris% to the current longterm interest rate, usually the interest rate on government bonds. The figure most often used for the mar%etris% premium -r m r f is a historical
mar%et ris% premium of about >.A, based on wor% by Ibbotson R "in$uefield./B
(778NDIL (
Figure / is a representation of an industry life cycles as it goes through three distinct stages of growth.
The initial pioneering stage -super growth period, epansion stage -superdeclining growth, and maturity stage -normal or constant growth are characteri!ed by varying growth patterns in sales, profit, or other measurements.
FI@);8 / Industrial ?ife 4ycle
"T(@8 I "T(@8 II "T(@8 III "T(@8 I'
7ioneering 8pansion 5aturity
-"uper -"uper -4onstant
Decline
;8?(TI<N"*I7 68T988N '(?)(TI<N, F;88 4("* F?<9", (ND DI"4<)NT ;(T8 '(?)8 <F ( FI;5 '(?)8 <F 8S)ITO 7<;TI<N <F T*8 FI;5 T;(DITI<N(? (DJ)"T8D 4(7IT(? 6)D@8TIN@ 7;8"8NT '(?)8 -98I@*T8D ('8;(@8 4<"T <F 4(7IT(? )N?8'8;8D )N?8'8;8D ?8'8;8D F<;5 <F F;88 4("* F?<9" F;88 4("* F?<9" T(L "*I8?D N8T F;88 4("* F?<9"
;8?8'(NT INT8;8"T (ND D86T INT8;8"T (ND F;<5 T< 8S)ITO *<?D8;"
F;88 4("* "8;'I48 N<T D86T "8;'I48 INT8;8"T -INT8;8"T R D86T "8;'I48
F?<9" IN4?)D8D N<T IN4?)D8D D8D)4T8D
F<;5 <F 98I@*T8D )N?8'8;8D 68F<;8 ;8S)I;8D
DI"4<)NT ('8;(@8 4<"T <F T(L 4<"T ;(T8 <F
;(T8 4<"T <F 8S)ITO <F D86T ;8T);N <N
DI"4<)NT ('8;(@8 4<"T <F T(L 4<"T ;(T8 <F
;(T8 4<"T <F 8S)ITO <F D86T ;8T);N <N
4(7IT(? ?8'8;8D8S)ITO
REFERENCES
(rnott, ;.D. /=>A. The use and misuse of 4onsensus 8arnings, Journal of 7ortfolio 5anagement //+/>0.
?ittel, I.5.D. /=0. *iggledy 7iggledy @rowth, <ford + Institute of "tatistics.
5ichael 4. Jensen, &The Ta%eover 4ontroversy+ (nalysis and 8vidence&, in John 4. 4offee and et.al, ed, Knights, ;aiders, and Targets, New Oor%, <ford )niversity 7ress, /=>>, p.10/.
;obert 4. *iggins, &*ow 5uch @rowth 4an a Firm (fford,& Financial 5anagement, Fall /=, 77. /.
NOTES
+* It is reasonable to assume that as the firm goes from high growth to stable growth, the relationship between capital spending and depreciation will change. In the highgrowth phase, capital spending is li%ely to be much larger than depreciation. *owever, this
difference should narrow as the firm enters its stable growth phase.
,* This ad2ustment is often ignored in other definitions of Free 4ash Flow, presumably on grounds of immaterially.
REFERENCES
(rnott, ;.D. /=>A. The use and misuse of 4onsensus 8arnings, Journal of 7ortfolio 5anagement //+/>0.
?ittel, I.5.D. /=0. *iggledy 7iggledy @rowth, <ford + Institute of "tatistics.
5ichael 4. Jensen, &The Ta%eover 4ontroversy+ (nalysis and 8vidence&, in John 4. 4offee and et.al, ed, Knights, ;aiders, and Targets, New Oor%, <ford )niversity 7ress, /=>>, p.10/.
;obert 4. *iggins, &*ow 5uch @rowth 4an a Firm (fford,& Financial 5anagement, Fall /=, 77. /.
NOTES
+* It is reasonable to assume that as the firm goes from high growth to stable growth, the relationship between capital spending and depreciation will change. In the highgrowth phase, capital spending is li%ely to be much larger than depreciation. *owever, this
difference should narrow as the firm enters its stable growth phase.
,* This ad2ustment is often ignored in other definitions of Free 4ash Flow, presumably on grounds of immaterially.
-* differing views on these items would results in different pro2ections of future cash flow and hence value. 6asing sales forecasts solely on past performance is li%e driving your car by loo%ing in the rearview mirror. *owever, a total separation of the future from the past is illogical. Future performance of most businesses is influenced by what occurred in the immediate pat.
.* For more etensive discussion, see ;ichard 6realy and "tewart 5yers, 7rinciple of 4orporate Finance - New Oot%+ 5c@raw *ill, /==B
@* (shton and D. (t%ins, CInteraction of 4orporate Financing and Investment Decision Implication for 4apital 6udgeting+ ( Further comment, Journal <f Finance, December /=>.
* J. 5iles and ;. 8!!ell, C The 9eighted (verage 4ost of 4apital, 7erfect 4apital 5ar%ets and 7ro2ect ?ife+ ( 4larification, Journal <f Financial and Suantitative (nalysis, "eptember /=>B.
<* This section is heavily drawn from (swath Damodaran, Investment 'aluation, /==.
* The original model developed by 5olodovs%y and et.al, was based on dividends streams. The author has modified this model based on free cash flows for valuation proposes
J* 5olodovs%y, Nicholas, 4. 5ay, and ". 4hottiner, & 4ommon "toc% 'aluation+ Theory and Tables,& Financial (nalyst Journal, 5arch (pril /=A, pp. /BH/01.
