Real options: valuation of the option to invest including corporate tax and information costs.

Levyne, Olivier

ABSTRACT

The valuation of the option to invest has been established by Dixit and Pindyck (1994). Using their methodology (Diffusion process, Ito's lemma, partial differential equation, boundary conditions), it is possible to determine the value of a project, including costs of information and corporate tax. The underlying assumptions of the formula are that input and output define Brownian geometrical motions, with the same Wiener increment, and that the manufactured activity is dependent on the selling price of the products. All of this enables to obtain a valuation formula of the real option to invest, which is compatible with Dixit and Pindyck's results.

JEL Classification: G20, G31

Keywords: Real options; Information costs

I. INTRODUCTION

An important part of the recent financial literature is dedicated to the contribution of the models, based on real options, to assess a project of investment and determine the optimal timing to invest, considering the perspectives of a project's future cash flows. The goal of this paper is to present a model which suggests integrating the costs of information and taxation into capital budgeting based on the use of the real options. This approach joins in a double lineage. Indeed, it consists at first in supposing, in the continuation of Merton (1987) that the investor has to engage costs to analyze the appropriate information for the project. Furthermore, this approach is based on the methodology of Dixit and Pindyck (1994), who consider that the price of a product defines a Brownian geometrical motion. The value of the project, which consists in manufacturing the aforementioned product, then arises from the resolution of an equation in the partial derivatives stemming from the application of the Ito's lemma and from usual boundary conditions (value matching and smooth pasting). Besides, it integrates the value of the option to invest or to delay the date of the investment, which can be specifically determined.

In this context, this paper is divided into three parts. The first part presents the formalization of the value of a project of investment in the presence of costs of information and corporate tax. The second part is centered, in this context of imperfect information and taxation, on the determination of the value of the option to invest at the most convenient date. The decision to realize the project of investment obliges the company to exercise this option. Then it bears a sunk cost, which corresponds to the value of the option that is de facto definitively given up. The third part proposes a series of numerical simulations, which underlines the characteristics of this approach of capital budgeting.

II. VALUE OF A PROJECT INCLUDING CORPORATE TAX AND INFORMATION COSTS

The model, which is presented hereafter, is based on the principle according to which the output price Pt of a product, as well as the input price Ct, defines a Brownian geometrical motion. So:

d[P.sub.t] = [[alpha].sub.p][P.sub.t]dt + [alpha][P.sub.t]d[B.sub.t] (1)

and

d[C.sub.t] = [[alpha].sub.p][C.sub.t]dt + [alpha][C.sub.t]d[B.sub.t] (2)

where [[alpha].sub.p] and [[alpha].sub.c] represent respectively the trend of the evolution of the selling price of the product and its unit production cost. Besides, [alpha] corresponds to the instantaneous volatility for the production, which is related to the project and d[B.sub.t]is the increment of a Wiener process. This allows integrating the uncertainty, which is inherent to the market of the product.

Equation (1) allows deducting that:

dln[P.sub.t] = ln[P.sub.t] - ln [P.sub.0] = ln [P.sub.t]/[P.sub.0] = ([[alpha].sub.p] - [[sigma].sup.2]/2)t + [sigma][B.sub.t] (3)

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

And finally,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

We now suppose that Rt is the after tax cash flow generated by the project at the date t, [Q.sub.t] is the quantity produced at the same date t, and t is the corporate tax rate. From then on, [R.sub.t] = ([P.sub.t] - [C.sub.t][Q.sub.t] x (1-[tau]).

Assuming that the manufactured quantity depends on the selling price of product, it is henceforth possible to note [Q.sub.t] = [P.sub.t.sup.b]. So,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Hence:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Or still

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

From then on, by applying Taylor's formula to [R.sub.t] which is a function of both variables [B.sub.t] and t and by truncating the expression in the order 2, we have

d[R.sub.t] = [partial derivative][R.sub.t]/[partial derivative]t dt + [partial derivative][R.sub.t]/[partial derivative][B.sub.t] d[B.sub.t] + 1/2 [[partial derivative].sup.2][R.sub.t]/[partial derivative][B.sup.2.sub.t] [(d[B.sub.t]).sup.2] (11)

As far as d[B.sub.t] is supposed to be the increment of a Wiener process d[B.sub.t] = [epsilon] [square root of dt], where [epsilon] is a normally distributed random variable with a zero mean and a standard deviation of 1. Moreover, [(d[B.sub.t]).sup.2] = [[epsilon].sup.2]dt and Var([epsilon]) = E([[epsilon].sup.2]) - [[E([epsilon])].sup.2], where E is the mean and Var is the variance.

So, as [E([epsilon])] = 0, it can be deducted that E([[epsilon].sup.2]). Consequently, Besides E([[epsilon].sup.2]dt) = [(dt}.sup.2]Var([epsilon].sup.2]), which converges towards 0 when dt aims towards 0. Then, it can be gathered that [[epsilon].sup.2]dt is equal to dt when dt is very small. So,

d[R.sub.t] = [partial derivative][R.sub.t]/[partial derivative]t dt + [partial derivative][R.sub.t]/ [partial derivative][B.sub.t] d[B.sub.t] + 1/2 [[partial derivative].sup.2][R.sub.t]/[partial derivative][B.sup.2.sub.t] dt (12)

Consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

where f(t) = 1/2 [(b + 1).sup.2][[sigma].sup.2] + F'(t)/F(t) and is different from 0.

To determine the V value of the project, an arbitrage portfolio can be constituted. It consists in buying the project and in selling n units of cash flow, n being determined so that the portfolio is risk free. The holding of the asset corresponding to the project enables to receive the Rdt income on the brief interval of time (t, t+dt). Besides, because of the short position on a unit of cash flow a dividend has to be paid to the holder of the long position. By considering a [delta dividend yield, the dividend paid on the (t, t+dt) interval of time by the holder of a short position on n units of cash flow stands is equal to n[delta]Rdt. So, the holder of the arbitrage portfolio receives, on the (t, t+dt) interval of time, a (R - n[delta]R)dt net dividend, of which Rdt because he has a long position on the project and -n[delta]Rdt because of his short position. Moreover, he realizes a capital gain, which is equal to dV[R] - nd[R.sub.t].

The formalization of the amount of the capital gain can be obtained by applying directly the lemma of Ito in V, which is a function of both variables [R.sub.t] and t. So,

dV(R, t) = [partial derivative]V/[partial derivative]t dt + [partial derivative]V/[partial derivative]R dR + 1/2 [R.sup.2] [(b + 1).sup.2] [[sigma].sup.2] [[partial derivative].sup.2]V/[partial derivative][R.sup.2] dt (14)

with dR = R x f(t) x dt + R (b + 1)[sigma] x d[B.sub.t]

By considering that the project can be delayed in perpetuity: [partial derivative]V/[partial derivative]t = 0. So,

dV(R, t) = [partial derivative]V/[partial derivative][R.sub.t] dt + 1/2 [R.sup.2] [(b + 1).sup.2] [[sigma].sup.2] [[partial derivative].sup.2]V/[[partial derivative].sup.2] dt

= V'(R)[R (b + 1)[sigma] x d[B.sub.t] + R x f(t) x dt] + 1/2 [R.sub.2] [(b + 1).sup.2][[sigma].sup.2]V"(R)dt (15)

To simplify the writings, will be noted below. Consequently,

dV(R) - nd[R.sub.t] = V'(R)[R (b + 1)[sigma] x d[B.sub.t] + R x f(t) x dt] + 1/2 [R.sub.2] [(b + 1).sup.2][[sigma].sup.2]V"(R)dt - n[R x f(t) x dt + R(b + 1)[sigma] x d[B.sub.t]

={Rf(t)[V'(R) - n] + 1/2 [R.sup.2] [(b + 1).sup.2] [[sigma].sup.2]V"(R)}dt + R(b + 1)[sigma][V'(R) - n]d[B.sub.t] (15)

Besides, by choosing n=V'(R), the global payment of the arbitrage portfolio's owner is equal to

(R - n[delta]R)dt + dV(R) - nd[R.sub.t] = [R - [delta]RV'(R) + 1/2 [R.sup.2] [(b + 1).sup.2][[sigma].sup.2]V"(R)dt]. (16)

On principle, return of an arbitrage portfolio is equal to the r risk free rate. However, considering the [[lambda].sub.V] costs of information, which have to be paid to study the project, and the [[lambda].sub.R] costs of information, which are related to the cash flows analysis, the project return must be equal to (r + [[lambda].sub.V]) and the return of the future cash flows must be equal to (r + [[lambda].sub.R]). It will be supposed, afterwards, that [[lambda].sub.V] > [[lambda].sub.R]. These parameters represent sunk costs, which must be engaged before realizing a project of investment. That is why it is advisable to integrate them into any discount calculation.

On the dt brief interval of time, the project return is (r + [[lambda].sub.V])V(R)dt and the return of the n--or V'(R)--sold cash flows return is (r + [[lambda].sub.R])Rdt. Then:

[R - [delta]RV'(R) + 1/2 [R.sup.2] [(b + 1).sup.2][[sigma].sup.2]V"(R)]dt = (r + [[lambda].sub.V]V(R)dt - RV'(R) (r + [[lambda].sub.R])dt (17)

The grouping of the terms of Equation (17) and the simplification by dt drives to the following differential equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

By using the principle according to which the function B [??] AR [??] satisfies the equation

1/2 [R.sup.2] [(b + 1).sup.2][[sigma].sup.2]V"(R) + (r + [[lambda].sub.R] - [delta])RV'(R) - (r + [[lambda].sub.V])V(R) = 0, (19)

it is possible to write down the associated quadratic equation

1/2 [R.sup.2] [(b + 1).sup.2][[sigma].sup.2]A([??] - 1)[R.sup.[??]-2] + (r + [[lambda].sub.R] - [delta])RA[??][R.sup.[??]-1] - (r + [[lambda].sub.V])A[R.sup.[??] = 0. (20)

Or, by simplifying by B A[R.sup.[??]]:

1/2 [(b + 1).sup.2][[sigma].sup.2]A[??]([??] - 1) + (r + [[lambda].sub.R] - [delta])[??] - (r + [[lambda].sub.V]) = 0. (21)

The general solution of Equation (21) has the following shape:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

where [??] and [??} are the both roots of the quadratic Equation (21). The solving of Equation (21) enables to find that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

So, if the project did not benefit from a perpetual option to invest, its value would be equal to

V(R) = R/[delta]+[[lambda].sub.V]-[[lambda].sub.R]. (25)

III. DECISION OF INVESTMENT AND VALUE OF THE OPTION TO INVEST INCLUDING CORPORATE TAX AND INFORMATION COSTS

This second part of is centered on the determination of the value of the option to invest which allows delaying the investment up to the most convenient date. This option must be exercised when the critical values reached by the project and the cash flows are respectively [V.sup.*] and [R.sup.*]. In that case, the invested amount is noted I.

The option value can be obtained by constituting again an arbitrage portfolio. This one consists in acquiring an option to invest in the project and in selling n units of cash flows generated by the aforementioned project, n being determined so that the portfolio is risk free. Besides, because of the short position, [delta]Rdt has to be paid for each unit of cash flow, which has been sold, where [delta] is the dividend yield. Furthermore, the capital gain on the portfolio is

dF(R) - nd[R.sub.t] = {Rf(t)[F'(R) - n] + 1/2 [R.sup.2] [(b + 1).sup.2][[sigma].sup.2]F"(R)}dt + R(b + 1)[sigma][F'(R) - n]d[B.sub.t] (26)

By choosing n=F'(R), the amount which is received by the owner of the arbitrage portfolio of is equal to

-n[delta]Rdt + dF(R) - nd[R.sub.t] = -F'(R)[delta]Rdt + dF(R) - nd[R.sub.t] =[1/2 [R.sup.2][(b + 1).sup.2][[sigma].sup.2]F"(R) - [delta]RF'(R)]dt (27)

On principle, the return of the arbitrage portfolio is the r risk free rate. However, considering the [[lambda].sub.F] costs of information, which are related to the option to invest at the convenient date and the [[lambda].sub.R] costs of information, which are related to the cash flow analysis, the return of the option must be equal to (r + [[lambda].sub.F]) and the return of the cash flows is (r + [[lambda].sub.R]). In this context

[1/2 [R.sup.2][(b + 1).sup.2][[sigma].sup.2]F"(R) - [delta]RF'(R)]dt = (r + [[lambda].sub.F])F(R)dt - RF'(R)(r + [[lambda].sub.R])dt. (28)

This allows obtaining the second order homogeneous differential equation:

1/2 [R.sup.2][(b + 1).sup.2][[sigma].sup.2]F"(R) + (r + [[lambda].sub.R] - [delta])RF'(R) - (r + [[lambda].sub.F)F(R) = 0 (29)

The general solution to Equation (29) is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [A.sub.1] and [A.sub.2] are real unknown constants which have to determined and where [[beta].sub.1] and [[beta].sub.2] are the solutions of the quadratic equation:

1/2 [(b + 1).sup.2][[sigma].sup.2] [beta]([beta] - 1) + (r + [[lambda].sub.R] - [delta])[beta] - (r + [[lambda].sub.F]) = 0 (30)

Its resolution drives to the following solutions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (32)

As far as F(0)=0, [A.sub.2]=0. Consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (33)

Assuming that the I investment is the strike price of the option, it is possible to determine the [R.sup.*] critical value of the cash flows by using the usual boundary conditions, issued from the methodology of Dixit and Pindyck (1994). The value matching condition allows writing:

F([R.sup.*]) = V ([R.sup.*]) - I (34)

So, by using Equations (25) and (33), Equation (34) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

The smooth pasting condition enables to write:

F'([R.sup.*]) = V'([R.sup.*]) (36)

So, by deriving both members of Equation (35) with regard to R, Equation (36) becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

From then on, by dividing Equation (35) by Equation (37), we obtain:

[R.sup.*]/[[beta].sub.1] = [[R.sup.*]/[delta] + [[lambda].sub.V] - [[lambda].sub.R] - I]([delta] + [[lambda].sub.V] - [[lambda].sub.R])

[R.sup.*]/[[beta].sub.1] = [R.sup.*] - ([delta] + [[lambda].sub.V] - [[lambda].sub.R])I

[R.sup.*] = [[beta.sub.1]/[[beta.sub.1] - 1] ([delta] + [[lambda].sub.V] - [[lambda].sub.R])I (38)

By substituting Equation (38) into Equation (37), it is possible to get A1. So,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40)

Finally, by substituting of Equation (38) into Equation (25), we have

[V.sup.*] = [[beta].sub.1]/[[beta].sub.1]-1 x I (41)

In other words, according to the operational conclusions of Dixit and Pindyck (1994), a project can be undertaken provided that the value of its cash flows is equal to a multiple of the amount of the envisaged investment. The value of [[beta].sub.1] in Equation (23) is different from that of the formula established by Dixit and Pindyck (1994). However, the by replacing b, [[lambda].sub.R] and by [[lambda].sub.V] 0, Equation (41) comes down to Dixit and Pindyck's formula In that case, the critical value of the investment stemming from Equation (41) and that stemming from the formula obtained by Dixit and Pindyck (1994) are identical.

IV. SIMULATIONS

The results of the second part of this paper allow formulating a decision of investment rule under uncomplete information. In the lineage of Dixit and Pindyck (1994), it emerges that an investment must be realized when the amount R cash flows is superior to the critical level [R.sup.*]. Correlatively, if R is lower than [R.sup.*] then the V(R) value of the project is lower than the sum of the amount of the investment I and of the value F(R) of the option of waiting for the optimal timing to invest.

The figure 1 below shows that the value of the option of wait is an increasing function of R. This figure is built by considering a r risk free rate equal to 3.5 %, a dividend yield, a [delta] = 1.5% dividend yield, a [[lambda].sub.R] = 1% cost of information about the cash flows, a [[lambda].sub.F] = 2.5% cost of information about the option, a [[lambda].sub.V] = 5% cost of information about the project, b = -2, and an investment I = 1. Besides, three assumptions of volatility have been taken into account: [sigma] = 20%, [sigma] = 25% and [sigma] = 30%.

[FIGURE 1 OMITTED]

Table 1 below recapitulates the various values of F(R) which have been represented graphically:

Table 1
Value of F(R) according to R and the volatility

[??] R

 0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0

20% 0 11 30 55 84 118 155 195 239 285 334
25% 0 10 25 45 67 92 119 147 178 210 243
30% 0 9 23 39 57 77 98 120 143 168 193

Figure 2 below shows that the critical [R.sup.*] value of the cash flows is also an increasing function of the level of volatility.

[FIGURE 2 OMITTED]

This figure is built by considering a r risk free rate equal to 3.5%, a [delta] = 1.5% dividend yield, a [[lambda].sub.R] = 1% cost of information about the cash flows, a [[lambda].sub.F] = 2.5% cost of information about the option, a [[lambda].sub.V] = 5% cost of information about the project, b = -2, and an investment I = 1. Table 2 below recapitulates the various values of [R.sup.*] represented graphically:

Table 2
Value of [R.sup.*] according to the volatility

 [sigma]

10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
0,1 0,2 0,2 0,3 0,4 0,5 0,6 0,7 0,9 1,1

Figure 3 below shows the impact of the evolution of the s volatility and of dividend yield on the critical value [V.sup.*] of the project. This figure is based on a r risk free rate equal to 3.5%, a [[lambda].sub.R] = 1% cost of information about the cash flows, a [[lambda].sub.F] = 2.5% cost of information about the option, a [[lambda].sub.V] = 5% cost of information about the project, b = -2, and an investment I = 1. Besides, three assumptions of dividend yield are taken into account: [delta] = 1%, [delta] = 1.5%, [delta] = 2%.

[FIGURE 3 OMITTED]

It emerges that the [V.sup.*] critical value of the project is an increasing function of the volatility. Consequently, the progress of the volatility is translated by a reduction of investments. Besides, any increase of the dividend yield increases the [V.sup.*] critical value of the project from which the company can invest.

Table 3 below recapitulates the various values of [V.sup.*] represented graphically.

Figure 4 below shows that the F(R) value of the option of waiting and the V(R) value of the project are increasing functions of R and decreasing functions of the dividend yield which takes the following values: [delta] = 1%, [delta] = 5% and [delta] = 2% The figure 4 is based on a r risk free rate equal to 3.5 %, a [[lambda].sub.R] = 1% cost of information about the cash flows, a [[lambda].sub.F] = 2.5% cost of information about the option, a [[lambda].sub.V] = 5% cost of information about the project, b = -2, and an investment I = 20. It emerges from this graph that when the dividend yield increases, the [R.sup.*] critical amount of the cash flows decreases. [R.sup.*] corresponds to the abscissa of the tangential point of the representative curves of the F(R) value of the option of waiting and the V(R)-I net present value of the project. So, when [delta] = 1%, then [R.sup.*]=6.2 ; when % [delta] = 1.5%, then [R.sup.*]=5.8 and when [delta] = 2% then [R.sup.*]=5.6.

[FIGURE 4 OMITTED]

Table 4 below recapitulates the various values of F(R) represented graphically.

Besides, Table 5 displays the values of V(R)-I which are represented by dotted lines on Figure 4.

Figure 5 below illustrates the impact of an increase in the r risk free rate on the F(R) value of the option to invest. The figure is based on a 1.5% dividend yield, a 40% volatility, a [[lambda].sub.R] = 1% cost of information about the cash flows, a [[lambda].sub.F] = 2.5% cost of information about the option, a [[lambda].sub.V] = 5% cost of information about the project, b = -2, and an investment I = 1.

[FIGURE 5 OMITTED]

Three assumptions of the future cash flows are taken into account: R=0,25, R=0,50 and R=0,75. It emerges that the F(R) value increases with the r risk free rate. So, a higher interest rate increases the cost of opportunity of the immediate investment and is translated, de facto, by a decrease in investments. The table 6 below recapitulates the F(R) various values of represented graphically:

Table 6
F (R) value according to the R cash flows and to the r risk free rate

[bar.R] r

 0% 5% 10% 15% 20% 25%

0,25 3,5 3,6 3,6 3,7 3,8 3,8
0,50 8,7 8,3 8,1 8,1 8,1 8,1
0,75 14,8 13,5 13,0 12,8 12,7 12,6

[bar.R] r

 30% 35% 40% 45% 50%

0,25 3,9 3,9 3,9 4,0 4,0
0,50 8,2 8,2 8,2 8,3 8,3
0,75 12,6 12,6 12,7 12,7 12,7

Figure 6 below shows that the value of the option of waiting is an increasing function of the [[lambda].sub.F] costs of information on the option. This graph is built by considering a r = 4% risk free rate, a 2% dividend yield, a [[lambda].sub.R] = 1% cost of information about the cash flows, a [[lambda].sub.V] cost of information about the project, b = -2, and an investment I = 1. Three level assumptions of the cash flows are taken into account: R=0,25, R=0,50 and R=0,75. It emerges from this graph that the increase in the [[lambda].sub.F] cost of information about the option is translated by an increase in the value of the option of waiting. In other words, the cost of opportunity of the immediate investment increases with the [[lambda].sub.F] cost of information about the option, which is translated by a decrease in investments.

[FIGURE 6 OMITTED]

It also emerges from this graph that the value of the option is an increasing function of the R cash flows.

Table 7 below recapitulates the various F(R) values represented graphically.

All in all, the consideration of the costs of information within the framework of the decision of investment does not question the spirit of the practical conclusions of Dixit and Pindyck (1994). So, a project can be undertaken if its cash flows are at least equal to a multiple of the investment to be achieved. This multiple integrates the costs of information into the sense of Merton's CAPM (1987). Besides, in the theoretical assumption where the investor is free of charge of complete information about the envisaged project, the value of the option to invest and the critical value of the cash flows stemming from this model are identical to those obtained by Dixit and Pindyck (1994).

REFERENCES

Aliber, R., 1970, "A Theory of Foreign Direct Investment", in Charles P. Kindleberger, ed, The International Corporation, Cambridge, Mass: MIT Press.

Adler, M. and B. Dumas, 1984, "International Portfolio and Corporation Finance: A Synthesis", Journal of Finance, p. 925-984, June.

Adler, M. and B. Dumas, 1984, "Exposure to Currency Risk: Definition and Measurement", Financial Management, p. 41-50, summer.

Bellalah, M., 1990, "Quatre essais sur l'evaluation des options : dividendes, volatilite des taux d'interet et information incomplete", Doctorat de l'Universite de Paris Dauphine, juin.

Bellalah, M. and B. Jacquillat, 1995, "Options Valuation with Information Costs: Theory and Tests", Financial Review, p.617-635, August.

Bellalah, M., 1999, "The Valuation of Futures and Commodity Options with Information Costs", Journal of Futures Markets, September.

Bellalah, M., 2001, "A Reexamination of Corporate Risks under Incomplete Information", International Journal of Finance and Economics, p.59-67, 6.

Bellalah, M., 2001, "Markets Imperfections, Information Costs and the Valuation of Derivatives: Some General Results", International Journal of Finance, volume 13, p.1895-1928.

Bellalah, M., 2001, "Irreversibility Sunk Costs and Investment under Incomplete Information", R&D Management, volume 31, p.127-136.

Black, F., 1974, "International Capital Market Equilibrium with Investment Barriers", Journal of Financial Economics, 1, p.337-352.

Choi, J.B., 1989, "Corporate International Investment", Journal of international business studies, p.145-155, spring.

Coval, J. and T.F. Moskowitz, 1999, "Home Bias at Home: Local Equity Preference in Domestic Portfolios", Working Paper, University of Michigan.

Dixit, A.K. and R.S. Pindyck, 1994, "Investment under Uncertainty", Princeton University Press, Princeton, New Jersey.

Dornbusch, R., 1980, "Exchange Rate Risk and the Macroeconomics of Exchange Rate Determination", NBER Working, N 493, June.

Kang, J. and R. Stulz, 1997, "Why Is There a Home Bias? An Analysis of Foreign Equity in Japan", Journal of Financial Economics, 46, p.3-28.

Merton, R., "An Equilibrium Model with Incomplete Information", Journal of Finance, 42, p.483-510.

Solnik, B., 1974, "An Equilibrium Model of the International Capital Market", Journal of Economic Theory, 1, p.500-524, August.

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Olivier Levyne

Professor of Finance, Institut Superieur du Commerce de Paris, 22 boulevard du Fort de Vaux, 75017 Paris, France olied@tiscali.fr

Table 3
Value of [V.sup.*] according to the [??] volatility and the
[??] dividend yield

[??] [sigma]

 5% 10% 15% 20% 25% 30%

1,0% 2,5 2,7 3,1 3,5 4,1 4,7
1,5% 2,1 2,3 2,6 3,0 3,5 4,0
2,0% 1,8 2,0 2,3 2,6 3,0 3,5

[??] [sigma]

 35% 40% 45% 50% 55% 60%

1,0% 5,4 6,2 7,1 8,1 9,2 10,4
1,5% 4,6 5,3 6,0 6,9 7,8 8,8
2,0% 4,0 4,6 5,3 6,0 6,8 7,6

Table 4
F(R) value according to the R cash flows
and of the [delta] dividend yield

[??] R

 0 0,5 1,0 1,5 2,0 2,5 3,0

1,0% 0 5 12 19 27 35 44
1,5% 0 4 10 16 23 30 38
2,0% 0 3 8 14 20 26 33

[??] R

 3,5 4,0 4,5 5,0 5,5 6,0

1,0% 53 62 71 80 90 100
1,5% 46 54 62 71 80 89
2,0% 40 48 55 63 72 80

Table 5
V (R)-I value according to the R cash flows
and the [delta] dividend yield

[??] R

 0 0,5 1,0 1,5 2,0 2,5 3,0

1,0% -20 -10 0 10 20 30 40
1,5% -20 -11 -2 7 16 25 35
2,0% -20 -12 -3 5 13 22 30

[??] R

 3,5 4,0 4,5 5,0 5,5 6,0

1,0% 50 60 70 80 90 100
1,5% 44 53 62 71 80 89
2,0% 38 47 55 63 72 80

Table 7
F (R) value according to the R cash flows and to the
[[lambda].sub.F] cost of information about the option

[bar.R] [[lambda].sub.F]

 0 5% 10% 15% 20% 25%

0,25 4 4 4 5 5 6
0,50 8 9 13 17 22 29
0,75 13 17 25 36 52 73

[bar.R] [[lambda].sub.F]

 30% 35% 40% 45% 50%

0,25 7 8 10 11 13
0,50 38 49 62 77 96
0,75 100 136 181 240 313