ABSTRACT Using DNPV for Valuing Invesmens in he Mining Secor by R. David Espinoza 1 and Javier Rojo 2 The reliance on classical valuaion mehods such as he ne presen value (NPV) has resuled in someimes conroversial asse valuaions in he mining secor. A complemenary mehod, ermed decoupled ne presen value (DNPV), is used o evaluae mining invesmen opporuniies. The proposed valuaion mehod decouples he ime value of money from he risk associaed wih he invesmen opporuniy. A simplified mining invesmen projec is analyzed using radiional NPV echniques and evaluaed wih wo alernaive echniques such as Modern Asse Pricing (MAP) and he proposed DNPV. The proposed DNPV mehodology provides a consisen valuaion mehod free from he problems ypically associaed wih he applicaion of NPV mehods and, more imporanly, i allows a seamless inegraion of he projec risk assessmen performed by echnical expers and he risk managemen implemened by business execuives ino he financial evaluaion of he projec. INTRODUCTION The mos popular valuaion mehods used in he mining indusry are he Ne Presen Value (NPV) echnique followed by is close relaive he inernal rae of reurn (IRR). These mehods consis of reducing fuure cash flows by a single facor ha grows exponenially wih ime. This facor is known as he risk adjused discoun rae (RADR) because he effec of ime is adjused for risk. The main problem of combining ime value of money (represened by he risk free rae) and risk in a single facor when calculaing he NPV of an invesmen is ha i arificially makes he value of cash flows ha occur far in he fuure negligible and over emphasizes he value of earlier cash flows. Thus, he resuls of such an analysis can be misleading, seering corporaions o adop and/or governmen eniies o accep design and operaion decisions ha can be derimenal o sociey and shareholders alike in he long erm. For mining invesors, NPV mehodologies produce high volailiy in he valuaion of long lived mines (30-40 years) which are comparable o he volailiy associaed o commodiy spo prices. The commens voiced by Dean Gehring (Presiden & CEO of Rio Tino Minerals) a he 2015 mining conference in Denver Colorado furher reinforces his poin: I hink we re going o see how mines are valued change. The way ha we ve all been valuing mines for years now is, you know, basically on ne presen value-ype analysis, which is a very, very common undersanding in a Wesern culure. Well, here s a lo of culures ha don ge ha, and, in fac, o hem, i s value desroying, and I hink ha as we go forward, we re no always going o design mines around ne presen value. There ll be somehing else ha we look a ha ells us wha really is he rue value of ha operaion. 1 Geosynec CAT (Capial, Asses & Transacions), www.geosynec-ca.com, Washingon DC Office 2 Susainabiliy Sraegic Advisors Ld., www.susainabiliy-sa.com, London, UK 1
Alhough he shorcomings of he NPV mehods have been widely recognized by many indusry expers (e.g., Salahor, 1998; Laughon e al., 2000; Samis e al., 2006; Guj and Garzon, 2007), and alernaives have been proposed, NPV is sill by far he valuaion mehod of choice. Recenly, a valuaion mehod o assess he value of long-erm infrasrucure projecs was inroduced (Espinoza and Morris, 2013; Espinoza, 2014). The proposed mehod, ermed decoupled ne presen value (DNPV), addresses many of he shorcomings of he NPV mehod while reaining he simpliciy in is presenaion ha has led o is populariy. The DNPV mehodology consiss in evaluaing each of he key risks associaed wih a projec and for each risk calculaing a synheic insurance premium ha would be demanded by a hypoheical risk neural insurance company. This hypoheical insurance policy premium, designed o proec he projec's cash flows from a shorfall below he expeced values in he even of an adverse oucome of a given risk (e.g., reducion of revenues due o commodiy prices volailiy, increase in expenses due o echnical difficulies), is ermed he cos of risk. The cos of risk is subraced from he projec's expeced cash flows. The subracion of all he relevan synheic insurance premiums leaves he remaining projec cash flows virually riskless and he DNPV valuaion for he projec can hen be obained by discouning he ime value of he remaining riskless cash flows using he applicable risk free rae. In some insance such insurance insrumens are available in he marke place and herefore here are observable prices available (e.g., a-he-money pu opions for commodiies or specially srucured insurance policies) ha can be used in he DNPV calculaion. In mos pracical applicaions, such financial insrumens are no available for porions (or any) of he period under consideraion so he risk of he projec becoming unfavorable (i.e., he revenues being lower and/or he associaed expendiures being higher han anicipaed) is borne by he invesor. Considering risks as coss ha affec cash flows is a more naural progression of a wellesablished business pracice of buying/selling insurance producs o obain/provide proecion agains insurable risks. For hose risks ha have no been ransferred o an insurance company and/or hedge provider, he invesor becomes he risk bearer agains unfavorable oucomes and he esimaed cos of risk heir compensaion for aking on such risks. In summary, one of DNPV s main feaures is he decoupling of he ime value of money and risk, which allows discouning fuure cash flows using he risk-free rae while accouning for risk as a cos o he projec. This feaure is paramoun for valuing long-erm asses as well as liabiliies (e.g., asse reiremen obligaions) as hese are no reduced o negligible values by he process of discouning by an arificially high RADR. Furhermore, he use DNPV can faciliae he use of he capive insurance concep o manage risk for large mining conglomeraes. THE PERILS OF USING RISK ADJUSTED DISCOUNT RATES Common o mos mining valuaion analysis is he selecion of a consan RADR o calculae he NPV of an invesmen. If he seleced RADR is greaer han he projec's currency risk-free rae, 2
hen he discoun rae has been adjused for risk. The seemingly innocuous assumpion of using a consan RADR hroughou he invesmen period can have a significan effec in invesmen decisions paricularly for long erm invesmens wih long-erm fuure liabiliies such as hose of mining. Risks ha can affec invesmen cash flows can be from many differen sources and evolve over ime in many differen ways. These risks are ypically classified in he lieraure as: (i) sysemaic (i.e., priced, non-idiosyncraic, non-diversifiable, public, marke); and (ii) non-sysemaic (i.e., unpriced, idiosyncraic, diversifiable, privae, non-marke). Sysemaic risk is associaed wih he ype of asse o be creaed (e.g., commodiy prices for mining projecs). Non-sysemaic risk is a projec specific risk and could be echnical (e.g., he amoun of ore mineral available in a geological formaion and is grade) as well as non-echnical (e.g., changes in he local ax code, changes in environmenal regulaions). The classical ex book expression o calculae he NPV of an invesmen considering all he associaed risks lised above using discree cash flows is: NPV C~ T 0 (1 r) (1) where T is he mauriy (i.e., invesmen period), C ~ is he expeced value of a sream of uncerain fuure ne cash flows (C ), and r is RADR ha lumps ime value of money and he risks described above. The approximae coninuous represenaion of NPV as a funcion of ime is given by: C~ (1 r) C~ e r (2) The classical equaion o esimae he discoun rae o accoun for marke risk is given by he Capial Asse Pricing Model (CAPM): r r r r ) (3) f ( m f where r m is he expeced rae of reurn of he overall marke (e.g., he S&P500); r f is he riskfree rae; and (i.e., he company bea) is a parameer ha measures he sysemaic risk of he asse relaive o he marke, and he difference (r m - r f ) is known as he sysemaic risk premium (r p ). Consisen wih Robichek and Myers (1966), he ime value of money is accouned for by he firs erm (i.e., he risk-free rae) whereas sysemaic risk is accouned for by he second erm in Equaion (3). To accoun for non-sysemaic risks posed by one-off projecs, an addiional risk premium (r ns ) is included o lump several non-sysemaic risks ogeher (e.g., Froh and Sein, 1998; Samis e al., 2006). Hence, Equaion (3) is simply modified as: 3
where r s r r r r (3a) f s r r ) represens he sysemaic (i.e., marke). Alhough modern porfolio ( m f heory predicaes ha invesors should no demand a risk premium for non-sysemaic risks, such predicamen is only valid for sufficienly liquid securiies ha can be raded in he open marke where eliminaion of non-sysemaic risk can be easily achieved hrough diversificaion. For invesmen in real projecs, as shown in Equaion (3a), invesors ypically demand a compensaion o ake on non-sysemaic risks, and congruen wih sysemaic risks, his compensaion akes he form of an addiional risk premium added o he risk free rae. The implici assumpion in he use of Equaion (3) is ha sysemaic and non-sysemaic risks are governed by he same sochasic processes. Alhough his simple approximaion is consisen wih popular represenaions of marke risk and i is easy o implemen in a discouned cash flow model, because of he sensiiviy of NPV o he selecion of he discoun rae, paricularly for long erm projecs, he impac of such an assumpion can be significan. Using he coninuous represenaion of NPV, he righ-hand side of Equaion (2), o explore he influence on he projec NPV of he apparenly innocuous simplificaion of adding risk premiums o he risk-free rae o accoun for sysemaic and nonsysemaic (i.e., marke and non-marke) risks, he value of he invesmen can be expressed as: ns C e r C e ( rf rs rns) C e rf e rs e rns (4) or NPV ( C, r) NPV( C, r ) F F (5) f s ns where NPV(C, r f ) represens he ime value of money; and F s r s e and F ns r ns e represen he risks facors ha vary from 1 o 0 (see Figure 1) and accoun for sysemaic (i.e., marke) and non-sysemaic (i.e., non-marke) risks, respecively. Risk facors equal o 1 indicaes ha here is eiher no risk (i.e., r s = r ns = 0) or ime = 0. Risk facors equal o 0 indicaes ha risks are infinie (i.e., r s = r ns = ) or ime is infinie =. Thus, he projec NPV can be inerpreed as he cash flow a ime discouned using he risk free rae o accoun for he ime value of money and furher adjused (reduced) o accoun for marke risk (F s ) and non-marke (nonsysemaic) risks (F ns ). I follows from Equaion (5) ha, independen of he acual sochasic processes ha bes describe he cash flows variaion wih ime, adjusing he risk-free discoun rae for risk (i.e., adding risk premiums) implies ha, regardless of he risk source, risk facors are always represened by exponenially decaying funcions of ime 3. As discussed before, his assumpion is no accurae excep for very few paricular cases (Bhaacharya, 1978; Halliwel, 3 This is anoher issue associaed wih RADR. Increasing he RADR o accoun for marke or non-marke risk makes he impac of negaive cash flows (e.g. losses, expendiures, coningency liabiliies) ha are far in he fuure arificially negligible. 4
2001 and 2011; Giacoo, 2007) and brings excessive volailiy o he valuaion of long erm asses. Furhermore, i follows from Equaion (5) ha any error in he esimae of he discoun rae will have he same funcional form as F s (or F ns ) and hus i will exponenially decay wih ime. This paricular feaure (i.e., exponenial decay) indicaes ha he magniude of his simple assumpion is no rivial paricularly for long-erm projecs wih lagging long-erm liabiliies such as mining projecs and depending upon he magniude of he assumed discoun rae, he cumulaive error can be significan. Figure 1 below shows he reducion facors curves for ypical RADRs (3%, 5% and 8%) for a period of 60 years. Table 1 shows specific reducion facors for 10-yr, 20-yr, 30-yr and 60-yr periods. As shown in his able, for a 60-year period, he reducion facors vary from 17% o 1%. Thus, assuming 8% insead of 3% (i.e., a 5% error in he esimae of he RADR) would resul in nearly a 95% cumulaive error over a 60 year period. As a resul, he presen value of a $1 billion liabiliy sixy years down he road would be a $10 million liabiliy (for an RADR of 8%) insead of $170 million (for an RADR of 3%). The arificial reducion of he long-erm liabiliies no only poses a looming financial liabiliy o fuure generaion of sakeholders bu also dicaes behavior of how mining aciviies are performed oday as his arificially reduced liabiliy would resul in an unrealisically higher valuaion of he mining asse. For he same reasons, he fuure profiabiliy of a mining asses can be severely undermined by he use of inappropriae risk premiums for he projec. For example, he presen value of a $10mn consan cash flow for he nex 60 years can increase from $123 million for a RADR of 8% o $276mn for a RADR of 3%, represening a difference of more han a 120%. The wo examples above illusrae he magniude of he valuaion differences achieved by selecing a no carefully enough pondered RADR. Unforunaely, risk premiums errors of his magniude are no uncommon even for cases where risks are seemingly well undersood wih significan amoun of daa available. For insance, Fama and French (1997) showed ha he use of expressions similar o Equaion (3) o esimae an indusry wide cos of equiy for marke risk only can be very imprecise and sandard errors of more han 3% per year are ypical. This finding does no bode well for he cos of equiy of individual firms (le alone specific projecs) as errors in esimaes of cos of equiy are likely o be significanly larger when non-sysemaic risks are accouned for. Because of he errors in he selecion of he discoun rae and he propagaion of hese errors wih ime, he presen value of cash flows from long-erm invesmen projecs are likely o be divergen from a properly risk adjused valuaion. Alhough modeling risk as an exponenial decay funcion makes i mahemaically more racable allowing for he derivaion of closed form soluions for simplified cases and he sraighforward applicaion of NPV, mahemaical expedience and convenience are no valid reasons o assume ha all projec risks can be represened by exponenial funcions. 5
Figure 1 Discoun Facor Variaion wih Time Table 1 Reducion Facors Mineral Exracion Period RADR 10-yr 20-yr 30-yr 60-yr 3% 74.4% 55.4% 41.2% 17.0% 5% 61.4% 37.7% 23.1% 5.4% 8% 46.3% 21.5% 9.9% 1.0% The classical NPV mehod as originally inroduced is a op-down approach ha resuled from he process of acquiring capial in he form of equiy and deb and mandaing ha all projecs mus earn he firm weighed average cos of capial (WACC). Essenially, he discoun rae in he classical NPV is more concerned wih he sources of funding han he projec iself (i.e., i is exogenous o he projec) and represens he demand of equiy invesors and deb holders. Consisen wih Equaion (3), if WACC is greaer han he risk free rae, hen i represens he company average invesmen risk. If he projec is deemed o have a risk profile differen from he firms overall risk, business execuives heurisically modify he company s WACC o accoun for he projec idiosyncraic risk. However, he coninued widespread pracice of heurisically adjusing he discoun rae o accoun for idiosyncraic risk and correlae he discoun rae o he overall risk of he projec has resuled in he deep-rooed believe ha higher risk can be accouned for by simply increasing he RADR. Because he effec of hese adjusmens reduces exponenially wih ime, i follows ha errors in he selecion of his parameer are magnified as he duraion of he invesmen projec increases. I is clear ha financial difficulies will arise 6
when funding for pos reclamaion care (PRC) beyond an originally budgeed imeframe is required paricularly when he mine has long sopped generaing any revenues. Therefore, o accrue funds more accuraely during he revenue-generaing life of he mining faciliy such ha an environmenally sound mine reclamaion program followed by a poenially very long erm PRC could be implemened, a valuaion procedure ha do no combine he ime value of money and risk is needed. From his discussion, i is clear ha he limied validiy of an exponenially decaying funcion o represen all sources of risk, regardless of he source, size o qualiies, makes i difficul o perform a meaningful analysis on long erm projecs. THE SEPARATION OF RISK AND TIME VALUE OF MONEY The Cerainy Equivalen Mehod (CEM) proposed by Robichek and Myers (1966) was he firs aemp o separae he ime value of money and risk. Despie is robusness, because of he difficulies o obain he necessary parameers o define he invesor risk preferences, he CEM has no been widely applied neiher in academia nor in pracice. Myers (1977) was he firs o connec he concep of opions o value growh opions embedded in poenial corporae invesmens and inroduce a promissory new valuaion mehod ha could address he main drawbacks associaed wih NPV: The real opion valuaion (ROV) mehod. In subsequen work, Myers (1984) ranslaed he logic of financial opions o address issues of capial budgeing and sraegic planning decisions and considered an exension of financial opion pricing models o he valuaion of nonfinancial asses (i.e., projecs). Since he publicaion of Myers (1977) seminal paper, ROV has been oued as he valuaion ool ha can ake ino accoun no only he changing naure of risk bu also he managerial flexibiliy embedded in invesing in real projecs while accouning for he ime value of money by discouning he cash flows using he risk free rae, effecively separaing he ime value of money from risk. Brennan and Schwarz (1985) were he firs o use ROV for valuing invesmens in naural resources such as mining. Since hen, numerous publicaions evaluaing invesmen in naural resources have been compleed (e.g., Paddock e al., 1988; Smih and Nau, 1995; Moel and Tufano, 1999; Corazar e al. 2001). However, mos of he ROV applicaions found in he lieraure are focused on capuring he value of managemen s flexibiliy o expand, conrac, defer and/or abandon projecs wih lile emphasis in he abiliy of ROV o model he changing naure of risk even for cases where here is lile o no flexibiliy. As a resul, mos applicaions have been mosly academic and difficul o implemen for pracical cases and more imporanly, is resuls difficul o convey o decision makers. Taking he advanage of he fac ha opions adjuss fuure cash flows for risk such ha hey can be discouned using he risk free rae, Salahor (1998) and Laughon (1998a) inroduced a concep ermed Modern Asse Pricing (MAP) o value invesmens in he oil and naural gas indusry. Alhough MAP is essenially derived from he work performed by Black and Scholes 7
(1973), he proponens of MAP focus he abiliy of opion pricing o adjus for risk raher han on is abiliy o model managerial flexibiliy and idenify he opimal policy. MAP uses informaion from fuure commodiy prices o esimae he value of fuure cash flows and adjus for he risk associaed o he volailiy of commodiy prices (Salahor, 1998; Laughon 1998a). To demonsrae he applicabiliy of MAP, sandard financial models ypically found in discouned cash flow (DCF) analysis were used for several mining invesmen opporuniies (Guj and Garzon, 2007; Shafiee e al. 2009; Samis and Davies, 2014). Alhough MAP focuses on risk adjusing he cash flows, i can also be used o model managerial flexibiliy (Laughon 1998a and 1998b). Hence, in heory MAP and ROV conceps are essenially he same and hey could be used inerchangeably. However, for his aricle, he erm MAP is used refer o he process inroduced by Salahor (1998) and furher promoed by Laughon (2000) ha is o use i as a valuaion ool for projecs wih lile o no managerial flexibiliy in which he only risk accouned for is marke risk (e.g., volailiy of fuure commodiy prices, foreign currency). Alhough MAP consiued a good aemp o rid he indusry from he problems associaed wih NPV providing a mehod ha is relaively easy o implemen, here are several drawbacks wih MAP, namely: (i) I only prices marke risk (e.g., commodiy risk; currency risk); and (ii) here are no provisions o include non-marke risks in a consisen manner. Galai (1977) was one of he firs o poin ou ha call opions can be viewed as a package of an insurance produc and an asse. Exending his concep o he valuaion of long-erm infrasrucure invesmens, Espinoza and Morris (2013) presened a mehod ha separaes he ime value of money. The mehod is ermed decoupled ne presen value (DNPV) and i is based on he sound concep of CEM and he synheic insurance concep which is used o quanify he risk of he projec (no he invesor s risk preference). DNPV considers risk as a cos o he projec (similar o paying for an insurance policy) and he price of risk is subraced from he poenial revenues (if he risk is associaed wih he revenues) or added o he cos (if he risk is associaed wih he cos). Considering risk as a cos iem o he projec is more in line wih he well-esablished business pracice of buying insurance producs o proec agains cerain insurable risks as well as wih he insurance concep uilized o describe call opions (Galai, 1977). The value of he invesmen is hen calculaed following a radiional cash flow mehod discouned a he risk-free rae because risks have already been accouned for separaely as a cos o he projec. In wha follows, a mining invesmen projec is described in his paper o illusrae how he DNPV mehod can be applied o mining invesmen opporuniies and how he synheic insurance for specific risks can be calculaed. Deails of he demonsraion for he differen insurance premiums used in he example can be found elsewhere (Espinoza 2014) whereas a pracical applicaion on an acual renewable invesmen projec evaluaion is given by Espinoza and Rojo (2014). 8
A MINING EXAMPLE CASE (SAMIS ET AL., 2003) Descripion To show he capabiliies of he proposed DNPV mehod applied o he mining indusry, an example of a greenfield copper mine projec aken from Samis e al (2003) is presened in his paper. Firs, he valuaion exercise performed by Samis e al. compleed using he informaion available a he ime back in 2003 is reproduced herein. Second, he value ha i would have been obained using he same informaion and DNPV mehodology is calculaed. The mining projec consised of a copper mine in a foreign counry wih an esimaed ore reserves of approximaely 400 million onnes and an average copper grade of 0.5%. The oal developmen cos was esimaed a $600 million a 2003 foreign exchange raes 4. The commodiy spo price (S), long-erm expeced spo price (S*) and foreign exchange (FOREX) rae a he ime of he analysis (i.e., 2003) were $0.8/lb, $0.85/lb, and $2 per foreign currency uni (F), respecively. The commodiy risk was assumed mean revering log normal process wih an assumed annual volailiy () of 20%, a median growh () of 0% and a mean revering facor () of 0.231. Equaion B.9 ogeher wih hese parameers are used o esimae he sandard deviaion as a funcion of ime. The assumed price of risk (i.e., he addiional reurn per uni of volailiy, ) was 0.4. The FOREX was assumed o follow a non-revering log normal process (i.e., geomeric Brownian moion) wih an annual volailiy of 40%. The nominal riskless rae for he hos counry was 12.5% wih an assumed inflaion of 7.5%. Similarly, he invesor s domesic nominal riskless rae was 3.0% wih an inflaion rae of 1.5% (i.e., he real rae is 1.5%). The projec cash flows were assumed o be subjeced o wo risks only: currency risk and commodiy price risk. Before performing an analysis using he proposed DNPV mehod, he resuls obained by Samis e al. (2003) were replicaed in Table 2 (albei wih somewha differen organizaion o faciliae he subsequen discussion) using he informaion summarized above ogeher wih he equaions presened in Appendix A. As summarized in Table 2, forward prices (Line 05) are calculaed using Equaions (A.2) hrough (A.6). Similarly, FOREX risk facors are presened in Line 06 in Table 2. To esimae he expeced operaing revenues, only he expeced commodiy prices (Line 03 of Table 2) and he expeced copper producion (Line 07) were needed. The esimaed copper producion rae was 20.4 million onnes (224.7 million pounds of copper) per year. 4 The example in Samis e al (2003) is described in erms of Foreign Moneary Unis (FMU) where he mine is locaed and domesic moneary unis (DMU) where he currency of he equiy s home counry is locaed. In his paper, he DMU is assumed o be he dollar currency ($) and he FMU is denoed by F. 5 See Appendix A for a descripion of each of hese parameers. 9
The projec oupu is seled in dollars whereas expenses are incurred in boh currencies. To faciliae he presenaion, he uninflaed OPEX and CAPEX (in real 2003 foreign and domesic currency) and operaing revenues are shown in he Table 2. The annual (real) operaing revenues (Line 08) is calculaed as he produc of he commodiy spo price (Line 03) imes he annual copper producion (Line 07) 6. The uninflaed (i.e., 2003 dollars) OPEX was esimaed a $0.4 per pound of copper which resuled in $89.87 (i.e., 224.7 0.4) million per year (Line 10 of Table 2): 25% of he OPEX in domesic currency (i.e., $22.47 million) and he remaining in foreign currency (i.e., 0.75 $89.7/2= F33.7 million). The annual foreign and domesic OPEX are shown in Lines 11 and 12 of Table 2. The projec required 4 years o develop. The corresponding CAPEX for he firs four years in real foreign and domesic (i.e., he invesors home counry) currency is also shown in Table 2 (Lines 14 and 15). Similarly, closure coss are assumed o be incurred in Year 24 (he las year of producion) in boh currencies and are shown Table 2 (Lines 16 and 17). The oal CAPEX is shown in Line 13. In wha follows, he mining invesmen opporuniy is evaluaed using hree differen valuaion echniques, namely NPV, MAP as proposed by Samis e al (2003) and DNPV. NPV Analysis To calculae he NPV of he proposed invesmen, Samis e al (2003) calculaed revenues using forecased expeced spo prices unadjused for risk. To accoun for currency risk, expenses in denominaed foreign currency were convered o nominal domesic currency using fuures 7. Thus, operaing revenues (in dollars) were obained by muliplying forecased mean spo prices (Line 03) imes copper producion (Line 07). Because forecased mean prices are in 2003 currency, he calculaed operaing revenues are also real. Nominal OPEX and CAPEX (Lines 20 and 22, respecively) in domesic currency (i.e., dollars) were obained by inflaing he expenses using he domesic inflaion rae of 1.5%. Similarly, nominal OPEX and CAPEX in foreign currency (Lines 19 and 21, respecively) were obained by inflaing he expenses using he foreign inflaion rae of 7.5%. Expendiures in foreign currency were hen ransformed o dollar denominaed fuures currency by adjusing he inflaed amouns by he FOREX facor (Line 06 of Table 2). The oal cos (he sum of Lines 19 hrough 22) in nominal dollars is presened in Line 18. The nominal Expeced Operaing Cash Flows (Line 24) is simply he Operaing Revenues (Line 09) minus he Toal Coss (Line 18). As discussed above, currency risk is accouned for by he use of fuures currency. Thus, he assumed RADR of 10% canno be properly jusified for a 6 Noe ha alhough Table 4 of Samis e al (2003) labels hese cash flows as nominal, hey represen 2003 currency (i.e., real) as changes of annual prices over ime are simply he resul of regression o he long-erm average of $0.85/lb expressed in real (2003) currency. 7 Typical NPV analysis would use boh revenues and expenses unadjused for risk and accoun for risk for boh FOREX and commodiy prices in he RADR. 10
greenfield projec beyond Weighed Average Cos of Capial (WACC) echniques bu i is inended o accoun for commodiy risk, and any oher risks impacing he expeced projec cashflows. To obain he projec discouned cash flows (Line 26 of Table 2), Samis e al (2003) risk adjused he resuling Operaing Cash Flows (Line 24) by muliplying hese quaniies by he coninuous represenaion of he risk facor (i.e., e -RADR ). The calculaed NPV using a RADR = 10% is $132.18 million (i.e., sum of Line 26 in Table 2). Comparison of Line 26 wih he las line of Table 4 of Samis e al (2003) indicaes ha he resuls were obained following he procedure described in his paper. The variaion of currency facor o esimae currency fuures conracs is presened in Line 06 of Table 2. As shown in his able, he currency facor decreases wih ime. Noe ha he use of fuure currency prices o accoun for foreign exchange risk in he MAP analysis appears o be inconsisen wih he noion of risk (i.e., higher risk, lower valuaion). In his example, expendiures in foreign currency makes he projec more valuable han if all expendiures were made in dollars. MAP Analysis To value he projec using MAP analysis, Samis e al. (2003) calculaed he operaing revenues using he values of fuures conrac prices insead of expeced commodiy prices. Concepually, his is similar o assuming ha fuure mining producion has been conraced a he calculaed fuure conrac prices, hus accouning for he risk associaed wih commodiy prices. Since copper fuures prices up o five years are ypically available, i is assumed he copper fuures daa from he firs five years in Table 2 corresponded o acual marke daa available a he ime and he remaining fuures prices from years 6 hrough 24 forecased using Equaion A.6 (Figure 2). To ha end, he risk adjused operaing revenues (Line 09) were calculaed as he mineral risk discoun facor (Line 04) imes he operaing revenues (Line 09). The mineral risk discoun facor (Equaion A.5) was derived by linking expeced and fuure prices. Risk adjused ne cash flow was calculaed as risk adjused operaing revenues (Line 09) minus he oal nominal expenses (Line 18) which are already risk adjused o accoun for currency risk. The valuaion using he MAP mehod (i.e., reducing he revenue cash flows by a facor and discouning he resuling cash flows a he risk free rae using he coninuous form of he discoun) is $262.42 million (he sum of Line 25 in Table 2). Noe ha he valuaion obained is higher han he $132.18 million NPV valuaion. The $262.42 million valuaion would imply he use of an 8.3% RADR for he NPV mehodology. In essence, he MAP analysis does no appear o inroduce significan addiional effor as compared o he NPV analysis bu raher focuses only on a single risk facor: he volailiy of commodiy prices. Alhough MAP analysis is a sep in he righ direcion as i promoes he separaion he ime value of money from risk, he mehod has several shorcomings ha limis is effeciveness. The mehod essenially assumes ha risk in commodiy prices are accouned for by using fuure prices in he cash flow analysis. In heory, his approach would be correc if: (i) he only risk associaed o he revenue side were due o commodiy fuure prices; and (ii) he 11
enire fuure producion shown in Table 2 is conraced a he esimaed fuure prices hus eliminaing he risk of commodiy prices. However, as applied, he MAP approach may creae inconsisencies depending upon wheher or no acual fuure conracs are used. For insance, he MAP analysis does no differeniae beween an invesmen proposiion where here is purchase agreemen in place and he oal producion is conraced a he esimaed fuure prices wih a similar invesmen proposiion where none of he producion is conraced. In boh cases, he same fuure prices would be used bu in he firs case, he invesmen would be obviously less risky. In he case when he risk is associaed wih expendiures, he use of fuures conracs o accoun for foreign currency risks may creae addiional inconsisencies in he valuaion analysis as discussed above. As fuure currency decreases wih ime, he value of fuure expenses are increasingly reduced. The impac of such an inconsisency is vividly illusraed in a companion paper when he value of PRC liabiliies are aken ino consideraion in he valuaion. In addiion, he MAP mehod does no accoun for oher han marke risks. As in he case of CAPM, MAP assumes ha invesors should no be compensaed for aking diversifiable risk, herefore, MAP does no accoun for non-marke risk. DNPV Analysis To make a direc comparison of he proposed DNPV wih boh NPV and MAP analysis, he cash flow analysis presened in Table 2 is evaluaed considering only he wo marke risks discussed in he original analysis (i.e., commodiy and exchange rae risk). Using Equaion B.10 o calculae copper fuure prices, daa for he firs five years was curve-fied and a convenience yield equal o 3.0% was obained. Alhough fuure prices are no used in he DNPV mehodology, he forecased values for years 6 hrough 24 is presened in Figure 2 for comparison purposes only. Using equal o 3.0% along wih Equaion B.9, a-he-money pus and calls were esimaed using Equaions B.1 hrough B.4. Assuming ha currency fuures follow a geomeric Brownian moion (GBM) process, i can be shown ha he is equal o (r f -r) (i.e., 9.5%). Table 3 shows he annual revenues as well as CAPEX and OPEX in real 2003 currency along wih he cos of risks associaed wih commodiy and currency prices calculaed using he equaions summarized in Appendices A and B, respecively. As predicaed by he DNPV mehodology, regardless he source of risk, risk always adversely affecs he value of he projec (Espinoza and Morris, 2013). For he wo risks analyzed, a drop in commodiy prices would adversely affec he revenue side of he equaion whereas an increase in he value of foreign currency agains he invesor s currency would adversely affec he expendiures (i.e., i would ake more dollars o buy he same services). Thus, for his invesmen opporuniy, a pu opion would represen he cos of risk associaed wih a drop in commodiy prices (Line 13 in Table 3) whereas a call opion on he value of he foreign currency would represen he cos of risk of appreciaion of he foreign currency (Line 14 in Table 3). The decoupled ne cash flows (Line 15 of Table 3) is obained by subracing hese coss of risk (Lines 13 and 14, Table 3) from he ne cash flows in real dollars (Line 11). Discouning he DCCF by he real domesic rae (i.e., 1.5%), he calculaed DNPV is $189.78 million. 12
Figure 2 Forecased Fuure Commodiy Prices The power on DNPV is no only in is abiliy o model marke risks such as commodiy and currency risks bu also is abiliy o evaluae he projec echnical risks and inegrae hem ino he financial analysis in a clear, ransparen and consisen manner. Alhough commodiy and currency risks are imporan sources of risk in a mining invesing projec, only considering hese risks o value mining invesmen opporuniies can lead o a significan overvaluaion. A summary of ypical risks ha could affec mining invesmens are presened in Table 4. To furher illusrae he power of he DNPV mehod and how hese risks affec he analysis, four addiional risks seleced from Table 4 (i.e., grade risk, OPEX risk, permanen shudown risk, and emporary shudown risk) are included in he financial evaluaion presened in Table 3. The cos of risks are compued using he equaions summarized in Appendix C and he resuls are included in Table 3 (Lines 17-20). The presen value of he oal cos of risk is $1,045.8 million. The percen conribuion of each risk is summarized in Table 3 and depiced in Figure 3. Figure 3 Risk Conribuion o Toal Risk 13
The decoupled ne cash flows for his case is obained by adding hese addiional coss o he cos of commodiy and currency risks and subracing he resuls from he ne cash flows. Discouning Line 21 by he real domesic rae (i.e., 1.5%), he calculaed DNPV is 46.12 million. This example shows how hese addiional risks can affec he value of he projec. For insance, even a small risk or permanen shudown due o expropriaion (0.1%) has a small bu noiceable effec in he invesmen valuaion (i.e., a presen value of $30.14 million). An increase of he likely of permanen shudown o 1% percen increases he cos of risk o $285.9, hus increasing he presen value of he oal cos of risk o 1,301.6 million (Figure 4) and reducing he DNPV o -$209.7 million. As shown in Figure 4, he risk of permanen shu down significanly increases from 1.9% of he oal cos of risk o 22%. This simple example shows how he increase of poliical insabiliy affecs he value of a projec. This risk can be miigaed by demanding o he hos counry a sovereign financial guaranee o be riggered in case of expropriaion. Figure 4 - Risk Conribuion o Toal Risk (for risk of permanen shudown equal o 1%) Crucially, DNPV provides a valuaion sysem o calculae echnical and operaional risks ha are largely disregarded by financial only mehodologies and consideraions (NPV, MAP, ROV) 14
Table 2 Mining Example from Samis e al. (2003) (in millions of dollars unless indicaed oherwise): NPV and MAP Analysis Time (yr) 0 1 2 3 4 5 6 7 8 21 22 23 24 01 Variance (Eq. A.2) 0.032 0.052 0.065 0.073 0.078 0.081 0.083 0.084 0.087 0.087 0.087 0.087 02 Median price [$/lb] (Eq. A.3) 0.810 0.818 0.825 0.830 0.834 0.837 0.840 0.842 0.850 0.850 0.850 0.850 03 Mean spo price [$/lb] (Eq. A.4) 0.8 0.823 0.840 0.852 0.861 0.867 0.872 0.875 0.878 0.887 0.887 0.887 0.887 04 Mineral Risk Discoun (Eq. A.5) 0.931 0.880 0.841 0.812 0.789 0.771 0.758 0.747 0.709 0.709 0.708 0.708 05 Forward Price [$/lb] (Eq. A.6) 0.766 0.739 0.716 0.698 0.684 0.672 0.663 0.656 0.629 0.629 0.629 0.628 06 FOREX Facor [$/F] (Eq. A.7) 2.000 1.819 1.654 1.504 1.368 1.244 1.131 1.029 0.935 0.272 0.247 0.225 0.205 07 Copper Producion (10 6 lbs) 224.67 224.67 224.67 224.67 224.67 224.67 224.67 136.56 08 Operaing Revenues (real) $194.805 $195.885 $196.695 $197.309 $199.321 $199.341 $199.357 $121.185 09 Risk Adjused Oper. Rev. (real) $153.667 $151.080 $149.020 $147.380 $141.359 $141.294 $141.243 $85.829 10 Toal OPEX (Real $) $89.87 $89.87 $89.87 $89.87 $89.87 $89.87 $89.87 $54.63 11 Foreign (F) F33.70 F33.70 F33.70 F33.70 F33.70 F33.70 F33.70 F20.48 12 Domesic ($) $22.47 $22.47 $22.47 $22.47 $22.47 $22.47 $22.47 $13.66 13 Toal CAPEX (Real $) $97.20 $178.25 $194.40 $113.45 $16.75 0 0 0 0 0 0 0 $50.00 14 Iniial (F) F36.45 F66.83 F72.90 F42.53 F6.08 15 Iniial ($) $24.30 $44.60 $48.60 $28.40 $4.60 16 Closure (F) F18.75 17 Closure ($) $12.50 18 Toal Cos (Nominal $) $97.20 $176.28 $190.16 $109.80 $16.10 $85.20 $84.36 $83.55 $82.77 $75.07 $74.66 $74.27 $86.05 19 Foreign relaed OPEX ($) $60.99 $59.78 $58.60 $57.43 $44.29 $43.41 $42.55 $25.35 20 Domesic relaed OPEX ($) $24.22 $24.58 $24.95 $25.33 $30.79 $31.25 $31.72 $19.57 21 Foreign relaed CAPEX ($) $72.90 $131.00 $140.08 $80.10 $11.22 $23.20 22 Domesic relaed CAPEX ($) $24.30 $45.27 $50.08 $29.71 $4.88 $17.92 23 Risk Adjused Ne Cash Flow -97.200-176.278-190.163-109.804-16.100 68.464 66.719 65.470 64.614 66.289 66.636 66.971-0.217 24 Expeced Operaing Cash Flow -97.200-176.278-190.163-109.804-16.100 109.602 111.523 113.146 114.542 124.251 124.682 125.085 35.139 25 Real Opions NPV -97.200-171.068-179.089-100.354-14.280 58.928 55.728 53.069 50.827 35.305 34.441 33.591-0.106 26 DCF NPV -97.200-159.503-155.692-81.345-10.792 66.477 61.205 56.187 51.467 15.215 13.815 12.541 3.188 15
Table 3 Mining Example (in millions of dollars unless indicaed oherwise): DNPV Analysis Time (yr) 0 1 2 3 4 5 6 7 8 21 22 23 24 01 Copper Producion (10 6 lbs) 224.67 224.67 224.67 224.67 224.67 224.67 224.67 136.56 02 Operaing Revenues (Real) $179.73 $179.73 $179.73 $179.73 $179.73 $179.73 $179.73 $109.25 03 Toal OPEX (Real) Millions 89.87 89.87 89.87 89.87 89.87 89.87 89.87 54.63 04 Foreign (F) 33.700 33.700 33.700 33.700 33.700 33.700 33.700 20.484 05 Domesic ($) 22.467 22.467 22.467 22.467 22.467 22.467 22.467 13.656 06 Toal CAPEX (Real) 97.2 178.25 194.4 113.45 16.75 0 0 0 0 0 0 0 50.00 07 Iniial (F) 36.45 66.825 72.9 42.525 6.075 08 Iniial ($) 24.3 44.6 48.6 28.4 4.6 09 Closure (F) 18.75 10 Closure ($) 12.5 11 Ne Cash Flows (Real) -$97.20 -$178.25 -$194.40 -$113.45 -$16.75 $104.94 $106.02 $106.83 $107.44 $109.45 $109.47 $109.49 $16.56 12 COST OF RISK 13 Commodiy (74.6%) $37.37 $39.79 $41.62 $41.85 $56.49 $57.08 $57.62 $35.33 14 Currency (11.7%) $14.73 $18.65 $11.25 $1.59 $8.51 $8.10 $7.64 $7.15 $2.44 $2.23 $2.04 $2.17 15 Decoupled Ne Cash Flows -$97.20 -$192.98 -$213.05 -$124.70 -$18.34 $59.12 $58.13 $57.34 $56.67 $50.52 $50.16 $49.83-20.94 16 ADD. COST OF RISK 17 Grade Risk (4.5%) $2.92 $2.93 $2.95 $2.96 $2.99 $2.99 $2.99 $1.82 18 OPEX Risk (5.2%) $3.49 $3.49 $3.49 $3.49 $3.49 $3.49 $3.49 $2.12 19 Temporary Shudown (2.9%) $0.75 $0.76 $0.76 $0.76 $0.77 $0.77 $0.77 $0.47 20 Permanen Shudown (1.2%) $3.32 $3.18 $3.03 $2.88 $0.69 $0.51 $0.31 $0.12 21 Decoupled Ne Cash Flows -$97.20 -$192.98 -$213.05 -$124.70 -$18.63 $47.77 $47.11 $46.57 $46.11 $42.58 $42.41 $42.27 -$25.46 16
Expendiures Revenue Table 4 Risk Idenificaion Analysis Source Parameer Poenial Risk Risk Magniude/Managemen CONCLUSIONS Ore Quaniy Mineral Grade Mineral prices Commodiy Producion Operaing Expendiures Capial Expendiures Reclamaion Cos Pos reclamaion care (PRC ) Cusodial Care Currency Less han esimaed from geological analysis Qualiy of he ore less han anicipaed Less han assumed in financial analysis. Temporary shudown (e.g., srikes); Permanen shudown (expropriaion, calamiy) More expensive o operae han anicipaed due o unforeseen condiions Higher CAPEX han anicipaed Expenses associaed o mine reclamaion cosing more han anicipaed Cosing more han anicipaed Taking longer han anicipaed (perpeuiy) Expendiures can be increased due a unfavorable change in he foreign exchange rae Moderae: risk assumed by he invesor. This risk can be reduced wih increased field exploraion Moderae: risk o be assumed by invesor. This risk can be reduced wih filed/lab exploraion Significan: risk can be shared among differen players. Porion of he producion could be conraced a prevalen fuure prices Moderae/high: Implemen a program ha moniors employee saisfacion, obain backed governmen guaranees o proec agains permanen shudown. Low: Operaions can be resrucured o improve efficiency. Low: Obain compeiive quoes from qualified vendors. High: Lack of reclamaion sandards may leave he door open for addiional mine reclamaion work High: Use well known echnologies; perform well designed pilo ess High: Generae income from closed propery o cover incidenals High: Ener ino fuures conrac agreemens, purchase equipmen when exchange raes are favorable A promising valuaion mehod called DNPV o value mining invesmen projecs is presened. DNPV inegraes echnical risk and marke risk and regardless of he sources of risk, risk is consisenly defined as a cos o he projec cash flows. Revenue risks reduces he value of he revenue whereas expense risks increases he overall expenses. Because DNPV requires projec risks be idenified and quanified, he effec of risk managemen measures (risk miigaion and risk allocaion) can be direcly evaluaed. A hypoheical example aken from he mining indusry is used o compare he DNPV valuaion wih popular echniques such as NPV and MAP. The inconsisencies in he analysis in boh mehods are discussed. The example is also used o clearly illusrae he negaive effec of using risk adjused discoun raes o value long erm mining invesmens. As discussed, he curren valuaion mehods, NPV in paricular, significanly discouns he value of fuure 17
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Samis M ad Davies G.A. (2014). Using Mone Carlo Simulaion wih DCF and Real Opions Risk Pricing Techniques o Analyse a Mine Financing Proposal. In. Journal Financial Engineering and Risk Managemen, Vol. 1, No. 3, 264-281. Shafiee S., Topal E., and Nehring M. (2009). Adjused real opion valuaion o maximize mining projec value-a case sudy using Cenury mine. Projec Valuaion Conference, Melbourne, 21-22 Arpil, pp. 125-134. Smih, J. and Nau, R. (1995). Valuing Risky Projecs: Opion Pricing Theory and Decision Analysis, Managemen Science, Vol. 41, No. 5, pp. 795-816. 20
APPENDIX A: MAP Formulaion Commodiy prices were assumed o follow a revering log-normal sochasic process given by (Salahor, 1998): 2 S ds 0. 5 ln Sd S dz (A.1) S * where S in he curren commodiy spo price; S * is he long-erm median spo price; is he shor-erm volailiy; is he shor-erm growh of he price medians, and is he rae of reversion. For his example, he parameers and were seleced o be in agreemen wih Samis e al. (2003), ha is 20%, 0, and 0.231, respecively. The variance (Var o ), median (M o ), and mean (E o ) of he commodiy spo price a ime T (S(T)) are given by he following expressions: M [ S( T)] o 2 2T Varo [ S( T)] 1 e (A.2) 2 S S exp( T ) * o T S exp 1 e * (A.3) E [ S( T)] M [ S( T)]exp 0. Var [ S( T)] (A.4) o o 5 The commodiy prices is assumed mean revering; hus, he risk discoun facor (R MR ) and he corresponding fuure price are given by (Salahor, 1998): o T R MR exp 1 e (A.5) Equaions (A.4) along wih (A.5) can be used o esimae he value of he commodiy fuure price as: o MR F E [ S( T)] R (A.6) The effec of he currency risk can be aken ino consideraion by muliplying Equaion (A.6) imes a currency risk facor (R FE ) which can be calculaed as (Hull, 2012, pp 115): R FE exp r r (A.7) F 21
where r F and r are he foreign currency and dollar risk free raes, respecively. APPENDIX B: COMMODITY AND CURRENCY COST OF RISK To calculae he projec DNPV, he downside poenial for boh he commodiy and currency risk needs o be esimaed. Equaions (B.1) and (B.2) shown below are he classical represenaion of an European call/pu opion developed by Black and Scholes (1973) which gives an invesor he righ, bu no he obligaion, o buy/sell a sock, bond, or any oher insrumen (S) a a specified price (called he exercise price, X) wihin a specific period of ime (T). P S C S T X r T f e N( d1) e N( d2) S X r T f T e N( d2) e N( d1) S where C and P are he prices of a call opion and a pu opion, respecively (i.e., he risk premium or price of risk) and he parameers d 1 and d 2 are calculaed as: ln( S / X) ( r ) T 0. 5 f d 1 d T 2 T (B.1) (B.2) (B.3) 2 d 1 T (B.4) T where is he dividend (or convenience yield) paid by he sock, bond, or any oher insrumen; is a measure of he volailiy (risk) of he sock; and he operaor N( ) is he cumulaive sandard normal disribuion funcion. Equaion (B.5) represens he sandard deviaion of he GBM sochasic process and i grows wih ime (i.e., risk increases he furher ino he fuure). Equaions (B.1) hrough (B.5) were derived for a GBM process of he form: ds ) T (B.5) ( Sd Sdz (B.6) where represens he sock expeced growh rae. Alhough Equaions (B.1) hrough (B.5) was derived for a GBM process, i has been shown ha similar expressions can be used o model nonlinear sochasic process (Espinoza, 2011). Thus, Equaion (A.1) can be rewrien as Equaion (B.6) by seing: Ln(s) (B.7) 2 * 0. 5 ln( S ) (B.8) 22
Equaions (B.6) hrough (B.8) represen he Ornsein-Uhlenbeck mean revering (MR) process. Differen from he GMB process, in he MR process, he convenience yield is no consan bu depends on he value of S. Approximae soluions for he opion prices can be obained from Equaions (B.1) hrough (B.4) wih he sandard deviaion given by Equaion (B.9) below insead of Equaion (B.5) o accoun for he fac ha he sandard deviaion does no increases boundless wih ime: 2 2 2 2 T T 1 e (B.9) Thus, because fuure prices can be observed from he fuures marke and can also be exrapolaed, i follows ha fuure prices (F T ) can be correlaed o he spo price wih an a-hemoney collar, ha is: F T S C( K,, ) P( K,, ) (B.10) o T T T T where S o is he curren spo price, K T is he srike price a ime T, (equal o S o ), is he commodiy convenience yield, C is he value of he call, and P is he value of he pu given by Equaions (B.1) and (B.2), respecively. Because Fuure prices are eiher known from marke daa or forecased from marke daa, Equaion (B.10) can be used o esimae he convenience yield. 23
APPENDIX C: OTHER COST OF RISKS Grade Risk To accoun for he echnical risk, he variabiliy in copper grades is assumed o be normally disribued wih a sandard deviaion equal o 0.02%. In his case, he downside is represened by he area o he lef side of he expeced value (i.e., he grades, and herefore revenues, lower han expeced). The probabiliy ha he copper grades is less han 0.44% (i.e., hree sandard deviaions less han he mean) is abou 1 in 740 (0.00135). The expeced downside for his copper grade is simply he cener of graviy of he downside (i.e., half of he bell disribuion) and i is calculaed a 1.5% of he esimaed revenues (Figure C.1). OPEX Risk Figure C.1 - Copper Grade Disribuion The variabiliy of he esimaed OPEX is assumed o be represened by a riangular disribuion wih a minimum, mos likely and a maximum of $0.35/lb, $0.40/lb and $0.54/lb, respecively. In his case, he downside is represened by he area o he righ side of he expeced value (i.e., he OPEX greaer han expeced). The expeced value of he OPEX for he assumed disribuion is $0.43. The esimaed cos of risk is calculaed o be $0.02 which corresponds o 3.88% of he expeced value (Figure C.2). 24
Figure C.2 OPEX Cos Disribuion Poliical Risk (Temporary Shudown Risk due o Srikes) Risks such as poliical risk can be modeled in a consisen manner using he proposed DNPV mehodology. For insance, if here is a chance ha mining operaions could be disruped a any ime (e.g., labor srike), his risk can be modeled as a binomial risk and does only affec he annual revenues. Assuming ha here is counry daa ha indicaes ha he likelihood of a srike is once every 10 years (i.e., = 10%) and he average srike lengh is 2 weeks. Then, he esimaed average producion loss in any given year would be 3.85% (2 weeks/52) of he expeced producion. This risk can be modeled as binomial. Figure C.3 represens his risk. As shown in his figure, he cos of risk is esimaed o be approximaely 0.385% of he oal revenues (his affecs only revenues no expenses, hey are coninued o be incurred). This risk can be considered independen and idenically disribued. Figure C.3- Temporary Shudown Risk Profile 25
Poliical Risk (Permanen Shudown Risk due o Expropiaion) A poenial poliical risk can be modeled as a Poison process (i.e., negaive jump) ha can resul in a permanen condiion as once he mine is expropriaed, i will never be reurned o is owners, hus affecing fuure poenial cash flows (his risk affecs boh revenues and expenses) from he ime he even akes place onward. The mehodology o esimae he cos of risk for his paricular risk is described in deail by Espinoza (2014). As described, he ineremporal relaionship of he cos of risk in a given year is capured by considering he remaining cash flows from ha year on discouned a a rae equal o r f + where r f is he risk free ineres rae and is he permanen shudown risk facor 8. Assuming Level I risk neuraliy as defined by Espinoza (2014), he risk facor can be esimaed as he expeced loss of remaining revenues. Assuming ha he counry where mining is aking place has a very small chance of expropriaion (0.01%) hen when he even akes place, he owners will no receive a fair compensaion for he expropriaed asse. Expropriaion risk can be esimaed from daa from mulilaeral organisms such as he World Bank, MIGA, and he likes. As in he previous case, his risk can be modeled as a binomial risk. As shown in Figure C.4, he risk facor is simply = 0.01% which can be used o esimae he cos of risk for each period. I follows ha as ime goes by, fuure remaining cash flows are reduced; hence, he risk associaed o expropriaion also decreases. Figure C.4- Permanen Shudown Risk Profile 8 Alhough he demonsraion included in Appendix A of Espinoza (2014) assumes consan revenues hroughou he period analyzed, i can be shown ha he same resul is obained assuming variable revenues. 26