# Monte Carlo simulation

## Contents

**Introduction**

Mining companies require significant funding to begin a project, and to obtain this financing, they must often turn to debt markets. Due to the substantial investment, it is essential that mining projects be evaluated in terms of economic viability. Mine management involves improving existing mine processes to reduce overall costs and maximize the Net Present Value (NPV) of a project. The purpose of any mine management team is to create value from a project to increase wealth for shareholders who have taken risk in investing their money into a project. In order to substantiate the economic evaluation of a project, they must determine the outcomes of risks taken in their mine planning^{[1]}.

**Risk**

There are numerous risks that mining company’s encounter that could have an effect on a project’s long-term economic viability. Such risks faced by the mining and metals industry are outlined in a report by Ernst and Young. These risks include productivity improvement, capital dilemmas, social license to operate, resource nationalism, capital projects, price and currency volatility, infrastructure access, sharing the benefits, balancing talent needs, and access to water and energy^{[2]}. Due to the uncertainty and variability of these risks, it is important for project planners to extensively utilize risk analysis techniques. Risk analysis allows planners to determine all the possible outcomes of a decision regarding finance, costs, forecasting models, and project management as well as the risks associated with each. This allows for better decision making^{[3]}.

**History**

Stanislaw Ulam created the Monte Carlo method in the late 1940s on the basis of determining the probability of winning a solitaire game. He began trying to solve the solitaire problem using combinational calculations which lead him to consider how problems regarding neutron diffusion might be represented as a succession of random operations. The Monte Carlo simulation is any technique of statistical data sampling used to approximate solutions to quantitative problems. Ulam, with the help of John von Neumann and Nicholas Metropolis, recognized the potential for the recently invented computer to automate sampling. He was eventually able to develop computer software algorithms and transform non-random problems into random forms that would facilitate their solution from statistical sampling. This work has changed the way statistical sampling is viewed to a formal methodology that is able to be used in a wide variety of problems. The methodology was named Monte Carlo after the casinos in Monte Carlo^{[4]}.

**Monte Carlo Assessment**

The Monte Carlo simulation is a mathematical simulation that allows for planners to account for risk in a quantitative way (Palisade Corporation, 2015). Monte Carlo simulation is a term that describes a computer simulation that uses random numbers generated by a program^{[5]}. Unlike a sensitivity analysis which involves changing the input variables one at a time, the Monte Carlo process involves changing two or more key input variables at the same time^{[6]}.

For the Monte Carlo simulation, fixed variables are replaced with random variables that create random variability in the model. The model is then recalculated numerous times, in which the random variable changes each cycle. The results from the model are analyzed and statistics from the information are derived^{[3]}.

Changing multiple input variables creates a much more accurate prediction of project risks rather than changing a single variable. Although, the challenge involved with changing multiple variables is the extremely large number of possible variable combinations and the different amounts of variation in each. Therefore, Monte Carlo analyses are performed on custom computers in order to accommodate the large number of possible iterations ^{[6]}.

**Input Parameters**

In order to perform the computerized Monte Carlo simulation and risk analyse effectively, the following conditions must be met^{[5]}:

- The management team must provide adequate time and resources to the Monte Carlo model builders and the risk analysts.
- All the conditions that affect the outcome of the project must be taken into account as the Monte Carlo process models the physical processes of the enterprise.
- The risks analyst must define relationships between the input variables affecting the evaluation variable.
- Strong coding practices must be utilized and the model must be documented at each stage so no one aspect is excluded.
- The Monte Carlo model must be tested against reality with the suitable data, therefore last year’s data must produce last year’s result.

First the model is tested, and calculates results that are appropriate to the situation. Next, the uncertainty factors are inputted into the model. In order to determine these different uncertainty factors, it is important to gain insight from experts in the industry along with economists. Once the uncertainties are determined, the simulation planner will choose the most appropriate density function from the software. These probabilities are then applied to the model which is step tested to determine if there are any errors, and corrections are made if needed ^{[5]}.

A graph is created for each input variable, showing the range and probability of the variable occurring. This process is repeated in order to create a graph of probability and range for each of the variables that are going into the financial model. From there, the probability curves and the financial model are inputted into the computerized software program where it performs the Monte Carlo simulation ^{[6]}.

**Monte Carlo Analysis Process**

Before starting a Monte Carlo analysis, the objectives of the model must be clearly identified along with the relationships between the variables. Monte Carlo simulates the changes of factors affecting the project risks through several random samplings and calculates how these changes influence the evaluation matrix. The model uses the key risk variables that influence the target function and each risk variable has its own probability distribution. With the use of random variables, the output is sampling values based on the distribution of each variable. The function is then calculated by inputting the sample values into the model. The process is then repeated numerous times.
The Monte Carlo simulation model uses the uncertainty that is associated with the probability model and gives a statistical estimate as an approximate solution of the original problem based on the amount of random sampling tests conducted ^{[7]}.

The methodology involves using a regression analysis to establish the variable value at a given time ‘T’. This technique involves going backwards: starting at the most mature option and going back to time zero (where T = 0). The numerical method by simulation is bound to make it less computationally time consuming ^{[1]}.

**Monte Carlo Analysis for Mining**

In order to run a Monte Carlo simulation for a mining project, the number of iterations must be determined along with the number of reports that need to be examined. Once this has been determined, the model can be coded with the following coding practices adopted: no hard coded data in the logic art of the model, use variable names rather than cell references, make use of multiple worksheets, and use charts frequently ^{[5]}.
In order to determine the economic viability of a project, there are many risk assessment metrics that can be used such as internal rate of return. The most common metric in financial evaluations is the NPV. This formula is shown below ^{[7]}:

Where, CIT is the cash inflow occurring at time, T, and COT is the cash outflow occurring at time, T, and N is the life of the mining project. This formula is used to create a standard for the project to discount cash flow ^{[7]}. When evaluating a project in terms of investment risk, the metric can be used if the NPV is greater than zero, but not if the NPV is less than zero as it will not be economically viable. When the project reaches the minimum acceptable rate of return, the NPV is equal to zero. When the project can achieve a higher income level, the NPV is greater than zero. In the case that the NPV is less than zero, this means that yield rate has not met the required yields ^{[7]}.

**Software**

The most common software used for Monte Carlo analysis for mining projects is called “@Risk”. The program allows for the analysis of risks in cost estimation and project scheduling. It also allows the users to import project schedules into Excel and use both the features of Excel and "@Risk" at the same time. Typically schedule data and cost data are stored and modelled in separate programs, which can make it difficult to effectively analyze the impacts of changes in one program, on the other. "@Risk" allows analysts to link cost data with project scheduling when both are imputed into Excel. This allows for the user to see the impacts of changing cost on the schedule, or the impacts of changing the schedule on the costs. This feature is critical for determining the economic viability of a mining project. This program also allows the user to use risk registers, such as adding delays in the project scheduling or cost as the risk events occur ^{[8]}.

The output of "@Risk" software is probabilistic Gantt charts and time-scaled reports and graphs. The probabilistic Gantt charts created display the expected start date and latest finish dates for the mining project directly on the Gantt chart. It also shows the critical index for each task which gives the managers the ability to determine the importance of the different tasks. The time-scaled reports and graphs show probability distributions which show a range of possible values for each time period in a project. This allows managers to see the distributions of costs or other variables over the lifespan of a project ^{[8]}.

**Challenges**

A major challenge regarding Monte Carlo analysis is that it is very difficult to perform the simulation effectively without software such as "@Risk". Without expensive software, the modelling becomes extremely complex and takes a long time to achieve results. It is very important for analysts to use a combination of tools to create the most realistic results^{[5]}.

Another major challenge is that there is discrepancy regarding the accuracy of the analysis when forecasting since it is not always clear regarding the distribution of factors. Analysts must consider the risk when choosing an appropriate distribution law for all the risk factors ^{[7]}.

**Monte Carlo Results**

The purpose of Monte Carlo simulation is to show the conditions that can cause either adverse or very favourable results. These results are at the tails of the outputted histogram ^{[5]}. The Monte Carlo output is a series of charts and reports that outline the result variable. For example, the graphs may output the internal rate of return of a mine evaluation with a standard deviation. Additional information that is provided is the certainty of the internal rate of return to lie between specific values^{[5]}.

**Monte Carlo Application**

The following shows an example using the Monte Carlo method in the case of a new approach on project evaluation, called the Real Options Approach, which has many advantages over the NPV method.
In order to determine the put option value for a current share price is $36 and the exercise price is $40. The risk free rate is 6% and simulated paths have been created from the risk-neutral processes. The following table shows the simulated price paths ^{[1]}:

Step 1: Determine the option value price at the maturity date which is shown in the table below ^{[1]}:

Step 2: Determine if it is necessary to keep the option at time 2. If the put value at time 3 is discounted at time 2, it may be possible to create a connection between the put value and the share price both at time 2. A second order polynomial regression is used to solve this equation. The following table was used for the regression and was created using discounted put values from time 3 to time 2 and the simulated value at time 2 ^{[1]}.

In doing a second order polynomial regression, the equation below describes the put value at time 2 which is based on the simulated share price at time 2 ^{[1]}.

Y=-42.69 + 3.27X – 0.055X^2

The regression is shown in the table below ^{[1]}:

In the process of substituting X with the share price, the evaluators gets the put value for time 3. Based on this information the following decision matrix can be created ^{[1]}:

The put option will be used on the three paths that are bolded above which are path 4, 6, and 7. From this information a new profit matrix is completed for time 2 and 3 ^{[1]}.

The same process is used until the profit matrix is completed. The final profit matrix is shown below ^{[1]}:

Therefore, if all the probable class flows are discounted at time zero, the put option value will be 4.35 which corresponds to the expected gain and takes into account all simulated paths ^{[1]}.

**Other Mining Applications**

There are various other applications that the Monte Carlo simulation method can be used for in the mining industry other than project evaluation and mine scheduling, including predicting blasting fragmentation and predicting block stability.

**Blasting Fragmentation**

The cost of operations such as loading, hauling and crushing along with drilling and blasting, are directly related to rock fragmentation and are a major contributing factor to the financial viability of a mining project. There are various factors that affect blast fragmentation such as rock mass properties, site geology, in-situ fracturing and blasting parameters. Based on a case study by Mario Morin and Francesco Ficarazzo, they use Kuz-Ram fragmentation model and Monte Carlo simulation as a tool to predict blasting fragmentation. The Monte Carlo method was developed to predict the entire fragmentation size distribution and the type and properties of explosives and drilling pattern required in open pit and quarry mining ^{[9]}.

The Monte Carlo simulation for this case study was based on two approaches, which were calculating the powder factor required to obtain a certain mean fragment size and oppositely, calculating the fragment size distribution while the powder factor is held constant ^{[9]}.

In the first approach, which is making the powder factor constant, the Monte Carlo simulator uses the powder factor as an input parameter. The results from the Monte Carlo simulation were used and compared to the fragments produced from the rock quarry. In order to conduct the simulation, the triangular distribution data is required for the study, which includes the lower bound, average, and upper bound for the frequency as well as values for the unconfined compressive strength, elastic modulus, discontinuity dip, dip direction and spacing. From this information along with the mass of explosive, the hole diameter, the bench height, length of hole, and powder factor are determined. The mean fragmentation size and standard deviation were determined. The value of the Roslin-Rammler exponent was calculated along with the burden height, blast hole spacing, drill hole diameter and drilling accuracy. Based on this information and increasing the rock fragment size from 0 to 300 cm, the cumulative distribution is produced, which is outlined below ^{[9]}:

The graph above shows the predicted percent passing versus the measured field passing at the rock quarry and they are almost the same, which prove that the Monte Carlo method is valid for predicting fragmentation size in blasting ^{[9]}..

The results of the overall study, were that the Monte Carlo simulator was able to produce results comparable to the real fragmentation data obtained from a blast quarry. It is expected that using the Monte Carlo simulator in the area of blasting in the future will help industry have a better understanding of the effects of rock mass and explosive properties on the rock fragmentation by blasting. With a better understanding, there will be improvements regarding blasting operations which should decrease the current costs and overall costs for open pit mines and rock quarries ^{[9]}..

One of the challenges regarding this simulation is the amount of field data that is required as it is difficult to fit certain parameters to triangular distributions when there is a small amount of data. Additionally collecting the data can be problematic as it is very costly and time consuming ^{[9]}..

**Block Stability**

In a case study by Oleg Nikitin, the Monte Carlo simulation is used to determine room-and-pillar stability parameters for an underground mine. The Monte Carlo method determined the probability of spontaneous collapse among the pillars as well as surface subsidence by the parameters of the registered collapsed mining blocks. The study took place in the Estonian oil shale mines where the pillars are arranged in a single grid, and the oil shale bed is 40-75 m. During the study, 16 spontaneous collapses in 13 blocks and 26 collapses in 18 blocks were received ^{[10]}.

In regards to the Monte Carlo sampling, the analysis was based on the assumption that the normal distribution of the pillars cross-section area, means a potential collapse of a mining block will occur. The data required for the simulation includes the standard deviation, mean and confidence interval for the pillar square, along with the standard deviation, mean and confidence interval for exploitation depth and the level of significance. Random values were generated for exploitation depth, H, and pillar cross-sectional area, S, to be used in Monte Carlo ^{[10]}.

The following graph outlines the data outputted from the Monte Carlo simulation in terms of exploitation depth and pillar cross-sectional area^{[10]}:

Each line represents a different parameter of sizes, age, and safety factor for the room and pillar design which can be seen in the table below ^{[10]}:

The following table summarizes the data for the significance factor of 1% ^{[10]},

The assumption made of the previous graph is that pillars are stable for a long period of time when the probability of destruction is less than 5% and room sizes do not get larger than 8.5 m in width. The results of the study using Monte Carlo simulation included ^{[10]}:

- Room and pillar stability parameters
- The probability of a spontaneous collapse of the pillars.
- The probability of surface subsidence by the parameters of the collapsed mining blocks.
- The method acts as an express-method for stability analysis and failure determination and is applicable for various geological conditions where the room-and-pillar mining system is used.
- The centers of potential collapse are determined.

**Conclusion**

The economic feasibility of a mining project can be assessed through Monte Carlo simulation. The simulation will produce results regarding the expected earnings of an investment project and will create reasonable references for investors. The Monte Carlo method allows for numerous risk factors to be evaluated, but a suitable distribution law should be chosen in order to achieve the most accurate results ^{[7]}.

The major advantage of using Monte Carlo method is that computation is easier with this method. In addition, the method can easily appraise the project value using numerous uncertainties that better reflect the mining reality. The Monte Carlo model has improved significantly over time as computational programs have improved. The computational upgrades have also reduced the simulation timing significantly. Additionally, the numerical method developed by Long staff and Schwartz introduced the least-squares method approach which has the advantage of handling multiple sources of uncertainties to reduce the computational needs ^{[1]}.

All of these advancements have improved the way financial risk is evaluated in the mining industry and has allowed investors to better educate themselves in order to improve their investment decisions.

**References**

- ↑
^{1.00}^{1.01}^{1.02}^{1.03}^{1.04}^{1.05}^{1.06}^{1.07}^{1.08}^{1.09}^{1.10}^{1.11}Lemelin, B. (2009). Mine Project Evaluation: A real Options Approach with Least-Squares Monte Carlo Simulations. Quebec City: University of Laval. - ↑ Ernst and Young. (2015). Business Risks Facing Mining and Metals. Toronto: Ernst and Young.
- ↑
^{3.0}^{3.1}RiskAmp. (2014). What is Monte Carlo Simulation. RiskAmp. - ↑ Holten, G. A. (2015). The Monte Carlo Method. Retrieved from Value-at-Risk: Theory and Practice: http://value-at-risk.net/the-monte-carlo-method/.
- ↑
^{5.0}^{5.1}^{5.2}^{5.3}^{5.4}^{5.5}^{5.6}Heuberger, R. (2005). Risk Analysis in the Mining Industry. - ↑
^{6.0}^{6.1}^{6.2}Mining Man. (n.d.). Mining Financial Basics #7 - Monte Carlo Simulation. Retrieved March 2, 2015, from Mining Man: http://www.miningman.com/Blog/May-2010/Monte-Carlo-Simulation-Analysis-Financial-Basics-7. - ↑
^{7.0}^{7.1}^{7.2}^{7.3}^{7.4}^{7.5}Wei, J. (2011). Mining Investment Risk Analysis Based on Monte Carlo Simulation. Management of e-Commerce and e-Government (ICMeCG). - ↑
^{8.0}^{8.1}@Risk for Project Managers. Retrieved from Palisade: http://www.palisade.com/projectriskmanagement/. - ↑
^{9.0}^{9.1}^{9.2}^{9.3}^{9.4}^{9.5}Ficarazzo, M. M. (2005, June 20). Monte Carlo Simulation as a Tool to Predict Blasting. Elsevier, p. 2005. - ↑
^{10.0}^{10.1}^{10.2}^{10.3}^{10.4}^{10.5}Nikitin, O. (2003). Mining Block Stability Prediction by the Monte Carlo Method. Tallin, Estonia: Tallin Technical University.