Cumulative cash flow and payback period

Quick refresher

The cumulative flow at the end of a year is the sum of the net cash flows (NCF) of all years up to and including it. Add flows in constant prices: the sum of nominal flows of different years mixes money of different purchasing power.
The lowest point of the curve is the maximum cash outlay (MCO): the money that must be found outside the project. For the example project of this library it is 140 million dollars, while the capital expenditure (the money spent on wells and facilities) is 160 million. The capital costs of the later years are paid from the revenue. If first oil is delayed, the financing requirement grows.
The payback period is the time from the start of the project to the moment when the cumulative flow becomes positive for good. For the example project it is 4.4 years from the start of the investment, or 2.4 years from first oil (the start of production). Always name the point from which it is counted.
The payback period shows how long the money stays at risk. It does not show the size, the return or the value of the project: it does not see anything that happens after payback.
The terminal cash surplus (TCS) is the end point of the curve, and it equals the NPV at a zero rate. It is not the maximum of the curve, because abandonment costs (closing the field down at the end) cannot be avoided. The ratio PIR = TCS/|MCO| shows the return per unit of investment, but it ignores time.
The discounted payback period is the moment when the project has returned the investment together with the required return on it. It is longer than the simple one (4.9 against 4.4 years at a rate of 10%). For an ordinary flow it exists only if the NPV at that rate is not negative.

Key relationships

\mathrm{CCF}_t = \sum_{k=1}^{t} \mathrm{NCF}_k

cumulative net cash flow at the end of year $t$

\mathrm{MCO} = \min_t \mathrm{CCF}_t

maximum cash outlay: the lowest point of the curve

t_{\text{pb}} = n + \dfrac{|\mathrm{CCF}_n|}{\mathrm{NCF}_{n+1}}

payback period; $n$ is the last year with a negative cumulative flow

\mathrm{TCS} = \mathrm{CCF}_T = \mathrm{NPV}(0)

terminal cash surplus: the end point of the curve

\mathrm{PIR} = \dfrac{\mathrm{TCS}}{|\mathrm{MCO}|}

profit-to-investment ratio

Common pitfalls

Adding flows in nominal money and calling the sum the “earnings” of the project.
Quoting a payback period without a starting point: from the start of the investment or from the start of production. The difference can be years.
Choosing the project with the shortest payback period without looking at the NPV.
Taking the sum of the capital expenditure for the financing requirement.
Taking the maximum of the curve as the result of the project, and forgetting the abandonment costs.

After this lesson you can

draw the curve of the cumulative cash flow, and find on it the maximum cash outlay, the payback period and the terminal cash surplus
explain how the maximum cash outlay differs from the sum of the capital expenditure, and why it grows when a project is delayed
calculate the payback period with interpolation within a year, and state the moment from which it is counted
calculate the profit-to-investment ratio (PIR) and explain why it ignores time
find the discounted payback period and link it with the net present value (NPV)
explain for which decisions these indicators are useful, and why they do not replace the NPV

The curve of the cumulative flow

In the lesson Project cash flow, a project is described by a series of yearly net cash flows (NCF). If you add them year after year, you get the cumulative cash flow. It is the “account” of the project without interest: how much money the project has taken from its owners, or returned to them, by a given moment.

Cumulative net cash flow

\mathrm{CCF}_t = \sum_{k=1}^{t} \mathrm{NCF}_k = \mathrm{CCF}_{t-1} + \mathrm{NCF}_t

where:

$\mathrm{CCF}_t$ cumulative net cash flow at the end of year $t$ : minus means that the project has taken money in total, plus means that it has returned more than it took, US$ million
$\mathrm{NCF}_k$ net cash flow of year $k$ : money in is “+”, money out is “−”, US$ million
$t$ number of the project year; the first year is the year of the first costs that still depend on the decision, —

Units: 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 slope of the curve equals the yearly NCF. While the yearly flow is negative, the curve goes down. When the flow becomes positive, the curve turns up. The turning point is its minimum.

For comparison, the discounted cumulative flow is shown below too. In it, each yearly flow is multiplied by its discount factor. This reflects the fact that late money is worth less than early money (Discounting and present value). Its end point is the net present value of the project (NPV; the lesson Net present value (NPV)). We need it at the end of this lesson. The main indicators are calculated from the ordinary, undiscounted flow.

Yearly and cumulative cash flow of the example project, US$ million (constant prices)

Year: Year 1
NCF: −50.0
Cumulative NCF: −50.0
Cumulative at 10%: −50.0

Year: Year 2
NCF: −90.0
Cumulative NCF: −140.0
Cumulative at 10%: −131.8

Year: Year 3
NCF: 32.8
Cumulative NCF: −107.2
Cumulative at 10%: −104.7

Year: Year 4
NCF: 78.4
Cumulative NCF: −28.8
Cumulative at 10%: −45.8

Year: Year 5
NCF: 78.4
Cumulative NCF: 49.6
Cumulative at 10%: 7.7

Year: Year 6
NCF: 65.6
Cumulative NCF: 115.2
Cumulative at 10%: 48.5

Year: Year 7
NCF: 52.8
Cumulative NCF: 168.0
Cumulative at 10%: 78.3

Year: Year 8
NCF: 33.6
Cumulative NCF: 201.6
Cumulative at 10%: 95.5

Year: Year 9
NCF: 20.8
Cumulative NCF: 222.4
Cumulative at 10%: 105.2

Year: Year 10
NCF: −12.0
Cumulative NCF: 210.4
Cumulative at 10%: 100.1

Year	NCF	Cumulative NCF	Cumulative at 10%
Year 1	−50.0	−50.0	−50.0
Year 2	−90.0	−140.0	−131.8
Year 3	32.8	−107.2	−104.7
Year 4	78.4	−28.8	−45.8
Year 5	78.4	49.6	7.7
Year 6	65.6	115.2	48.5
Year 7	52.8	168.0	78.3
Year 8	33.6	201.6	95.5
Year 9	20.8	222.4	105.2
Year 10	−12.0	210.4	100.1

The same project as in the lessons Project cash flow and Net present value (NPV). The cumulative values are at the end of the year. The last column is the cumulative discounted flow at a rate of 10%. The data in this example are assumed for illustration.

CalculatedCumulative cash flow of the example project without discounting and at a rate of 10%. The figure marks the maximum cash outlay (−140 million dollars), the simple and the discounted payback periods (4.4 and 4.9 years), and the end points of the curves: the terminal cash surplus (210.4 million dollars) and the NPV₁₀ (100.1 million dollars).

The curve has four characteristic points, and each gives its own indicator.

The minimum is −140 million dollars at the end of year 2. This is how much money the project has taken from its owners at the worst moment.
The crossing of zero is in year 5. By then the receipts equal everything that has been invested. The time to this moment is the payback period.
The maximum is 222.4 million dollars at the end of year 9, the last year of production.
The end point is 210.4 million dollars. This is the result of the project after all costs, including abandonment.

All these indicators come from simple addition, so it matters what is added. A sum of flows in nominal money (money of the day) mixes money of different purchasing power. Inflation “inflates” the late receipts, and the project looks better than it is (Inflation: nominal and real money).

Worked exampleThe same project in nominal money

Given (assumed data)

Flow in constant prices of year 1: −50.0; −90.0; 32.8; 78.4; 78.4; 65.6; 52.8; 33.6; 20.8; −12.0 US$ million
Inflation: 5% a year; for simplicity, all items of the flow grow with it

Solution

The nominal flow of a year is found by multiplying by $1.05^{t-1}$ . For year 2 it is $-90\cdot 1.05 = -94.5$ , and for year 5 it is $78.4\cdot 1.05^{4} = 95.30$ million dollars.
The cumulative nominal flow falls to −144.5 million dollars and ends at 291.6 million dollars, against −140 and 210.4 in constant prices.
The payback period from the nominal sums is $4 + 17.58/95.30 = 4.2$ years, against 4.4 in constant prices.

Answer: In nominal money the project “earns” 291.6 million dollars instead of 210.4, and “pays back” in 4.2 years instead of 4.4.

What it means. The difference is only inflation: the extra 81.2 million dollars add nothing to the purchasing power of the owners. This is why the indicators of the cumulative flow are calculated in constant prices. The exception is the financing plan. The bank and the company's treasury department need the sums that will actually have to be paid, that is, the nominal 144.5 million dollars, not 140. The data in this example are assumed for illustration.

The maximum cash outlay

The lowest point of the curve shows how much money the project takes from its owners at the worst moment. It is called the maximum cash outlay (maximum capital outlay, MCO; it is also called the maximum exposure). This is the amount that the owners must put into the project from other sources, their own or borrowed, before the project starts to pay for itself.

Maximum cash outlay

\mathrm{MCO} = \min_{t}\,\mathrm{CCF}_t

where:

$\mathrm{MCO}$ the minimum of the cumulative flow: a negative number; the financing requirement equals its absolute value, US$ million
$\mathrm{CCF}_t$ cumulative net cash flow at the end of year $t$ , US$ million

Units: 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.

After the minimum, the yearly NCF is positive: the revenue is enough for the running costs and for the remaining capital expenditure. From then on, the project pays for itself.

MCO is not the same as the sum of the capital expenditure, although both are called “the size of the investment”.

The financing requirement is usually smaller. Part of the capital expenditure falls in years when the project already earns revenue (new wells drilled after start-up, an expansion of facilities), and it is paid from that revenue. In the example project, the capital expenditure is 160 million dollars and the financing requirement is 140 million: the 20 million dollars of year 3 are covered by revenue.
It can also be larger than the capital expenditure before start-up, if the first months or years of production make a loss: the operating costs are already running, while output is still small.
The line between capital and operating costs is a matter of definition. Equipment that is leased does not enter the capital expenditure at all, although there is no production without it. MCO does not depend on the name of an item: it is calculated from the money.

So as a measure of financial commitments, MCO is more reliable than the sum of the capital expenditure. But it has a weakness too. It depends on how costs and income are placed in time, and it is especially sensitive to a delay in the start of production.

Worked exampleFirst oil one year later

Given (assumed data)

Project: from the examples of this library
What changed: production starts in year 4 instead of year 3; the capital expenditure (50, 90 and 20 million dollars) stays in years 1–3, and abandonment moves to year 11

Solution

In year 3 there is no revenue, so the remaining 20 million dollars of capital expenditure must be financed from outside. The cumulative flow falls to $-140 - 20 = -160$ million dollars instead of −140.
After that, the flow repeats the original one, shifted by a year. The first year of production, now year 4, brings 52.8 million dollars. This is the 32.8 million dollars of the original year 3, plus 20 million. In the original project, those 20 million went on capital expenditure in year 3. Here they were already paid a year earlier. The tax is the same, because depreciation starts with first oil.
Payback period: $5 + 28.8/78.4 = 5.4$ years, against 4.4.
The terminal cash surplus has not changed: it is 210.4 million dollars. The NPV at a rate of 10% has fallen from 100.1 to 77.5 million dollars.

Answer: A delay of one year raised the financing requirement from 140 to 160 million dollars (by 14%) and the payback period by a year. The final sum is the same, and the NPV is 23% lower.

What it means. The closer the delay is to the start-up, the more money has already been invested and the more the waiting costs. The terminal cash surplus does not notice this, because it ignores time. In practice a delay costs even more than in the example. While a facility stands idle, fixed costs go on, and the delay itself is often caused by a rise in the cost of the work. The data in this example are assumed for illustration.

CalculatedCumulative flow of the example project and of the same project with first oil one year later. The minimum falls from −140 to −160 million dollars, the payback moves from 4.4 to 5.4 years, and the end point stays the same, 210.4 million dollars.

The payback period

The payback period is the time in which the cumulative flow returns to zero: the receipts of the project have caught up with everything invested in it. An annual model gives only the values at the end of each year. So the moment of crossing zero is found by interpolation (a straight-line estimate) within the year.

Payback period from the yearly cumulative flow

t_{\text{pb}} = n + \frac{|\mathrm{CCF}_n|}{\mathrm{NCF}_{n+1}}

where:

$t_{\text{pb}}$ payback period, counted from the start of the first year of the project, years
$n$ the last year at whose end the cumulative flow is still negative, —
$\mathrm{CCF}_n$ cumulative flow at the end of year $n$ , US$ million
$\mathrm{NCF}_{n+1}$ net cash flow of the year in which the cumulative flow crosses zero, US$ million

Units: 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.

The fraction shows what part of year $n+1$ is needed to close what remains of the “debt” of the project. The bigger the flow of this year compared with what remains, the earlier in the year payback comes.

Worked exampleIndicators of the cumulative flow of the example project

Given (assumed data)

NCF in years 1–10: −50.0; −90.0; 32.8; 78.4; 78.4; 65.6; 52.8; 33.6; 20.8; −12.0 US$ million (constant prices)
Start of production: start of year 3

Solution

The cumulative flow at the end of each year: −50.0; −140.0; −107.2; −28.8; 49.6; 115.2; 168.0; 201.6; 222.4; 210.4 million dollars.
The minimum is −140 million dollars at the end of year 2: this is the MCO. The capital expenditure of the project is 160 million dollars, which is 20 million more.
The last year with a negative cumulative flow is year 4 (−28.8 million dollars), and the flow of year 5 is 78.4 million dollars: $t_{\text{pb}} = 4 + 28.8/78.4 = 4.37$ years from the start of the project.
From the start of production, $4.37 - 2 = 2.37$ years have passed.
The maximum of the curve is 222.4 million dollars at the end of year 9. The end point, after abandonment, is 210.4 million dollars.

Answer: MCO = −140 million dollars. The payback period is 4.37 years from the start of the investment, or 2.37 years from first oil. The terminal cash surplus is 210.4 million dollars.

What it means. The same numbers were obtained in the lesson on cash flow. Here they are treated as indicators. The phrases “the project pays back in 2.4 years” and “in 4.4 years” describe the same project. Only the starting point is different. The data in this example are assumed for illustration.

What the payback period shows and what it does not

The payback period answers one question: how long the invested money stays unreturned. It says nothing about the rest. This is easy to see on three projects with the same payback period.

Worked exampleThree projects with one payback period

Given (assumed data)

Project A: −100 in year 1, then +40 for three years
Project B: −100 in year 1, then +40 for eight years
Project C: −100; +40; +40; +40; +10 and −25 in year 6 (abandonment)
Units and rate: US$ million in constant prices; discount rate 10%

Solution

For all three, the cumulative flow at the end of year 3 is −20, and the flow of year 4 is +40: $t_{\text{pb}} = 3 + 20/40 = 3.5$ years.
Terminal cash surplus: A is 20, B is 220, C is 5 million dollars.
NPV at 10% (the first flow is at the origin, the others come after 1, 2, … years): A is −0.5, B is 113.4, C is −9.2 million dollars.

Answer: The payback period is the same, 3.5 years, but the NPV at 10% ranges from −9.2 to 113.4 million dollars.

What it means. Project A returns the invested money and almost nothing more: at 10% its NPV is negative. Project B goes on working for more than five years after payback. Project C “pays back” just as fast, but the late abandonment costs eat almost all the surplus. The payback period does not see these differences. The data in this example are assumed for illustration.

CalculatedCumulative flow of the three projects of the example. All cross zero after 3.5 years, and they end at +20, +220 and +5 million dollars.

What the payback period does not take into account:

Everything that happens after payback: how many years the project still works, how much it brings, and how it ends.
The time value of money. Within the payback period, all sums are added one to one. The discounted payback period corrects this partly (see the section below).
The size of the project. Replacing a pump and developing a field can pay back in the same time. In general, big and complex projects pay back more slowly, because their construction period is longer.
The required return. The payback period has no natural dividing line between “good” and “bad”. A company sets a threshold itself, such as “not longer than three years”, and such a threshold systematically cuts out long projects of high value.

Why is it used, then? The payback period shows how long money stays at risk. This matters when risk grows with time: the term of a contract is limited, subsoil use conditions may change, and the price in the far future is less well known. It also matters for a company with little free money, because money that comes back quickly can be put into the next project. Finally, this number is clear without explanations. So the payback period is shown next to the NPV and the IRR as an additional characteristic, but not instead of them.

The terminal cash surplus and the profit-to-investment ratio

The end point of the curve is the terminal cash surplus (TCS; it is also called the final surplus). It is everything that the project received, minus everything that it spent: capital and operating costs, taxes, and abandonment. It is the simplest measure of the profit of a project. At the same time, it is the NPV of the project at a zero discount rate.

TCS is exactly the end point, not the maximum of the curve. For the example project, the maximum is 222.4 million dollars and the result is 210.4. The difference is the abandonment costs. They cannot be avoided, and stopping the project earlier does not save this money: the duty to abandon wells and facilities arises together with them, not in the last year.

A big project has a big TCS, so to compare projects of different size the result is divided by the investment. The simplest such ratio is the profit-to-investment ratio (PIR). Both quantities are taken from the same curve of the cumulative flow.

Profit-to-investment ratio

\mathrm{PIR} = \frac{\mathrm{TCS}}{|\mathrm{MCO}|}

where:

$\mathrm{PIR}$ profit-to-investment ratio, —
$\mathrm{TCS}$ terminal cash surplus: the end point of the curve, US$ million
$\mathrm{MCO}$ maximum cash outlay: the minimum of the curve, US$ million

Units: 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.

How much net surplus each dollar invested at the worst moment brings in the end, on top of the return of the dollar itself. For the example project, $\mathrm{PIR} = 210.4/140 = 1.50$ . If the whole curve is divided by $|\mathrm{MCO}|$ , projects of different size can be drawn on one graph: each curve goes down to −1 and ends at the PIR value.

Worked exampleThe same PIR, different value

Given (assumed data)

Investment: $100 million in year 1 in each of the three projects
Project X: +$60 million a year for 5 years
Project Y: +$30 million a year for 10 years
Project Z: +$10 million a year for 30 years

Solution

For all three, MCO = −100 and TCS = $300 - 100 = 200$ million dollars, so PIR = 2.0.
Payback period from the start of the project: X is 2.7 years, Y is 4.3 years, and Z is 11 years.
NPV at a rate of 10%: X is 127.4, Y is 84.3, and Z is −5.7 million dollars.
Internal rate of return (IRR, the yearly return that the project earns on the money invested; the lesson Internal rate of return (IRR)): X is 52.8%, Y is 27.3%, and Z is 9.3%.

Answer: With the same MCO, TCS and PIR = 2.0, the projects are worth from −5.7 to 127.4 million dollars at a rate of 10%.

What it means. PIR, like TCS, does not distinguish money received after one year from money received after thirty. Project Z returns three times what was invested, but so slowly that at 10% it reduces the value of the company. So PIR is used only to compare projects with a similar distribution of money in time. The final choice is made by the NPV. The data in this example are assumed for illustration.

PIR, PI and other indices

Similar names are used for different ratios: the result to MCO, the result to the sum of the capital expenditure, and the discounted income to the discounted investment (the profitability index). The abbreviation PI is ambiguous. It stands both for the profitability index and for the productivity index of a well (how much oil a well gives for each unit of pressure drop), so it is spelled out at the first mention. The discounted “relative” of PIR is the NPV index from the lesson Net present value (NPV). The denominator also depends on the development concept (the plan for how the field is built and equipped). Suppose the oil treatment plant of the example project (the plant that prepares the oil for sale) is leased, not bought (example). Then the financing requirement falls from 140 to 80 million dollars, and PIR rises from 1.50 to 2.39, although the terminal surplus is smaller: 191.2 instead of 210.4 million dollars. Only indicators defined in the same way, for projects built in the same way, can be compared.

The discounted payback period

The simple payback period treats a dollar of year 5 as equal to a dollar of year 1. Take instead the cumulative discounted flow (the lesson on NPV). The moment at which it crosses zero gives the discounted payback period. By this moment the project has returned the investment together with a return equal to the discount rate, as if the money for the project had been borrowed at this rate.

Worked examplePayback period of the project at different rates

Given (assumed data)

Flow: the example project; origin in the middle of year 1, flows in the middle of each year
Rates: 0, 10 and 20%

Solution

At 10%, the cumulative discounted flow at the end of year 4 is −45.8 million dollars, and the discounted flow of year 5 is $78.4\cdot 1.1^{-4} = 53.5$ million dollars.
Payback period at 10%: $4 + 45.8/53.5 = 4.9$ years, against 4.4 without discounting.
At 20%, the crossing of zero moves into year 6: 5.7 years.
NPV at these rates: 210.4, 100.1 and 36.9 million dollars.

Answer: The payback period grows with the rate: 4.4 years at 0%, 4.9 at 10% and 5.7 at 20%.

What it means. The more expensive the money, the later the project returns it together with the interest. The discounted payback period is fairer than the simple one, but its main weakness is the same: it does not see the flows after payback. The data in this example are assumed for illustration.

The discounted payback period is linked with the NPV. Take an ordinary flow, with the investments first and the receipts later. After its minimum, the cumulative discounted flow only grows. So it reaches zero exactly when its end point, that is, the NPV, is not negative. In other words, the project pays back with the cost of money taken into account only if the NPV at this rate is positive. At a rate equal to the IRR, payback comes at the very end of the project. If there are large costs at the end, for example for abandonment, the curve may cross zero and return to it. This is why the definition has the words “for good”.

How to use these indicators

Indicators of the cumulative flow: what question they answer and what they do not see

Indicator: MCO
The question it answers: how much money must be found outside the project
What it does not see: how long it will be invested and what it will bring

Indicator: Payback period
The question it answers: how long the invested money stays at risk
What it does not see: everything after payback; the size and the value of the project

Indicator: TCS
The question it answers: how much money is left in the end
What it does not see: when it is received and how much had to be invested

Indicator: PIR
The question it answers: how much of the terminal surplus falls on each unit of investment
What it does not see: time: a fast and a slow project are equal

Indicator: Discounted payback period
The question it answers: when the investment is returned together with the required return
What it does not see: the flows after payback

Indicator	The question it answers	What it does not see
MCO	how much money must be found outside the project	how long it will be invested and what it will bring
Payback period	how long the invested money stays at risk	everything after payback; the size and the value of the project
TCS	how much money is left in the end	when it is received and how much had to be invested
PIR	how much of the terminal surplus falls on each unit of investment	time: a fast and a slow project are equal
Discounted payback period	when the investment is returned together with the required return	the flows after payback

For comparison: the NPV takes into account all the flows and the time they are received, but it shows neither the need for money nor the time of its return.

None of these indicators says whether a project creates value. The NPV at the company's rate answers that question. But the NPV is silent on how much money will be needed and when it will come back. This can decide the fate of a project: a good project for which financing cannot be found will not happen. So a project is described by a set of indicators: NPV and IRR for value and return, MCO for the need for money, and the payback period for the time at risk.

All these numbers come from one forecast of the cash flow, and they depend on its assumptions: the price, production, costs and timing. How each of them reacts to a change in the assumptions is checked by sensitivity analysis (Sensitivity analysis and scenarios). The example with the delay of first oil shows that they react differently. The final sum did not change, while the MCO, the payback period and the NPV changed a lot.

Terms and symbols

Cumulative cash flow: The sum of the net cash flows of a project from its start to a given moment.
Maximum cash outlay (MCO): The minimum of the cumulative flow. It is the money that must be found outside the project.
Payback period: The time from the chosen starting point to the moment when the cumulative flow becomes positive for good.
Terminal cash surplus (TCS): The end point of the curve of the cumulative flow. It equals the NPV at a zero rate.
Profit-to-investment ratio (PIR): The terminal cash surplus divided by the absolute value of the MCO.
Discounted payback period: The payback period found from the cumulative discounted flow at a given rate.

Common mistakes and misconceptions

Wrong

“The project pays back in two years, so it is a good one.”

In fact

A fast payback means only that the invested money comes back fast. If the project brings almost nothing after payback, its NPV at the company's rate may be negative.

Wrong

“The project needs 160 million dollars: that is the sum of the capital expenditure.”

In fact

The financing requirement is the minimum of the cumulative flow, not the sum of the capital expenditure. For the example project it is 140 million dollars, because the costs of year 3 are paid from revenue. With a delay in start-up, it will grow.

Wrong

“A payback period of 2.4 years for one project and 4 years for another: the first pays back faster.”

In fact

First find out the starting point. 2.4 years from first oil and 4.4 years from the start of the investment are the same project.

Wrong

“The maximum of the cumulative flow is the earnings of the project.”

In fact

The result of the project is the end point of the curve, after the abandonment costs. These costs cannot be avoided, and stopping early does not remove them.

Wrong

“PIR = 2 is better than PIR = 1.5.”

In fact

Only if the money comes in comparable times. PIR ignores time: a project with PIR = 2 that is stretched over thirty years can have a negative NPV.

In depth: advanced detail

Payback period and IRR: a rough link

Take the simplest project: one investment $I$ and equal yearly receipts $A$ for $N$ years. Its payback period from the moment of the investment is $I/A$ . The IRR of such a project is found from the condition that the present value of the receipts equals the investment. The annuity factor gives the sum of the discount factors of equal payments (Discounting and present value).

Link between the payback period and the IRR for equal receipts

\frac{I}{A} = \frac{1 - (1 + \mathrm{IRR})^{-N}}{\mathrm{IRR}}, \qquad \mathrm{IRR} \to \frac{A}{I}\ \text{as}\ N \to \infty

where:

$I$ investment at the initial moment, US$ million
$A$ equal yearly receipts at the end of each of the $N$ years, US$ million a year
$I/A$ payback period from the moment of the investment, years
$N$ life of the project, years
$\mathrm{IRR}$ internal rate of return, fraction per year

Units: 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.

One divided by the payback period is an upper limit of the IRR. It is reached only by an “eternal” project. The shorter the life of the project after payback, the further the IRR is from this limit.

Worked exampleA payback period of 4 years: what is the IRR?

Given (assumed data)

Payback period $I/A$: 4 years from the moment of the investment
Life of the project $N$: 5, 8, 10, 15 and 20 years

Solution

The upper limit: $1/4 = 25$ % a year, for an endlessly long project.
Solving the equation numerically: for $N = 5$ the IRR is 7.9%, for $N = 8$ it is 18.6%, and for $N = 10$ it is 21.4%.
For $N = 15$ it is 24.0%, and for $N = 20$ it is 24.7%.

Answer: With the same payback period of 4 years, the IRR runs from 7.9% to almost 25%, depending on how many years the project works after payback.

What it means. A short payback period is needed for a high return, but it is not enough by itself. The rule “IRR is about one divided by the payback period” is good only for long projects with even receipts. The data in this example are assumed for illustration.

Connections to other topics

Project cash flow — what the net cash flow is made of, and which flows belong to the project.
Cash flow model of an oil and gas project — how a cash flow is built year by year, and how the price, the timing and the leasing of equipment affect it.
Net present value (NPV) — the cumulative discounted flow, the NPV and the NPV index.
Internal rate of return (IRR) — the return of a project in per cent, and its pitfalls.
Inflation: nominal and real money — why flows are added in constant prices.
Choosing the discount rate — which rate to take for the discounted payback period.
Acid treatments: principle, economics and candidate selection — the payback period of a well treatment when the extra production falls with time.
Sensitivity analysis and scenarios — how the indicators react to price, costs and timing.

Check yourself

1
The cash flow of a project by year is −30; −50; +20; +40; +40; +30 million dollars. What are the MCO and the payback period from the start of the project?
1. A.MCO = −80, payback period 5 years
2. B.MCO = −80, payback period 4.5 years
3. C.MCO = −50, payback period 4.5 years
4. D.MCO = −60, payback period 3.5 years
Show the answer
Correct answer: B. MCO = −80, payback period 4.5 years
The cumulative flow at the end of each year is −30; −80; −60; −20; +20; +50. The minimum is −80 million dollars. The last year with a negative value is year 4 (−20), and the flow of year 5 is +40: $4 + 20/40 = 4.5$ years. The answer “5 years” comes from not interpolating within the year, and −50 is the largest yearly outflow, not the minimum of the cumulative flow.
2
Why is the maximum cash outlay usually smaller than the sum of the capital expenditure of a project? When can it be larger than the costs made before start-up?
Show the answer
Part of the capital expenditure falls in years after production starts, and it is paid from the revenue of the project itself. So less has to be found outside than the project spends on capital objects over its whole life. MCO can be larger than the costs before start-up when the first periods of production make a loss: the operating costs are already running, output is still small, and the cumulative flow goes on falling.
3
Project A pays back in 2 years and ends with a terminal flow of 5 million dollars. Project B pays back in 5 years and ends with 300 million dollars. The investments in both are equal. What can be said from these data?
1. A.A is better: it pays back faster
2. B.B is better: its final sum is bigger
3. C.Neither the payback period nor the final sum alone decides: the NPV at the company's rate is needed
4. D.The projects are equal in value
Show the answer
Correct answer: C. Neither the payback period nor the final sum alone decides: the NPV at the company's rate is needed
The payback period shows only the speed of the return of the investment, and the terminal flow shows only a sum without time. Project A has almost no value after payback, and for project B much depends on how its receipts are spread in time. The NPV at the company's rate decides. The payback period and the need for money add information about risk and financing.
4
A project has MCO = −200 million dollars and a terminal cash surplus of 500 million dollars. Find the PIR. What will change in this indicator if the same receipts are stretched from 8 to 25 years?
Show the answer
$\mathrm{PIR} = 500/200 = 2.5$ : each dollar invested brings in the end 2.5 dollars on top of its own return. If the receipts are stretched, the PIR will not change as long as the sums do not change, because it ignores time. The NPV and the IRR will fall a lot. This is the main weakness of the PIR.
5
How does the discounted payback period compare with the simple one, and what does its absence mean for an ordinary project?
1. A.It is longer than the simple one; if it does not exist, the NPV at this rate is negative
2. B.It is shorter than the simple one; its absence means nothing
3. C.It is longer than the simple one; if it does not exist, the project does not even return the money invested
4. D.They are equal at any rate
Show the answer
Correct answer: A. It is longer than the simple one; if it does not exist, the NPV at this rate is negative
Discounting reduces the late receipts more than the early costs, so the cumulative discounted flow crosses zero later. For an ordinary flow it reaches zero only if its end point, the NPV, is not negative. The project can still return the money invested, without regard to its cost: it has a simple payback period.
6
The start of production was delayed by a year, and the sums of costs and income did not change. How will the MCO, the payback period, the terminal cash surplus and the NPV change?
Show the answer
The MCO will grow in absolute value if the year of delay brings costs that used to be covered by revenue: now they must be financed from outside. The payback period will grow by about a year. The terminal cash surplus will stay the same: the sums are the same, and it does not take time into account. The NPV will fall: the receipts moved a year later, while the costs stayed in place. In the lesson's example it is −160 instead of −140 million dollars, 5.4 instead of 4.4 years, the same 210.4 million dollars, and an NPV that is 23% lower.

Updated: 2 October 2026