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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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SOCIETY FOR MINING, METALLURGY, AND EXPLORATION, INC.



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.

VOL. 308 TRANSACTIONS 2000



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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SOCIETY FOR MINING, METALLURGY, AND EXPLORATION, INC.



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.

SOCIETY FOR MINING METALLURGY. AND EXPLOR ATION INC.

59

VOL. 3 8 TRANSACTIONS

2



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-

TRANSACTIONS 2 VOL. 3 8

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SOCIETY FOR MINING METALLURGY AND EXPLORATION

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



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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TRANSACTIONS 2



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

TR NS CTIONS 2

VOL. 3 8

62

SOCIETY FOR

MINING. METALLURGYAND

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



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.

References

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