Chapter 9 The Economics of Valuation
The value of an asset is the risk adjusted presented value of all future net benefits.
Whether or not the market price of an asset will always reflect its value is a source of
contention among market participants and academics that study financial markets. Some
participants and academics believe that financial markets are efficient in the sense of
quickly eliminating mispricing (i.e., price deviating from value).1 We have seen how the
adherents of the Efficient Market Hypothesis (EMH) explain why any mispricings do not
last long. The EMH provides much of the micro foundations for Modern Portfolio
Theory (MPT), while the theory turns around and provides depth to the EMH (e.g.,
equality of risk adjusted rates of return according to CAPM). Even so, adherents of the
EMH and MPT may still be active traders.2 The trading takes place in assets and markets
where the assumptions of EMH and MPT are not likely to be fulfilled. Hence, the search
for alpha tends to be a search for less efficient markets where more assets tend to be
mispriced.
In recent years we have seen a renewed effort to empirically attack the EMH and
offer explanations for certain anomalies identified in the markets. We have seen that
much of this renewed effort has stemmed from participants and academics that fall under
the category of Behavioral Finance. Thus far, we have identified only the investment
strategy of technical analysis as being bolstered by Behavioral Finance. However, the
new approach may also lend support to those following ‘Fundamental Analysis’ – or, in
other words, attempting to determine the intrinsic value of assets. If market prices in
financial markets are not governed by the dictums of the EMH, then excess profits (or,
excess returns) are possible by identifying intrinsic value. The difficulty that arises for
followers of fundamental analysis will be the patience required for market prices to
eventually return to intrinsic value. The conundrum is that if one believes that market
prices deviate from intrinsic value, then why should one believe also that prices will
eventually return to this value? The technical analyst gets around this problem by
throwing out intrinsic value all together and focusing on market psychology instead.
The current chapter introduces the basics of valuation – i.e., the soul of
fundamental analysis. Although it is quite easy to make a general statement of what
determines value (‘intrinsic value’), it is quite another to put this into practice. Even
here, we will side-step many of the technical difficulties involved in order to provide an
introduction to the approach. The subsequent chapter will present a framework for
conducting a competitive analysis of the corporation’s market. It will be recalled from
It should be remembered that the terms ‘intrinsic value’, ‘true value’, and ‘fundamental value’ are all
used to refer to the value of an asset. The important point being that it – whatever you want to call it –
tends to regulate market price, even if the act of regulation takes a considerable amount of time. In this
sense, one may also use the terms ‘equilibrium price’, ‘long-run price’, or ‘natural price’ to refer to value.
However, the introduction of the word price may lead to some confusion.
2
Recall our previous distinction between ‘traders’ and ‘investors’. A trader has been defined as someone
attempting to make abnormal profits in the short run by identify mispricing. An investor, on the other
hand, is buying for the long-term and hopes to profit from the growth of the underlying asset. The term
trader could be replaced by ‘speculator’ – but the latter terms carries with it negative connotation and
requires far more discussion.
1
1
Chapter 3 that the classical economists assumed that competitive product markets would
achieve equilibrium where firms earned equal risk adjusted rates of return. However,
there was little discussion of how one should study actual competitive markets. We will
focus on a framework developed by Michael Porter as a method of organizing our studies
of actual market conditions.
Chapter 11 applies valuation techniques to a variety of situations. Valuation is
not limited to stocks (or, equity). In fact, valuation techniques are applied to a variety of
financial assets (e.g., bonds, etc.). Moreover, valuation becomes an important resource
for attacking several types of questions that arise. Much of Chapter 11 will be devoted to
valuing the entire firm. Firm valuation arises, for example, in take-over bids. The point
of the chapter will be to introduce the wide variety of ways one may apply the techniques
of valuation.
9.1
Cash is King – The Discounted Cash Flow (DCF) Method
Considering the generic definition of the value of an asset we can classify the
differences in valuation methods differ in terms of (i) what constitutes the net benefit, (ii)
how to arrive at estimates of the future net benefits, and (iii) how to adjust for risk. In
addition, valuing an asset will require a decision on how to ‘close’ the valuation. Closing
the valuation requires a recognition that the further into the future one attempts to
forecast, the more difficult – and, most likely, just plain wrong – the estimation will be.
The following chapter will help us to come to grips with this difficulty.
Formally, we can demonstrate the differences in valuation methods in the
following way.
(9.1)
V0 
NB3
NB1
NB2



1
2
(1  k1 ) (1  k 2 )
(1  k 3 ) 3
Where,
V0 is the current value of the asset
NBi is the net benefit in period (e.g., year) i
ki is the ‘appropriate’ discount rate
Setting aside the question of the ‘appropriate’ discount rate, we must recognize that there
are choices to make about what constitutes the net benefit. The Gordon Model was
introduced in Chapter 3 as one way to define the net benefit for a stock. There, the net
benefit was defined as the dividend paid to the holder of the stock. This was certainly a
reasonable definition of net benefit – after all, the ultimate benefit from holding a stock
was the dividend payment.3 However, what happens if a corporation chooses not to pay
3
Keep in mind that stock repurchases could be included as net benefits within the Gordon Model. In
recent years, stock buybacks have been a favored way to return cash to shareholders by many corporations.
An additional benefit occurs when corporations announce a buyback plan at a particular price. Hence, a
shareholder has an idea of how far the price of the stock can fall – thus, minimizing the downside risk of
owning shares in the corporation.
2
dividends? In fact, corporations that have investment opportunities more profitable than
shareholders would do well to retain their earnings.
Given that not all corporations pay dividends, we may want a more general
measure of net benefit. It may appear simplistic to replace dividends with earnings,
however this gets at something that was actually missing previously. Dividends were
direct benefits to the shareholder. However, we know that if a dividend is not paid, the
shareholder may still benefit indirectly. The stock represents a claim on the equity of the
corporation. Earnings can go one of two places. The earnings can either be paid out as
dividends or be kept within the corporation as retained earnings thereby increasing the
equity of the corporation. If one could trust the accountant’s measure of earnings, then
this could take the place of dividends. We know, however, that the accountant’s measure
of earnings – utilizing Generally Accepted Accounting Principles (GAAP) – is dependent
upon several assumptions and a general attempt to match expenses to the revenues they
generate. Hence, one may want to adjust earnings to reflect only the cash flow generated
and impacting the equity of the firm.
Free Cash Flow to the Firm (FCFF) can be viewed from the angle of its source or
its use. FCFF is generated (i.e., source) by the difference between cash flow from
operations and cash used to purchase operating assets (i.e., capital expenditures). The
FCFF can be used to purchase financial assets, meet financial obligations, or make
payments to shareholders (either in the form of dividends or stock buybacks). Figure 9.1
illustrates the generation and use of free cash flow. The cash flow from operations is
labeled C, capital expenditures I, flows to and from debt holders or simply the net cash
flow to debtholders and issuers F, and flows to and from shareholders or simply net cash
flow to shareholders d. We can now state the definition of free cash flow as the
following.4
FCFF = C – I = F + d
When the firm generates positive free cash flow, then the cash must end up somewhere.
Thus, a positive free cash flow can go towards paying dividends or buying back shares
from shareholders (d) or towards paying interest on financial obligations or purchasing
financial assets in the form of either bank accounts or bonds (F). A negative free cash
flow (C < I) will require the firm to sell some of its financial assets (e.g., decrease a bank
account), issue new debt, or issue new equity.
4
The development of free cash flow here follows that of Penman (2007).
3
Financial Markets
The Firm
Debtholders
C
Net
Operating
Assets
(NOA)
F
or
Debt issuers
Net
Financial
Assets
(NFA)
d
Shareholders
I
Figure 9.1
Source: financial Statement Analysis and Security Valuation, 3 rd edition, Stephen H. Penman, p. 246.
The valuation for a stock requires that only Free Cash Flow to Equity (FCFE) be
used. This is the cash flow that could be used to pay dividends. Before deriving the
FCFE it is worthwhile to follow the implications of the accounting identities. Suppose
we subtract F from both sides.
C–I–F=d
Hence, net payments to shareholders must be the difference between free cash flow and
net payments to debtholders and issuers. It is necessary to break into the F a bit in order
to understand what is happening. The net cash flow to debtholders and issuers (F) is the
difference between the change in net financial assets and net financial income. The net
financial assets (NFA) is the difference between financial assets (e.g., bank deposits,
4
bonds held, etc.) and financial obligations (e.g., bank loans, bonds issued, etc.).5 The
change in net financial assets during a period of time records the net effect of paying
down obligations or issuing new obligations. Net financial income (NFI) is the
difference between interest earned on financial assets and interest paid on financial
obligations.
F = ΔNFA – NFI
Substituting the above definition back into the previous we arrive at a statement for the
net payments made to shareholders.
C – I – ΔNFA + NFI = d
We are now ready to state the Free Cash Flow to Equity (FCFE). The change in
net financial assets must be broken up into the change in Financial Assets (FA) and the
change in Financial Obligations (FO).
ΔNFA = ΔFA – ΔFO
Upon substitution we arrive at the following.
C – I – (ΔFA – ΔFO) + NFI = d
By taking the change in financial assets to the right hand side we get the FCFE.
FCFE = C – I + ΔFO + NFI = d + ΔFA
The far right hand side records the total amount that could be paid to shareholders. If all
is paid, then there will be no change in financial assets. If less than what could be paid to
shareholders is actually paid, the cash must end up increasing the firm’s financial assets.
A typical case might help to clarify the situation. Assume the following:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
The firm generates $120 in operating cash flow
The firm spends $20 on new operating assets
The firm pays down a bank loan by $30
The firms pays $15 in interest
The firm receives $5 in interest from a bank account
The firm pays out $40 in dividends
The firm adds $20 to its bank account
The free cash flow to the firm (FCFF) amounts to $100 ($120 - $20 from 1 and 2). The
reduction of the bank loan means that the firm’s financial obligations have been reduced
by $30 – a negative change. The net financial income is the difference between the
5
Financial obligations may include preferred stock as well. Preferred stock is a hybrid security of equity
and debt. Preferred stock is equity promising to pay a particular dividend. The promise of the dividend
makes preferred stock much more like a debt instrument.
5
interest received ($5) and interest paid ($15), or -$10. The free cash flow to equity is
$60.
FCFE = C – I + ΔFO + NFI = $120 - $100 -$30 - $10 = $60
This is the amount that could be paid out to shareholders. In this example only $40 is
actually paid out to shareholders, what happens to the remaining $20? The remaining
$20 is added to its bank account, thereby increasing its financial assets.
FCFE = d + ΔFA = $40 + $20
The $20 may eventually be used to make capital expenditures, pay off more debt, or
anything else the firm chooses. For now, the $20 must be placed somewhere – here, it is
in a bank account.6
The Free Cash Flow to Equity (FCFE) constitutes the net benefit in the
Discounted Cash Flow (DCF) method of valuation. The FCFE must be forecasted and
the results discounted (i.e., present value) in order to arrive at the value of equity.
(9.2)
V0 

FCFE3
FCFEi
FCFE1
FCFE2






1
2
3
i
(1  k1 ) (1  k 2 )
(1  k 3 )
i 1 (1  k i )
Presumably, the further into the future the less confidence we will have in our forecasts.
In order to avoid attempting to forecast far into the future, we will close the valuation by
arriving at a terminal value. For example, suppose we believe forecasting 3 years into the
future is best.
(9.3)
V0 
FCFE3
FCFE1
FCFE2
T



1
2
3
(1  k1 ) (1  k 2 )
(1  k 3 )
(1  k 3 ) 3
The T stands for the terminal value. A common method of closing DCF valuation is to
assume that the terminal value is based on a constant growth rate. For example, we may
have chosen to forecast three years out because the firm has exceptional opportunities,
which will lead to high growth in FCFE. However, after 3 years elapse, we believe that
competition will eliminate the exceptional opportunities leaving FCFE to grow at a
constant and sustainable rate. We know from the Gordon Model that an assumption of
constant growth reduces to a simple mathematical expression.
(9.4)
T
FCFE4
kg
6
The definition of FCFE given here is actually identical to that given in Darmodaran (p. 89). The
apparent difference between the two is reconciled by noting that Net Income in Darmodaran has already
been adjusted for the Net Financial Income (NFI) – in the example given, the $10 in net interest payments
would have been already subtracted from free cash flow to arrive at net income.
6
The expression for value under the assumption of a 3 year forecast can now be stated
completely.
V0 
(9.4)
FCFE3
 FCFE4 
FCFE1
FCFE2
1





1
2
3
3 
(1  k1 )
(1  k 2 )
(1  k 3 )
(1  k 3 )  k  g 
Or, the DCF can be stated in general.
T
V0  
(9.5)
i 1
FCFEi
1

i
(1  k i ) (1  kT ) T
 FCFET 1 


 kg 
Example of DCF Valuation. Assume the firm expects that a new product line will
generate excess growth in FCFE over the next 4 years. After the fourth year, the firm
expects FCFE to grow at a constant 5%. The cost of (equity) capital – or, appropriate
discount rate – is thought to be 12%. You have forecasted the FCFE for the next 5 years
to be the following.
Year
1
2
3
4
5
FCFE
$100
$150
$195
$225
$235
In order to value the equity in the firm, we simply apply equation (9.5)
T
V0  
i 1

FCFEi
1

i
(1  k i ) (1  kT ) T
 FCFET 1 

 
 kg 
100
150
195
225
1  235 






1
2
3
4
(1.12)
(1.12)
(1.12)
(1.12)
(1.12) 4  .12  .05 
 89  120  139  143  2134  2,624
In short, we have the equity portion of the firm at $2,624. If, for illustrative purposes, the
firm had 100 shares outstanding, then we arrive at a $26.24 value per share.
The example points to a critical feature of a typical DCF valuation. Notice that
nearly 80% of the value comes from the terminal value. In practice, the terminal value
does in fact tend to explain much of the total valuation (70% is a fairly typical
proportion). The assumption about the terminal value (e.g., constant growth rate)
becomes extremely important – though often, this gets less attention due to all the hard
work going into coming up with accurate forecasts of the FCFE.
7
The DCF method is a rigorous approach to valuation. It has been favored by
many for years. However, in practice, the method tends to get watered-down or
approximated.7 A complete DCF valuation requires one to not only estimate earnings but
go further and estimate cash flows, then cash flows to equity. Estimating FCFE can be
quite time consuming and requires a very in-depth knowledge of the inner workings of
the firm. In arriving at our definition of FCFE we made a rather large leap at the very
beginning when starting with cash flow from operations (C). If one were to begin by
estimating earnings (i.e., net income) – which is typical – then several adjustments would
need to be made to get to the cash generated in those earnings. For example, sales
revenue comes in the form of cash sales and credit sales (appearing as accounts
receivable). The credit sales will need to be taken out of the revenue – all non-cash forms
of revenue will be subtracted from earnings. Expenses will include non-cash transactions
as well. Typically, the two largest categories of non-cash expenses will be depreciation
(i.e., expensed tangible assets) and amortization (i.e., expensed intangible assets).
Because depreciation and amortization are non-cash expenses, they will need to be added
to earnings. In short, if one begins by estimating earnings, then any accruals associated
with revenues and expenses will need to be adjusted. The fun does not stop there.
Recall, free cash flow deducts capital expenditures as well. Thus, in order to forecast
FCFE, one will need to forecast capital expenditures. Often this is done by estimating the
capital expenditures required to keep the existing capacity of the firm – unless there is an
indication from management of some significant change in the future. Analysts will
often produce full valuation models – forecasting each of the three financial statements –
for the core stocks they cover, but use truncated versions for related stocks. Part of this is
due to time constraints, but the other part is due to the level of knowledge one must have
on a corporations to complete a full valuation.8
One may note that Darmodaran’s cash flow definition is likely a bit of a mix. Darmodaran begins his
definition of FCFE with Net Income (then subtracts reinvestment and change in debt). By starting with Net
Income, Darmodaran has let accrual based accounting seep into the cash flow definition (that is, unless he
is making an implicit assumption that accruals have already been taken out – which he may, but he
certainly does not draw attention to it). The significant point that Darmodaran seems to be in subtracting
the reinvestment needs (i.e., planned capital expenditures), which is fine – kind of. In fact, if Darmodaran
has not adjusted Net Income (specifically adding back in depreciation and amortization), then the firm will
be penalized twice for making the same capital expenditure (once at the beginning and another time when
depreciation gets expensed).
8
[[[NOTE: If Darmodaran will not be used next time, then DCF will need quite a bit of beefing-up. What
is needed? 1. a “real” example – GIS might be a good one to use, but should also do one with a high
growth corp. (maybe LULU); 2. Cover the various closures in more detail; 3. must cover the various
definitions of FCFE in more detail; 4. the “real” example should show how to go from the actual financial
statements to FCFE; 5. at some point it should be noted that the change in cash must equal the change in
liab + change in equity – change in non-cash assets. 6. how much of this could be done in the previous
chapter covering the financial statements and accounting principles? 7. at what point does this chapter get
turned into two – one for DCF and one for AB? Would that mean adding even another chapter to just deal
with the comparables? The comparables could be pushed into the competition chapter – might make sense
anyway, could really draw out how the comparables method is subject to market sentiment (BF, charts,
etc.) rather than intrinsic value – you actually get more of ‘relative’ intrinsic value and back to Samuelson’s
comment on micro efficient/macro inefficient.]]]
7
8
In practice most DCF models begin with net income (i.e., earnings) and make
adjustments for accruals (i.e., non-cash transactions). Darmadaron (xxx – other, big,
book) provides a practical and rigorous definition of Free Cash Flow to Equity (FCFE).
FCFE =
Net Income
– (Capital Expenditures – Depreciation)
– (Change in non-cash working capital)
+ (New Debt Issued – Debt Repayments)
The second line subtracts net capital transactions (notice, depreciation is really an
addition to FCFE because it is a non-cash expense). The third line subtracts out the
change in non-cash working capital – defined as current assets minus current liabilities –
to capture short-term accruals. The last line adds net cash from additional financial
obligations. This definition of FCFE does in fact match the previous definition.
However, it has the advantage of using terms that appear on actual financial statements.
Theoretically, one would need to forecast the income statement and balance sheet for the
corporation in order to utilize the definition, as well as the capital expenditures. In
practice, it is possible to obtain information on the net debt issued from the corporation’s
annual report or conference call, or based on historical debt ratios. Forecasts of the
change in non-cash working capital could certainly be made based on historical data.
The depreciation during the coming years is obtainable from financial statements
(though, this may require reading the footnotes). The capital expenditures would remain
to be forecasted, which always presents difficulty for this valuation method. However,
the work required to arrive at reasonable forecasts of capital expenditures would certainly
be useful in understanding the competitive position and strategy of the corporation.
9.2
Normal Isn’t Good Enough - Abnormal Earnings (AE) Method
Discounted Cash Flow (DCF), with time permitting along with an in-depth
knowledge of the corporation, is likely the most rigorous valuation method.
Conceptually, there is still a bit of lingering doubt about notion of value creation. The
DCF seems to identify value creation with the attainment of cash. Standard accounting
practices, though certainly subject to manipulation, identifies value creation with its
definition of earnings (or, net income). For valuation purposes, the notion that value is
created when earnings occur seems intuitively appealing. The Abnormal Earnings (AE)
approach presented here goes a step forward in defining value added as the key net
benefit for the valuation model.
We can illustrate the basic insight of the Abnormal Earnings (AE) method with a
simple example. Suppose you deposited $100 into a mutual fund account. You required
a 5% rate of return on the deposit. At the end of one year, the account had grown to
$105. Now, if you could go back in time prior to making the deposit, you may value the
account at $100. However, suppose the account had grown to $108 by the end of the
year. Again, if you could go back in time prior to making the deposit, what would the
value of the account been? Of course, the answer is $102.86 (just finding the present
9
value of $108 a year from now with a 5% discount rate). We can break up the process of
determining value in the following way.
V  $100 
Actual  Re quired
$8  $5
 $100 
 $100  $2.86  $102.86
(1  .05)
(1  .05)
When the account earned just the required rate of return (i.e., 5%) then no additional
value was created. However, when the account earned an excess (or, abnormal) rate of
return, then value was in fact created and added to the original $100. We can think of the
value of the equity in the corporation as a rather larger savings account. The book value
of the equity (that is, the accountant’s definition of the value of equity as assets minus
liabilities) will only increase if the corporation earns more than the required return.
The Book value (B) of a corporation’s shareholder’s equity can be treated much
the same as a large savings account. The accountant’s measure of value is similar to the
old classical treatment of value discussed in Chapter 3. The accountant measures most
assets, liabilities, and shareholders’ equity at historical costs.9 That is, for example, the
value of an asset is determined by what it cost at the time of purchase. The cost of a
physical asset is at least partially dependent upon the costs of production of the asset
(hence, the similarity to the classical approach). As stated in Chapter 3, from the
perspective of financial economics, or finance, the value of an asset depends upon its
future benefits. We have seen this already with the Discounted Cash Flow where the
future benefit was defined in terms of the free cash flow to equity (FCFE). The abnormal
earnings (AE) approach takes a similar view, but with a different definition of future
benefit.
The abnormal earnings (AE) valuation method finds that the accountant’s
definition of earnings is the best representation of value creation. As discussed
previously, the accountant’s definition of earnings (or, net income) attempts to match
revenue with the expenses associated with generating those revenues (i.e., the so-called
matching principle of accrual accounting). However, like the saving account example,
the creation of value (i.e., earnings or interest in the above example) does not necessarily
translate in value added. Earnings will only lead to value added if those earnings exceed
the required return. We can begin to develop the abnormal earnings (AE) by noting the
clean surplus identity.
(9.6)
B1  B0  E1  D1
Where
Bi is the book value of equity in period i
Ei is the earnings in period i
Di is the dividend payment in period i.
9
Accounting principles do treat some assets and liabilities at market costs, implying of course that equity
may change as well since by definition Equity = Assets – Liabilities.
10
The clean surplus assumes that the book value of equity will not change for any other
reason than the difference between earnings and dividends (i.e., the retained earnings
within the period). A dirty surplus would allow for changes in the book value of equity
for some types of changes in market prices of financial assets, currency translations, and
derivatives.10
The Abnormal Earnings (AE) approach was a variant of the Dividend Discount
Model (DDM) in Chapter 3.11 We can rearrange equation (9.6) to solve for the dividend
(D),
(9.7)
D1  B0  E1  B1  E1  ( B1  B0 )
and replace the dividend with (9.7).
V0 
E  ( B1  B0 ) E 2  ( B2  B1 )
D1
D2

  1

 
1
2
(1  k )
(1  k )
(1  k )1
(1  k ) 2
(9.8)


i 1
Ei  ( Bi  Bi 1 )
(1  k ) i
We now use a little algebra trick (note, if you would rather not follow the algebraic
manipulations feel free to skip down to the end result of equation 9.15). The following
expression is a mere identity (you can check this for yourself by multiplying through be
1+k).12
(9.9)
B0
kB0
 B0 
1
(1  k )
(1  k )1
We can use this identity by substituting it into the first term on the far right hand side of
(9.8).
10
For large corporations, the types of changes listed in fact do occur quite often. Multinational
corporations always face the issue of translating some of their accounts from one currency to another.
Financial corporations nearly always have to adjust some of their financial assets, and therefore equity, for
changes in market prices. The abnormal earnings approach to valuation can be done to allow for a dirty
surplus. The mechanics merely require us to switch to comprehensive earnings and comprehensive income
where these changes are accounted for. Most annual financial statements for large firms show what
comprehensive income would be in the detailed shareholders’ equity statement.
11
The Gordon Model presented in Chapter 3 was simply another variation of the Dividend Discount
Model. In order to go from the Dividend Discount Model to the Gordon Model one just assumes the
dividend will grow at a constant rate forever.
12
It should be noted that we are following the derivation by James English in his excellent little book
Applied Equity Analysis (2001). Similar derivations appear in other places – e.g., Penmann (2007) –
though with less clarity.
11
(9.10)
E1  ( B1  B0 ) E1  B1
B0
kB0
E  B1


 1
 B0 
(1  k )
(1  k ) (1  k ) (1  k )
(1  k )
Recall, the definition of Return on Equity (r) as the earnings (i.e., net income) divided by
book value of equity at the beginning of the period. Rearranging the definition of Return
on Equity (r) gives us a way to state earnings.
(9.11) Ei  ri B0
Using the definition of Return on Equity, we can arrange the last expression in (9.10).
(9.12)
kB0
r B  kB0
E1  B1
B1
 B0 
 B0  1 0

(1  k )
(1  k )
(1  k )
(1  k )
We have arrived at an alternative expression for the first term on the far right hand side of
(9.8).
E1  ( B1  B0 ) E 2  ( B2  B1 )

 
(1  k )1
(1  k ) 2
(9.8’)
r B  kB0
B1
E  ( B2  B1 )
 B0  1 0

 2

1
1
(1  k )
(1  k )
(1  k ) 2
V0 
If you have stuck with the derivation this far, then I assume you want to see the rest (it
won’t take long now). We now apply the little algebra trick to the second period.
(9.13)
B1
B1
kB1


2
1
(1  k )
(1  k ) (1  k ) 2
We can substitute (9.13) into the last term expressed in (9.8’) after rearranging.
(9.14)
E2  ( B2  B1 ) E2  B2
B1
E  B2
B1
kB1


 2


2
2
2
2
1
(1  k )
(1  k )
(1  k )
(1  k )
(1  k ) (1  k ) 2
Substituting this back into equation (9.8’) and we find that something drops out.
V0  B0 
r1 B0  kB0
B1
E  B2
B1
kB1

 2


1
1
2
1
(1  k )
(1  k )
(1  k )
(1  k )
(1  k ) 2
(9.8’’)
 B0 
r1 B0  kB0 r2 B1  kB1
B2



1
2
(1  k )
(1  k )
(1  k ) 3
12
Hopefully you see what will happen if we simply repeat this procedure for the third time
period, then the fourth, the fifth, etc. The last expression of (9.8’’) keeps dropping out.
Upon doing this for the infinite series, we finally arrive at our goal.
(9.15) V0  B0 

r1 B0  kB0 r2 B1  kB1
ri Bi 1  kBi 1




B


0
1
2
(1  k )
(1  k )
(1  k ) i
i 1
This is the Abnormal Earnings (AE) valuation model – we’ll start a new paragraph to
discuss it.
The Abnormal Earnings (AE) model of (9.15) looks a bit like the expression we
stated for our saving account at the beginning of this section. The value of the asset (e.g.,
stock) is equal to the book value of equity at the present time period plus the present
value of any excess earnings. For any time period, we can state actual and required
earnings as the following.
Actual Earnings = ri Bi 1
Required Earning = kBi 1
Thus, the difference between actual and required is the excess – or, in this case, abnormal
– earnings. Of course, we can factor out the book value of equity at the beginning of
each period to restate the model.

(9.15’) V0  B0  
i 1
(ri  k ) Bi 1
(1  k ) i
This expression shows us that in order for value to be added to equity, the firm must earn
abnormal earnings. It is not enough, in other words, just to create value (or, have positive
earnings) the firm must add value by achieving a return on equity exceeding the cost of
capital.
The abnormal earnings valuation model has a certain intuitive appeal. Value is
enhanced by growth and reinvestment (i.e., an increase in the book value, Bi-1), but only
if excess returns (i.e, ROE>k) are being earned. In addition, the intuition of the AE
model creates a conceptual bridge between the old classical backward looking view of
value and the financial forwarding looking view from Chapter 3. Recall, the classical
view was that value was determined by costs of production (or, in the extreme form, by
the labor time required in production). The AE model recognizes that part of value is
determined by the past. The current book value of equity depends upon costs and past
decisions.13 However, the historical costs are only part of the story. Value should also
13
Recall, equity is simply net assets (assets minus liabilities). Accounting principles require most assets
and liabilities to be recorded at historical costs. Thus, the book value of equity ends up, for the most part,
being a record of the historical costs of net assets.
13
depend upon what future benefits the asset will generate (i.e., forward looking valuation).
Here again the classical view had something to offer. If market prices rose, for example,
above the intrinsic value, then firms in that market would earn abnormal (or, excess)
returns. Eventually, new competitors would enter the market, increasing supply and
driving down the market price until the abnormal returns were eliminated. The AE
model simply takes the classical view of the competitive process in product markets and
draws out the implications for the value of the assets (thus, equity) operating in those
markets. To illustrate, suppose two firms had identical balance sheets but one had
exclusive rights to produce a particular product, while the other operated in a fiercely
competitive market. The classical economists would certainly see that the firm with
exclusive rights would be able to earn abnormal returns, while the other firm would earn
only the competitive rate of return (e.g., cost of capital). However, they did not seem to
take the next step in seeing that this would mean the value of the assets and equity of the
firm with exclusive rights would be greater than the other firm. The AE model rectifies
this oversight of the classical economists.14
An additional appeal of the AE valuation model is in its application. The model
utilizes financial statements prepared under Generally Accepted Accounting Principles
(GAAP).15 Adjustments to accrual accounting are not required with this approach. On
the other hand, this reminds us that the AE model is simply a rearrangement of the
Dividend Discount Model (DDM). The rearrangement provides the intuition behind the
DDM. In practice, both models require the same forecasting for a complete valuation.
The additional intuition is most helpful when closing the model. Three cases can be used
to illustrate bringing closure to the model.
Case 1. Competitive Equilibrium.
Assume that due to market conditions, the analyst believes the firm will achieve
abnormal returns for the next two years after which it will just earn the cost of (equity)
capital on any investments. The analyst will need to forecast earnings and the payout
ratio (i.e., fraction of earnings paid out as dividends – the remaining will be retained
14
We should not be too hard on the old classical economists. It has been accepted that one can define the
term value as one likes. If one would like to define the term value as the number of labor hours required to
produce a commodity, then so be it. The point is to determine whether the particular definition helps in
economic analysis. For some types of analysis a cost of production definition of value may help very
much. We are suggesting here that it only partially helps when the analysis concerns assets – both real
(physical) and financial. In short, it is not mere semantics that are at stake here, but rather what definition
of value helps shed light on understanding asset markets.
15
There are, however, ways in which a rearrangement of the financial statements could lead to better
valuations. Analysts using AE may in fact reformulate all three of the main financial statements by
grouping accounts in terms of net operating assets and net financial assets. This type of grouping can, and
probably should, be used with all valuations models. Accountants have tended to group accounts according
to time (e.g., current assets and long-term assets). This particular grouping may help those (e.g., a loan
officer at a bank) most interested in understanding the credit worthiness of the firm and/or potential for
bankruptcy. For valuation purposes, it is often more useful to understand the division between operating
and financial activities. The cash flow statement comes close to this sort of group, but falls short with its
definition of investing activities.
14
within the firm and added to equity for next period) for two years only under the AE
model.
V  B0 
AE1
AE2

(1  k ) (1  k ) 2
The assumption of reaching a competitive equilibrium in the product market implies that
the return on equity (r) equals the cost of capital (k) and no abnormal earnings, thus the
valuation stops.
Case 2. Steady-State Abnormal Earnings
Assume the analyst believes that the firm’s position is quite strong even after the two
years. The analyst may forecast out two years, then choose to capitalize a constant
amount of abnormal earnings.
V  B0 
AE1
AE2
1  AE3 




2
(1  k ) (1  k )
(1  k ) 2  k 
This would be a case in which the firm has some permanent competitive advantage in its
market. The competitive advantage may arise due to a patent, first-mover, or anything
else. The discussion of competitive analysis in the next chapter will take up these issues.
Case 3. Abnormal Earnings Growth
The analyst may have reason to believe that after an extraordinary two year period, the
competitive position will only increase for the firm. For example, abnormal earnings are
projected to grow at a constant rate (g).
V  B0 
AE1
AE2
1


2
(1  k ) (1  k )
(1  k ) 2
 AE3 


k g
Keep in mind that abnormal earnings may continue to grow either because the return on
capital increases or growth in book value (as long as r exceeds g).
A numerical example may help to solidify the ideas. Suppose you are given the
following data and forecasts for a particular corporation.
Current Book Value of Equity = $1,000,0000,0000
Shares Outstanding = 10,000,000
Earnings Forecast for Year 1 = $250,000,000
Earnings Forecast for Year 2 = $200,000,000
Payout Ratio = 40%
Cost of Capital = 10%
15
We may use the numbers as given or translate them into per share terms. In per share
terms we have the following.
Book Value per share = $100
Earnings per share Year 1 = $25
Earnings per share Year 2 = $20
Thus, we are forecasting a 25% return on equity for the first year. What about the second
year? It is not 20%! Given an assumed 40% payout ratio, we are assuming that 60% of
earnings (or, $15) will be retained within the corporation and added to the book value of
equity. Thus, in year two, the book value per share will be $115 for a return on equity of
17.4%. Now, we can take the three cases in turn, making additional assumptions where
necessary.
Case 1. Competitive Equilibrium
V  100 
(.25  .10)100 (.174  .10)115


(1.10)
(1.10) 2
 100  13.63  7.03  $120.67
Case 2. Abnormal Earnings are a constant $4 per share.
V  100 
15
8.51
1  4 




2
(1.10) (1.10)
(1.10) 2  .10 
 100  13.63  7.03  33.06  $153.73
Case 3. Abnormal Earnings will grow at a constant rate of 3% after two years.
V  100 
15
8.51
1


2
(1.10) (1.10)
(1.10) 2
4




 .10  .03 
 100  13.63  7.03  47.23  $167.89
The value of the stock is influenced by the assumptions made to close the model.
However, compared with the DCF method, the AE closure assumptions carry
significantly less weight for determination of value. The simple reason is that much of
the value is already contained in the beginning book value.16
16
{{{Note: Kind of an abrupt ending to the section. Possible additions: 1. recap what would need to be
forecasted. 2. show how to use the ratios to forecast. 3. big question is whether or not to show an
example or three using actual corp data. No – need the competitive analysis stuff first in order to do
forecasting; Yes – go back in time and use actual data, thus get a valuation from a previous year. 4. do a
16
9.3
Down and Dirty Valuation – The Comparables Method
The previous valuation methods avoid using market prices.17 The cost of these
valuation methods is the time, information, and patience required. The use of
comparables is (very) often used as an alternative method to a complete valuation. This
method comes in a variety of forms. For example, Jim Cramer of CNBC’s Mad Money
often employs a quick and dirty comparable method whereby value (or, sometimes, a
price target) is found in the following way.
P
V  Em  E  
E
Knowledge of the current, or expected, earnings is combined with some measure of what
the market should be willing to pay for those earnings. If one believes, for example, that
the corporations is average or above average in the sector or the market as a whole, then
one could safely use the average price-earnings ratio in the sector or market to price the
earnings. Several techniques have been introduced in order to make this method more
rigorous. However, the general idea of this method can be seen quite clearly in Cramer’s
use. The weakness of this method, always to be kept in mind, is that value is influenced
by current market sentiment. We do not, therefore, arrive at a value of the asset (e.g.,
corporation) independent of current market prices.
The Gordon Model provides the most straightforward way to derive the various
ratios typically employed in the comparables method.18 We will use only two of the
possible ratios to illustrate the method, others can be derived similarly. Recall the
Gordon Model is a special case of the Dividend Discount Model, whereby dividends are
assumed to grow at a constant rate.
comparison with DCF/FCFE? Possible. Benefit would be to make clear that DCF/FCFE is in fact the best
method. The previous section does not bring this out enough – saying it is the most rigorous doesn’t do it
justice. The problem is once you say it’s the best, then why bother learning AE? Also, the way things
stand it sounds as if these two are competing – they are, but not the way it sounds. Could show, by
example, that the two come to the same valuation if applied consistently --- how messy would that get, and
how far off would that take the discussion? The problem is that leaving things as is makes DCF come
across as bs and AE is so much better. This is certainly the wrong impression to convey. What about
doing an excess returns cash flow model? Could be done fairly quickly – but, so what? If you’re going to
do excess returns then AE is simpler and more efficient.???}}}
17
We will see that market prices may in fact seep into the valuation. We have avoided any attempts to
identify where the actual number to be used for the cost of capital actually comes from. The following
chapter takes up this question. One method of identifying the cost of (equity) capital to be used in the
valuation models may in fact be influenced by market prices.
18
The price-book ratio can easily be derived from the AE model – simply divide thru by the book value.
However, the important drivers for the ratio will be the same as in the Gordon Model. The price-earnings
ratio can, with a certain amount of difficulty, be derived from AE as well. There is actually some benefit
from deriving price-earnings from the model. In addition to the usual drivers, the AE model demonstrates
that price-earnings depend significantly on the growth of abnormal earnings. This may in fact help to
explain some of the odd empirical results obtained in using price-earnings.
17
(9.16) P0 
D1
kg
Value (V) has been replaced with price (P) for ease of exposition only. The priceearnings ratio is found by simply noting that dividends are paid out of earnings and
assuming a constant payout ratio (b).
(9.17) P0 
P
D1
bE1
b

 0 
kg kg
E1 k  g
It is possible to break the price-earnings ratio down further, but (9.17) will be far enough
for now. We see that the drivers of the price-earnings ratio are the payout ratio (b),
growth of dividends and earnings (b), and the cost of (equity) capital (k). The price-book
ratio is derived by noting that earnings can be stated as the product of book value (B) and
return on equity (r).
(9.18) P0 
b * rB0
P
D1
bE1
b*r


 0 
kg kg
kg
B0 k  g
The price-book ratio is driven by the same factors as the price-earnings ratio with the
addition of the return on equity (r). Again, the drivers can be broken down further into
various ratios.19
Intuitively, we can quickly compare a corporation’s ratio (either P/E or P/B) and
drivers with others in the same sector or with corporations with similar attributes (e.g.,
risk factors, debt-levels, balance sheets, stage, etc.). A corporate with lower than average
price-book but higher than average return on equity (other drivers being relatively
similar) compared with others in its sector could be considered undervalued. A similar
process could be conducted for price-earnings ratios. More rigorously, we could run a
regression on the ratio with various drivers as independent variables. For example,
General Mills operates within Processed & Prepackaged goods sector of the Consumer
Goods industry. In November of 2007, General Mills traded at a price-book ratio of 4.16
while earning a return on equity of 23.82%. Choosing 15 related corporations within the
sector and running a regression on price-book with only return on equity as the
independent variable, we arrive at the following regression.
P
 2.16  13.55 * ROE
B
General Mills’ return on equity can be put into the regression to arrive at a predicted
price-book ratio of 5.39. Thus, General Mills is said to be undervalued because it is
currently trading at a lower price-book than justified. It is certainly possible to extend
19
It is often useful to break-down the return on equity into at least the product of the net profit margin,
asset turnover, and equity multiplier – each which could be broken down even further.
18
this method to include more than one independent variable (e.g., growth, cost of capital,
payout ratio). Though, the basic procedure would remain the same. A similar procedure
could be conducted for the price-earnings ratio.
The method of comparables continues to be popular in practice. Partly, this is due
to the ease of obtaining data and running regressions. Another reason for its popularity is
that market sentiment is included in the valuation. The other valuation methods attempt
to bypass any market prices and therefore market sentiment. For long term investors, or
for many other purposes (e.g., banking), this is most likely preferable. However, if one is
valuing for the short-term, then inclusion of current market conditions may in fact be
superior (again, in certain type of banking activities – especially investment banking for
initial public offerings – the current state of the market should be included). A bull or
bear market can last for several years (e.g., the last great bull market in the U.S. was from
1982-1999, the current one is in its fifth year) and not including the state of the market
may mislead the analysis. Regardless, it must be recognized that the inclusion of market
sentiment in the method of comparables is a departure from traditional valuation models.
9.4
Conclusion
{{{need to write. The comparables section was kept short because Darmadaron covers
the same thing in greater detail.}}}
19