• product price $10/gu
• recovery 90 per cent, and therefore
• recoverable revenue $9.00/gu
• variable cost $22.50/t
Variable cost break-even = $10/gu 90$22.50/t % # = 2.5 gu/t
and total cost break-even = $10/gu 90$60.00/t % # = 6.7 gu/t
The grade is expressed as grade units per tonne (gu/t); for example, per cent (or product tonnes per ore tonne) for base metals, or grams per tonne for precious metals. The metal price has been converted from the price per unit of metal as normally quoted to a price per grade unit to simplify the break-even calculation.
Figure 3.1 shows these break-even relationships graphically. Variable and total unit costs and unit revenues are plotted as straight-line functions of material grade. Costs per tonne are assumed to be independent of grade and are therefore the same at all grades, being $22.50 and $60.00/t, as in the data above. The revenue received increases at the rate of $9/t/gu, as calculated above. As expected, the revenue line crosses the cost lines at 2.5 and 6.7 gu/t, the break-even grades calculated.
Figure 3.2 also shows these break-even relationships graphically but this time as net contributions, positive and negative. The costs already noted have simply been deducted from the revenue. At zero grade with no revenue generated, the contribution is negative and numerically equal to the cost per tonne. The contribution based on variable costs increases linearly at the rate of $9/t/gu from –$22.50/t at zero grade. Similarly, the contribution based on total costs increases at the rate of $9/t/gu from –$60/t at zero grade. As expected, the net contribution lines cross the zero contribution axis at 2.5 and 6.7 gu/t, again the break-even grades calculated previously.
FIGURE 3.1
Graphical representation of break-evens: revenue = cost.
1 0 2 3 4 5 6 7 8 9 10 0 20 40 60 80 100 $/tonne
Grade (grade units) Variable Cost ($22.5/t)
Net Revenue ($9/t/gu)
Total Cost ($60/t) Total Cost Break-even
(6.7 gu/t)
Variable Break-even
CHAPTER 3 | Break-even Analysis
A more GenerAl definition of BreAk-even – equAl net BenefitS
The simple definition of break-even, as previously described, is the grade at which revenue obtained is equal to the costs of producing that revenue. A more general definition is the grade at which the benefit from one course of action equals the benefit from an alternative course of action.Using this definition, the concept of the ore–waste break-even becomes the grade at which the benefit obtained from treating a block of rock as ore is the same as treating it as waste. This leads us to what is commonly known as a cut-over grade, where the benefit from treating ore by one process is the same as treating it by another. Typically this is applied to two distinct metallurgical processes, one with relatively high costs and recoveries, the other with relatively low costs and recoveries.
The equal benefits break-even occurs when:
Price × grade × recovery1 – cost1 = price × grade × recovery2 – cost2 Rearranging the terms of this formula, we obtain:
Break-even grade [gu/t] = product rice (recovery recovery )p cost cost# 1- 2 1- 2
An equal net benefits ore–waste break-even calculation
Let us reconsider the break-even calculation example, the key features of which are reproduced here:
• product price $10/gu
• recovery 90 per cent, and therefore
• recoverable revenue $9.00/gu
• variable cost $22.50/t
Assume now that the variable costs comprise the following components: FIGURE 3.2
Graphical representation of break-evens: net contribution = 0.
1
0 2 3 4 5 6 7 8 9 10
$/tonne
Grade (grade units)
–80 –40 –60 –20 0 20 100 80 60 40
Variable Contribution Total Contribution
Total Cost Break-even (6.7 gu/t) Variable Break-even
• treatment $10/t of ore
• mining $12.5/t of ore
• strip ratio 1.5 t of waste per tonne of ore
(therefore 2.5 t of rock mined per tonne of ore) • thus, mining cost $5/t of rock
Therefore, the variable cost incurred by each tonne of rock if classified as:
• ore: $10 (treatment) + $5 (mining) = $15/t total
• waste: $5/t (mining only)
Break-even occurs when:
Price × grade × recoveryore - costore = price × grade × recoverywaste - costwaste Break-even grade = product rice (recovery recovery )p cost# ore-costwasteore- waste
$10$#15 5( .0 9 0 0$ . )
= - -
= $10/$9 = 1.11 gu/t
Figure 3.3 shows graphically the net benefits of classifying material as ore or waste. The net benefit of waste is negative – a cost of $5/t at all grades of material. The benefit of ore increases linearly at the rate of $9/t/gu from –$15.00/t at zero grade. At zero grade with no revenue generated, the net benefit is negative and equal to the cost per tonne of material that is both mined (at $5.00/t) and treated (at $10.00/t). The benefit- of-ore line crosses the benefit-of-waste line at a grade of 1.11, as calculated. Note that the net contribution from ore does not become positive until the grade exceeds 1.67,
FIGURE 3.3
Graphical representation of ore–waste break-evens – equal benefits.
1 0 2 3 4 5 6 7 8 9 10 –20 0 20 40 60 80 $ / tonne
Grade (grade units)
Contribution (Ore) Contribution (Waste)
Ore/Waste Break-even 1.1 1 gu/t
CHAPTER 3 | Break-even Analysis at which point the revenue covers the costs of both mining and treatment. However, given the assumption that the mine is operating, the $5.00/t mining cost is a committed or sunk cost and the net revenue from treating a tonne of rock as ore need only cover the $10.00/t cost of treatment to make it worthwhile doing so. As noted in the preceding chapter, there is no guarantee that the value obtained when using the appropriate cut- off will be positive.
Our simple break-even calculation generated a value of 2.5 gu/t, compared with the equal benefits value of 1.11. What causes this difference? The distinctions in rationale and the arithmetic of the two calculations are self-evident, but what is perhaps more important for our understanding of cut-off derivation is the recognition that mines’ cost systems often report all costs as a cost per tonne of ore, which may be satisfactory for many purposes but can be dangerously misleading when used for cut-off derivation without critical thought. As previously indicated, it is essential that costs be allocated to the physical activities that actually cause them to be incurred.
It can be seen from the derivation of the mining cost of $5.00/t of rock above that the $12.50/t total mining cost for ore is made up of $5.00 for the one tonne of rock that is classed as ore, plus $7.50 for the associated 1.5 t of rock classed as waste. The split of the costs of mining the total 2.5 t of rock between ore and waste is a function of the cut-off applied. If the cut-off were such that 0.5 t was ore and 2.0 t was waste, the cost per tonne of ore would be reported as $25.00/t. Conversely, if 2.0 t was ore and 0.5 t was waste, the cost per tonne of ore would be reported as $6.25/t. It is now evident that in the first simple calculation above, where the reported cost per tonne of ore was used in the break-even calculation, one of the main component costs used to determine the cut-off is itself a function of the cut-off. The value used is based on a prior assumption about what the cut-off is, generating the 1 t of ore/1.5 t of waste split.
This introduces a circularity into the calculation. If a set of iterative calculations were conducted, the process might eventually converge to the right answer; however, that would necessitate making use of the tonnage versus cut-off relationships for the block of rock to identify the ore–waste split at the cut-off calculated, from which one could recalculate the mining cost per tonne of ore, the break-even and so on. This iterative process is described in more detail later in this chapter.
There is no problem with this in principle, but in the development of cut-off theory we are at this stage considering break-even calculations that have been identified as 1D, considering only financial factors (prices and costs). The necessity to consider the geology, as expressed by the tonnage versus cut-off relationships, means we need to go beyond the 1D models’ limitations and bring other parameters into the analysis. At first glance, this implies that the simple break-even cut-off model is inadequate for such a situation, and so it is if costs are expressed only per tonne of ore.
By using the more general definition and calculation of break-evens, however, and by correctly allocating costs to rock and ore, the circularity of having costs dependent on cut-off in the break-even calculation has been removed. A break-even cut-off grade can be derived by a simple 1D break-even calculation.