Net present value (NPV)

Quick refresher

To calculate an NPV you need three things: the cash flow by year, the discount rate, and the origin (the date to which all flows are brought) with a convention for when in the year the flows arrive. The result is written together with the rate: NPV₁₀ means the NPV at 10%.
The NPV is the end point of the cumulative discounted flow. At a zero rate it equals the undiscounted result of the project.
As the rate rises, the NPV of an ordinary project (costs first, income later) falls. The rate at which NPV = 0 is the internal rate of return (IRR), the yearly return that the project itself earns.
The rule: NPV > 0 means that the project earns more than the rate and increases the value of the company. NPV < 0 means that it earns less, even if it brings in money when undiscounted.
Moving the origin changes the size of the NPV by a factor of $(1+i)^{\Delta}$ , but not its sign. NPVs of projects can be compared only at the same rate, origin and money units.
The NPV does not show the size of the investment, the efficiency of the capital, the time to get the money back, or the risk. So we add the financing requirement (how much money must be found), the NPV index, the IRR and sensitivity analysis (a check of how the result changes when the assumptions change).

Key relationships

\mathrm{NPV}(i) = \sum_{t} \frac{\mathrm{NCF}_t}{(1+i)^{t}}

the sum of the discounted net cash flows; $t$ is counted from the chosen origin

\mathrm{NPV}(0) = \sum_{t} \mathrm{NCF}_t

at a zero rate, the undiscounted result of the project

\mathrm{NPV}_{t_0+\Delta} = \mathrm{NPV}_{t_0}\,(1+i)^{\Delta}

moving the origin $\Delta$ years later; for $\Delta < 0$ , earlier; the sign of the NPV does not change

\mathrm{NPV}(\mathrm{IRR}) = 0

the definition of the internal rate of return

\mathrm{NPVI} = \mathrm{NPV}_i / |\mathrm{MCO}_i|

NPV index: the value created per unit of the maximum discounted financing requirement

Common pitfalls

The standard NPV() function of a spreadsheet discounts the first flow of the range by a whole period. The flow of the year taken as the origin is added separately.
Nominal flows are discounted at a nominal rate, and flows in constant prices at a real rate. They must not be mixed.
Comparing NPVs of projects at different rates, with different origins or in different money units.
Subtracting loan interest from the cash flow and also discounting at a rate that already includes the cost of capital. This counts it twice.
Thinking that a large NPV means a small risk, or that the company will have enough money for the project.

After this lesson you can

calculate the NPV from a yearly net cash flow at a given rate and origin
build the cumulative discounted flow and find from it the NPV, the discounted financing requirement and the discounted payback period
build the NPV profile, explain how the NPV depends on the rate, and estimate the IRR from the profile
apply the NPV decision rule, and explain its meaning through the “bank account of the project”
list what the NPV does not show, and which indicators are added to it

What the NPV is

The cash flow of a project is a series of sums of different years. To get one number that answers the question “is the project worth its money?”, all the sums are moved to one moment of time with the discount rate and added up with their signs (plus for money in, minus for money out). The word “net” in the name means that the investments have already been subtracted: the sum includes both the negative flows of the construction years and the positive flows of the production years.

Net present value

\mathrm{NPV} = \sum_{t=0}^{T} \frac{\mathrm{NCF}_t}{(1+i)^{t}}

where:

$\mathrm{NPV}$ net present value at the moment of the origin, US$ million (in the units of the cash flow)
$\mathrm{NCF}_t$ net cash flow assigned to moment $t$ : money in is “+”, money out is “−”, US$ million
$t$ time from the origin to the moment of the flow, years
$T$ time of the last flow of the project, including abandonment (closing the field down), years
$i$ discount rate, fraction per year

Units: 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).

The NPV is the single sum that, at the date of the origin, is worth the same as all the future flows of the project together. A positive NPV means that the project returns the money invested with a return $i$ , and on top of that it creates value equal to the NPV.

For the calculation, we set three things: the cash flow by year, the discount rate, and the origin with a convention for the moment of the flows within the year. A change in any of them changes the NPV, so the result is written together with the rate: NPV₁₀ or NPV [10%]. In Russian-language project documents, the same indicator is usually called net discounted income (abbreviated ChDD in Russian). The name “net present value” is also used there.

Calculating the NPV step by step

The order of the calculation

1
Collect the cash flow
NCF by year after tax, including abandonment (Project cash flow). All sums are in the same money, for example in constant prices of the base year.
2
Choose the origin and the convention
For example, the middle of the first project year, with flows in the middle of each year: the exponents are $t = 0, 1, 2, \ldots$
3
Calculate the factors
$DF_t = (1+i)^{-t}$ for each year.
4
Discount
The discounted cash flow (DCF) of a year: $\mathrm{DCF}_t = \mathrm{NCF}_t\cdot DF_t$ .
5
Add up
The NPV is the sum of the DCF over all years. The running total of the DCF, year by year, shows when the project moves into plus once the value of money is taken into account.

We apply this to the project from the lesson Project cash flow: a small oil field, with two years of construction, seven years of production and abandonment in year 10. The flow is in constant prices, the rate is 10% (real), the origin is the middle of the first year, and the flows are assigned to the middle of each year.

Calculation of the NPV at a rate of 10%, US$ million (constant prices)

Year: 1
$t$: 0
NCF: −50.0
$DF = 1.1^{-t}$: 1.0000
DCF: −50.0
Cumulative DCF: −50.0

Year: 2
$t$: 1
NCF: −90.0
$DF = 1.1^{-t}$: 0.9091
DCF: −81.8
Cumulative DCF: −131.8

Year: 3
$t$: 2
NCF: 32.8
$DF = 1.1^{-t}$: 0.8264
DCF: 27.1
Cumulative DCF: −104.7

Year: 4
$t$: 3
NCF: 78.4
$DF = 1.1^{-t}$: 0.7513
DCF: 58.9
Cumulative DCF: −45.8

Year: 5
$t$: 4
NCF: 78.4
$DF = 1.1^{-t}$: 0.6830
DCF: 53.5
Cumulative DCF: 7.7

Year: 6
$t$: 5
NCF: 65.6
$DF = 1.1^{-t}$: 0.6209
DCF: 40.7
Cumulative DCF: 48.5

Year: 7
$t$: 6
NCF: 52.8
$DF = 1.1^{-t}$: 0.5645
DCF: 29.8
Cumulative DCF: 78.3

Year: 8
$t$: 7
NCF: 33.6
$DF = 1.1^{-t}$: 0.5132
DCF: 17.2
Cumulative DCF: 95.5

Year: 9
$t$: 8
NCF: 20.8
$DF = 1.1^{-t}$: 0.4665
DCF: 9.7
Cumulative DCF: 105.2

Year: 10
$t$: 9
NCF: −12.0
$DF = 1.1^{-t}$: 0.4241
DCF: −5.1
Cumulative DCF: 100.1

Year: Total
$t$: —
NCF: 210.4
$DF = 1.1^{-t}$: —
DCF: 100.1
Cumulative DCF: —

Year	$t$	NCF	$DF = 1.1^{-t}$	DCF	Cumulative DCF
1	0	−50.0	1.0000	−50.0	−50.0
2	1	−90.0	0.9091	−81.8	−131.8
3	2	32.8	0.8264	27.1	−104.7
4	3	78.4	0.7513	58.9	−45.8
5	4	78.4	0.6830	53.5	7.7
6	5	65.6	0.6209	40.7	48.5
7	6	52.8	0.5645	29.8	78.3
8	7	33.6	0.5132	17.2	95.5
9	8	20.8	0.4665	9.7	105.2
10	9	−12.0	0.4241	−5.1	100.1
Total	—	210.4	—	100.1	—

The total of the NCF column is the undiscounted sum, and the total of the DCF column is the NPV₁₀. The data in this example are assumed for illustration.

Worked exampleNPV of a small field at a rate of 10%

Given (assumed data)

Net cash flow 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
Discount rate: 10% (real)
Origin: the middle of year 1, flows in the middle of each year ( $t = 0 \ldots 9$ )

Solution

The factors $DF_t = 1.1^{-t}$ are 1; 0.9091; 0.8264; …; for year 10, $DF_9 = 0.4241$ .
For example, year 4 ( $t = 3$ ): $\mathrm{DCF} = 78.4\cdot 1.1^{-3} = 58.9$ million dollars. So 78.4 million dollars in three years are worth the same as 58.9 million today.
The sum of the discounted flows (the table): $\mathrm{NPV}_{10} = 100.1$ million dollars. Without discounting, the sum is 210.4 million dollars.
The cumulative discounted flow falls to −131.8 million dollars (without discounting, to −140) and turns positive in year 5. This is about 4.9 years from the start of the investment, against 4.4 years without discounting (if the flow of a year arrives evenly).

Answer:

\mathrm{NPV}_{10} \approx 100.1

million dollars > 0: at a rate of 10% the project is worth doing.

What it means. The project returns all the investment with a return of 10% a year and, on top of that, creates about 100.1 million dollars of value, in money at the date of the origin. More than half of the undiscounted result has been “eaten” by the cost of money: the income of the project comes in years 3–9, and the costs in the first two years. The data in this example are assumed for illustration.

The cumulative discounted flow

If you add the DCF year after year, you get the cumulative discounted flow. It has the same shape as the ordinary cumulative flow (Cumulative cash flow and payback period), but it takes the value of money into account. Three characteristics of the project can be seen on it.

The end point is the NPV at the given rate. At a zero rate it equals the undiscounted result of the project.
The minimum is the maximum discounted financing requirement (maximum cash outlay, MCO): how much money, in its equivalent at the date of the origin, the project needs at the worst moment.
The crossing of zero is the discounted payback period: the moment when the project has returned the investment together with a return $i$ on it.

CalculatedCumulative flow of the example project without discounting (the curve “0%”), at a rate of 10% and at a rate equal to the IRR (≈ 29.6%). The end points of the curves are the NPV at these rates: 210.4, 100.1 and 0 million dollars.

As the rate grows, the curve goes down: the end point (the NPV) falls, and payback comes later. At 20% it comes after about 5.7 years. At a rate equal to the IRR, the curve ends exactly at zero: the project only returns the investment with this return. Note that the minimum shrinks in absolute value (131.8 million dollars at 10%, against 140 without discounting). The costs of the second year are expressed in money at the date of the origin, and 90 million dollars paid in a year are worth 81.8 million today at 10%. The real payments do not get smaller because of this. For financing planning, the undiscounted sum is needed.

NPV at different rates: the NPV profile

The graph of the NPV against the discount rate is called the NPV profile. For an ordinary project, with costs first and income later, it falls. As the rate rises, the positive flows of the distant years lose more present value than the negative flows of the first years.

NPV of the example project at different rates, US$ million

Rate $i$: 0%
NPV: 210.4

Rate $i$: 5%
NPV: 147.3

Rate $i$: 10%
NPV: 100.1

Rate $i$: 15%
NPV: 64.4

Rate $i$: 20%
NPV: 36.9

Rate $i$: 25%
NPV: 15.5

Rate $i$: 30%
NPV: −1.3

Rate $i$: 35%
NPV: −14.7

Rate $i$: 40%
NPV: −25.3

Rate $i$	NPV
0%	210.4
5%	147.3
10%	100.1
15%	64.4
20%	36.9
25%	15.5
30%	−1.3
35%	−14.7
40%	−25.3

The origin is the middle of year 1, with flows in the middle of each year. The data in this example are assumed for illustration.

CalculatedNPV profile of the example project. At a zero rate the NPV equals the undiscounted result (210.4 million dollars), at 10% it is 100.1 million dollars, and the curve crosses the rate axis at the IRR ≈ 29.6%.

At $i = 0$ the profile starts from the undiscounted result of the project, 210.4 million dollars.
At the company's rate, the profile gives the NPV on which the decision is made.
At the crossing with the rate axis is the internal rate of return (IRR): the rate at which the NPV equals zero (Internal rate of return (IRR)). For the example project it is about 29.6%.

Worked exampleEstimating the IRR from the NPV profile

Given (assumed data)

NPV at 25%: 15.5 million dollars
NPV at 30%: −1.3 million dollars

Solution

The NPV changes sign between 25 and 30%, so the IRR lies between them.
Linear interpolation: $\mathrm{IRR} \approx 25 + 5\cdot\dfrac{15.5}{15.5 + 1.3} = 29.61$ %.
The exact solution of the equation $\mathrm{NPV}(i) = 0$ by a numerical method, for example by halving the interval again and again (bisection): $\mathrm{IRR} = 29.57$ %.

Answer: IRR ≈ 29.57%: the project stays worthwhile at any discount rate below this.

What it means. Interpolation along the chord (the straight line between the two points) slightly overstates the IRR, because the profile is convex and lies below the chord. For a narrow interval of rates the error is small. The IRR shows the “safety margin” of a project against the rate, but this indicator has pitfalls, which the lesson Internal rate of return (IRR) explains. The data in this example are assumed for illustration.

The decision rule and the meaning of the NPV

NPV > 0: the project returns the investment with a return above the rate $i$ and increases the value of the company. It is worth accepting if the risk and the need for money are acceptable.
NPV = 0: the project earns exactly $i$ . It is neither better nor worse than an alternative with that return.
NPV < 0: the project earns less than $i$ , and the money is better invested where that return is available. This is not necessarily a loss: the undiscounted result can be positive.

Why the NPV is value. Imagine that the project is financed from a bank account. A negative balance costs $i$ a year, and a positive balance earns $i$ a year. At the end of the project, the account will hold exactly $\mathrm{NPV}\cdot(1+i)^{T}$ , where $T$ is the time from the origin to the last flow. So the NPV is the equivalent, at the date of the origin, of what is left after all the invested money has been returned together with the return $i$ on it. The calculation for the example is in the advanced section.

From this comes the use of the NPV in buying and selling assets. The NPV at the buyer's rate is the highest price that the buyer can pay at the date of the origin and still get the return that the buyer requires. So when a deal is valued, the origin is put at the date of the deal.

Choosing between mutually exclusive options (only one of them can be built), for example two development schemes for one field: at one rate and with one origin, choose the option with the larger NPV. The profiles of the options can cross, so at another rate a different option may be the better one, just as when present values are compared (Discounting and present value).

When there is not enough money for all the worthwhile projects, the large NPV of one big project can lose to the sum of the NPVs of several small ones with the same total need for money. Then we look at the efficiency of the investment: the NPV index (NPVI).

NPV index

\mathrm{NPVI}_i = \frac{\mathrm{NPV}_i}{|\mathrm{MCO}_i|}

where:

$\mathrm{NPVI}_i$ NPV index at rate $i$ , —
$\mathrm{NPV}_i$ NPV at rate $i$ , US$ million
$\mathrm{MCO}_i$ the minimum of the cumulative discounted flow at the same rate: the maximum discounted financing requirement (a negative number), US$ million

Units: 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).

How much value is created by each unit of money that the project “freezes” at the worst moment. For the example, $\mathrm{NPVI}_{10} = 100.1/131.8 = 0.76$ . A similar indicator is the profitability index: the ratio of the discounted income to the discounted investment. Only indices defined in the same way can be compared.

The origin, the conventions and the money units

Moving the origin $\Delta$ years later multiplies all the DCFs, and so the NPV, by $(1+i)^{\Delta}$ . Moving it earlier divides them. The size changes, and the sign never does. So the conclusion “accept or reject” does not depend on the origin. But the NPVs of different projects can be compared only with the same origin. The same applies to the convention for the moment of the flows within the year.

Worked exampleOne project, several NPVs

Given (assumed data)

Project: from the example above, NPV₁₀ = 100.1 million dollars with the origin in the middle of year 1
Rate: 10%

Solution

Origin at the start of year 1, half a year earlier: $100.1/1.1^{0.5} = 95.5$ million dollars.
Flows at the end of the year, origin at the start of year 1 (this is how the standard NPV() function of a spreadsheet works): $100.1/1.1 = 91.0$ million dollars.
The decision date is two years before the start of year 1, with flows in the middle of the year: $95.5/1.1^{2} = 78.9$ million dollars.

Answer: Depending on the origin and the convention, the NPV of the same project ranges from 78.9 to 100.1 million dollars, but it is always positive.

What it means. All these numbers are correct. They answer different questions: how much the project is worth on one date or another. An error arises when they are mixed, for example when the NPV of one project at the decision date is compared with the NPV of another at the start of construction. The data in this example are assumed for illustration.

Which money the NPV is in. The NPV is expressed in the same units as the cash flow. If the flow is in constant prices of a base year, the NPV is in the prices of that year too, and the date of the origin is a separate parameter. The statement “NPV₁₀ = 100.1 million dollars in base-year prices, at the middle of project year 1” is correct: the year of the prices and the date of discounting need not be the same. Use a rate of the same kind as the money: real for constant prices, nominal for nominal money (Inflation: nominal and real money).

What the NPV shows and what it does not

The NPV and the indicators that complement it

Question: Does the project create value at the accepted rate?
Does the NPV answer it?: yes: this is the main answer of the NPV
What to add: —

Question: How much money must be invested, and how much can be lost at the worst moment?
Does the NPV answer it?: no
What to add: the maximum financing requirement (Cumulative cash flow and payback period)

Question: How efficiently is the money used?
Does the NPV answer it?: no: the NPV is an absolute figure, and it is larger for large projects
What to add: the NPV index, the IRR

Question: When will the investment come back?
Does the NPV answer it?: no
What to add: the payback period, including the discounted one

Question: What is the return of the project in per cent?
Does the NPV answer it?: no
What to add: the IRR (Internal rate of return (IRR))

Question: How stable is the result against the oil price, the costs and the reserves?
Does the NPV answer it?: no: it is one number for one set of assumptions
What to add: sensitivity analysis and scenarios (Sensitivity analysis and scenarios), expected monetary value (Expected monetary value and preference theory)

Question: Are the strategic benefits counted: experience, infrastructure, access to a region?
Does the NPV answer it?: only if they are expressed in the cash flow
What to add: a qualitative assessment, analysis of development options

Question	Does the NPV answer it?	What to add
Does the project create value at the accepted rate?	yes: this is the main answer of the NPV	—
How much money must be invested, and how much can be lost at the worst moment?	no	the maximum financing requirement (Cumulative cash flow and payback period)
How efficiently is the money used?	no: the NPV is an absolute figure, and it is larger for large projects	the NPV index, the IRR
When will the investment come back?	no	the payback period, including the discounted one
What is the return of the project in per cent?	no	the IRR (Internal rate of return (IRR))
How stable is the result against the oil price, the costs and the reserves?	no: it is one number for one set of assumptions	sensitivity analysis and scenarios (Sensitivity analysis and scenarios), expected monetary value (Expected monetary value and preference theory)
Are the strategic benefits counted: experience, infrastructure, access to a region?	only if they are expressed in the cash flow	a qualitative assessment, analysis of development options

The NPV is necessary for a decision, but it is not enough. A decision is made on a set of indicators and with risk in mind. The value of the NPV has meaning only together with the assumptions from which it was obtained: the price, the production profile, the costs and the rate.

Terms and symbols

Net present value (NPV): The sum of the discounted net cash flows of a project at the date of the origin.
Discounted cash flow (DCF): The NCF of a year multiplied by the discount factor.
Cumulative discounted flow: The sum of the DCF from the start of the project to a given year. Its end point is the NPV.
NPV profile: The NPV as a function of the discount rate.
IRR: The internal rate of return is the rate at which the NPV equals zero.
NPV index (NPVI): The NPV per unit of the maximum discounted financing requirement.
Discounted payback period: The time in which the cumulative discounted flow becomes positive.

Common mistakes and misconceptions

Wrong

“NPV = 100 million dollars means that the project will bring 100 million of profit.”

In fact

This is the value on top of the return of all the investment at the rate $i$ , expressed in money at the date of the origin. The undiscounted result of the same project is 210.4 million dollars. These are different quantities.

Wrong

“A negative NPV means that the project makes a loss.”

In fact

It means that the project earns less than the discount rate. Undiscounted, the project may bring in money, but it is worse than an alternative with a return of $i$ .

Wrong

“Of two projects, the one with the larger NPV is always better.”

In fact

Only with the same rate, origin and money units, and if there is enough money for either of them. When capital is limited, look at the efficiency of the investment too (the NPV index), and when the risks differ, look at the risk as well.

Wrong

“The NPV() function of a spreadsheet gives the right answer at once.”

In fact

It assumes that the first flow of the range is a whole period away from the origin. If that is not so, correct the result: add the first flow separately, or calculate through explicit factors.

Wrong

“If we move the origin to the start of production, the NPV will be larger and the project will be better.”

In fact

Only the date to which the value is expressed changes: the NPV is multiplied by $(1+i)^{\Delta}$ . The project itself does not change, and projects can be compared only with the same origin.

In depth: advanced detail

The project as a bank account

Let all the flows of the project pass through an account on which both a negative and a positive balance are multiplied by $1+i$ every year. The balance at the time $T$ of the last flow is all the flows compounded (grown with interest) up to that moment: $B_T = \sum_t \mathrm{NCF}_t\,(1+i)^{T-t} = (1+i)^{T}\sum_t \mathrm{NCF}_t\,(1+i)^{-t} = (1+i)^{T}\,\mathrm{NPV}$ . In the same way, the balance in any year in between equals the cumulative discounted flow compounded up to that year, so their signs always agree.

Worked exampleThe balance of the project account at 10% a year

Given (assumed data)

Flow: from the example of the NPV calculation
Rate on the account: 10% a year, on both a negative and a positive balance

Solution

After year 1 the balance is −50 million dollars. Every next year it is the balance × 1.1 plus the NCF of the year. After year 2: $-50\cdot1.1 - 90 = -145$ million dollars. This is the deepest minus. Without interest it would be −140.
The balance becomes positive in year 5: $-61.0\cdot1.1 + 78.4 = 11.3$ million dollars. The debt, together with its interest, has been repaid, as the discounted payback period showed.
At the end of year 10, the account holds 236.1 million dollars.
The NPV compounded over the same 9 years: $100.13\cdot1.1^{9} = 236.1$ million dollars. The sum is the same.

Answer: The balance of the account at the end of the project equals the NPV compounded at 10%: 236.1 million dollars.

What it means. The NPV is the “extra” money of the project, expressed at the date of the origin: everything above what is needed to return the investment and to pay 10% a year for the use of the money. If the rate reflects the real cost of money for the company, this is the sum by which the project increases its wealth. The data in this example are assumed for illustration.

If the rate changes from year to year

Sometimes the rate is set differently for different periods, for example when the risk of the project or the terms of financing change. Then the discount factor of year $t$ is a product of yearly factors: $DF_t = \prod_{k=1}^{t} (1+i_k)^{-1}$ . The NPV formula stays the sum of $\mathrm{NCF}_t\cdot DF_t$ . But the NPV profile against a single rate, and the IRR, lose their direct meaning: a return cannot be compared with a rate that changes in time by using one number. In practice this approach is rarely used, and it is always stated explicitly.

Connections to other topics

Project cash flow — what the net cash flow, which is discounted, is made of.
Discounting and present value — discount factors, the conventions for the moment of payment, the origin.
Cumulative cash flow and payback period — undiscounted indicators: the financing requirement, the payback period, the terminal surplus.
Internal rate of return (IRR) — the rate at which the NPV equals zero, its meaning and its pitfalls.
Choosing the discount rate — which rate to choose, and how it affects decisions.
Sensitivity analysis and scenarios — how the NPV reacts to the price, the costs and the reserves.
Expected monetary value and preference theory — how to take the probabilities of success and failure into account.

Check yourself

1
The cash flow of a project is −100 now ( $t = 0$ ), +60 in a year, and +60 in two years. What is the NPV at a rate of 10%?
1. A.3.8
2. B.−4.1
3. C.20.0
4. D.4.1
Show the answer
Correct answer: D. 4.1
$\mathrm{NPV} = -100 + 60/1.1 + 60/1.21 = -100 + 54.55 + 49.59 = 4.13$ . 20 is the undiscounted sum. 3.8 comes from discounting the first flow by a year too, as the NPV() function of a spreadsheet does: $4.13/1.1$ .
2
The NPV of a project at 10%, at the date of the start of construction, is 50 million dollars. What will it be if the flows are brought to a date one year earlier?
1. A.It may become negative
2. B.50 million dollars
3. C.55 million dollars
4. D.45.5 million dollars
Show the answer
Correct answer: D. 45.5 million dollars
Moving the origin a year earlier divides the NPV by $1.1$ : $50/1.1 = 45.5$ million dollars. The sign never changes, so the conclusion on whether the project is worthwhile is the same.
3
Why does the NPV of an ordinary project fall as the discount rate rises, while the NPV of a flow of the kind “receipts first, payments later” may rise?
Show the answer
In an ordinary project the negative flows are early and are discounted little, and the positive flows are late and are discounted a lot. As the rate rises, more of the positive is lost, and the NPV falls. If the flow is the other way round, the NPV may rise. Suppose money is received in advance and large payments, such as abandonment, come later. Then a rise in the rate reduces the present value of the late payments more, and the NPV rises. Such a flow is in essence like a loan.
4
The NPV of a project is +12 million dollars at 8% and −5 million dollars at 12%. What is the IRR, roughly?
1. A.about 9.2%
2. B.it cannot be estimated from these data
3. C.about 10.8%
4. D.about 11.5%
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Correct answer: C. about 10.8%
Linear interpolation: $8 + 4\cdot 12/(12 + 5) = 8 + 2.8 = 10.8$ %. Because the profile is convex, the exact IRR is a little lower than this estimate.
5
Project A: NPV 60 million dollars with a maximum discounted financing requirement of 400 million. Project B: NPV 35 million with a requirement of 100 million, and there are three more independent projects like it. The company has 400 million dollars. What should it choose?
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Four projects of type B take the same 400 million and give an NPV of 140 million dollars, against 60 for project A. The NPV index shows the same: 0.35 for B against 0.15 for A. When capital is limited, a larger NPV of one project does not mean a better choice. When capital is unlimited, it pays to accept all the projects with a positive NPV.
6
The NPV profiles of two development options of a field cross at a rate of 12%. At lower rates the “slow” option has the larger NPV, and at higher rates the “fast” one. The company's rate is 10%. What should it choose?
1. A.The “slow” one: at the company's rate its NPV is larger
2. B.The “fast” one: it has the higher IRR
3. C.It does not matter: the profiles cross
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Correct answer: A. The “slow” one: at the company's rate its NPV is larger
The decision is made by the NPV at the company's rate. At 10%, the “slow” option has the larger NPV. The higher IRR of the “fast” option comes from the fact that its income arrives earlier, and its NPV falls more slowly as the rate rises. But at 10% it is the “slow” option that creates more value.

Updated: 19 October 2026