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RATIONAL PRICING OF INTERNET COMPANIES
REVISITED
September 2000
Eduardo S. Schwartz
Anderson School at UCLA
Mark Moon
Fuller & Thaler Asset Management
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capital expenditures and decreases with depreciation. This feature of the model is
important for Internet companies that require large investment in fixed assets.
Fourth, we attempt to improve the bankruptcy condition in the model by allowing
the cash balances to become negative. This allows for future equity and debt financing.
The optimal financing is that which maximizes the value of the firm.
Fifth, we suggest a number of simplifying assumptions that considerably facilitate
the practical implementation of the model. One of these is to set all the speeds of
adjustment in the model equal to one another and derive them from the half-life of the
company to becoming a normal firm. Another is to have only one market price of risk
in the model by assuming that the growth rates in revenues and variable costs are
orthogonal to the market returns.
Sixth, we relate the half-life of the deviations to analysts (or the evaluators)
expectations about future revenues. This comes from the fact that the speed of
adjustment of the growth rate in revenues is the most critical one for valuation purposes.
The expanded model, then, has six state variables, three of which are stochastic
and three of which are deterministic (although path dependant). The three stochastic
variables are revenues, the growth rate in revenues and variable costs. The three
deterministic variables are the amount of cash available, the loss-carry-forward and the
accumulated Property, Plant and Equipment. We solve the problem by Monte Carlo
simulation which can easily deal with this number of state variables and the complex
path-dependencies of the problem.
In Section 2 we present the model with all the new features described above in its
continuous-time version. Since the model is solved by simulation using an interval of
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time that can be quite long, such as one quarter or one year, Section 3 discusses in detail a
discrete time approximation. Section 4 provides an illustrative example of the approach
by pricing Exodus Communications stock and Section 5 presents comparative statics with
respect to some of the key parameters. Section 6 concludes.
2. CONTINUOUS-TIME MODEL
Consider an Internet company with instantaneous rate of revenues (or sales) at
time t given by R(t). Assume that the dynamics of these revenues are given by the
stochastic differential equation:
(1) 1)()()()( dztdtt
tRtdR +=
where )(t , the drift, is the expected rate of growth in revenues and is assumed to follow
a mean reverting process with a long-term average drift . That is, the initial very high
growth rates of the Internet firm are assumed to converge stochastically to a more
reasonable and sustainable rate of growth for the industry to which the firm belongs.
(2) 2)())(()( dztdtttd +=
The mean-reversion coefficient () affects the rate at which the growth rate is expected to
converge to its long-term average, and )/(2ln can be interpreted as the "half-life" of the
deviations in that any growth rate is expected to be halved in this time period.
The unanticipated changes in revenues are also assumed to converge
(deterministically) to a more normal level whereas the unanticipated changes in the
expected growth rate are assumed to converge (also deterministically) to zero.
(3) dtttd ))(()( 1 =
(4) dtttd )()( 2 =
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The unanticipated changes in the growth rate of revenues and the unanticipated changes
in its drift may be correlated:
(5) dtdzdz 1221 =
The costs at time t have two components. The first is a variable component,
which is assumed to be proportional to the revenues. The second is a fixed component.
(6) FtRttCost += )()()(
The variable costs parameter (t) in the cost function is also assumed to be
stochastic reflecting the uncertainty about future potential competitors, market share, and
technological developments. It follows the stochastic differential equation:
(7) 33 )())(()( dztdtttd +=
The mean-reversion coefficient (3) describes the rate at which the variable costs are
expected to converge to their long-term average, and 32ln )/( can be interpreted as the
"half-life" of the deviations in that any deviation is expected to be halved in this time
period. The unanticipated changes in variable costs are also assumed to converge
(deterministically) to a more normal level
(8) dtttd ))(()( 4 =
We also allow for correlation between unanticipated changes in variable costs and
both revenues and growth rates in revenues:
(9) dtdzdz1331
= (10) dtdzdz2332
=
The after tax rate of net income to the firm, Y(t), is then given by
(11) )1))(()()(()( ctDeptCosttRtY =
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The corporate tax rate c in (11) is only paid if there is no loss-carry-forward (i.e.
if the loss-carry-forward is positive the tax rate is zero) and the net income is positive.
The dynamics of the loss-carry-forward are given by:
(12) dL(t) = - Y(t)dt if L(t) > 0
dL(t) = Max [- Y(t)dt , 0] if L(t) = 0
The accumulated Property, Plant and Equipment at time t, P(t), depends on the
rate of capital expenditures for the period, Capx(t), and the corresponding rate of
depreciation, Dep(t). Planned capital expenditures, CX(t), are known for an initial
period and after that they are assumed to be a fraction CR of revenues. Depreciation is
assumed to be a fraction DR of the accumulated Property, Plant and Equipment.
(13) dttDeptCapxtdP )}()({)( =
(14) ttfortCXtCapx = )()(
ttfortRCRtCapx >= )(*)(
(15) )(*)( tPDRtDep =
Then, the amount of cash available to the firm, given by X(t), evolves according
to:
(16) dttCapxtDeptYtrXtdX )}()()()({)( ++=
The untaxed interest earned on the cash available is included in the dynamics of the cash
available to make the valuation results insensitive to when the cash flows are distributed
to the security-holders. Since in the risk neutral framework we discount risk adjusted
cash flows at the risk free rate of interest, we also need to accumulate cash flows at the
same risk free rate. For all purposes, except for determining the bankruptcy condition,
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this is equivalent to discounting the cash flows generated by the firm when they occur as
opposed to at the horizon T.
To avoid having to define a dividend policy in the model, we assume that the cash
flow generated by the firms operations remains in the firm and earns the risk free rate of
interest. This accumulated cash will be available for distribution to the shareholders at an
arbitrary long-term horizon T, by which time the firm will have reverted to a normal
firm.
The firm is assumed to go bankrupt when its cash available reaches a
predetermined negative amount, X
*
. The purpose of this is to allow for future financing.
The optimal amount of new financing in the future is the one which maximizes the
current value of the firm, though we recognize that in many practical situations firms go
bankrupt before their value reaches zero. To obtain the optimal amount of new financing
we decrease X* until firm value is maximized.
The objective of the model is to determine the value of the Internet firm at the
current time. According to standard theory this value is obtained by discounting the
expected value of the firm at the horizon under the risk neutral measure (the equivalent
martingale measure) at the risk free rate of interest, which for simplicity is assumed to be
constant1. The value of the firm at the horizon T has two components. First, the cash
balance outstanding and second, the value of the firm as a going concern, the value which
is assumed to be a multiple M of the EBITDA at the horizon T:
(17) ])]}()([)({()0(rT
Q eTCostTRMTXEV+=
1 It would be easy to incorporate stochastic interest rates into our framework.
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The model has three sources of uncertainty. First, there is uncertainty about the
changes in revenues, second, there is uncertainty about the expected rate of growth in
revenues and third, there is uncertainty about the variable costs. We will assume that only
the first source of uncertainty has a risk premium associated with it and later on we will
relate this risk premium to the beta of the stock. Under some simplifying assumptions
(see for example Brennan and Schwartz (1982)), the risk adjusted processes for revenues
can be obtained from the true processes as in:
(18) *1)()]()([)(
)(dztdttt
tR
tdR +=
where the market price of factor risk is constant. This market price of risk is equal to
the correlation of the revenue process and the return on aggregate wealth (proxied by the
market portfolio) times the standard deviation of the return on aggregate wealth (proxied
by the market portfolio). Since the rate of growth in revenues process and the variable
cost process do not have risk premiums attached to them the true and the risk adjusted
processes are the same.
Most analysts and investors are more interested in the price of a share than in the
value of the whole company. To obtain the price of a share we need to examine the
capital structure of the company in detail. We need to know how many shares are
currently outstanding and how many shares are likely to be issued to current employee
stock option holders and convertible bondholders. We also need to know how much of
the cash flows will be available to the shareholders after coupon and principal payment to
the bondholders.
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To simplify the analysis, we assume that if the firm survives options will be
exercised and convertible bonds will be converted into shares.2 This means that in each
of the paths of the simulation where the company does not go bankrupt, we adjust the
number of shares to reflect the exercise of options and convertibles. To obtain the cash
flows available to shareholders from the cash flows available to all security-holders
(which determine the total value of the firm), we subtract the principal and after-tax
coupon payments on the debt and add the payments by option holders at the exercise of
the options. Since we are assuming that the firm pays no dividends, the exercise of the
options and convertibles occurs at their maturity. If all option holders exercise their
options optimally, the above procedure overvalues the stock by undervaluing the options
and convertibles, since there might be some states of the world where the firm survives
but it is not optimal to exercise the options or convert the convertible.
In addition, it is well known, that for diversification purposes, employee stock
options are frequently exercised before maturity, if they are exercisable, to allow for the
sale of the underlying stock. Also, even if they are in the money, not all the options will
be exercised since many of the employees will leave the firm before their options become
vested. However, over time the firm will likely issue additional employee stock options
to existing and new employees. If the number of shares to be issued at exercise and
conversion is small relative to the total number of shares outstanding, the impact of these
issues on the share value is likely to be very small.
Implicit in the model described above is that the value of the firm and the value of
the stock at any point in time are functions of the value of the state variables (revenues,
expected growth in revenues, variable costs, loss-carry-forward, cash balances and
2 The Longstaff and Schwartz (2000) approach could be used to obtain a more precise exercise strategy.
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accumulated Property, Plant and Equipment) and time. That is, the value of the stock can
be written as:
(18) ),,,,,,( tPXLRSS
Applying Itos Lemma to this expression we can obtain the dynamics of the stock value
(19)
ddSdRdSdRdSdSdS
dRSdtSdPSdXSdLSdSdSdRSdS
RR
RRtPXLR
++++
++++++++=
2
212
21
2
21
The volatility of the stock can be derived directly from this expression
(20) 2313122222 222 222)()()(
SVS
SSS
SSS
SS
SS
SS
S RRRRRR +++++=
The partial derivatives of the stock value with respect to the level of revenues, to
the expected rate of growth in revenues, and to the variable costs are obtained by
simulation3. Since the volatility of the expected growth rate of revenues (0) is one of the
most critical parameters in the valuation model and is difficult to estimate, equation (20)
is used in the implementation of the model to imply 0 from the observed volatility of the
stock.
Finally, using expression (19) and the continuous time return on the market
portfolio, and noting that only the revenue process has a risk premium associated with it,
we can write the beta of the stock as:
(21)222
)()(
M
R
M
MRMR
M
SM t
S
RSt
S
RS
===
Expression (21) establishes the relation between the market price of risk in the model and
the beta of the stock, and can be used to infer the value of the market price of risk, .
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3. DISCRETE-TIME APPROXIMATION OF THE MODEL
The model developed in the previous section is path-dependent. The cash
available at any point in time, which determines when bankruptcy is triggered, depends
on the whole history of past cash flows. Similarly, the loss-carry-forward and the
depreciation tax shields, which determine when and how much corporate taxes the firm
has to pay, are also path-dependent. These path-dependencies can easily be taken into
account by using Monte Carlo simulation to solve for the value of the Internet company.
In the implementation of the model we assume that all the mean reversion
coefficients are equal and their unique value is inferred from the expected half-life of
their deviations. To perform the simulation we use the discrete version of the risk-
adjusted processes:4
(22)})(]
2
)()()({[ 1
2
)()(
ttt
ttt
etRttR+
=+
(23) 22
)(2
1)1()()(
t
eetett
ttt
++=+
(24) 3
2
)(2
1)1()()(
t
eetett
ttt
++=+
where
(25) )1()( 0tt eet
+=
(26) tet = 0)(
(27) )1()( 0tt eet
+=
3 The initial value of the revenues (the rate of growth in revenues and the variable costs) is perturbed toobtain new values of the equity from which these derivatives are computed.
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Equations (25) - (27) are obtained by integrating (3), (4) and (8) with initial values 0 ,
0 and 0 . 1 , 2 and 3 are standard correlated normal variates.
Note that as time grows the volatilities in the model converge to:
(28) =)(
(29) 0)( =
(30) =)(
and the growth rate in revenues converges to
(31) =)(
This implies that the revenue process converges to
(32) 1)(
)(dzdt
R
dR +=
The net income after tax is still given by equation (11) where all variables are
measured over the period t . Similarly, the discrete versions of the dynamics of the
loss-carry-forward, the accumulated Property, Plant and Equipment, and the amount of
cash available are immediate
Depending on the situation, the unit of time t chosen for the analysis is one quarter
or one year. The advantage of using the latter is that it allows for smoothing of seasonal
effects and is more consistent with typically annual longer-term analyst projections.
4. ILLUSTRATIVE EXAMPLE
In this section we apply the model described in previous sections to value Exodus
Communications. Exodus is one of the leading providers of Internet web-page hosting
4 Note that there is a typo in the equation corresponding to (23) in Schwartz and Moon (2000).
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for companies with high-traffic web sites. It was founded in 1994 and by August 2000 it
had a market capitalization of over 22 billion dollars.
The results obtained from our valuation approach, like those of any other valuation
approach, depend critically on the assumptions we make about future revenues, rates of
growth in revenues, and costs. In what follows we describe how we estimate the
parameters needed to run the Monte Carlo simulation from the past and present data
available for Exodus, and from analysts forecasts relating to the company. Since most
forecasts of capital expenditures, revenues and costs are available on an annual basis, all
our simulations are done on an annual basis.
Revenue Dynamics
As the starting value for the revenue simulation we take the actual revenues for
1999 of $242.2 million. The initial volatility of revenues is obtained from quarterly data
from the first quarter of 1997 to the second quarter of 2000, and annualized to give 0.135.
This volatility is assumed to decrease with time to the long-term volatility of revenues of
0.065. The half-life of this deviation and of all the others in the model are assumed to be 2
years. Later on we explain in more detail this last assumption.
Growth Rate of Revenues Dynamics
The initial growth rate in revenues is taken to be 120% per year which is what
analysts expect it to be from 1999 to 2000. It is assumed that in the long run, when
Exodus becomes a normal firm, this growth rate will decrease to 5% per year. The
initial annual volatility of the growth rate, which is one of the critical parameters in the
model, is unobservable. We imply it from the volatility of the stock. Figure 1 shows the
weekly implied volatility of Exodus options from September 25, 1998 to August 4, 2000.
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The average implied volatility of the stock during this period was 96.2%. Figure 2
illustrates the relation between the model price and volatility of the stock and the
volatility of the growth rate of revenues. As an input to the model we choose the
volatility of the growth rate of revenues, 0.23, which gives a model stock volatility which
approximates the market volatility. Note that the corresponding level of the stock price is
$49.92.
Variable Costs Dynamics
Using actual data for 1998 and 1999, and analyst estimates for 2000 to 2009,
Figure 3 reports the results of regressing cash costs on revenues. From these results we
take 0.56 as the initial variable cost as a fraction of revenues, and $382 million as the
annual fixed cost. Since it is unlikely that the firm will continue to have such large profit
margins forever, we assume that in the long run variable costs will increase to $0.75 per
dollar of revenue. Similarly, we assume that the initial volatility of variable costs of 0.06
will decrease in the long run to 0.03.
Half-Life of Deviations and Correlations
As mentioned earlier, to simplify the analysis we have assumed that all the mean
reversion processes in the model have the same speed of adjustment coefficient. But only
the one which determines how fast the initial growth rate in revenues reverts to the long-
term rate has a significant effect on valuation. This is not surprising. If we start with an
initial growth rate of 120% and we assume that in the long run it decreases to a more
normal 5%, then the speed at which it decreases will have a tremendous impact on future
revenues. Figure 4 shows the expected revenues up to year 10 (year 2009) for different
assumptions about the half-life parameter. Note that if we increase the half-life from the
5 The valuation results are not sensitive to this parameter.
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assumed 2 years to 2.2 years, the expected revenues in year 2009 increases from $18.2
billion to $23.8 billion. Clearly, this should have a major impact on valuation. Finally
note that the analysts expectation of eventual revenues of $30.3 billion seems too large
in our view.
In this illustration we have assumed that the three stochastic processes are
uncorrelated. In the next section we look at the effect of possible correlations between
the variables.
Balance Sheet Data
The cash and marketable securities available after two bond issues on July 6, 2000
was $1,720.2 million and the loss-carry-forward at the end of the second quarter of 2000
was $337.9 million. The accumulated property, plan and equipment at the end of 1999
was $390.6 million.
The number of shares outstanding after the stock split of June 20, 2000 was
412.36 million shares. In addition the company had 110.3 million employee stock
options outstanding with a weighted average life of 8.9 years and a weighted average
exercise price of $7.71.6 Finally, the firm had six debt issues and two convertible bond
issues for a total outstanding amount of $2,547 million. This information, together with
the maturity, coupon and conversion ratios (for the convertibles) is used to determine the
price of the stock starting from the total value of the firm.
Capital Expenditures and Depreciation
Analysts have made forecasts of the capital expenditures of Exodus for the next 9
years starting form $1,350 million in year 2000 and decreasing to $600 million in year
2008. We use these as inputs to the model and starting from 2009 we assume that capital
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In all cases we use 10,000 simulations with steps of one year and up to a horizon
of 20 years. At the horizon we assume that the value of the firm is a multiple of ten times
earnings before interest, taxes, depreciation and amortization (EBITDA).
Simulation Results
Using the parameters described above, the model stock price for Exodus
Communications that we obtain is $49.92, with model volatility and beta that closely
match (by construction) observed volatility and beta of the stock. The model (risk
neutral) probability of bankruptcy is 3.4% with default occurring in years 5 to 9 (0.2% in
year 5, 0.8% in year 6, 1.2% in year 7, 0.7% in year 8, and 0.6% in year 9).
On August 9, 2000 the market price of Exodus at the close was $54.75. This is
approximately 10% above the model price.
Figures 7, 8 and 9 show the frequency distribution of revenues, growth rates in
revenues and variable costs, respectively, at the end of year 3 implied by the above
parameters. As can be seen from the figures there is substantial uncertainty about the
state variables of the model even three years in the future. (Note that with only 10,000
simulations the frequency distributions do not appear as smooth as they really are.)
5. SENSITIVITY ANALYSIS
In this section we perform some sensitivity analysis on the more critical or
controversial parameters of the model.
Correlations
Consider the situation in which the firm is in a competitive environment such that
increases in profit margins (i.e. decreases in variable costs) are associated with decreases
in growth rates in revenues. This implies a positive correlation between variable costs
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and growth rates in revenues. We run the program with the same data described above,
but assuming a correlation of 0.8. The market price of risk and the volatility of the
growth rate had to be adjusted slightly to match the volatility and the beta of the stock.
The model price decreased only slightly to $49.60, but the (risk neutral) probability of
default decreased substantially to 1.9%. The effect of this correlation on prices increases,
however, with the volatility of variable costs.
Variable Costs
As expected long-term variable costs have a big effect on prices. If long-term
variable costs are increased from 0.75 to 0.80, the model stock price goes to $38.91 and if
they are decreased to 0.70, the model stock price goes to $60.35. In both cases small
adjustments are needed in the market price of risk to match the beta of the stock. This
result is not surprising: a 20% increase or decrease in the profit margin has a similar
effect on model stock prices.
Bankruptcy Level
Changes in the amount of future financing allowed has a very small effect on
stock prices since we seem to be close to the optimal amount of future financing. If we
increase the financing constraint from $3 billion to $3.5 billion the stock price increases
only to $50.00 and if we decrease it to $2.5 billion the stock price decreases only to
$49.82. These differences are well within simulation error.
Half-Life of Deviations
As mentioned earlier and as Figure 4 shows, the speed of adjustment has an
important impact on expected future revenues and therefore on valuation. If we increase
the half-life parameter from 2 to 2.2 years, the stock price increases to $73.55 (recall that
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expected revenues in year 10 increase from $18.2 billion to $23.8 billion). If we decrease
the parameter to 1.8 years the stock price decreases to $33.39 (the expected revenues in
year 10 decrease to $13.9 billion).
6. CONCLUSION
In this article we have substantially extended and perfected the model developed
in Schwartz and Moon (2000) for pricing Internet companies. We introduce uncertainty
in costs and we take into account the tax effects of depreciation. We use the beta and the
volatility of the stock to infer two critical unobservable parameters of the model, the
market price of risk and the volatility of growth rates in revenues. We improve the
bankruptcy condition by allowing future financing by the firm. Finally, we facilitate the
implementation of the model by assuming the half-life of all processes is the same and
suggest that it can be estimated from evaluators expectations of future revenues.
Like any other valuation method, our real options approach to value Internet
companies depends critically on the parameters used in the estimation. The next step in
the implementation of the model would be to use cross-sectional data of many Internet
companies to estimate some of the parameters. Finally, since the growth rate in revenues
is not observable learning models could be used to update this critical factor in the
model when new information on revenues is revealed.
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References
Brennan, M.J. and E.S. Schwartz, (1982), "Consistent Regulatory Policy Under
Uncertainty," The Bell Journal of Economics, 13,, 2, 507-521 (Autumn).
Longstaff, F.A. and E.S. Schwartz, (2000), Valuing American Options by Simulation: A
Simple Least-Square Approach, Review of Financial Studies (forthcoming).
Schwartz, E.S. and M. Moon, (2000), Rational Pricing of Internet Companies,
Financial Analysts Journal 56:3, 62-75 (May/June).
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Figure 1
Exodus: Implied Volatility (%)
0
50
100
150
200
250
24-Jul-98 01-Nov-98 09-Feb-99 20-May-99 28-Aug-99 06-Dec-99 15-Mar-00 23-Jun-00 01-Oct-00
Implied
Volatility
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Figure 2
Exodus Communications: Valuation
40
42
44
46
48
50
52
54
56
58
0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28
Volatiltiy of the Revenue Growth Rate
StockPrice
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
Volatility
Stock Price
Volatility
0.97
$49.92
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Figure 3
EXDS: Cash Costs vs. Revenues
Not Including CAPX
1998A, 1999A, E[2000-2009]
y = 0.5623x + 382.4
R2
= 0.9991
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
18,000
20,000
0 5,000 10,000 15,000 20,000 25,000 30,000 35,000
Annual Revenues ($MM)
AnnualCostsinEBITDA($MM)
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Figure 5
EXDS returns vs. S&P500 returns
(weekly data)
y = 2.7758x + 0.031
R2
= 0.2686
-0.500
-0.400
-0.300
-0.200
-0.100
0.000
0.100
0.200
0.300
0.400
0.500
-0.150 -0.100 -0.050 0.000 0.050 0.100 0.150
S&P500 Return
EXDSReturn b
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Figure 6Exodus Communication: Valuation
46
47
48
49
50
51
52
53
54
0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.41 0.42
Market Price of Risk
StockPrice
2
2.2
2.4
2.6
2.8
3
3.2
Beta Stock Pric
Beta2.74
$49.92
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Figure 7
Frequency of Revenues in Year 3
0
100
200
300
400500
0 2000 4000 6000 8000 10000
Revenues
Frequency
Mean=3300
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Figure 8
Frequency of Growth Rates in Revenues in Year 3
0
50
100
150
200
250
300
350
400
0 0.2 0.4 0.6 0.8 1
Growth Rate in Revenues
Frequency
Mean=0.46
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Figure 9
Frequency of Variable Costs in Year 3
0
50
100
150
200
250
300
350
0.5 0.6 0.7 0.8 0.9
Variable Costs
Frequency
Mean=0.68