# Box: Algebraic Description of the Alternative Methods of Valuing Mineral Resources

Current rent method I (Based on average return to capital):

GR = TR - COE

RR = GR - (rNS+DEP)

$\delta$ r = RR/QE

VR = $\delta$ r(QRES)

DEPL = $\delta$ r(QE)

VA = $\delta$ r(QADD)

REVAL = VA(t) - VA(t-1) + DEPL - VA

Current rent method II (Based on value of capital stock): *

$\delta$ GR = GR/QE

V = $\delta$ GR(QRES)

VR = V - NS

$\delta$ r = VR/QRES

Net present discounted value: *

$\Phi$ = $\sum ^T^_j=1_ \left\{1/T\left(1+i\right)^j-1/2^\right\}$

$\delta$ r = $\Phi$ [(V-NS)/(QRES)]

Replacement cost: *

bf = [(QE/QRES)/((QE/QRES)+r)]

$\delta$ r = bf[(TR-COE)/Q] - (ADD/Q)

Transaction price: *

$\delta$ GR = (TV/TQ)

$\delta$ r = $\delta$ GR - (NS/QRES)

[3]

Definitions:

Aggregate value measures:

TR = total revenue

CO = other extraction expenses, including compensation of employees, materials consumed, and overhead cost allocated to current production

GR = gross rent

RR = resource rent

NS = net stock of capital valued at current replacement cost

TV = value of purchased reserves during the year

V = value of the proved reserves (resource and fixed capital values)

VR = value of the resource stock

VA = value of the annual additions

DEP = depreciation

DEPL = value of the annual depletions

REVAL = the effect of price changes on the value of the stock

ADD = the annual exploration and development expenditures for drilling oil and gas wells in fields of proven reserves (including overhead costs allocated to development)

$\Phi$ = Net discounted present value factor

Quantity measures:

QE = quantity of the resource extracted during the year

QRES = stock of reserves

QADD = Quantity of resources added to reserves during the year (through new discoveries, extensions of existing sites, or revisions in estimated reserves)

TQ = quantity of proved reserves purchased during the year

Per unit measures:

$\delta$ GR = gross rent per unit (GR / Q)

$\delta$ r = resource rent per unit

Rates and other items:

r = real rate of interest, or discount rate

N = Life span of a resource (e.g., well or mine), R/Q

j = current year

T = life of asset (NIPA convention)

a = reserve decline rate, Q/R

bf = barrel factor

* DEPL, VA, REVAL for all methods are computed using the same formulas as presented for current rent method I.

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