Equations: Oil and gas project economics

any units: a ratio of relative changes is dimensionless

Holds when: changes are small; for large changes, use the constant-elasticity model (below) everything else — incomes, technology, policy — stays unchanged elasticity depends on the time allowed: after months its size (ignoring the sign) is much smaller than after years

dimensionless; $s$ is a fraction

Holds when: demand and supply have constant elasticities: $Q = A\,P^{\varepsilon_d}$ and $Q = B\,P^{\varepsilon_s}$; the disruption multiplies supply at any price by $1 - s$ no stocks are released, and spare capacity enters supply only in the way $\varepsilon_s$ describes this is a teaching model: it shows roughly how large the effect is and in which direction it goes, but it is not a price forecast

prices in one currency, indices from one series

Holds when: the index describes the purchasing power of the money of whoever makes the comparison; for prices in dollars, this is the US consumer price index the annual average price is divided by the annual average index

reserves and production in the same units; for gas and oil together, in barrels of oil equivalent (gas converted into barrels of oil by its energy content)

Holds when: reserves of one category, usually proved, at the start and end of the period the period is a year or, better, several years the organic and full ratios are calculated and shown separately

years

Holds when: production is constant; in reality it declines, so the reserves are produced over a longer time, and the production “plateau” (the period of steady output) is shorter

US$ million / million bbl = $/bbl

Holds when: costs and additions over the same years, usually three to five years only companies with the same methodology of accounting for costs and reserves are compared

money for the same period

Holds when: state which investment has been deducted: after sustaining capex, the flow shows what the company can pay out without shrinking; after all capex, it shows how much is left to the company this year

dollars; production in barrels per day

Holds when: decline and capital intensity are constant; new capacity is brought on at the start of the year a rough estimate of the size: in reality new wells also begin to decline in their very first year

reporting currency

Holds when: the asset is carried at actual cost; revaluation is not considered here over the whole life, depreciation adds up to the initial cost minus the salvage value; the scheme decides only how this sum is spread over the years

any consistent units

Holds when: the life $N$ and the salvage value are estimated in advance and reviewed at least once a year; a review changes the depreciation of future years, and past years are not recalculated

any consistent units

Holds when: the balance never reaches zero, so at the end of the life it is written off in full, or the method switches to straight-line write-off of the balance

any consistent units: production and reserves in the same units

Holds when: reserves are not revised; on a revision the remaining cost is divided by the new remaining reserves, so only future years change (advanced section) the asset serves while it is being produced from: when production stops, there is no depreciation

reporting currency

Holds when: the smallest group of assets that generates independent cash flows is tested — for oil, usually a field or a group of fields with shared infrastructure under the standard, value in use is calculated from pre-tax flows and a pre-tax rate; the fair value of oil and gas assets is often estimated from the after-tax flow — as a buyer would see it under IFRS an impairment loss can be reversed if prices have recovered (except for goodwill, the extra price paid for a business above the value of its assets), but not above the value there would have been without the impairment

one currency and one date for all the terms

Holds when: non-controlling interests (shares in subsidiaries that belong to outside owners) and preferred shares, if there are any, are added to EV the obligations included in debt are the same for the companies being compared — for example, leases or an abandonment provision — otherwise the multiples are not comparable EV relates to the same assets as the measure in the denominator of the multiple

one currency, one price scenario, one rate and one date for all the parts

Holds when: each flow is counted once: if abandonment, taxes or overheads are already in an asset's flow, they are not subtracted again the parts are added only if the assets are independent; shared infrastructure and mutual benefits require a common flow $\sum_j V_j - PV_{\text{OH}}$ is the value of the business (EV), and NAV is what is left for the shareholders

any currency; current prices

Holds when: physical deterioration is taken as proportional to age; in practice it is also assessed from the condition of the object external (economic) obsolescence — the loss of value because the object has nothing to do — is taken into account separately, by comparison with the income the object can earn

any consistent units

Holds when: without revisions the formula coincides with $D_t = C_0\,q_t/R$; new investment is added to $B_{t-1}$, and the reserves it opens up are added to the remaining reserves

any consistent units

Holds when: the flow grows at a constant rate forever — the company keeps finding new opportunities that pay back the investment the flow of each year arrives at its end; with mid-year flows, as in the examples of the lesson, the result is multiplied by $(1 + i)^{0.5}$

any currency, the same for all items; all amounts refer to the same period

Holds when: the period includes all incremental receipts and payments of the project, assigned to the moment of their actual payment costs and taxes are written as positive amounts and are subtracted; depreciation is not in the formula because it is not a payment all items are expressed in the same monetary units: one currency and one kind of prices — nominal or constant

any currency; numerator and denominator in the same units

Holds when: the rate of inflation is constant; if it changes, divide by the product of the annual indices $(1+f_1)(1+f_2)\cdots$ for costs, use a suitable price index, general or specific to the sector; the oil price is set by the market and may not follow inflation

US$ million per year; all rows in the same kind of money (constant prices or nominal) and in one currency

Holds when: the same formula as in the lesson on cash flow, except that revenue is expanded into volume times price and abandonment is separated from capital expenditure each row is assigned to the year of actual payment; depreciation is not in the formula, because it is needed only to calculate the tax if there are several products, revenue is summed over the products

any consistent set: bbl/d and bbl per year, t/d and t per year, m³/d and m³ per year

Holds when: $q_t$ is the average flow rate over the year; if only the rates at the start and the end of the year are known, an average is taken that allows for the shape of the decline curve in the year of start-up and the year of shutdown, the actual number of days of operation is used instead of 365 $u$ allows for planned repairs and downtime; it is taken from the operating experience of similar facilities

volume and price in consistent units; the price in the same kind of money as the whole model

Holds when: transport is counted once: either deducted from the price or shown as a line of operating expenditure the price is a scenario assumption, not a prediction; the model must allow it to be changed in a single cell in a model in constant prices the price is set in constant prices: an unchanged nominal price means that in real terms it declines (Inflation: nominal and real money)

US$ million divided by million bbl gives $/bbl

Holds when: costs are divided into a fixed part and a part proportional to production all sums are in constant prices of one year

US$ million per year divided by $/bbl gives million bbl per year

Holds when: only what depends on the decision to stop is compared: next year's revenue and costs. Past capital expenditure and depreciation do not take part in the comparison income tax takes a share of positive profit and does not change the sign of the result, so it is not in the formula abandonment costs lie ahead under either decision; only their timing depends on it (see the advanced section) the formula holds while the denominator is positive: if the price after royalty is below the variable costs, every barrel loses money

any currency; the rate and the number of periods in consistent units of time

Holds when: interest is calculated only on the original sum and earns nothing in later periods the rate is constant for the whole term there are no additional deposits or withdrawals within the term

any currency; an annual rate with a number of years, a monthly rate with a number of months

Holds when: the interest stays invested and earns income at the same rate the rate is constant; if it changes, the factors of the individual periods are multiplied: $(1+i_1)(1+i_2)\cdots(1+i_n)$ interest is calculated once per period to which the rate refers

any currency

Holds when: the compounding periods are equal in length the interest of each period is capitalised

dimensionless fractions

Holds when: all interest is capitalised one-off fees and bonuses are not part of the comparison; if there are any, we account for them separately, in the full cash flow

any currency; the rates as fractions per year

Holds when: interest is calculated and capitalised continuously the rate is constant for the whole term

any currency, the same for all payments

Holds when: all payments are counted from one origin and assigned to moments by one convention one rate for all periods and for balances of either sign

any currency

Holds when: the growth rate is constant; if it changes, the annual factors are multiplied this is a forecast assumption, not a law: market prices, such as the oil price, do not change so smoothly

fractions for the same period

Holds when: all three quantities refer to the same period and are effective rates inflation is measured by a price index suited to the task

any currency; $PV$ and $FV$ in the same units, the rate and the time in matching time units (a yearly rate with years)

Holds when: The rate is constant over the whole term. If it changes, multiply the factors of the individual years: $PV = FV/[(1+i_1)(1+i_2)\cdots(1+i_n)]$. Interest is added once in each period of the rate (once a year for a yearly rate). The future sum is given. Any uncertainty in it is dealt with separately. $PV$ and $FV$ are in the same kind of money (both nominal or both in real terms), and the rate matches them.

any currency, the same for all payments

Holds when: All payments are in the same kind of money, and the rate matches it. The time of every payment is counted from the same origin, and the whole series follows one convention for when in the year a payment falls. The rate is the same for all years.

any currency; the rates and the length of the investment in matching time units

Holds when: There is one payment now and one repayment, with the accumulated return, in $n$ years. The rates $g$ and $i$ are constant over the whole term.

any currency; the rate as a fraction

Holds when: The rate changes only a little, so a linear (straight-line) approximation is good enough. All payments of the series have the same sign. For a series with payments of different signs, the weighted time loses its clear meaning.

any currency; $A$ and $PV$ in the same money units, and the rate is the effective rate for one payment period (the rate that really applies over that period)

Holds when: The payments are equal and come without gaps, once in each period of the rate: a yearly rate means yearly payments. The first payment is one period after the time origin and the last one $n$ periods after it. Other timings are handled with the factors of the next section. The rate is constant for the whole term. At $i = 0$ the factor equals $n$. The payments and the rate are in the same kind of money: a real rate (inflation taken out) for constant prices, and a nominal rate for nominal money, the money of each year (Inflation: nominal and real money).

$PV$ and $A$ in the same monetary units, $i$ a fraction per year

Holds when: A solution exists only if $i\,PV < A$. Otherwise the receipts do not even cover the interest, and the investment is never recovered. A fractional $n$ is a smooth (continuous) estimate. With end-of-year payments, the investment is recovered with the payment of the year whose number is the next whole number above $n$.

any currency; the rate effective per contribution period

Holds when: The contributions are equal and made at equal intervals. The income earned is added to the fund and earns the same rate itself. The sum is the value at the moment of the last contribution. To get its value at a later date, multiply it by $(1+i)^{k}$, where $k$ is the number of periods after the last contribution.

any currency; $g$ and $i$ are fractions for the same period

Holds when: The rate $g$ is constant for the whole term. For a declining series, $d$ is the fraction by which the payment falls in a year. At $g = i$ the formula is undefined, but the limit is finite: $PV = n A_1/(1+i)$. Growth exactly cancels discounting. A perpetual series ($n \to \infty$) is worth $A_1/(i - g)$ only if $g < i$. For $g \ge i$ the sum is infinite, which shows that the assumption of eternal growth is unrealistic.

US$ million = $/bbl × million bbl (bbl is the abbreviation for barrel); the rate as a fraction per year

Holds when: The price is constant in the money in which the calculation is made: in constant prices, a constant real price and a real rate. Revenue and production are placed at the same moments and discounted from one time origin, as in NPV (time origin and conventions). $Q_t$ are the volumes sold: production minus the oil that the producer uses itself, and minus losses.

fractions per year; prices in $/bbl, volumes in million bbl

Holds when: The rate of price growth is constant. The real rate is growth above general inflation (escalation). The price of year $t$ is $P_0(1+e_r)^{t}$, with the same time origin as for production.

$q_i$ is a volume per year, $D$ and $j$ are in 1/year, $T$ in years; $Q_\text{disc}$ in the same volume units

Holds when: The decline is exponential from the start of production, and the time origin is the start of production ($t = 0$). Discounting is continuous: $j$ corresponds to the effective annual rate $i$. Production stops at the flow rate $q_\text{lim}$. Without a limit ($T \to \infty$), $Q_\text{disc} = q_i/(D + j)$.

million bbl per year; divide by 365 to get the daily production

Holds when: The production years, the rate and the convention for the timing of flows are the same as for the profile. Equivalence is understood at a constant price: both productions bring the same present value of revenue.

the rates are fractions per year; $E$ and $D$ are in the same currency

Holds when: The weights are market or target shares of capital, not book (accounting) shares. The project has a risk similar to the average of the company. A project that is clearly more or less risky needs its own rate. The rate is applied to the project flow after tax, without loans, interest and repayment of debt. The cost of debt is already in the rate itself. The tax saving on interest arises only if the company has taxable profit and the interest may be deducted. All the rates are of one kind: usually nominal and in one currency, like market rates.

fractions per year; all rates in one currency

Holds when: Investors are diversified and ask for a premium only for market risk. Beta is estimated from the share price history of the company or of similar companies, and the premium from the long-term history of the market. Both estimates are imprecise. For emerging markets, a country risk premium is often added to the result. This is a practical addition, not a part of the original model.

dimensionless; the rates are fractions per year

Holds when: The rate with the premium is constant in time. Both rates are of the same kind, nominal or real, and in the same currency. The flow is the expected (average) value, not an optimistic case.

dimensionless; the index values are from one series

Holds when: The values are comparable: from one series with one base, both yearly averages or both on the same date of the year. The index suits the task: for drilling costs, an industry index can behave differently from the CPI.

any currency; the indices are from one series

Holds when: The index matches the currency of the sum: sums in dollars are converted with a dollar price index, such as the US CPI, and sums in tenge with the Kazakhstan index. The second form holds for constant inflation. If inflation changes, divide by the product of the yearly factors $(1 + f_1)(1 + f_2)\cdots$. An industry index gives the constant prices of a single item. This already takes escalation into account (see the section below).

fractions for the same period; the rates are effective rates

Holds when: Inflation is measured with an index that suits the currency and the task. In forecasts, $f$ is the expected inflation. If actual inflation turns out different, the real return of deals with a fixed nominal rate changes too.

the currency of the flow; the rates in fractions per year

Holds when: The base year of the constant prices coincides with the origin. Otherwise the two NPVs differ by a constant factor, a conversion into the prices of another year. The nominal rate and the nominal flow are built on the same assumption about inflation. All items of the flow, including taxes, grow together with general inflation. Where this is not so, see the section on taxes.

fractions per year; prices in any currency

Holds when: The rates are constant. If they change, multiply the yearly factors. Escalation is an assumption of the forecast. We base it on industry indices and the terms of contracts, and we do not simply carry past rates forward.

the currency of the cash flow, in constant prices of one base year

Holds when: NCF includes all items of the project after tax, including the abandonment costs, and it does not include loans and the interest on them. All flows are expressed in constant prices of one base year. Nominal sums of different years must not be added. The value is taken at the end of the year: by then all payments of the year have been made. There is no discounting: a dollar of the first year and a dollar of the tenth year are added one to one.

the currency of the cash flow

Holds when: The project flow does not include loans and interest: MCO shows the need for money before the decision on where to get it. An annual model sees the position only at the end of each year. If big payments fall at the start of the year, the minimum inside the year is deeper. For the financing plan, a quarterly or monthly model is used. For comparing projects, use constant prices. For the financing plan, use nominal sums.

years; the flows are in one currency and in constant prices

Holds when: The flow of year $n+1$ arrives evenly through the year. An annual model cannot say anything more exact. After the crossing of zero, the cumulative flow stays positive. If late costs push it below zero again, the first crossing does not count as payback. Time is counted from the start of the first year of the project. Another starting point must be stated.

dimensionless; the numerator and the denominator are in one currency and in constant prices

Holds when: Both quantities are taken from one curve of the undiscounted cumulative flow in constant prices. The project has a period of negative cumulative flow. Otherwise the denominator loses its meaning. The time at which the money is received is not taken into account.

money in one currency and in constant prices; the rate in fractions per year

Holds when: One investment and equal receipts at the end of each year. In real projects the production falls, and the link is only approximate. If $N \le I/A$, the receipts do not exceed the investment, and there is no positive IRR.

the currency of the cash flow, for example US$ million in constant prices of the base year

Holds when: NCF includes all the incremental flows of the project after tax (the extra money that the project brings or takes), including abandonment. It does not include financing flows: loans, interest and their repayment. The rate matches the kind of money: a real rate for constant prices, a nominal rate for nominal money. The rate is constant. If it changes from year to year, the factors are multiplied year after year. All flows are assigned to moments by one convention, the end or the middle of the year (Discounting and present value).

dimensionless; the numerator and the denominator are in the same money, at one rate and with one origin

Holds when: NPV and MCO are calculated from one cash flow, at one rate and with one origin. The project has a period of negative cumulative flow (otherwise the denominator loses its meaning).

US$ million divided by million bbl gives $/bbl; the rate is a fraction per year

Holds when: The costs and the production are discounted at one rate, from one origin and by one convention for the moment of the flows. The calculation is in constant prices with a real rate, and $c$ is a constant real price. A nominal calculation is covered in the advanced section. Payments to the state are not included: these are costs before tax.

US$ million; prices in $/bbl, $Q_\text{disc}$ in million bbl

Holds when: The price is constant in the money of the calculation. All quantities are at one rate and with one origin. There are no taxes. With taxes, the role of $c$ is played by the break-even price (the next section), and the slope of the NPV against the price is reduced by the share of the royalty and the tax.

all the discounted sums are at one rate $i$ and from one origin; $r$ and $T$ are fractions

Holds when: The royalty is a constant share of revenue, the income tax is a constant share of profit, and at the price $P^{*}$ the profit is positive in every year of production. Otherwise the price is found by trial in the model. The capital expenditure reduces the tax through depreciation, the operating expenditure reduces it in the year it is paid, and abandonment does not reduce the tax, as in the assumed regime of the examples. All sums are in constant prices at a real rate.

any currency; $A$ and $C$ in the same money, and the rate is effective for the period of payment

Holds when: The payments are equal and arrive at the end of each year, the first one a year after the date to which the investment is assigned. For other payment moments, the factor is recalculated in the same way as the annuity factor (payment in advance, mid-year payment, delayed start). The rate is constant. In constant prices it is a real rate, and the charge is constant in real terms. In a nominal calculation it is a nominal rate, and the charge is constant in nominal money and falls in real terms (Inflation: nominal and real money). At the end of its life the asset is worth nothing. The residual value is treated separately (see the section on equivalent annual cost). Income tax is not included: this is a charge before tax (tax is in the advanced section).

shares of the investment a year

Holds when: The same as for the capital recovery factor. The contributions to the fund for the return of capital earn the same return $i$.

any currency; $t$ of all the sums is counted from one origin

Holds when: The investments and the payments are assigned to moments by one convention, for example the middle of the year. If payments are made for $n$ years in a row and the first is a year after the chosen date, the denominator, brought to this date, equals $a_n(i)$. Then the formula becomes $A = C\cdot \mathrm{CRF}_n(i)$ with the investments brought to this date.

US$ million divided by million bbl gives $/bbl

Holds when: The charge is paid in the years of production, and $q_\text{eq}$ is calculated for the same years (level-equivalent production). With level production $q$, the denominator equals $q$, and $c_K = A/q$. This is part of the costs before tax. The operating costs per barrel are added separately.

any currency; all sums in the same money

Holds when: At the end of its life, each option is replaced by the same one as many times as needed, endlessly or up to the common multiple of the lives. Only then are equal yearly costs comparable between options. The options provide the same service: the same volume, the same reliability. If not, the difference is taken into account in the flow. With constant operating costs $O$, the formula simplifies to $\mathrm{EAC} = \mathrm{CRF}_n(i)\,C + O - S/s_n(i)$.

any currency; the rate is a fraction per year

Holds when: Over the life, depreciation adds up to the investment: $\sum D_t = C$. The return is charged on the balance of the previous year at the same rate $i$ that is used for discounting.

any currency; $T$ is a fraction, and $i$ is a fraction per year

Holds when: The margin is taxed every year, and the depreciation deductions are used in the year they arise: the owner has enough profit. All sums are in the same money at one rate. If tax depreciation is calculated from the historical cost in nominal money, its real value is lower under inflation (where inflation is not neutral).

$K$ in dollars, $q_i$ in bbl a year, $D$ and $j$ in 1/year, $T$ in years; $c_K$ in $/bbl

Holds when: The production rate falls exponentially from the start, and the investment is made at the moment production starts. Discounting is continuous. Without an economic limit ($T \to \infty$), $c_K = K\,(D + j)/q_i$.

the currency of the cash flow; IRR is a fraction per period between the flows (a fraction per year for annual flows)

Holds when: The flow is the project's net cash flow after tax, without loans and the interest on them. Otherwise the result is the return on the owners' own capital, not the return of the project. An ordinary flow (first minuses, then pluses: one change of sign) has exactly one root, and it is positive if the sum of the flow is greater than zero. If the sign changes several times, there can be several roots or none (see the section below). All flows are placed at moments by one convention. Shifting all the flows in time by the same amount does not change IRR.

rates are fractions; NPV and its derivative are in the currency of the flow

Holds when: The NPV profile is smooth and not horizontal near the root. The starting guess is close enough to the root. If there are several roots, the method finds the one to which the starting rate leads.

the currency of the cash flow

Holds when: Interest is added at one rate, IRR, on a balance of either sign. The flows are one period apart, and IRR refers to this period.

the currency of the flow; MIRR is a fraction per period

Holds when: The positive flows are invested at the reinvestment rate until the end of the project. The negative flows are covered in advance by money invested at the finance rate. This is how the MIRR function of a spreadsheet defines it. Other variants exist, so state the rates next to the result.

US$ million in constant prices; the rate is real, a fraction per year

Holds when: Both forecasts are at the same prices, costs, tax rules and rate. The forecast “without the project” is the expected development of the field without this decision, not its current state. Both forecasts contain all the lines that the project changes, including tax, the period of operation and abandonment.

US$ million; $\Delta Q_t$ in million bbl, $m$ in US$/bbl

Holds when: The simplified regime of the example project: royalty is a share of revenue, income tax lets the investment be depreciated, and abandonment does not reduce the tax (payments to the state in the model). In both cases every year of production is profitable, and there are no carried-forward losses. Otherwise the tax is calculated from the full forecasts. The price, the royalty rate and the variable costs per barrel are the same in both cases.

US$ million; $\Delta Q_\text{disc}$ in million bbl, $m$ in US$/bbl

Holds when: The same as for the incremental flow under a simple tax regime. Fixed costs and the time of abandonment do not change. For pure acceleration $\sum_t \Delta Q_t = 0$, so $\Delta Q_\text{disc}$ is also zero at $i = 0$.

dimensionless; all sums at one rate and one time origin

Holds when: The denominator holds only what is in short supply: usually the investment of the decision year. The late shortfall, operating costs and taxes go into the numerator. The projects are independent and divisible. Then choosing them in order of falling PI gives the largest total NPV. Indivisible projects are checked by trying all the combinations, and limits in several years are handled by optimisation methods. Mutually exclusive variants of one project are compared not by PI but by NPV, for example through the incremental flow “larger variant minus smaller variant” (IRR and NPV when choosing between options).

$N$ and $\Delta Q_\text{disc}$ in the same volume units; $D$ and $j$ in 1/year

Holds when: The decline is exponential before and after the treatment, the reserves do not change, and there is no economic limit. Discounting is continuous, from the moment of the treatment.

any consistent currency; in the examples, US$ million in constant prices

Holds when: All flows are discounted at one rate $i$, the one that the investor requires from an investment with this risk. The higher the rate, the smaller the rent. The rent is calculated at the moment of the decision to develop. Money already spent on exploration is not in it. The full-cycle rent is calculated with that money included (full-cycle cost). The equality of the split holds at any rate: in every year, the flow before payments is equal to the sum of the payments to the state and the investor's flow. Production and costs do not depend on the terms of the split. In reality, a high payment on revenue stops production sooner and reduces the rent itself.

US$ million, US$/bbl, million bbl; rates as fractions

Holds when: Royalty is charged on revenue and is deducted from the income tax base, so each point of the rate costs the investor the share $(1 - T)$ of its present value. The income tax base stays positive in all years of production. If losses appear, the tax saving is carried forward to future years, and the formula gives too high a limit rate. The price is constant, and production and costs do not depend on the royalty rate.

US$ million (or barrels, by dividing by the price)

Holds when: This is how the simple production sharing of the example works. Real agreements have different limits for different costs, uplifts on recoverable costs (extra percentages added to them), and special rules for bonuses and abandonment. What is not recovered this year is carried to the next: $K_t^{\text{r}} = K_{t-1}^{\text{r}} + C_t + O_t - \mathrm{CO}_t$.

US$ million

Holds when: In the example, the share is constant and there are no other payments. Often it is made to rise in steps, with the volume of production or with the ratio of the contractor's accumulated income to its costs.

any currency unit; the rate and the delay in consistent units of time

Holds when: all the flows of the project move, including the investment. This happens when the waiting falls before the decision to develop. if the investment has already been made and only the revenue waits, the loss is larger. It is calculated by moving the revenue alone. the costs of waiting itself (keeping the team, renting equipment) are not included. They make the delay even more costly.

the project and actual values in the same units and for the same period

Holds when: indicators with a range or a limit (pressure, gas–oil ratio, voidage replacement) are compared with the boundary of the range. What matters is whether the indicator went past it. the deviation is calculated for each indicator. The Code ties the 10% threshold to the conclusion of the analysis that the project must be changed.

SI; the gas volume at the standard conditions to which gas is referred in Kazakhstan

Holds when: the burning is complete: every carbon atom becomes CO₂. The CO₂ already in the gas passes through the flare and is also counted. in reality a flare does not burn all the gas, and unburned methane is a much stronger greenhouse gas, so the formula understates the harm from a flare. the gas is treated as ideal. At atmospheric pressure the correction is a fraction of a per cent.

money in constant prices and in one unit (in the example, US$ million a year: $P\,V$ and $c\,m_{\mathrm{CO_2}}$ are converted from dollars to millions); gas volumes at 20 °C and 101.325 kPa

Holds when: the investment is converted into an equal annual capital charge. Taxes are not included. the gas sold replaces the same volume of gas that would otherwise be produced and burned elsewhere, so the emission avoided equals the emission of the flare. If the replacement is partial, or the gas it replaces has less carbon, the emission avoided and society's gain are smaller. If the gas is injected into the reservoir, the emission is avoided without this assumption, but there is no revenue. the value of the gas to society is taken as equal to the price $P$.

any currency unit

Holds when: the other terms of the bids are equal, and the price decides the winner. the reduction is conditional: the payment is at the bid price, not at the conditional price.

any currency unit

Holds when: the shares $s_k$ of the contracts have already been calculated by the unified methodology. Here they are only added up with the contract values as weights. goods are not part of this requirement: it applies to works and services (Article 28).

any consistent currency unit; in the examples, US$ million in constant prices

Holds when: both takes are calculated for the whole life of the project. Bonuses and other payments before production are in $G_t$, not in the costs. the denominator of $\mathrm{GT}_\text{disc}$ is the rent (resource rent). For a project that is almost unprofitable, it is close to zero, the take goes beyond 100% and loses its meaning. Such a project is judged by the investor's NPV. if a state company takes part in the project, $G_t$ includes its cash flow, with its investment counted with a minus sign.

fractions

Holds when: each next tax is charged on what is left after the previous ones. The order does not matter for the formula. For example, the royalty is deducted from the base of the rent tax, and both from the base of the income tax. In the Kazakhstan excess profit tax, on the other hand, CIT is deducted from its base. the project already pays all these taxes, and the increase does not move it to another tier of a scale or change the deductions (depreciation, carried-forward losses). the increase comes with no extra costs: it is an increase of the flow before payments, not of revenue.

fractions per US$ million

Holds when: $\Pi > 0$, and the payments change smoothly with the flow. At the steps of scales the take changes with a jump. the same is true for the take in the rent, if present values are used instead of sums.

fractions

Holds when: a year in which the costs are not yet recovered and the recovery limit is reached. There are usually no income taxes in such years, so they are not in the formula. for a concession with no production sharing, $c = 1$, and ERR equals the payments on revenue.

US$ million; $R$ and the shares are dimensionless

Holds when: in regime B, $s_0 = 0.30$ at $R_0 = 1$ and $s_1 = 0.70$ at $R_1 = 2$. the sums are not discounted: $R = 1$ means that the contractor has got back what it put in, but not yet any return on the capital. abandonment is not in $R$, because it is not recovered.

US$ million; rates are fractions

Holds when: a positive balance is taxed in the same year, so only a negative one is carried to the next year. in regime C, $e = 0.40$ and $u = 0.15$. The tax does not count abandonment. if $u$ equals the investor's rate, the tax is neutral (see the advanced section on the neutral tax).

tenge or any currency unit; fractions

Holds when: net income up to 25% of the deductions is not taxed (Article 800). Each rate is applied only to the part of the income in its own tier (Articles 807, 808). the tax period is the calendar year. The return is filed by 31 March, and the tax is paid within 10 days after that date (Articles 809–811). the tax is not paid under production sharing agreements signed before 2009 that passed the mandatory tax expertise (a state review of the tax terms), or under contracts approved by the President (Articles 755, 799), or under the alternative tax.

US$ million; rates are fractions

Holds when: the uplift $u$ on the carried-forward balance equals the investor's rate $i$. the balance finally becomes positive, and after that there are no negative years. Otherwise their loss would have to be paid back or carried backwards. if $u > i$, the tax is milder than a neutral one and takes less than $e\,\mathrm{NPV}^{*}$. If $u < i$, it is harsher: it also taxes part of the normal return.

dimensionless: the per cent change of NPV for a one per cent change of the parameter

Holds when: one parameter changes and the others stay at their base values the change is small, so the dependence can be taken as linear; for large changes we recalculate the model $\mathrm{NPV}_0$ is not close to zero: for a project at the edge of break-even the elasticity grows without limit and loses its meaning

prices in US$/bbl; the sum $\sum_t Q_t/(1+i)^{t}$ is the discounted production, million bbl

Holds when: only the revenue, the royalty and the income tax depend on the price; production and costs stay the same when the price changes at all prices from $P^{*}$ to $P_0$ the project pays income tax in every year of production; if loss-making years appear at a low price, the dependence is no longer a straight line, and the answer is found by trial the royalty is proportional to revenue and the tax rate is constant; for sliding rates (rates that change with the price or the output) and for production sharing (where the state and the company split the oil) the formula does not fit the other parameters are at their base values; the break-even price refers to the chosen measure and rate, here to NPV at rate $i$

US$ million; $d$ in years

Holds when: the capital expenditure keeps its amounts and dates, and everything else (production, operating expenditure, payments to the state, abandonment) moves by $d$ years unchanged the calculation is in constant prices; tax depreciation (the yearly write-off of the capital cost for tax) starts from first oil and moves with it the extra expenses during the standstill and the rising cost of the work are not counted; in practice they make a delay even more expensive

US$ million; all the present values at one rate and from one origin

Holds when: the royalty is a constant share of revenue, the income tax is a constant share of profit, and profit is positive in every year of production the capital expenditure reduces tax only through depreciation, so its tax return comes later and is worth less the changes do not touch the production profile; when production changes, we take the change of revenue and of variable costs instead of $\Delta P$

the money units of the NPV, for example US$ million in constant prices

Holds when: the outcomes exclude each other and together cover all the possibilities, so the probabilities add up to one all the NPVs are calculated at one date (the date of the decision), at one rate and in the same kind of money the NPV includes only the future flows that depend on the decision; costs already spent are not included (what belongs to a decision) the risk of failure is counted by the probabilities; the discount rate is not raised for the same risk (Choosing the discount rate)

US$ million at the date of the decision

Holds when: there are two outcomes: a discovery that is developed, and a dry well; the size of the discovery is taken as known (different sizes are covered in the section below) $V$ and $C$ refer to one date; if the moment to which the NPV of the project is brought comes $t$ years after the decision, the NPV is multiplied by $(1 + i)^{-t}$ (moving the time origin) the tax effects of the cost of the well are not counted (or $C$ is already taken after tax)

dimensionless

Holds when: the same as for the EMV of a well: two outcomes, with $V$ and $C$ at one date.

US$ million at the date of the decision

Holds when: the partner pays for the whole well, and the sides carry the development costs in proportion to their shares both sides assess $P_g$ and $V$ in the same way

dimensionless

Holds when: the outcomes of the wells are independent: the prospects share no play risk (a play is a group of prospects in the same geological setting: if a key ingredient is missing there, all of them fail) the chance of success is the same for all wells; with different chances the distribution is found by going through the outcomes the whole programme is drilled without revising it after the first results

$x$, $R$ and CE are in the same money units

Holds when: the attitude to risk is the same at any wealth of the company: adding the same guaranteed sum to every outcome adds that sum to CE too $R$ is set for the company as a whole and is revised together with its size and financial position when $R$ is much larger than the spread of the outcomes, the function is almost a straight line on this interval, and CE almost equals EMV

US$ million

Holds when: exponential utility function the spread of the outcomes is small compared with $R$ ($\sigma \ll R$); for a normally distributed NPV the formula is exact at any spread

dimensionless

Holds when: it follows from $\mathrm{CE}(\alpha) \approx \alpha\,\mathrm{EMV} - \alpha^2\sigma^2/(2R)$ and inherits the limits of that approximation the share divides both the costs and the income in proportion (without paying for the well on behalf of a partner)

US$ million at the date of the first decision (any units, the same for the whole tree)

Holds when: the attitude to risk is neutral: a chance node is valued by EMV; how to allow for risk aversion is shown in the advanced part all the flows are brought to one date at one rate and in one kind of money the probabilities of each chance node add up to one and take into account everything known on the path to it in each decision node we choose only by what is already known at that moment

US$ million at the date of the decision

Holds when: the state becomes known before the decision, without errors the information is free and does not delay the decision; its price and time are counted separately the attitude to risk is neutral

US$ million at the date of the decision

Holds when: two outcomes: a discovery worth $V$, or a dry well; $V > C$ without information we choose the better of “drill” ($P_g V - C$) and “walk away” (0)

dimensionless

Holds when: the events $H$ and $\bar H$ exclude each other and cover all the possibilities the detection and false-alarm rates refer to objects of the same type as the one being assessed: an indicator that is reliable in one play may mean nothing in another for a negative answer the formula is the same, with $P(-\mid H) = 1 - P(+\mid H)$ and $P(-\mid \bar H) = 1 - P(+\mid \bar H)$

US$ million at the date of the decision

Holds when: the price of the data is not subtracted: VOI is compared with the price the data do not delay the decision; if they do, the first term is discounted by the time of the delay the attitude to risk is neutral

US$ million at the date of the decision

Holds when: without data, drilling pays; after a positive answer we drill, and after a negative one we do not (if neither answer changes the decision, VOI = 0) at $s = 1$ and $f = 0$ the formula turns into EVPI $= (1 - P_g)\,C$

dimensionless

Holds when: the same as for Bayes' theorem for several indicators, the likelihood ratios are multiplied only if the indicators are independent in each state of nature; two attributes of the same seismic record are usually dependent

production in units of volume (or mass), the rate as a fraction per year

Holds when: the same conventions about the time origin and the moment of the flows as in NPV (the time origin of NPV): here it is the middle of project year 1, and the flows are in the middle of each year, $t = 0, 1, 2, \ldots$ the rate is real if the prices and costs are taken in constant prices

the units of the quantity itself (million bbl, US$ million …)

Holds when: the three numbers are the estimates P90, P50 and P10 of one distribution the distribution is not too skewed: for a log-normal distribution with a P10/P90 ratio of 4 the error of the mean is less than 1%, with a ratio of 10 it is about 3.6%, and with a ratio of several tens the rule clearly understates the mean for NPV the rule is used as an approximate averaging over the distribution of one main parameter: the three cases are its P90, P50 and P10. If NPV changes with this parameter in one direction, the NPVs of the cases are also the P90, P50 and P10 of NPV

units of NPV

Holds when: $f$ flattens as $X$ grows (it is concave) on the whole range of $X$: each next increase of $X$ adds less than the previous one; for a function that steepens as $X$ grows (a convex one) the sign of the inequality is reversed equality holds only for a linear $f$ or when there is no spread

$\sigma_\text{NPV}$ is in US$ million; $a\sigma_X$ and $b\sigma_Y$ are also in US$ million

Holds when: within the ranges, NPV ≈ $\mathrm{NPV}_0 + a\,\Delta X - b\,\Delta Y$, a linear approximation $X$ raises NPV and $Y$ lowers it, hence the minus before the last term; if both act in the same direction, the sign is plus $\rho$ is the linear correlation; a link set through ranks or normal values gives a number close to it