# 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 Carlow 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 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]}.

**References**

- ↑ 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}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}Wei, J. (2011). Mining Investment Risk Analysis Based on Monte Carlo Simulation. Management of e-Commerce and e-Government (ICMeCG).