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Effects of mineral price models on
mineral project evaluation
G
Ansong and P K Achireko
Abstract
The authors present a new mineral price m odel comp are
it with time series and naive mod els and analyze the effects of
the forecasting mode ls on mineral project evaluation. Min-
eral commodity prices are volatile which means that the
results of evaluation to ols that do not treat the stochasticity of
metal prices rigorously may be misleading . But in mine
valuation commodity price foreca sts a re required to ass ess
the economic viability of apr oje ct. In this study a new mineral
forecasting method called the MNDRVG -MFNN-RM model
which incorpo rates randomness neural networks and regres-
sion models is introduced. The MNDRVG-M FNN-RM mod el
a naive method and time series model we re used to forecast
gold prices for tw successive years for the evaluation of a
proposed open-p it mine. The MNDRVG -MFNN-RM mo del
yielded the best results among the three methods. It produced
the true optimum pit limits and an optimum pit value sligh tly
less than the true optimum value. The main novelty of the
methodology is the simulation and rigorous analysis of the
randomness property as sociated w ith mineral prices to re-
duce the estimation and forecast errors an important contri-
bution to mineral venture evaluation mine planning and
design.
Introduction
Mineral project evaluation consists of an array of analyti-
cal and judgmental techniques and processes that can define
for an investor the value, viability and uncertainty associated
with aproject in a given economy (Gentry, 1980; Gocht et al.,
1988). The evaluation provides the basis for decisions about
project acquisition, financing, taxation and regulation. Min-
eral project evaluation is interdisciplinary in nature. It re-
quires knowledge from many fields such as geology, mining,
engineering, mineral processing, economics, finance, envi-
ronmental and regulatory departments. The decision-making
combines the vision of the developer, the organizing talents of
the manager, the analytical ability of the economist and the
technical capability of the engineer, together with the math-
ematics of finance (Sprague and Whitaker, 1986). Mineral
marketing considerations are very important in determining
annual profit margins and returns on mineral project, which
form the basis for all communications relating to raising
capital for financing the project.
In mine valuation, commodity price forecasts are required
to assess the economic viability of a project and must cover a
sufficient time period to capture the trend and volatility in
prices within a mining business cycle. In the volatile economic
environment of most mineral ventures, feasibility studies and
development should respond adequately to changes that affect
mineral project value. The extent of variation in metal prices
must be dealt with in the best quantitative way possible. In
recent times, mineral markets have become increasingly so-
vhisticated. Commoditiesfrom mineral ventures often involve
intricate contractual agreements, where performance at speci-
fied quality, quantity and cost is required to attain projected
profit margins. Future uncertainties in metal prices are a
significantconcern in most mineral ventures. ~luctuat ions f
commodity prices in the spot markets can and do cause project
failure if they are not properly taken into account.
However, the projection of mineral prices in the mineral
industrv is still not standardized. There e a number of wavs
mineral prices are determined and used in mineral ventures.
Most analysts simply use mineral prices deemed most likely
to occur upon the assumption that certain events will defi-
nitely prevail. While some use current prices, others recom-
mend strongly the use of average price for the past 5 to 35
years, as that particular time period is deemed to cover a
definite business cycle (Lewis and Clark, 1967).
Basicallv. naive methods and econometric models are
some of the most popular mineral-price projection method-
ologies used in the mineral industry. However, each of these
methodologies has demerits that make them unsatisfactory for
most investment analyses in mining. Econometric models for
minerals pricing have been of limited value for the evaluation
of investment decisions due to timing (Gentry and O'Neil,
1984). Whereas mining investment requires price estimates
for more than five years where the values of the explanatory
variables are unknown, econometric models require knowl-
edge of the explanatory variables one or two periods (nor-
mally quarters or years) prior to the desired forecast date.
Therefore, the business experience and a sound understanding
of mineral economic conditions are always valuable aids to
the engineer in estimating future mineral prices.
In this study, a new mineral forecasting method called the
MNDRVG-MFNN-RM model. which incorporates random-
ness, neural networks and reg;ession modeis is introduced.
G
Ansong
is assistant professor with the Department of Accounting, Saint Mary s University, Halifax, Nova Scotia,
Canada; P.K.
Achireko
is the founder and president of Achreko Consulting Ltd., Ottawa, Ontario, Canada.
Nonmeeting paper number 99-350. Original manuscript submitted for consideration September 1999. Revised
manuscript accepted for publication October 2000. Discussion of this peer-reviewed and approved paper is invited
and must be submitted to SME prior to Sept. 30, 2001.
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The MNDRVG-MFNN-RM model, a naive method and time-
series model are used to forecast gold price for two successive
years for the evaluation of a proposed new open-pit mine.
Optimized pit layouts determination is one of the most impor-
tant tasks in the overall open pit mine-design process, which
has to be solved right at the very beginning of mine planning.
These layouts must continuously be readjusted throughout the
life of the mine due to changing database. Optimized pit limits
define the size and shape of mineable reserves and the associ-
ated waste materials to be excavated based on the technical,
economic and safety constraints. They also provide informa-
tion for evaluating the economic potential of a mineral de-
posit, project acquisition, financing, taxation, regulation and
the formulation of long-, intermediate- and short-range mine
plans. Open pit limits are also used to determine the bound-
aries within which surface structures such as processing
plants and mine offices should be located to avoid interruption
in the long-term mine plans. Lerch and
Grossman (1965)
published one of the most important mathematical algorithms
for optimizing open pit limits, based on dynamic program-
ming and graph theory. The two-dimensional Lerchs-
Grossman s algorithm is what is used to determine the pit
limits in the evaluation.
MNDRVG M FNN RM Model
The factors that determine mineral prices are very compli-
cated, largely because of their intricate interdependence. Spot
and future mineral prices are the result of such myriad factors,
which are very difficult to coalesce into a traditional math-
ematical model for calculations. More often than not, only a
subset of factors is employed to define a function to forecast
commodity prices. Because the resulting forecasting formu-
lae ignore many factors, the forecasts are often at variance
with actual data. Neural networks, however, have the poten-
tial to handle the problem of a large set of variables if
sufficient data are available to train the network.
Because of the ever-changing socio-politico-economic
factors in the world, mineral price and transaction data con-
tinually experience enormous perturbations that lead analysts
to ponder over what and how to use the data in designing
mineral forecast models. The values of the currencies in which
mineral prices are quoted change with inflation. Data on
mineral transactions in the eastern bloc, for example, were
unavailable before the collapse of the former U.S.S.R, and all
transactions between Council for Mutual Economic Assis-
tance (COMECON or CMEA) member countries were done
in the COMECON market, but the situation has changed.
Therefore, historical mineral price data must be truncated to
develop a better price-forecasting model that reflects modern
socio-politico-economic realities. The data for validating
realistic commodity price models, however, are still scant.
The paucity of data is dealt with here by employing a
multivariate normally distributed random variable generator
(MNDRVG) to generate additional data for each of the factors
that influence gold price. The data generated are used to
estimate a regression model that provides the basis for fore-
casting mineral commodity prices.
In this study, the following nine variables are identified as
affecting the world market price of gold:
annual gold production,
annual gold consumption,
annual average monthly low gold prices,
annual average monthly high gold prices,
inflation,
SOCIETY FOR MINING METALLURGY AND EXPLORATION INC.
interest rates,
gold loan transactions,
gold sales by central banks and
socio-politico-economic factors.
It is assumed that the world inflation and interest rates are
already reflected in the annual gold price and that only data
reported for periods with consistent socio-politico-economic
trend are used to forecast the future prices. Primarily gold
producers, whoearn instant cash flow by selling the borrowed
gold and repaying the loan at some point in the future out of
their gold mine production, use loan transactions. The markets
usually react negatively to news of large loan agreements
because more gold is added to the market. However, large loan
agreements are seldom. Consequently, their impact on the
overall gold price is negligible.
Sales of gold from central banks. Countries facing
particular volatility in their economic circumstances may
wish to consider the level of gold in their reserves. Lack of
clarity about central bank policies and intentions has led to
unjustified fears of large-scale and continuing gold sales,
which open the door to speculative activity that further de-
presses the gold price. Monetary authorities hold gold re-
serves to diversify their asset portfolios because a strategy of
diversification will normally provide a less volatile return
than one based on a single asset. A gold reserve is one way to
secure the economy of a country. Gold is a unique asset in that
it is no one else s liability. It is not directly influenced by the
monetary and fiscal policies of any individual country. Its
status cannot, therefore, be undermined by inflation in a
reserve currency country, nor is there any risk of repudiation
of the liability. Gold has maintained its value in terms of real
purchasing power in the long run and is thus particularly
suited to form part of central banks reserves.
How much gold to sell is a matter for countries and central
banks to decide in the light of their particular circumstances.
Consequently, it is not easy to predict and quantify the sales.
However, large gold sales by central banks are rare. Where the
impact is deemed significant the effect can be captured in the
following analysis by including the sales from central banks
in the annual gold supply (the variable
PR in the regression
described below).
Input variables to models. For the present research the
period considered to exhibit a consistent socio-politico-eco-
nomic trend and therefore appropriate as a source of data
spans the fifteen years from 1980 to 1994. The gold price is
assumed to be forecasted by a linear combination of the
following four variables:
world annual gold production,
world annual gold consumption,
world annual average monthly low gold prices and
world annual average monthly high gold prices.
It is assumed that these variables follow a multivariate
normal distributi~n.~lso, the limited data available are
assumed to be appropriate in defining the random distribution
of the commodity price.
Analysis based on information obtained from the website of
the World Gold Council.
The random variable genera tor Ghosh and Kulatilake, 1987 ,
which was emp loyed in this study, assumes normality.
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GP.
nput
R,
Data
.
H.
L
M N D R V G
generation of more data)
Pr ice Forecast
with use of MR M)
Figure 1 low diagram for price model.
Input
Hidden Output
layer
layer layer
Figure
2
Multilayer feed-forward neural network.
MNDRVG Model
To obtain high accuracy and realistic estimates from the
MFNN values, additional data are needed to train the network.
Also, more data are required to estimate the regression model.
The method employed in generating the extra data is the
MNDRVG model, which takes into account the uncertainty of
the factors and the spread of the data about the mean. Below
is a succinct description of the methodology involved:
Theory and algorithms for generation
A q-dimensional random variable Z
=
(ZL,
..
Zq is said to
be distributed as a q-variate normal distribution with means
given by them vector and covariances given by the covariance
matrix x f the joint density f(Z) is given by Eq. (1) (Rao,
1973).
where
A
is a positive definite matrix and
IA is the determinant of inverse of E
Zandx an also be expressed respectively in Eqs. (2) and 3)
as
where
is a q x 1 vector of independent univariate normal
variables, each with mean zero and unit variance; and
L is a lower triangular matrix satisfying Eq. 3).
The production of operating and prospective new mines
can be estimated from their production schedules. A careful
and detailed study of consumption trends in end-use markets
can permit enlightened estimates to be made about likely
levels of demand. The world annual gold production and
consumption are always available in books such as the Cana
dian Mineral Y ear Book. Average annual monthly high and
low gold prices depict the range of fluctuation of the average
annual gold price in a year. Hence, the variability of the
average annual monthly high and low gold prices with average
annual gold price is of statistical significance. A multiple
regression model consequently expresses the sum total of the
relationship between average annual gold price and the afore-
mentioned four variables.
MNDRVG MFNN RM Gold Price Modeling
The stochastic gold price is modeled via three main mod-
els, namely, through the use of a multivariate normally distrib-
uted random variable generator (MNDRVG) (Ghosh and
Kulatilake, 1987), multilayer feed-forward neural networks
(MFNN) and a multiple regression model (RM) as depicted in
Fig. 1.
The MNDRVG model is used to generate additional data
for each of the factors that influence gold price. The generated
data are used as input for the multiple regression model to
estimate the coefficients of the respective factors. The MFNN
model is used to predict the average annual monthly low and
high gold prices, which are stochastic and unavailable during
project evaluation. The predicted average annual monthly low
and high gold prices, and the world annual production and
consumption during the predicting year are substituted in the
multiple regression equation to estimate the mineral price.
Ang and Tang (1984) have recommended Box and Muller's
(1958) method to generate random values for normally dis-
tributedrandom variables. Box and Muller (1958) have shown
that if
U l
and
U2
are two independent standard uniform
random variables, then the functions in Eqs. (4a) and (4b)
constitute a pair of statistically independent standard normal
variates. The relationships given in Eqs. (4a) and (4b) are used
to obtain a total number of q independent univariate standard
normal variates from independent standard uniform variates
to form (Ghosh and Kulatilake, 1987).
Once the mean and variance-covariance matrix of the
observed data of the multivariate factors is computed, Eqs.
(4a) and (4b) are used to generate the desired number of data
points.
MFNN Model for average annual monthly high
and low gold prices.
The method of MFNN is used to predict the average-annual
monthly high and low gold prices. MFNN has many identical
nodes, with computational features that enable it to transform
perceived signals into new transmittable signals. MFNN can
learn and implement arbitrary complex inputloutput map-
pings or decision surfaces separating pattern classes. Figure 2
illustrates the MFNN and its components. The network learn-
ing is based on repeated representations of the training samples.
In a MFNN, the nodes are grouped into input, hidden and
output layers by the network. The output layer yields the
computation results. The hidden or intermediate layers follow
the input layer.
Xp
= x x x2 ... x . is the presented input
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pattern,
w
s the weight from node
i
to node
j
and 0 . is the
actual outputs for pattern s on nodej.
For the neural network theory and algorithm, we recom-
mend Rumelhart et al., 1986 and Beale et al., 1990 as a good
source.
R
Regression model
In standard econometrics and regres-
sion analysis (Pindyck and Rubinfeld, 1991 Hamilton 1994),
the multiple linear regression model can be written as
where
Y is the N
x
1 vector of observations of the dependent
variable,
Xis the N x k matrix of observations of the k-independent
variables,
p i s the k
x
1 vector of parameters and
is N x 1 vector of observations for the disturbance term.
Given the (usual) classical assumptions of the linear model,
the best3 forecast of Y conditioned on X, is P Y, where P =
X(X?X)-I XTis the orthogqnal projection of Y onto the span of
X. PY is also written as XB where
P=
(XTX)-IX?Y is the least-
squares estimate of p. In this paper, Y = GP = gold price, and
X= 11, PR, C,H L] is the set of independent variables defined,
respectively, as the constant term, the worldannual gold price,
annual gold production, annual gold consumption, average-
annual monthly high gold price and average-annual monthly
low gold price, so that the model is given by:
where
is an error term and
P1,p2 p3 p4and p5are the regression coefficients (PI is a
constant term)4.
Naive method
There are different types of naive models used in mineral
price forecasting: no-change model same change model and
average-price-over- business-cycle model. In the no-change
model, the spot price at any given time is assumed to be as
good as any future price estimate. With the same-change
model, future prices are estimated by using regression tech-
niques to fit a linear trend to historical data. The average-
price-over- business-cycle model stipulates that the average
price for the past 25 to 35 years is the best estimate of the future
price, as that particular time period is deemed to cover a
definite business-cycle (Lewis and Clark, 1967).
Time series
A very popular method for modeling stationary time series
is the autoregressive integrated moving average (ARIMA)
method, popularly known as the Box-Jenkins (BJ) methodol-
ogy. The BJ method involves fitting ARIMA(@,
I?
A) to a
The best linear unbiased estimator is (BLUE).
Th e choice if linearity is based o n parsimony. It turns out
howeve r that the linear specification and least-squares are
sufficient to produce very good forecasts. Th e econometric
literature is replete with methods that deal with situations
where spec ification is not appropriate and where the classi-
cal assumptions are violated; fortunately howeve r it was
not necessary to apply su ch methods.
Figure
3
lock grades for two-dimensional section (glt).
time series data, where @denotes he number of autoregressive
terms,
r
the number of times the series has to be differenced
before it becomes stationary, and A the number of moving
average terms. Given the values of @ r, and A, one can tell
what process is being modeled. The first step in applying the
BJ method is to identify the appropriate values of @
T
and A.
Correlogram and partial correlograrn aid in this task. The next
step is to estimate the parameters of the autoregressive and
moving average terms included in the model. This task is now
routinely handled by several statistical packages. In this
paper, the TSP statistical software package is used. After
estimation, the next step is forecasting. One of the reasons for
the popularity of the ARIMA modeling is its success in
forecasting. In many cases, the forecasts obtained by this
method are more reliable than those obtained from the tradi-
tional econometric modeling, particularly for short-term fore-
casting. Of course, each case must be checked. In the present
case, to preview the results, the ARIMA forecasts were not as
successful as the forecasts from the MNDRVG-MFNN-RM
methodology introduced in this paper.
Data and information for validation of mineral
prices modelsand evaluation of mineral deposit
In this work, gold prices obtained from all the mineral price
models are used in the evaluation using data from an actual
gold mine. The Lerchs-Grossmann algorithm is applied to a
two-dimensional section of the Star Gold Project5 o evaluate
its economic potential and the results are compared.
Optimized pit limits are required for the two-dimensional
section of the Star gold deposit, depicted in Fig. 3. The block
dimensions are 8 x 8 x 8 m (26 x 26 x 26 ft) and the weighted
tonnage factor of material is 2.76 t/m3 (172 lb/ft3). The mill
recovery efficiency is 95%. The average cost for mining ore
and waste, the pit wall slope and the administrative overheads
are $4.45/t ($4.04/st), 45 and 6%, respectively, on gross
revenues. The coordinates of all the blocks at their centers of
gravity and grades are depicted in Fig. 3. Furthermore, accord-
ing to the mining statistics (Canadian Mineral Yearbook), the
production of gold would increase by 3% of 1994 production,
The actual name and location of this project cannot be revealed
for confidential reasons.
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Table
1
old Prices data from 1980-1994.
Average
Monthly Monthly annual gold
Produc tion PR), Consumption C), high
(H),
low
L),
price
GP),
Year t t US US US
Price forecasting using the
models
MNDRVG-MFNN-RM Model. Fif-
teen data points from 1980 to 1994, re-
ported in Table 1 (Kitco Minerals and
Metals, 1997) for world annual gold pro-
duction, annual gold consumption, aver-
age annual gold prices, and average-an-
nual monthly high and low gold prices,
are used as input data in the MNDRVG
model to generate 1,000 sets of data via a
FORTRAN 77 program. The means of
the determinant variables of the actual
and generated data, depicted in Table 2,
are close to the actual parameter values
used for the generation.
From the generated data, world annual
gold production andconsumption are used
as inputs, while the world average annual
monthly high and low gold prices are
used as outputs to train the neural net-
works depicted in Fig. 4. After training,
world annual gold production and con-
as input in the RM model in which the
Actual Data 1,638.62 1,908.64 466.23 354.08 400.59
world average-annual gold price is re-
Generated D ata 1,642.09 1,914.40 467.03 354.61 401
gressed on the world annual gold produc-
sumption of the desired forecasting year
Table
he means of actual and generated data.
Average Average
Produc tion, Consumption, mon thly high, month ly low, Price,
Type of data t t US US US
Annual Gold
annual gold production and annual gold
Production
P
vera e Annual Monihly
production in Table 1 (1980-1994) as
Annual Gold
High to l d Pr ice
input, the monthly high and monthly low
Consumption
Average Annua l
Monthly
gold prices were predicted to be
Low Gold Price
US 388.43/ozandUS 369.04/0~,espec-
tively. The error and momentum term for
n p u t
Hidden O u t p u t
complete training were 0.01 and 0.36,
layer layer layer
respectively. According to Eq. (7), the
gold price is forecasted to be US 383.25/
oz. The value obtained for R-squared is
Figure 4
FNN model for average-annual monthly high and low gold price
99.1%.6 This means that the regression
prediction.
model is able to explain 99.1 of the data
generated, indicating a high degree of
consum~tionwouldincreaseb~
%of 1994consum~tion.The
accuracy for the model. When all the original 15 data sets for
identified S O C ~ O - p ~ l i t i ~ ~ - e ~ ~ n ~ m i ~ycle data for use in the
the independent variables used to generate the 1,000 addi-
mineral-~riceorecast model spans the period 1980 to 1994.
tional data are substituted in the multiple regression equation,
Each mineral price model is used to forecast gold prices for
1995 and 1996,and these forecasts are then used to evaluate the
Tech nically the R-squared referred
to
here is the adjusted R-
economic viability of the mineral deposit depicted in Fig. 3.
squared.
are used as inputs to predict the corre-
sponding world average-annual monthly
high and low gold prices.
The 1,000generateddata are now used
tion, annual gold consumption, average-
annual monthly high gold price and aver-
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INC.
Table
3 M
results.
Estimated
Variable name Variable Coefficient
coefficient
Error
t-statlstic
Constant P P 19.390 7.838 2.474
Production PR PP 0.01 1 0.008 1.424
Consumption C P -0.009 0.006 -1.522
Monthly High H P 0.349 0.01 5 23.645
Monthly Low
L
s 0.617 0.038 16.104
R2 0.991
F 27,206.7
age-annual monthly low gold price. From
TSP statistical software, the estimated
coefficients, R-squared, F- and t-statis-
tics obtained from the RM for all the four
parameters in the model are as tabulated
in Table 3.
Equation (7) is the resulting equation
using the regression coefficients in Table 3.
Gold Price
=
19.390 0.01 1PR
0.009C 0.349H 0.617L
(7)
Using the MFNN model illustrated in
Fig. 4, with the 15 data values of world
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it is found that the regression model predicts the gold price
with very high precision. The results are as depicted in Table
4. The actual prices and the predicted ones are almost the
same.
The estimated regression, Eq. (7), is used to forecast the
gold price for 1995, which is outside the estimation period.
The results, which are presented in Table 5, are based on
multiples of 50 data points. This allows one to investigate the
effect of the number of generated data on the forecast. The
value of R~ for each multiple of 50 data points was approxi-
mately 99%. The 1995 actual data7 on world annual gold
production, consumption, average-monthly high price, aver-
age-monthly low price and average-annual price were 2272.10
t, 3008.00 t, 391.03, 376.64 and 384.17, respectively8.
The results in Table 5 indicate that the price of gold does not
vary significantly with the increase in the number of data
generated.
For the given data set, the least error, 0.24%, of the
predicted price is obtained with 700 generated data. This is
closely followed by 650 and 600 generated data that have
prediction errors of 0.25% and 0.26%, respectively. Hence,
600 to 700 generated data are enough to give an accurate
mineral price forecast. However, in forecasting mineral prices
for mineral project evaluation, the RM of the number of
generated data that has the highest R-squared value and the
least error in forecasting the actual prices used in constructing
the RM should be used. The correlation between the predicted
and the actual price data used in the model is approximately
99%. When the observed fifteen data points for the socio-
politico-economic cycle were used in constructing the RM
model, the prediction error was 2.73%. Thus, expanding the
data points using the MNDRVG model provides a more
accurate result than using only the actual fifteen data points in
the forecast model. In addition, with only the fifteen observed
data points, it would be impossible to train a neural network
to obtain accurate MFNN predictions for average-annual
monthly high and low gold prices to substitute in the RM
model for prediction.
Time series and naive method
Data for gold prices spanning the period 1833 to 1994 were
used in the time series model to forecast the mineral prices for
1995 and, data spanning the period 1833 to 1995 was used to
forecast the 1996 gold price. The results are shown in Tables
6 and 7, respectively.
In using the time series model, it turns out both
ARIMA(0, 1 0) and ARIMA(4,0,0) yielded virtually the same
forecasts, and these forecasts, from Tables 6 and 7 are very
good because they are close to the actual values. The fact that
a simple random walk fits the data with a small measurement
error, which is what
ARIMA(O,l,O) describes, is consistent
with the results from the huge literature on stock market
prices. The literature indicates that stock prices essentially
follow random walk stochasticprocesses with a drift (Campbell
et al., 1997).
In forecasting the mineral prices using the naive model, the
averages of annual-average gold prices for
25,30 and 35 years
yield, respectively, US 293.64, US 243.91 and US 220.06.
From Canadian Mineral Year
Book
published by the Min-
istry of Natural Resources.
8
Note that the figures for the average monthly high and low
prices were generated from the neural network methodolog y
previously described. This is because it is assumed that at
the time of the forecast these prices would not be known.
Table
ctual and predicted prices.
Actual price Model predicted price
Year US US
1980 61 4.38 614.78
1981 459.22 469.24
1982 375.52 369.81
1983 423.52 428.95
1984 360.63 347.42
1985 317.35 31 3.42
1986 367.58 371.93
1987 446.66 437.58
1988 436.45 434.84
1989 381.27 384.64
1990 383.72 380.80
1991 362.34 371.82
1992 343.86 346.13
1993 360.06 361.45
1994 384.1 384.88
Table 5
rediction of 1995 average-annual gold price
using model.
Number of 1995 price, Predicted price, Error, Error,
data US
US
US
50 384.1
381.89 2.28
0.59
100 384.17
383.04 1.13 0.29
150 384.1 382.66
1.51 0.39
200 384.17 382.72
1.45 0.38
250 384.1
382.21
1.96 0.51
300 384.17 382.50 1.67
0.43
350 384.1 382.61 1.56 0.41
400 384.17 382.93
1.24 0.32
450 384.17 383.00 1.17 0.30
500 384.17 382.96 1.21 0.31
550 384.17
383.1
1.07 0.28
600
384.1 383.19
0.98 0.26
650 384.17 383.20 0.97 0.25
700
384.1 383.25 0.92 0.24
750
384.1
383.05
1.12 0.29
800
384.17 383.02 1.15 0.30
850
384.17 383.01 1.16 0.30
900
384.17
382.91
1.26 0.33
950 384.1 382.95 1.22 0.32
1 000 384.1 382.96 1.21 0.31
Table 6 Models results for 1995 prediction.
Time series, Naive method,
US US
Observed
384.17 384.1
Predicted
380.82 293.64
Table
7
Models results for 1996 prediction.
Time series, Naive method,
US US
Observed
387.69 387.69
Predicted
380.98 307.57
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Figure 5 -The optimum pit limits using actual price.
Figure
7-The optimum pit limits using time series model.
Figure 6
The optimum pit limits using MNDRVG-
MFNN-RM.
Figure
8 -The optimum pit limits using naive method.
Table 8 Models results for 1995 prediction.
MNDRVG-MFNN-RM, Time Series, Naive Method,
US US US
Observed 384.1 384.1 384.1
Predicted 383.25 380.82 293.64
Pit Value 289 130
286 260 167 190
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62
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EXPLORATION.
INC.
Project evaluation of the open pit mine
The economic block value of each of the gridded-blocks in
Fig.
4
is calculated using each of the mineral prices forecasted
by the three models. Then two-dimensional Lerchs-
Grossmann's algorithm was applied to find the optimum pit
limits and optimum pit value. The actual average annual gold
price in 1995 was also used in finding the optimum pit limits
and optimum pit value. The average annual gold prices for
1995 and 1996 forecasted by all the three models and their
respective optimum pit value are summarized in Tables
8
and
Table
9 Models results for
1996
prediction.
MNDRVG-MFNN-RM, Time Series, Naive Method,
US US US
Observed 387.69 387.69 387.69
Predicted 386.93 380.98 307.57
9. The economic block values in Figs. 5 to 8 are in thousands.
Using the actual average-annual gold price for the year
1995 resulted in an optimum pit value of 290,190.00. The
pits outlines for each mineral price model used in the evalua-
tion are as shown in Figs. 5 through 8. The bigger-headed
arrows on the diagrams indicate the path of the optimum pit
outline.
As depicted in Figs 5 through
8
the MNDRVG-MFNN-
RM model yields an optimum pit outline that is equal to the
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actual one and an optimum pit value which is slightly less than
the true optimum value for the 1995 year. The time series
model also yields an optimum pit outline that is similar to the
actual one but optimum pit value less than that produced by
actual price and MNDRVG-MFNN-RM model. The naive
mineral price model yields optimum pit outlines and value
entirely different from the true one, and those produced by
MNDRVG-MFNN-RM and time series model. The naive
model predictions are abysmally lower than the actual prices
as well as the values produced by MNDRVG-MFNN-RM and
time series, irrespective of the number of years used in
averaging. The averages of annual-average gold prices for 25,
30 and 35 years yielded are, respectively,
US 293.64, US
243.91 and US 220.06. As illustrated in Fig. 5, the true
optimum pit value is 290,190
onclusion
The results obtained by the gold price model indicate that
a realistic gold forecasting can be made with an identified
socio-politico-economic cycle data. The data can be aug-
mented and used in the model to give the desired forecast price
with the highest precision. Based on the results, it can be
inferred that world annual gold production, annual gold con-
sumption, average-annual monthly low gold prices and aver-
age-annual monthly high gold prices are critical forecasting
variables of mineral prices. Even though the study is done
with average-annual values, it can be adapted to daily, weekly
and monthly values if desired.
As seen from the study, the MNDRVG-MFNN-RM model
yields true optimum pit limits and an optimum pit value
slightly less than the true optimum value. This provides a very
good mineral price forecast that can be used in mine invest-
ments evaluation. Even though the time series also produces
a true optimum pit outline, it yields
n optimum pit value that
is less than the true optimum pit value. The naive model yields
both nonoptimized pit limits and values, and, hence, it should
not be used in mine investments evaluation. In fact, the naive
model does not solve the problem of which number of years'
prices to average and use in mine investment.
The different optimum values produced by each mineral
price method generates capital acquisition and project financ-
ing problems that can crop up if the naive and time series
methods, which produce lower mineral prices and hence
lower optimum pit values, are used. Consequently, improper
definition of the venture may result in higher than expected
capital investment requirements resulting in complete dissat-
isfaction of project financiers who may decide against putting
in more money. Ventures developed with huge capital up front
may be severely debt-ridden as a result of unexpected higher
interest rates. Underestimation of project capital and operat-
ing costs may result in huge project cost overruns that could
cripple a project's viability.
With high uncertainties in metal prices in today's markets,
investors in mineral projects will be misled by the results of
evaluation tools that do not treat the stochasticity of metal
prices rigorously. The MNDRVG-MFNN-RM price model is
set up to deal rigorously with metal price uncertainties in
mineral project evaluation, and this presents one of the most
viable metal price forecasting methods for the economic
evaluation of mineral deposits. The results from the model
show that data of a particular socio-politico-economic cycle
are efficient for price modeling. Indeed, understanding the
socio-politico-economic situation is of prime importance in
mineral price forecasting. Furthermore, in forecasting min-
eral prices for mineral project evaluation, one should use the
regression equation for the number of generated data that has
the highest R2 value and the least error in forecasting the actual
prices. Fluctuations of commodity prices in the spot markets
can cause project failure if not handled with care.
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