Using Real Options for the Evaluation of Venture Projects

153

Gadjah Mada International Journal of Business – May-August, Vol. 18, No. 2, 2016

Gadjah Mada International Journal of BusinessVol. 18, No. 2 (May-August, 2016): 153-185

* Corresponding author’s e-mail: [email protected]

ISSN: 1141-1128http://journal.ugm.ac.id/gamaijb

Using Real Options for the Evaluation of Venture Projects

Alexander Baranov,1 and Elena Muzyko2*

1 Faculty of Economics, Novosibirsk National Research State University; Institute of Economics andIndustrial Engineering, Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russia

2 Faculty of Business, Novosibirsk State Technical University, Novosibirsk, Russia

Abstract: This paper considers the peculiarities of the application of the real options method for assess-ing the economic efficiency of venture investments in innovative projects from the venture fund’s posi-tion. The results of the practical use of the author’s approach for the evaluation of venture investmentswith real options are analyzed. The paper shows the applicability of the real options concept to thevaluation of the effectiveness of venture capital investments. The use of the real options method raisesthe accuracy of the estimation and enhances the instruments of the venture fund in evaluating the eco-nomic efficiency of innovative projects.

Abstrak: Artikel ini membahas kekhasan aplikasi metode real options untuk menilai efisiensi ekonomisinvestasi ventura dalam berbagai proyek inovatif dari posisi dana ventura. Hasil penggunaan praktikpendekatan penulis terhadap evaluasi investasi ventura dengan real options dianalisa. Artikel ini menunjukkanaplikabilitas konsep real options untuk penilaian keefektifan investasi modal ventura. Penggunaan metodereal options meningkatkan akurasi estimasi dan menambahkan instrumen dana ventura dalam mengevaluasiefisiensi ekonomis proyek inovatif.

Keywords: financial options; innovative project; real options; uncertainty; venture capital in-vestment

JEL classification: G24; G32; M13

Baranov and Muzyko

154

Introduction

With current financial theories, tradi-tional approaches to the assessment of theefficiency of investment projects very oftenshow their limitations, since most of them arenot suitable for projects implemented in con-ditions with high risk and uncertainty. Tradi-tional discounted cash flow analysis (the NPV-method) is based on the assumption that aftermaking a decision on starting an investmentproject, the management should follow theinitially chosen strategy even in adverse cir-cumstances. However, after starting theproject’s implementation, the company’s man-agement can change the initial plan, for ex-ample, to expand or constrict the scope ofthe project, change the “inputs” or “outputs”of the project, abandon further implementa-tion of the project, or “freeze” it for a certainperiod of time, once new information be-comes available.

In this regard new methods of valuinginnovative projects become particularly impor-tant. Among such instruments is the real op-tions method. The most important feature ofthis method is its compliance with the rapidlychanging economic environment in which acompany operates, as well as taking into ac-count the company’s managerial flexibility inmaking decisions. The real options theory canexplain the fact, which is known from prac-tice, that investors often do not abandonprojects with a negative NPV (Net PresentValue), as the situation may change for thebetter, and the real option, which is incorpo-rated in the innovative project, can be used.As a result, the net present value of the projectwill be positive.

The real options concept has developedas a result of the transfer of risk managementinstruments, that use option contracts, fromthe financial sector to the real sector of the

economy. The real options theory is an alter-native view on investments and projects’ ef-fectiveness valuations. The basis of the theoryis the assumption that it is possible to presentan investment project schematically as a finan-cial option.

A financial option is a contract, which givesthe buyer (or the owner) the right, but not theobligation, to buy or sell an underlying asset orinstrument at a specified strike price on orbefore a specified date. A real option is an op-tion on a non-financial asset. The underlyingasset of the real option is the cash flow of theinvestment project.

As venture capital investments are char-acterized by high risks and high returns, andthey often have a phased nature, the traditionalNPV method can be supplemented by othermethods, which allow a more accurate valua-tion of the effectiveness of risky projects tobe obtained. Meanwhile the existing valuationmodels of real options have certain limitationsin their application for the purpose of assess-ing venture investments.

In the article the following results wereobtained:

1. The peculiarities of the application of thereal options method for assessing the eco-nomic efficiency of venture investments ininnovative projects from the venture fund’sposition were determined.

2. The author’s modification of the real op-tions method for the valuation of venturecapital investments from the venture fund’sposition was approbated using data on ven-ture projects in the industrial sector in Rus-sia. The practical effectiveness of this ap-proach was demonstrated.

3. The application of the real options methodto assess the efficiency of investmentprojects in Russia is very limited. Unfortu-nately, this method is not widely used by

155


Russian financial managers. It is knownfrom experience that only one investmentcompany uses this method (“Laboratory ofinvestments “LABRATE”). Therefore oneof the objectives of our research was theextension of the practical use of the realoptions method for project analysis in Rus-sia. So, for this case we have taken Russianinnovative project in the industrial sectorand have estimated it’s effectiveness by theorder of the initiators.

Literature Review

In Russia there are separate works onventure capital financing and real options. Butno research has been done concerning theapplication of the real options method to theventure capital financing of innovative projectsup to now. We will analyze the work of for-eign scientists concerning this problem, un-fortunately the amount of such research workis quite limited.

According to which way is chosen ofusing the real options method in venture capi-tal investing, the papers analyzed by us can besplit into four groups. We will begin our analysiswith considering the papers of the first group.Botteron and Casanova (2003) proposed anoption pricing model that made it possible toevaluate the flexibility acquired by a venturecapitalist when he staged his investment pro-cess. The authors consider the start-up valueas a value of two options, namely, a Europeancall-option and a European binary call-option.

The formula developed by Black andScholes (1973) to value a European call-op-tion and the formula proposed by Geske(1979) to value a two-stage compound Euro-pean call-option can be applied only in the caseof the constant volatility of the underlyingasset’s value. Hsu in his work (2002) modified

these formulae to evaluate options with time-dependent volatility. In this paper the processof decision-making by a venture capitalist toinvest in the staged manner is analyzed in thelight of the principal-agent problem. A ven-ture capitalist can invest up front in the formof a one-time fixed amount or he can cut hisinvestment into several stages. Staged invest-ment is treated as a compound European call-option with time-dependent volatility. To as-sess the value of this option Hsu (2002) modi-fied the Geske’s formula. Venture financinginvolving a one-time fixed amount is treatedas a straight European call-option but withtime-dependent volatility. To assess the valueof this option the author modified the Black-Scholes formula (Hsu 2002).

Gong et al. (2006) proposed a model ofthe elementary compound options valuation.They introduced time-dependent volatility intothe model of valuing a multistage compoundreal option based on the model of valuing amultistage compound real option with con-stant volatility, as proposed by Lin (2002).Willner (1995) proposed to value a start-up asa real option. To determine the dynamics ofchanging the underlying asset value, the Pois-son process instead of the Winner process wasused in the model.

Let us make a critical analysis of the pa-pers. Botteron and Casanova’s (2003) paperstated that a venture project (a start-up is noth-ing else than a project) was an underlying as-set. In our opinion the underlying asset is notthe whole venture project but a portion ofthe shares of the investee company, since aventure capitalist owns only a portion of theshares but not the whole project (start-up).

In his paper, Hsu (2002) does not givean interpretation of the Black-Scholes formulaelements. In all the reviewed papers the analy-sis is made in terms of a venture project as a

Baranov and Muzyko

156

whole. The cash flows of the venture capital-ist and the cash flows of the project itself arenot separated. No approbation of the pro-posed models of assessing the options valuebased on real innovation projects with ven-ture financing is made in the papers consid-ered above. A common feature of the papersof the first group is a high degree of math-ematization while any economic interpretationof the parameters included in the proposedmodels of valuing real options in venture fi-nancing is not given.

Let us consider the second group of papers. Intheir paper, Huixia and Tao (2010) used a bi-nomial model of option valuation in the caseof venture financing, with the focus being onAmerican-style options. However, the pecu-liarities of venture financing are not analyzed.It is not explained why it is the binomial modelthat is chosen for the venture financing case.Any approbation of the binomial model basedon real data from innovative projects with ven-ture financing is not made. Seppa andLaamanen (2001) arrived at the conclusion thatthe binomial model was more appropriate forthe valuation of venture capital investmentsthan the traditional method of discounted cashflow.

Let us now analyze the third group of papers.In Li’s (2008) paper, the decision to stage theinvestment process is viewed as a choice be-tween holding an option on investing later (aninvestment delay) and investing now to ob-tain an option on further staged investment.The authors Tong and Li (2010) claimed thatwhen making initial investments, venture capi-talists obtain an option on expansion, an op-tion on the project abandonment and an op-tion on an investment delay. We believe thatin the venture financing of innovative projects,all these three types of real options could becombined into one type, namely the com-pound call option on staged investment, which

will reflect the specificity of venture financ-ing.

Wadhwa and Phelps (2010) in their pa-per characterized corporate venture financingas a two-stage compound option. An initialventure capital investment created a com-pound growth option. Forming a strategic al-liance with a portfolio company was viewedas exercising the second stage of this option.

Let us move to the last or fourth group ofpapers. Vanhaverbeke et al. (2008) analyzedthe advantages of external corporate venturefinancing as “open innovation practices” interms of the real options. They claimed that itwas necessary to take into account the valueof the real options but they do not explainhow to assess this value or by using what mod-els.

The authors of the papers in the fourthgroup (Vanhaverbeke et al. 2008; Kulatilaka andToschi 2011) wrote about the usefulness ofthe real options theory in the analysis of ven-ture financing, but they confined themselvesto detailed verbose reasoning without makingany analysis of models of the real options valu-ation that emerged in the venture financingof investment projects. Besides, they do notmake any calculations

Also let’s make a brief review of otherstudies, in which the authors apply Geske’sformula to compound real options valuationsin multi-stage investments. Carr (1988), build-ing on Margrabe (1978) and Geske (1979),provided the valuation of sequential exchangeoptions. Lee and Paxson (2000 a) suggested atwo-point confined exponential extrapolationmethod for the American option, by extend-ing Geske and Johnson’s (1984) compoundoption approach. The value of such an Ameri-can sequential exchange option is given in Leeand Paxson (2000 b). Lee and Paxson (2001)in their work modeled the stages of R&D ex-

157


pense and the ultimate discovery using realsequential (compound) exchange Americanoption models.

Herath and Park (2002) investigated amulti-stage project setting, where each invest-ment opportunity derived revenues from dif-ferent markets but shared common techno-logical resources. They extend the binomiallattice framework to model a multi-stage in-vestment as a compound real option. Jensenand Warren (2001) in their paper examinedthe practicalities of applying the real optionstheory to valuing research in the service sec-tor, where the relationship between researchand subsequent business benefits is less easilydiscerned than in most previous applicationsof the options theory in, e.g. the pharmaceu-tical industry. The paper used a compoundoptions model, the Geske model, based on athree-phase lifecycle consisting of research,development and deployment. All these pa-pers had a high degree of mathematization,while any economic in-depth interpretation ofthe parameters included in the proposed mod-els of valuing the real options was not given.

Having made a critical survey of thestudies in which the real options method wasused in analyzing the venture financing of in-vestment projects we can make the followingconclusions. In Russia not many researchers arenow working on this problem. However, stud-ies concerned with the use of the real optionsmethod for the evaluation of the efficiencyof investment projects with venture financ-ing, carried out by foreign researchers are alsoscarce. In these papers any economic inter-pretation of the parameters included in theproposed models of the real options valua-tion that emerge in the venture financing ofinvestment projects is often not given. In allof the reviewed papers the analysis has beendone from the point of view of the invest-

ment project as a whole. In our opinion, it isnecessary to separate the cash flows of theventure fund and the cash flows of the projectitself. A venture fund has its own cash flows,which are different from those of the entireproject under analysis. An approbation of theproposed models based on real innovationprojects with venture financing is not made.To our mind the case for a venture capital in-vestment is a compound real call option withtime-dependent volatility.

Methods

Venture capital investments are charac-terized by high risk levels and high uncertainty,and they often have a phased nature. More-over, each phase of a venture investment hasits own risk level. So, the model, which couldbe used for the valuation of the real optionsin the venture financing of innovative projectsshould take into account the different levelsof risk at the different phases of the invest-ment.

Geske (1979) created a formula for theevaluation of the two-stage compound Euro-pean call-option. But Geske’s formula can beapplied only in the case of the constant vola-tility of the underlying asset’s value. Hsu inhis work (2002) modified this formula to evalu-ate options with a time-dependent volatility(the modified Geske’s formula).

In this paper we used the modified ver-sion of Geske’s formula, but we have changedthe sense bearing understanding of the entryparameters of the model. We believe that whenassessing the value of the real options in aventure capital investment, it should be takeninto account that there are differences betweenthe cash flow of the entire innovative projectand the cash flow of the venture fund. Forinstance, the venture fund’s investment may

Baranov and Muzyko

158

be less than the investment for the entireproject. Positive cash flows for the venturefund are less than the positive flows for theentire innovative project. Thus, we are con-sidering the economic efficiency of ventureinvestments in innovative projects from theventure fund’s position. Figure 1 shows a sche-matic representation of the venture investmentbuying a share purchase call option at thebeginning of the investment process.

Let’s appraise the efficiency of the in-novative project from the point of view ofthe venture fund, both by the traditionalmethod of the discounted cash flow (a stan-dard calculation) and using the real optionsmethod. We will now describe the suggestedprocedure in detail.

The first step is to determine the innova-tive project’s efficiency from the viewpoint ofthe venture fund using the traditional methodof discounted cash flows. This step includesan estimation of the cash flows of the ven-ture fund, it’s net present value and the inter-

nal rate of return (NPVv and IRRv respectively)for different variants of the venture fund’sshare in the charter capital of the investeecompany (starting from 25% with a step upof 4%, for shares of 29%, 33%, 41%, 45%,49%) and different values of the price-earn-ings ratio for the shares (P/E = 2, 3, 4, 5, 6,7).

The second step is to determine the inno-vative project’s efficiency from the viewpointof the venture fund by using the real optionsmethod (an estimation taking into account thevalue of the compound call option). This stepincludes the following: An estimation of theinternal rate of return of the venture fundIRRv

option and the net present value of the ven-

ture fund NPVvoption

taking into account thevalue of the compound call option as an ad-ditional cash flow, which appears at the verymoment of the venture fund’s “exit” from thebusiness of the investee company.

The compound call option for the ven-ture fund with a changed sense bearing un-

Figure 1. A Schematic Representation of a Venture Investment by Buying a Share Pur-chase Option at the Beginning of the Investment Process

To is the time when a ven-

ture fund buys a com-pound call option at the

price Iv

0 to purchase a

portion of the shares ofthe investee company atthe time T

1.

T1 is the time of purchasing a

portion of the shares of theinvestee company at the price

Iv

1, i.e. the exercise of an exter--

nal option, which means the pur-chase of an internal option onmaking a profit on selling theshares of the investee companyat the time T

2.

T2 is the time of selling

the shares held by theventure fund, i.e., gettingassets in the form of theprofit from selling theshares of the investeecompany at the price

Iv

2.

159


derstanding of the entry parameters could beestimated with a developed version of theGeske model (see formula (10.1), Appendix10). As the size of the paper is limited thedetailed definitions of all parameters are givenin Appendix10.

Now we will describe the technique ofthe compound call option’s estimation in de-tail. The critical value of the investeecompany’s equity shares at the moment T

1 can

be found from the equation (10.2) (see Ap-pendix 10).

It should be said that as the venture fundusually has a portfolio of investment projects,stopping its investment, at moment T

1, in the

project will allow the venture fund to reallo-cate its limited resources to other projects. Inour interpretation, the value V

T1 is the evalua-

tion of the business at the moment T1:

.....................(1)

To obtain the VvT1

the value VT1

is mul-tiplied by the fund’s share. There is no analyti-cal solution of the equation (10.2). So, to findthe figure for the critical value of the investeecompany’s equity shares at moment T

1 it is

necessary to use an optimization method. Suchoptimization procedures as Newton’s methodand the conjugated gradient method are real-ized in Microsoft Excel. Since these methodsgive practically the same result, we can use ei-ther of them.

Selling the shares at moment T2 the ven-

ture fund will lose the current period’s profitin proportion to its share of the charter capi-tal in the investee company. We consider thisfigure to be its implicit costs and the exerciseprice of the internal call option:

I2v = NPAT

total Exit * S................(2)

where NPATtotal Exit

is the net profit (total) inthe year of the “exit” by the venture fund fromthe business it invested in; S is the share ofthe venture fund in the charter capital of theinvestee company.

The current value of the underlying as-set in our interpretation is the current valueof an equity share of the investee company,which is owned by the venture fund (Vv). It isthe evaluation of the income that the venturefund will get from the sale of its shares.

The risk-free interest rate r in our calcu-lations will be 7 percent. This figure is takenas an average of deposit rates for alternativeassets like long-term deposits in the largest andsafest Russian banks as on 19.09.2011 (“TheRosselkhozbank,” “The Sberbank of Russia,”“Gazprombank,” and the “VTB Group”).

1

is the variability index of the NASDAQ Bio-technology Index (NBI) over a 7 year period(from 14.10.2004 to 14.10.2011) (NASDAQBiotechnology Index. URL:http://w w w . n a s d a q . c o m / d y n a m i c /nasdaqbiotech_activity.stm. Date of access:09.09.2011). It was also decided to use the NBIindex over a 7 year period and not just for thelast year only, because this 7 year time periodincluded the World financial crisis (2007, 2008and 2009). The standard deviation of theNASDAQ Biotechnology Index equals 15.44percent. The variation coefficient of theNASDAQ Biotechnology Index will be 12.78percent. Thus,

1 = 12.78 percent.

The compound call option value is prac-tically independent on the

2 values. No sig-

nificant impact of 2 on the value of the com-

pound call option is explained mathematicallyfrom the formula (10.1, Appendix 10). In equa-tion (10.1)

2 is included in the first additive

component as a parameter, that influences thevalue of N

2:

EPNPATV TT /11

Baranov and Muzyko

160

2 2 22 1 1 1 1 2 2( , ; )N h l

.

Let’s convert the sum: 2 21 1 2 2l .

We put the value of l (see formula (10.4), Ap-pendix 10) and get the fraction:

In the obtained fraction the parameter

2 enters both the numerator and denomina-

tor. This explains the contradictory influenceof

2 on the value of the compound call op-

tion. Furthermore, the parameter 2 enters the

formula for calculating the compound calloption [see (10.1) with a positive sign (the firstcomponent) or a negative one (the secondcomponent)]. Thus, the increase of

2 in the

first component is compensated by its growthin the second component N

2. However, the

parameter 2 influences the venture fund’s

decision about its investment as it is a param-eter of the equation (10.2).

When calculating the compound calloption’s value (see formula (10.1)) it is neces-sary to calculate the two-dimensional standardnormal distribution functions:

N2(h,l;)

as well as the standard normal distribution

function ).(1 hN The two-dimensional normal

distribution function is defined by the prob-ability density f (x,y):

where a, b are mathematical expectations ofstochastic variables x, y;

x ,

y are the mean

square deviations of stochastic variables x, y;

x,y is the correlation coefficient of the sto-

chastic variables x and y.

The normal distribution with the math-ematical expectation 0 and the standard de-viation 1 is called the standard normal distri-bution. The two-dimensional standard normaldistribution function F (x,y) takes the form:

.........................................................(4)

where (x,y) is a two-dimensional random vari-able;

x,y is the correlation coefficient of the

random variables x and y. The calculation ofthe two-dimensional standard normal distri-bution functions including double integralscan be made using the Maple 14 programpackage. The one-dimensional standard nor-mal distribution function can be calculatedusing the Microsoft Excel statistical function.

Results and Discussion

The author’s modification of the realoptions method for determining the valuationof venture capital investments from the ven-ture fund’s position was approbated using datafrom a real venture project in the industrialsector in Russia. The data from this real com-pany were used to construct a financial model,to make calculations and to draw conclusions.The company’s management kindly providedus with the initial data for our calculations,with the condition that we will not disclosethe company’s name and the name of its prod-ucts.

This company uses innovations in itsmanufacturing process and is going to imple-ment an innovative project. The core purposeof this project is to organize the production

)1(2

1exp

12

1),(

2,

2

yxyx

yxf

12

1),(

)2()1(2

1

2

222

dxdyeyxFl yxyxh

),;,( 2221

211

212 lhN

N2(h,l;)

2 2 2 21 1 2 2 1 1 2 22 2

2 1 1 2 2 2

2 2 2 2 2 21 1 2 2 1 1 2 2 1 1 2 2

1 1ln ( ) ln ( )

2 ( ) 2V V

r rI I

2 2 2 21 1 2 2 1 1 2 2

2 1 1 2 2 2

2 2 2 2 2 21 1 2 2 1 1 2 2 1 1 2 2

1 1ln ( ) ln ( )

2 ( ) 2V V

r rI I

,)()()(

2)(

2

2

2

2

yyxx

bybyaxax

........(3)

161


of its innovative products using its own pro-duction facilities. The project will increase thecompany’s profits, as the increase in its pro-duction capacity will ensure the growth ofsales associated with its introduction of newproducts onto the market, and increase themarket share of its existing products.

The total amount of financing requiredfrom external sources to implement the projectis 232,000,000 Rubles (7,282,000 USD). Sincethe subject of our analysis is the venture fundlet us consider the financing of the project bythe venture fund only. It is assumed that theventure fund will invest on a staged basis:35,000,000 Rubles (1,100,000 USD) in 2009and 197,000,000 Rubles (6,182,000 USD) in2010. The company-initiator of the project inits turn will invests the following resources:Its intangible assets (Russian, European andAmerican patents), a well-known brand name,the know-how (technology and formulationof the products), its unique equipment (thecompany-initiator was actively involved in thedevelopment of this equipment, with theknow-how being partly realized in this equip-ment). The cash flow forecast for the entireinnovative project is presented in Appendix1. The calculated indicators of the economicefficiency of the entire project by the tradi-tional discounted cash flow method (the NPV-method) are given in Appendix 2. We haveappraised the efficiency of the innovativeproject from the point of view of the venture fundby the traditional method using the NPVmethod (a standard calculation) and using thereal options method.

Let us determine the innovative project efficiencyfrom the viewpoint of the venture fund by the tradi-tional NVP method (a standard calculation). Frompractical experience it is known that a venturefund, as a rule, considers the possibility of in-vesting in a project beginning with 25% plus

one share of the investee company (purchas-ing a blocking parcel of shares). The blockingparcel of shares allows their owners to imposea veto on the Board of Directors’ decisions.The venture fund has its own cash flows, whichare separate from the general cash flows forthe entire project. The composition of theventure fund’s financial flows is shown in Table10.1, Appendix 10.

Let us take eight variant methods of cal-culating the cash flows of a venture fund. Wewill calculate the venture fund’s cash flows,and the indicators of the fund’s investmenteffectiveness IRRv and NPVv, for differentvariants of the venture fund’s share in the eq-uity capital of the investee company. The nec-essary calculations for every variant will bemade using various values of the P/E (Price-Earnings ratio for the shares): For P/E = 2, 3,4, 5, i.e., the rate of return is 50 percent, 33.3percent, 25 percent, and 20 percent per yearwhile for P/E = 6 and 7 the rate of return is16.7 percent and 14.3 percent per year respec-tively.

Now let us calculate the cash flows,NPVv and IRRv of the venture fund for thedifferent years of the venture fund’s exit fromthe business: In the years 2018, 2017, 2016,2015, 2014 and 2013. The venture fund willexit from the investee company in the yearwhen the IRRv values of the venture fund areat their maximum.

The results of the standard calculationof the venture fund’s internal rate of returnIRRv for the different years of the fund’s exitfrom the business of the investee companywith different levels of shareholding by thefund, and different P/E values are given inAppendix 3. This calculation is referred to asthe “standard” because it is made without tak-ing into account the value of the compoundcall option.

Baranov and Muzyko

162

The calculation of the cash flows for theNPVv and IRRv of the venture fund for thedifferent years of “exit” have demonstratedthat the venture fund must exit the investedbusiness in 2018 because this year sees thehighest internal rate of return for the venturefund. The venture fund will vary its share inthe equity capital of the investee companyfrom 25 percent to 49 percent because the cur-rent owners of the company want to retainthe majority shareholding.

Let us calculate the cash flows of theventure fund and the returns on investmentIRRv and NPVv for the year of “exit” 2018for different levels of shareholding by the ven-ture fund in the equity capital of the investeecompany (starting from 25% with a step of4%, for shareholdings of 29%, 33%, 41%,45%, 49%) and different values of the price-earnings ratio for the shares (P/E = 2, 3, 4, 5,6, 7) using the traditional NPV-method. Toobtain the net present value for the fund NPVv

we shall discount the cash flows at the ratesof 20 percent, 30 percent and 35 percent,which are widely used for appraising projects

in Russia by venture capitalists. The results ofthe standard calculation of the venture fund’sinternal rate of return IRRv for the venturefund’s exit from the business of the investeecompany in 2018 are shown in the left part ofthe Table 4.1, Appendix 4.

Let us analyze the obtained results. Weknow from our practical experience that ac-ceptable internal rates of return to the fundstart at 20 percent. According to our calcula-tions the IRRv acceptable to the fund is ob-served with a 25 percent shareholding, and aP/E = 7: IRRv = 20 percent. Although thisreturn, at only 14.3 percent, is rather low.

With the share of the venture fund inthe total equity of the investee company at 29percent the IRRv equals 20 percent or moreonly when the P/E = 6 (IRRv = 20%) or theP/E = 7 (IRRv = 22%). However the obtainedinternal rates of return, while acceptable, arestill rather low (“lower limit”). AcceptableIRRvs for the fund with the P/E = 5 and P/E= 6 are achieved with a shareholding at either45 percent or 49 percent: For a 45 percentshare at P/E = 5, the IRRv = 25 percent, at P/

Figure 2. The Correlation between the Internal Rates of Return for the Venture Fund(IRRv) and the Shares Held by the Fund

163


E = 6, the IRRv = 27 percent; with a 49 per-cent shareholding at P/E = 5, the IRRv = 26percent, and at P/E = 6 the IRRv = 29 per-cent. With the fund’s shareholding at 41 per-cent and a P/E = 6 the IRRv = 26 percent.Thus, the higher the shareholding of the eq-uity capital of the investee company by theventure fund and the higher the P/E ratioare, the higher is the internal rate of returnfor the venture fund (see Figure 2).

To calculate the NPVv of the fund let usget the net present value of the fund’s cashflows according to the so called “venture” dis-count rates of 20 percent, 30 percent and 35percent. With the IRRv of the venture fundless than the discount rate, the NPVv of theventure fund is negative. The results of thestandard calculation of the venture fund’sNPVv are given in the left part of Table 5.1,Appendix 5.

We see that the positive NPVv of theventure fund starts with the fund’sshareholding at 29 percent. With a P/E = 6,the NPVv is 370,000 Rubles; at a P/E = 7, theNPVv is 27,652,000 Rubles. With the fund’sshareholding at 33 percent, the NPVv of thefund is positive only when the P/E = 6 (NPVv

= 27,892,000 Rubles) and at P/E = 7 (NPVv

= 58,937,000 Rubles). But the value P/E = 7,that is, a return of 14.3 percent per year israther low. Also the positive NPVv of the ven-ture fund for shareholdings of 29 percent, 33percent, 37 percent, 41 percent, 45 percent and49 percent is only when the discount rate is20 percent, which is the bottom limit for a“venture” rate.

For the discount rates of 30 percent and35 percent the NPVv is negative. The NPVv ispositive with the discount rate equal to 30percent only when the fund’s shareholding is49 percent and the P/E = 7 (NPVv =8,825,000 Rubles). Thus, on true “venture”

terms acceptable to the fund, the NPVv ofthe fund is negative, that is, the project doesnot provide sufficient returns to the venturefund and should be rejected by the investmentcommittee.

Let us evaluate the innovative project from theventure fund’s position using the real option’s method.The zero moment is the year 2009: Iv

0=

35,000,000 Rubles (according to the cash flowforecast, for the implementation of the projectin 2009 an external investment of 35,000,000Rubles is required) (see Appendix 1). This sumof 35,000,000 Rubles is needed to pay for theadministrative expenses and approvals (moni-toring of the project, the investment agree-ment and the land lease); the design (feasibil-ity study (a booklet), a working draft, engineer-ing works, and (gas) energy supplies) and build-ing construction (pre-payment, basement andground floor construction). Thus, the periodof expiration of the compound (external) calloption T

1 is 1 year. The period of expiration

of the internal call option T2 is 9 years. As we

are calculating the value of the compound calloption for the venture fund at the moment ofinvestment, that is, at the moment when adecision to invest has been taken, t is the start-ing moment: t = 0.

= T

1 -

t= 1 year;

= T

2 -

T

1= 9

-

1= 8 years;

= T

2 -

t=

+

= 9 years

Thus, 2 = T

2 – T

1 is the period of time

the venture fund stays in the business of theinvestee company;

1 is the moment before

the fund makes an investment in the company,in return for a share of its charter capital. Inthe case of the expiration of the compound(external) call option by the venture fund atthe moment T

1 it will make an investment Iv

1

in the amount of 197,000,000 Rubles. Thepresent value of I

1discv discounted will be equal

to 184,112,000 Rubles.

Baranov and Muzyko

164

Let us take one example – for the fund’sshareholding of 49 percent and a price-earn-ings ratio for the shares P/E = 6, the venturefund’s investment at the moment T

2 will be

(see Formula 2):

Iv2 = NPAT

total

in 2018 * the fund’s share =

591,235,000 Rubles * 0.49 = 289,705,000Rubles. The present value of Iv

2disc will be

157,580,000 Rubles.

Vv is the value of the underlying assetof the internal call option at the moment ofits expiration, that is, in the year 2018, at thenet present value at the moment of evalua-tion. The assets the venture fund acquires theright to buy at the moment T

1 are nothing but

the venture fund’s income it can get at themoment T

2 after selling the shares bought at

the moment T1. The value Vv is nothing but

the terminal value of the project for the ven-ture fund TERv in the year of the “exit” ofthe fund from the business of the investeecompany (in the year 2018). It is the evalua-tion of the income that the venture fund willget from the sale of its shares. For example,for the fund’s shareholding of 49 percent with

the price-earnings ratio for the shares P/E =6 the Vv will be:

Vv= 485,412,000 Rubles * 0.49 * 6 +47,570,000 Rubles= 1,474,682,000Rubles.

The net present value Vv at the moment zerowill be 802,129,000 Rubles.

Let us calculate the compound call op-tion value using formula (10.1) (Appendix 10)for various values of

2. Figure 3 shows the

graph of the compound call option value’sdependence on the

2 values. It is seen on the

diagram that the compound call option valueis practically independent of the

2 values.

Figure 4 shows the graph of the depen-dence of the threshold value of the investeecompany’s equity shares at the moment T

1,V

T1,

on the different 2 values.

The diagram shows, that with the reduc-tion in uncertainty and risk of the operationsof the investee company over the time period(T

1, T

2),

2, the threshold value of the investee

company increases. This could be explainedin the following way: According to Black and

Figure 3. The Diagram of the Compound Call Options’s Value Dependence on the 2

Values

165


Sholes’ formula, with the decrease in the levelof uncertainty the option value will decline.In other words, in order to reach the same levelof profit with a lower level of uncertainty andat a lower option value at the moment T

1, the

price of the investee company’s businessshould be higher. In this case the decrease inthe option value, as a result of the decrease in

2, is compensated for by the higher value of

the business at the moment T1.

This result, in our opinion, shows thecontradiction between the traditional methodsof investment effectiveness’ evaluation and thereal options approach. In the first case a posi-tive decision on investment will be more likelyin the case of the lower volatility of the un-derlying asset, and, therefore, with the loweruncertainty of the development of the project.In the second case the high level of uncer-tainty is considered as a factor contributing tothe growth of the assets’ value, generated as a

result of the investment project’s implemen-tation. Since the compound call option valueis practically independent of the ó

2 values, let

us take the maximum ó2 value (the “worst”

case) as a value of the riskiness of the investeecompany’s operations during the time period(T

1, T

2), under the assumption that the risk level

will diminish over time. Thus, 2 =10.22%.

Let us make a variant calculation of thecompound call option value for the venturefund with different levels of the fund’sshareholdings in the investee company’s char-ter capital. The results of calculating the com-pound call option values for the different lev-els of the venture fund’s shareholdings areshown in Appendices 6–9. The results of cal-culating the value of the compound call op-tion for the various shareholdings by the fundin the charter capital of the investee companyat different values of P/E are represented inFigure 5.

Figure 4. The Diagram of the Dependence of the Threshold Value of the InvesteeCompany’s Equity Shares at the Moment T

1 on the

2 Values

Baranov and Muzyko

166

Let us calculate the internal rate of re-turn of the venture fund IRRv and the netpresent value of the venture fund NPVv tak-ing into account the value of the compound call optionas an additional cash flow of the venture fund, whichappears at the moment T

2 (in 2018), that is, at

the moment of the “exit” of the venture fundfrom the business of the investee company.The calculated results of the internal rate ofreturn (IRRv) for the venture fund, taking intoaccount the compound call option value areshown in the right part of Table 4.1, Appen-dix 4. The calculated results of the net presentvalue for the venture fund (NPVv), taking intoaccount the compound call option value areshown in the right part of Table 5.1, Appen-dix 5.

Let us compare the calculations of NPVv

and by the traditional method of the dis-counted cash flow and by taking into accountthe value of the compound call option. Fig-ure 6 represents the dependence of the inter-nal rate of return of the venture fund on thefund’s shareholding in the standard calculation.

Figure 7 shows the dependence of the inter-nal rate of return of the venture fund on thefund’s shareholding in calculations taking intoaccount the value of the compound call op-tion.

Figure 8 shows the net present value ofthe venture fund with 49% of the charter capi-tal of the investee company at various P/Evalues and the discount rate at 20 percent. Fig-ure 9 demonstrates the net present value ofthe venture fund with 49% of the charter capi-tal at various P/E values and the discount rateat 35 percent. The standard calculation of theinternal rate of return of the venture fund andthe calculation with the real option are pre-sented in Figure 10.

So, we can observe that when incorporating the estimate of the real option,the internal rate of return and the net presentvalue of the venture fund improve. In the cal-culations using the real options method, theinnovative project has a positive value andshould be funded by a venture capital inves-tor.

Figure 5. The Value of the Compound Call Option for Various Shares of the Fund in theEquity Capital of the Investee Company at Different P/Es, in Thousands ofRubles

167


Figure 6. The Dependence of the Internal Rate of Return of the Venture Fund IRRv onthe Fund’s Share (A Standard Calculation)

Figure 7. The Dependence of the Internal Rate of Return of the Venture fund IRRv on theFund’s Share (A Calculation Taking Into Account the Value of the CompoundCall Option)

Baranov and Muzyko

168

Figure 8. NPVv of the Venture Fund with 49 Percent of the Equity Capital at Various P/E Values and the Discount Rate at 20 percent

Figure 9.NPVv of the Venture Fund with 49 percent of the Equity Capital at Various P/EValues and the Discount Rate at 35 percent

-5 0,0 00

0

5 0,0 00

10 0,0 00

15 0,0 00

20 0,0 00

25 0,0 00

30 0,0 00

35 0,0 00

40 0,0 00

45 0,0 00

P /E = 2 P /E = 3 P/ E = 4 P /E = 5 P /E = 6 P /E = 7

NPV

v, in thousands of Rubles

stand ard c alc ulat ion of N PVv (in th ousan ds of Rubles)

c alculat ion of NP Vv taking in to acco un t th e value of th e c om po und call o ption ( in tho usands o f Rub les)

-120,000

-100,000

-80,000

-60,000

-40,000

-20,000

0

20,000

40,000

60,000

P/E = 2 P/E = 3 P/E = 4 P/E = 5 P/E = 6 P/E = 7

NPV

v, in thousands of Rubles

standard calculation of NPVv (in thousands of Rubles)

calculation of NPVv taking into account the value of the compound call option (in thousands of Rubles)

NP

Vv ,

in t

ho

usa

nd

s o

f R

ub

les

NP

Vv ,

in t

ho

usa

nd

s o

f R

ub

les

169


Conclusion

1. The peculiarities of the application of thereal options method for assessing venturecapital investments in innovative projectswere revealed. The options approach isapplicable to the evaluation of innovativeprojects by venture capitalists, but only tak-ing into account the specific features ofventure investments.

2. The author’s modification of the real op-tions method from the point of view of itsapplication to the venture capital’s financ-ing of innovative projects and its approba-

tion using data on venture projects werepresented. The entry parameters of themodified version of Geske’s formula wereconstrued from the venture fund’s position.

3. The practical viability of the author’s ap-proach was demonstrated. The use of thereal options method raises the accuracy ofthe estimation and enhances the instru-ments of the venture fund in evaluating theeconomic efficiency of innovative projects.This will allow the practical use of the realoptions method for analyzing projects byventure funds to be expanded.

Figure 10. The Standard Calculation of IRRv and Calculation of IRRv Taking Into Ac-count the Value of the Compound Call option at P/E=6 for the Fund’sShareholding at 24 percent and 49 percent

0%

5%

10%

15%

20%

25%

30%

35%

40%

for venture fund's share 24%

for venture fund's share 49%

17%

29%

22%

36%

standard calculation of IRRv

calculation of IRRv taking into account the value of the compound call option

Baranov and Muzyko

170

References

Barone-Adesi, G., and R. Whaley. 1987. Efficient analytic approximation of American option values.Journal of Finance 42: 301 - 320.

Black, F., and M. Scholes. 1973. The pricing of options and corporate liabilities. Journal of Political Economy81 (3): 637 - 659.

Botteron, P., and J. Casanova. 2003. Start-ups defined as portfolios of embedded options. Research Paper 85(May): 1-14. FAME – International Center for Financial Asset Management and Engineering.

Carr, P. 1988. The valuation of sequential exchange opportunities. Journal of Finance 58: 1235 - 1256.

Geske, R. 1979. The valuation of compound options. Journal of Financial Economics 7(1): 63 - 81.

Geske, R., and H.E. Johnson. 1984. The American put option valued analytically. Journal of Finance 39:1511 - 1524.

Gong, P., Z.-W. He, and J.-L. Meng. 2006. Time-dependent Volatility Multi-stage Compound Real OptionModel and Application. Journal of Industrial Engineering and Engineering Management, February: 1 - 14.

Herath H. S.B., and C.S. Park 2002. Multi-Stage Capital Investment Opportunities as Compound RealOptions. The Engineering Economist 47 (1): 1 - 27.

Hsu, Y. 2002. Staging of venture capital investment: A real options analysis. Working Paper (May), Univer-sity of Cambridge, JIMS: 1 - 47.

Huixia, Z., and Y. Tao. 2010. Venture capital decision Model Based on Real Option and Investor Behavior.Economics and Management School, Wuhan University, China: 221 - 225.

Jensen, K., and P. Warren. 2001. The use of options theory to value research in the service sector. R&DManagement 31 (2): 173 - 180.

Kulatilaka, N., and L. Toschi. 2011. An Integration of the Resource Based View and Real Options Theory forInvestments in Outside Opportunities. Available at SSRN: http://ssrn.com/abstract=1541865. Date ofaccess: 02.08.2011.

Lee, J., and A. Paxson. 2000a. Confined exponential approximations for the valuation of American op-tions. Working Paper. Manchester Business School.

Lee, J., and A. Paxson. 2000b. Analytic approximations for American exchange options. Working Paper.Manchester Business School.

Lee, J., and A. Paxson. 2001. Valuation of R&D real American sequential exchange options. R&D Manage-ment 31 (2): 191 - 201.

Li, Y. 2008. Duration analysis of venture capital staging: A real options perspective. Journal of BusinessVenturing 23: 497 -5 12.

Li, Y., and J. T. Mahoney. 2011. When are venture capital projects initiated? Journal of Business Venturing 26:239 - 254.

Lin, W. T. 2002. Computing a multivariate normal integral for valuing compound real options. Review ofQuantitative Finance and Accounting 18 (2): 185 - 209.

Margrabe, W. 1978. The value of an option to exchange one asset for another. Journal of Finance 33: 177 -186.

171


NASDAQ Biotechnology Index. URL:http://www.nasdaq.com/dynamic/nasdaqbiotech_activity.stm. Dateof access: 09.09.2011.

Perlitz M., T. Peske, and R. Schrank. 1999. Real options valuation: The new frontier in R&D projectevaluation? R&D Management 29 (3): 255 - 269.

Seppa, T. J., and T. Laamanen. 2001. Valuation of venture capital investments: Empirical evidence. R&DManagement 31: 215 - 230.

Tong, W. T., and Y, Li. 2010. Real options and investment mode: Evidence from corporate venture capitaland acquisition. Organization Science Forthcoming. Available at SSRN: http://ssrn.com/ab-stract=1529692. Date of access: 23.07.2011.

Vanhaverbeke, W., V. Van de Vrande, and H. Chesbrough. 2008. Understanding the advantages of openinnovation practices in corporate venturing in terms of real options. Creativity and Innovation Man-agement 17(4): 251 - 258.

Wadhwa, A., and C. Phelps. 2010. An option to ally: A Dyadic Analysis of Corporate Venture CapitalRelationships”. In Proceedings of the Atlanta Competitive Advantage Conference (ACAC), 02/2010. DOI:10.2139/ssrn.1553322.

Willner, R. 1995. Valuing start-up venture growth options. In Trigeorgis, L. (Ed.), Real Options in CapitalInvestment. New York: Praeger: 221 - 239.

APP

EN

DIX

1, T

able

1.1

. Cas

h F

low

For

ecas

t of

the

Ent

ire

Pro

ject

(in

tho

usan

ds o

f R

uble

s)

Sour

ce: T

he a

utho

r’s

calc

ulat

ion

resu

lts o

btai

ned

by th

e in

nova

tive

pro

ject

fina

ncia

l mod

el

Ind

icat

ors

20

09

2010

20

11

2012

20

13

2014

20

15

2016

20

17

2018

Cas

h f

low

fro

m o

per

atio

ns

28,6

67

10,8

35

59,4

17

103,

749

44,8

34

40,8

27

77,1

52

93,9

47

114,

628

140,

129

Net

pro

fit

37,6

62

77,2

53

110,

511

140,

309

184,

208

240,

604

327,

332

398,

607

485,

412

591,

235

Plu

s: D

epre

ciat

ion

79

9 88

0 3,

081

9,75

1 9,

754

9,75

4 9,

754

9,75

4 9,

754

9,75

4

Min

us:

Incr

ease

in w

ork

ing

cap

ital

-2

4,76

7 45

,501

54

,174

46

,312

55

,157

66

,781

80

,908

98

,091

11

9,01

1 14

4,50

6

Val

ue a

dd

ed t

ax (

VA

T)

pay

able

to

th

e b

udge

t 34

,562

21

,797

0

0 93

,970

14

2,74

9 17

9,02

5 21

6,32

2 26

1,52

7 31

6,35

4

Inve

stm

ent

Inve

stm

ent

in f

ixed

ass

ets

wit

ho

ut

VA

T

2,58

9 36

,114

23

1,77

4 67

8 0

0 0

0 0

0

Fin

anci

ng

35

,000

19

7,00

0 -7

,571

-1

0,83

0 -1

3,75

0 -1

8,05

2 -2

3,57

9 -2

5,70

4 -2

7,82

9 -2

9,95

4

Div

iden

d p

aym

ents

0

0 7,

571

10,8

30

13,7

50

18,0

52

23,5

79

32,0

79

39,0

63

47,5

70

Equ

ity

inve

stm

ent

mad

e b

y th

e ve

ntu

re f

und

35

,000

19

7,00

0 0

0 0

0 0

0 0

0

Net

cas

h f

low

61

,078

17

1,72

1 -1

79,9

27

92,2

41

31,0

84

22,7

75

53,5

73

61,8

69

75,5

64

92,5

58

Mo

net

ary

fun

ds

on

th

e p

roje

ct

op

erat

or’

s ac

cou

nt

at t

he

beg

inn

ing

of

the

per

iod

10

,202

71

,280

24

3,00

1 63

,074

15

5,31

5 18

6,39

9 20

9,17

4 26

2,74

7 32

4,61

6 40

0,18

0

Mo

net

ary

fun

ds

on

th

e p

roje

ct

op

erat

or’

s ac

cou

nt

at t

he

end

o

f th

e p

erio

d

71,2

80

243,

001

63,0

74

155,

315

186,

399

209,

174

262,

747

324,

616

400,

180

492,

738

Baranov and Muzyko

172

Indi

cato

rs

Tot

al

2009

-201

8 20

09

2010

20

11

2012

20

13

2014

20

15

2016

20

17

2018

1 2

3 4

5 6

7 8

9 10

11

12

In

vestm

ent i

n fix

ed as

sets,

with

out

VA

T 27

1,791

3,

225

36,1

14

231,

774

678

0 0

0 0

0 0

Inve

stmen

t in

work

ing

capi

tal i

ncre

ase

610,

766

0 0

0 46

,312

55

,157

66,78

1 80

,908

98

,091

11

9,011

14

4,506

To

tal i

nves

tmen

t 88

2,55

7 3,

225

36,1

14

231,

774

46,9

90

55,15

7 66

,781

80,9

08

98,0

91

119,0

11

144,5

06

Posit

ive c

ash

flow

incr

ease

(net

pro

fit

plus

dep

recia

tion

min

us V

AT

paya

ble

to th

e bu

dget

) as c

ompa

red

to 2

011

afte

r com

miss

ioni

ng th

e pl

ant

430,

892

0 0

0 36

,469

-1

3,60

0 -5

,983

44,4

69

78,4

47

120,0

47

171,0

43

Tota

l pos

itive

and

neg

ative

flow

s with

re

gard

to th

e te

rmin

al va

lue

relat

ive to

P/

E =

4 (fo

r the

end

of th

e pe

riod)

1,

913,

275

-3,2

25

-36,

114

-231

,774

-10,5

21

-68,

757

-72,

764

-36,

439

-19,

644

1,03

6 2,

391,

477

Tota

l pos

itive

and

neg

ative

flow

s W

ITH

OU

T re

gard

to th

e te

rmin

al va

lue

-451

,665

-3

,225

-3

6,11

4 -2

31,77

4 -1

0,521

-6

8,75

7 -7

2,76

4 -3

6,43

9 -1

9,64

4 1,

036

26,53

7

Boo

k va

lue

of fi

xed

asse

ts a

s of

the

end

of 2

018

(term

inal

val

ue)

207,

551

Ter

min

al v

alue

rela

tive

to P

/E

2,36

4,94

0

P/E

(the

com

pany

’s pr

ofit

per y

ear i

s to

be1/

4 of

the i

nves

tmen

t, i.e

. 25%

) 4

The

inte

rnal

rate

of r

etur

n (I

RR)

with

re

gard

to th

e te

rmin

al va

lue

29,85

%

num

ber o

f the

per

iod

0

1 2

3 4

5 6

7 8

9

NPV

(with

rega

rd to

the

term

inal

valu

e)

912,

554

-3,2

25

-33,

751

-202

,440

-8,5

88

-52,

455

-51,

880

-24,

281

-12,

233

603

1,30

0,80

5

Disc

ount

multip

lier:

1.0

0 0.9

3 0.8

7 0.

82

0.76

0.

71

0.67

0.62

0.

58

0.54

Payb

ack

perio

d w

ith re

gard

to

disc

ount

ing

20

qua

rters

(5 y

ears

)

Payb

ack

perio

d w

ithou

t reg

ard

to

disc

ount

ing

25 q

uarte

rs (6

yea

rs an

d 1

quar

ter)

APP

END

IX 2

, Tab

le 2

.1. I

ndic

ator

s of t

he E

cono

mic

Effe

ctiv

enes

s of t

he E

ntir

e Pr

ojec

t (in

thou

sand

s of

Rub

les)

173


Baranov and Muzyko

174

APPENDIX 3, Table 3.1. Venture Fund’s Internal Rate of Return IRRv for Different Yearsof the Fund’s Exit from the Business of the Investee Company, % (standard calculation)

P/E

Year of fund’s exit from the business

2018 2017 2016 2015 2014

Venture fund’s share 24%

P/E = 2 5 2 -2 -8 -17

P/E = 3 9 7 4 -2 -9

P/E = 4 12 10 8 3 -3

P/E = 5 15 13 12 7 1

P/E = 6 17 16 15 11 6

P/E = 7 19 18 17 14 9


P/E = 2 5 3 -1 -8 -16

P/E = 3 10 7 4 -1 -8

P/E = 4 13 11 9 4 -2

P/E = 5 15 14 12 8 2

P/E = 6 18 17 15 11 7

P/E = 7 20 19 18 15 10


P/E = 2 7 5 1 -5 -13

P/E = 3 12 10 7 2 -5

P/E = 4 15 14 12 7 1

P/E = 5 18 17 15 11 6

P/E = 6 20 19 18 15 11

P/E = 7 22 22 21 18 15


P/E = 2 9 7 4 -2 -10

P/E = 3 14 12 10 5 -2

P/E = 4 17 16 14 10 5

P/E = 5 20 19 18 14 10

P/E = 6 22 22 21 18 14

P/E = 7 24 24 24 21 18


P/E = 2 11 9 6 0 -7

P/E = 3 15 14 12 7 1

P/E = 4 19 18 16 12 8

P/E = 5 22 21 20 17 13

P/E = 6 24 24 23 21 18

P/E = 7 26 26 26 24 22

175


APPENDIX 3, Table 3.1. Continued

P/E

Year of fund’s exit from the business

2018 2017 2016 2015 2014


P/E = 2 13 11 8 2 -5

P/E = 3 17 16 14 9 4

P/E = 4 20 20 18 15 10

P/E = 5 23 23 22 19 16

P/E = 6 26 26 26 23 21

P/E = 7 28 28 28 27 25


P/E = 2 14 12 10 4 -2

P/E = 3 19 17 16 11 6

P/E = 4 22 21 20 17 13

P/E = 5 25 25 24 22 19

P/E = 6 27 27 28 25 23

P/E = 7 29 30 31 29 28


P/E = 2 16 14 11 6 0

P/E = 3 20 19 17 13 9

P/E = 4 23 23 22 19 15

P/E = 5 26 26 26 24 21

P/E = 6 29 29 30 28 26

P/E = 7 31 32 32 31 30

Baranov and Muzyko

176

APPENDIX 4, Table 4.1. Standard Calculation of the Venture Fund’s Internal Rate ofReturn IRRv and IRRv Calculations Taking into Account the Value of the CompoundCall Option for the Venture Fund’s “Exit” from the Business in 2018

P/E Standard

calculation Calculation taking into account the value of the compound call option

Venture fund’s share 24% P/E = 2 5% 5% P/E = 3 9% 9% P/E = 4 12% 14% P/E = 5 15% 19% P/E = 6 17% 22% P/E = 7 19% 25%





177



P/E Standard

calculation Calculation taking into account the value of the compound call option



Venture fund’s share 49% P/E = 2 16% 17% P/E = 3 20% 24% P/E = 4 23% 29% P/E = 5 26% 33% P/E = 6 29% 36%

Baranov and Muzyko

178

APPENDIX 5, Table 5.1. Standard Calculation of the Venture Fund’s NPVv and NPVv

Calculations Taking into Account the Value of the Compound Call Option for the Ven-ture Fund’s “Exit” from the Business in 2018, in thousands of Rubles

P/E Standard calculation

Calculation taking into account the value of compound call option

r = 20% r = 30% r = 35% r = 20% r = 30% r = 35% Venture fund’s share 24%

P/E = 2 - 124,346 - 145,779 - 149,963 - 124,345 - 145,779 - 149,962 P/E = 3 - 101,768 - 134,794 -142,141 - 99,513 - 133,696 - 141,359 P/E = 4 - 79,189 - 123,808 - 134,319 - 59,998 - 114,470 - 127,670 P/E = 5 - 56,611 - 112,822 - 126,497 - 15,015 - 92,583 - 112,086 P/E = 6 - 34,033 - 101,836 -118,675 30,141 - 70,612 - 96,442 P/E = 7 - 11,455 - 90,850 -110,853 75,297 - 48,640 - 80,798

Venture fund’s share 25% P/E = 2 - 121,228 - 144,081 - 148,672 - 121,226 - 144,080 - 148,672 P/E = 3 - 97,709 - 132,638 - 140,525 - 94,398 - 131,027 - 139,377 P/E = 4 - 74,190 - 121,194 - 132,377 - 51,740 - 110,271 - 124,599 P/E = 5 - 50,671 - 109,751 - 124,229 - 4,794 - 87,428 - 108,335 P/E = 6 - 27,152 - 98,307 - 116,081 42,244 - 64,541 - 92,039 P/E = 7 - 3,633 - 86,863 - 107,933 89,282 - 41,654 - 75,743

Venture fund’s share 29% P/E = 2 - 108,758 - 137,288 - 143,512 - 106,777 - 136,324 - 142,826 P/E = 3 - 81,476 - 124,014 -134,060 - 71,754 - 119,283 - 130,692 P/E = 4 - 54,194 - 110,739 - 124,609 - 18,466 - 93,355 - 112,231 P/E = 5 - 26,912 - 97,464 - 115,157 36,092 - 66,809 - 93,330 P/E = 6 370 - 84,190 - 105,706 90,656 - 40,260 - 74,427 P/E = 7 27,652 - 70,915 - 96,254 145,220 - 13,711 - 55,524

Venture fund’s share 33% P/E = 2 - 96,288 - 130,495 - 138,351 - 95,911 - 130,311 - 138,221 P/E = 3 - 65,243 - 115,389 - 127,596 - 46,915 - 106,472 - 121,247 P/E = 4 - 34,198 - 100,284 - 116,841 14,887 - 76,401 - 99,836 P/E = 5 - 3,153 - 85,178 - 106,086 76,978 - 46,190 - 78,325 P/E = 6 27,892 - 70,073 - 95,330 139,068 - 15,979 - 56,815 P/E = 7 58,937 - 54,967 - 84,575 201,158 14,232 - 35,304

Venture fund’s share 37% P/E = 2 - 83,818 - 123,702 - 133,191 - 82,358 - 122,991 - 132,685 P/E = 3 - 49,010 - 106,765 - 121,132 - 21,314 - 93,289 - 111,537 P/E = 4 - 14,202 - 89,829 - 109,073 48,248 - 59,443 - 87,438 P/E = 5 20,606 - 72,892 - 97,014 117,864 - 25,570 - 63,320 P/E = 6 55,415 - 55,956 - 84,955 187,480 8,303 - 39,203 P/E = 7 90,223 - 39,019 - 72,896 257,096 42,175 - 15,085

Venture fund’s share 41% P/E = 2 - 71,348 - 116,909 - 128,030 - 67,623 - 115,096 - 126,740 P/E = 3 - 32,777 - 98,141 - 114,668 4,474 - 80,016 - 101,762 P/E = 4 5,795 - 79,374 - 101,305 81,607 - 42,486 - 75,041 P/E = 5 44,366 - 60,606 - 87,943 158,749 - 4,951 - 48,315 P/E = 6 82,937 - 41,839 - 74,580 235,892 32,584 - 21,590 P/E = 7 121,508 - 23,071 - 61,217 313,034 70,118 5,135

Venture fund’s share 45% P/E = 2 - 58,878 - 110,115 - 122,870 - 51,649 - 106,598 - 120,366 P/E = 3 - 16,544 - 89,517 - 108,203 30,300 - 66,724 - 91,975 P/E = 4 25,791 - 68,919 - 93,537 114,967 - 25,528 - 62,643 P/E = 5 68,125 - 48,320 - 78,871 199,636 15,668 - 33,311 P/E = 6 110,459 - 27,722 - 64,205 284,304 56,865 - 3,978 P/E = 7 152,793 - 7,123 - 49,538 368,973 98,062 25,355

Venture fund’s share 49% P/E = 2 - 46,408 - 103,322 - 117,709 - 34,693 - 97,622 - 113,651 P/E = 3 - 310 - 80,893 - 101,739 56,133 - 53,430 - 82,185 P/E = 4 45,787 - 58,464 - 85,769 148,327 - 8,571 - 50,245 P/E = 5 91,884 - 36,034 - 69,799 240,522 36,288 - 18,306 P/E = 6 137,982 - 13,605 - 53,830 332,716 81,146 13,634 P/E = 7 184,079 8,825 - 37,860 424,911 126,005 45,574

APP

END

IX 6

, Tab

le 6

.1. C

alcu

latio

n of

the

Com

poun

d C

all O

ptio

n Va

lues

for t

he V

entu

re F

und’

s Sha

res a

t 24

perc

ent

and

25 p

erce

nt w

ith D

iffer

ent P

/E V

alue

sIn

put p

aram

eter

s: I 0v =

35,

000,

000

Rubl

es; I

1v = 1

84,1

12,0

00 R

uble

s; I 2v =

77,

182,

000

Rubl

es (f

or sh

are 2

4%);

I 2v =

80,3

98,0

00 R

uble

s (fo

r sha

re 2

5%);

r =7%

; 1 =

12.

78%

; 2 =

10.

224%

; T1 =

1 y

ear;

T 2 = 9

yea

rs;

1 = 1

year

; 2 = 8

year

s;

= 9

year

s.

P/E

va

lues

vV

, th

ousa

nd

Rubl

es

v TV1

, th

ousa

nd

Rub

les

vV

,

thou

sand

R

uble

s

h l

ρ N

2(h, l

, ρ)

12 1

h

22 2

12 1

l

);

,(

22 2

12 1

12 1

2

l

hN

Cv,

thou

sand

R

uble

s

Ven

ture

fund

’s sh

are

24%

P/E

= 2

139,4

09

37,08

1 54

,768

-3.3

7224

3067

3.

7046

6935

9

0.

4042

2604

2

0.0

0037

2793

-3

.2444

4306

7

4.

0208

2909

8

0.00

0588

403

2,7

P/E

= 3

202,7

76

55,62

2 54

,768

-0.4

4039

7503

4.

8897

9768

6

0.

4042

2604

2

0.3

2982

4615

-0

.3125

9750

3

5.

2059

5742

5

0.

3772

9323

7

6,32

9

P/E

= 4

266,1

44

74,16

3 54

,768

1.687

4240

44

5.

7499

1856

9

0.

4042

2604

2

0.9

5423

9077

1.8

1522

4044

6.

0660

7830

8

0.

9652

5525

7

53

,862

P/E

= 5

329,5

12

92,70

4 54

,768

3.358

5919

46

6.

4254

4815

0.

4042

2604

2

0.9

9960

8297

3.4

8639

1946

6.

7416

0788

9

0.

9997

5520

8

11

6,74

3

P/E

= 6

392,8

80

111,2

44

54,7

68

4.7

3489

4368

6.

9817

8543

1

0.

4042

2604

2

0.9

9999

8904

4.8

6269

4368

7.

2979

4517

0.

9999

9942

1

18

0,10

9

P/E

= 7

456,2

47

129,7

85

54,7

68

5.9

0492

6511

7.

4547

4289

2

0.

4042

2604

2

0.9

9999

9998

6.0

3272

6511

7.

7709

0263

2

0.

9999

9999

9

24

3,47

6

Ven

ture

fund

’s sh

are

25%

P/E

= 2

145,2

18

38,62

7 57

,509

-3

.115

5435

43

3.

7046

7243

3

0.

4042

2604

2

0.0

0091

8031

-2

.9877

4354

3

4.

0208

3217

3

0.

0014

0522

6

7

P/E

= 3

211,2

26

57,94

0 57

,509

-0.1

8367

6652

4.

8898

0938

2

0.

4042

2604

2

0.4

2713

3559

-0

.0558

7665

2

5.

2059

6912

1

0.

4777

2003

5

9,

293

P/E

= 4

277,2

34

77,25

3 57

,509

1.944

1266

71

5.

7499

2289

6

0.

4042

2604

2

0.

9740

5991

2.0

7192

6671

6.

0660

8263

6

0.9

8086

386

63

,008

P/E

= 5

343,2

42

96,56

6 57

,509

3.615

2833

44

6.

4254

4794

3

0.

4042

2604

2

0.

9998

4999

3.7

4308

3344

6.

7416

0768

3

0.

9999

0911

2

12

8,75

9

P/E

= 6

409,2

50

115,8

80

57,5

09

4.9

9157

8168

6.

9817

8215

2

0.

4042

2604

2

0.9

9999

9701

5.1

1937

8168

7.

2979

4189

2

0.

9999

9984

7

19

4,76

6

P/E

= 7

475,2

58

135,1

93

57,5

09

6.1

6162

1972

7.

4547

4432

8

0.

4042

2604

2

0.9

9999

9999

64

6.2

8942

1972

7.

7709

0406

7

0.99

9999

9998

4

26

0,77

4

179


APP

END

IX 7

, Tab

le 7

. Cal

cula

tion

of th

e C

ompo

und

Cal

l Opt

ion

Valu

es fo

r the

Ven

ture

Fun

d’s S

hare

s at 2

9 pe

rcen

t and

33 p

erce

nt w

ith D

iffer

ent P

/E V

alue

s

Inpu

t par

amet

ers:

I 0v = 3

5,00

0,00

0 Ru

bles

; I1v

= 18

4,11

2,00

0 Ru

bles

; I2v =

93,

262,

000

Rubl

es (f

or sh

are 2

9%);

I 2v = 1

06,1

26,0

00 R

uble

s (fo

r sha

re 3

3%);

r=

7%;

1= 1

2.78

%;

2 = 1

0.22

4%; T

1 = 1

yea

r; T 2 =

9 y

ears

; 1 = 1

year

; 2 =

8 ye

ars;

3 =

9 ye

ars.

P/E

valu

es

vV

, th

ousa

nd

Rubl

es

v TV1

, th

ousa

nd

Rubl

es

vV

, th

ousa

nd

Rub

les

h l

ρ N

2(h, l

, ρ)

12 1

h

22 2

12 1

l

);

,(

22 2

12 1

12 1

2

l

hN

Cv,

tho

usan

d Ru

bles

Vent

ure f

und’

s sha

re 29

%

P/E

= 2

168,4

52

44,80

7 68

,841

-2

.2002

7446

4

3.7

0464

5057

0.4

0422

6042

0.

0436

4838

4

-2

.0724

7446

4

4.0

2080

4797

0.0

6003

7743

5,

560

P/E

= 3

245,0

21

67,21

0 68

,841

0.

7315

9625

9

4.8

8978

3555

0.4

0422

6042

0.

7677

9244

6

0.8

5939

6259

5.2

0594

3294

0.8

0493

9026

27

,287

P/E

= 4

321,5

91

89,61

3 68

,841

2.8

5942

592

5.7

4990

7716

0.4

0422

6042

0.

9978

7795

4

2.

9872

2592

6.0

6606

7455

0.9

9859

2392

10

0,27

3

P/E

= 5

398,1

60

112,0

17

68,84

1

4.530

5791

5

6.4

2543

1371

0.4

0422

6042

0.

9999

9705

9

4.

6583

7915

6.7

4159

111

0.9

9999

8406

17

6,82

5

P/E

= 6

474,7

30

134,4

20

68,84

1

5.90

6888

123

6.9

8177

1299

0.4

0422

6042

0.

9999

9999

8

6.0

3468

8123

7.2

9793

1039

0.9

9999

9999

25

3,395

P/E

= 7

551,2

99

156,8

24

68,84

1

7.07

6927

954

7.4

5473

1869

0.4

0422

6042

0.

9999

9999

9999

22

7.2

0472

7954

7.7

7089

1608

0.9

9999

9999

9997

03

32

9,964

Vent

ure f

und’

s sha

re 33

%

P/E

= 2

191,6

87

50,98

7 80

,762

-1

.4277

5304

7

3.7

0464

0818

0.4

0422

6042

0.

0766

8142

3

-1

.2999

5304

7

4.0

2080

0558

0.0

9680

8516

1,0

59

P/E

= 3

278,8

18

76,48

0 80

,762

1.

5041

3588

7

4.8

8978

6677

0.4

0422

6042

0.

9337

2667

1

1.6

3193

5887

5.2

0594

6416

0.9

4865

3477

51

,438

P/E

= 4

365,9

48

101,9

74

80,76

2

3.63

1929

374

5.7

4989

6216

0.4

0422

6042

0.

9998

5934

1

3.7

5972

9374

6.0

6605

5955

0.9

9991

4951

13

7,76

2

P/E

= 5

453,0

79

127,4

67

80,76

2

5.30

3097

264

6.4

2542

5797

0.4

0422

6042

0.

9999

9994

3

5.4

3089

7264

6.7

4158

5537

0.9

9999

9972

22

4,892

P/E

= 6

540,2

09

152,9

61

80,76

2

6.67

9385

202

6.9

8175

7223

0.4

0422

6042

0.

9999

9999

9986

54

6.8

0718

5202

7.2

9791

6962

0.999

9999

9999

4877

312,

022

P/E

= 7

627,3

40

178,4

54

80,76

2

7.84

9436

507

7.4

5472

243

0.4

0422

6042

0.99

9999

9999

9996

7.9

7723

6507

7.7

7088

2169

0.9

9999

9999

9999

93

39

9,153

Baranov and Muzyko

180

APP

EN

DIX

8, T

able

8.1

. Cal

cula

tion

of th

e C

ompo

und

Cal

l Opt

ion

Valu

es fo

r the

Ven

ture

Fun

d’s

Shar

es a

t 37

perc

ent a

nd 4

1 per

cent

with

Diff

eren

t P/E

Val

ues

Inpu

t par

amet

ers:

I 0v = 3

5,00

0,00

0 Ru

bles

; I1v

= 18

4,11

2,00

0 Ru

bles

; I2v =

118

,989

,000

Rub

les (

for s

hare

37%

); I 2v =

131

,853

,000

Rub

les (

for s

hare

41%

); r

= 7%

; 1=

12.

78%

; 2 =

10.

224%

; T1 =

1 y

ear;

T 2 = 9

yea

rs;

1 = 1

year

; 2

= 8

year

s;

3 = 9

year

s.

P/E

valu

es

vV

, th

ousa

nd

Rubl

es

v TV1

, th

ousa

nd

Rub

les

vV

, th

ousa

nd

Rubl

es

h l

ρ N

2(h, l

, ρ)

12 1

h

22 2

12 1

l

);

,(

22 2

12 1

12 1

2

l

hN

Cv,

tho

usan

d R

ubles

Vent

ure f

und’

s sha

re 3

7%

P/E

= 2

214,9

22

57,16

7 93

,270

-0

.763

9752

64

3.

7046

6407

8

0.4

0422

6042

0.

2224

4044

9

-0

.636

1752

64

4.

0208

2381

7

0.2

6233

0946

4,

099

P/E

= 3

312,6

14

85,75

1 93

,270

2.1

6790

2913

4.

8898

0558

8

0.4

0422

6042

0.

9849

1668

5

2.2

9570

2913

5.

2059

6532

8

0.

9891

5351

77

,731

P/E

= 4

410,3

06

114,3

34

93,27

0

4.295

7121

47

5.

7499

2149

3

0.4

0422

6042

0.

9999

9128

9

4.4

2351

2147

6.

0660

8123

2

0.9

9999

5144

17

5,269

P/E

= 5

507,9

98

142,9

18

93,27

0

5.966

8724

58

6.4

2544

801

0.4

0422

6042

0.999

9999

9872

5047

6.0

9467

2458

6.7

4160

775

0.

9999

9999

9443

858

27

2,96

1

P/E

= 6

605,6

89

171,5

02

93,27

0

7.343

1568

28

6.

9817

7799

3

0.4

0422

6042

0.999

9999

9999

8436

7.4

7095

6828

7.

2979

3773

3

0.

9999

9999

9999

811

37

0,65

2

P/E

= 7

703,3

81

200,0

85

93,27

0

8.513

2042

07

7.

4547

4161

4

0.4

0422

6042

0.99

9999

9999

9844

8.6

4100

4207

7.

7709

0135

3

0.

9999

9999

9999

811

46

8,34

4

Vent

ure

fund

’s sh

are

41%

P/E

= 2

238,1

57

63,34

7 10

6,365

-0.1

8555

3711

3.7

0465

881

0.4

0422

6042

0.

4263

9458

8

-0

.057

7537

11

4.0

2081

855

0.

4769

7163

10

,454

P/E

= 3

346,4

10

95,02

1 10

6,365

2.746

3158

09

4.

8897

9682

1

0.4

0422

6042

0.

9969

8617

3

2.8

7411

5809

5.

2059

5656

1

0.9

9797

4123

10

4,548

P/E

= 4

454,6

63

126,6

95

106,3

65

4.8

7412

0508

5.

7499

1089

2

0.4

0422

6042

0.

9999

9944

9

5.0

0192

0508

6.

0660

7063

2

0.9

9999

9716

21

2,774

P/E

= 5

562,9

16

158,3

69

106,3

65

6.5

4527

8028

6.

4254

3628

2

0.4

0422

6042

0.99

9999

9999

045

6.6

7307

8028

6.

7415

9602

1

0.9

9999

9999

9796

4

32

1,027

P/E

= 6

671,1

70

190,0

42

106,3

65

7.9

2158

5085

6.

9817

7543

5

0.4

0422

6042

0.99

9999

9999

9855

8.0

4938

5085

7.

2979

3517

5

0.

9999

9999

9999

853

42

9,28

1

P/E

= 7

779,4

22

221,7

16

106,3

65

9.0

9161

7645

7.

4547

3306

6

0.4

0422

6042

0.99

9999

9999

9996

9.2

1941

7645

7.

7708

9280

6

0.

9999

9999

9999

996

53

7,53

3

181


APP

END

IX 9

. Tab

le 9

.1. C

alcu

latio

n of

the

Com

poun

d C

all O

ptio

n Va

lues

for t

he V

entu

re F

und’

s Sha

res a

t 45

perc

ent

and

49 p

erce

nt w

ith D

iffer

ent P

/E V

alue

sIn

put p

aram

eter

s: I 0v =

35,

000,

000

Rubl

es; I

1v = 1

84,1

12,0

00 R

uble

s; I 2v =

144

,717

,000

Rub

les (

for s

hare

45%

); I 2v =

157

,580

,000

Rub

les (

for s

hare

49%

); r =

7%

; 1=

12.

78%

; 2=

10.

224%

; T1=

1 ye

ar; T

2= 9

yea

rs;

1= 1

year

; 2=

8 ye

ars; 3

= 9

year

s.

P/E

va

lues

vV

, th

ousa

nd

Rub

les

v TV1

, th

ousa

nd

Rub

les

vV

, th

ousa

nd

Rub

les

h l

ρ N

2(h, l

, ρ)

12 1

h

22 2

12 1

l

22

11

22

11

22

(, ;

)

Nh

l

Cv ,

tho

usan

d Ru

bles

Vent

ure

fund

’s sh

are

45%

P/E

= 2

26

1,39

2 69

,528

12

0,049

0.

3243

1606

7 3.

7046

5447

9 0.

4042

2604

2 0.

6271

4178

7 0.

4521

1606

7 4.

0208

1421

9 0.

6744

0486

7 20

,287

P/E

= 3

38

0,20

6 10

4,29

2 12

0,049

3.

2561

7846

7 4.

8897

8961

3 0.

4042

2604

2 0.

9994

3492

9 3.

3839

7846

7 5.

2059

4935

2 0.

9996

4269

5 13

1,471

P/E

= 4

49

9,02

1 13

9,05

5 12

0,049

5.

3839

9511

8 5.

7499

0851

5 0.

4042

2604

2 0.

9999

9995

9 5.

5117

9511

8 6.

0660

6825

4 0.

9999

9998

2 25

0,28

1

P/E

= 5

61

7,83

5 17

3,81

9 12

0,049

7.

0551

4732

8 6.

4254

3175

8 0.

4042

2604

2 0.

9999

9999

9933

389

7.18

2947

328

6.74

1591

497

0.99

9999

9999

9182

5 36

9,09

5

P/E

= 6

73

6,64

9 20

8,58

3 12

0,049

8.

4314

3912

9 6.

9817

6474

5 0.

4042

2604

2 0.

9999

9999

9998

541

8.55

9239

129

7.29

7924

484

0.99

9999

9999

9985

3 48

7,90

9

P/E

= 7

85

5,46

4 24

3,34

7 12

0,049

9.

6014

8989

7 7.

4547

2973

5 0.

4042

2604

2 0.

9999

9999

9999

954

9.72

9289

897

7.77

0889

475

0.99

9999

9999

9999

6 60

6,72

4

Vent

ure

fund

’s sh

are

49%

P/E

= 2

28

4,62

6 75

,708

13

4,320

0.

7780

4773

4 3.

7046

5981

5 0.

4042

2604

2 0.

7817

0974

9 0.

9058

4773

4 4.

0208

1955

4 0.

8174

8629

6 32

,877

P/E

= 3

41

4,00

2 11

3,56

2 13

4,320

3.

7099

3167

4.

8898

0365

3 0.

4042

2604

2 0.

9998

9585

6 3.

8377

3167

5.

2059

6339

0.

9999

3781

9 15

8,41

2

P/E

= 4

54

3,37

8 15

1,41

6 13

4,320

5.

8377

4392

5.

7499

2077

7 0.

4042

2604

2 0.

9999

9999

3 5.

9655

4392

6.

0660

8051

6 0.

9999

9999

8 28

7,78

7

P/E

= 5

67

2,75

4 18

9,27

0 13

4,320

7.

5089

0608

8 6.

4254

4804

5 0.

4042

2604

2 0.

9999

9999

9934

227

7.63

6706

088

6.74

1607

784

0.99

9999

9999

9215

6 41

7,16

3

P/E

= 6

80

2,12

9 22

7,12

4 13

4,320

8.

8851

9487

8 6.

9817

7981

5 0.

4042

2604

2 0.

9999

9999

9998

541

9.01

2994

878

7.29

7939

554

0.99

9999

9999

9985

3 54

6,53

8

P/E

= 7

93

1,50

5 26

4,97

8 13

4,320

10

.055

2427

3 7.

4547

4362

5 0.

4042

2604

2 0.

9999

9999

9999

954

10.18

3042

73

7.77

0903

364

0.99

9999

9999

9999

6 67

5,91

4

Baranov and Muzyko

182

183


APPENDIX 10, Table 10.1. Definition of the Parameters

Parameters Definition of the parameters

0≤DIVv(t)<∞ dividends paid by the investee company to the venture fund in year t; positive cash flow of the venture fund

0≤SHKv ≤1 equity share of the venture fund in the charter capital of the investee company

0≤ div(t) ≤1 part of the investee company’s net profit in the previous year (t-1), used in year t for dividend payout

0≤PERv(t)<∞ interest paid to the venture fund by the investee company in year t on extended loan; positive cash flow of the venture fund

0≤LRv(t)<∞ repayment of the loan extended by the venture fund to the investee company in year t; positive cash flow of the venture fund

0≤Lv(t)<∞ the loan extended by the venture fund to the investee company in year t; negative cash flow of the venture fund

0≤Iv(t)<∞ equity investment transferred by the venture fund to the investee company in year t; negative cash flow of the venture fund

0≤TERv(T)<∞ terminal value determined as valuation of the returns the venture fund will get in the last year T of its investment from the sale of its shares in the investee company; positive cash flow of the venture fund

0<r<∞ discount rate acceptable for the venture fund

0≤NPAT(T)<∞ net profit of the investee company in the year T

0≤NPAT(T-1)<∞ net profit of the investee company in the year preceding the “exit” of the venture fund from the business

0<To<∞ initial moment of time, when the venture fund purchases the compound call option (the initial payment which allows starting the project’s implementation)

0<T1<∞ moment of time, when venture fund purchases the shares of the investee company (expiration of the compound (external) call option)

0<T2<∞ moment of time, when venture fund sells its shares of the investee company (expiration of the internal call option)

0<τ1<∞ moment before the fund makes an investment into the company in return for a share of its charter capital

0<τ2<∞ period of time the venture fund stays in the business of the investee company

0<τ<∞ period of expiration of the internal call option

0<P/E<∞ price-earnings ratio for the shares, in venture funds’ financial calculations often P/E=3, 4, 5, 6

0≤Vtν <∞ value of the shares of the investee company, which belongs to the venture fund, at the

moment t

0<Iov <∞ cost of purchasing at the moment T0 the compound call option – is the initial payment

which allows starting the project’s implementation

Baranov and Muzyko

184



0≤I1v <∞ price of the purchase of the shares by the venture fund at the moment T1 – is the

purchase of an internal call option for buying an asset at the moment T2 with an exercise price I2

v (the exercise price of the compound (external) call option). 2

1

* 2 *1 1 2 2 2 1( ) ( ) .............(10.2)rv

ТI V N l I e N l

0≤Iv2 <∞ sum of implicit costs of the venture fund – the loss of its share of the net profit for the

last year of the fund’s participation in the business of the investee company (exercise price of the internal call option)

σ1 ≥0 risk level of operations of the investee company over the time period (0,T1)

σ2 ≥ 0 risk level of operations of the investee company over the time period (T1,T2)

0≤Сventure fund <∞ value of the compound call option at the current moment t, which belongs to the venture fund:

0≤Vν<∞ current value of the equity shares of the investee company, which belong to the venture fund

r≥0 risk-free interest rate

1);,(0 2 lhN two-dimensional standard normal distribution function

);(, lh upper limits of the integrals of the two-dimensional standard normal distribution function;

);(* l

value of l at the moment T1

-1≤ρ≤1 correlation coefficient of stochastic variables;

1)(0 1 hN one-dimensional standard normal distribution function

0V such a value of the investee company’s equity shares at the moment Т1 (1TV ), which

conforms to the equation (10.2) (the threshold figure of the value of the investee company’s equity at the moment Т1)

10 TV

value of the investee company’s equity shares at the moment Т1, which belong to the venture fund

20 TV

value of the investee company’s equity shares at the moment Т2, which belong to the venture fund

( , )v

IRR internal rate of return for the venture fund, in financial calculations only a positive IRRv is of interest for an investor

( , )V

NPV net present value for the venture fund

( , )V

optionNPV net present value for the venture fund taking into account the value of compound call option

2 21 1 1 1 1

1(ln ) / ....(10.3);

2

v

v

Vh r

V

2 2 2 21 1 2 2 1 1 2 2

2

1(ln ( )) / .....(10.4)

2

v

v

Vl r

I

2 2 21 1 1 1 2 2/ ( ) ......(10.6)

1* 2 22 2 2 2 2

2

1(ln ) / ......(10.5)

2T

v

Vl r

I

12 2 22 1 1 1 1 2 2 2 2 1 1( , ; ) ( , ; ) ( )....(10.1)rventure fund rC V N h l I e N h l I e N h

185




( , )v

optionIRR internal rate of return for the venture fund taking into account the value of compound call option, in financial calculations only a positive IRRv is of interest to an investor

ExittotalNPAT0 total net profit of the investee company in the year of “exit” of the venture fund from the invested business

0≤S≤1 share of the venture fund in the charter capital of the investee company

I1discv≥0 discounted value of I1

v

I2discv≥0 discounted value of I2

v

0≤f (x,y)<+∞ probability density

-∞<(x,y)<+∞ values of random variables

a, b = 0 (for standard normal distribution)

mathematical expectations of stochastic variables x, y

σx≥0 mean square deviation of stochastic variable x

σy≥0 mean square deviation of stochastic variable y

-1≤ρx,y≤1 correlation coefficient of stochastic variables x and y

Documents

Using Real Options for the Evaluation of Venture Projects