NPV Calculation

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MiningMath's internal math for NPV Calculation

The following video explains more about the NPV calculation made by MiningMath’s algorithm. The understand of these steps might be useful for users working on projects with variable mining costs, which are not yet smoothly implemented on the UI.

Video 1: NPV calculation.

Discount rate per period

The discount rate (%/year) is provided by the user in MiningMath’s interface, as depicted in the figure below.

Discounted cash flow for annual time frames

In a usual scenario period ranges are defined by annual time frames, as depicted in Fig-2.

In this case, the annual discount rate multiplier (annual_multiplier) to return the discounted cash flow is performed as follows:

$\text{annual_multiplier}(t) = $ $\frac{1}{(1 + \text{input discount rate})^t}$

The table below exemplifies one case for 10 periods.

Period	Process 1	Dump 1	NPV (Discounted) M$	Annual multiplier	Undiscounted NPV M$
1	P1	Waste	1.2	0.909	1.320
2	P1 +5%	Waste	137.9	0.826	166.859
3	P1 +5%	Waste	132.5	0.751	176.358
4	P1 +5%	Waste	105.4	0.683	154.316
5	P1 -5%	Waste	89	0.621	143.335
6	P1 -5%	Waste	92	0.564	162.984
7	P1 -5%	Waste	91.3	0.513	177.918
8	P1 -10%	Waste	52.3	0.467	112.110
9	P1 -10%	Waste	54.3	0.424	128.037
10	P1 -10%	Waste	12.1	0.386	31.384

Table 1: Example of annual multiplying factors and undiscounted cash-flows for a 10% discount rate per year. Process 1 exemplifies the use of different economic values per period.

In details, Table 1 lists:

the NPV (discounted) resulting from a 10 yearly period with a 10% discount rate per year.
the annual discount rate (annual_multiplier) for each period; and
the undiscounted NPV as the result of the discounted NPV divided by the annual_multiplier.

Discounted cash flow for custom time frames

MiningMath allows the creation of scenarios in which period ranges are defined with custom time frames (months, trienniums, decades, etc.), as depicted in Figure 3.

In this case, the discount rate is still provided in years on the interface. However, the discount rate per period follows a different set of calculations.

In the example on Figure 3, the first range is composed of 12 periods of one month each. To calculate their discount rate, a timeframe factor for each period t is defined first. This is basically the value associated with each timeframe, as depicted in Figure 3, or a custom value given by the user. Another required information is the year in which t occurs. For instance, the 2nd month occurs in year 1, while the 13th month occurs in year 2.

$ TF(t)= \begin{cases} 1,& \text{if}\, t\, \text{is in years}\\ \frac{1}{12},& \text{if}\, t\, \text{is in months}\\ 3,& \text{if}\, t\, \text{is in trienniums}\\ etc.& \end{cases} $

$ \text{year_of}(t) = \text{year in which}\, t\, \text{occurs} $

The discount rate of periods in which timeframes correspond to less than one year (low_multiplier) is, then, given by the formula:

$ \text{low_multiplier}(t) = \text{annual_multiplier}(\text{year_of}(t)^x) $

$ \text{where}\, x = \sum_{i=1}^{t}TF(i) $

Table 2 shows an example of 12 periods of one month each, with:

timeframe factor;
cumulative timeframe factors;
corresponding annual_multiplier for each period;
the low_multiplier of each period; and finally,
the undiscounted cash flow (discounted cashflow divided by the low_multiplier)

Period	Type	Process 1	Dump 1	NPV (Discounted) M$	Time frame	Years rounded	Cumulative factor	annual multiplier	low multiplier	Undiscounted NPV M$
1	Month	P1	Waste	10.8	0.0833	1	0	0.90909	0.99209	10.88612
2	Month	P1	Waste	13.6	0.0833	1	0.08333	0.90909	0.98424	13.81776
3	Month	P1	Waste	11.4	0.0833	1	0.16666	0.90909	0.97645	11.67490
4	Month	P1 +5%	Waste	12.9	0.833	1	0.25	0.90909	0.96873	13.31641
5	Month	P1 +5%	Waste	11.2	0.0833	1	0.33333	0.90909	0.96107	11.65373
6	Month	P1 +5%	Waste	13	0.0833	1	0.41666	0.90909	0.95346	13.63452
7	Month	P1 -5%	Waste	11.2	0.0833	1	0.5	0.90909	0.94592	11.84033
8	Month	P1 -5%	Waste	14.4	0.0833	1	0.58333	0.90909	0.93844	15.34467
9	Month	P1 -5%	Waste	11.8	0.0833	1	0.66666	0.90909	0.93101	12.67437
10	Month	P1 -10%	Waste	14.7	0.0833	1	0.75	0.90909	0.92365	15.91517
11	Month	P1 -10%	Waste	13.6	0.0833	1	0.83333	0.90909	0.91634	14.84165
12	Month	P1 -10%	Waste	12	0.0833	1	0.91666	0.90909	0.90909	13.20000

Table 2: Example of discount rates calculated per month and undiscounted cash flows with a 10% discount rate per year. Process 1 exemplifies the use of different economic values per period.

Timeframes greater than a year

In the example on Figure 3, the last period ranges are composed of 3 trienniums. To calculate the discount rate of a triennium, the annual_multiplier of each year contained in it is computed and then averaged. Hence, the discount rate of periods in which timeframes are greater than one year (great_mutiplier) is given by the formula:

$ \text{great_multiplier}(t) = $ $\frac{\sum_{y=\underline{t}}^{\overline{t}}\text{annual_multiplier}(y)}{\overline{t} – \underline{t} + 1} $

$ \text{where}\, \underline{t}\, \text{is the first year in period}\, t\, \text{and}\, \overline{t}\, \text{is the last year in period}\, t $

Table 3 shows the example of 3 trienniums occuring after 14 periods (12 months and 2 years). The annual_multiplierof each year contained in each triennium is listed. The average of these factors return the great_multiplier of each triennium, which, in turn, divides the discounted cash flow to calculate the undiscounted one.

Period	Type	Process 1	Dump 1	NPV (Discounted) M$	Time frame	great multiplier	Undiscounted NPV M$	Year rounded	annual multiplier
15	Triennium	P1	Waste	378.4	3	0.62280	607.57584	4	0.68301
								5	0.62092
								6	0.56447
16	Triennium	P1 -5%	Waste	120.8	3	0.46792	258.16321	7	0.51318
								8	0.46650
								9	0.42409
17	Triennium	P1 -10%	Waste	4.2	3	0.35155	11.94688	10	0.38554
								11	0.35049
								12	0.31863

Table 3: Example of undiscounted cash flows and discount rates calculated per triennium with annual discount rate of 10%. Process 1 exemplifies the use of different economic values per period.

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NPV Calculation

MiningMath's internal math for NPV Calculation

Discount rate per period

Discounted cash flow for annual time frames

Discounted cash flow for custom time frames

Timeframes greater than a year

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NPV Calculation

Timeframes greater than a year​

CONTENTS

Timeframes greater than a year