Sensitivity analysis and scenarios

Quick refresher

A decision is judged only by the future flows that depend on it. Costs that are already spent, and costs that are already committed, do not affect the choice. We allow for risk with simple methods (a premium on the discount rate, a limit on the payback period), with analysis around the base case, and with expected value. The payback period is the time needed to get the invested money back.
We calculate sensitivity from the base case: we change one parameter and recalculate the whole model. In the example project of the library (NPV at 10% is $100.1 million) a price 10% lower reduces NPV by 28%, and capital expenditure (CAPEX, the money spent to build the field) 10% higher reduces it by 13%.
NPV is a small difference between large discounted sums (sums brought back to today's money). So it changes more than the parameter does: the project has “leverage”. The measure of this leverage is the elasticity: how many per cent NPV changes when the parameter changes by 1%.
Every parameter has its own plausible range, and the range is often not symmetric. The spider diagram shows the slopes and the ranges on one chart. The tornado chart ranks the parameters by how far NPV swings (from its lowest to its highest value) and shows which data to improve first.
The break-even value of a parameter is the value at which NPV is zero when the other parameters stay at their base values. For the example project it is a price of $32.4 per barrel (bbl), against the base price of $50/bbl. If another parameter changes, the break-even value moves.
Parameters change together, so we build scenarios: consistent sets of assumptions. The “worst case”, made of all the extreme values, shows what the project must withstand, but it is not a likely estimate. Only simulation gives probabilities, and it usually shows that the base case is not the average result.

Key relationships

E_x = \dfrac{\Delta\mathrm{NPV}/\mathrm{NPV}_0}{\Delta x/x_0}

elasticity of NPV with respect to the parameter $x$

P^{*} = P_0 - \dfrac{\mathrm{NPV}_0}{s_P}

break-even price, as long as NPV depends on price linearly

s_P = (1 - r)(1 - T)\sum_t \dfrac{Q_t}{(1+i)^{t}}

slope of NPV against price: the discounted production after royalty (a share of revenue paid to the state) and income tax

\mathrm{NPV}_{\text{delay}}(d) = \dfrac{\mathrm{NPV}_0 + PV_C}{(1+i)^{d}} - PV_C

first oil $d$ years later, with the dates of the capital expenditure unchanged

Common pitfalls

Testing all parameters by the same ±10% and calling this an assessment of risk.
Concluding that the project is safe because no single parameter on its own makes NPV negative.
Taking the length of a tornado bar for a probability, and the “worst case” for a cautious estimate.
Quoting a break-even price without the measure and the rate: at a rate of 0%, at 10% and for a single year of production these are three different numbers.
Taking the base case for the average expected result.
Including costs that have already been spent in the calculation of the decision.

After this lesson you can

separate the cash flows that belong to a decision from sunk costs (money that is already spent and cannot be recovered), and name the groups of methods that bring risk into a decision
calculate the sensitivity of NPV to price, production, costs and timing by changing one parameter at a time, and express it as an elasticity (the percentage change of NPV when the parameter changes by 1%)
draw and read spider and tornado charts, and explain why every parameter has its own range
find the break-even value of a parameter (the value at which NPV is zero) and the project's margin of safety against it, and state which measure and which rate the value refers to
build consistent scenarios, and explain why the “worst case” is a stress test (a check of what the project can survive) and not a likely estimate
explain what simulation adds, and why the base case is usually not equal to the average result

What belongs to a decision, and how a decision allows for risk

An economic calculation reduces an investment decision to a comparison of cash flows. The decision may be whether to drill a well, whether to build a second phase, or whether to extend production. Before we ask how far we can trust these flows, we must decide which of them matter at all.

Only the flows that depend on the decision belong to it: the future flows and the incremental ones, which the decision adds or removes (which flows belong to a project). Here is an example. A well is taking longer to drill than planned. Its budget was US$12 million, it has already cost US$15 million, and another US$4 million is needed to reach the planned depth. To decide whether to go on drilling, we act as if the US$15 million had never been spent. Money that is already spent and cannot be recovered is called a sunk cost. We compare the remaining US$4 million with what the well can bring in. Future payments that can no longer be avoided do not affect the choice either. An example is the payment for ordered equipment, when the order cannot be cancelled without paying in full. There is only one exception. If past costs still reduce a future tax, this saving is a future flow, and we count it.

That is why the economics of one and the same field is calculated in two ways. The “full-cycle” calculation includes all the costs from the very start, with exploration and appraisal (the drilling and testing that follow a discovery, to show how large it is). It shows whether the whole enterprise paid off, and it is needed when new exploration work is planned. The “point-forward” calculation includes only what still lies ahead. We make the next decision from it: whether to appraise a discovery, whether to start building the field, or whether to drill more wells. A project that loses money over the full cycle may still be worth continuing. Investments in a producing field are judged by what they change in its production and costs (Acceleration projects and incremental analysis).

Which flows to count is the first question. The second is how far we can trust them. At the moment of the decision we know neither the reserves nor the price nor the costs exactly. New data, partners and insurance do not remove this uncertainty completely (Risk reduction and diversification). The methods that bring uncertainty into the calculation fall into four groups.

How an investment decision allows for risk

Group of methods: Simple
What they do: raise the discount rate (the yearly return that the investor demands), limit the payback period, compare outcomes by simple rules
Where they are covered: this lesson; Choosing the discount rate; Cumulative cash flow and payback period

Group of methods: From the base case
What they do: take the best estimate of each parameter and see how the result reacts when the parameters differ from it: one parameter (sensitivity), several (scenarios), all at once (simulation)
Where they are covered: this lesson; Probabilistic volumetrics by Monte Carlo simulation

Group of methods: Expected value
What they do: weigh the outcomes by their probabilities, build decision trees
Where they are covered: Expected monetary value and preference theory; Decision trees and the value of information

Group of methods: Portfolio
What they do: assess not a single project but a set of the company's projects
Where they are covered: Risk reduction and diversification

Group of methods	What they do	Where they are covered
Simple	raise the discount rate (the yearly return that the investor demands), limit the payback period, compare outcomes by simple rules	this lesson; Choosing the discount rate; Cumulative cash flow and payback period
From the base case	take the best estimate of each parameter and see how the result reacts when the parameters differ from it: one parameter (sensitivity), several (scenarios), all at once (simulation)	this lesson; Probabilistic volumetrics by Monte Carlo simulation
Expected value	weigh the outcomes by their probabilities, build decision trees	Expected monetary value and preference theory; Decision trees and the value of information
Portfolio	assess not a single project but a set of the company's projects	Risk reduction and diversification

The methods do not exclude one another. Sensitivity analysis is done almost always, and the others are added as the sums at stake and the uncertainty grow.

Simple methods: the payoff matrix, the payback period, a premium on the rate

Simple methods need no probability distributions and no large calculation. They are crude, and it helps to understand what exactly each of them leaves out.

The payoff matrix writes a decision down in its most compact form. The rows are the actions that the decision-maker can choose. The columns are the states of nature: things that do not depend on the decision-maker and are not known in advance, such as the oil price or the size of the reserves. The cells hold the money result of each combination.

To choose an action from the matrix, we need a rule. The simplest rules need no probabilities. The pessimist's rule (in decision theory, the Wald criterion, or maximin) compares the worst outcomes of the actions and picks the action whose worst outcome is the best. The optimist's rule (maximax) does the same with the best outcomes. Minimax regret (the Savage criterion) looks at the regret: the difference between the best outcome that was possible in a given state of nature and the outcome of the chosen action. It shows how much we failed to gain, looking back. We choose the action whose largest regret is the smallest.

As an example, take the decision on how to sell the oil of the library's example project. We can sell it at the price that the market gives. We can also agree a fixed price with the buyer in advance, which is a forward contract. Such a price is lower than the expected one, but it does not depend on the market.

Payoff matrix: NPV of the project at a rate of 10%, US$ million

Action: Sell at the market price
Price 35: 15
Price 50: 100
Price 65: 185
Worst outcome: 15

Action: Fix the price for the whole volume
Price 35: 77
Price 50: 77
Price 65: 77
Worst outcome: 77

Action: Fix the price for half of the volume
Price 35: 46
Price 50: 89
Price 65: 131
Worst outcome: 46

Action	Price 35	Price 50	Price 65	Worst outcome
Sell at the market price	15	100	185	15
Fix the price for the whole volume	77	77	77	77
Fix the price for half of the volume	46	89	131	46

The oil price is in US$/bbl, one price for the whole production period; the fixed price is $46/bbl. The outcomes are calculated by the model of the project (Cash flow model of an oil and gas project) and rounded to whole numbers. The data in this example are assumed for illustration.

Regret matrix, US$ million

Action: Sell at the market price
Price 35: 62
Price 50: 0
Price 65: 0
Largest: 62

Action: Fix the price for the whole volume
Price 35: 0
Price 50: 23
Price 65: 108
Largest: 108

Action: Fix the price for half of the volume
Price 35: 31
Price 50: 11
Price 65: 54
Largest: 54

Action	Price 35	Price 50	Price 65	Largest
Sell at the market price	62	0	0	62
Fix the price for the whole volume	0	23	108	108
Fix the price for half of the volume	31	11	54	54

Regret is the best outcome in the same column of the payoff matrix minus the outcome of the action.

Worked exampleHow to sell the oil: three rules without probabilities

Given (assumed data)

Project and measure: the library's example project; the outcome is NPV at a rate of 10%, US$ million, rounded to whole numbers
States of nature: the oil price for the whole production period: $35, $50 or $65 per barrel
Actions: sell at the market price; fix the price at $46/bbl in advance for the whole volume; fix it for half of the volume

Solution

Outcomes. Selling at the market gives 15, 100 or 185 million dollars, depending on the price. Fixing the whole volume gives 77 million dollars at any price. Fixing half of the volume gives 46, 89 or 131 million dollars.
The pessimist's rule. The worst outcomes of the three actions are 15, 77 and 46 million dollars. The best of them belongs to fixing the whole volume.
The optimist's rule. The best outcomes are 185, 77 and 131 million dollars. The choice is to sell at the market.
Regrets. At a price of $35/bbl the best action would have been to fix the whole volume. Selling at the market falls short by $77 - 15 = 62$ million dollars, and fixing half of the volume falls short by 31. At a price of $65/bbl the best action would have been to sell at the market. Fixing the whole volume falls short by $185 - 77 = 108$ million dollars, and fixing half of the volume by 54.
Minimax regret. The largest regrets of the three actions are 62, 108 and 54 million dollars. The smallest of them belongs to fixing half of the volume.
None of the rules asked how likely each price is. Suppose the probabilities of the prices $35, $50 and $65/bbl are 25, 50 and 25%. The average result of selling at the market is $0.25\cdot15 + 0.5\cdot100 + 0.25\cdot185$ $= 100$ million dollars, against 77 for fixing the whole volume. With probabilities of 50, 40 and 10% the same calculation gives 66 million dollars. That is less than 77, so fixing the price becomes the better choice.

Answer: The pessimist's rule chooses to fix the whole volume, the optimist's rule chooses to sell at the market, and minimax regret chooses to fix half of the volume. None of the three answers depends on the probabilities of the prices.

What it means. The three rules gave three different answers, and each of them matches a type of behaviour. The pessimist pays for safety: a guaranteed 77 million dollars instead of 100 in the base case. People act like this when the worst outcome cannot be allowed, for example if at a low price the company would have nothing to pay its loan with. The optimist gives up protection. Minimax regret looks for the action that we will regret least whatever happens, and it usually leads to a compromise. The rules share one weakness: they do not need probabilities, so they treat a price that happens once in twenty years in the same way as an ordinary one. The matrix is useful as a record of the problem: the actions, the states and the outcomes. To choose from it in a sensible way, we must add probabilities (Expected monetary value and preference theory). The data in this example are assumed for illustration. In practice it is rare to be able to fix a price for years ahead and for the whole volume.

The payback period as a measure of risk. The longer the invested money stays unreturned, the more can change in that time. So companies limit the payback period, and where conditions are less stable they limit it more strictly (what the payback period shows and what it does not show). This kind of filter protects only against what builds up with time: a revision of taxes and contract terms, or prices and exchange rates moving far from the forecast. It is powerless against uncertainty that does not depend on time. Whether the reservoir turns out to be as it was drawn on the map, and whether the equipment reaches its design capacity, will be known in the first months. A short payback period promised in case of success does not make that success any more likely.

This measure has two variants. The payback volume is the production that must be reached to pay back the investment. For the library's example project it is about 4.9 million bbl out of 12. It shows what part of the reserves must be produced for the project to return the money put into it. The bailout time is the moment after which the project can be stopped without a loss. Here we count the money from selling the equipment and the cost of abandonment (closing the field down at the end of its life).

A premium on the discount rate. The most common method is to demand a higher return from riskier investments. A premium is added to the discount rate, a different one for each country, kind of activity or stage of the project (the hurdle rate, the lowest return a project must promise, and the premiums). The method is simple, but crude. The premium does not act on the risk itself but on time. It reduces distant flows strongly and near flows hardly at all, and it makes late costs, such as abandonment, “cheaper”. And a risk that is resolved in the first year, such as whether an exploration well turns out dry, is almost invisible to the premium.

The high calculated return of the risky projects that were accepted is a result of selection, not a property of risk: a higher rate only filters out the projects that cannot meet it. Besides, it is the return for the case when everything goes to plan. The rate does not say how often the plan fails.

The base case and sensitivity to one parameter

The base case is a run of the cash-flow model in which every parameter has its best estimate: the most likely production profile, the accepted price, the cost estimate (Cash flow model of an oil and gas project). It is not a forecast but a point of reference. The methods of this group ask what happens to the result if the assumptions turn out different.

From here on, the base case in this lesson is the library's example project. It produces 12 million bbl of oil (about 1.64 million t) over seven years. The price at the field is $50/bbl (about $366 per tonne). The capital expenditure (CAPEX) is $160 million, and the operating expenditure (OPEX, the running costs) is $6 million a year plus $5/bbl. The royalty (a share of the revenue paid to the state) is 10% of revenue, and the income tax is 20%. The NPV of this project at a real rate of 10% is $100.1 million (Net present value (NPV)). (A real rate is a rate with inflation taken out, used together with constant prices.) The prices, the costs and the tax regime are assumed and do not describe any country.

Sensitivity analysis is simple. We change one parameter by a set amount and leave the others at their base values. Then we recalculate the model and write down the new result. We recalculate the whole model, not one row: together with the price, the royalty and the income tax change, and together with production, the variable costs change. To compare the parameters with each other, we express the changes as fractions of the base values and take their ratio. This ratio is the elasticity.

Elasticity of NPV with respect to a parameter

E_x = \frac{\Delta\mathrm{NPV}/\mathrm{NPV}_0}{\Delta x/x_0}

where:

$E_x$ elasticity of NPV with respect to the parameter $x$ , —
$\mathrm{NPV}_0$ NPV of the base case (the index 0 means the base case, not a zero rate), US$ million
$\Delta\mathrm{NPV}$ change of NPV after the parameter changes, US$ million
$x_0$ base value of the parameter, units of the parameter
$\Delta x$ change of the parameter, units of the parameter

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

Holds when

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

An elasticity of 3 means: the parameter got worse by 1%, and NPV fell by 3%. The sign shows the direction. For price and production the elasticity is positive, and for costs it is negative.

Worked exampleSensitivity of NPV: one parameter at a time, by 10%

Given (assumed data)

Base case: the library's example project, $\mathrm{NPV}_0 = 100.1$ million dollars at a rate of 10%
Change: each parameter in turn by 10% in the unfavourable direction; first oil one year later

Solution

A price of $45/bbl instead of $50/bbl: NPV = 71.7 million dollars, a change of −28.4 million dollars, or −28.4%. Elasticity: $-28.4/(-10) = 2.84$ .
Production 10% lower in all years: NPV = 74.9 million dollars (−25.2%), elasticity 2.52. It is smaller than for price, because the variable costs fall together with production.
Capital expenditure 10% higher: NPV = 87.5 million dollars (−12.6%), elasticity −1.26.
Operating expenditure 10% higher: NPV = 94.8 million dollars (−5.3%), elasticity −0.53.
First oil one year later, with the dates of the capital expenditure unchanged: NPV = 77.5 million dollars (−22.6%), although the sum of the cash flow over the project has not changed.

Answer: NPV responds most to the price (elasticity 2.84) and to production (2.52). A year of delay costs almost as much as 10% of production.

What it means. All the elasticities but one are greater than one in absolute value (ignoring the sign): NPV changes more than the parameter does. But an equal step of 10% is only a way to compare the slopes. It says nothing about which parameter is more likely to deviate, and by how much. For that we need ranges, which are covered below. The data in this example are assumed for illustration.

Ranges of the parameters, the spider diagram and the tornado chart

A step of ±10% shows the slopes but not the risk, because parameters differ in how far they can move from their base values. During the life of a project the oil price can become several times higher or lower, while the operating costs of a proven technology are known fairly exactly. The range is also often not symmetric. A cost estimate is more likely to be exceeded than to be saved, and first oil can be delayed but can hardly come much earlier than planned. So each parameter gets its own plausible range, taken from data, from similar projects and from expert estimates. In a working model, the list includes everything that is set on the assumptions sheet and known inexactly. Besides price, production and costs, this means the exchange rate, inflation, tax rates and tariffs.

Ranges of the parameters and NPV at their edges, US$ million

Parameter: Oil price
Range: $35–65/bbl
NPV: worst edge: 14.8
NPV: best edge: 185.4
Swing: 170.6

Parameter: Production
Range: −20% … +15%
NPV: worst edge: 49.6
NPV: best edge: 138.0
Swing: 88.5

Parameter: Capital expenditure
Range: −10% … +30%
NPV: worst edge: 62.2
NPV: best edge: 112.8
Swing: 50.5

Parameter: Time of first oil
Range: on time … a year later
NPV: worst edge: 77.5
NPV: best edge: 100.1
Swing: 22.6

Parameter: Operating expenditure
Range: −15% … +20%
NPV: worst edge: 89.6
NPV: best edge: 108.1
Swing: 18.5

Parameter	Range	NPV: worst edge	NPV: best edge	Swing
Oil price	$35–65/bbl	14.8	185.4	170.6
Production	−20% … +15%	49.6	138.0	88.5
Capital expenditure	−10% … +30%	62.2	112.8	50.5
Time of first oil	on time … a year later	77.5	100.1	22.6
Operating expenditure	−15% … +20%	89.6	108.1	18.5

NPV at a rate of 10%. Each time one parameter changes and the others stay at their base values (the base NPV is $100.1 million). The rows are in order of decreasing swing (the difference between the best and the worst NPV). The ranges are assumed and chosen for the example; in a working calculation they have to be justified.

The spider diagram shows these relationships on one chart. The parameters have different units: dollars per barrel, millions of barrels, millions of dollars. So the horizontal axis shows not the values themselves but the change of each from its base value, in per cent. The vertical axis is NPV.

CalculatedSpider diagram of the example project: NPV at a rate of 10% against the change of each of the four parameters. The lines meet at the point of the base case. The length of each line is the accepted range of the parameter.

Here is how to read the diagram. All the lines pass through the point of the base case. The slope of a line is the sensitivity: the steeper it is, the more NPV depends on the parameter. The length of a line is the range of the parameter. A parameter is dangerous if its line is both steep and long, and the point where the line crosses zero is the break-even value. In the example project no line reaches zero. This means that no single parameter, within its range, makes NPV negative. A delay of start-up cannot be put on this diagram, because it has no base value from which to count per cent, so it is shown separately.

The tornado chart keeps from each line of the spider diagram only the two end points. For each parameter we draw a horizontal bar from the NPV at the unfavourable edge of the range to the NPV at the favourable edge. A vertical line marks the base NPV. We place the bars from top to bottom in order of decreasing length. The result is a funnel that narrows downwards and looks like a twister.

CalculatedTornado chart of the example project: the swing of NPV when each parameter changes within its range. The vertical line is the base NPV, $100.1 million.

The tornado chart answers practical questions.

What to refine first. The top rows show which data are worth paying for. If production volume matters most, we need appraisal wells and seismic surveys. If price matters most, we need to look at the sales terms and at hedging (fixing the price in advance with a financial contract). If capital expenditure matters most, we need a deeper study of the project and of the contracts.
What to manage. The company cannot influence the oil price, but it can influence timing and costs to some extent. While the project is carried out, we keep these parameters under control.
What not to argue about. Parameters with short bars can be left at their base values, and we need not spend discussion time on them.

Break-even values and the margin of safety

The question can be turned around. Instead of “what happens to NPV if the price falls by 10%?”, we ask “at what price does NPV become zero?”. The value of a parameter at which NPV is zero, with the other parameters at their base values, is called break-even. For the price it is the break-even price. The distance from the base value to the break-even value is the project's margin of safety against this parameter.

In general, we find the break-even value by trial: we change the parameter and recalculate the model until NPV becomes zero. For the price, we can often write the answer at once, because NPV depends on it almost linearly.

Break-even price when NPV depends linearly on price

P^{*} = P_0 - \frac{\mathrm{NPV}_0}{s_P}, \qquad s_P = (1 - r)(1 - T)\sum_t \frac{Q_t}{(1+i)^{t}}

where:

$P^{*}$ break-even price: the price at which NPV at rate $i$ is zero, US$/bbl
$P_0$ price of the base case, US$/bbl
$\mathrm{NPV}_0$ NPV of the base case at rate $i$ , US$ million
$s_P$ slope: how much NPV changes when the price changes by 1 US$/bbl, US$ million per 1 US$/bbl
$r$ share of revenue that goes to payments on revenue (royalty), fraction
$T$ income tax rate, fraction
$Q_t$ production (sales) of the year that is $t$ years from the time origin, million bbl
$i$ discount rate, fraction per year

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

Holds when

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

Each dollar of price brings the project the discounted production multiplied by the share that remains after royalty and income tax. NPV divided by this slope shows by how many dollars the price can fall.

NPV when first oil is delayed

\mathrm{NPV}_{\text{delay}}(d) = \frac{\mathrm{NPV}_0 + PV_C}{(1+i)^{d}} - PV_C

where:

$\mathrm{NPV}_{\text{delay}}(d)$ NPV when first oil is delayed by $d$ years, US$ million
$\mathrm{NPV}_0$ NPV of the base case at the same rate $i$ , US$ million
$PV_C$ present value of the capital expenditure, $\sum_t C_t/(1+i)^{t}$ , US$ million
$d$ delay of first oil, years
$i$ discount rate, fraction per year

Units: US$ million; $d$ in years

Holds when

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

The sum $\mathrm{NPV}_0 + PV_C$ is the present value of all the flows of the project except the capital expenditure. A delay pushes it back by $d$ years, while the investments stay where they are. If we set NPV to zero, we get the longest delay that the project can stand, $d^{*}$ : $(1+i)^{d^{*}} = (\mathrm{NPV}_0 + PV_C)/PV_C$ .

Worked exampleBreak-even values of the example project

Given (assumed data)

Base case: price $50/bbl, $\mathrm{NPV}_0 = 100.1$ million dollars at a rate of 10%
Discounted production $\sum_t Q_t/1.1^{t}$: 7.899 million bbl
Present value of the capital expenditure $PV_C$: 148.3 million dollars
Royalty and income tax: 10% of revenue and 20% of profit

Solution

Slope against price: $s_P = 0.9\cdot0.8\cdot7.899 = 5.687$ million dollars for each dollar of the price of a barrel.
Break-even price: $P^{*} = 50 - 100.1/5.687 = 32.4$ $/bbl. A trial on the model gives the same: at this price the project still pays income tax in all the years of production. The margin of safety is 35% of the base price.
Production (by trial): NPV becomes zero if production in all years is 0.60 of the base, that is 40% lower.
Costs (by trial): the capital expenditure must rise to 1.79 times its base value, and the operating expenditure to 2.9 times.
A delay of one year: $\mathrm{NPV}_{\text{delay}}(1)$ $= (100.1 + 148.3)/1.1 - 148.3$ $= 77.5$ million dollars. The longest delay: $d^{*} = \ln 1.675/\ln 1.1$ $= 5.4$ years.
If the capital expenditure turns out 30% higher, the break-even price rises to $39.1/bbl, and with a delay of one year, to $35.0/bbl.

Answer: The project withstands a price down to $32.4/bbl, a loss of 40% of production, a rise of the capital expenditure to 1.79 times its base value, or a delay of first oil by 5.4 years, each one taken separately.

What it means. The margin for each parameter looks large as long as the others stand at their base values. But the margins are not added up: they are used up together. A cost overrun of 30% already moves the break-even price from $32.4/bbl to $39.1/bbl. A break-even value is not a forecast but a yardstick: we compare it with what the company considers possible. The data in this example are assumed for illustration.

CalculatedNPV of the example project against the oil price, with the base capital expenditure and with expenditure 30% higher. The break-even price moves from $32.4/bbl to $39.1/bbl.

The words “break-even price” do not define anything by themselves: the answer depends on which measure is set to zero and at which rate. One and the same project has several such prices, and they answer different questions.

Three “break-even prices” of one project

What is set to zero: NPV at a rate of 10%
Price, US$/bbl: 32.4
What it means: the project pays back the investment and earns a return of 10%; usually this number is called the break-even price of the project

What is set to zero: The sum of the cash flow without discounting
Price, US$/bbl: 25.6
What it means: the project pays back the money invested but earns no return on it

What is set to zero: The cash flow of the last year of production
Price, US$/bbl: 13.9
What it means: for a field that is already producing, it pays to keep working rather than to stop; past investments do not count (economic limit)

What is set to zero	Price, US$/bbl	What it means
NPV at a rate of 10%	32.4	the project pays back the investment and earns a return of 10%; usually this number is called the break-even price of the project
The sum of the cash flow without discounting	25.6	the project pays back the money invested but earns no return on it
The cash flow of the last year of production	13.9	for a field that is already producing, it pays to keep working rather than to stop; past investments do not count (economic limit)

At a rate of 15% the first row would give $36.4/bbl. The last row is the cost per barrel in year 9 divided by the share of revenue that is left after royalty. The data in this example are assumed for illustration.

Scenarios: several assumptions at once

In real life, parameters do not wait for their turn. The reserves turn out smaller, the cost estimate larger, the start-up is delayed, and all this happens in one project. A scenario is a set of values of several parameters that change at the same time and consistently. It describes one possible picture of the future, not a mechanical mix of numbers.

Three scenarios of the example project

Measure: Price, US$/bbl
Low: 40
Base: 50
High: 60

Measure: Production
Low: −15%
Base: 12 million bbl
High: +10%

Measure: Capital expenditure
Low: +20%
Base: 160 US$ million
High: −5%

Measure: Operating expenditure
Low: +10%
Base: base
High: −5%

Measure: First oil
Low: one year later
Base: in year 3
High: in year 3

Measure: NPV at 10%, US$ million
Low: −31.0
Base: 100.1
High: 197.1

Measure: IRR, %
Low: 5.1
Base: 29.6
High: 46.2

Measure: Payback period, years
Low: 7.2
Base: 4.4
High: 3.8

Measure: Financing requirement, US$ million
Low: 192
Base: 140
High: 133

Measure	Low	Base	High
Price, US$/bbl	40	50	60
Production	−15%	12 million bbl	+10%
Capital expenditure	+20%	160 US$ million	−5%
Operating expenditure	+10%	base	−5%
First oil	one year later	in year 3	in year 3
NPV at 10%, US$ million	−31.0	100.1	197.1
IRR, %	5.1	29.6	46.2
Payback period, years	7.2	4.4	3.8
Financing requirement, US$ million	192	140	133

The payback period is counted from the start of the investment, without discounting. The financing requirement (the maximum cash outlay) is the lowest point of the cumulative cash flow (Cumulative cash flow and payback period). IRR is the internal rate of return, the yearly return that the project earns (Internal rate of return (IRR)). The scenarios are assumed.

Worked exampleTogether worse than separately

Given (assumed data)

The “low” scenario: price $40/bbl, production −15%, capital expenditure +20%, operating expenditure +10%, first oil one year later
Base NPV at 10%: 100.1 million dollars

Solution

One at a time: only a price of $40/bbl gives NPV 43.3. Only production −15% gives 62.2. Only capital expenditure +20% gives 74.9. Only operating expenditure +10% gives 94.8. Only a delay of one year gives 77.5 million dollars. None of these calculations gives a negative NPV.
All five changes together: NPV = −31.0 million dollars.
If we simply add up the five separate losses, we get 147.9 million dollars and an NPV of −47.8 million. The model gives a smaller loss. The reason is that a low price and low production reduce the same revenue ( $0.8\cdot0.85 = 0.68$ , not 0.65), and the delay pushes back a flow that is already reduced.
Four parameters, each 10% in the unfavourable direction (price, production, capital expenditure and operating expenditure): NPV = 31.7 million dollars, which is 68% lower than the base.
All parameters at the unfavourable edges of the ranges from the tornado table: NPV = −80.0 million dollars. At the favourable edges: 257.4 million dollars.

Answer: The “low” scenario gives an NPV of −31.0 million dollars, although none of its deviations on its own makes the project loss-making.

What it means. In the “low” scenario the price, the production and the costs do not reach the edges of their ranges. The price is 20% below the base while its range goes to 30%, and production is 15% below while its range goes to 20%. Only the delay is at its edge: one year. An analysis that changes one parameter at a time cannot see such an outcome. It is built that way. A calculation at the edges of all the ranges, the “worst case”, is useful as a stress test: can the company survive such an outcome, and will it have enough money? We cannot take it as a likely estimate. For all five parameters to be at their edges at the same time takes a rare coincidence. For the same reason, the product of all the pessimistic values does not give a cautious estimate of the hydrocarbons in place (uncertainty of the volumetric estimate). The data in this example are assumed for illustration.

It helps to try scenarios and sensitivity on a live model. The investors section of this website has an economic model of a project. You can change the assumptions in it and see at once how the results respond, including the results for price scenarios.

All parameters at once: simulation

Sensitivity changes one parameter and a scenario changes several, but both leave the main question unanswered: how likely is a given result? The answer comes from simulation. We recalculate the project model thousands of times. Each time we use a new set of values of all the uncertain parameters, chosen at random but in agreement with what is known about them. The result is not one value of NPV but its distribution: a picture of how likely each value is.

The order of the calculation is the same as in the probabilistic estimate of hydrocarbons in place, except that the cash-flow model takes the place of the volumetric formula (how the calculation works). Parameters that are known accurately enough stay constant. Each uncertain parameter gets a distribution. For each recalculation, which is called a trial, we choose a set of values at random, calculate the model and record the NPV. The same lesson explains how a random number becomes a parameter value and how to choose the type of distribution. It also shows how plain random drawing differs from layered drawing (Latin hypercube sampling, LHS), and how many trials are enough.

In project economics, two features are added to this.

Asymmetric distributions. A cost estimate is more often exceeded than saved. Deadlines are more often missed than beaten. Production more often falls short of the plan than exceeds it. We describe three estimates of a parameter, the smallest, the most likely and the largest, by a triangular distribution, and for costs it is skewed, with a long tail towards the larger values. The most likely value is then smaller than the mean.
Links between parameters. A cost overrun usually comes together with a delay of start-up. Contractor services follow the oil price up and down. Linked parameters cannot be drawn independently. We choose one at random and calculate the second from it, or we set a correlation between them (dependence between parameters). Otherwise combinations that never occur appear in the calculation, and exactly the dangerous combinations become too rare.

Worked exampleThe distribution of NPV of the example project

Given (assumed data)

Oil price: triangular distribution: 35, 50, 65 US$/bbl (the smallest, the most likely and the largest value)
Production, as a fraction of the base profile: triangular: 0.80, 1.00, 1.15
Capital expenditure, as a fraction of the base: triangular: 0.90, 1.00, 1.30
Operating expenditure, as a fraction of the base: triangular: 0.85, 1.00, 1.20
First oil: one year later if the capital expenditure is exceeded by more than 10%
Number of trials: 10,000

Solution

The first trial. The random numbers 0.5701, 0.5202, 0.6998 and 0.6989 give a price of $51.1/bbl, a production fraction of 0.991, a capital expenditure fraction of 1.110 and an operating expenditure fraction of 1.055. The overrun is more than 10%, so start-up is delayed by a year. The model gives NPV = 64.3 million dollars.
After 10,000 trials: the mean NPV is 79.4 million dollars, and the standard deviation (a measure of how widely the results are spread) is 43.2 million dollars.
NPV is negative in 2.7% of the trials.
With a probability of 90%, NPV is not less than 24 million dollars. With a probability of 50%, it is not less than 78. With a probability of 10%, it is not less than 137 million dollars.
NPV is below the base 100.1 million dollars in 69% of the trials. Start-up is delayed in 33% of the trials.
Trials in which NPV is worse than in the “low” scenario (−31.0 million dollars): 17 out of 10,000.

Answer: The mean NPV is about 79.4 million dollars, against 100.1 in the base case. The probability of a negative NPV is about 2.7%.

What it means. The base case turned out to be optimistic: in two trials out of three the result is worse than it. The reason is the asymmetric ranges. A cost estimate is more often exceeded than saved, production more often falls short of the plan than exceeds it, and a delay can happen only in one direction. A set of the most likely values gives neither the most likely result nor the average one. The second conclusion is about the “low” scenario. When the price, the production and the costs are drawn independently of each other, as here, such a coincidence of failures happens less often than once in 500 trials. If failures in reality come together, the “tail” of the distribution (the rare, very bad results) is heavier, and we must put these links into the model explicitly. The result of a simulation is no more accurate than the distributions put into it. The data in this example are assumed for illustration.

CalculatedResult of the simulation: the probability that the NPV of the project is not less than a given value (10,000 trials). The zero NPV, the mean and the base case are marked.

The distribution of NPV is the basis for the next step: weighing the outcomes by their probabilities and comparing decisions by expected value (Expected monetary value and preference theory, Decision trees and the value of information).

Terms and symbols

Base case: A run of the model with the best estimates of all the parameters. It is the point of reference of the analysis, but not necessarily the average result.
Sensitivity: The change of a project measure when one parameter changes and the others stay the same.
Elasticity $E_x$: The relative change of NPV divided by the relative change of the parameter.
Spider diagram: A chart of NPV against the relative change of each parameter. The lines meet at the point of the base case.
Tornado chart: A chart of bars that show the swing of NPV for each parameter, placed in order of decreasing length.
Break-even value: The value of a parameter at which the chosen measure (usually NPV at a given rate) is zero, with the other parameters at their base values.
Scenario: A consistent set of values of several parameters that describes one possible picture of the future.
Payoff matrix: A table of money results for each combination of an action and a state of nature. Regret is the difference between the best outcome in a given state and the outcome of the chosen action.
Simulation: Recalculating the model many times with randomly chosen values of all the uncertain parameters. It gives a distribution of the result.

Common mistakes and misconceptions

Wrong

“No single parameter makes NPV negative, so the project is reliable.”

In fact

The parameters change together. In this lesson's example, four parameters, each made worse by only 10%, reduce NPV by 68%. And the “low” scenario, in which the price, the production and the costs do not reach the edges of their ranges, gives a negative NPV.

Wrong

“Let us test all the parameters by ±10%, and the picture of risk is ready.”

In fact

An equal step shows slopes, not risk. The price can move by tens of per cent, and the operating costs by only a few per cent. For the cost estimate and the timing the range is asymmetric. Each parameter needs its own justified range.

Wrong

“The longest bar of the tornado chart is the most likely risk, and the worst case is a cautious estimate.”

In fact

Neither the chart nor the scenario contains probabilities. A bar shows the swing for the assigned range, and all parameters reaching the edges of their ranges at the same time is a rare coincidence.

Wrong

“The base case is what we will get on average.”

In fact

The base case is made of the most likely values, and the ranges are asymmetric. In this lesson's example the mean NPV from the simulation is $79 million, against $100 million in the base case. In 69% of the trials the result is worse than the base case.

Wrong

“The break-even price is $32/bbl, so at $35 the project pays.”

In fact

Only if everything else goes as in the base case. With a cost overrun of 30%, the break-even price of the same project is already $39/bbl. And the number itself belongs to a certain measure and rate: without them it cannot be compared with other numbers.

In depth: advanced detail

Where the slopes come from: discounted sums

In a model with constant rates of royalty and income tax, each sensitivity is a discounted sum multiplied by the share that remains with the project after the payments to the state. This lets us check the model “on paper” and understand what each slope depends on.

Change of NPV for small changes of price and costs

\Delta\mathrm{NPV} = (1 - T)\left[(1 - r)\,PV_Q\,\Delta P - \Delta PV_O\right] - \Delta PV_C + T\,\Delta PV_D

where:

$\Delta\mathrm{NPV}$ change of NPV; an increase has a “plus” sign, US$ million
$PV_Q$ discounted production, $\sum_t Q_t/(1+i)^{t}$ , million bbl
$\Delta P$ change of price, US$/bbl
$\Delta PV_O$ change of the present value of the operating expenditure, US$ million
$\Delta PV_C$ change of the present value of the capital expenditure, US$ million
$\Delta PV_D$ change of the present value of the tax depreciation of this capital expenditure, US$ million
$r,\ T$ share of royalty in revenue and the income tax rate, fractions

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

Holds when

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

A dollar of revenue brings the project $(1 - r)(1 - T)$ dollars, and a dollar of operating expenditure costs it $(1 - T)$ dollars. A dollar of capital expenditure costs more. Tax returns its share not at once but through depreciation, and in present-value terms this return is worth less.

Worked exampleChecking the sensitivities without recalculating the model

Given (assumed data)

Present values at 10%: production 7.899 million bbl; capital expenditure 148.3; operating expenditure 66.0; depreciation 110.3 million dollars
Price, royalty, tax: $50/bbl; 10% of revenue; 20% of profit
Variable costs: $5/bbl

Solution

Price −10% (5 dollars): $0.8\cdot0.9\cdot7.899\cdot5 = 28.4$ million dollars.
Production −10%: from a barrel, after royalty, variable costs and tax, $(50\cdot0.9 - 5)\cdot0.8 = 32$ dollars remain. The loss is $0.1\cdot32\cdot7.899 = 25.3$ million dollars.
Operating expenditure +10%: $0.8\cdot0.1\cdot66.0 = 5.3$ million dollars.
Capital expenditure +10%: $0.1\cdot148.3 = 14.8$ million dollars. The tax returns $0.2\cdot0.1\cdot110.3 = 2.2$ million dollars. The total is 12.6 million dollars. That is 0.85 dollars of lost NPV for each present-value dollar of cost, against 0.80 for operating expenditure.
The recalculation of the model gives the same losses: 28.4, 25.3, 5.3 and 12.6 million dollars.

Answer: The losses of NPV are 28.4, 25.3, 5.3 and 12.6 million dollars. The formulas and the model agree.

What it means. The agreement is a check of the model itself: if the recalculation had given a different number, there would be an error or a hidden link in some row. The reason for the ranking is also visible. The length of a tornado bar follows the discounted sum that the parameter touches (revenue 394.9 million dollars, capital expenditure 148.3, operating expenditure 66.0) and the share of each change that the state bears. With a different tax regime, the order of the parameters can be different. The data in this example are assumed for illustration.

Two parameters at once: price and capital expenditure

Between the analysis of one parameter at a time and a full simulation there is an intermediate method: a table for two parameters, usually the two top rows of the tornado chart. It shows how the break-even value of one parameter depends on the other.

NPV at 10%, US$ million: oil price and capital expenditure

Capital expenditure: −10%
$35/bbl: 27.5
$40/bbl: 55.9
$50/bbl: 112.8
Break-even price: $30.2/bbl

Capital expenditure: base
$35/bbl: 14.8
$40/bbl: 43.3
$50/bbl: 100.1
Break-even price: $32.4/bbl

Capital expenditure: +15%
$35/bbl: −4.1
$40/bbl: 24.3
$50/bbl: 81.2
Break-even price: $35.7/bbl

Capital expenditure: +30%
$35/bbl: −23.2
$40/bbl: 5.4
$50/bbl: 62.2
Break-even price: $39.1/bbl

Capital expenditure	$35/bbl	$40/bbl	$50/bbl	Break-even price
−10%	27.5	55.9	112.8	$30.2/bbl
base	14.8	43.3	100.1	$32.4/bbl
+15%	−4.1	24.3	81.2	$35.7/bbl
+30%	−23.2	5.4	62.2	$39.1/bbl

The other parameters are at their base values. The data in this example are assumed for illustration.

The boundary between positive and negative NPV runs diagonally across the table: every 10% of cost overrun raises the break-even price by about $2.2/bbl. For three or more parameters a table no longer works, and scenarios and simulation take over.

Connections to other topics

Net present value (NPV) — the measure whose sensitivity is studied, and the decision rule based on it.
Cash flow model of an oil and gas project — the model that is recalculated at every change of the assumptions, and its rows.
Cumulative cash flow and payback period — the payback period and the financing requirement, measures that are also compared across scenarios.
Choosing the discount rate — the premium on the rate for risk, its effect on distant flows, and the sensitivity of NPV to the rate.
Internal rate of return (IRR) — the project's margin against the discount rate.
Probabilistic volumetrics by Monte Carlo simulation — how a random draw works, how many trials are needed, and what P90, P50 and P10 mean.
Expected monetary value and preference theory — how to weigh outcomes by their probabilities and compare decisions by expected value.
Decision trees and the value of information — sequential decisions and the value of extra information.

Check yourself

1
The NPV of the base case is 80 million dollars. With a price 10% lower it is 56 million dollars. What is the elasticity of NPV with respect to price?
1. A.24
2. B.3.0
3. C.0.3
4. D.2.4
Show the answer
Correct answer: B. 3.0
NPV changed by $(56 - 80)/80 = -30$ %, and the price by −10%. Elasticity: $-30/(-10) = 3.0$ . A change of the price by 1% changes NPV by 3%. The number 24 is the change in millions of dollars, not an elasticity.
2
US$40 million has already been spent on the appraisal of a discovery. The NPV of development, from the future flows, is +US$25 million. Should the field be developed, and why?
Show the answer
It should. A decision is judged by the flows that depend on it. The US$40 million that was spent cannot be recovered either by developing the field or by refusing, and development adds US$25 million of value to the company. To subtract the past costs and get “−15” means turning down a profitable decision because of money that it does not affect. The exception is when the past costs still reduce a future tax: this saving is part of the future flows.
3
The NPV of a project depends linearly on the price: at $50/bbl it is 60 million dollars, and every dollar of price changes it by 4 million dollars. What is the break-even price?
1. A.$65/bbl
2. B.$46/bbl
3. C.$35/bbl
4. D.$15/bbl
Show the answer
Correct answer: C. $35/bbl
$P^{*} = 50 - 60/4 = 35$ $/bbl. The answer is right as long as the dependence stays linear (as long as the project pays income tax in all the years of production) and the other parameters are at their base values.
4
On a tornado chart the top bar is the oil price, the second is the production volume, and the bottom one is the operating expenditure. What follows from this for the preparation of the project, and what does the chart not show?
Show the answer
This gives an order of importance: the price affects the result most, then production, and the operating expenditure can be left at its base value. The company cannot influence the price, so it checks which price the project can stand and thinks about hedging. The production volume can be refined with appraisal wells. The chart does not show the probabilities of the deviations, because the author of the calculation sets the ranges. It also does not show what happens when several parameters change at the same time.
5
A well can be completed (prepared for production) in two ways. The expensive completion gives an NPV of +30 million dollars if the reservoir turns out highly productive and −10 million dollars if it turns out poorly productive. The simple completion gives +12 and +4 million dollars. What will the pessimist's rule and minimax regret choose?
1. A.Both rules choose the simple completion
2. B.The pessimist's rule chooses the expensive completion, minimax regret chooses the simple one
3. C.The pessimist's rule chooses the simple completion, minimax regret chooses the expensive one
4. D.Both rules choose the expensive completion
Show the answer
Correct answer: C. The pessimist's rule chooses the simple completion, minimax regret chooses the expensive one
The worst outcomes are −10 for the expensive completion and +4 for the simple one, so the pessimist's rule chooses the simple one. Now the regrets. If the reservoir is highly productive, the best outcome is 30, and the simple completion falls short by $30 - 12 = 18$ . If it is poorly productive, the best outcome is 4, and the expensive one falls short by $4 - (-10) = 14$ . The largest regret is 14 for the expensive completion and 18 for the simple one, so minimax regret chooses the expensive one. Neither rule asked how likely a highly productive reservoir is. This is their main weakness.
6
A calculation with the most likely values of all the parameters gave an NPV of 100 million dollars, and simulation gave a mean NPV of 79 million dollars. Is there an error here?
Show the answer
Most likely, there is not. The base case is made of the most likely values, but the ranges of the parameters are asymmetric. Costs more often exceed the estimate than fall below it, production more often falls short of the plan, and delays happen in one direction only. So the mean over all the possible outcomes is lower than the result of the base case. To compare decisions by expected value, we must use the mean and not the base case.

Updated: 19 October 2026