In order to address the perceived shortcomings of the classic deterministic approach towards the capital valuation of assets within an organisation, the issue of quantifying the stochastic valuation of the discounted cash flows was presented
earlier, by means of the introduction of the real options approach. In order to address the perceived lack of modelling uncertainty into the classic discounted cash flow horizon of the asset in question, several researchers (Raychaudhuri 2008; Rozycki 2011; Platon & Constantinescu 2014; Samis, Davis & Laughton; Shaffie & Jaaman 2015) have identified the Monte Carlo Simulation (MCS) approach as a means of valuing probabilistic risk into the capital valuation of assets.
Lifland (2015:56) stated that by coupling the DCF and MCS together, a closer approximation of the true NPV will be realised as a result of the incorporation of the appropriate stochastic variables. Raychaudhuri (2008:91) defined the Monte Carlo simulation as a type of simulation that relies on repeated random sampling and statistical analysis to compute the probabilistic outcome of the future events under investigation. This method of simulation can be very closely associated to random experiments in which the specific result or outcome is not known in advance. Monte Carlo simulation can therefore be considered as being a methodical way or means of carrying out so-called what-if analyses. Brealey, Myers and Allen (2011:249) recognised the value adding benefit of employing the Monte Carlo Simulation approach by stating its advantages over the traditional application of sensitivity analysis. Monte Carlo simulation enables one to simultaneously consider all the probabilistic outcomes for a given scenario by including all the respective independent variables as opposed to a sensitivity analysis which only considers the manipulation of one independent variable at a time.
Dhiensiri and Balsara (2014:95) shared these sentiments and stated that the Monte Carlo simulation is a powerful analytical tool which provides more useful information compared to conventional sensitivity analysis or scenario analysis. It is therefore argued that the Monte Carlo simulation provides an unbiased estimation of the expected NPV outflows on a probabilistic basis and also takes cognisance of the interaction of the respective independent variables comprising the cash flow model under consideration.
The Monte Carlo simulation can therefore be succinctly defined as being an analytical technique that employs random numbers and probability distributions to investigate or study problems that involve uncertainty. In order to obtain a basic
understanding of the mathematical principles involved in Monte Carlo simulation, a brief overview of the core mathematical principles involved are explained as follows: Consider a function ( ) where is a variable with a probability density function . The expected value of:
( ), ̅
can be evaluated by the following integral where subscript denotes all possible values of .
( ) ( ) = ̅ (2.34)
In reality however, it may be mathematically difficult or impossible to derive directly the value of , however, the Monte Carlo method can be used to obtain an estimate of ̅ by randomly generating the values of , (where ranges from 1 to ), according to the probability density function, ( ). Then, the estimated value of ̅ and can be obtained through:
1
( ) (2.35)
If s are properly generated according to the probability density function ( ), then should provide an unbiased estimator of ̅, according to the law of large numbers. In addition to this, we can obtain the variance of the estimate from equation 2.36.
=( − 1)1 [ ( ) − ] (2.36)
In addition, according to the Central Limit Theorem, the distribution converges to a standardised normal distribution as increases.
− ̅ √ − 1
(2.37)
Raychaudhuri (2008:91) explains that in a Monte Carlo simulation, a statistical distribution is obtained for each of the independent variables, viz., input parameters of the discounted cash flow model. Thereafter, numerous random samples from each distribution which represents the input values of the model are obtained. For each input parameter, a set of output parameters are obtained. The value of each
output parameter is obtained from the outcome of a number of random simulations that are carried out. Once the output variables have been obtained, statistical analysis is carried out on the aforementioned results. In addition to this aspect can sampling statistics of the output parameters is used in order to characterise the output variation. Given the mathematical complexity and the number of iterations required to perform a reliable Monte Carlo analysis of the respective data, a statistical software package (XL STAT) has been employed in order to carry out the required analysis.
Dhiensiri and Balsara (2014:98) concluded by stating that Monte Carlo simulation is a powerful analytical tool that has a wide range of application in finance and economics and stated three major advantages of incorporating the aforementioned as follows:
(i) Monte Carlo simulation can be employed when in a situation where the expected outcome is difficult or impossible to achieve;
(ii) Monte Carlo simulation also provides an expected range of possible outcomes to a given probabilistic scenario; and
(iii) Monte Carlo Simulation provides an unbiased estimation of the expected NPV and the other key input variables.
In essence, Monte Carlo simulation is utilised to model components of the asset’s cash flow that are impacted by uncertainty. The result of the Monte Carlo simulation of both the actual and future cash flows is that one can evaluate the range of possible outcomes in respect of the predicted net present value (NPV’s) and the corresponding volatility associated with the asset in question. Based upon these findings, one can therefore ascertain or estimate the probability that the NPV will be greater than zero.
Notwithstanding the fact that numerous advantages as discussed earlier are evident with Monte Carlo method, some disadvantages are also evident and need to be taken cognisance of. Zhao (2013:26) highlighted some of the disadvantages as follows:
(i) Certain simulations can be very complex. Given the nature of the independent variables being considered;
(ii) Obtaining the required probability distributions functions for the variables being modelled can be a challenge; and
(iii) Solutions are not exact since the accuracy depends upon the number of repeated simulations employed in order to produce the required output.