Keywords: real options, Monte Carlo simulation, financial valuation

This paper studies how sensitive real option valuations are to incorrect assumptions about the stochastic process followed by the state variables. We design a valuation model which combines Monte Carlo simulation and dynamic programming and provides an appropriate framework to evaluate the effect of estimation errors on both the value of real options and their critical frontier. Although the model is flexible enough to value American-type options contingent on a wide range of stochastic processes, we focus on the analysis of the effect of stochastic jumps. We apply our model to the valuation of an investment in the car parts industry documented in previous literature. Our results clearly show that underestimating this type of jumps might lead to substantial misjudgements in a firm's decision-making processes. For instance, it may lead to profitable projects being rejected when jump diffusion is low, or negative expanded net present value projects being accepted.

I. Introduction

In the spring of 2000, Tom E. Copeland predicted that the real option valuation, henceforth ROV, would eventually replace the discounted cash flow (DCF) model in under ten years [START_REF] Copeland | New developments in valuation, in Strategic Finance in the 21st Century (15[END_REF]. This forecast may seem an inconceivably short space of a time, yet it entails a process of change which has lasted over 30 years, beginning in 1977 with Stewart C. Myers' proposal. This thirty year period should be more than enough time for the adoption of a valuation technique which is theoretically superior to any other known model, even more so when other more costly innovations have successfully been implemented in a significantly shorter period of time.1 Among the reasons put forward for the limited application of ROV, [START_REF] Newton | Application of OPT to R&D[END_REF] pointed to the operational complexity of its valuation techniques, [START_REF] Myers | Fischer Black´s Contributions to Corporate Finance[END_REF] hinted at a lack of understanding of its underlying philosophy on the part of managers, and [START_REF] Lander | Challenges to the Practical Implementation of Modelling and Valuing Real Options[END_REF] referred to the failure of mathematical tools to fulfil certain requirements. An even greater problem needs to be considered, namely the paradoxical lack of flexibility arising from the lack of any general model -no matter how complex its understanding or application may be-to be employed in the valuation of, if not all, then at least the more common real options.

Whereas the DCF formula may be applied directly to virtually all investment opportunities, ROV is bereft of any similar formula. What is more, ROV comprises a combination of analytical formulae and numerical techniques, each of which is appropriate to the valuation of a specific option on a particular underlying asset. Not This paper analyses the consequences arising from the use of ROV techniques inappropriate to the nature of the stochastic process for the state variable. We specifically investigate how sensitive values of American-type real options are to the effect of unusual discontinuities of the state variable. Valuation is approached by a Monte Carlo simulation model which is inspired by the proposal of Grant, Vora and[START_REF] Grant | Simulation and the Early-Exercise Option Problem[END_REF]1997) and [START_REF] Ibáñez | Monte Carlo Valuation of American Options Through Computation of the Optimal Exercise Frontier[END_REF] for financial options.

Our model considers the specific nature of real investments which requires simultaneously determining the values of both real options and their underlying assets, from the state variable on which its cash flows depend. The main contribution of our model is that it directly estimates the values of the state variable that define the optimal exercise frontier, and thereby provides a clear decision rule to the holder of the option. Namely, comparing observed values of the state variable and the critical value at each exercise date it is possible to know whether exercise is recommended. For our analysis purposes, this feature represents an advantage compared to other simulation-dynamic programming methods, such as the powerful regression based procedure proposed by [START_REF] Longstaff | Valuing American Options by Simulation: A Simple Least-Squares Approach[END_REF]. 2 We apply our model to the valuation of a foreign direct investment (FDI) undertaken by a Tier-One multinational supplier of automobile components, whose real options are well outlined in [START_REF] Azofra | Las Opciones Reales en la Industria de Componentes del Automóvil[END_REF]. The results of our valuation highlight the important consequences arising from mistakes in the estimation of the 2 Our model's advantage decreases with the number of early exercise opportunities, since determining each critical value requires simulating new paths for the state variable, and hence the model costs -in terms of time and computing resources-grow exponentially. -4-state variable stochastic process. Specifically, these findings underline the existence of significant differences in the value of call and put options when the foreseen evolution of the state variable advocates consideration of random discontinuities. The nature and scale of the errors depend both on the influence of the continuous motion as well as on jump diffusion, the greatest differences being observed when the continuous influence is lower and the jump diffusion is higher. Excluding discontinuities in these cases entails the appearance of bias in the evaluation which may lead to erroneous investment decisions being taken.

The rest of the paper is structured as follows. The next section addresses the problem of evaluating American options on discontinuous stochastic processes. Our valuation proposal is outlined in section three. Section four deals with the analysis of the valuation results for options to expand and contract embedded in an investment case in the car parts industry. The final section concludes and discusses relevant implications.

II. Valuing American real options contingent on discontinuous processes

One challenge currently facing the real option approach is the development of a general model to enable valuation of a wide range of both call and put options, involving multiple exercise dates depending on the evolution of stochastic processes of an extensive nature.

The chance to exercise options at more than one future date is probably one of the most common features of corporate investment. For this reason, the same arguments put forward by the advocates of ROV when criticising DCF [START_REF] Mcdonald | The Value of Waiting to Invest[END_REF][START_REF] Lee | Capital Budgeting Under Uncertainty: The Issue of Optimal Timing[END_REF][START_REF] Pindyck | Irreversibility, uncertainty and investment[END_REF], are now applicable to "European" ROV. It is just as difficult to imagine a corporate investment opportunity whose -5-exercise may not be postponed in the least, as it is to imagine a growth or abandonment option with only one future date. 3As for the stochastic evolution of the state variable, most ROV models assume geometric Brownian processes. These are processes which have been widely used to describe the evolution of financial asset and commodities prices, but are hard to upscale to other types of state variables on which real options may depend.

Variables such as demand, output prices or input costs, may adapt better to mixed processes, which combine continuous Brownian motion with random discontinuous jumps. Depending on each variable, these discontinuities could be brought about by a change in customers' preferences, technological progress, or competitors' actions.

The occurrence of random jumps complicates -and indeed may render impossible-valuation of American options by means of traditional solving techniques: closed-form solutions, analytical approximations and lattice approaches. [START_REF] Merton | Option Pricing When Underlying Stock Returns Are Discontinuous[END_REF] derives the analytical valuation formula for the European option when the price of the underlying asset follows a mixed process, comprising a continuous geometric Brownian motion contingent on discrete Poisson jumps. Merton's proposal enables us to value options whose exercise is restricted to the expiration date, but may not be used in the case of American options.

Analytical approximations together with the most common numerical procedures -the binomial model [START_REF] Cox | Option Pricing: A Simplified Approach[END_REF]) and finite differences [START_REF] Brennan | Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis[END_REF]-allow us to consider the possibility of early exercise, although their computational application proves more complex when By contrast, models based on Monte Carlo simulation [START_REF] Boyle | Options: A Monte Carlo Approach[END_REF] may be applied to the case of multiple state variables regardless of the type of stochastic evolution to which they are linked. The greater flexibility inherent in this method revolves around the fact that valuation is undertaken by directly determining the process of the underlying asset, meaning that the partial differential equation (PDE)

need not be solved. The drawback is that the direct application of Monte Carlo simulation is not suitable for valuing American style options. At least this is what was believed to be true until fairly recently. 4Restrictions in simulation valuation are due to the very nature of this technique. Since exercising an option at a specific date prevents its subsequent exercise at a later date, the strategy which dictates optimal exercise of an American option relies not only on the prior evolution of its underlying variables, but also on their future values. Considering future events can only be performed through procedures involving backward induction, such as dynamic programming which resolves binomial and trinomial trees or finite difference procedures which do likewise with partial differential equations. By contrast, the Monte Carlo method is a forward induction process, generating future values for the variables based on previous values and therefore offering a suitable technique for assets whose cash flows at a given moment do not depend on subsequent events, as is the case of European options (Cortazar, 2001). In order to overcome this restriction, recent research has proposed combining simulation with some backward induction procedure that may lead to a valuation model applicable to both European and American type options, whatever the number of state variables and the nature of the stochastic processes. The earliest attempt to apply simulation in American style options valuation may be found in [START_REF] Tilley | Valuing American Options in a Path Simulation Model[END_REF] 5 , who proposes a model for valuing financial options dependent on the stochastic evolution of a single state variable coinciding with its underlying asset. Tilley proposes sorting the simulated values of the underlying asset for each exercise date into groups or "bundles" for which a single value of keeping the option alive until the next period is assigned, as the mean of the continuation value of the whole of these paths. 6Tilley's approach has been followed by a growing number of papers that propose different combinations of simulation and backward induction procedures for valuing American-type financial derivatives. The resulting models approximate the early exercise frontier or conditional expectation function for the derivative.

Prominent within this approach are the works of [START_REF] Barranquand | Numerical Valuation of High Dimensional Multivariate American Securities[END_REF] and [START_REF] Raymar | Monte Carlo Estimation of American Call Options on the Maximum of Several Stocks[END_REF], who propose the use of a partitioning algorithm on the unidimensional space of the cash flows yielded by the option, rather than considering the multidimensional space of the underlying assets defined in [START_REF] Tilley | Valuing American Options in a Path Simulation Model[END_REF]. [START_REF] Grant | Simulation and the Early-Exercise Option Problem[END_REF] as well as [START_REF] Ibáñez | Monte Carlo Valuation of American Options Through Computation of the Optimal Exercise Frontier[END_REF] directly estimate the values of the state variables for which the value of keeping the As an alternative proposal, Broadie andGlasserman (1997, 2004) and [START_REF] Broadie | Enhanced Monte Carlo Estimates for American Options Prices[END_REF] advocate use of non-recombinatory simulated trees7 and stochastic meshes to determine two estimates of the option value, one biased "upward" and another biased "downward", both asymptotically unbiased and convergent towards the certain value. Finally, [START_REF] Longstaff | Valuing American Options by Simulation: A Simple Least-Squares Approach[END_REF] opt for least square regressions as a method for approaching the expected value of keeping the option alive at each decision point.

In the light of this type of proposal, recent corporate finance literature has welcomed Monte Carlo simulation procedures for valuing real options. The [START_REF] Barranquand | Numerical Valuation of High Dimensional Multivariate American Securities[END_REF] model and its subsequent development by [START_REF] Raymar | Monte Carlo Estimation of American Call Options on the Maximum of Several Stocks[END_REF] have been used to value American-type options where the state variables evolve following conventional mean reverting and geometric Brownian processes. Such is the case of [START_REF] Cortazar | Monte Carlo Evaluation Model of an Undeveloped Oil Field[END_REF] who resolve optimal timing of oil reserves, and Cortazar (2001) who evaluates optimal operations in a copper mine. Other papers apply the [START_REF] Longstaff | Valuing American Options by Simulation: A Simple Least-Squares Approach[END_REF] algorithm in valuing real options linked to patents and R+D projects [START_REF] Schwartz | Patents and R&D as Real Options[END_REF][START_REF] Miltersen | R&D Investments with Competitive Interactions[END_REF], dot-com companies [START_REF] Schwartz | Rational Pricing of Internet Companies[END_REF][START_REF] Schwartz | Rational Pricing of Internet Companies Revisited[END_REF] and pharmaceutical companies (León and Piñeiro, 2003).

III. The model

After reviewing previous literature, we address the problem of valuing real optionsboth growth (call) as well as abandonment (put)-of limited duration, T O , which may -9-be exercised at one or more future points, τ < 2τ < ... < Nτ = T O , during the also limited life of the underlying investment, T.8 This investments yields cash flows dependent on an uncertain variable, S t , whose evolution over time we suppose to follow a mixed process, comprising a geometric-Brownian type motion subject to random jumps spread over a Poisson variable. This implies that the infinitesimal variation of the state variable dS t is given by:

( ) ( ) dq S dz S dt S k dS t t t t 1 - + + - = π σ λ α
where α and σ represent, respectively, the expected drift and volatility of the continuous motion; λ is the mean frequency of discontinuous jumps per unit of time;

(π-1) and k are, respectively, the random variable measuring the size of the proportional jump and its mean value; 9 and dz t and dq t represent stochastic Wiener and jump processes, which we assume to be independent and characterised by their usual expressions:

(0,1) , N ξ dt dz → ⋅ = ξ [ ] λ λ λ Poisson q dt dt dq → ⎩ ⎨ ⎧ ⋅ = ⋅ - = = , prob 1 1 prob 0
Regarding the discontinuous variation, we assume each jump size to be independent and log(π) to follow a normal distribution with mean µ π and deviation

σ π , such that: [ ] 1 2 exp 1 2 - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = - = π π σ µ π E k
Discontinuities are linked to "unusual" and significant events, which give rise to upward and downward variations in the uncertain variable, while the continuous -10-process -the geometric Brownian motion-is linked to the idea of "normal" events.

Following common practice, we suppose the direction of the jump to be unknown a priori and hence the effect of the jump in the trend to be zero, and therefore

µ π = -σ π 2 /2 y k = 0
The method we propose to value both European and American style real options requires calculating, at each early exercise point, τ < 2τ < ... < Nτ = T O , the successive "critical" values S τ *, S 2τ *, ..., S Nτ * of the state variable. These critical values represent the exercise frontier, since for these values the payoff resulting from exercising the option is equal to the payoff of maintaining it until the next period.

The exercise frontier is used to estimate both the option and underlying project values for a series of simulated paths. Solving the valuation problem is thus performed in two stages: i) an initial phase which consists of estimating the successive "critical" values by combining Monte Carlo simulation with dynamic programming; and ii) a second phase which involves determining the value of the option from the previous critical values and which merely requires the use of simulation techniques.

III.I. Stage one: Estimating the critical value frontier

Assuming complete markets, the expression for the future equilibrium value for the state variable is the following: 10

( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + ∆ + ∆ - - = ∑ = q i i t z t z t r S S 1 2 0 2 0 2 5 . 0 exp π π σ σ σ σ δ
where r and δ represent, respectively, the continuous risk-free rate of return and the convenience yield, 11 z 0 represents a standard normal random variable linked to the 9 Hence, the mean growth rate caused by the discrete jumps is λk.

10 This expression derives from applying the risk-neutral valuation approach to the "twin" financial asset, which is perfectly correlated to the state variable. -11-continuous diffusion process; z i are standard normal independent variables determining the size of each jump and, as mentioned, q reflects the number of discrete jumps generated by a Poisson distribution with frequency λ.12 Each of the paths of the state variable determines a pair of values for the underlying project contingent on the decision to exercise, ( )

i t i exercise t S V ,
, and to continue without exercise, ( )

i t i no t S V ,
, where the superscript i indicates the number of simulations performed. The value of the project when the option is exercised at t

(with t ≤ T O
) is obtained adding the cash flow generated at that point,

i exercise t F ,
, to the expected value of the project at the following period, [ ]

i exercise t V E , τ +
, also taking into account the exercise price, X, obtained or paid for the sale or purchase of the project, depending on whether it is a put or a call option:

( ) [ ] ) exp( , , , τ τ r V E F X S V i exercise t i exercise t i t i exercise t - + + ± = +
where τ is the time interval into which the lifespan of the option has been divided.

To calculate the value of the underlying project should the option not be exercised, we need to distinguish between the option expiration date and the remaining points at which early exercise is allowed. Hence, when t is the expiration date (t = T O ), the value of the underlying is the present value of cash flows derived from expiration of the option without exercise, [START_REF] Merton | Option Pricing When Underlying Stock Returns Are Discontinuous[END_REF] we assume the risk associated to the discontinuous jump of the state variable to be diversifiable. The risk-neutral simulation would then show a continuous modified drift, r-δ, rather than the initial α. This is the equivalent of subtracting from the continuous drift the risk premium of the corresponding asset (Trigeorgis, 1996: 102). The value of the underlying would be calculated from the expected value of the project at the following period -including optimal decisions taken up to expiration of the option-and the cash flow generated at that point.

( ) [ ] ) exp( , , , τ τ r V E F S V i no T i no T i T i no T O O O O - + = + 11 Following
( ) [ ] ) exp( , , τ τ r V E F S V i t i no t i t i no t - + = +
We initiate the estimation of the critical values series by simulating M values of the state variable at the option expiration date,

M i T O S ,..., 2 , 1 =
, taken from the initial value, S 0 , and according to the stochastic process assumed. 13 By definition, the critical value at the option expiration date Each of these paths enables us to estimate both cash flows derived from the exercise and non-exercise generated from the option expiration date up to the end of the project (T O , T). Discounting each of these cash flows at the option expiration date, T O , yields the corresponding contingent value of the investment, 13 The simulation may begin at any moment and for any value of the state variable [START_REF] Grant | Simulation and the Early-Exercise Option Problem[END_REF]. However, when dealing with American options, whose optimal exercise at each moment depends on future expectations, the first critical value to be calculated must correspond to expiration.

M i T O S ,..., 1 = -τ
, from which other K paths are simulated up to the end of the project -values of

j i T O S , , j i T O S , τ + , j i T O S , 2τ + , … , j i T S , for i=1, 2, …, M and j=1, 2, …, K.
These paths in turn serve to determine the cash flows generated from that date and for both exercise and non-exercise cases.

The optimization process requires not only considering whether the option is exercised or not at T O -τ, but also possible exercise at a later date, which will mainly affect the future expected value of the project. If the option is exercised at T O -τ, the expected value of the project at the following point, [ ]

i exercise T O V E ,
, must be calculated considering the exercise which has already occurred, in turn preventing any new exercise decisions at subsequent dates. However, if the option is not exercised at T Oτ, the expected value of the project at the following period requires comparing the simulated value of the variable In this case, simulation involves estimating a sufficient number of state variable paths from the current moment to the project expiration date and we estimate the moment of optimal exercise along each path in accordance with the optimal early exercise frontier. Finally, we obtain the present value of the option discounting the resulting payoff form each path, and then taking the average of all the paths.

IV. Valuation of an investment in the car parts industry

In order to evaluate both interest in flexibilising valuation of real options and the proposed simulation model, we analyse the results of its application to the case of an investment undertaken by a Tier-One multinational supplier of automobile components. A previous valuation of this investment and its real options, using the -15-binomial model, is documented in [START_REF] Azofra | Las Opciones Reales en la Industria de Componentes del Automóvil[END_REF]. The project involves a foreign direct investment in the acquisition of production capacity in Brazil. As evidenced in Azofra et al. (op. cit.), the project is one of those opportunities which despite yielding a negative NPV is undertaken due to their strategic value, which is shown to emerge from its real options.

Through this commitment, the investing firm aimed to cater for the demand for car parts manufacturers in Brazil and, at the same time, achieve a strategic foothold in the South American car parts market so as to access future growth opportunities. The initial outlay of 38 million dollars was enough to meet sales forecasts for the initial five years, leading to the supply of components for some 500,000 vehicles during the first year of operations and nearly 850,000 units in each of the next four.

Table 1 shows the expected cash flows for these five years. The corresponding NPV varied between a loss of seven million dollars, in the case of a five year period and capital cost of 13.66%, and a loss of 21 million dollars, in the case of perpetual life and a capital cost of 28.4%. In either case, DCF valuation advised against the investment.

[ Taking the information from this case as a reference, we explore the consequences of neglecting possible discontinuities in the future evolution of the state variable when valuing American-type options. Our analysis focuses on valuing options to expand and contract the initial size of the project. The expiration date of these options coincides with the end of the fourth year of operations and may be exercised at three different dates, at the end of the second, third and fourth annual periods.

The option to expand the project resembles a call option with a strike price equal to the cost of the required assets. Exercising the option involves an increase of 50% of both the market share and the maximum production capacity and entails an outlay equal to 40% of the initial investment. 14 The option to contract resembles a put option with a strike price equal to the book value of the assets affected.

Exercising the option entails the reducing the maximum production capacity by 50%.

The values of these options are estimated using two different stochastic processes for the state variable: a pure geometric Brownian motion, on the one hand, and a mixed process comprising the previous Brownian process and discontinuous jumps linked to a Poisson type motion, on the other. The values of the parameters used in the evaluation are: i) for the pure geometric Brownian process, we assayed with alternative volatilities of 7%, 13% and 20% and annual average growth rates of 0%, 7% and 15%; and ii) for the discontinuous part of the mixed process, the volatility values of the jump considered are 25%, 50%, 200%, 400% and 500% We also assume that the jumps belong to the category of diversifiable or nonsystematic risk. Following [START_REF] Azofra | Las Opciones Reales en la Industria de Componentes del Automóvil[END_REF], the initial value of the state variable is 1 629,000 vehicles, for which there is a maximum value acting as an absorbing barrier at 4 000,000 units. Besides, initial configuration of the project implies a maximum production capacity of 1 200,000 vehicles that can be modified by its options to expand and contract. Based on S&P-500, the beta coefficient of the state variable is 1.035, and the risk-free return and the market premium are 6.59% and 6.85%, respectively.

Simulation is undertaken in annual subintervals, valuing the possibility of early exercise of the option at each of the three exercise dates considered. The number of simulated paths to obtain the present value of each option amounts to 400,000, the result of 200,000 direct runs plus another 200,000 estimations using the "antithetical variates" technique 16 and M and K parameter values equal to 250.

IV.I. Valuing the option to expand

Table 2 shows the estimated values of the option to expand for the scenarios within the parameters established by the pure geometric Brownian process and the mixed process.17 These results highlight the significant differences in the value of the option and, therefore, the impact of possible mistakes made by conventional ROV models, -18-particularly when the influence of continuous motion is smaller. Hence, in the scenario with least volatility and continuous variation (α = 0% and σ = 7%), the value of the option to expand increases ten fold when a jump with a 50% dispersion is introduced, and is reduced by almost 63% when jump dispersion reaches 500%.

[Insert Table 2]

These results underscore the fact that the sign of the relation between the value of the option and the volatility of the jump depends on the latter's level. Thus, the value of the option increases in those scenarios in which jump dispersion is lowest, as a consequence of the rise in total volatility to which the variable is subject.

These differences are more significant in the case of lower continuous volatility (σ = 7%), since in this case the increase which the total volatility undergoes when there is a jump of equal dispersion is much greater, in relative terms, than at levels of permanently high volatility. Thus, for example, a jump dispersion equal to 100% implies a total volatility in the process of 53.9% or 50.5% depending on whether the standard deviation of the continuous motion is either 20% or 7% respectively. 18 As jump volatility increases, this relation is inverted and gains in significance at higher levels of continuous volatility. This result will remain the same provided the mean size of the jumps, k, is zero and does not affect the trend of the process.

This case implies that the average taken by the distribution of the jump logarithm inversely depends on the value allocated to the jump volatility (µ π = -σ π 2 /2) and, therefore, an increase in these values of over 100% reduces this average and thereby the value of the underlying project. Moreover, the upper limit of the cash flows imposed by the maximum production capacity reinforces the inverse reaction between the value of this option and jump volatility. As regards the continuous evolution of the state variable, the expected positive relation between the growth rate and the value of the option for the various levels of volatility and stochastic processes considered is confirmed. Evidently, the more favourable the scenario foreseen vis-à-vis the growth in car production the more likely and profitable will be the expansion of the initially planned investment. This is not the case, however, for the continuous volatility of the state variable, which evidences a negative relation with the value of the option to expand, which is contrary to the ceteris paribus relation established by the theory of financial

options. Yet, this result concurs both with the nature of the investment project and the discretisation method used for the state variable. On the one hand, the maximum production capacity of the project leads to an asymmetrical effect of the volatility on the net cash flow probability distribution. Greater volatility implies greater dispersion of the lower values, whereas the higher values remain bounded by the maximum production capacity and, as a result, the increase in volatility will diminish the value of expanding, particularly at those levels where growth of the state variable is highest. 19The assumption that the state variable evolves in the continuous field following a lognormal diffusion process leads to its relative variation being distributed normally with a tendency reduced 0.5 times the variance of the process.

As a result, the volatility parameter not only affects the deviation of future values, but also the expected value in its simulation. Hence, the increase in volatility not only widens the range of possible future values of the underlying, but also reduces its . 20 This positive trend ceases to be apparent in the higher levels of jump dispersion, since the joint volatility increase undergone by the state variable process delays optimal exercise of the option.

IV.II. Valuing the option to contract

The valuation results of the option to contract are shown in Table 3. Once again, different scenarios are considered with regard to the parameters of the pure geometric Brownian motion and the mixed process. The inclusion of discontinuous jumps impacts the value of the option, although the relative variation which this option undergoes is noticeably lower than in the option to expand. The lower relative influence in the option to contract is a consequence of the stochastic process nature.

Positive growth levels in the state variable mean that the likelihood of exercising the option hardly varies when there is an increase in the discrete volatility despite the fall in the value of the underlying.

[Insert Table 3]

20 It should be noted that, as the project presents a finite life span, approaching the option expiration implies less time to compensate for the cost of additional investment through the higher cash flows arising from expansion.

V. Conclusion

This paper identifies some potential problems in the application of ROV. Traditional ROV models work well when the state variable follows a Brownian-type continuous motion. As this may not always be the correct assumption about the stochastic process followed by the state variable, understanding how sensitive real option values are to failures becomes a critical issue.

We focus on the valuation of American options whose state variable evolves following a mixed process, comprising continuous random walks together with random discontinuous jumps. The presence of random discontinuities complicates valuation of American options in commonly used numerical techniques -binomial trees and finite differences-thus making simulation the only technique applicable with any degree of simplicity and generality. We specifically propose a flexible model by merging Monte Carlo simulation and dynamic programming which allows us to weigh up the potential errors when omitting the possible occurrence of random jumps.

We have applied the model to value a real investment case, whose valuation using a binomial model is outlined in [START_REF] Azofra | Las Opciones Reales en la Industria de Componentes del Automóvil[END_REF]. In general terms, the application results confirm that one critical issue in correctly implementing ROV is -23-correct estimation of the state variable process. We observe that random mean-zero discontinuities alter the value of the options to expand and to contract, particularly when the influence of the continuous variation is lower and jump dispersion is high.

Further, the joint effect of the jump together with the real restrictions caused by the maximum production capacity of the investment or the very nature of the state variable, magnify the differences identified in the estimations.

Since the influence of random jumps is not quantified in the evaluation obtained using conventional numerical techniques, estimating the expanded NPV may lead to errors similar to those which occur when using the much criticised DCF approach. In our case study, leaving out discontinuities entails the appearance of biases -downward when the jump dispersion is low and upward in the opposite casewhich may lead to inefficient investment decisions being taken.

At lower discrete volatility levels, undervaluing the option to expand may lead to sub-optimal deferral of exercising the right and even the rejection of profitable projects. Over-valuation, might could appear in traditional ROV models in the case of high jump dispersion may explain the naïve selection of negative expanded NPV projects. Finally, the probability of early non-optimal exercise of the option to contract increases with the relative influence of discrete volatility, leading to the possible rejection of subsequent positive cash flows.

Although in this work we only assess the influence of random jumps with a zero mean value, our results show that the influence of discontinuities on option values will be even greater and thus errors arising from conventional ROV models even more evident. Summing up, the differences observed allow us to justify the effort required to flexibilise ROV models when the investment opportunities 18,084.20 37,115.88 38,229.35 39,376.23

  model, perhaps the most flexible of all traditional models for valuing options, allows for direct treatment of stochastic processes other than continuous Brownian-type motions, or multiple sources of uncertainty.

  other than the geometric Brownian family and multiple sources of uncertainty.

  the following period matches the value of its immediate exercise at each exercise date.

  t is a point prior to expiration, t < T O , the project value is determined by considering the possibility of adopting a new contingent decision at a later point.

  the one for which the value of the underlying, assuming immediate exercise of the option, values requires simulating K paths of the state variable up to expiration of the underlying project, i=1, 2, …, M and j=1, 2, …, K, as Fig. 1 details: [Insert Fig. 1]

  critical value of the state variable at the option expiration date, we go back to the immediately prior exercise point, T O -τ, for which we repeat the process of estimating the critical value of the state variable, *

  to incorporate the possibility of adopting a new decision.Hence, determining the value of the project at T O -τ, assuming non-exercise at this date, merely involves adding the current cash flow to the discounted expected value of the project atT O , [ ] i T O V E, which is calculated by averaging the K simulated values for both exercise or non-exercise at T O cases. Finally, it just remains to for each of the prior dates where exercise is possible until the remaining values that make up the optimal exercise frontier are found. Logically, as we go back to the initial moment and although the logic for estimation always remains the same, the complexity and number of operations involved in determining critical values multiplies.III.II. Stage two: Estimating the current value of the optionHaving determined the critical values of the state variable at the different moments when early exercise is possible, * τ value of the American option may be estimated by conventional simulation as if it were a European option.

  with a single value of the expected number of jumps equal to 0.2 (λ = 0.2).15

  Finally, Fig. 2 shows the behaviour of the critical values at the three exercise dates depending on the parameter values of the stochastic process. These values logically reflect the relation between the value of the option and the previously described characteristics of the state variable evolution. The figure also highlights the expected positive trend of the critical values together with their lower dispersion as the option expiration approaches

  be seen that the greater the influence of the continuous component (high α and σ values) the lower is the influence of the jump in the option value. By contrast, when the effect of the continuous motion is barely relevant, the increase in the jump dispersion significantly reduces the value of the option. This result reflects the nature of the state variable, whose lower limit is bounded at zero such that the jump dispersion produces an asymmetric effect on the cash flow probability distribution. Jump volatility implies greater dispersion in the underlying project higher values, whereas lower values remain bounded and, thus, it diminishes the chances of optimal exercise of this option.The results also confirm the expected negative relation between the drift of the state variable and the value of the option for the different levels of volatility analysed. The less favourable the expected evolution for the state variable, the more valuable is any action aimed at reducing the size of the initial project. The same happens with the continuous volatility of the state variable, which displays a positive relation with the value of the option to contract, resulting not only from the expected asymmetrical increase of possible gains for the option, but also the "implicit" reduction of the mean simulated value of the underlying, already mentioned for the option to expand. Only when the state variable shows a flat trend (α = 0%), does the increase in volatility favour the value of the underlying asset without affecting the probability of exercise, once again due to the lower boundary inherent in the nature of the state variable. Finally, Fig.3shows the evolution of critical values at exercise dates in terms of the stochastic process parameters. The figure bears out the expected upward trend of the optimal exercise frontier and the lower dispersion of the critical values approaching expiration of the right. Given the finite lifespan of the investment, the present value of its cash flows is lower as the expiration date of the right approaches, and as a result, exercising the option will be advisable on a greater number of occasions. Only when the jump dispersion is extremely high -and thus the volatility of the joint process-can any reduction be seen in the critical values corresponding to the point prior to the expiration date indicating deferment in exercising the right.

Fig. 2 .

 2 Figures

Insert Table 1]

	evolution of the state variable, providing empirical evidence to support the relevance
	of real options when explaining the investment decision.
	F o r
	P
	e
	e r
	R e v
	i e
	Considering growth and flexibility options offers quite a different valuation w
	result. Azofra et al. (op. cit.) value European growth options, capabilities to
	redistribute resources towards more profitable uses, and early abandonment of the
	project -the latter as an American-type option-. Estimation is performed by applying
	a binomial model, in which the uncertain variable is defined as the number of cars
	produced in Brazil, which is assumed to follow a geometric Brownian process. The
	study confirmed the significance of these options in different scenarios for future
	-16-

Critical values for the option to contract

  Growth rate of state variable is 7%. All values are in thousands USD. Growth rate of state variable is 7%. All values are in thousands USD.

	3000	Critical Value at T	Critical value at T-1
	2500			Critical Value at T-2
		2000
	Critical Value	1500
		1000
		500
		0
		0%
	7% 13% 20%	25% F o r Jump Volatility 50%	75%	200%	400%	500%	7%	13% 20% Vol. Cont
	Fig. 3. Tables		P e r e
				R e v
					i e
					w
				-29-

Notes:

Notes:

Table 1 . Predicted Cash Flows

 1

		Year 1	Year 2	Year 3	Year 4	Year 5
	Revenues	31,009.74 64,419.80 64,024.38 64,042.57 64,061.30
	Cost of Goods Sold

  Values are in thousands USD. Source:[START_REF] Azofra | Las Opciones Reales en la Industria de Componentes del Automóvil[END_REF].

						40,557.52
	Personnel Costs	3,254.57	4,080.05	4,202.45	4,328.52	4,458.38
	Other Operating Costs	705.89	1,243.18	1,280.47	1,318.88	1,358.45
	General and Administrative Expenses 4,565.59	6,031.87	6,212.82	6,399.21	6,591.18
	Taxes	815.30	3,378.46	3,345.60	4,102.07	3,970.55
	Total Payments	27,425.55 51,849.43 53,270.69 55,524.91 56,936.08
	Net Free Cash Flow	3,584.20 12,570.38 10,753.69	8,517.65	7,125.22
	Notes:

Table 2 . Present value of the option to expand

 2

	Jump Volatility

In addition to its theoretical superiority, a number of papers have found empirical insights regarding the relevance of real option in market values. Such is the case of[START_REF] Paddock | Option Valuation of Claims on Real Assets: The Case of Offshore Petroleum Leases[END_REF];[START_REF] Berger | Investor valuation of the abandonment option[END_REF];[START_REF] Danbolt | Measuring growth opportunities[END_REF] or[START_REF] Andrés-Alonso | Real options as a component of the market value of stocks: evidence from the Spanish Stock Market[END_REF]; among others.

This does not reduce the advance of the ROV approach over traditional DCF models, but highlights interest in analysing the value to postpone the exercise of real options when possible. See Vandenbroucke (1999) for a comparison of the NPV approach and ROV approximations.

A good example of this view is the second edition of Hull's handbook on financial options[START_REF] Hull | Options, Futures, and other Derivative Securities[END_REF], which on page 334 explains that "one of the drawbacks of the Monte Carlo approach is that it may only be used for European style derivatives". In this same sense,[START_REF] Hull | Efficient Procedures for Valuing European and American Path-Dependent Options[END_REF] postulate that "Monte Carlo simulation cannot deal with early exercise since there is no way of knowing whether this is optimal when a specific price is reached at a given moment".

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Some authors cite Bossaerts' (1989) working paper as the earliest reference to the analysis of early exercise of American options through simulation.

This procedure entails certain significant drawbacks, such as the need to store all the simulated paths -a time-consuming exercise-as well as the complexity linked to the sorting process when dealing with multiple sources of uncertainty.

Unlike binomial and trinomial trees, the values that appear at each node are placed in the order in which they are generated and not following a hierarchic order.

The equidistance (τ) of the exercise dates, derived from this formula, is assumed for clarification and explanation purposes only and does not condition the model in any way. Logically, T O <T.

The simulation of the number of discrete jumps at a time interval, ∆t, is obtained from applying the

In order to simplify the analysis we have not considered the car-makers' control of the value chain and its implications for the option valuation. For a detailed study, seeAzofra et al. (op. cit.).

The parameter λ = 0.2 implies that, on average, only one discrete jump will occur during the five-year life span of the underlying project. We feel it is more interesting to show the valuation results assuming a highly volatile and low frequency process of random jumps, as opposed to multiple smaller jumps, which may prove hard to distinguish from continuous evolution itself. On the choice of parameters see http://www.puc-rio.br/marco.ind/stoch-a.html#jump-dif.

The technique of antithetical variates consists of generating two symmetrical observations at zero for each of the random simulations of the normal distribution.

It should be noted that the 0% level of the volatility of the discontinuous jump corresponds to the pure geometric Brownian process.

The estimation of the total volatility for the mixed process was performed using the expression obtained in[START_REF] Navas | Correct calculation of volatility in a Jump-Diffusion Model[END_REF], who amends the one initially obtained in[START_REF] Merton | Option Pricing When Underlying Stock Returns Are Discontinuous[END_REF].

In fact, the reader can note that this negative relation disappears in cases of zero jump volatility and α values equal to 0 and 7%.

available to the firm suggest considering exercise at more than one future date and the inclusion of stochastic processes other than pure geometric Brownian.

G r o w t h r a t e


Carlos Pestana Barros
email: cbarros@iseg.utl.pt

Pedro Garcia-Del-Barrio

Stephanie Leach
email: stephanie.leach@imperial.ac.uk

Pedro Garcia
email: pgarcia@cir.uic.es

Angel Barajas

Miguel Cardenal

Joaquin Serrano

Carlos Del Campo

Analysing the Technical Efficiency of the Spanish Football League First Division with a Random Frontier Model

Keywords: Football, Sports industry, Random frontier models, Policy implication

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Economic efficiency in sports is a theme that has attracted some research in recent decades. Two traditions are observed, the DEA -data envelopment analysis, which has been applied by Barros and Leach, 2006A;Haas, 2003A, 2004B, Barros and Santos 2005, 2003; Espitia-Escuer and [START_REF] Espitia-Escuer | Measuring the Efficiency of Spanish First-Division Soccer Teams[END_REF]Fizel andD'Itry, 1996,1997;[START_REF] Porter | Measuring managerial efficiency: The case of baseball[END_REF] and the stochastic frontier models, which have been applied by [START_REF] Barros | Incentive Regulation and Efficiency in Sports Organisational Training Activities[END_REF]Leach, 2006B, 2006C;[START_REF] Kahane | Production Efficiency and discriminatory Hiring Practices in the National Hockey League: A Stochastic Frontier Approach[END_REF][START_REF] Gerrard | A Resource-Utilization Model of Organisational Efficiency in Professional Sport Teams[END_REF]; [START_REF] Hoeffler | Measuring efficiency in the National Basketball Association[END_REF][START_REF] Dawson | Stochastic Frontier and the Temporal Structure of Managerial Efficiency in English Soccer[END_REF][START_REF] Carmichael | Production and efficiency in association football[END_REF][START_REF] Scully | Managerial efficiency and survivability in professional team sports[END_REF]and Zak, Huang and Siegfried (1979). The aim of this research is to combine sports and financial variables in the evaluation of the clubs' efficiency.

This paper uses the stochastic framework approach, but innovates in relation to previous research by adopting a random stochastic frontier model [START_REF] Greene | Distinguishing between heterogeneity and efficiency: stochastic frontier analysis of the World Health Organisation's panel data on national health care systems[END_REF][START_REF] Greene | Fixed and random effects in stochastic frontier models[END_REF]. This model enables the separation of the covariates in the cost function into homogenous and heterogeneous variables. The identification of heterogeneous variables is of prime importance for policy purposes, since heterogeneity in the cost function of the Spanish football league would result in generic policy procedures that would affect the different clubs asymmetrically.

Homogenous stochastic frontier models are common in many research fields, for example, [START_REF] Haghiri | Assessing the impact of economic liberalization across countries: a comparison of dairy industry efficiency in Canada and USA[END_REF] apply this type of model to the dairy industry; in [START_REF] Cullinane | A stochastic frontier model of the productive efficiency of Korean container terminals[END_REF], it is applied to seaports; [START_REF] Mahadevan | How technically efficient are Singapore's manufacturing industries[END_REF] applies it to manaufacturing in Singapore; [START_REF] Lothgreen | Specification and estimation of stochastic multiple-output production and technical inefficiency[END_REF] applies it to the health services sector; and [START_REF] Street | Relative performance evaluation of English acute Hospital sector[END_REF] apply it to hospitals. However, in the sports context, the random frontier model has not previously been applied. 1 Also associate lecturer at the University Pompeu-Fabra (pedro.garcia@upf.edu). This author gratefully acknowledges financial support from the Ministerio de Ciencia y Tecnologia (SEJ2004-04649, Spain). The motivation for our research stems from the fact that while efficiency in sporting terms is observable on the field of play, the financial efficiency of clubs is far less transparent, being accessible for observation only in the club's financial report.

Thus, the comparison of sporting and financial results is of value when evaluating the efficiency of a football league club. Moreover, whilst Deloitte & Touche publish an annual financial report on Spanish football, among other leagues, reports are not compiled for every European country. Without the publication of a pan-European standardised report, researchers encounter difficulties in gathering the data sets needed to compare sporting performance with its financial underpinning.

The contribution of this paper to sports research is based on two aspects: first, it adopts an innovative stochastic frontier approach, the random frontier model, which to the authors' knowledge, has not previously been applied to sports. The random frontier model assumes that the variables are possibly heterogeneous, overcoming the hypothesis of homogeneity of the variables applied to earlier frontier models. Secondly, this is the first time that a stochastic frontier model has been applied to the Spanish Football League First Division, in relation to which prior analysis had been carried out with a DEA model (Espitia-Escuer and [START_REF] Espitia-Escuer | Measuring the Efficiency of Spanish First-Division Soccer Teams[END_REF].

The paper is organised as follows: In Section 2, the contextual setting is described; Section 3 presents the literature survey; in Section 4, the model is explained; in Section 5, the data and results are presented; Section 6 discusses the results, and in Section 7, the concluding remarks are made.

Contextual Setting

The first division (known as the Primera Liga) of the Spanish National Football League is renowned as one of the strongest leagues in the world, since many of the best Another consequence is that various scientific studies have used data from the Spanish Primera Liga, for instance, [START_REF] García | The Determinants of Football Match Attendance Revisited: Empirical Evidence from the Spanish Football League[END_REF] study the determinants of attendance of the Spanish league; Espitia-Escuer and [START_REF] Espitia-Escuer | Measuring the Efficiency of Spanish First-Division Soccer Teams[END_REF] study the efficiency of the Primera Liga teams using DEA (Data Envelopment Analysis), which is a non-parametric technique, while [START_REF] Ascari | Spanish Football[END_REF] analyse the financial crisis of Spanish professional football as a whole.

On the other hand, the main characteristic of the Primera Liga is that two teams, Real Madrid and F.C. Barcelona, are global brand giants, with the means to buy many of the top players, boast the largest numbers of supporters (Barcelona regularly play at home in front of a maximum capacity 110,000 spectators) and usually achieve the first two positions in the league, as in the 2004-2005 season.

Table 1 below presents information on the financial situation of the Primera Liga (in millions of Euros). In Table 2, we compute the average league position, revenues and wages for each club that played in the Primera Liga during the period analysed.

Table 1. Summary of financial information in the Spanish PLF Spanish PLF 1996[START_REF] Slack | Understanding Sport Organisation: the application of organisational theory[END_REF][START_REF] Coelli | An Introduction to Efficiency and Productivity Analysis[END_REF][START_REF] Audas | Organizational Performance and Managerial Turnover[END_REF]2000[START_REF] Carmichael | Production and efficiency in association football[END_REF]20022003

Literature Survey

Sports efficiency is commonly analysed as an aspect of sports economics and management [START_REF] Slack | Understanding Sport Organisation: the application of organisational theory[END_REF]. There are two contemporary approaches to measure efficiency: firstly, the econometric or parametric approach and second, the nonparametric. Besides these two approaches, we observe other papers relying on ratio analysis to address the same issue.

Among the papers which have used the econometric frontier, which is of particular relevance to the present research, Zak, Huang and Siegfried (1979) frontier. [START_REF] Scully | Managerial efficiency and survivability in professional team sports[END_REF] analysed measures of managerial efficiency for professional baseball, basketball and American football coaches, with a deterministic and a stochastic econometric frontier. A survival analysis was used to measure the coaching tenure probability in these sports. Extending the analysis of efficiency in sports, [START_REF] Ruggiero | Technical efficiency in major League Baseball[END_REF] analysed the efficiency of baseball teams with panel data. [START_REF] Hoeffler | Measuring efficiency in the National Basketball Association[END_REF] analysed the stochastic frontier of American basketball with cross-section data. Audas, Dobson and Goddard (2000) analysed involuntary and voluntary managerial job-termination, with hazard functions for English professional football. [START_REF] Hadley | Performance evaluation of National Football League teams[END_REF] analysed the performance of the American NFL, using a Poisson regression model. [START_REF] Dawson | Stochastic Frontier and the Temporal Structure of Managerial Efficiency in English Soccer[END_REF] analysed the managerial efficiency of English football managers with an econometric stochastic frontier [START_REF] Carmichael | Production and efficiency in association football[END_REF] analysed the efficiency of the English Premiership clubs with residuals. Gerrard (2001) analysed the production function of coaches working in the English Premier League with win-ratios for the period of 1992 to1998. [START_REF] Kahane | Production Efficiency and discriminatory Hiring Practices in the National Hockey League: A Stochastic Frontier Approach[END_REF] investigated the efficiency of the USA Hockey League's discriminatory hiring practices with a stochastic frontier model.

Among the papers which have taken the non-parametric approach, we mention Fizel andD' Itri (1996, 1997), who applied the DEA analysis to measure the managerial efficiency of college basketball teams to assess the conflicting theses concerning the impact of managerial succession on organisational performance, and [START_REF] Porter | Measuring managerial efficiency: The case of baseball[END_REF], who analysed the managerial efficiency of baseball managers with a nonparametric approach. [START_REF] Barros | Incentive Regulation and Efficiency in Sports Organisational Training Activities[END_REF] adopting the approach of this paper.

Theoretical Framework

Our framework is based on two strands of literature: models of industry efficiency and stochastic frontier models.

Models of Industry Efficiency

Two competing models of industry efficiency exist in the literature. Firstly, the strategic-group theory [START_REF] Caves | From entry barriers to mobility barriers: Conjectural decisions and contrived deterrence to new competition[END_REF], which justifies differences in efficiency scores as being due to differences in the structural characteristics of units within an industry, which in turn lead to differences in performance. In the case of Spanish football clubs, units with similar asset configurations pursue similar strategies with similar results in terms of performance [START_REF] Porter | The structure within industries and companies' performance[END_REF]. While there are different strategic options to be found among the different sectors of an industry, not all options are available to each club, due to mobility impediments, causing a spread in the efficiency scores of the industry.

The second model is the resource-based theory [START_REF] Barney | Firm resources and sustained competitive advantage[END_REF][START_REF] Rumelt | How much does Industry matter?[END_REF][START_REF] Wernerfelt | A resource-based view of the firm[END_REF], which justifies different efficiency scores on the grounds of heterogeneity in relation to the resources and capabilities on which the clubs base their strategies. These resources and capabilities may not be perfectly mobile across the industry, resulting in a competitive advantage for the best-performing clubs. Purchasable assets cannot be considered to represent sources of sustainable profits. Indeed, critical resources are not available in the market. Rather, they are built up and accumulated on the club's premises, their non-imitability and nonsubstitutability being dependent on the specific traits of their accumulation process. The difference in resources thus results in barriers to imitation [START_REF] Rumelt | How much does Industry matter?[END_REF] and in the football managers'/club executive managements' inability to alter their accumulated stock of resources over time. In this context, unique assets are seen as exhibiting inherently differentiated levels of efficiency; sustainable profits are ultimately a return on the unique assets owned and controlled by the football clubs [START_REF] Teece | Dynamic capabilities and strategic management[END_REF].

Random Frontier Models

In this paper, we adopt the stochastic cost econometric frontier approach. This approach, first proposed by [START_REF] Farrell | The Measurement of Productive Efficiency[END_REF], came to prominence in the late 1970s as a result of the work of [START_REF] Aigner | Formulation and estimation of stochastic frontier production function models[END_REF], [START_REF] Battese | Estimation of a production frontier model with application to the pastoral zone of eastern Australia[END_REF] and [START_REF] Meeusen | Efficiency estimation from a Cobb-Douglas production function with composed error[END_REF].

The frontier is estimated econometrically and measures the difference between the inefficient units and the frontier by the residuals. This is an intuitive approach based on traditional econometrics. However, when we assume that the residuals have two components (noise and inefficiency), we have the stochastic frontier model. Therefore, the main issue is the decomposition of the error terms. The general frontier cost function proposed by [START_REF] Aigner | Formulation and estimation of stochastic frontier production function models[END_REF] and [START_REF] Meeusen | Efficiency estimation from a Cobb-Douglas production function with composed error[END_REF] is the following:

(

1) 1,2, t N, 1,2, i ; ). ( T it u it v e it X C it C … = … = + = F o r P e e r R e v i e w 9
Where C it represents a scalar cost of the decision-unit i under analysis in the t-th period; X it is a vector of variables including the input prices and the output descriptors present in the cost function, [START_REF] Varian | Production efficiency: The case of professional basketball[END_REF].

The error term v it is the one that is traditional of the econometric models, assumed to be independently and identically distributed, that represents the effect of random shocks (noise) and is independent of u it .

The inefficient term u it represents the technical inefficiencies and is assumed to be positive and distributed normally with zero mean and variance 2 u . The positive disturbance u it is reflected in a half-normal independent distribution truncated at zero, signifying that each club cost must lie on or above its cost frontier. This implies that any deviation from the frontier is caused by management factors controlled by the football clubs.

The total variance is defined as

2 2 2 u v + =
. The contribution of the error term to the total variation is as follows:

) 2 1 /( 2 2 + = v
. The contribution of the inefficient term is:

) 2 1 /( 2 2 2 + = u
. Where 2 v is the variance of the error term v, 2 u is the variance of the inefficient term u and is defined as

v u =
, providing an indication of the relative contribution of u and v to =u+v.

Because estimation procedures of equation ( 1) yield merely the residual , rather than the inefficiency term u, this term in the model must be calculated indirectly [START_REF] Greene | Econometric Analysis[END_REF]. In the case of panel data, such as that used in this paper, [START_REF] Battese | Prediction of firm-level technical efficiencies with a generalised frontier production function and panel data[END_REF] However, the inefficiency can also be due to the heterogeneity of the clubs. In order to take this heterogeneity into account, we consider the following random effects model:

it it it i it u v w c + + + + = x ' ) ( 0 (2)
where the variables are in logs and i w is a time invariant, firm-specific random term that captures firm heterogeneity.

To estimate the model, the random coefficient model requires the identification condition that the random components of the coefficients be uncorrelated with the explanatory variables. A second issue concerns the stochastic specification of the inefficiency term u. For the latter, we assume the half-normal distribution.

For the estimation of the parameters of this model, we construct the likelihood function using the approach proposed by [START_REF] Greene | Fixed and random effects in stochastic frontier models[END_REF]. With the previous assumptions, the conditional density of c it given i w is:

it i it it it it i it w c w c f x ' ) ( , 2 ) | ( 0 + = = (3)
Where is the standard normal distribution and the respective cumulative distribution function. The parameters and 2 were defined before. Conditioned on i w , the T observations for enterprise i are independent and therefore, the joint-density for the T observations is:

= = T t it it i iT i w c c f 1 1 2 ) | ,..., ( (4)
The unconditional joint density is obtained by integrating the heterogeneity out of the density: The log likelihood, i i L log , is then maximised with respect to the parameters 0 , , , and any parameters appearing in the distribution of w i . The integral in (5) will be intractable. However, if we consider that equation ( 5) can be rewritten in the equivalent form:

i i w T t it it iT i i dw w g c c f L i ) ( 2 ) ,..., ( 1 1 = = = (5)
= = = T t it it w iT i i i i E c c f L 1 1 2 ) ,..., ( (6)
we propose to compute the log likelihood by simulation. Averaging the function in ( 6) over sufficient draws from the distribution of w i will produce a sufficiently accurate estimate of the integral in (5) to allow estimation of the parameters (see, [START_REF] Greene | Distinguishing between heterogeneity and efficiency: stochastic frontier analysis of the World Health Organisation's panel data on national health care systems[END_REF][START_REF] Greene | Fixed and random effects in stochastic frontier models[END_REF]. The simulated log likelihood is:

= = = = R r T t ir it ir it N i s w w R L 1 1 1 0 | | 2 1 log ) , , , , ( log (7)
where includes the parameters of the distribution of w i and w ir is the rth draw for observation i. Based on the panel data, Table 3 presents the maximum likelihood estimators of model (1) as found in other authors' recent studies [START_REF] Greene | Distinguishing between heterogeneity and efficiency: stochastic frontier analysis of the World Health Organisation's panel data on national health care systems[END_REF][START_REF] Greene | Fixed and random effects in stochastic frontier models[END_REF]

Data and Results

Since the early 1990s, most of the Spanish clubs have adopted corporate status, thereby being enforced to publish their financial accounts regularly. Yet, the task of gathering our panel was not an easy one. Some clubs do not publish the information punctually, while another four clubs (Barcelona, Real Madrid, Athletic de Bilbao and Osasuna)

have retained club status. Hence, given these limitations of data availability, we have restricted our analysis to the period 1995/96-2004/05. In almost all the cases, the data was obtained directly from the clubs' financial reports. To estimate the cost frontier, we used an unbalanced panel data on the Primera Liga for the years from 1995/96 to 2004/05 (159 observations). Frontier models require the identification of inputs (resources) and outputs (transformation of resources).

Several criteria can be used. First, one empirical criterion is availability of data. Second, the literature survey is a way to ensure the validity of the research and therefore, another criterion to take into account. The last criterion for measurement selection is the professional opinions of managers in the industry. In this paper, we follow these three criteria. Based on the data span available, we estimate a stochastic generalised Cobb-Douglas cost function. We have transformed the variables according to the description column in Table 3. We adopt the traditional log-log specification to allow for the possible non-linearity of the frontier. The rationale for using capital-premises and capital-investment requires justification.

Financial resources and premises are needed to develop their activity. Therefore, in order to capture the specificity of this activity, we must disentangle these two types of capital

Results

In this study, we estimate a stochastic generalised Cobb-Douglas cost function with three input prices (one price of labour and two prices of capital), and three outputs (sales, points and attendance). Linear homogeneity in input prices is imposed, restricting the parameters in the estimated function. The model is as follows:

) ( 7 int 6 5 2 4 1 3 2 1 0 it U it V it LogAtten it s LogPo it LogSales it LogPK it LogPK it LogPL Trend it ogCost L + + + + + + + + + = (7)
where PL, PK1 and PK2 are the prices of labor, capital-premises and capitalinvestment, respectively. This is the cost frontier model, known as the Error Components Model in [START_REF] Coelli | An Introduction to Efficiency and Productivity Analysis[END_REF], because it accounts for causes of efficiency controlled by the club management. The variables have been defined and characterised in Table 3. Table 4 presents the results obtained for the stochastic frontier, using the Gauss program.

The regularity conditions require that the cost function be linearly homogeneous, non-decreasing and concave in input prices [START_REF] Cornes | Duality and modern economics[END_REF]. Considering the number of observations and exogenous variables, the Cobb Douglas model with a half-normal distribution was chosen and statistically supported by the data. The error components model is then adopted [START_REF] Coelli | An Introduction to Efficiency and Productivity Analysis[END_REF]. Table 4 presents the results obtained for the stochastic frontier adopting a half-normal distribution specification for the costs function frontier. We also compute the Chi-square statistic that serves as a general specification test of adding variables to the model. Therefore, it is concluded that the addition of variables by the heterogeneous frontier model is supported by the test, signifying that the heterogeneous frontier better describes the data set. Finally, in order to decide if the frontier model is better than the cost function, the Sigma square and lambda variables of the cost frontier model are statistically significant, which means that a traditional cost function is unable to capture adequately all the dimensions of the data set.

Moreover, the random cost function specified above fits the data well, since both the R-squared value and the overall F-statistic from the initial ordinary least-squares estimation used to obtain the starting values for the maximum-likelihood estimation are higher than the standard cost function, presented for comparative purposes.

The value of parameter lambda is positive and statistically significant in the stochastic inefficiency effects. We also verify that the coefficients of the variables have the expected signs, with the cost increasing with the trend, signifying that there was no technological improvement to drive costs down in this market during the period studied.

Furthermore, the cost increases with the price of labour, price of capital-premises and attendance. These are statistically significant coefficients. However, the price of capitalinvestment and sales, despite being positive, are statistically insignificant in the standard frontier, but become statistically significant in the random frontier models. This signifies that the random frontier better captures the dynamics in this data set. The significant random parameters vary along the sample. The identification of the mean values of random parameters means that the price of capital-finance, points and attendance are heterogeneous and therefore, a strategy to control costs must take into account this heterogeneous characteristic of the sample. Hence, a common policy can be defined for the sample based on the average values of the homogeneous variables, but no common policy can cater for all of the clusters identified in the heterogeneous variables. Different policies for the different segments among the Spanish clubs by heterogeneous variables are required. The model does not identify how many clusters exist in the sample and only identifies their heterogeneous nature. However, market knowledge and other techniques can be applied to identify the clusters. The scale parameters of the heterogeneous variables are statistically significant, meaning that the heterogeneity of the variables is statistically supported.

Discussion

How do we interpret these results? First, we conclude that random frontier models describe the Spanish football league more accurately than homogenous frontier models. This is the main result of the present paper. The implication of this result is that a common (+government?) policy is unable to embrace all of the clubs, since heterogeneity exists relative to the price of capital-finance, points won and attendance.

Therefore, any policy targeting any of these heterogeneous variables has to be tailored by clusters. What is the explanation for these findings? This is an intuitive result, since In the Primera Liga, at least three clusters can be identified: the top group comprises Real Madrid and Barcelona; The middle group includes the clubs that challenge for qualification for the lucrative European competitions, or survive in the top flight; and the last cluster that consists of the weakest clubs which are destined to battle against relegation. These clusters can be distinguished from each other on the basis of the price of capital-investment, points and attendance. This finding also signifies that the Spanish clubs are relatively homogenous in terms of the price of labour, price of capitalpremises and sales. Relative to labour, we observe that competition by resources drives the market and translates into homogenous dynamics. In addition, relative to capitalpremises, a certain level of investment is found to be a pre-requisite in this market, translating into homogenous behaviour. Relative to sales, it is unexpected homogenous.

What differs substantially is the price of capital-investment, points and attendance. Why are sales homogenous, in this context? Probably because there are other factor in action in this market that forces the sales to be homogenous, despite attendance. This factor can be the municipal funds allocated to Spanish clubs, which are not taken into account in the present research, since this information is not displayed in the club accounts.

Second, the trend is positive which signifies that costs increase along the time. This is an expected result for football. Football is not driven by technology improvements; therefore a negative sign is not expected for the cost frontier. However, this result is problematic since it signifies that costs are always increasing in this activity, which results in financial distress (Ascagni and Gagnepain, 2006). Third, the lambda inefficient parameter signifies that on average 80% of the costs are imputable to inefficiency, according to the homogenous frontier. However, this value translates into 0.25 within the heterogeneous frontier, signifying that the heterogeneity translates into inefficiency in the homogenous frontier models [START_REF] Greene | Fixed and random effects in stochastic frontier models[END_REF]. Furthermore, the sigma is similar in both models, signifying that the average inefficiency changes little between homogenous and heterogeneous frontier models.

Finally, in this context, unique assets are seen as exhibiting inherently differentiated levels of efficiency; sustainable profits are ultimately a return on the unique assets owned and controlled by the football clubs [START_REF] Teece | Dynamic capabilities and strategic management[END_REF]. In addition, the strategic-groups theory [START_REF] Caves | From entry barriers to mobility barriers: Conjectural decisions and contrived deterrence to new competition[END_REF], which justifies different efficiency scores on the grounds of differences in the structural characteristics of units within an industry, explains part of the efficiency differences observed in the Primera Liga football clubs.

With regard to comparisons between this paper and similar research undertaken in other research fields, there are several models that estimate a parameters model, allowing for heterogeneity. Traditional models include the heterogeneity as an individual effect ( [START_REF] Mundlak | Empirical production free of management bias[END_REF]Farsi et al., 2005). Other Models that are continuous in the sense that they allow each club to have a different technology, estimate the model parameters allowing for heterogeneity, as does the random parameter model [START_REF] Greene | Fixed and random effects in stochastic frontier models[END_REF](Greene, , 2006) ) which is adopted in the present paper, and the local maximum likelihood estimation [START_REF] Kumbhakar | Nonparametric stochastic frontiers: A local maximum likelihood approach[END_REF]. Other methods are discrete, in the sense that they create several groups and that estimate as many technologies as there are groups, such as the latent frontier model [START_REF] Orea | Efficiency measurement using a stochastic frontier latent class model[END_REF]. Therefore this paper is only comparable to [START_REF] Greene | Fixed and random effects in stochastic frontier models[END_REF]Greene ( , 2006)).

  particular the Latin Americans, are contracted by Spanish clubs. As a result, the top Spanish teams are frequently successful in the lucrative European competitions 2 .

  analysed the incentive regulation on sports organisational training activities, disentangling technical and allocative efficiency with DEA. Haas (2003A) analysed the efficiency of the USA Major Soccer League with Barros and Santos (2003) estimated a Malmquist index for Portuguese sports organisational training activities. In summary, we find no published paper

  Having estimated two competing Cobb-Douglas models, the homogeneous Cobb-Douglas frontier model and the heterogeneous Cobb-Douglas frontier model, the likelihood test enables the selection of the most appropriate functional form, which, in the present case, is the heterogeneous frontier model. The likelihood test is a statistical test of goodness-of-fit between two competing models. It compares models with different numbers of parameters. On comparison, the likelihood test has a chi-square distribution higher for the heterogeneous frontier than the standard frontier. Therefore, it is concluded that the heterogeneous frontier model describes better the data set than the Cobb-Douglas model.

  not homogenous. There are small and large clubs, located in large cities and small towns, and so on. These visible characteristics translate into the variations in points gained in the league, wide variations in stadium attendances and different dimensions of financial debts, resulting in different clusters among the clubs.

  proposed a simple framework for the comparative evaluation of Spanish Primera Liga football clubs and the rationalisation of their operational activities. The analysis is based on a stochastic frontier model that allows for the incorporation of multiple inputs and outputs in determining the relative efficiencies and the inclusion of heterogeneity observed in the data. Several interesting and useful managerial insights and implications from the study are discussed. The general conclusion is that, on average, the heterogeneous frontier model better captures the dynamics observed in the data.The result suggests that resources, scale economies and organisational structure are major factors in determining a unit's efficiency. More investigation is needed to confirm the present research.

Table 2 .

 2 Clubs' Statistics for the Primera Liga (Averages for seasons1995/96-2004/05)

	2004

  used the conditional expectation of u it , conditioned on the realized value

		F o r
						P
									e
										e r
										R e v
										i e
										w
	of the error term	it	=	(	it v	+	u	it	)	, as an estimator of u it . In other words, [ u it E /	it	]	is the
	mean productive inefficiency for the i the sport club at any time t.
										10

Table 3 :

 3 Descriptive Statistics of the Data

	Variable	Description	Minimum	Maximum	Mean	Standard
						deviation
	Log Cost	Logarithm of
		operational cost in Euros	6.6685	8.9475	7.4633	0.4104
		at constant prices
		2000=100
	Log PL	Logarithm of price of
		workers, measured by dividing total wages by	4.61378	6.8152	5.7316	0.3782
		the number of
		employees
	Log PK1-	Logarithm of price of
	premises	capital-premises, measured by dividing	0.00453	0.3959	0.0689	0.0486
		the amortisations by the
		value of the total assets
	Log PK2-	Logarithm of price
	investment	capital-investment,
		measured by dividing	3.07E-06	2.1188	0.2438	0.3603
		the cost of long-term
		investment by the value
		of the long term debt
	Log Sales	Logarithm of the sales in Euros at constant prices	5.6367	8.3703	7.2507	0.4537
		2000=100
	Log points	Logarithm of the number of points	1.4313	1.9542	1.7216	0.0988
		obtained in the league
	Log Atten	Logarithm of the total	3.9469	4.9410	4.4003	0.2302
		attendences

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 4 :

 4 Stochastic Cobb-Douglas panel cost frontier (dependent variable: Log Cost)

	Variables	Random Frontier model	Non Random Frontier Model
	Non-random parameters	Coefficients	Coefficients
		(t-ratio)	(t-ratio)
	Constant ( 0 )	1.0380	1.194
		(5.480)	(1.442)
	Trend ( 1 )	0.0269	0.0270
		(5.709)	(2.680)
	Log PL( 2 )	0.6993	0.6809
		(19.610)	(5.232)
	Log PK1 ( 3 )	0.5401	0.5513
		(5.141)	(2.248)
	Log PK2( 4 )		0.0490
			(0.409)
	Log Sales( 5 )	0.0540	0.0521
		(2.018)	(0.461)
	Log Point ( 6 )		0.2350
			(0.793)
	Log Atten( 7 )		0.2881
			(1.694)
		Mean for Random Parameters
	LogPK2 ( 2 )	0.6022
		(3.957)
	LogPoints( 6 )	0.1975
		(2.219)
	LogAtten ( 7 )	0.3388
		(6.256)
	Scale Parameters for Dists. Of Random Parameter
	LogPK2 ( 2 )	1.4281
		(10.414)
	LogPoints( 6 )	0.0202
		(4.459)
	LogAtten ( 7 )	0.0115
		(6.453)
	Statistics of the model
	Sigma of u	0.1362	0.1225
		(29.032)	(1.079)
	Lambda	0.2532	0.8094
		(2.706)	(2.132)
		75.169	72.010
	Log likelihood
	Chi Square	144.338	132.214
	Degrees of freedom	3	3
	Probability	0.0002	0.0005
	Observations	159	159
	t Statistics in parentheses are below the parameters, those followed by * are significant
	at 1% level.

F.C.Barcelona won the European Champions League, while Sevilla C.F. won the UEFA Cup at the end of the 2005/06 season. These are the two European-level club competitions.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript