مسابقه ثبت اختراع در تنظیم گزینه های واقعی : استراتژی سرمایه گذاری، ارزیابی، CAPM بتا، و نوسانات بازگشت

In this paper, we study financial properties of R&D intensive firms through a continuous-time real-options patent-race model. Numerical analysis in this study shows that intense competition drives a firm to invest more aggressively, which then pushes up its cost of capital and return volatility while introducing negative return correlation with its competitor. Furthermore, we find that a firm's position in competition has important impacts on its financial properties. For instance, a firm's cost of capital is a non-monotonic function of its relative position in the race. In addition, the relationship between cash flow uncertainty and investment can be negative when a firm is far ahead or far behind, or positive when firms are close in the race.

This paper uses a continuous-time real-options methodology to develop a duopoly patent-race model, which is applicable for examining financial properties of R&D intensive firms such as firms in the business of creating new software, inventing NANO or WI-FI technology, or innovating new drugs. Many of these firms are early stage, private, venture capital backed startups, with limited available financial data.1 Therefore, investors, financial analysts, and other market participants often have difficulty analyzing their financial properties. For instance, how do these firms make investment decisions? How are they valued? How are their cost of capital and the pattern of their return volatility and return correlation with competitors determined? This paper aims to answer these questions and develop a model to serve as guidance for future empirical work. R&D intensive firms face three primary types of uncertainty during the innovation process, namely, cash flow, technological feasibility, and competitive uncertainties. As a firm typically does not receive any cash inflows until the creation/innovation of a new product/technology is completed (like a biotechnological project), there is cash flow uncertainty. If a firm is uncertain about its technological ability or feasibility of its project (like a new software product), then it faces technological uncertainty. R&D intensive firms typically operate in highly competitive environments. A patent, like any intellectual property protection, is granted to the first inventor. So, firms face competitive uncertainty resulting from their changing relative position in a patent race. As a result, each firm has to strategically interact with its competitors when making investment decisions. We build these three uncertainties into our duopoly model in which two firms compete to invent a new technology. Each of the two firms chooses an investment rate at which it develops the new technology. The winner of the race is awarded a patent, from which point it receives a sequence of cash flows, which we value by using the Capital Asset Pricing Model (CAPM). The loser of the race receives nothing. A duopoly competition for a patent is modeled as a stochastic game in which there are three publicly observable state variables: the value of the patent and the expected cost-to-completion of each of the two firms. Each state variable is governed by its own source of risk. The value of the patent is the present value of the cash flows received upon winning, which follow a geometric Brownian motion subject to both systematic and idiosyncratic risks. The overall uncertainty underlying the patent value is referred to as “cash flow uncertainty,” which gives the firms real options to delay their investment and wait for more information about the profitability of the patent. The position of each firm in the race is described by a state variable called the expected “cost-to-completion,” which measures the expected amount of money a firm needs to succeed in developing the new technology and winning the patent race (the “follow-on” capital). Each firm's expected cost-to-completion follows a diffusion process governed by its own idiosyncratic technological risk, which is a special case of the technology described by Pindyck (1993).2 Given a firm's individual technological uncertainty, it has an incentive to learn more about the difficulty of the project and to invest in order to resolve this uncertainty (“learning by doing”). A firm's investments may lead to technological improvements as well as setbacks. Faced with competitive uncertainty, a firm receives either preemptive threats to invest or incentives to withhold investments depending on the relative position of this firm to its competitor. In an environment with multiple sources of uncertainty, when making strategic investment decisions, a firm has to strike a balance among the real options to wait, the incentives to invest or withhold investments, and the preemptive threats to invest. We employ the full information Markov Perfect equilibrium in our model. Each firm chooses an investment rate constrained by its money “burn rate” to maximize its firm value given the other firm's investment rate. Because there is no closed-form solution to our continuous-time model, we solve it by using numerical approximations. Using a lattice method, we develop a discrete-time implementation on a 201×61×61 grid with 600 monthly decisions, upon which we examine the investment strategy, valuation, CAPM beta, return volatility, and return correlation of the two firms in the race. We report these results for various relative positions of the firms throughout the race. Through numerical analyses, we find that competition drives the firms to over-invest relative to a case of joint monopoly, which then pushes up the firms’ CAPM beta and return volatility, and introduces negative return correlation between the two competitors. With regard to a firm's CAPM beta, and thus its cost of capital, our numerical analyses depict that it depends on the firm's relative position to its competitor in a non-monotonic way. A firm's beta is closer to the beta of the underlying patent when it gets closer to winning the race because its investment option becomes deep-in-the-money and the leverage induced by investment expenditures declines. The firm's beta becomes higher when its position is closer to its competitor, because intense competition drives the two firms more eager to invest which increases their leverage. The firm's beta becomes low again when its investment option becomes deep-out-of-the-money as it lags far behind and stops investing. The numerical analyses also show that annual return volatility of R&D intensive firms can be in excess of 100%, which is quite high compared to the range of 20–40% per annum for a typical firm.3 Further decomposition in our numerical example reveals that this high level of return volatility is largely attributable to the firm's idiosyncratic technological risks. These results are consistent with recent empirical findings on venture capital-backed startups by Cochrane (2005). Another interesting result is that competition introduces negative return correlation. When both firms are not investing, their returns are perfectly positively correlated by the common factor of patent value. When one of the firms or both are investing aggressively as competition intensifies, a progress of one firm means a setback for the competitor. As a result, this firm enjoys positive returns whereas the competitor suffers negative returns. The return correlation between the two firms thus turns negative. In addition, we conduct a comparative-static analysis of such parameters as cash flow uncertainty, technological uncertainty, and dividend yield to study how the leader and the follower react differently to the change in these key underlying parameters. One interesting finding is that the relationship between cash flow uncertainty and investment can be either negative or positive; it is negative when a firm is far ahead or far behind and positive when firms are close in the race. This finding contributes to the latest literature on strategic real options by showing the impact of a firm's position in competition on the relationship between uncertainty and investment.4 In all, our sensitivity analysis shows that a firm's relative position to its competitor is an important control variable for studying the investment strategies of R&D firms, which has been ignored in previous empirical studies. We provide a discussion of empirical proxy for this control variable for future studies. This paper unifies two lines of research, the literature of patent race in economics and that of investment under uncertainty using standard real-options techniques, and contributes to the latest literature on strategic real options by showing how a firm's position in competition affects its financial properties. Along the literature of patent race, technological competition has been investigated first through stationary games under uncertainty by Loury (1979), Dasgupta and Stiglitz (1980), and Lee and Wilde (1980), and then through dynamic games under uncertainty but without explicit strategic interactions pioneered by Reinganum, 1981 and Reinganum, 1982, or through dynamic games with strategic interactions but without uncertainty pioneered by Fudenberg et al. (1983) and Harris and Vickers (1985). Finally, strategic interactions and technological uncertainties are combined within a dynamic structure, as in Judd (2003), Grossman and Shapiro (1987), and Harris and Vickers (1987). Models in this line of research all lack the cash flow uncertainty since the patent value is set as a constant, in which case, a real-options methodology has not been applied. The value of the option to delay investments embedded in cash flow uncertainty has been ignored. As a result, the investment strategies derived in those papers lack the “delay” feature. On the other hand, the traditional real-options literature, represented by McDonald and Siegel (1986), Pindyck (1988), and Dixit (1989), highlights cash flow uncertainty.5 Therefore, the real-options premium of delaying investments, which has been ignored in the literature on the patent race, is the main finding of this category of research. However, those papers have typically ignored strategic interactions between firms and regard firms as either monopolists or entities in a perfectly competitive market.6 Our continuous-time model combines real options with strategic interactions, which contributes to a recently growing body of literature on strategic real options represented by Kulatilaka and Perotti (1998), Grenadier (2002), and Weeds (2002).7 Most existing models assume risk-neutrality precluding the analysis of CAPM beta and return volatility of firms evolved in competition. In addition, technological uncertainty has also often been ignored. Childs and Triantis (1999) examine dynamic R&D investment policy and valuation for a firm with multiple R&D projects, which can be run in parallel or in sequential to each other. They provide a thorough analysis of the interactions across different projects. In our model, both firms are assumed to have a single project. Novy-Marx (2007) investigates optimal investment decisions of heterogeneous firms in a competitive, uncertain environment and shows that the strategic equilibrium real-options premia are significant. Grenadier (1999), Lambrecht (2000), and Lambrecht and Perraudin (2003) study option exercise games under incomplete information. In our model, we assume that the two firms have complete information for simplicity. Our continuous-time model is also related to parallel works by Miltersen and Schwartz (2004) and Garlappi (2004). Miltersen and Schwartz (2004) studies R&D spending with competitive effects and multiple sources of uncertainty from a welfare perspective. Firms compete in both innovation and production stages, while the innovation stage is emphasized in our paper. Miltersen and Schwartz illustrate an example in which firms tie at the start of competition. In comparison, we provide results for various relative positions of the firms throughout the race, which allows us to analyze how relative positions of firms in competition affects their financial properties. Garlappi (2004) studies the dynamics of firms’ risk premia in a duopoly patent race. In his model, firms jump over a series of hurdles before winning the patent race. He provides analytical solutions to a two-stage game and numerical analysis for a five-stage game. Our numerical approximation is a 61-stage game. This more general scenario allows us to analyze firms’ costs of capital when a firm is well ahead or behind. We find a firm's CAPM beta is a non-monotonic function of its position relative to its competitor in contrast to monotonically increasing function found in Garlappi's model. We also introduce technological setbacks, which is omitted in his model. This enables us to study what happens to the cost of capital when a firm realizes more time and more money than previously expected have to be spent to complete the project. We find that a firm's setbacks increase its own cost of capital and decrease its competitor's cost of capital when competition is fierce. The paper is organized as follows. Section 2 describes the model set-up. Section 3 describes the numerical implementation. Section 4 discusses the equilibrium investment strategy, valuation, CAPM beta, return volatility, and return correlation of firms involved in a patent race. Section 5 provides a comparative-static analysis on parameters of cash flow uncertainty, dividend yield of the patent value, and technological uncertainty. Section 6 concludes with a discussion of further research.

In this paper, we develop a continuous-time real-options patent-race model to analyze financial properties of R&D intensive firms. The numerical analysis provided in this study shows that competition causes over-investment and value dissipation relative to the case of joint monopoly, which then pushes up CAPM beta and return volatility, and introduces negative return correlation between the competing firms. These results imply that venture capitalists including competing firms in the portfolio cannot only improve efficiency by mitigating over-investment and value dissipation, but can also reduce portfolio risk via hedging as the returns of competing firms could appear negative correlation. In addition, we find that a firm's position in competition is an important factor in the determination of its financial properties. In particular, the CAPM beta, and thus the cost of capital of an R&D firm, is a non-monotonic function of its relative position to its competitor. The cost of capital is higher when a firm intensively competes with its competitor than when well ahead or well behind. We also find that, as another example, the relationship between cash flow uncertainty and investment can be negative when a firm is far ahead or far behind, or positive when firms are close in the race. So, the relative position of competing firms in a race should be taken as an important control variable in future empirical studies of financial properties of R&D intensive firms. To extend the model, it could be interesting to add explicit learning process about profit flows of the patent and investment opportunities for a firm and its competitor. It may be also worthwhile to study a firm's investment behavior and other financial properties in a competitive environment with the presence of technological spillovers. While competition motivates a firm to invest, technological spillovers may motivate a firm to withhold investments so as to obtain a free ride on investment efforts of its rival. We leave these extensions to future research.