The relation between implied and realised probability density

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The relation between implied and realised probability density

    The Evaluation of Venture Capital

    As an Instalment Option:

    Valuing Real Options Using Real Options

    By Mark Davis, Walter Schachermayer

    # and Robert Tompkins

    20 October 2003

    This piece of research was partially supported by the Austrian Science Foundation (FWF) under grant SFB#10 (Adaptive Information Systems and Modelling in Economics and Management Science).

     # Corresponding Author, Hochschule für Bankwirtschaft, Sonnemannstraße 9-11, D-60314, Frankfurt am Main, Germany , Telephone: 49-69-154008-718, Facsimile: 49-69-154008-728, email: .

    The Evaluation of Venture Capital

    As an Instalment Option:

    Valuing Real Options Using Real Options


    Many start-up companies rely upon venture capitalists to begin operations. Typically, after the initial injection of funds, additional funding is provided as the firm reaches certain performance targets. The payment of the first funding round is comparable to an initial option premium. Further payments are contingent claims: the right but not the obligation to continue financially supporting the project. If at any point, the venture capitalist ceases to pay, the project is assumed to end. Therefore, the venture capitalist can be thought of injecting funds that not only keep the project alive but also retain the right to pay the remaining payments in the future.

    We interpret this type of corporate finance transaction as a multiple stage compound option, which is also known as an instalment option. In previous work by Davis, Schachermayer and Tompkins (2001, 2002), instalment options on traded assets were considered. In these papers, it is shown that static portfolios of simple European options can be formed that yield no arbitrage bounds on the value of instalment options.

In this research, we extend the analysis to venture capital contracts. For imitator (clone)

    projects, we derive bounds on how much a venture capitalist should initially invest in such start-up firms. Under suitable assumptions, such bounds rely solely upon no arbitrage arguments. Upper and lower bounds can be enforced by constructing portfolios of European options on firms in the same industry. To the best of our knowledge, the relationship between the financing of imitator start-up companies and instalment options has not been identified previously.

JEL classifications: G24, G32, C52

    Keywords: Venture Capital, Capital and ownership structure, instalment Options, Replicating Portfolios, No-arbitrage Bounds, Real Options



    Venture Capital (VC) is an important source of funding for the development of innovative start-up companies. Yet, the finance literature provides little guidance as to how much the venture capitalist should pay to fund new projects. Cossin, Leleux & Saliasi (2002) state: “Valuing early-stage high-technology growth-oriented companies is a challenge to current valuation methodologies.” The traditional approach to project analysis requires the forecasting of future cashflows and discounting these to present value at some risk adjusted rate. However, the structure of VC funding could be more appropriately analysed in terms of Real Options, as subsequent injections of funds are contingent on the project reaching performance targets.

    Most of the literature on VC has focused on the contractual design of the investment contracts, with particular emphasis on skewing the distribution of the payoffs in favour of the VC investors. Research has examined the role of contracting design in such transactions and principal agent conflicts. Another line of research has concentrated

    1 Recently some work has appeared on the empirical analysis of existing VC projects.

    identifying the possible application of real options methodology to the evaluation of VC transactions. So far, the pricing of VC real options is at a nascent stage and this research aims to address this issue.

    The VC investor evaluates the project and, assuming favourable prospects, provides initial funding and further funding as time goes on, depending upon the firm reaching certain performance targets. If at any time (when such decisions for further funding are made), the firm has not met the targets, the VC investor can abandon the project and potentially receive some portion of the recovery value of the firm. For those projects (e.g. Internet firms) without fixed assets, there is no recovery value. We will consider this case here, without loss of generality.

    We contend that this sort of VC funding is an instalment option. Such instalment options are multiple-period compound options. The initial introduction of project funding is comparable to the premium payment of an instalment option. Subsequent injections of funds into the project resemble further instalment premium payments. In previous


    research by Davis, Schachermayer & Tompkins [DST] (2001, 2002) tight bounds on the value of these options were deduced from the prices of European options (for traded assets) using the no arbitrage principle. Not only does this provide a tractable method for the pricing of instalment options, but also the bounds are robust to model assumptions such as the assumed price process (typically Geometric Brownian Motion with constant volatility). Furthermore, there is no need to estimate model parameters such as volatility or dividend yields if there is a liquid market for European options (with sufficiently long exercise times and comparable exercise prices). These results imply that such options can be super- and sub-replicated by standard European claims (which really exist and thus the

    sub-title to this paper). Therefore, the amount of initial funding provided by VC investors can be precisely determined and is preference-free.

    In this research, we consider venture capital projects that are in a strong sense (made precise in Definition 1 below) imitators (or clones) of existing firms in some

    industry. Under the assumption that European options are traded for the existing firms (at appropriate strike prices and maturities), the Venture Capitalist can use this information to decide whether or not to fund a new project. This is achieved by comparison of the expected cash flows and future investment payments of the venture project to that of an instalment option on an existing firm in the same industry. We show that the VC investor can realise an arbitrage profit by funding the VC project and at the same time selling a portfolio of European options. This will occur if the required initial payment for the venture project is lower than the cost of the European option strategy. This provides a lower boundary on the amount that the VC investor would be willing to pay initially. Unlike the usual inclusion of investor risk preferences in the pricing of Real Options, this lower bound for the initial investment is preference-free.

    The upper boundary is not determined by arbitrage, in a strict sense. However, if the VC investor is a profit-maximising rational agent, the upper boundary for initial project funding must be the upper boundary of the instalment option on the related firm. As this boundary is also enforced by a static portfolio of European options (assumed to be observable), this will be preference-free and model independent. In DST (2001), such


    bounds are typically within 5% of the theoretical value of the instalment option. Therefore, this research provides substantial guidance for the pricing of imitator VC projects.

    This paper is organised as follows. In the first section, we will review the venture capital literature and concentrate on the evaluation of venture capital as a real option. This is followed by a discussion of instalment options pricing and the DST (2001) bounds with static portfolios of European options. Then, we show that the bounds for an instalment option on an existing firm are also the bounds for an imitator VC project when both the existing firm and the venture project share the same sources of external risk. Finally, conclusions and suggestions for further research appear.


    A number of papers have studied what venture capital firms do and theorise how they add value. Examples of these include Gorman & Sahlman (1989), Hellman and Puri (2000) and Lerner (1995). These papers examine what the VC investors tend to do after the initial investment in the firm. For example Kaplan & Strömberg (2001, 2002) and Gompers (1995) focus on the implications of contractual terms of VC arrangements. Their objective is to test various theories of the investor / principal agent conflict. Another line of research has been to use evidence from surveys of VC investment partnerships to describe the characteristics of these investments [see MacMillan, et al. (1985, 1987) and Fried and Hisrich (1994)].

     Faced with valuation uncertainty, Sahlman (1990) suggests that the coping mechanism is to either design investment contracts which materially skew the distribution of the payoffs from the project to the VC investors or involve the active participation of the VC investor to assure that the project has the professional managerial expertise to succeed.

    Sahlman (1990) identifies three key facets of the investment contract that skew payoffs in favour or the VC investor; (1) the staging of the commitment of capital, (2) the


    use of convertible securities instead of straight common shares and the associated senior claims on the assets of the firm in case of failure and (3) anti-dilution provisions to secure the VC investor’s equity position in the new firm. Of these mechanisms, he concludes that staged capital infusions are the most potent control mechanism that a venture capitalist can employ. Cossin, Leleux & Saliasi (2002) examine the economic value of these legal features in a Real Option context.

    The usual sequence of events in VC funding is that an entrepreneur either has previously developed a project (with prior revenues) or plans to start a new venture (without prior revenues). The entrepreneur approaches a VC partnership and seeks funding for this project. After submission of an appropriately detailed business plan and analysis by the VC investors, a funding proposal is made. Typically, a total amount of funding is approved for the project (committed funds) and payment is made in stages (or funding rounds). The first stage allows the project to begin and then at fixed points of time in the future, if certain performance targets are reached, the VC investors introduce additional funding. Ultimately, when the firm has reached sufficient size and has established a track record, the company is sold to the capital markets as an Initial Public Offering (IPO). The VC investors may not introduce additional funds to the project because certain performance targets have not been met. Also, as the project develops, competitive firms may enter the market place, copy the idea and the exclusivity value of the project is reduced. We will only consider the former case in this research.

    For their investment, the VC investors obtain an equity share in the project and expect to profit when the IPO is launched. According to a survey of recent VC projects in the United States, Kaplan & Strömberg (2001) [Table 1], show that almost all VC investors receive convertible preferred stock in the firm when they pay in the funds. Optional redemption and put provisions are commonly used to strengthen the liquidation rights of the VC's investments. When discussing the expected profit from VC investment, Kaplan & Strömberg (2001) find that the median IPO stock price is 3.0 times greater than the cash infusion (the estimated value of the company) in the initial financing round


    (payment of the instalment options). Over a four-year horizon, this works out to a return of 31% per year (page 12).

    VC projects also vary depending on how the level and timing of additional funding are initially defined. Kaplan & Strömberg (2001) state, “Even though redemption rights

    are the part of the VC contracts that most resemble debt, there are other ways that a VC investor can force a liquidation of badly performing firms. The most important mechanism is through staging of the investment [see Bolton & Scharfstein (1990), Neher (1999) provides a model of staging based on Hart & Moore (1998)]. We distinguish between two different forms of staging: ex ante (or within-round) and ex post (or between round) staging.”

    Kaplan & Strömberg (2001) further state, “In an ex-ante staged deal, part of the VC

    projects committed funding is contingent on financial or non-financial milestones (internal targets). This essentially gives the VC investor the right to liquidate the venture in the bad state of the world. Even though not all VC financings are explicitly staged ex ante, most of them are implicitly staged ex post, in the sense that even when all the funding in the initial round is released immediately, future financing will be needed to support the firm until the IPO.” As Cossin, Leleux & Saliasi (2002) show, “ex-ante”

    funding or as they call it “Contingent Pre-Contracting” further funding is theoretically a

    better approach than the simple “right of refusal”, which is an informal commitment of

    additional funds, as and when they are needed. Given this, we will restrict our analysis to those cases when future funding levels are explicitly set ex-ante.

    “Of particular concern to VC investors is the liquidation cash flow rights that are

    2 By providing less funding in a given round, and assigned upon the failure of the venture.

    hence shortening the time until the next financing round, the VC arrangement increases the ability to liquidate the venture if performance is unsatisfactory. Gompers (1995) analyses ex post staging using Venture Economics data. Time between financing rounds decreases with industry R&D intensity and market-to-book ration; it increases with industry tangible asset ratios”. [Kaplan & Strömberg (2001) pages 28-29 and footnote



    VC projects also vary in terms of the nature of the proposed firm. According to Hellman & Puri (2000), these firms can be split into innovator or imitator firms. The difference is that innovator firms are launching a new product or service that has not been offered previously. For imitator firms, such a product or service has been introduced previously and the new firm contends that it can provide this more cost effectively or efficiently. In this instance, we have a frame of reference for comparison: existing firms in this industry. For our purposes, innovator and imitator firms are distinguished by how performance targets are set by Venture Capitalists (for additional introduction of funding). Innovator firms have targets set by internal performance, such as the development of patents or the successful completion of research projects. Imitator firms have targets set by external performance, which include sales targets, cashflows or attainment of predetermined market share levels. In the latter case, the success of these firms is assumed to be driven by “external” sources of risk that are general to the industry they belong to. This implies that both existing firms in the industry and the new imitator firm share the same sources of risk.

    In this research, we will restrict ourselves to case of imitator firms with ex ante funding. Our rationale for this choice is twofold. Firstly, Hellman & Puri (2000) contend that: “ For imitators the provision of funds may be the most important aspect of venture capital., whereas for innovators, the product market dimension can be more important.” (page 963). As we are interested in the value of the real option (the precise amount of initial funds) and not the impacts of expertise by the VC investor, we will restrict our analysis to these. Secondly, under our rather restrictive assumptions that both imitator firms and existing firms share exactly the same sources of risk, we can derive precise no

    arbitrage bounds.


    Such ex ante funding (like our instalment options), is according to Kaplan & Strömberg (2002) “[such] staging, on the other hand, does not seem to be related to internal risk, but instead to the amount of risk external to the firm. This suggests that the driving force for


ex ante staging is not asymmetric information, but rather the option to abandon the

    project, which will be more valuable in volatile environments.” (page 24). Berger, Ofek and Swary (1996) have considered such an abandonment option in such a context.

    As our research will only examine the initial amount that should be paid for VC projects, we contend that we are justified in restricting our analysis to imitator firms. Hellman and Puri (2000) show that in real VC practice, funding is of prime importance to imitator firms. Furthermore, in their sample of Silicon Valley start-up firms, “We find

    that the presence of venture capital increases the amount of funds raised by imitators, by not by innovators.” (page 979).

    Kulatiliaka and Perotti (1998) point out that for both innovator and imitator firms, the value of the option must take into account competing firms. In financial options markets, actions taken by one investor will not affect other investors. For example, the holder of an option on a financial asset has the exclusive right to exercise that option, and exercise by one agent does not effect the exercise decision by other firms. The agent has monopoly over the exercise opportunity. From Bruun & Bason (2001) “Not always so in real options analysis. When a firm is undertaking, in example, an R&D investment, it is in effect purchasing a option on possible commercialisation or further development. But a competitive firm can make similar investments and thus exercise by one firm will affect the market value of the option for other firms - possibly drive it to zero”. In this research,

    we will not assume such a feedback effect. We assume that the decision by the VC investor to exercise does not affect competitive projects that may be introduced during the tenure of the project.

    Bruun & Bason (2001) have applied real options to VC investments. For a broad based review of the literature on real options see Dixit & Pindyck (1994), Trigeorgis (1996) and Lander and Pinches (1998). Growth options and VC are seen as the same thing.

    The obvious starting point for the pricing of the VC real option is the Black & Scholes (1973) model. Benaroch & Kauffman (1999) and Panayi & Trigeorgis (1998)

    3. However, as is pointed out by Bruun & have used the BS (1973) model in this context


    Bason (2001), “There is a growing body of evidence, however, that the assumptions underlying the standard Black Scholes option pricing model are either too simplistic, or downright false when it comes to pricing options on many real assets.” (Page 3). They go on to suggest that modifications of the Black & Scholes (1973) model as proposed by Merton (1973, 1976) and by Cox and Ross (1976) may be more appropriate.

    The Merton (1976) jump diffusion model is of particular interest, due to the discrete nature of VC project evaluation and the fact that such projects experience jump-like behaviour (as technical breakthroughs are achieved in the project). Pennings and Lint (1997) have argued that a more appropriate approach to modelling VC & R&D projects would be a jump process, and this would lead to an intuitive economic interpretation of the volatility. Changes in price are likely to be caused by technical discoveries and the arrival of information affecting the particular project (e.g. competitor entry), and these occurrences often happen at discrete intervals.

    They formulate such a model with the value of the firm driven by stochastic jumps, plus a deterministic drift term. They successfully apply this to the analysis of R&D projects at Philips [see a discussion in Lint & Pennings (1998)]. A similar model has been presented by Willner (1995) where the impacts of jumps are decreasing with time (the continued existence of the firm).

    A particular challenge for real option modelling is that the underlying asset must be defined. For the articles previously referenced, the underlying asset is assumed to be the overall value of the firm. This is assumed to follow Geometric Brownian Motion (GBM). However, Angelis (2000) extends the Black & Scholes (1973) model by substitution of predictions of revenues and costs rather than the value of the project. She also assumes that such revenues and costs conform to Arithmetic Brownian Motion (ARM) rather than assuming that the process follows GBM. Thus, this approach harkens back to Bachelier's (1900) original option pricing model.

    A number of authors have proposed models with multiple state variables. For financial options, the striking price is fixed at the beginning of the contract. In VC & R&D projects, this is not always the case. A fixed strike price in this instance suggests


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