Patent Valuation under Spatial Point Processes with Delayed and Decreasing Jump Intensity

1 Introduction

This paper provides a contribution to the literature on patents valuation. Assigning economic value to patents is relevant for patentees and at social level. Probably due to this twofold relevance, the economic valuation of patents has attracted much attention and efforts among researchers, insomuch that extensive documentation has been provided since the second half of the 1980s. In this paper, we move from the common assumption that a patent is an option over a technology so that modeling can proceed as a derivative contract where the patent value is linked to the properties of the underlying asset. Indeed, a patent gives the holder the exclusive right (option) on an invention. The right can be renewed to the expiration date T or be dropped before the statutory limit, if the holder decides not to pay the renewal fee. The real options approach is based on the definition of an underlying asset the evolution of which drives the patent value. In agreement with industrial economics theory, the underlying asset is given by the cash flow associated to the patent. Specifically, the fluctuations in the underlying asset represent the quantitative translation of the qualitative characteristics of the economic environment. Under this assumption, the underlying is able to capture the uncertainty affecting supply and demand of goods, the production of which is entitled through patent ownership, research programs from other firms, the attainment of patents, the changes in the regulatory environment, etc. These factors are not associated to a continuous time horizon, but rather they occur at discrete points in time, causing the underlying state to undergo a jump (for instance, it is quite unrealistic to assume that changes in the laws protecting patents occur continuously). This fact implies the presence of jumps in the dynamics of the patent value.

In spite of the relevance that jumps can take place in the real world, to the best of our knowledge, very little has been said in the literature, because most of the models are formalized in continuous time.

Within the real options approach, we will present a very general model meant to provide a unifying theoretical framework where the existing models of patent valuation can be considered as special cases. The generality of the model also allows for unconsidered theoretical hypotheses, taking a step forward toward a more realistic model. As we will see, this can be achieved by incurring the cost of adopting an unusual process: the Spatial Mixed Poisson Process, SMPP.

Is this cost worth incurring? The answer must be given both in terms of theoretical advances the model can generate and in terms of economic implications.

As to the theoretical aspects, the approach can produce several new results with respect to the existing literature. First, it allows tackling the problem of the value of patents in the presence of both negative and positive jumps in the underlying.^[1] Second, the renewal process is not contemplated in that part of the literature which models (only negative) jumps through a Poisson process.^[2] At the same time, the renewal process has been extensively modeled in the literature, but dropouts have been modeled under the “no jumps” assumption. Our general model puts together the two strands of literature, modeling the jumping value of a patent in the presence of a renewal schedule. Specifically, the results generated by the model can be easily compared to those accruing from the studies related to the patent valuation with renewal schedule (from Pakes (1986) onward). In this respect, an explicit connection between real options and the patent renewal schedule literature is proposed. Third, we introduce the realistic feature that not any jump can kill the patent, but only a large enough jump can do it. This feature is consistent with the good news principle, as first spelled out by Bernanke (1983) and more recently taken up by Boyarchenko (2004) and the literature therein. The principle can be stated as follows. Suppose that a negative jump is followed by a positive one. If the firm disinvested, after the unlucky jump, but then the cash flow increased, the firm cannot invest again because of investment irreversibility. Hence, it was better not to disinvest. In our specific case, disinvestment is represented by not paying the renewal fee, once the patentee misses one fee the patent cannot be resumed. Hence, the assumption that not any jump kills the patent is consistent with the good news principle. By the same token, the same argument applies for the bad news principle in case of investment decisions.^[3]

Fourth, there is a random delay in the transmission of the jump from the underlying state to the patent value. Fifth, as the patent gets older, namely approaching the final date T, its value becomes less sensitive to the jumps in the underlying asset. That is because the propagation can take place progressively rather than abruptly and also because some jumps in the underlying asset do not have enough time to propagate to the patent value. Therefore, we can realistically affirm that the sensitivity of the patent value decreases with respect to the time of the jumps. Sixth, the adoption of the SMPP avoids the unpleasant hypothesis of no expiration date (Schwartz and Moon 2000;Bloom and Van Reenen 2002). Finally, the adoption of SMPP allows to take into account the dependence between time and size of jumps in the cash flow. We will turn to all these points in more details when introducing them formally.

These theoretical advances spawn relevant economic and policy implications. The main message from the model is that, in principle, the valuation can drastically differ and policy recommendations can be mistakenly inferred if one neglects the possibility of positive jumps, and the delay and decay in jumps propagation. Neither a sensible comparison of patent rights can be made between countries and sectors. Hence, if the value of patent rights is inadequately estimated, biased incentives will be fostered.

It is important to highlight that the cost to incur in terms of algebra to enjoy the benefits is not sunk, in the sense that the Mixed Poisson Processes on the line and the infinite-server queue models are widely known and used (see e.g. Grandell 1997); and also a spatial setting can be useful in modeling several practical situations (e.g. spatial queues, see Cinlar 1995).

From a purely technical point of view, the dynamics of the underlying state is assumed to evolve in discrete time and both sizes and times of the jumps are assumed to be stochastic variables. This set up allows to obtain an approximation formula of patent value by providing a computation of the total size of its jumps in a given period. Since patents are supposed to jump with a certain delay with respect to the occurrence of the event, the jumps that will occur “tomorrow” are partially due to the jumps in the economy that occur “today”. Hence, to value a patent tomorrow, one must have the knowledge of the jumps in the economy already occurred, the effect of which will be exerted tomorrow. Moreover, the information set to be used for the estimation contains also a sample of how external events impact on the underlying asset.

The remaining part of the paper is organized as follows. Section 2 briefly reviews the real options literature on patents valuation. Section 3 presents the set up of the model describing the dynamics of the underlying asset, constrained by some economic and statutory terms which patents are subject to. Section 4 presents the valuation mechanism for valuing a patent at a given point in time. Finally, Section 5 summarizes the results and provides some conclusive remarks. The main definitions and the key results on SMPPs are contained in the Appendix.

2 Review of the literature

The literature on patent valuation is very broad. In this short review, we will restrict our attention to the contributions which tackle the problem with a real options perspective. Special attention will be paid to those contributions in which the underlying asset is supposed to undergo possible jumps.

As patents have the three characteristics of (i) partial irreversibility of the investment undertaken, (ii) uncertainty of future payoffs, (iii) possibility to delay the actions, a prominent part of the literature has regarded the problem of patent valuation as one of the (real) options valuation. In this respect, some important papers are worth mentioning. Pakes (1986) deals with the analysis of future cash flow generated by a patent in presence of a renewal fee schedule and solves the optimal stopping problem of the abandonment of a patented project. The main parameters of Pakes’ model are estimable from aggregate data on proportions renewed and the author applies the model to some Western countries. The estimation of patents value in a renewal fee model is also faced by Pakes and Simpson (1989) who, differently from Pakes, adopt a parametric estimation strategy. Bloom and Van Reenen (2002) still formulate an estimable real options model, but taking advantage of citations data, rather than renewal, and estimate the impact of patents on firm-level productivity and market value. The analysis they propose is eminently empirical, as that of Laxam and Aggarwal (2003), where the value of a specific patent related to a telecommunication technology is assigned by means of the real options approach. Baudry and Dumont (2006) is a prominent Pakes (1986)’s follower. The authors regard at renewal schedules as a tool to evaluate patents in a more general context than that presented by Pakes and important policy implications are derived. Specifically, renewal fees can be used to discourage low-value patents, reducing the burden of patent offices. Indeed, the authors introduce a binomial tree framework, hence a discrete time framework in which the underlying asset takes on jumping values, to make their results comparable to a wide number of studies related to financial options.

In spite of the relevance that jumps can take place in the real world, to the best of our knowledge, very little has been said in the literature. Some exceptions are worth mentioning: Pennings and Lint (1997) is the first contribution dealing with patent evaluation in a real options framework in presence of jumps of stochastic amplitude obeying a Poisson law. The option value of the proposed model depends on the expected number and size of the jumps. The model is also verified with an empirical application to a company operating in the electronic field. We refer the reader to this paper for a review of earlier contributions on real options and jumps in other areas, different from patents. Schwartz (2004) models changes in regulatory environment as a possible jump in the Poisson process that the net cash flow generated by the project can undergo. In the spirit of McDonald and Siegel (1986), he considers only negative jumps affecting the value of the patent in two ways: First, reducing the expected rate of capital gain in the underlying asset, which reduces the value of the option. Second, increasing the variance of percentage changes in the underlying asset over finite time intervals, and this tends to increase the value of the option. The net effect is a reduction in the value of the option (patent). Pennings and Sereno (2011) are close to Schwartz (2004) for the application to pharmaceutical companies and assume that the underlying asset is driven by a mixed jump-diffusion law. The authors build a simple model, where the intervention of jumps is limited to the representation of the binary event of occurrence (or not) of technological failure. Also in this case, the arrival times are of Poisson type.

In a context of strategic interactions, the real options approach has been extensively used (Hsu and Lambrecht (2007) and Meng (2008) and the literature therein), and jumps in a Poisson process are used to capture other firms patent attainment (Miltersen and Schwartz 2004;Lambrecht and Perraudin 2003;Weeds 2003;Schwartz and Moon 2000). According to a different approach, Baudry and Dumont (2006) give a definition of a very broad class of discrete time stochastic processes to describe the evolution of the rent generated by a patent. In such a way they model regulatory interventions, such as a change in the schedule of renewal fees, highlighting how the profile of renewal fees can reduce applications from worthless patents.

In keeping with the idea of Baudry and Dumont to use a new stochastic process, in the next section we will make use of the SMPP. To the best of our knowledge, few studies are present in the literature implementing SMPP processes in a real options context. The first contribution is due to Cerqueti, Foschi, and Spizzichino (2009) who develop a real options-like model with SMPP applied to insurance–reinsurance issues. Cerqueti (2013) proposes a similar setting to the problem of available stock of natural resources, whereas the contribution of Cerqueti and Ventura (2013) is closer to our paper, as they deal with a (would-be) patent-related problem, still in the real options language with SMPP. Specifically, Cerqueti and Ventura (2013) model the R&D phase before patent attainment and focus on competition and strategic interactions in a race context. In their work, the real options approach hinges on the optimal time for taking out a patent, as well as on discussing the payoff of the R&D phase. Differently, the model presented in the next section focuses on the valuation of a patent, right after the R&D phase.

3 The model

The model for the valuation of a patent presented hereafter is in accordance with the real options approach. The value of a patent, denoted by Ct, is supposed to be driven by the evolution of an underlying state. According to the literature, the underlying state is represented by the cash flow. We stress that the evolution of patent value can drastically change in the presence of impulsive events. A technological improvement or the introduction of laws modifying the protection rules can be reasons for a jump in the dynamics involved in the patent valuation process. Therefore, as already said in the Introduction, we rely on an underlying process able to provide a quantitative description of the qualitative shocks occurring in the economic environment, adopting a discrete time framework. To this respect, it is interesting to note that some models of patent valuation approximate the results obtained in continuous time with an approximate formulation in discrete time, recognizing the latter to be more realistic. “[…] The decision to abandon the project is evaluated at discrete points in time, instead of continuously. This would seem to be a more reasonable assumption when analyzing R&D projects […]” (Schwartz 2004, 52).

To make the model as clear as possible, we present it in a stepwise form.

4 Estimation of the patent value

This section is devoted to provide a mechanism for the computation of the patent value at a given time t.

First of all, a discussion on the randomness of the process involved in the patent value is needed.

There is a clear correlation between γi and the (τi,ξi) couple. This fact does not allow to consider R as a stochastic process on the line, and we need to treat R as a Spatial Point Process. The following Assumption will stand in force hereafter:

For the formal definition of SMPP, see Definition 1 in the Appendix.

The statement of Assumption 1 is important for two reasons. First of all, it contributes to construct a model for simultaneously computing the number and the size of the jumps in the economy and, consequently, the number and the size of the jumps in the patent value dynamics. Secondly, according to some recent results, SMPPs can guarantee the invariance of the stochastic structure between R and N. More precisely, if R is a SMPP, then the process N is a SMPP too (see the Appendix). This is a key result for the estimation procedure.

In order to proceed, we now need to define a time interval

with T′≤T. The interval I represents the period under scrutiny, and it plays a key role in providing the estimate we are dealing with.

We now define the regions I≡I×[−a,a] and J≡I×[−b,b]. Moreover, in agreement with a common used mathematical notation, we introduce △⊆R2 and denote by R(Δ) and N(Δ) the number of the elements of the Spatial Point Process R and N respectively that are contained in Δ. Further, let us define the random subset of indexes {i1,…,iK}⊂N, such that

We notice that K≡R(I).

The stopping time τ∗ is located at random jumping time τiKˉ, where Kˉ∈N or, alternatively, τ∗=+∞. We denote by Iτ∗ the restriction of the counting spatial process R in the set I up to the stochastic threshold τ∗. Formally, we have

and K∧Kˉ≡R(Iτ∗).

Equally, let us denote the random subset of indexes {i1,…,iK˜}⊂N, such that

where Jτ∗ denotes the propagation on the process N in the set J of the restriction of R due to the presence of the stopping time τ∗. We notice that K˜≡N(Jτ∗).

Furthermore, property Πt implies the existence of an index K˜∗≤K˜ such that K˜∗<j∗≤K˜∗+1. We denote as Jτ∗,∗ the further restriction on Jτ∗ driven by property Πt. We have

We also consider the random subset of indexes {i1′,…,iK′′}⊆{i1,…,in∧Kˉ}n=1,…,K such that the delay λi, with i∈{i1′,…,iK′′}, falls in the set J. We have a conditioning on the set Iτ∗. Taking into account property Πt, we finally define:

We denote K′∧K˜∗≡N(Iτ∗)(Jτ∗,∗). Evidently, P(N(Iτ∗)(Jτ∗,∗)≤R(Iτ∗))=1.

We now apply the approach of Cerqueti, Foschi, and Spizzichino (2009) to provide an approximation of the expected value of the patent at time t in formula [7]. The mechanism is based on the knowledge of the both the jumps in the economy and in the cash flow occurring over the time interval I. Define the time interval H=[T′,T], the region H=H×[−b,b] and also consider t∈H.

Let Υ={j1,…,jK′′} be the random subset of indexes such that {(λj1,γj1),…,(λjK′′,γjK′′)}=Hτ∗,∗∩N. As usual, Hτ∗,∗ denotes the restrictions on the process N in H due to the presence of the threshold τ∗ for R and the property Πt for N. By definition of the process N, we can also write K′′≡N(Hτ∗,∗).

We omit hereafter the subscripts τ∗ in I and τ∗,∗ in J and H to have a less cumbersome notation. Nevertheless, the computed quantities have to be intended under the threshold conditionτ∗for the process R and under the propertyΠtfor the process N.

In this framework, the random amount CtJ can be written as follows:

We will approximate the conditional expectation of CtJ given:

1. the number R(I) of jumps in the economy during the interval I;

2. the number N(J) of the jumps in the cash flow, and therefore in the patent during I;

3. the number of jumps in the economy occurred over I propagated to the patent in the same time interval. We denote this quantity as N(I)(J).

We consider two partitions Δ and Ψ of H as follows:

where Hs(k):=H×(cs−1(k),cs(k)], with s=1,…,k, c0(k)=−b,ck(k)=b and, for each k, {cs(k)} is increasing with respect to s;

where Gv(h):=(tv−1(h),tv(h)]×[−b,b], with v=1,…,h, t0(h)=T′,th(h)=T and, for each h, {tv(h)} is increasing with respect to v.

A further refined partition of H can be obtained by the intersection of the partitions defined in eqs. [10] and [11]. We have

Fix the four integers s,v,h,k. We denote by bs,v(k,h) the expected number patent jumps observed in the time interval (tv−1(h),tv(h)] with size (cs−1(k),cs(k)], conditioned on the previous history in period I, i.e.

The next result provides a closed form expression to compute bs,v(k,h), for any (k,h)∈N2 and (s,v)∈{1,…,k}×{1,…,h}. Evidence is given in Cerqueti, Foschi, and Spizzichino (2009).

Proposition 1 allows to get an upper and a lower approximation of E[CtJ|R(I)=n′,N(J)=n′′,N(I)(J)=m].

Consider the following sequences:

For any h,k∈N, we have

By eq. [14], taking the limit, we can conclude that

Assume that Fλ∗ is the (marginal) distribution function of λ∗.

We have

5 Conclusions and further research

This paper is a generalization of the existing models for valuing a patent in a real options framework. Briefly speaking, the generalization consists in accounting for positive and negative random jumps, inducing possible delays in the transmission mechanism between the jumps in the underlying process and the corresponding jumps in the patent value, considering a decay process of the intensity of the shocks occurred to the economy and considering also the dependence between size and time of the jumps in the cash flow due to jumps in the economy. In this respect, if a shock intervenes in the economy when the patent is close to expiration, ceteris paribus, it will affect the patent with lower intensity.

We have suggested that the inclusion of these hypotheses is essential to add realism to the valuation process. In turn, this step also brings significant improvements, such as a solution to patent valuation in a closed form, without assuming the absence of expiration and considering the presence of non-killing jumps. It follows that patent evaluation can greatly differ whether our model is adopted or a more naive approach is followed. Many factors account for that difference. Empirical works (Schankerman 1998;Lanjouw, Pakes, and Putnam 1998) report that the distribution of the value of patents is highly skewed, there are many patents with low value and just a few with a high one. Our model sensibly assumes that when a low-valued patent is taken out by a competing firm, the high-valued patents undergo a non-deadly jump. Therefore, the number of these jumps is quite far from being negligible and, consequently, the cumulated change in the high-valued patent is not marginal too. Moreover, the delay in the transmission mechanism is such that the patents value is accidentally equal under the two models, and the differences in both sign and magnitude can be hardly predicted.

Giving an estimate of the value of patent rights through a naive model, from a policy point of view, can lead to biased results that hinder a meaningful comparison between sectors and countries. If policy recommendations and interventions are based on a biased comparison, misleading incentives may be put into play.

From a theoretical point of view, it is worth stressing that the dependence of the γ s on the τ s prevents to deal with processes on the line and motivates the assumption that R is a SMPP. In fact, consider the familiar simple model described as follows: (i) the process {τi}i∈N is a one-dimensional Mixed Poisson Process; (ii) the ξ s are i.i.d. and independent of the τ s; (iii) the γ s are i.i.d. and independent from the process R.

Then, N has the same stochastic structure described by the process R identified by conditions (i) and (ii). In our specific case, as condition (iii) does not hold, R and N do not have the same structure. To state an invariant property between R and N and, at the same time, maintaining the validity of conditions (i)–(iii), we need to abandon the usual Mixed Poisson framework and let the process R be a SMPP. This allows us to maintain the model as general as possible and to proceed with the estimation.

Another key point that would be worthwhile tackling in a succeeding study is the empirical evaluation of patent rights under the present approach, through an econometric estimate. A natural comparison between our model and Pennings and Lint (1997) can be then put forward: indeed, Pennings and Lint (1997) model can be viewed as a benchmark model, and our framework as an extension. Specifically, in Pennings and Lint (1997) the cash flow process does not include delays and time variations of the jump times. However, Pennings and Lint do not deal with SMPPs, and so their paper is radically different from the present one. Having said this, it is important to note that simulating a SMPP represents a challenging issue, which has never been treated in the literature. We are currently working in this direction.