The Flaw of Averages in Mine Project Evaluation

By Dr. Luis A. Martínez Tipe, PhD Director General & Principal Researcher, CAIDTech Originally published: September 3, 2015

Fuente: Savage, S. (2000). The Flaw of Averages. reproduced with permission.

INTRODUCTION Traditionally, mine organisations use various types of quantitative methods to estimate profit and loss associated with a proposed mine project. Among all these measures of profitability, the Net Present Value (NPV) that is based on the Discounted Cash Flow (DCF) technique – which is based on expected values and a risk adjusted discount rate – is the most widely used in the mining industry.

One consequence of using expected values when estimating cash flows is that the resulting NPV value is also assumed to be an expected value, which may not be reflecting the real value of the project, leading to incorrect decisions.

The problem with traditional evaluation techniques based on the DCF is that in cases involving uncertainty and non-linear processes, in our case the mine optimisation/evaluation process, single estimated (average) values are often of little use because of their lack of accuracy in describing an uncertain process over time.

In other words, as it is shown in Figure 1, serious trouble can arise when a single number is substituted for a distribution of probabilities. That is, if the expected value, E{X}, of the uncertainty variable, X, is input into the non-linear process F(.), the resulting output, F(E{X}), will not be the same as the expected value of the resulting outputs, E{F(X)}, generated by inputting the entire distribution of values, i.e., F(E{X}) ≠ E{F(X)} (In finance this process is known as the Jensen’s inequality).

Figure 1. The mining view of the Jensen’s inequality indicating the flaw of averages.

Professor Savage, from Stanford University, refers to this problem as ‘the flaw of averages’ (Savage, 2002a, 2002b, 2003), which states that plugging average values of uncertain inputs into a non-linear process does not result in the average value of the process, i.e., F(E{X}) ≠ E{F(X)}. He explains this concept with the following example (see Figure 2):

"Consider the state of a drunk, wandering back and forth on a busy highway. His average position is the centreline of the highway. Therefore it appears that the state of the drunk at his average position is alive. However, it is clear that the average state of the drunk is dead."

Figure 2. The drunk guy who does not know about the flaw of averages.

A ROAD TO IMPROVEMENT IN MINE PROJECT EVALUATION An analogous situation happens when evaluating a mine project. That is, when evaluating a mine project it is common to use expected single (average) values for representing all the mine variables that are input into the non-linear mine optimisation process. The final output of this practice is a single estimated value for each of the project indicators, which are assumed to be the average values to be obtained – which is not totally true.

Despite current technology have been developed to overcome the complexity of the mine evaluation problem, showing to be very efficient in dealing with a specific part of the problem – specially in the day by day operation , or short term period, it still have not been able to solve the complete problem, i.e., considering, appropriately, all sources of uncertainty. The reason that current technology cannot solve the mine problem evaluation appropriately is that these techniques have been developed in isolation, and are based on average values as input.

If we think a bit about the probability of occurrence that all the input values, e.g., grades, prices, costs, recoveries, etc., used when evaluating a mine project will happen at the same time and over the life of the mine, we will realise that it is, for sure!, less than 100%. So we know there is uncertainty but decide to ignore it, because we prefer to deal with numbers instead of distribution of probabilities.

Uncertainty should not be a “scary” word, and consequently ignored when evaluating a mine project, but a “must” to include and analyse, understand, and manage in order to maximise project value – we just need a strategy for this.

Remember uncertainty not only brings risk but also opportunity for improvement.

For example, I could ask you all (assuming there are more than one people reading this article and considering an even number of readers) to go into a bet with me, for $0.5, where I can predict if the price of gold can go up or down tomorrow. If I am wrong I will pay each of you $50, but if I am right each of you will pay me $50.  Assuming you accept the bet, I have two strategies to follow:

1. Provide you all with a prediction, let’s say the price will go up, and wait till tomorrow to see if I was correct or wrong. In this case, the uncertainty for me is big, I could say 50/50 chance (assuming a pure random movement of the gold price) to either win or lose a lot of money. Definitely will not do this.

2. Now, instead of doing one prediction I will do the following. To one half of you, I will tell the price of gold will go up, and to the other half I will tell the price will go down. Now, my uncertainty has been minimised – the reason for this is that with this strategy I do not care if the price tomorrow goes up or down – this is because whatever happens tomorrow with the gold price one half of the readers will pay the other half, however I still will do some money, i.e., the money you all gave me for entering into the bet. In finance this is known as hedging.  

   
   

Evaluating a mining project is also a sort of bet we get into where at specific periods (normally yearly periods) we have to predict ore tonnes, metal productions, recoveries, costs, prices, and other economic and operational targets. The true is that we do not know with certainty what will happen at the end of each year, but we could implement strategies (see for example Figure 3) that could protect us from the risk of not achieving our targets, while opening opportunities for improvement.

Figure 3. Example of a strategy map over time.

SOME COMMENTS – DEALING WITH UNCERTAINTY IN MINING

Recognising that uncertainty will be always present in a mine operation, it is important for the mining industry to adopt a more proactive approach when evaluating mining projects, instead of a reactive one. The early stages of the planning of a mining project (i.e., before completion of the feasibility study) usually offers the best possibilities to explore alternatives, assess the risks and implement strategic changes in order to minimize total (operating) costs project. To achieve this, the mining industry needs to:

• Switch (upgrade) the way of thinking from numbers to distribution of probabilities, and

• New technology (i.e., software tools and processes) able to manage uncertainty, appropriately, when evaluating mining projects.

  
  

The Integrated Valuation/Optimisation Framework (IVOF), is a holistic process/tool that considers uncertainty when evaluating mine projects. This is because the IVOF is an advanced data analytic process – engine that uses advanced quantitative simulation – optimisation and real options techniques. Editor's note: This article was written in 2015. Since then, CAIDTech has developed and applied the probabilistic frameworks described here across multiple mine projects in Latin America and Australia, integrating geological variability, operational dynamics and economic uncertainty into a single quantitative model. Learn more at [caidtechnology.com]