Acceleration projects and incremental analysis

Quick refresher

A project is valued by the difference between the forecasts “with the project” and “without the project”: $\Delta \mathrm{NPV} = \mathrm{NPV}^{\text{with}} - \mathrm{NPV}^{\text{without}}$ , at the same rate and the same time origin. Royalty, costs, tax and abandonment are calculated in full in each forecast and then subtracted.
Pure acceleration does not change total production: $\sum_t \Delta Q_t = 0$ . Without discounting, it only spends money. Value appears only because an early barrel is worth more than a late one. Its measure is the gain in discounted production $\Delta Q_\text{disc}$ (the barrels counted at their value today).
The NPV profile of pure acceleration is a hump. At a zero rate NPV is negative, it rises as the rate grows, and then it falls again. There are two IRRs, or none. A rising real oil price lowers the value of acceleration, and a falling one raises it.
IRR flatters acceleration. The upper IRR is higher the smaller the investment is compared with the revenue that is moved, and at a high rate a late shortfall (the later loss of production) is hardly visible. We decide by NPV at the company's rate.
When the budget is limited, we rank projects by the profitability index $\mathrm{PI} = 1 + \mathrm{NPV}/PV(I)$ . The denominator holds only what is in short supply, usually the investment of the decision year. Indivisible projects (you do all of one or none of it) are checked by trying combinations.
Even a small addition of reserves is usually worth more than the acceleration itself. In the example of the lesson, 0.3 million bbl of new oil add 6.78 million dollars, and the acceleration of 0.7 million bbl adds 1.33 million dollars.

Key relationships

\Delta \mathrm{NCF}_t = \mathrm{NCF}_t^{\,\text{with}} - \mathrm{NCF}_t^{\,\text{without}}

incremental cash flow

\Delta \mathrm{NCF}_t = (1 - T)(m\,\Delta Q_t - \Delta F_t) - \Delta C_t - \Delta A_t + T\,\Delta D_t

incremental flow under a simple tax regime, $m = P(1 - r) - v$

\Delta Q_\text{disc} = \sum_t \frac{\Delta Q_t}{(1+i)^{t}}

incremental discounted production; for pure acceleration $\sum_t \Delta Q_t = 0$

\mathrm{PI} = 1 + \frac{\mathrm{NPV}}{PV(I)}

profitability index: value per unit of limited investment

Common pitfalls

Comparing production “before” and “after” instead of the forecasts “with the project” and “without the project”.
Forgetting the late shortfall: oil that was produced earlier cannot be produced later.
Ranking projects by IRR, or taking the larger of two IRRs.
Putting the late shortfall or the operating costs in the denominator of the profitability index.
Calculating the incremental tax as the tax rate multiplied by the incremental profit, when one of the forecasts has losses or special tax rules.

After this lesson you can

build an incremental cash flow from two full forecasts, “with the project” and “without the project”, and check it
tell acceleration of production from added reserves, and estimate what it is worth to move production in time
explain the shape of the NPV profile of an acceleration project, and how its value depends on the rate, on the investment and on the change of the price over time
explain why IRR overstates the appeal of acceleration projects, and what to use instead
rank projects under a limited budget by the profitability index, and check the choice by the total NPV

The incremental flow: “with the project” minus “without the project”

The principle is known from the lesson on cash flow. We value the difference between two futures, “with the project” and “without the project”, and not “before” and “after” (which flows belong to the project). In a producing field, there is real work behind this principle: we have to build two full forecasts that differ only in the project.

How to build the incremental flow

1
Forecast “without the project”
This is not the current production extended by a straight line. It is what will happen if the project is not done. Production falls as it would have fallen. The work that will be done in any case, such as planned repairs and the required injection, stays in the forecast. This case is called “without the project” (do-nothing), or the case with the minimum of work (do-minimum).
2
Forecast “with the project”
It is built with the same model of the reservoir and the wells, and at the same prices, costs and taxes. Only the project differs. If the two forecasts come from different methods, the difference will show the difference between the methods as well.
3
Full calculation of both cases
Each case is run through the whole cash flow model (Cash flow model of an oil and gas project): royalty (the state's share of the revenue), fixed and variable costs, income tax with depreciation and with losses carried forward to later years, and the time of abandonment.
4
Difference by year
The yearly flows of the case “without the project” are subtracted from those of the case “with the project”. This is done for all the lines, not only for production. The result is the incremental cash flow.
5
Check
The NPV of the incremental flow must be equal to the difference between the NPVs of the two cases at the same rate and the same time origin: the present value of a difference is the difference of the present values.

Incremental cash flow and its NPV

\Delta \mathrm{NCF}_t = \mathrm{NCF}_t^{\,\text{with}} - \mathrm{NCF}_t^{\,\text{without}}, \qquad \Delta \mathrm{NPV} = \sum_t \frac{\Delta \mathrm{NCF}_t}{(1+i)^{t}} = \mathrm{NPV}^{\text{with}} - \mathrm{NPV}^{\text{without}}

where:

$\Delta \mathrm{NCF}_t$ incremental cash flow of the year that is $t$ years from the time origin, US$ million
$\mathrm{NCF}_t^{\,\text{with}},\ \mathrm{NCF}_t^{\,\text{without}}$ net cash flow of the whole field with the project and without it, US$ million
$\Delta \mathrm{NPV}$ net present value of the project: the change in the value of the field, US$ million
$i$ discount rate, fraction per year

Units: 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.

$\Delta \mathrm{NPV}$ shows by how much the project changes the value of the field. It is positive if the project earns more than the rate $i$ on the money put into it.

Variable costs change in proportion to the gain in oil and fluid production. In a reservoir that produces a lot of water, extra oil brings extra water with it.
Fixed costs change if the project lengthens or shortens the life of the field: the number of years in which they are paid changes.
Abandonment moves together with the last year of production. Bringing it closer is a cost of the project, and putting it off is a gain (abandonment in the model).
Tax changes twice: the new investment is depreciated, and the gain in profit is taxed. If one of the cases has losses or special rules, the tax is calculated in full for each case (the advanced section).
Shared capacity. If the processing facilities are fully loaded, the new production pushes out the production of other wells, and the gain is smaller than the output of the new well.

Suppose the tax regime is simple: royalty is a share of revenue, income tax is a share of profit, and every year is profitable in both cases. Then the incremental flow can be written with one formula. It shows where the value of the project comes from. The difference of the full calculations must agree with it, and this is a check of the model.

Incremental flow under a simple tax regime

\Delta \mathrm{NCF}_t = (1 - T)\,(m\,\Delta Q_t - \Delta F_t) - \Delta C_t - \Delta A_t + T\,\Delta D_t

where:

$\Delta Q_t$ production gain of year $t$ (negative in the years of shortfall), million bbl
$m$ margin per barrel before income tax: $m = P(1 - r) - v$ , the price minus royalty and variable costs, US$/bbl
$\Delta F_t$ change in fixed operating costs, US$ million
$\Delta C_t$ additional capital investment, US$ million
$\Delta A_t$ change in abandonment costs (in the simplified regime they do not reduce tax), US$ million
$\Delta D_t$ additional tax depreciation, US$ million
$T$ income tax rate, fraction

Units: 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.

The value of the project is the margin on the production gain, reduced by tax, minus the investment, with part of the tax coming back through depreciation. Fixed costs and abandonment enter only if the project changes the life of the field.

Worked exampleTwo infill wells in the project of the examples

Given (assumed data)

Project: the example project of the lessons: production of 1.6, 2.4, 2.4, 2.0, 1.6, 1.2 and 0.8 million bbl in years 3–9, price US$50/bbl, royalty 10%, variable costs US$5/bbl, income tax 20% with the investment depreciated over 5 years
Decision: in year 5, drill two infill wells (wells between the old ones) for US$3 million
Production with the wells: 2.4, 1.9, 1.0 and 0.3 million bbl in years 6–9 instead of 2.0, 1.6, 1.2 and 0.8. The plateau (the period of steady, highest production) lasts a year longer, and then production falls faster. Over the life of the project the total is the same 12 million bbl.
Rate and time origin: 10% (real); the middle of year 5 is the decision date; the flows are in the middle of the year

Solution

The production gain in years 6–9: +0.4, +0.3, −0.2 and −0.5 million bbl. The sum is zero: the wells produce the same oil sooner.
The margin per barrel before income tax is $m = 50\cdot0.9 - 5 = 40$ dollars, and after tax $0.8\cdot40 = 32$ dollars. The depreciation of the wells is $3/5 = 0.6$ million dollars a year in years 5–9. It saves $0.2\cdot0.6 = 0.12$ million dollars of tax a year.
The incremental flow by the formula: year 5 $-3 + 0.12 = -2.88$ ; year 6 $32\cdot0.4 + 0.12 = 12.92$ ; year 7 $32\cdot0.3 + 0.12 = 9.72$ ; year 8 $-32\cdot0.2 + 0.12 = -6.28$ ; year 9 $-32\cdot0.5 + 0.12 = -15.88$ million dollars. The difference of the two full model calculations is the same, year by year (table below).
Without discounting: $-2.88 + 12.92 + 9.72 - 6.28 - 15.88 = -2.40$ million dollars. The project loses the investment, less the tax saving.
The flows discounted at 10% to the middle of year 5: −2.880, 11.745, 8.033, −4.718 and −10.846 million dollars. Their sum is $\Delta \mathrm{NPV} = 1.334$ million dollars.
Check: the NPV of the whole field at 10%, at the middle of year 1, is 101.045 million dollars with the wells and 100.134 without them. The difference is 0.911 million dollars, or $0.911\cdot1.1^{4} = 1.334$ million dollars at the middle of year 5.

Answer: The wells add 1.334 million dollars of value at the decision date for an investment of 3 million dollars, although they do not give a single new barrel.

What it means. All the gain comes from the fact that 0.7 million bbl are produced two to three years sooner, while the 3 million dollars are paid at once. Without discounting, the project loses money (−2.40 million dollars). The value of acceleration exists only because early money is worth more than later money. The data in this example are assumed for illustration.

Incremental flow of the infill wells by year, US$ million

Year: 5
$\Delta Q$ , million bbl: 0.0
Change in margin before tax: 0.00
Change in tax: −0.12
$\Delta \mathrm{NCF}$: −2.88

Year: 6
$\Delta Q$ , million bbl: 0.4
Change in margin before tax: 16.00
Change in tax: 3.08
$\Delta \mathrm{NCF}$: 12.92

Year: 7
$\Delta Q$ , million bbl: 0.3
Change in margin before tax: 12.00
Change in tax: 2.28
$\Delta \mathrm{NCF}$: 9.72

Year: 8
$\Delta Q$ , million bbl: −0.2
Change in margin before tax: −8.00
Change in tax: −1.72
$\Delta \mathrm{NCF}$: −6.28

Year: 9
$\Delta Q$ , million bbl: −0.5
Change in margin before tax: −20.00
Change in tax: −4.12
$\Delta \mathrm{NCF}$: −15.88

Year: Total
$\Delta Q$ , million bbl: 0.0
Change in margin before tax: 0.00
Change in tax: −0.60
$\Delta \mathrm{NCF}$: −2.40

Year	$\Delta Q$ , million bbl	Change in margin before tax	Change in tax	$\Delta \mathrm{NCF}$
5	0.0	0.00	−0.12	−2.88
6	0.4	16.00	3.08	12.92
7	0.3	12.00	2.28	9.72
8	−0.2	−8.00	−1.72	−6.28
9	−0.5	−20.00	−4.12	−15.88
Total	0.0	0.00	−0.60	−2.40

The difference of two full calculations of the project model. Margin is revenue after royalty minus operating costs. $\Delta \mathrm{NCF}$ = change in margin − change in tax − investment. The investment of US$3 million is in year 5. In year 10 the flows of both cases are the same. The data in this example are assumed for illustration.

CalculatedCumulative production of the field without the infill wells and with them. The curves diverge from year 6. At the end of year 7 the wells are ahead by 0.7 million bbl. At the end of year 9 the curves meet again at the same 12 million bbl: the wells borrow oil from the last years.

Acceleration or added reserves: where the value comes from

Incremental production is of two kinds. Added reserves are oil that would never be produced without the project: a new layer of oil-bearing rock, an undeveloped block, better sweep of the reservoir by waterflooding (water injected to push the oil towards the wells). Acceleration is oil that would be produced anyway, but later. Examples are a well between old wells in a well-connected reservoir, well stimulation (treating the reservoir to raise the flow into the well), a more powerful pump, and a lower wellhead pressure thanks to a compressor. Pure acceleration does not change total production: the sum of the gains over the years is zero. Most real projects are a mixture: a little new oil and a lot of acceleration.

How can oil that will be produced anyway have any value? Only through time. At a constant price, the discounted revenue is the price multiplied by the discounted production (discounted production). Moving barrels to earlier years increases the discounted production, although the reserves are the same. This gain is what an acceleration project “sells”.

Value of the incremental production at a constant margin

\Delta \mathrm{NPV} = (1 - T)\,m\,\Delta Q_\text{disc} - PV_C + T\,PV_D, \qquad \Delta Q_\text{disc} = \sum_t \frac{\Delta Q_t}{(1+i)^{t}}

where:

$\Delta Q_\text{disc}$ incremental discounted production, million bbl
$\Delta Q_t$ production gain of year $t$ , million bbl
$m$ margin per barrel before income tax, US$/bbl
$PV_C$ present value of the project's investment, US$ million
$PV_D$ present value of the tax depreciation of this investment, US$ million
$T$ income tax rate, fraction
$i$ discount rate, fraction per year

Units: 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 $i = 0$ .

Acceleration creates value if the margin on the incremental discounted production, after tax, is larger than the net investment. At a zero rate, pure acceleration is left with costs only. As the rate rises, early barrels weigh more and more against later ones, until discounting starts to cut down the early gain itself.

Worked exampleWhat moving 0.7 million bbl is worth

Given (assumed data)

Production gain of the wells: +0.4, +0.3, −0.2 and −0.5 million bbl in years 6–9 (the example above)
Margin and tax: US$40/bbl before tax, income tax 20%
Investment: US$3 million at the middle of year 5; tax depreciation of US$0.6 million a year in years 5–9

Solution

The incremental discounted production at the middle of year 5 is the gains of years 6–9 divided by $1.1^{1} \ldots 1.1^{4}$ : $0.36364 + 0.24793 - 0.15026 - 0.34151 = 0.1198$ million bbl.
The margin on it after tax: $32\cdot0.1198 = 3.834$ million dollars.
The tax saving from depreciation: $0.12\cdot(1 + 1.1^{-1} + \ldots + 1.1^{-4}) = 0.12\cdot4.1699 = 0.500$ million dollars.
$\Delta \mathrm{NPV} = 3.834 - 3 + 0.500 = 1.334$ million dollars, as in the example above.

Answer: Moving 0.7 million bbl to two or three years earlier is worth as much as 0.1198 million bbl of new oil sold at once. That is about a sixth of the volume that is moved.

What it means. The value of acceleration depends most strongly on the rate. A company with a higher hurdle rate (the lowest return it accepts) values early barrels more, and the same project is better for it, up to a certain limit. This is also the reason for the unusual shape of the NPV profile in the next section. The data in this example are assumed for illustration.

The NPV profile, the price and added reserves

The NPV profile of an acceleration project does not look like the profile of an ordinary project (NPV profile). At a zero rate NPV is negative: only the investment is left. As the rate rises, the late shortfall is discounted more strongly than the early gain, and NPV rises. At very high rates, discounting cuts down both the gain and the shortfall, and NPV falls again towards the value of the first flow: the investment of the decision year minus the tax saving of the same year. The profile crosses the rate axis twice: the project has two IRRs (several IRRs or none).

NPV of the infill wells at different rates, US$ million at the middle of year 5

Rate $i$: 0%
NPV: −2.40

Rate $i$: 5%
NPV: −0.25

Rate $i$: 10%
NPV: 1.33

Rate $i$: 15%
NPV: 2.50

Rate $i$: 20%
NPV: 3.34

Rate $i$: 50%
NPV: 5.06

Rate $i$: 100%
NPV: 4.23

Rate $i$: 300%
NPV: 0.80

Rate $i$: 500%
NPV: −0.50

Rate $i$	NPV
0%	−2.40
5%	−0.25
10%	1.33
15%	2.50
20%	3.34
50%	5.06
100%	4.23
300%	0.80
500%	−0.50

NPV is zero at rates of about 5.7 and 403%. At an infinitely high rate it approaches the first flow, −2.88 million dollars. The data in this example are assumed for illustration.

CalculatedNPV profile of the infill wells; the rate is on a logarithmic scale. NPV is negative up to 5.7%, positive up to 403% and negative again beyond that. The dot is NPV at the company's rate of 10%.

Acceleration pays only while an early barrel is worth more than a late one. If the real oil price rises, late barrels are worth more, and the gain from acceleration melts away. If the price falls, the gain grows. If the price grows at a constant rate $e_r$ a year, we can discount the revenue the same way as the barrels, but at the changed rate $(1+i)/(1+e_r) - 1$ (if the price changes). When the price grows at $e_r = i$ a year, moving a barrel in time does not change the discounted revenue at all. But the investment and the early variable costs are still there.

NPV of the infill wells for different changes of the real oil price, rate 10%

Real oil price: falls by 2% a year
NPV, US$ million: 1.99

Real oil price: constant
NPV, US$ million: 1.33

Real oil price: rises by 2% a year
NPV, US$ million: 0.61

Real oil price: rises by 4% a year
NPV, US$ million: −0.18

Real oil price	NPV, US$ million
falls by 2% a year	1.99
constant	1.33
rises by 2% a year	0.61
rises by 4% a year	−0.18

The price is US$50/bbl in year 5 and then changes at a constant rate. The variable costs and the tax are as in the examples. NPV is at the middle of year 5. NPV is zero when the price grows by about 3.6% a year. The data in this example are assumed for illustration.

Added reserves change the picture more strongly than the rate or the price. Suppose the same wells, besides the acceleration, recover another 0.3 million bbl of oil that would otherwise stay in the reservoir. Production in years 8 and 9 is then 1.1 and 0.5 million bbl instead of 1.0 and 0.3. The incremental flow is −2.88, 12.92, 9.72, −3.08, −9.48 million dollars, and NPV at 10% is 8.11 million dollars. The 0.3 million bbl of new oil give 6.78 million dollars, and the acceleration of 0.7 million bbl gives only 1.33. The undiscounted total would become positive (7.2 million dollars), the cumulative flow would change sign once, and the positive IRR would be the only one, about 408%.

Why IRR flatters acceleration projects

The flow of an acceleration project is an investment and a loan at the same time. The investment is US$3 million for the wells. The loan is from the future: the barrels produced in years 6–7 are “paid back” by the field with the shortfall of years 8–9. IRR values the whole flow at one rate, and at a high rate this loan is almost free. At a rate of 400%, a shortfall three or four years away weighs less than 1% of its size. So the upper IRR of an acceleration project mostly shows just one thing: how many times the early gain is larger than the investment.

Hence the paradox. For pure acceleration to have a positive NPV, the investment must be small compared with the revenue that is moved. Then its upper IRR is unavoidably huge. A high IRR here is not a sign of great value. It follows from the very nature of the project. The lower IRR misleads in the other direction: it is below the company's rate, although NPV is positive. And Newton's method arrives exactly at this lower root from a starting guess of 10%, which is the default of the IRR function of a spreadsheet (the example in the lesson on IRR).

Worked exampleWhat the different measures show for the infill wells

Given (assumed data)

Incremental flow: −2.88, 12.92, 9.72, −6.28, −15.88 US$ million in years 5–9 (the example above)
Company's rate: 10%

Solution

The roots of the equation $\mathrm{NPV} = 0$ , found by halving the interval on the ranges 0–20% and 20–1000%, are 5.7% and 403%. Newton's method from a starting guess of 10% arrives at 5.7%.
If we forget the shortfall of years 8–9 and count the gain as new oil, the flow becomes −2.88, 12.92, 9.72, 0.12, 0.12 million dollars. NPV at 10% is 17.07 million dollars instead of 1.33, and IRR is 414% instead of 403%.
If the wells cost 1.5 million dollars, the upper IRR would be 857% and NPV would be 2.58 million dollars. At 4.5 million dollars they would be 245% and 0.08 million dollars.
The modified internal rate of return (MIRR) at finance and reinvestment rates of 10% (modified IRR): the negative flows discounted to year 5 are −18.445 million dollars, and the positive flows compounded to year 9 are 28.958 million dollars. $\mathrm{MIRR} = (28.958/18.445)^{1/4} - 1 = 11.9$ %.

Answer: NPV of 1.33 million dollars and MIRR of 11.9% say the same thing: the project is profitable, but only a little. The two IRRs, 5.7 and 403%, say nothing definite. The upper one hardly notices the shortfall and reacts mainly to the size of the investment.

What it means. The forgotten shortfall raised NPV almost 13 times, but it raised the upper IRR only from 403 to 414%, less than 3% of its size. At a rate of about 400%, late flows weigh almost nothing, so IRR does not show an error that NPV shows at once. And at an investment of 4.5 million dollars the upper IRR is still 245%, although NPV is almost zero. The IRR of such a flow tells us how fast the investment is returned, not how much value is created. The data in this example are assumed for illustration.

Count the changes of sign of the incremental flow. If there are two, build the NPV profile over a wide range of rates, instead of looking for one root from one starting guess.
Decide by NPV at the company's rate, and show the efficiency of the investment by the profitability index (next section).
If you need a return in per cent, use the modified IRR with explicit finance and reinvestment rates: its answer is unique.
Check whether the flow has a late shortfall. An acceleration project without a shortfall is a sign of an error in the forecasts. Most likely, the production of the new wells was simply added to the forecast “without the project”, without reducing the production of the neighbouring wells, or “before” and “after” were compared.

Selection and ranking: NPV and the profitability index

The first step is selection. Projects that do not earn the company's rate, that is, projects with NPV < 0, are dropped. For ordinary projects this is the same as the condition IRR ≥ $i$ (Internal rate of return (IRR)). For acceleration projects it is not. They can have two IRRs, and both can be above the rate, both can be below it, or they can lie on different sides of it. If there were enough money for all the projects that pass the selection, there would be no second step: we would accept every project with a positive NPV. But the yearly investment budget is usually smaller than the sum of all the good proposals, and then we choose a set of projects.

If the investment budget is limited, this is called capital rationing. The goal is the largest total NPV for the given investment. For this we compare the projects by the value they create per unit of the limited resource. This measure is the profitability index (PI; do not confuse it with the productivity index of a well, which is also written PI):

Profitability index

\mathrm{PI} = \frac{PV(\Delta \mathrm{NCF} + I)}{PV(I)} = 1 + \frac{\mathrm{NPV}}{PV(I)}

where:

$\mathrm{PI}$ profitability index, —
$I$ capital investment of the project (positive amounts) to which the budget limit applies, US$ million
$PV(I)$ its present value, US$ million
$\Delta \mathrm{NCF} + I$ the incremental flow without this investment: everything else, including the late shortfall and taxes, US$ million
$\mathrm{NPV}$ present value of the incremental flow, US$ million

Units: 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).

PI > 1 exactly when NPV > 0, so for selection it adds nothing to NPV. Its job is to give the order when money is short: how much value each invested dollar brings. The NPV index from the lesson Net present value (NPV) is built in a similar way, but it divides NPV by the largest discounted need for financing. For projects whose whole investment is made at once, it is almost equal to $\mathrm{PI} - 1$ .

Worked exampleFour projects and a budget of US$10 million

Given (assumed data)

Investment budget of the decision year: US$10 million
Infill wells (pure acceleration): US$3 million; flow −2.88, 12.92, 9.72, −6.28, −15.88
Near-wellbore treatments (mostly acceleration): US$1.5 million; flow −1.5, 4.0, 0.9, −1.4, −1.6
Sidetrack at a neighbouring field (new reserves): US$4 million; flow −4, 2.2, 1.9, 1.6, 1.3, 1.0, 0.7
Waterflood expansion (new reserves): US$5 million; flow −5, 0.6, 1.5, 1.9, 1.9, 1.7, 1.5, 1.3, 1.1, 0.9
Conditions: incremental flows after tax, US$ million by year starting from the year of the decision; rate 10%, flows in the middle of the year, time origin the middle of the decision year

Solution

NPV, IRR and the profitability index are in the table below. By IRR the order is: wells (403% if we take the upper IRR), treatments (170.7%), sidetrack (35.8%), waterflood (23.1%).
Selection by IRR: the wells, the treatments and the sidetrack, $3 + 1.5 + 4 = 8.5$ million dollars. There is no money left for the waterflood (5 million dollars). The total NPV is $1.33 + 0.74 + 2.68 = 4.75$ million dollars.
Selection by the profitability index: the sidetrack (1.67) and the waterflood (1.59), 9 million dollars. The treatments (1.49) and the wells (1.44) no longer fit. The total NPV is $2.68 + 2.97 = 5.65$ million dollars.
Trying all 15 combinations of the four projects and checking the budget gives the same best set: 5.65 million dollars.

Answer: Selection by IRR loses about 0.9 million dollars of value, which is 16% of what is possible. It prefers the acceleration projects, whose IRR is huge and whose profitability indexes are the lowest of the four.

What it means. Acceleration projects return money quickly, but they create little value for each dollar invested. Projects that add reserves return money more slowly, but they create more. If the budget were not limited, it would be worth accepting all four projects: the NPV of each is positive. A budget limit makes the value per dollar important, not the percentage return. The data in this example are assumed for illustration.

Four projects at a rate of 10%

Project: Infill wells
Investment, US$ million: 3.0
NPV, US$ million: 1.33
IRR: 5.7% and 403%
PI: 1.44

Project: Near-wellbore treatments
Investment, US$ million: 1.5
NPV, US$ million: 0.74
IRR: 170.7%
PI: 1.49

Project: Sidetrack
Investment, US$ million: 4.0
NPV, US$ million: 2.68
IRR: 35.8%
PI: 1.67

Project: Waterflood expansion
Investment, US$ million: 5.0
NPV, US$ million: 2.97
IRR: 23.1%
PI: 1.59

Project	Investment, US$ million	NPV, US$ million	IRR	PI
Infill wells	3.0	1.33	5.7% and 403%	1.44
Near-wellbore treatments	1.5	0.74	170.7%	1.49
Sidetrack	4.0	2.68	35.8%	1.67
Waterflood expansion	5.0	2.97	23.1%	1.59

The infill wells have two IRRs. The budget is US$10 million. The data in this example are assumed for illustration.

Indivisible projects. If the next project in order of falling PI does not fit into the budget, a different set may be better. With a small number of projects, we try all the combinations.
Several years. If the investment is limited in more than one year, one index is not enough. We solve it as an optimisation problem, with a limit for each year.
A ratio, not a sum. The index is a ratio, and a small project with a high return per dollar beats a large one. For choosing between independent projects when money is short, this is as it should be. For choosing a development option of one field it is not: the options exclude each other, and a smaller option with a better index can create less value.
Not only money. If drilling rigs or specialists are in short supply, the denominator must be exactly this shortage: NPV per rig-month or per person-year.

Terms and symbols

Forecast “without the project”: The expected development of the field without this decision, with the work that will be done in any case; not the current production extended by a straight line.
Incremental cash flow: The difference between the flows “with the project” and “without the project” in all items, including tax and abandonment; its NPV is the difference between the NPVs of the two cases.
Pure acceleration: A project that does not change total production: $\sum_t \Delta Q_t = 0$ ; its value lies only in moving production in time.
Late shortfall: The fall in production at the end of the life, by which an early gain from acceleration is paid for.
Incremental discounted production $\Delta Q_\text{disc}$: The sum of the production gains multiplied by the discount factors; for pure acceleration it is zero at a zero rate.
Profitability index (PI): $1 + \mathrm{NPV}/PV(I)$ : value per unit of limited investment; it is used to choose projects when the budget is short.
Capital rationing: The situation when there is not enough money for all the projects with a positive NPV; the goal is the largest total NPV for the given budget.

Common mistakes and misconceptions

Wrong

“Two new wells give 1100 bbl/d, and this is their contribution.”

In fact

The contribution is the difference between the forecasts “with the project” and “without the project”. In a well-connected reservoir, part of what a new well produces is oil that the neighbouring wells would have produced. In the example of the lesson, the two wells add 0.7 million bbl in the first two years, but not a single barrel over the whole life.

Wrong

“The IRR of the wells is 403%, so this is the best project in the portfolio.”

In fact

The flow has two IRRs, and the upper one shows only how fast a small investment is returned. The value of the wells is 1.33 million dollars, and their profitability index is the lowest of the four projects of the example.

Wrong

“If reserves do not grow, the project creates no value.”

In fact

It does, if the rate is positive and the price does not rise too fast: early barrels are worth more than late ones. But this value is small, and it is sensitive to the rate, the price and the size of the investment.

Wrong

“The shortfall at the end is also a cost, so let us put it in the denominator of the profitability index.”

In fact

The denominator is the resource in short supply, that is, the investment of the decision year. The shortfall reduces the numerator. Otherwise the index of the wells in the example would fall from 1.44 to 1.07 at the same NPV.

Wrong

“Let us compare the production after the treatment with the production before it.”

In fact

Without the treatment, production would also fall. The gain is the difference between forecasts, not between “before” and “after”. A comparison with the old production rate takes no account of this fall, nor of the faster fall of the production rate after the treatment, and it does not see the shortfall at the end at all.

In depth: advanced detail

Acceleration of a well with exponential decline

For a single well with an exponential decline, the value of pure acceleration can be found with one formula. Suppose the remaining reserves of the well $N$ are produced at a nominal decline rate $D_1$ (the share of the production rate that is lost each year), and after a treatment at a rate $D_2 > D_1$ . The reserves are the same, so the initial rate grows in proportion to the decline rate: $q_i = N D$ . With no economic limit (the rate below which production stops paying), the discounted production is $N D/(D + j)$ , where $j = \ln(1+i)$ (exponential decline). It is the reserves multiplied by the present-volume factor $D/(D + j)$ (the share of the reserves that is left after discounting).

Gain in discounted production from accelerating an exponential decline

\Delta Q_\text{disc} = N\left(\frac{D_2}{D_2 + j} - \frac{D_1}{D_1 + j}\right) < N\,\frac{j}{D_1 + j}

where:

$\Delta Q_\text{disc}$ gain in the discounted production of the well, bbl
$N$ remaining reserves of the well, the same before and after the treatment, bbl
$D_1,\ D_2$ nominal decline rate before and after the treatment, 1/year
$j$ continuous discount rate $\ln(1+i)$ , 1/year

Units: $N$ 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.

The gain is the difference between the present-volume factors of the reserves. The right-hand side is a limit: even if all the reserves were produced at once, acceleration could not give more than discounting takes away from them now. So the best candidates for acceleration are slowly declining wells: discounting takes the most from their reserves.

Worked exampleA treatment that doubles the production rate

Given (assumed data)

Remaining reserves of the well: 300 thousand bbl
Decline rate: $D_1 = 0.2$ 1/year before the treatment: the initial rate is 60 thousand bbl a year, about 164 bbl/d. $D_2 = 0.4$ 1/year after it: the rate is twice as high
Margin and cost of the treatment: US$40/bbl before income tax; the treatment costs US$0.5 million
Rate: 10%, $j = \ln 1.1 = 0.09531$ 1/year

Solution

The present-volume factors of the reserves: $0.2/0.29531 = 0.6773$ and $0.4/0.49531 = 0.8076$ .
The gain in discounted production: $300\cdot(0.8076 - 0.6773) = 39.1$ thousand bbl. The gain before tax is $40\cdot39.1 = 1564$ thousand dollars.
NPV before tax: $1564 - 500 = 1064$ thousand dollars.
The limit of the gain: $300\cdot(1 - 0.6773) = 96.8$ thousand bbl, or about 3.87 million dollars.

Answer: A treatment that only makes the well produce the same 300 thousand bbl twice as fast is worth about 1564 thousand dollars before tax. At a cost of 0.5 million dollars it gives an NPV of about 1064 thousand dollars. No acceleration of this well can give more than 3.87 million dollars.

What it means. The formula shows that the value of acceleration has an upper limit. An economic limit changes the answer a little, in both directions. It cuts off the “tail” of both cases, but a fast-declining well has less oil left at the limit, and part of the acceleration turns into added reserves. In practice a treatment almost always adds a little to the reserves as well, and this part is valued separately. The data in this example are assumed for illustration.

Tax in an incremental calculation

The formula for the incremental flow with the factor $1 - T$ is correct while every year is profitable in both cases and the tax is a constant share of profit. In a producing field this is not always so.

Losses of past years. If the field has losses carried forward (to be set against future profit) without the project, the gain in profit is first absorbed by them, and the tax on it appears later. The incremental tax is then smaller than the rate multiplied by the incremental profit, and the project is worth more than the simplified formula shows.
A loss-making year with the project. If the investment of the project turns a year that would have been profitable without it into a loss-making one, the tax deduction is put off to later years. The simplified formula shows it at once and overstates the value.
Taxes that depend on profitability. If the tax rate or the government share grows with the profitability of the field, the incremental profit is taxed at the marginal rate (the rate on the last extra dollar), not at the average one (Petroleum fiscal systems and taxation).
Separate accounting by area. In some tax regimes, the deductions of one area or contract cannot be set against the income of another (ring-fencing). The costs of a project in a new area reduce tax only there.

In all these cases, the incremental tax is found in the same way as the incremental flow: the tax is calculated on the full forecasts “with the project” and “without the project”, and the two are subtracted. In the example of the lesson, both cases are profitable in every year. The smallest taxable base with the wells, 5.4 million dollars, falls in year 9. So the simplified formula and the full calculation agree.

Connections to other topics

Project cash flow — the principle of “with the project” and “without the project”, sunk costs and side effects.
Cash flow model of an oil and gas project — the model that calculates both forecasts: the tax regime, the economic limit, abandonment.
Net present value (NPV) — the NPV profile, the decision rule and the NPV index.
Internal rate of return (IRR) — several IRRs, the modified IRR and the incremental flow of mutually exclusive options.
Annuities and discounted production — discounted production, the rate for barrels when the price rises, and exponential decline.
Unit costs per barrel — why the average cost per barrel is not the cost of an incremental barrel.
Sensitivity analysis and scenarios — the calculation “from the current point” and how stable the result is to the assumptions.
Choosing the discount rate — the hurdle rate, on which the value of acceleration depends.
Annual capital charge — the annual capital charge and the comparison of options with different service lives.
Water and gas coning — when too fast a withdrawal lowers the recovery and acceleration turns into a loss of reserves.

Check yourself

1
A project moves 100 thousand bbl of production from year 6 to year 1; the reserves do not change. The margin is US$30/bbl before tax, the rate is 10%, and the flows are at the end of the year. What is the move worth, before the costs of the project?
1. A.Nothing: the reserves are the same
2. B.US$3.0 million
3. C.US$1.31 million
4. D.US$1.03 million
Show the answer
Correct answer: D. US$1.03 million
The moved barrels bring $0.1\cdot30 = 3.0$ million dollars in year 1 instead of year 6: $3.0\cdot(1.1^{-1} - 1.1^{-6}) = 3.0\cdot(0.90909 - 0.56447) = 1.03$ million dollars. The 3.0 is the revenue without discounting. “Nothing” is a conclusion that ignores the time value of money. The 1.31 comes from assuming that the early revenue arrives at once and not a year later: $3.0\cdot(1 - 1.1^{-6})$ .
2
Company A values projects at a rate of 6%, and company B at a rate of 15%. For which of them are the infill wells of the lesson worth more? Can a sale of such a project from A to B suit both sides?
Show the answer
At 6%, the NPV of the wells at the decision date is 0.11 million dollars, and at 15% it is 2.50 million dollars. Acceleration is worth more to the company with the higher rate, because early money is worth more to it. So a deal can suit both sides. At a price between these values, A gets more than the project is worth to itself, and B pays less than the project is worth to it. This is a property of acceleration projects: for an ordinary project, NPV only falls as the rate rises.
3
The budget is US$5 million. Project P needs US$2 million and gives an NPV of US$1.2 million. Q needs US$3 million and gives US$1.5 million. R needs US$5 million and gives US$2.3 million. The projects are indivisible and independent. What should we choose?
1. A.All three
2. B.P and R
3. C.P and Q
4. D.Only R: it has the largest NPV
Show the answer
Correct answer: C. P and Q
The profitability indexes: P is $1 + 1.2/2 = 1.6$ , Q is $1 + 1.5/3 = 1.5$ , and R is $1 + 2.3/5 = 1.46$ . P and Q together use exactly US$5 million and give an NPV of US$2.7 million, which is more than R alone (2.3). P and R, or all three, do not fit in the budget.
4
The effect of a treatment was estimated like this: before it, the well gave 50 t/d, and after it 80 t/d. The gain of 30 t/d was multiplied by 365 days and by the margin to get a yearly gain. What is wrong with this calculation?
Show the answer
It compares “before” and “after”, but what is needed is “with the treatment” and “without it”. Without the treatment the rate would also fall, and in a year it would be below 50 t/d, so the gain in the first year may even be more than 30 t/d. But after the treatment the rate falls too, often faster. If the treatment only speeds up the withdrawal of the same reserves, then after a few years the production with it will be lower than without it. This shortfall is not in the calculation at all. The right way is two forecasts of production by year and the difference of the full cash flows.
5
The forecast says that the real oil price will grow by 10% a year, and the company's rate is also 10%. What is pure acceleration of production worth, at constant costs per barrel?
1. A.Moving barrels gives nothing by itself, and the investment and the early costs make the project loss-making
2. B.More than at a constant price
3. C.It cannot be valued: the formula for the rate for barrels does not work
4. D.The same as at a constant price
Show the answer
Correct answer: A. Moving barrels gives nothing by itself, and the investment and the early costs make the project loss-making
If the price grows at the discount rate, the barrels are discounted at the rate $(1.1/1.1) - 1 = 0$ . The discounted revenue of a barrel is the same in any year, and moving production does not change it. What remains is the investment and the variable costs, which acceleration also moves to earlier years, so NPV is negative. The faster the expected price grows, the less acceleration is worth. In the example of the lesson, the NPV of the wells falls to zero at a price growth of about 3.6% a year.
6
Why do the infill wells of the lesson have two IRRs, while the same wells with added reserves of 0.3 million bbl have one?
Show the answer
Pure acceleration has a negative undiscounted total (−2.40 million dollars), and its cumulative flow changes sign twice: −2.88, then plus, then minus again towards the end. So the NPV profile starts below zero, rises and falls again: two crossings. With added reserves the undiscounted total is positive (7.2 million dollars), the cumulative flow stays positive after the first year until the end, and there is one positive IRR.

Updated: 17 October 2026