A comparison of pricing models for mineral rights: Copper mine in China

Resources Policy 65 (2020) 101546

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A comparison of pricing models for mineral rights: Copper mine in China Chang Xiao a, c, Ionut Florescu b, *, Jinsheng Zhou c a

China Development Bank, No.18 Fuxingmennei Street, Xicheng District, Beijing, 100037, China School of Business, Stevens Institute of Technology, 1 Castle Point Terrace, Hoboken, NJ, 07030, USA c School of Humanities and Economic Management, China University of Geosciences, 29th Xueyuan Road, Haidian District, Beijing, 100083, China b

A R T I C L E I N F O

A B S T R A C T

Keywords: Mineral rights Mining valuation Futures Real options

Purchasing mineral rights to allow a corporation to extract minerals is a large investment and potentially very profitable. Generally, the assessment and valuation process is conducted by an official organization at the na­ tion’s governmental level. From a corporation perspective it is thus crucial to calculate an accurate value of the right to exploit a mineral product or what we call “mineral right”. In this work we compare three valuation methods and we compare their results in case of pricing a copper mine in China. We explain and apply the official valuation method based on a simple Discounted Cash Flow (DCF). We present what we believe is a more accurate valuation based on the term structure of future mineral prices: the so called modified Discounted Cash Flow (mDCF) method. Finally, we contrast with a method based on real options which considers the mineral right as a swaption. We implement two real options models: the classical Black model and a more modern Miltersen and Schwartz model. We highlight and discuss differences and commonalities between the three valuation methods. Finally, we make a recommendation for future financial reporting based what we learn from the three methods.

2010 MSC: 91B32 91B62 91B70

1. Introduction Generally, a company purchases the right to extract minerals through a government contract or license from the sovereign state which owns the territory. We shall refer to such a contract as the “mineral right”. This work focuses on methods used for pricing this mineral right. We shall exemplify the methods on a case study of pricing the mineral right of a copper mine in China. We focus on China since it is one of the countries with rich mineral resources, and at the same time the pricing methodology is less known than other states. A mining company that wants to develop and exploit a mine has two avenues to do so. First, the company may bid for the mineral right in an auction organized by the relevant state or provincial department. Sec­ ond, if the mineral right is already established, the company may pur­ chase said mineral right from the entity which owns it. In the first scenario only the initial pricing methodology is important while in the second scenario it is very important to have an up to date pricing methodology. Mining projects are large investment projects sometimes in excess of a billion dollars and once established it is hard to abandon. Therefore, having an accurate valuation model to price the mineral right or the remaining value of the mineral is important for both buyers and sellers of the mineral right.

Generally, the discounted cash flow (DCF) method is the most commonly used method to evaluate mineral rights. The method, initially proposed by Rappaport (1986), is used in China as the official valuation model. We discuss the official valuation model in section 2.1. Generally, the DCF method is suitable for projects with a stable cash flow. As we shall see, the DCF makes a series of assumptions about the numbers used in calculation. These numbers are issued by relevant departments of China and they do not change over the years. For example, Liu and Xie (2010) point out that using the same fixed value for the discount rate when valuating mineral rights for all types of minerals is unreasonable. In order to address the issues in the official valuation model, we introduce in section 2.2 a modified discounted cash flow (mDCF) method thus named to distinguish it from the official valuation method. In this method we use a term structure model to estimate expected future prices for the mineral product as well as the future term structure of interest rates. We use these prices directly and we produce what we believe a more up to date representation of the mineral right price. The Dual Risk Model (DRM) (Cramer, 1955; Albrecher et al., 2008; Ng, 2009) may be thought as an extension to the DCF model. In this model uncertainty comes not only from the random cash flow but also from the time of the payments, often modeled using a Poisson process. The DRM has recently been used to value pharma and petroleum

* Corresponding author. E-mail address: [email protected] (I. Florescu). https://doi.org/10.1016/j.resourpol.2019.101546 Received 9 January 2019; Received in revised form 5 November 2019; Accepted 18 November 2019 Available online 26 December 2019 0301-4207/© 2019 Elsevier Ltd. All rights reserved.

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companies, these companies have fixed cost but uncertain revenue. In our case, the copper mine produces monthly revenue from the mineral extraction. In fact, the monthly quantity of mineral extracted is gener­ ally planned well ahead. The value of the mineral extracted is typically the largest source of randomness. Thus, the DRM actually reduces to the mDCF model we employ in section 2.2. In section 2.3 we consider the mineral right as a payer swaption. We use two models to price it: the classical Black (1976) model, and a more modern model due to Miltersen and Schwartz (1998). In this method the mineral right is valuated as an instrument that pays a fixed amount and received a floating amount (the mineral value). Treating the mineral right as an option is referred in literature as a “real option” valuation method. Such methods have been implemented in literature though never as a swaption. Guj and Garzon (2007) price a nickel mine using a vanilla European call option using a Geometric Brownian Motion to model the underlying value of the mineral, a model introduced in Blais et al. (2005). Rebiasz et al. (2017) use the same Black-Scholes option pricing framework to price the value of a copper mine. Shafiee et al. (2009) use a binomial tree method to price a zinc mine as a European call with the underlying modeled using the DCF models of Colwell et al. (2002). In section 3, we use the DCF method, the modified DCF method and the real options approach to price the mineral right of a copper mine in China. The maturity of the mineral right is 20 years. All the information about the copper mine is obtained from the Ministry of Land and Re­ sources of the People’s Republic of China and the mining company which has an interest in purchasing the mineral right. We discuss the results and conclude in Sections 4 and 5.

Table 1 Notations introduced in equations (2) and (3).

t¼0

R¼

CIn ¼ ASP⋅Qn G þ RES

COt ¼ FAIt þ WCIt þ MCt þ OCt þ TAXt

(3) t ¼ 1; 2; 3; …; n

R1 ¼

APMonth APMonth Max Min � 100% Month APMin

R2 ¼

5 1X APpart APpart i i 1 � 100%: part 4 i¼2 APi 1

Table 2 The relationship between the time period of historical price data used to calculate average selling prices and the fluctuation range.

(2)

t¼n

(5)

Here, APMonth and APMonth are the maximum and minimum value of Max Min monthly average selling price respectively. To calculate R2 , the 60 months prior to the appraisal are divided into 5 equal periods (years). The APpart is the average selling price of the ith year. i We further note that the price of copper futures traded in China include the tax which is not a part of income. The ASP value used in the cash inflow equations (2) and (3), needs to reflect this adjustment. The empirical calculations presented in Section 3 exemplifies the adjustment. The formulas presented in this section are specific to calculating the price of mineral right in China. However, the calculation in equation (1) based on the Discounted Cash Flow method is used in many other countries, such as the United States (Bhappu and Guzman, 1995),

(1)

1

R1 þ R2 � 100%; 2

where

where P is the value of the mineral right, CFt is the cash flow at time t, CIt is the cash inflow at time t, COt is the cash outflow at time t. The parameter r is a discount rate which is generally larger than the risk-free interest rate. This discount rate parameter is a single constant which is specified in the official documentation (Chinese Association of Mineral Resources Appraisers, 2010). The cash inflow CI represents the income obtained from the sales of the extracted mineral product. The cash outflow CO includes but is not limited to construction costs, maintenance costs and operation costs. The official documentation does not have clear guidelines or formulas and thus we introduce the next equations to reflect the official docu­ mentation (which is verbose). CIt ¼ ASP⋅Qt G t ¼ 1; 2; 3; …; n

the average selling price of the mineral product the production level of the mineral product for period t the ore grade of the mineral product the recovery of residual value and working capital, the fixed asset investment for period t, the working capital investment for period t the maintenance cost for period t the operation cost for period t the tax for period t

The range calculation R is specified in the official documentation and for simplicity we express the calculation using:

In China, the official documents published by the department responsible for valuation of mineral contain descriptions of the DCF method presented next. We translated the official documentation and translated the very lengthy and wordy document to precise mathemat­ ical formulas. To our knowledge, this work is the first to detail this valuation process. The DCF method for pricing the mineral right is mathematically formulated as follows: n X CFt CIt COt ¼ ð1 þ rÞt ð1 þ rÞt t¼0

ASP Qt G RES FAIt WCIt MCt OCt TAX

1. For the mine with an estimated service life shorter than or equal to 3 years, the mineral product selling price is calculated as the average selling price for the last 12 months prior to the appraisal date. 2. For the mine with an estimated service life longer than 3 years and shorter than or equal to 5 years, the mineral product selling price is calculated as the average selling price for the last 24 months prior to the appraisal date. 3. For the mine with an estimated service life longer than 5 years, the calculation is done in two steps. First, the method estimates a fluc­ tuation range (R) of the historical selling price for the last 60 months prior to the appraisal date. Second, once the value of R is calculated, Table 2 is used to determine the time period used in the calculation of the average selling price.

2.1. Model 1: the official valuation model used in China

n X

Definition

is described. The method uses historical selling price data for the mineral product in the following way:

2. A description of models used for pricing mineral rights

P¼

Abbreviation

(4)

Table 1 presents the definition of all abbreviations used above. In the official documentation (Chinese Association of Mineral Re­ sources Appraisers, 2010), a model to estimate the average selling price 2

The fluctuation range (R)

The time period

1%–5% (including 5%) 5%–10% (including 10%) 10%–30% (including 30%) 30%–50% (including 50%) >50%

the the the the the

last 12 months of the appraisal date last 36 months of the appraisal date last 60 months of the appraisal date last 96 months of the appraisal date last 120 months of the appraisal date

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Canada (Moyen et al., 1996), Chile (Botín et al., 2011) etc. Each country uses specific guidelines when estimating the cash inflow and outflow in equation (1).

mineral product for period t, G is the ore grade of the mineral product, VATRt is the value-added tax rate for period t. TRt is the tariff rate for period t. We calculate the value of the cash inflows at t using: � � � � � Fyuan Qt G t �F 0 E CI ’t jF 0 � ¼ E ð1 þ VATRt Þð1 þ TRt Þ�

2.2. Method 2: the modified DCF (mDCF) method There are two issues in the official valuation method. First, the calculation of cash inflow is based on the historical selling price of the mineral. However, the inflow of cash (CI) is coming once the mineral right is purchased, thus the calculation should be based on an expected mineral selling price at future times. To this end, using the market value of future commodity prices may produce a more accurate result. Second, the official valuation uses a constant for the discount rate. This known constant is published in the official documentation. However, in general the discount factor is a random variable and again modeling it as a expectation of a stochastic process should improve the official valuation. Previous literature constructs a more accurate proxy for the discount rate using financial statements data. In this approach, financial in­ dicators are extracted from quarterly financial statements data then substituted into the traditional DCF method. For example, Capinski (2006) replaces the discount rate with the cost of equity. Cooper and Nyborg (2006) argues that when valuing a leveraged firm, the discount rate used in the DCF method should consider the leverage policy of the firm. Based on the assumptions in Miles and Ezzell (1980), the authors propose 4 alternative assumptions about debt policy and 4 valuation methods with different discount rates: the weighted average cost of capital (WACC), the cost of equity, the cost of debt and the unlevered cost of capital. The results obtained by the authors using the 4 methods of estimating the discount rate are similar. The financial statements data used previously is quarterly data. The data published in financial statements is in fact dependent on future prices of the mineral product. We think it is important to re-evaluate the future mineral product value on a daily basis. Further, the data extracted from financial statements does not contain specific information about the particular mine being priced, only on the general resource being mined (possibly) by multiple excavations. In our approach, we calculate the mineral right using a stochastic model and we calibrate these pro­ cesses to observed prices daily. Stochastic processes have been used in literature to estimate cash inflow. Samis et al. (2007) propose a stochastic discounted cash flow model to valuate a gold and copper mine. In their model, the taxable income is considered as a stochastic variable and it is used to estimate the value of cash inflow. We are extending the framework proposed therein to all processes involved. We would like to point out as well Jinfa and Biting (2017) who price R&D projects using stochastic processes. This work is relevant to our approach since we consider the price of the mineral right as a particular example of an R&D project. Specifically, the authors argue that a R&D project value may be divided into the tradi­ tional DCF part and a growth option which in their case uses a tradi­ tional Black-Scholes model. In their paper, the early time of R&D investment is priced as a growth option which gives investors the right to obtain the future growth of project income. This idea is similar with our approach based on real options and presented in the next section. The mDCF method based on the term structure model. The modified DCF method presented next uses a term structure model to calculate the value of the mineral rights. We note that the futures prices of the mineral product (in China) include the value-added tax and the tariff (import tax). These components should not be included in the cash inflow and we need to exclude them from the calculations. Specifically, the value of the mineral right is calculated using equation (1), where the Cash Inflow (without residual value) (CI’ ) at each time t is modified as follows: CI ’t ¼

Fyuan Qt G t ; ð1 þ VATRt Þð1 þ TRt Þ yuan

Here Ft

t ¼ 0; 1; 2; …; n

where we use the traditional conditional expectation notation and F 0 is the filtration (information available) at the time 0 when the pricing is done (Florescu, 2014). In general, the quantities appearing in the con­ ditional expectation may not be separated. However, if we assume that the ore grade G and the tax rates: VATRt and TRt are known at time 0 when the pricing is done (i.e., measurable with respect to F 0 ) we can calculate this expectation. We also assume that the mining quantity Qt is a yearly constant throughout the exploitation period. Under these as­ sumptions the conditional expectation becomes: � � � Qt G �F 0 E Fyuan t ð1 þ VATRÞð1 þ TRÞ Therefore, the only quantity we need to estimate for the cash inflow is the expected future price of the mineral product expressed in yuan. The cash outflow is paid at time t in the future, thus it needs to be estimated as well. To this end we discount the quantity using China treasury futures. Equation (1) becomes: � n � X E½CIt jF 0 � COt RES P¼ (7) þ n t ð1 þ CTYn Þ ð1 þ CTYt Þ t¼0 where CTYt is the China treasury yield with maturity t years, RES is the recovery of residual value and working capital. Calculating the Cash Inflow (CI) using a term structure model. To yuan � calculate the expected mineral price E½Ft �F 0 � in equation (7), we use a three factor model following Cortazar and Schwartz (2003). We prefer this particular model since it contains long-term factors which are suitable for a long-term mining investment. To present the model denote the spot price of the mineral product as S, the instantaneous convenience yield as y and the long-term spot price return of the mineral product as v. The model dynamics is expressed as: dS ¼ ðv y λ1 ÞSdt þ σ1 Sdz1 dy ¼ ð κy λ2 Þdt þ σ 2 dz2 dv ¼ ðaðv vÞ λ3 Þdt þ σ 3 dz3

(8)

with dz1 dz2 ¼ ρ12 dt;

dz1 dz3 ¼ ρ13 dt;

dz2 dz3 ¼ ρ23 dt:

In this model κ is the speed of reversion for y (the convenience yield) and the long term expected value for y is 0. Similarly, the expected value for the long-term spot price return v is ðv0 þ vtÞe t . The σ 1 , σ 2 and σ 3 are the volatilities for S, y and v respectively. The model was initially introduced in Schwartz (1997). The model is expressed under the equivalent martingale measure Q. For completeness we provide the details of the formulas we will be using. We can use the change of variables: b y¼yþ

λ2 κ

b v¼v

λ1 þ

v� ¼ v þ λ1

λ2 κ λ2 κ

λ3

thus the system dynamics in (8) become:

(6)

is the copper price at time t, Qt is the production level of the 3

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dS ¼ ðb v by ÞSdt þ σ1 Sdz1 dby ¼ κby dt þ σ2 dz2 dbv ¼ aðv� b v Þdt þ σ 3 dz3

filtering method introduced in Del Moral et al. (2001) to calibrate the unknown variable distributions given the parameter values. In this work we couple the two problems by solving a minimization problem for two variables (b y and b v ) and using the optimal values found to solve the 9 parameters minimization problem for the θ parameters. This coupled process is repeated until the difference between two successive minima is lower than a small tolerance level. We use the set of calibrated pa­ rameters each day to calculate a mineral right price specific to that day. This procedure produces much more stable parameter estimates from day to day. Data issues. In the futures markets in China the longest maturity for most of mineral futures contracts is under a year. In our valuation we need long maturity contracts (up to 20 years) for calibration. To this end, we construct the term structure for future prices using data from western exchanges. Specifically, we obtain the term structure of copper futures in China (expressed in yuan) using the London Metal Exchange (LME) prices. We convert the prices in yuan using the term structure of forward exchange rate for the USDCNY (US dollar to yuan). The longest maturity of forward contracts on the exchange rate USDCNY is 5 years, thus we need to calculate the term structure for USDCNY. We are following Campa and Goldberg (2005), Sarno (2005), Dal Bianco et al. (2012), Amer (2014):

(9)

This change of variables is needed since the model will be calibrated using the futures prices observed in the market under the objective probability measure P. Based on the stochastic process in (9) the futures on the mineral product must satisfy the partial differential equation (Cortazar and Schwartz, 2003): 1 2 2 1 1 σ S FSS þ σ 22 Fbyby þ σ 23 Fbvbv þ σ 1 σ 2 ρ12 SFSby þ σ1 σ3 ρ13 FSbv þ σ2 σ3 ρ23 Fbybv þ 2 1 2 2 ðb v

b y ÞSFS þ ð κb y ÞFby þ aðv�

b v ÞFbv

FT ¼ 0; (10)

subject to the terminal boundary condition: (11)

FðS; by ; b v ; T ¼ 0Þ ¼ S:

Here T is the maturity date of the commodity futures contract priced. This model has 9 parameters. These parameters are: (12)

θ ¼ fκ; a; v; σ 1 ; σ 2 ; σ3 ; ρ12 ; ρ13 ; ρ23 g

Cortazar and Schwartz (2003) solve the PDE analytically, and we reproduce their formula for the price of the futures contract: � 1 e κT 1 e aT FðS; by ; bv ; TÞ ¼ Sexp b y þb v κ a

σ1 σ2 ρ12 κ

2

κT þ e

κT

� σ2 1 þ 23 4κ

e

2κT

þ 4e

κT

þ 2κT

FERðTÞ ¼ logb ðT þ cÞ þ d

where FERðtÞ is the expected forward exchange rate at time T, and A, b, c and d are parameters. The future price of the mineral product in yuan is,

� 3

� σ2 av� þ σ1 σ3 ρ13 þ aT þ e aT 1 þ 33 e 2aT þ 4e aT þ 2aT 2 a 4a σ2 σ3 ρ23 � 2 aT þ a2 e κT þ κa2 T þ κ2 aT þ κae κT þ κae κ e κ2 a2 ðκ þ aÞ �� κae ðκþaÞT κ2 a2 κa

FðTÞyuan ¼ FðS; b y ; bv ; TÞ⋅FERðTÞ

3

�

Mi X fθg

Fij ðb y ðti Þ; b v ðti Þ; ti ; θÞ

F’ij

�2

i ¼ 1; 2; …; N

(16)

This concludes the implementation details. We have all the elements: 9 parameters in equations (12) and (4) parameters in equation (15) needed to apply the modified Discounted Cash Flow method (mDCF) (equation (7)).

(13)

aT

2.3. Method 3: the real options approach

Theoretically, this equation is all we need to estimate the futures price. However, implementing these formulas for the mDCF method requires some futher clarification. Implementation details. We note that in Cortazar and Schwartz (2003) the authors observe that some of the 9 parameters appear together in the pricing formula for the futures. They group these parameters in an intelligent way and create a new formula with only 7 parameters. This simplification is useful when calibrating the model to real data since the optimization problem will have two less variables. However, in this manuscript we compare three different pricing methods. To price real options (section 2.3) we need to calculate options written on the futures contract. The option formula used requires the whole set of 9 parameters and thus in our calibration method we need to estimate all 9 parameters. The model parameters are calibrated every day. In a particular day ti with i ¼ 1; …; N we observe Mi contracts, each with a different maturity. We note that the values b y and b v in equation (13) are realizations of unobserved stochastic drift processes and therefore unknown. In prac­ tical implementations they are often treated as extra parameters in the optimization problem resulting in a 11 parameter problem: min

(15)

Myers (1977) first proposed the concept of real options. Put in very general terms, a real option is any methodology that applies option pricing theory to value rights related to a financial investment. The real options approach was used to price mining projects since the late 1970s. In such work researchers consider mining management de­ cisions as embedded options. For example, delaying investment has been used in (Titman, 1985; McDonald and Siegel, 1986; Ingersoll and Ross, 1992), temporary closure in (Brennan and Schwartz, 1985), scaling up or down production in (McDonald and Siegel, 1985; Asad and Dimi­ trakopoulos, 2012), abandoning the mine in (Myers and Majd, 1990; Abdel Sabour and Wood, 2009; Haque et al., 2014). If a company owns the particular mineral right, they can choose to take more than one kind of decisions. The value of such multiple management decisions has been expressed as a compound option in Herath and Park (2002); Baranov and Muzyko (2015). In most manuscripts, the price of mineral right is expressed as a vanilla European option and is directly calculated most often using the Black-Scholes formula (Yu and Dai, 2005; Li, 2009). The results obtained using this very direct approach are several orders of magnitude greater than the results obtained using the DCF method. Sometimes, the numbers are 10 times larger. This observed discrepancy was argued in Guj and Garzon (2007) to represent the monetary value of the price risk discount. Liu (2014) argues that the huge differences is attributable to the management flexibility when exploiting the mineral right. We believe the reason for the difference is not economic but rather purely mathematical. Specifically, in option pricing theory derivative prices are expressed under the equivalent martingale measure and therefore the model parameters should be calibrated using option data. However, options data does not exist in the context of mineral rights. Instead, one needs to rely on the underlying instrument data to calibrate

(14)

j¼1

where we denoted with Fij the model price and F’ij the market price of the respective futures contracts (maturities Tj and times ti ). This approach often leads to instability and large variation in the parameter estimates from day to day. In Florescu and Viens (2008) we introduce an estimation procedure which distinguishes between the stochastic realizations and parameters. The method involves a two step 4

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parameters. This, in fact amounts to calibrating parameters under the real probability measure. Consequently, the price of market risk and the price of volatility risk (when using stochastic volatility models) are pa­ rameters that are hard to estimate. Most papers just use 0 for these and thus not subtracted from the drift parameters. This may artificially in­ crease the drift parameters of the model thus leading to increased valuation numbers. The real option approach based on the term structure model. In our approach we use swaptions as primary contract type to value mineral rights. This is in contrast with the traditional approach which uses Eu­ ropean option contracts. We next explain why we believe this is a more appropriate contract type for valuing mineral rights. The process to extract the mineral and obtain revenue from the mine may be divided into two distinct stages. In the first stage, the company acquires the mineral right and builds the needed mining infrastructure to start extraction. To do so it needs to secure a loan or sufficient funds that need to be repaid later. In China this stage may take up to 2 years (24 months) but in all countries the process is similar. Most investment loans have a grace period until the investment is completed and repayment of the loan only starts after the investment is completed. In the second stage, which starts after the completion of the first stage the company starts to exploit the mine and receives revenue from selling the mineral product. It however needs to repay the loan in addition to paying the running costs of exploiting the mine. For simplicity, we lumped all running costs into the fixed leg of the swap. During the first stage the company may decide to forego the in­ vestment and not to start the exploitation (in financial terms the swap contract). An option to enter a swap is called a swaption. The mining company that acquires the mineral right may be viewed as owning a European payer swaption. If the company exercises the swaption at the maturity of the first stage, it enters into the second stage where the mine is exploited. A contract that gives the owner the right to pay the fixed leg and receive the floating leg is called a European payer swaption. Euro­ pean payer swaptions are essentially priced as a series of call options (Clelow and Strickland, 1998). In this work we use two swaption models to price the mineral right using a real options framework. The first model we use is the traditional Black (1976) model: SWNðtÞ ¼ △τ

the same way as in equation (8). The Black model uses the geometric Brownian motion as the dynamics for the future price and we calibrate the parameters of the process to the observed future prices. The second model used is the Miltersen and Schwartz (1998). The underlying future price is the three factor model presented in Cortazar and Schwartz (2003). Under this model the swaption price may be expressed as: SWN ¼ △τ

i¼1

d1 ¼

d2 ¼

KNðd2Þ (18)

σi �

σ 2i 2

logðFð0; Ti Þ=KÞ þ αi

σi

where σ2 is �

σ 3 ρ13

σ2i ¼ σ21 t þ 2σ 1

a

� t

2�

σ2

þ

κ2

tþ

e

e

� e tþ 2

σ 23

þ � 2σ2 σ3 ρ23 t aκ

a

e

κTi

aTi

eat a

2κTi

e2κt

2aTi

e2at

ðeκt κ

1Þ

e

��� 1

σ 2 ρ12 κ

� 1 4e 2κ � 1 4e 2a aTi

ðeat a

� t

e

κTi

ðeκt

� 1Þ

aTi

ðeat

� 1Þ

1Þ

þ

e

ðaþκÞTi

κTi

ðeκt κ

ðeðaþκÞt aþκ

1Þ

��

� 1Þ (19)

and α is � � � � σ 1 e at σ3 e aTi ðeat 1Þ 1 e at e aTi ðeat e αi ¼ 3 σ 1 ρ13 t þ t þ a 2a a a a a �� � σ2 ρ23 e κTi ðeκt 1Þ 1 e at e κTi ðeκt e at Þ t þ κ aþκ κ a

Ffswap G Nðd1Þ ð1 þ VATRÞð1 þ TRÞ Xn Fð0; Ti ÞPðT; Ti Þ i¼1 Xn ¼ PðT; Ti Þ i¼1

� Þ

Here, SWN is the value of the swaption, the notation for all parameters have the same meaning as in the Black model above (equation (17)). The underlying forward price Fð0; Ti Þ is obtained using the Cortazar and Schwartz (2003) model. The strike price of the options is calculated as the average cost per metric ton of the mineral product. We estimate this cost using:

KNðd2Þ

(17)

Fð0; Ti Þ ¼ S0 erTi

K¼

. � � � log Ffswap K þ σ 2fix 2 ðT pffiffiffiffiffiffiffiffiffiffi d1 ¼ σ fix T t

at

(20)

n X Pðt; Ti ÞCðTi ÞQTi

CðTi Þ ¼

�

d2 ¼ d1

Fð0; Ti Þ eαi Nðd1Þ ð1 þ VATRÞð1 þ TRÞ � logðFð0; Ti Þ=KÞ þ αi þ σ 2i 2

CðTi Þ ¼

i¼1

Ffswap

n X Pð0; Ti ÞCðTi ÞGQTi

tÞ

MC þ OC þ TAX þ FAI Pn t¼1 Qt

RES

(21)

where MC, OC, TAX and FAI are total maintenance cost, total operation cost, total tax and total fixed assets investments. RES is the residual value of the fixed assets, Qt is the production (metric ton) of the mineral product in year t, n is the maturity of the mineral right.

pffiffiffiffiffiffiffiffiffiffi t

σ fix T

Here SWNðtÞ is the value of the swaption at time t, n is the number of reset periods of the swap, the ith transaction occurs at time Ti , the maturity of the swaption is T, △τ is the reset period, Pð0; tÞ is the zero coupon bond price, CðTi Þ is the value of the call option which maturity is Ti , QTi is the production (metric ton) of the mineral product from Ti 1 to Ti . The value Ffswap is the forward swap commodity futures price, G is the ore grade of the mineral product, VATR is the value-added tax rate of the mineral product, TR is the tariff rate (import rate) of the mineral product, K is the average cost per metric ton of the mineral product, Fð0; Ti Þ is the mineral futures price at time t ¼ 0 for the contract which maturity is Ti . Other parameters in the model, such as σ 1 , are defined in

3. Empirical comparison of the three mineral right valuation methodologies In this section we compare the official DCF method, the modified DCF method and the real options approach to estimate the value of the mineral right for a copper mine in China. The information about the copper mine studied was obtained from the Ministry of Land and Re­ sources of the People’s Republic of China. The income and cost data of the copper mining company in China is obtained from a private corpo­ ration which will remain anonymous for this study. The China treasury yield data is obtained from China Central Depository & Clearing Co., Ltd 5

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(CCDC). The data for the forward exchange rate of USDCNY is obtained from the Bank of China. The case study we consider in this empirical example is a copper mine with service time of 18 years and maturity of the mineral right of 20 years. The mine has estimated reserves of 133,107.54 metal metric tons of copper and the average ore grade is 1.39%. The construction period of the mine is about 2 years which essentially means that if the company purchases the mineral right, their first cash inflow will happen after the end of the 2-year construction period. The mining company can produce 21,600 metric tons of copper concentrate annually with the exception of the last year when they can produce only 8640 metric tons. The ore grade of the copper concentrate is 20%. The fixed assets in­ vestment (construction cost) is divided into 2 parts. The first part is 206,785,700 yuan and will be invested at the beginning of the first year. The second part consisting in 94,184,300 yuan will be invested at the beginning of the second year. The working capital is 36,471,300 yuan and will be invested at the beginning of the third year. The expected residual value which can be recovered is 81,616,800 yuan. The expected maintenance costs and operation costs are fixed and the annual total cost is 137,601,800 yuan except the last year of the exploitation when it is 55,040,720 yuan. The value-added tax rate is 17% and the import tariff rate of copper is 0 (in China case). Referring to Chinese Association of Mineral Resources Appraisers (2010), the official DCF method uses a discount rate of 8%.

respective day. The values (in yuan) of the mineral right obtained by the 4 models mentioned above is shown in Table 4 below. We can observe from these numbers that when pricing the mineral right using the real options methodology we obtain much larger values. This is a general observation in line with the real option methodology and a trend that is observed throughout the study. 3.2. The valuation of the same mineral right on other appraisal dates In order to have a complete picture of the performance of the models considered, we value the same mineral right every day from January 3, 2014 to October 30, 2017. We plot the results obtained in Fig. 3. On August 11, 2015, the central bank of China reformed the for­ mation of the onshore Chinese yuan central parity rate against the U.S. dollar. This is a change that the official Chinese news outlets called the “major improvment” in the exchange rate formation. Due to this reform the banks in China started to take into consideration the closing rate on the inter-bank foreign exchange market of the previous day and in turn, this had a major impact on the forward exchange rate market. To take this change into account, we divided the appraisal dates into 2 time periods. The first time period spanning from January 3, 2014 to May 26, 2015 is before the date of the foreign exchange rate calculation change. The second time period is after the change and spans from September 15, 2015 to October 30, 2017. For the period when the rate was changed (May 27, 2015 to September 14, 2015) we could not secure data thus that period is left blank in the image. We can see in Fig. 3 that the valuation based on real option approach is much larger than the Discounted Cash flow methods. There are two reasons for this. The first reason is due to data. As we noted the copper futures have a flat rate beyond 5 years. However, when we calibrate the model we cannot impose a flat rate to the resulting futures. Often that makes a large difference when pricing payments that are taking place 15–20 years in the future. In particular the difference is even more pronounced for the Black model since the expected term structure is

3.1. The valuation of the mineral right on a particular appraisal date To better understand the calculation we first compare the three valuation models on a particular day (Oct 30, 2017). This date is an arbitrary chosen date without any news of any significance. To extract the term structure of future copper prices there are about 123 copper futures contracts traded in LME for distinct maturities. The majority of the contracts has no trading volume and furthermore the settlement price of LME copper futures contracts with maturities longer than 5 years are usually identical. Fig. 1 exemplifies the term structure for three different days and the situation is similar for all days we investigated. In our calibration, we use the LME copper futures contracts with maturity shorter than 5 years. The mDCF parameters (section 2.2) are estimated using equation (14). For the real options approaches (section 2.3), appraisers evaluate the cost of the exploitation on an annual frequency. Thus we use a △τ ¼ 1. For the Black model, we use the realized volatility value which was 0.2 for that day. All values of the parameters estimated on May 8, 2017 are summarized in Table 3. Fig. 2 presents the calibration results for the

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simply S0 eðr σ =2ÞT . Thus if the constant r σ2 is positive the future price will grow exponentially as the time to maturity gets larger. The second reason why the real option approach gives much larger numbers was mentioned before. Mathematically, options are priced under the equivalent martingale measure since only under this measure the option price is a martingale. We need this property to express the current price of the option as an expectation of the payoff at maturity. This makes it necessary to calibrate the model parameters using option market data. However, this isn’t possible in the context of real options which lack such market data. Instead, we calibrate the model parame­ ters to the underlying futures data which are expressed under the real measure. To obtain risk neutral parameters we would have to also es­ timate the parameters λ1 , λ2 , and λ3 in Cortazar and Schwartz (2003) however at this time there isn’t a clear methodology to estimate the market price of risk and the market price of volatility risk. Generally, to obtain these parameters we still need some type of option data and recent work is just beginning to approach this methodology (see the Recovery Theorem, Ross (2015)). 4. Discussion 4.1. Relationship between copper futures listed on Shanghai futures exchange (SHFE) China and London Mercantile exchange (LME) In our empirical study, we use copper futures price listed on LME to calculate the term structure of copper futures in China. Fig. 4 compares the spot price (in dollars and without value-added tax) of copper in Jiangxi province, China (red) and the LME (black). Although the correlation between the prices is 0.9869, we can easily see that sometimes the difference in the two prices is relatively large.

Fig. 1. The term structure of LME copper futures price on 3 days: 09/08/2017, 09/28/2017 and 10/30/2017. 6

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Table 3 The values of the parameters used on the date 5/8/2017. The DCF method

The term structure model

Parameters

Values

r

0.08

Qt (2 < t < 20)

21600

Qt (t ¼ 20)

8640

G

0.2

VATR

0.17

TR

0

Unit

Parameters

Values

Parameters

Values

κ

1.8956

△τ

1

year

metric ton/year

a

0.3

n

20

year

matric ton

v�

0.03

t

0

year

σ1

0.4

T

2

year

σ2

0.1181

K

6487.18

yuan

σ3

0.0088

σfix

0.2

FAI0

206,785,700

yuan/year

FAI1

94,184,300

yuan/year

WKI

36,471,300

yuan/year

COt (2 < t < 20)

1

ρ12

Unit

Unit

1

ρ13

0.3

ρ23

137,601,800

yuan/year

A

0.5483

COt ðt ¼ 20Þ

55,040,720

yuan

b

10.9841

RV

81,616,800

yuan

c

1.9169

Construction period R1

2 68.44

year

d

6.5012

R

30.88

R2

The real options approach

6.68

Fig. 2. (a) The estimated term structure of copper futures prices and the observed copper futures price (LME) on May 8, 2017. (b) The observed and the estimated term structure of the forward exchange rate of USDCNY on May 8, 2017. Table 4 The values (in yuan) of the mineral right obtained by the 4 models on May 8, 2017. B. for Black model. M. & S. for Miltersen and Schwartz model. \Date

Official

mDCF

ROA (B.)

ROA (M. & S.)

May 8, 2017

125,299,775

133,576,119

1,222,860,545

535,760,254

The difference in prices come from two factors. First, the obvious dif­ ference from using the exchange rate to calculate the price in dollars of the spot copper in the Jiangxi province. We note that after the reform date (August 11, 2015) the prices are more aligned than before the re­ form date. Second, before October 2016 the demand of Copper in China was low. At the same time the copper reserves were high and this sup­ ply/demand unbalance created comparably lower spot prices in the Chinese markets. Starting with Oct 2016, the Chinese companies saw an increased demand for copper and we can see that reflected in both the rising copper spot price as well as in the difference in price between the two exchanges. We have mentioned the reform in the formation of the onshore Chinese yuan central parity rate against the U.S. dollar dating August 11, 2015. After this date, the central bank of China allowed the foreign

Fig. 3. The values of the mineral right obtained from the valuation models. The M.& S. stands for the Miltersen and Schwartz model, while B. stands for the Black model.

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4.2. A comparison of the DCF valuation methods In Fig. 6 we display the calculation result based on the official valuation model (DCF) and the mDCF method. We also plot the spot copper price in China (in yuan) to display a reference that is important for the mining rights valuation. Looking at the mDCF method we see that generally the direction of the valuations is keeping in line with the spot price and that the value oscillates around the official valuation method. Analyzing the official method results we point out two issues we believe are important. First, we can see that the trend of the spot price and the value from the official valuation model are usually not in the same direction. The correlation between the spot copper price and the official valuation model before and after the rate reform is 0.0455 and 0.2393 respec­ tively. For instance, we can see in Fig. 7 that between March 18, 2016 and August 18, 2016, the spot price of copper is relatively stable while the official valuation of the copper mine is declining. Since a major component in the official valuation method is the spot price of copper, the resulting mineral right value should stay relatively stable as well. Second, if we look at the values obtained using the official valuation method (Fig. 6) we see that the daily results seem to jump at seemingly random times. Upon investigation, the reason is related to the official valuation method particularly to the fluctuation range of the historical average copper price. The fluctuation range is defined as a type of average difference between maximum and minimum copper price (equation (5)). We can see in the summary Table 2 on page 6 that if the fluctuation range falls below 30%, the time period of price data used to calculate the average price will suddenly change from 96 months to 60 months. This change will drastically modify the average copper price which in turn will produce a jump in the mineral right value. A worse situation happens when the fluctuation range happens to hover around 30% (or any other cutoff value). In this situation the resulting mineral right value will exhibit jumps. Fig. 8 shows the relationship between the fluctuation range and the resulting value of the mineral right. We can see that as the fluctuation range value (red) crosses the dashed line cutoffs (30% and 50%) the official model valuation (black line) jumps. In contrast to the official valuation model, we believe that the calculation based on the mDCF method is able to reflect the price in­ formation of the copper futures on the appraisal date. The correlation between the spot copper price and the value obtained from the mDCF method before and after the exchange rate reform is 0.5945 and 0.9763 respectively. The values obtained from the mDCF model have no jumps. However, there is a major drawback of the mDCF method, specif­ ically the high volatility of the valuation results. Clearly, this is due to the frequent changes in the future price term structure and thus the model will not allow mining companies to keep relatively stable

Fig. 4. The spot copper price of Jiangxi province, China and LME from January 3, 2014 to October 30, 2017.

exchange rate of USDCNY to become more volatile and its effects may be seen in Fig. 5. This figure shows the settlement prices of 3 months LME and SHFE copper futures contracts. Before the date of the reform, due to the strict foreign exchange control, the prices (in yuan and without value-added tax) of copper futures listed on SHFE are lower on the average by 1004 yuan than their counterpart on LME. After the reform the average price difference is 251 yuan. The price correlation between the copper leading contracts which are traded in SHFE and LME are 0.9853 (before the improvement) and 0.9986 (after the improvement). In our work we use LME future prices for calibration. The reason is that LME lists longer maturity contracts and this produces better parameter calibration. We have investigated the difference if we use SHFE prices instead. Indeed due to the lack of data for long maturities the term structure curve may be very different for long maturities and in turn this will produce very unrealistic numbers. If we examine Fig. 5 we see a side by side comparison of the term structure curves calibrated using LME data (black) and SHFE data (red). We can see that since the maturities available in SHFE are very short the fit is not good especially for long term maturities. In fact the fitted curve using the SHFE data produces such large values that they do not even fit on the left image. We present the full curve with the actual prices in the right image. We should also mention that the SHFE fit is in dollars for ease of comparison and the values were obtained at a particular day (Jan 3, 2014). The particular day considered takes place before the reform in the exchange rate thus the difference in futures prices is considerable.

Fig. 5. The comparison of copper price term structure by calibrating parameters from the LME and SHFE data on January 3rd, 2014. 8

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Fig. 6. The values obtained from DCF methods and the spot copper price in Jiangxi province, China.

Miltersen and Schwartz model results using a flat term structure after 5 years. This confirms that the Black model is overestimating the future values for long term maturities. We believe the results also indicate that the Milterstein and Schwarts is a more appropriate model for the term structure of mineral futures.

financial statements. From this point of view, the official valuation model produces a stable price based on a historical average of spot prices and this is better. However, we suggest the mining companies use the mDCF model to produce a realistic value which may be averaged over a period of time. This value may be used in financial statements and we believe it to be an important reference for investors.

5. Conclusion

4.3. A comparison of real option valuation methods

In most countries the official valuation model for mineral rights is based on a discounted cash flow (DCF) method. This method typically use some measure of selling price of the mineral extracted projected into the future. As discussed, the official valuation model has 2 main disad­ vantages. First, the changes in the value produced does not keep up with the trends in the mineral future price. Second, the calculation results may jump quite significantly even though there is no significant change in the actual spot price of the mineral product. In this work we translate and detail official valuation method. We present what we believe to be a better alternative which we call the mDCF method. Finally, we compare with a methodology based on real options. We present the contract as a swaption which we believe is a more appropriate financial instrument. The mineral right is considered as an swaption and the process of exploitation after a certain

In Fig. 9 we present the official valuation model results, and the two real option methods presented in section 2.3. We also graph the spot copper price to have a baseline for the real option valuation methods. We see that as expected the 2 real options methods are dependent on the spot price movements. From Fig. 9, we see that the trend of the values obtained from the 2 models is similar. The values obtained from the real options methods are much higher than either the official method or the mDCF method. We also see that after the financial reform prices get closer but this may not be a statistically significant result. We refer to section 3.2 for a discussion of the possible reasons why the real option results obtained are much larger than any DCF method. We noted there that one of the reasons may be the discrepancy in the term structure for long maturity futures. We continue the discussion here by looking at the difference in prices when imposing a flat term structure after 5 years. Specifically, we are considering two fundamental models for the underlying term structure - the geometrical Brownian motion in the Black model and the three factor model of Cortazar and Schwartz (2003) for the Miltersen and Schwartz model. As we see in the market data, generally the futures with maturities longer than 5 years have the same exact price with the 5 year maturity prices. However, neither model can impose such constraint. The problem is more prom­ inent in the Black model which essentially has an exponentially increasing term structure. In Fig. 10 we present the mining right valuation results using the full term structure model (dashed lines) as well as imposing a flat term structure after 5 years (plotted using solid lines). We can see two interesting results. In general the values obtained using a flat term structure after 5 years are lower than the values obtained using the full model. This confirms that for longer maturities the models produce values that are generally larger than the observed future prices. The second observation is that the Black model results are much closer to the

Fig. 7. The trends of spot copper price in Jiangxi province, China and the calculation results obtained from the official valuation model from March 2016 to August 2016. 9

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Fig. 8. The calculation result based on the official valuation model and the value of the fluctuation range.

Fig. 9. The calculation results based on the term structure model and the spot copper price in Jiangxi province, China. The B. stands for the Black model, while M.& S. stands for the Miltersen and Schwartz model.

construction period is modeled using a swap. The use of swaptions is novel for the management science literature. We use two models suit­ ably modified for the mineral rights problem: the Black model and the Miltersen and Schwartz model. All the valuation models presented in this work may be used to value mineral right. However, transferring (selling) mineral rights in China is only possible after a one-year exploitation period, essentially three years after the company purchased the mineral right. Therefore, the mining company has to exercise the two year swaption before they can transfer the mineral right thus effectively negating the option inherent in the swaption. Consequently, if the company is obligated to exercise the option the contract essentially becomes a forward start swap contract or more specifically the mDCF method presented in this paper. Even though the swaption method is not extremely relevant for China we believe it is important to understand its use, especially for

other countries where transferring the mineral right is possible at any time. The official valuation documentation for Canada (CIMVAL), Australia (VALMIN), and South Africa (SAMVAL) mention the real op­ tion method as an acceptable alternative (see Table 2 on page 22 in 2003 version of CIMVAL (2003)). The documentation does not mention any specific model or reference. We believe that it is important to study such methods in specific model context and understand the consequences if they are used. When comparing the mDCF method with the official valuation method we believe the mDCF has some advantages. First, because of the low sensitivity to price changes, the official valuation model is not applicable to price a mine which is going to be re-sold (transferring mineral rights). Second, using an average value based on the mDCF method would produce more realistic values in their financial state­ ments, thus producing a more accurate share value for the investors than 10

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Fig. 10. The value of the mineral right obtained by using non-flat term structure model and the model with partial flat term structure.

the official DCF valuation. We need to mention the main drawback of the mDCF when compared with the official valuation method. mDCF produces a value that changes every day and is dependent on the expectation of the mineral value futures quoted on the market. This is clearly not good for tax purposes and official budgets that need to have a known budget planned in advanced. Thus the method is recommended as a necessary tool for mining companies when deciding on bidding for mineral rights in an official auction.

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