RATIONAL PRICING OF INTERNET COMPANIES REVISITED

September 2000

Eduardo S. Schwartz Anderson School at UCLA

Mark Moon Fuller & Thaler Asset Management

RATONAL PRICING OF INTERNET COMPANIES REVISITED 1. INTRODUCTION In Schwartz and Moon (2000) we developed a model for pricing Internet companies using real options theory and modern capital budgeting techniques. The novelty of the approach is that uncertainty about the key variables which determine the value of an Internet company play a central role in the valuation. In particular, we considered uncertainty in both the revenues and the rates of growth in revenues. In this article we expand and improve the model in several directions. First, we introduce a third stochastic variable to the model, variable costs. These are allowed to follow a mean reverting process with volatility that also mean reverts deterministically. This feature is important since many Internet companies have not yet been profitable but, presumably, are expected to be profitable in the future. Second, we use the fact that the theoretical framework allows us to compute the “beta” of the stock as means of inferring the risk premium in the model. The estimation of the market price of risk in the model, an unobservable parameter, was one of the most challenging aspects of the approach. We can now use the beta of the Internet stock to infer this parameter. Third, we explicitly take into account capital expenditures and depreciation when calculating the net after tax cash flows. This is done by introducing a third deterministic path-dependant state variable - Property, Plant and Equipment – which increases with

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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 evaluator’s) 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)

dR (t ) = µ (t )dt + σ (t )dz1 R(t )

where µ (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 µ . 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) dµ (t ) = κ ( µ − µ (t ))dt + η (t )dz 2 The mean-reversion coefficient (κ) affects the rate at which the growth rate is expected to converge to its long-term average, and ln(2 )/κ can be interpreted as the "half-life" of the deviations in that any growth rate µ 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) dσ (t ) = κ 1 (σ − σ (t ))dt (4) dη (t ) = −κ 2η (t )dt

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The unanticipated changes in the growth rate of revenues and the unanticipated changes in its drift may be correlated: (5) dz1 dz 2 = ρ12 dt 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) Cost (t ) = γ (t ) R(t ) + F The variable costs parameter γ(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) dγ (t ) = κ 3 (γ − γ (t ))dt + ϕ (t )dz 3 The mean-reversion coefficient (κ3) describes the rate at which the variable costs are expected to converge to their long-term average, and ln( 2)/κ 3 can be interpreted as the "half-life" of the deviations in that any deviation γ 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) dϕ (t ) = κ 4 (ϕ − ϕ (t ))dt We also allow for correlation between unanticipated changes in variable costs and both revenues and growth rates in revenues: (9) dz1 dz 3 = ρ13 dt

(10) dz 2 dz 3 = ρ 23 dt

The after tax rate of net income to the firm, Y(t), is then given by (11) Y (t ) = ( R(t ) − Cost (t ) − Dep(t ))(1 − τ c )

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The corporate tax rate τc 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) dP (t ) = {Capx (t ) − Dep (t )}dt (14) Capx (t ) = CX (t )

Capx(t ) = CR * R(t )

for

t≤t

for t > t

(15) Dep (t ) = DR * P (t ) Then, the amount of cash available to the firm, given by X(t), evolves according to: (16) dX (t ) = {rX (t ) + Y (t ) + Dep (t ) − Capx (t )}dt 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 firm’s 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: − rT ] (17) V ( 0 ) = E Q {( X (T ) + M ∗ [ R (T ) − Cost (T )]} e

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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)

dR(t ) = [ µ (t ) − λ σ (t )]dt + σ (t )dz1* R(t )

where the market price of factor risk λ 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. 9

accumulated Property, Plant and Equipment) and time. That is, the value of the stock can be written as: (18) S ≡ S ( R, µ , γ , L, X , P, t )

Applying Ito’s Lemma to this expression we can obtain the dynamics of the stock value

(19)

dS = S R dR + S µ dµ + S γ dγ + S L dL + S X dX + S P dP + S t dt + 12 S RR dR 2 + 1 2

S µµ dµ 2 + 12 S γγ dγ 2 + S Rµ dRdµ + S Rγ dRdγ + S µγ dµdγ

The volatility of the stock can be derived directly from this expression 2 (20) σ S = (

SR S

S

S

σR) 2 + ( Sµ η ) 2 + ( Sγ ϕ ) 2 + 2

S R Sµ S2

Rσηρ12 + 2

S R Sγ S2

Rσγρ13 + 2

Sγ Vµ S2

ηϕρ 23

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 (η0) 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 η0 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) β =

σ SM RS R ρ RM σ M σ (t ) RS R λσ (t ) = = σ M2 S σ M2 S σ M2

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, λ.

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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 riskadjusted processes:4 {[ µ ( t )−λ σ ( t ) −

(22)

R(t + ∆t ) = R(t )e

σ (t )2 ]∆t +σ ( t ) ∆t ε1} 2

(23) µ (t + ∆t ) = e −κ∆t µ (t ) + (1 − e −κ∆t ) µ +

(24) γ (t + ∆t ) = e −κ∆tγ (t ) + (1 − e −κ∆t )γ +

1 − e −2κ∆t η (t )ε 2 2κ 1 − e −2κ∆t ϕ (t )ε 3 2κ

where (25) σ (t ) = σ 0 e −κt + σ (1 − e −κt ) (26) η (t ) = η0e −κt (27) ϕ (t ) = ϕ 0e −κt + ϕ (1 − e −κt ) 3

The initial value of the revenues (the rate of growth in revenues and the variable costs) is perturbed to obtain 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 σ 0 , η 0 and ϕ 0 . ε 1 , ε 2 and ε 3 are standard correlated normal variates. Note that as time grows the volatilities in the model converge to: (28) σ (∞) = σ (29) η (∞) = 0 (30) ϕ (∞) = ϕ and the growth rate in revenues converges to (31) µ (∞) = µ This implies that the revenue process converges to dR (∞) = µ dt + σ dz1 R (∞) The net income after tax is still given by equation (11) where all variables are (32)

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 longterm 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. 14

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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expenditures will be 2% of revenues. In accordance with the firm practice we assume that the annual depreciation allowance is 25% of the accumulated property, plant and equipment of the previous year. Environmental and Risk Parameters The corporate tax rate is taken to be 35% and the risk free rate of interest 6%. The annual volatility of the market portfolio used in the model to relate the market price of risk to the beta of the stock is assumed to be 15%. The market price of risk in the model corresponding to the revenue process is an unobservable parameter.

We imply it from the beta of the stock. Figure 5 shows the

regression used to estimate beta using weekly stock returns for Exodus and the S&P500 from March 27, 1998 to August 4, 2000. The estimated beta is 2.78. Figure 6 reports the relation between the market price of risk parameter and the stock price and beta. A market price of risk of 0.37 gives a beta for the stock of 2.74, which is very close to the estimated beta for Exodus. For this market price of risk the corresponding stock price is once again $49.92.7 Given its substantial capital expenditure plans, Exodus has future financing plans of the order of $2.5 billion. Therefore, we assume that Exodus has future financing possibilities of $3 billion. Increasing or decreasing this amount by $500 million has little effect on the valuation. Simulation Parameters

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The computer program actually uses the more detailed employee stock options data. This is no coincidence, since the volatility of the revenue growth and the market price of risk are implied simultaneously to give the stock volatility and beta. 7

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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 evaluator’s 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 (%) 250

Implied Volatility

200

150

100

50

0 24-Jul-98

01-Nov-98

09-Feb-99

20-May-99

28-Aug-99

21

06-Dec-99

15-Mar-00

23-Jun-00

01-Oct-00

Figure 2 Exodus Communications: Valuation 1.4

58

56 1.3 54 1.2

$49.92

1.1

50

48

1

46

0.97 0.9

44 0.8 42

40 0.18

0.19

0.2

0.21

0.22

0.23

0.24

Volatiltiy of the Revenue Growth Rate

22

0.25

0.26

0.27

0.7 0.28

Volatility

Stock Price

52

Stock Price Volatility

Figure 3 EXDS: Cash Costs vs. Revenues Not Including CAPX 1998A, 1999A, E[2000-2009]

20,000 18,000

y = 0.5623x + 382.4 2 R = 0.9991

Annual Costs in EBITDA ($MM)

16,000 14,000 12,000 10,000 8,000 6,000 4,000 2,000 0 0

5,000

10,000

15,000

20,000

Annual Revenues ($MM)

23

25,000

30,000

35,000

Figure 4 Estimated Revenues 35000

30000

Revenue

25000

Analyst H-L=2.0 H-L=2.2 H-L=1.8

20000

15000

10000

5000

0 2000

2001

2002

2003

2004

2005 Year

24

2006

2007

2008

2009

Figure 5 EXDS returns vs. S&P500 returns (weekly data) 0.500 0.400 0.300 y = 2.7758x + 0.031 2 R = 0.2686

EXDS Return

0.200 0.100 b 0.000 -0.100

-0.200 -0.300 -0.400 -0.500 -0.150

-0.100

-0.050

0.000 S&P500 Return

25

0.050

0.100

0.150

Figure 6 Exodus Communication: Valuation 54

3.2

53 3 52 $49.92

2.8

50

2.6

2.74

49 2.4 48 2.2 47

46 0.32

0.33

0.34

0.35

0.36

0.37

0.38

Market Price of Risk

26

0.39

0.4

0.41

2 0.42

Beta

Stock Price

51

Stock Price Beta

Figure 7 Frequency of Revenues in Year 3

Frequency

500 400 300

Mean=3300

200 100 0 0

2000

4000

6000

Revenues

27

8000

10000

Frequency

Figure 8 Frequency of Growth Rates in Revenues in Year 3 400 350 300 250 200 150 100 50 0

Mean=0.46

0

0.2

0.4

0.6

Growth Rate in Revenues

28

0.8

1

Figure 9 Frequency of Variable Costs in Year 3 350

Frequency

300 250 Mean=0.68

200 150 100 50 0 0.5

0.6

0.7 Variable Costs

29

0.8

0.9