TO OPEN OR NOT TO OPEN—OR WHAT TO DO WITH A CLOSED COPPER MINE

Journal of Applied Corporate Finance W I N T E R 2 0 0 2 V O L U M E 1 5 . 2

To Open Or Not To Open ---- Or What to Do With a Closed Copper Mine by Jane McCarthy and Peter H.L. Monkhouse, BHP Billiton

63ACCENTURE JOURNAL OF APPLIED CORPORATE FINANCE

TO OPEN ORNOT TO OPEN—OR WHAT TO DO WITHA CLOSED COPPER MINE

by Jane McCarthy andPeter H.L. Monkhouse,BHP Billiton

Deciding how—or even whether—to operate a long-term asset can

be a real test of a company’s attitude towards real options. Whenthe alternatives are as varied as making cash payments to maintainthe ability to operate the asset (holding the option), relinquishing

the asset (abandoning the option), operating the asset (exercising the option), orselling the asset (transferring the option), management faces a difficult businessdecision. We examined this management decision for a higher-cost copper minethat had ceased operations, but was not yet permanently closed, because of theprevailing copper price.1 The basic decision can be outlined in simple terms:

continue to make annual care and maintenance expenditures so that themine can be re-opened if the copper price increases;

save the annual care and maintenance payments by permanently closing themine, but in so doing incur a large up-front payment for environmentalrehabilitation of the site;

re-open and begin to operate the mine; orsell the mine.We looked at the decision through both a conventional discounted cash flow

(DCF) lens and a real options lens. While the real options approach is theoreticallysuperior to the conventional DCF approach, at the practical level the inputassumptions to the real options approach must be realistic if this theoreticalsuperiority is to be translated into better management decisions. Too many realoptions applications make unreasonable simplifying assumptions in the name oftransparency and simplicity that undermine the theoretical benefits of theapproach and prevent their being translated into better management decisions.

1. While the mine valued in this paper was initially based on a real operation, a number of simplifying assumptionswere made in the financial model that changed the character of the operation such that the mine presented in this paperis hypothetical. However, the authors believe that the mine described in this paper is still representative of many higher-cost operations.

64JOURNAL OF APPLIED CORPORATE FINANCE

In this paper two real (American) option valuesare calculated simultaneously—an abandonmentoption and a deferral option.2 Further, this is done inthe presence of a number of real-world complexitiesthat are normally “assumed away,” including a finitemineral resource, non-zero lead times for switchingbetween modes of operation, a significant productionramp-up schedule, project-specific time-varying op-erating and capital costs, and time-varying produc-tion due to variations in ore grade and tonnes mined.We also used futures data to develop a plausibleprobabilistic forecast for copper prices. The introduc-tion of this complexity—or richness, depending uponyour point of view—is likely to increase both theaccuracy of the real options analysis and seniormanagement’s confidence in the results.

When real option values are computed using theframework outlined in this paper, what results is avalue for the asset that reflects the value of themodeled real options: the deferral and abandonmentoptions. But we begin our analysis by asking thequestion: What is the value of the asset if all the realoptions are “turned off”? The answer to this questionis not the same as the answer computed fromconventional DCF analysis. Therefore, this paperdiscusses three project valuation approaches: (1) theconventional DCF approach; (2) the project valuewith the value of the real options turned off, whichwe call the forward-curve or risk-neutral valuationapproach; and (3) the real options approach. The realoptions approach is simply the forward-curve ap-proach with the value of the real options added in.Understanding the differences between these threevaluation approaches is informative in its own right,and leads us to consider the additional—and impor-tant—effects of operating leverage and the probabi-listic behavior of copper prices.

The structure of the paper is as follows. The nextsection will describe the key economic parametersfor the different decisions that management canmake. We then give an overview of the three differentvaluation approaches. Next, we analyze the manage-ment decision with conventional DCF techniques.The project is then examined using the forward-curve, or risk-neutral, approach. A key element ofthis approach involves extrapolating the copper priceforward curve. The real options approach is thenconsidered, and this is relatively straightforward

given the previous analysis. Finally, we compare therecommendations of the different approaches, de-scribe the eventual management decision, and drawsome conclusions.

THE ECONOMICS OF THE ALTERNATIVEDECISIONS

Continuing to hold the real option means payingongoing maintenance costs for plant and equipmentof US$5 million per year (after tax), and eventuallyfacing environmental clean-up costs. The clean-upcosts are reduced in net present value terms bydeferral. Due to regulatory constraints, the maximumlife of this deferral option—the length of time man-agement can continue to simply “hold” the mine ona care and maintenance basis—is estimated at twoyears.

Relinquishing the real option involves perma-nently closing the mine and immediately rehabilitat-ing the site. To permanently close the mine todaywould cost US$34.3 million, which is the presentvalue of the environmental clean-up costs, less anyproceeds from the sale of some plant and equipmentitems, all discounted at the risk-free interest rate.When the mine is operating or being held on a care-and-maintenance basis, management has the optionto permanently close the mine. Accordingly, the lifeof the abandonment option corresponds to the life ofthe mine.

Exercising the real option means re-opening themine. Once the decision is made to re-open the mine,it takes six months to commence copper productionand another six months to ramp up to full production.Start-up costs are estimated at US$12 million. Onceproduction has commenced, the mine life is esti-mated at nine years. It has average copper productionof 72,000 tonnes per annum, average cash operatingcosts in real terms of 75 US cents per pound after by-product credits, and profits are subject to a corporatetax rate of 30%. The cash operating costs includesustaining capital but exclude the initial capitalrequirement to start up the mine.

Selling the real option requires valuing the mineto determine if the price offered for the mine isreasonable. The valuation results are presented in thefollowing sections. To simplify the analysis, the mineis assumed to be all equity financed.

2. A seminal paper in this area is M. Brennan and E. Schwartz, “EvaluatingNatural Resource Investments,” Journal of Business, Vol. 58 (1985), pp. 135-157.

65VOLUME 15 NUMBER 2 WINTER 2003

ALTERNATIVE VALUATION APPROACHES

The standard valuation technique of using DCFanalysis with a single rate to discount all project cashflows has a number of well-known shortcomings astypically applied to resource projects:3

it does not allow for mean reversion in commodityprices;

it does not allow for the effects of operatingleverage; and

it ignores the value of managerial flexibility, or realoptions.

Before discussing these three issues in more detail,we would like to stress that they can and do come intoplay in the real world. For example, because the standardDCF valuation approach fails to address mean rever-sion, it systematically underestimates the risk premiumof short-life assets, thereby leading to overvaluation.Conversely, the risk premium of long-life assets istypically overestimated, leading to undervaluation.

By ignoring operating leverage, the standardvaluation approach underestimates the risk of projectswith high operating costs (low margins)—that is, highoperating leverage—and that therefore have morevolatile cash flows compared with projects withlower operating leverage. Conversely, the risk pre-mium of assets with low operating costs is overesti-mated. If the effects of mean reversion and operatingleverage are taken together, it means that low-cost,long-life assets (“trophy assets”) are systematicallyundervalued using conventional DCF techniques,and high-cost, short-life assets tend to be overvalued.The effects of mean reversion and operating leveragecan be handled by using the forward-curve, or risk-neutral, valuation approach described later in thisarticle.

Another example of where these issues comeinto play in the real world is in the area of plantoptimization. Failing to properly address mean rever-sion and managerial flexibility results in the underes-timation of the optimal mine life, which in turn resultsin the construction of plants whose annual capacity

is too large, thereby wasting shareholder capital.These issues of mean reversion and the failure toincorporate managerial flexibility can be handled,together with operating leverage, by using the realoptions valuation approach outlined in this paper.

Mean Reversion

Mean reversion is the term used to describe thefact that commodity prices in general tend to oscillatearound an estimated long-run, or mean, price thatmay well change over time, if only due to inflation.In other words, price increases tend to be followed byprice declines as the price of the commodity revertsto its long-run level. This reversion to a long-run priceis observed in historical price data and, based onfutures prices, it is expected to continue in the future.Supply and demand responses to deviations from thelong-run price provide the mechanism by whichmean reversion is likely to continue. Thus, the keyquestion for most commodities is not the existencebut rather the strength of mean reversion.

The significance of mean reversion in commod-ity prices is that it implies a saturation of price risk, asillustrated in the following simple example. Supposeyou are making a presentation to a group of stockmarket analysts and one of them asks what you expectthe copper price to be in ten years. You might respond,“I am 95 per cent sure that prices will be betweenUS$0.65 and US$1.00 per pound in today’s dollars.”Another analyst might then ask what you think thecopper price will be in 11 years’ time. You couldplausibly give the same response. In other words, thereis no incremental price risk in moving from year 10 toyear 11. Put another way, there is a saturation of copperprice risk by year 10. If we believe that commodity pricerisk is a significant contributing factor to a company’sbeta or systematic risk, this means that the discount ratefor the project should decline over time. But at lowerlevels of mean reversion, such that expected deviationsfrom the mean become greater, the decline in thediscount rate over time will be less pronounced.4

3. See, for example, The Energy Journal, Vol. 19 (1998), No. 1, particularlythe article by G. Salahor, “Implications of Output Price Risk and OperatingLeverage for the Evaluation of Petroleum Development Projects.”

4. In the extreme case, when the strength of mean reversion approacheszero, the mean-reverting process effectively becomes geometric Brownianmotion. A price process described by a geometric Brownian motion probabi-listic model can be thought of as a permanent price-shock model in that a largedeviation from the prevailing price becomes the new benchmark price, fromwhich future fluctuations can be expected to occur. By contrast, in a mean-reverting price model any large price shock from the mean price can beexpected to dissipate over time, as the price returns to the long-run, or mean,price. The use of a constant discount rate is consistent with a price process thatfollows geometric Brownian motion.

Low-cost, long-life assets (“trophy assets”) are systematically undervalued usingconventional DCF techniques, and high-cost, short-life assets tend to be overvalued.


Operating Leverage

The intuition behind adjusting for operatingleverage is simple. If Company A owned a single low-cost resource project and had significant debt, wewould readily estimate a “geared” or levered beta toreflect the debt load and use it to calculate the rate atwhich we discount the company’s equity cash flows.But what about Company B, which owns a singlehigh-cost resource project and has zero debt? In thiscase we would typically discount the equity cashflows of Company A at a higher rate than the equitycash flows of Company B, even if the overall cashflows in both cases were identical and the volatilityof the overall cash flows (stemming from changes inprices, for example) were identical. That is, we wouldexplicitly take into account the effects of financialleverage through the discount rate.

In the same way, adjusting for operating lever-age should entail using a lower discount rate for high-margin projects than for low-margin projects to reflectthe lower risk of the cash flows from high-marginprojects. While simple to say, the difficulty comes inquantifying this effect. It can be shown with relativelysimple algebra that the effects of operating leveragewill be captured in project evaluation by usingdifferent discount rates for the revenue stream and thecost stream. However, the conventional DCF valua-tion technique uses the same discount rate on allproject cash flows, mainly because of the difficulty inestimating the different discount rates for the differentcash flow streams. Accordingly, the effects of operat-ing leverage are not captured in the conventionalvaluation approach.

As we will show, however, futures data allow usto effectively discount the revenue stream separately. Thechallenge then becomes estimating suitable discountrates for the cost streams. In the absence of a betteralternative, the Capital Asset Pricing Model (CAPM)framework can be used to estimate the appropriatediscount rate for costs, although finding a suitable proxyfor costs is difficult. While it is tempting to assume thatthe risk of operating costs is entirely diversifiable, this isunlikely to be a reasonable assumption. Thinking ofenergy inputs and labor costs leads one to the conclusionthat operating costs are likely to contain some systematicor non-diversifiable risk. While undoubtedly more com-plicated, estimating a separate discount rate for the coststreams allows us to quantify the effects of operatingleverage. At the very least, this is better than simplyignoring the effect, as many companies do.

Capturing the Effects of Mean Reversion andOperating Leverage

When thinking about mean reversion and oper-ating leverage it is important to recognize that thetypical single corporate discount rate will be about rightfor a typical project. But once a project is atypical—interms of life or operating margins—the typical singlecorporate discount rate will be incorrect. A goodanalogy is the well-known error of using a company-wide discount rate for projects that differ markedly interms of risk from the company’s general operations.

Overcoming the effects of mean reversion andoperating leverage in the conventional DCF approachcould be done by varying the discount rate with timeand with the level of operating leverage. Unfortu-nately, in the conventional framework there is nocommonly accepted method by which the discountrate as applied to project cash flows can be varied inthis way. However, using the market-derived forwardcurve for commodity prices to value the revenuestream allows us to capture the effects of meanreversion in a fairly robust manner. In practice, thismeans discounting the forward-curve-derived rev-enues (production volume times forward-curve com-modity prices) at (approximately) the risk-free rate,since the forward curve already reflects price risk. Anddiscounting the cost stream at its own, risk-adjusteddiscount rate captures the effect of operating leverage.This risk-neutral or forward-curve approach is thestandard approach when valuing financial derivativesand real options. It is most straightforward to applywhen certainty-equivalent prices (futures prices) areavailable, which they are for many commodities.Otherwise, fundamental judgments about certainty-equivalent cash flows or risk premiums must be made.

DISCOUNTED CASH FLOW ANALYSIS

Conventional DCF analysis involves forecastingfuture copper prices, or expected spot prices, anddiscounting the resulting free cash flows at the un-levered cost of equity capital, which we will assumeis 12.5% per annum. The projected spot prices, asshown in Table 1, can be thought of as a typicalcorporate price forecast.

Because of the forecast of rising expected spotcopper prices, the conventional valuation approachresults in a slightly higher estimate of value if themine is assumed to re-open in one year’s time.Delaying the re-opening is assumed to cost an


additional US$5 million per annum, after tax, to coverthe cost of care and maintenance. More fully, Table2 shows the NPVs calculated for various times of re-opening. Under the conventional approach andbased on today’s price forecast, the mine would bevalued at US$9.7 million and would be re-opened inaround one year’s time, when the copper price isforecast to be US$0.89 per pound.

THE FORWARD CURVE, OR RISK-NEUTRAL,APPROACH

The use of risk-neutral pricing techniques tovalue financial derivatives and real options is wellknown. To make the analysis clear, we initiallyapplied the risk-neutral, or certainty equivalent,pricing technique to value the project in the sameway as the conventional DCF valuation technique.The risk-neutral approach adjusts for risk in theestimates of the net cash flows, and discounts therisk-adjusted net cash flows at the risk-free rate ofinterest. The challenge in this approach is estimatingthe risk-adjusted cash flows, including revenues,costs, and tax payments. Real options are not valuedin this section, although the techniques developedhere will greatly facilitate the valuation of flexibilityoptions in the next section.

Certainty-Equivalent Revenues

Risk-adjusted revenues can be obtained rela-tively easily by noting that risk-adjusted copper pricesare available from futures data, albeit for maturitiesthat extend only to 27 months. By assuming thatproduction risk is diversifiable, or non-systematic, theestimated production levels can be multiplied by theprojected certainty-equivalent copper prices to obtainan estimated risk-adjusted revenue stream. This risk-adjusted revenue stream is then discounted at the risk-free interest rate to calculate a present value.

The adjustment for risk in the risk-neutral rev-enue stream means that the forward curve price issystematically lower than the expected spot priceused in calculating the risky revenue stream in theDCF valuation approach. The difference is a mea-sure of the systematic risk of the copper price. Thetwo price curves—the expected spot curve used inthe DCF analysis and the forward curve used in therisk-neutral valuation approach (expressed in natu-ral log terms)—are shown in Figure 1. Another wayof thinking of the two different price curves is to notethat in valuing the revenue stream one uses aweighted average cost of capital to discount theexpected spot curve, whereas one uses the risk-freeinterest rate to discount the forward curve. The useof different discount rates only makes sense if therisk-neutral price curve is less than the expectedspot curve.

One challenge is to extrapolate the copperfutures price data beyond 27 months. At its simplest,this could be reasonably done using a pencil andruler, because at the longer maturities, the futurescurve—if the copper price is expressed in natural logterms, as in Figure 1—is effectively a horizontal line.This approach would be sufficient for the risk-neutralmine valuation described in this section. However,for the purpose of valuing real options, it is necessaryto model fully the underlying stochastic (probabilis-tic) price process.

An important assumption for all valuation work—and for real options valuation in particular—is theassumed stochastic price process. Two commonlyassumed processes are geometric Brownian motionand mean reversion. While geometric Brownianmotion has an appealing simplicity, it does notaccord with intuition, nor does it accurately modelcertain aspects of observed futures data. In contrast,a mean-reverting price process is a far better assump-tion for commodities like copper, and this is what hasbeen assumed in our analysis.

TABLE 1ASSUMED FUTURESPOT PRICE OF COPPER(US$/POUND)

t=0 yr t=1 yr t=2 yr t=3 yr t=4 yr t=5 yr t=6 yr t=7 yr t=8 yr t=9 yr t=10 yr

0.81 0.89 0.97 0.98 1.00 1.01 1.03 1.04 1.06 1.07 1.09

Immediate re-opening Re-open in 1 year’s time Re-open in 2 years’ time

NPV at t=0 9.6 9.7 1.7

TABLE 2DCF MINE VALUESAT VARIOUSRE-OPENING TIMES(US$ MILLIONS)

Failing to properly address managerial flexibility results in the underestimation ofthe optimal mine life, which in turn results in the construction of plants whose

annual capacity is too large, thereby wasting shareholder capital.


Mean-reverting price processes come in a varietyof flavors—depending, for example, on the numberof underlying stochastic variables modeled. While aone-factor (one stochastic variable) mean-revertingprice process is relatively simple, it does not modelthe observed volatility term structure of futures dataparticularly well. A two-factor (two stochastic vari-ables) mean-reverting price model fits the futuresdata far better and is also the simplest price processthat accords with intuition and observed futures data.Accordingly, the analysis in this paper uses a two-factor mean-reverting price process.5 The two factorsof the copper price model can be thought of as:

a factor that represents the long-run trend price,which itself varies stochastically, albeit less severelythan the short-term factor or the spot price; and

a factor that represents short-term deviations froma long-run trend price.

A more complex stochastic price model could beused, but the challenge lies in estimating the para-meters for the assumed model. In our case we needto estimate values for seven variables (and two initialconditions) that together define our risk-neutralforecast copper price. Our method for estimatingthese parameters and initial conditions is described inmore detail in the box insert.

Certainty-Equivalent Costs

Unfortunately, a forward curve does not exist forunit operating costs, so as a practical matter we areforced to use the CAPM to estimate certainty-equiva-lent operating costs. Within the CAPM framework, weassumed that operating costs have a beta of 0.4,corresponding to an asset beta for a typical miningoperation. In the absence of better information, thisis considered a reasonable proxy for the systematicrisk of operating costs. Assuming a market riskpremium of 5% per annum implies that costs arediscounted at a two-percentage-point premium (0.4x 5%) over the risk-free interest rate. The expectedcosts are converted to certainty-equivalent costs, orrisk-neutral costs, using this estimated risk premiumby algebraically equating the present value of theexpected costs, using the risk-adjusted discount ratecalculated above, to the present value of certainty-equivalent costs, calculated using the risk-free rate ofinterest. The approach is conceptually similar to theuse of the two copper price curves described above.

Capital cost risk is assumed to be diversifiable andas such future capital costs are discounted at the risk-free rate of interest. Working capital payments are alsodiscounted at the risk-free rate. Certainty-equivalent

FIGURE 1DCF AND FORWARDCURVE PRICE FORECASTS,AND THEINTERPRETATIONOF THE VALUES OFTHE Y-INTERCEPT ANDSLOPE OF THE LONG-RUNFORWARD CURVE(NATURAL LOGARITHM)

Notes: (1) The then-current price (at t=0) of 4.40 corresponds to a copper price of 81 US cents per pound.(2) The then-current y-intercept (at t=0) of 4.36 corresponds to a copper price of 78 US cents per pound.

5. The particular price model assumed in this paper is discussed in moredetail in E. Schwartz and J. Smith, “Short-term Variations and Long-TermDynamics in Commodity Prices,” Management Science, July 2000, pp. 893-911.

Log

of

Cu

No

min

alP

rice

(U.S

.Cen

tsp

erP

ou

nd

)

Years

4.80

0 1 24.30

4.70

4.60

6 83 4

Expected Spot Curve

Y-intercept = 4.36

4.50

4.40

5 7 9

Asymptotic Slope (m) = 0.7%

10

Forward Curve


taxes are computed in nominal terms in the usualmanner by:

using the forward curve to calculate certainty-equivalent revenues, as described above;

using the CAPM to compute certainty-equivalentcosts, as described above; and

calculating “certainty-equivalent” depreciation sincecapital cost risk is assumed to be diversifiable.

Certainty-Equivalent Valuation

The above assumptions give an NPV under theforward-curve approach of negative US$54.8 mil-lion. This compares with the conventional NPVvaluation of positive US$9.6 million. Note that theestimated cost of permanently closing the mine imme-diately was calculated at US$34.3 million. The analysis

was repeated assuming the mine was to be re-openedin one and two years’ time. As for the DCF case,delaying the re-opening is assumed to cost an addi-tional US$5 million per annum, after tax, to cover thecost of care and maintenance (see Table 3).

This analysis suggests that the mine should beclosed immediately on a permanent basis as theliability is (only) US$34.3 million. The major reasonsfor the differences between the two valuation ap-proaches are the:

different assumed price forecasts, as shown inFigure 1. The DCF price forecast, or the expectedspot price forecast, was derived independently fromthe futures prices that formed the basis for the forwardcurve, or risk-neutral, approach;

effects of mean reversion; andeffects of operating leverage.

Analyzing histori-cal futures data toparameterize theassumed stochas-tic price process isa task well suitedto Kalman filter-ing,* a recursiveprocedure for op-timizing param-eter estimates. Themost recent para-meter estimatesfrom the Kalmanfilter would nor-mally be used inany valuation work.However, confidence in these parameter estimateswould increase if they were shown to have beenrelatively stable over time. While the stability of indi-vidual variables can be examined, it is more useful toexamine two key parameters that have a large influ-ence on asset valuations: the level (y-intercept) andslope of the long-run forward curve. These parametersare defined graphically in Figure 1 in the text.

To test the stability of these two key variablesover time, the Kalman filter was stopped every weekand the predicted parameters noted. It is as though

we were turningthe clock backand filtering thefutures data at apoint of time inthe past. We thenmoved throughtime another weekand ran the filteragain to get newand updated pre-dicted parameterestimates. Themovement of thesehistorical estimatesof the level (y-in-tercept) and the

slope of the (natural logarithm of the) long-runcopper forward curve are shown graphically above.

The historical data suggest that the slope of thenatural logarithm of the long-run forward curve isaround zero, in nominal terms, and over the last fiveyears the y-intercept of the long-run copper forwardcurve has been around 80 US cents per pound. The(recent) relative stability of these parameters gives ussome confidence that the Kalman filter estimates areplausible, and hence that our estimates of certainty-equivalent copper prices are reasonable.

* See A. Harvey, Forecasting, Structural Time Series Models, and the Kalman Filter (Cambridge, UK: Cambridge University Press, 1989).

ESTIMATING THE PARAMETERS OF THE PRICE MODEL

The conventional DCF valuation technique uses the same discount rate on all projectcash flows, mainly because of the difficulty in estimating the different discount ratesfor the different cash flow streams. Accordingly, the effects of operating leverage are

not captured in the conventional valuation approach.

y-i

nte

rcep

t (U

.S.C

ents

per

Po

un

d)

Slop

e

Slope

y-intercept

1.6%

1.2%

0.8%

0.4%

0.0%

120

100

80

60Jul-93 Jul-94 Jul-95 Jul-96 Jul-97 Jul-98 Jul-99 Jul-00

VALUES OF Y INTERCEPT AND SLOPE OF LONG-RUN FORWARD CURVE OVER TIME


The above analysis depends on the assumption thatthe risk in the costs is equivalent to a premium of 2.0%per annum. Alternatively, we can turn the analysis on itshead and ask how risky the costs would have to be forthe DCF and certainty-equivalent methods to be consis-tent with each other. The answer is a risk premium of4.10% per annum, corresponding to a beta of 0.82. Thissimple analysis highlights the sensitivity of the valuationresults in general, and this assumption in particular, in thecase of a high-cost operation as modeled here, wherethe present value of the operating costs over the life ofthe operation is some US$900 million.

THE REAL OPTIONS APPROACH

In addition to the estimated copper price for-ward curve required for the risk-neutral valuationapproach, the valuation of real options using the two-factor price model requires a number of additionalparameters, including long- and short-term commod-ity price volatility and the strength of mean reversion.Historical estimates of these parameters over time areshown in Figure 3. The methodology for estimatingthese parameters is described in the box insert.

The valuations in this paper assume a copperprice short-term volatility of 18.6% (over the last fiveyears this parameter has varied between 18.6% and

22.4%), a long-term volatility of 14.0% (over the lastfive years this parameter has varied between 13.5%and 15.1%), and a strength of mean reversion of 0.72(over the last five years this parameter has variedbetween 0.65 and 0.72). The parameter estimateschosen correspond to the then most recent estimates.

An estimate of 0.72 for the strength of meanreversion corresponds to a half-life of about one year.This means it takes about one year for the expected priceto reach the mid-point between the current price and thelong-run price. Or, put another way, it takes about fouryears for the effects of a price disturbance to dissipate.

Once the price process is fully specified, the nextstep is to calculate real option values. To do this, it isnecessary to explicitly model the stochastic commod-ity price process. Because of the mean reversioncharacteristics of the price process, we used a trinomialtree approach, as illustrated in Figure 4. The origin ofthe tree (So) corresponds to the initial estimate of oneof the two factors, or stochastic variables, namelyeither the short-term deviation from the long-runcopper price or the long-run copper price itself.

The trinomial tree approach assumes that, ateach node of the tree, the underlying stochastic variablecan move either up (with probability pu) or down (withprobability pd) or stay at the same level (with probabil-ity 1 – pu – pd). The fixed probabilities assigned to each

FIGURE 3KEY COPPER MODELPARAMETERS

TABLE 3FORWARD CURVE MINEVALUES AT VARIOUSRE-OPENING TIMES(US$ MILLIONS)

Immediately In 1 Year’s Time In 2 Years’ Time

NPV at t=0 assuming re-opening (54.8) (53.3) (51.0)

NPV at t=0 assuming permanent closure(discounting at the risk-free rate) (34.3) (37.4) (40.3)

strength of mean reversion

short-term volatility

long-term volatility

Rate o

f Mean

Reversio

n

Vo

lati

lity

0.30

0.20

0.15

0.10

0.8

0.6

0.4

0.2

0.0Jul-93 Jul-94 Jul-95 Jul-96 Jul-97 Jul-98 Jul-99 Jul-00

0.25


of these alternatives depend on the incremental timestep chosen and the volatility of the underlyingvariable.

Because we used a two-factor price model, weneeded to generate two trinomial trees (one for eachfactor) and to combine them to ensure simultaneityand correlation between the factors. This procedureresults in a three-dimensional price tree.6 In evaluatinga two-year deferral option, as described in this paper,the price tree must extend for two years, and becausewe have used time increments of one month, the treecontains 24 time steps. The real option is evaluatedusing an iterative dynamic programming approach.This approach involves starting by valuing the optionat maturity, at the nodes on the last time step of the tree,where its value only depends on the differencebetween the commodity price and the exercise price,and then working backwards through the tree calcu-lating values at all the nodes until obtaining the valueat time zero, corresponding to the value of the assetinclusive of the real options.

In fact, in a separate analysis we have extendedthis approach so as to value multiple real optionsover the mine’s nine-year life, including options toexpand or contract operations at specified times, orto switch use (such as by permanently closing theoperation).7 This can greatly increase the complex-ity of the problem and the number of steps requiredfor its solution. Even for the simpler case describedin this paper, it was necessary to include complica-tions such as a finite mineral resource, a significantproduction ramp-up schedule, project-specific time-varying operating and capital costs, and time-varying production due to variations in ore grade andtonnes mined, in order to approximate a realisticand useful model. The model was programmed inVisual Basic, and the results are shown in Table 4.The value of the option is relatively robust to theunderlying assumption about the risk premium incosts. If a risk premium of 4.1% were used insteadof 2.0%, the option value would still be about US$50million.

TABLE 4SUMMARY PROJECTVALUES(US$ MILLIONS)

Value

NPV assuming re-opening (54.8)

Two-year deferral option value with an abandonment option for project’s life 65.3

Total project value(with two-year deferral option and an abandonment option for project’s life) 10.5

6. The procedure for evaluating real options using trinomial trees isdiscussed in more detail in J. Hull, Options, Futures & Other Derivatives, 4th ed.(Upper Saddle River, NJ: Prentice-Hall, Inc., 2000).

FIGURE 4TRINOMIAL TREEILLUSTRATINGTHE EVOLUTION OFTHE PRICE PROCESS

7. The evaluation of multiple real options is discussed in L. Trigeorgis, RealOptions: Managerial Flexibility and Strategy in Resource Allocation (Cambridge,MA: The MIT Press, 1996).

T=1T=0 T=3T=2

S0

S u0

S0

S d0

S u0

S0

S d0

S u02

S d02

S u0

S0

S d0

S u02

S d02

S u03

S d02

Too many real options applications make unreasonable simplifyingassumptions in the name of transparency and simplicity that undermine thetheoretical benefits of the approach and prevent their being translated into

better management decisions.


Critical Prices

The key output of the real options approach, ofcourse, is the answer to the question as to what actionmanagement should take regarding the currentlynon-operational mine. The above values allow us toevaluate the sale decision and determine whether toaccept an offer to acquire the mine. Based on thisanalysis, we should be willing to accept any offerover US$10.5 million. Coincidentally, this is almostidentical to the value obtained by the original DCFanalysis. Perhaps more important, the real optionsframework also allows us to calculate at what copperprice the mine should be re-opened and at what pricethe mine should be permanently closed (that is, thecritical prices), thereby addressing the remainingmanagement decisions.

The critical copper price for immediately open-ing the mine is the price at which the value of theunopened mine with options equals the value of theopened mine, including any option value. Similarly,the critical copper price for permanent closure is theprice at which the value of the unopened mine withoptions equals the value when permanently closed.

With a two-factor copper price model, thecritical price is actually a function of two variables:the y-intercept of the long-run forward curve and theshort-term price deviation. The critical prices havebeen calculated using the trinomial tree methoddescribed above. Figure 5 shows the critical prices

for re-opening and permanently closing the coppermine. Another way of thinking about Figure 5 is thatit shows the various optimal states of the mine(operating, care and maintenance, and permanentclosure) for a variety of copper “prices,” given that themine is initially non-operational but not permanentlyclosed.

This analysis suggests that management shouldconsider re-opening the mine when the spot copperprice is around US$1.11 per pound, and considerpermanently closing the mine when the y-intercept ofthe long-run forward curve is around US$0.56 perpound.8

The above graph shows that the copper price atwhich the mine is permanently closed depends almostsolely on the y-intercept of the long-run forward curve.However, the re-opening decision is more dependenton the spot copper price, due to the relatively imme-diate receipt of revenues. These findings accord withintuition; the focus in the permanent closure decisionshould be on the long-run copper price only.

DISCUSSION OF THE DIFFERENT APPROACHES

Based on the spot price and an un-levered costof equity capital (the conventional DCF approach),the mine was valued at US$9.7 million, and wewould be planning to re-open the mine in around oneyear’s time when the spot copper price is forecast tobe 89 US cents.

FIGURE 5CRITICAL PRICE ENVELOPEFOR PERMANENTLYOPENING/CLOSINGTHE MINE WITHA TWO-YEAR DEFERRALOPTION AND ANABANDONMENT OPTIONFOR THE PROJECT’S LIFE

Notes: (1) The data values shown on the graph are the corresponding spot prices.(2) The y-intercept numbers assume a slope of the long-run forward curve of 0.7%.(3) These data assume the mine is initially non-operational but not permanently closed.

8. The copper prices quoted correspond to zero short-term price deviation.That is, the forward curve is not in backwardation or contango. Were the forward

copper prices to be in backwardation (or contango), the critical spot prices forre-opening the mine and permanent closure of the mine would alter.

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Under the forward curve, or risk-neutral, valua-tion approach, the mine was valued at negativeUS$34.3 million, and the analysis suggests that themine should be permanently closed.

Assuming a two-year deferral option and anabandonment option for the project’s life, the realoptions value was estimated at US$65.3 million andhence the total project was valued at US$10.5 million.Further, the real options approach suggests that themine be kept non-operational. It should be re-openedif the copper price reaches approximately US$1.11 perpound, and permanently closed when the y-interceptof the forward curve is around US$0.56 per pound.

The question of which methodology is best, andwhy, is more complex. The differences in value andrecommendations between the conventional DCFapproach and the forward-curve approach highlightwhat is already well known, namely that valuationmethods are extremely sensitive to price forecasts. Atone level, the better valuation methodology hingeson one’s preference for price forecasting methodolo-gies—whether by an extrapolated forward curve ora corporate crystal ball. At a more theoretical level,however, the differences in the two approaches aredriven by different estimates of risk premiums result-ing from considerations of mean reversion andoperating leverage. The large difference in this casearises because the conventional DCF valuation meth-odology underestimates the risk premiums of a short-life, high-cost operation. In the case of a long-life,low-cost asset, the forward-curve approach couldgive a substantially higher asset value. Note thatacross a portfolio of operations, however, the twoapproaches will tend to give similar answers. One’spreference between the conventional DCF approachand the forward-curve approach thus turns in part onone’s view of the treatment of mean reversion andoperating leverage. This paper argues that only theforward-curve approach allows for these effects.

The real options approach goes a step beyondthe forward-curve approach because it values flex-

ibility in a relatively rigorous way. Accordingly, it canbe considered the best valuation approach in that itovercomes the identified weaknesses of the other twoapproaches. Of course, the real options approach andthe forward-curve approach are only as good as theirinput assumptions, particularly those regarding theextrapolation of the certainty-equivalent price curveand the volatility structure of prices.

CONCLUSION

For the hypothetical mine presented in thispaper, it is our belief that management’s intuitionwould lead to the decision to keep the mine non-operational but not permanently closed, despite therecommendation of the conventional DCF analysisthat suggested that the mine be re-opened in a year’stime. A sophisticated real options analysis wasundertaken and its conclusions supportedmanagement’s likely intuition, reinforcing the con-tention that the real options framework is superior tothe conventional DCF approach.

While the real options framework is undoubt-edly superior in a theoretical sense, at a practicallevel its weaknesses include complexity and areliance on a number of key assumptions. Theseassumptions include the validity of the extrapolatedforward curve for forecasting prices as well as thelevel of systematic risk inherent in production,operating costs, capital expenditure requirements,and working capital requirements. Such assump-tions should probably be explicitly considered inany conventional DCF analysis, but they tend in-stead to get rolled up into the assumptions relatingto the project discount rate. If capital budgetinganalysis is to provide meaningful input to the morecomplex decisions made in business, it may benecessary to introduce the additional complexity ofreal options analysis on a more consistent basis.This would appear essential for any substantial andcomplex capital expenditure commitment.

JANE MCCARTHY

is Manager, Portfolio Risk Management, of BHP Billiton.

PETER H.L. MONKHOUSE

is Vice President, Business Strategy, Carbon Steel Materials, ofBHP Billiton.

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TO OPEN OR NOT TO OPEN—OR WHAT TO DO WITH A CLOSED COPPER MINE

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