Risk reduction and diversification

Quick refresher

For a company, risk is not only the probability that a project fails but also the size of the possible loss compared with its money. The same well is a small stake for a large company and an all-in stake for a small one.
A company that repeats equal bets and can afford $m$ of them is ruined with a probability $\psi_1^{\,m}$ , where $\psi_1 = q + p\,\psi_1^{\,g+1}$ . If EMV ≤ 0, ruin is certain. If EMV > 0, its probability falls with the number of stakes that the capital can bear. By halving the stake, when the capital is a multiple of the stake, the company squares the probability of ruin (multiplies it by itself).
The EMVs of projects always add up, but the spreads do not: for $n$ equal independent shares, the standard deviation relative to the mean is $\sqrt n$ times smaller.
Prospects of one play are linked by a common risk, and the probability that all of them fail does not fall below $1 - P_\text{play}$ , however many traps are drilled. We diversify between plays and basins, not only between traps.
A farm-out does not create EMV but divides it together with the risk. Both sides gain when the risk goes to the one for whom it is smaller. A promote is the share of costs that the partner pays above its working interest. A carry is the payment by the partner of all the owner's costs at the work stage.
Insurance, a fixed price from a contractor, project finance and hedging are a transfer of risk for a fee. On average they cost more than carrying the risk oneself, and they are justified when the possible loss is large for the company.

Key relationships

\psi_m = \psi_1^{\,m}, \qquad \psi_1 = q + p\,\psi_1^{\,g+1}

the probability of ruin with capital for $m$ equal stakes; $g = V/C - 1$

\sigma_\Sigma^2 = \sum_i w_i^2\sigma_i^2 + 2\sum_{i<j} w_i w_j\,\rho_{ij}\,\sigma_i\sigma_j

the spread of a portfolio with working interests $w_i$

\rho = \dfrac{P_\text{loc}\,(1 - P_\text{play})}{1 - P_\text{play}\,P_\text{loc}}

the correlation of the outcomes of two prospects of one play

\mathrm{EMV}_\text{partner} = w\,P_g V - s\,C

the partner gets the share $w$ and pays the share $s$ of the cost of the well

Common pitfalls

Believing that a positive EMV protects against ruin.
Calling a portfolio diversified when all its prospects are in one play.
Comparing farm-out deals only by EMV, with no account of the spread of the outcomes and the company's money.
Insuring and hedging “just in case”: protection costs money, and it is justified when the loss is large for the company.
Adding up the standard deviations or the P90 of projects as ordinary numbers.

After this lesson you can

explain how the risk of a company differs from the risk of a project, and name four ways to reduce it: information, transfer, sharing and diversification
calculate the probability of ruin of a company that repeats equal bets, and explain why a positive EMV does not protect against ruin
estimate the EMV, the spread and the probability of complete failure of a portfolio of independent prospects and of prospects of one play (a group of prospects in the same geological setting), and explain why the common risk of a play cannot be removed by diversification
calculate the EMV and the certainty equivalent of both sides of a farm-out deal with a promote or a carry, and choose working interests in several blocks
explain insurance, a fixed-price contract with a contractor, project finance and hedging of the price as a transfer of risk for a fee, and estimate who gains from it

The risk of a company, and four ways to reduce it

For a geologist, the risk of exploration is the probability that the trap is empty, and uncertainty is the spread of the volume if there is an accumulation (risk and uncertainty of a prospect). For an economist, any inexactness of knowledge becomes a risk when money depends on it (uncertainty and risk of a project). For a company a third question is added: what will happen to the company itself in the worst outcome? The same prospect is one of hundreds of bets for a large company, and for a small one it is a bet whose loss leaves it with no money for the next step.

The risks of an exploration company and what reduces them

Source: Geological
Example: a dry well, a small discovery
What reduces it: data before drilling, partners, a portfolio of independent prospects

Source: Costs and timing
Example: drilling problems, growth of the prices of services
What reduces it: an estimate from similar wells (analogues) and a contingency reserve (Drilling time and cost), a fixed price from the contractor

Source: Accidents
Example: a blowout, a spill, a fire
What reduces it: prevention, insurance

Source: Market
Example: a fall of the oil price, the exchange rate
What reduces it: hedging, low costs, different links of the value chain

Source: Regulatory and political
Example: a change of taxes and of the terms of work
What reduces it: terms of the contract, different countries (Market, financial and regulatory uncertainty)

Source: Financial
Example: no money for the compulsory work programme
What reduces it: working interests, partners, project finance

Source	Example	What reduces it
Geological	a dry well, a small discovery	data before drilling, partners, a portfolio of independent prospects
Costs and timing	drilling problems, growth of the prices of services	an estimate from similar wells (analogues) and a contingency reserve (Drilling time and cost), a fixed price from the contractor
Accidents	a blowout, a spill, a fire	prevention, insurance
Market	a fall of the oil price, the exchange rate	hedging, low costs, different links of the value chain
Regulatory and political	a change of taxes and of the terms of work	terms of the contract, different countries (Market, financial and regulatory uncertainty)
Financial	no money for the compulsory work programme	working interests, partners, project finance

An incomplete list: each company makes it for its own portfolio.

Information: We buy, before the decision, data that make the probabilities and ranges more exact: a seismic survey, an appraisal well, studies, an analysis of analogues. Information does not change the subsurface, but it lets us avoid bad bets. Whether it is worth its money is calculated by the value of information (Decision trees and the value of information).
Transfer: We put the consequences, for a fee, on someone who is ready to carry them: an insurer, a fixed-price contractor, a bank in project finance, a market participant in price hedging. The average result of the company usually gets worse: the one who transfers pays a premium.
Sharing: We take a share in a project, not the whole project: joint operation, a farm-out to a partner. Both the costs and the income are shared.
Diversification: We spread money over many projects whose outcomes are not linked. The average does not change, and the total becomes more predictable.

One more tool does not reduce risk but prepares for it: scenario planning (the section at the end of the lesson). In numbers, the attitude of a company to risk is described by the utility function and the certainty equivalent (attitude to risk). Below they serve as a yardstick, not as the subject of the lesson.

A positive EMV does not save from ruin

The EMV criterion assumes that there are many bets and that the average will have time to show itself (why companies do not simply maximise EMV). But the money of a small company runs out sooner. Imagine a company that, time after time, takes the same share in similar exploration prospects, and turns each discovery into money at once, by selling it or by raising finance against it. A series of dry wells can eat the capital before the first discovery, even if each bet pays on average. This is gambler's ruin.

It helps to measure capital in stakes: the cost to the company of one well. A dry well reduces the capital by one stake. A discovery increases it by $g = V/C - 1$ stakes: the company gets its share of the value of the discovery and has already paid its share of the well. The company is ruined when it does not have enough money for the next stake. The main property is that, downwards, the capital moves only by one stake at a time. To lose $m$ stakes, we need to “step down” $m$ times, and each step down does not depend on the previous one.

The probability of ruin with equal stakes

\psi_m = \psi_1^{\,m}, \qquad \psi_1 = q + p\,\psi_1^{\,g+1}, \qquad g = \frac{V}{C} - 1

where:

$\psi_m$ the probability of ever being ruined if the capital is enough for $m$ stakes, fraction
$\psi_1$ the probability of ruin with capital for one stake; it is also the probability of ever losing one stake from the present level; the smallest root of the equation on the interval from 0 to 1, fraction
$m$ how many stakes the capital can bear: the integer part of the ratio of the capital to the stake, —
$p,\ q$ the chance of success $P_g$ of a well, and the chance of failure $1 - P_g$ , fraction
$g$ the gain of a discovery in stakes, —
$V,\ C$ the value of a discovery at the date of the decision, and the cost of a well at a share of 100% (the ratio does not depend on the share), US$ million

Units: dimensionless

Holds when

all the stakes are equal: the same share in prospects of one type with the same $p$ and $V/C$ ; the outcomes of the wells are independent
$g$ is a whole number of stakes, as in all the examples of the lesson; for a fractional $g$ the formula is approximate
a discovery is turned into money at once at the value $V$ ; in reality development needs time and capital, and the probability of ruin is higher than calculated
the company goes on betting while it has money, with no limit on the number of wells; for a programme of a finite number of wells the probability of ruin is smaller
the equation for $\psi_1$ always has the root 1; when EMV > 0, that is $p(g + 1) > 1$ , there is also a smaller root, and that is the answer; when EMV ≤ 0 there is no smaller root, and ruin is certain

The first well is either dry (probability $q$ ), and then the stake is lost at once, or successful ( $p$ ), and then the capital has grown by $g$ stakes, and to lose one stake from the earlier level we must step down $g + 1$ steps. This gives the equation for $\psi_1$ . We solve it by iteration (repeated substitution): we start with $\psi_1 = q$ and put it into the right-hand side until the number stops changing. The probability of ruin depends not on money as such but on the number of stakes that the capital can bear. By halving the stake, the company doubles $m$ (or more, if the capital does not divide into the stake exactly; in the example below $m$ goes from 1 to 3), and $\psi_m$ falls to the square of its earlier value or lower.

Worked exampleA capital of 30 million dollars and prospects of type D

Given (assumed data)

Capital for exploration: 30 million dollars
Prospects: like block D below: $P_g = 0.12$ , a well of 20 million dollars, a discovery of 300 million dollars at a share of 100%; the EMV of the whole well is 16 million dollars
Options: to drill the wells whole, or to take 50, 25, 12.5 or 5% in each such prospect

Solution

The gain of a discovery in stakes: $g = 300/20 - 1 = 14$ . The equation: $\psi_1 = 0.88 + 0.12\,\psi_1^{15}$ .
Iterations from $\psi_1 = 0.88$ : 0.89764, 0.90375, 0.90630 … and so on up to $\psi_1 = 0.9084$ .
A whole well: the capital bears one stake. After a dry well, 10 million dollars are left, which is less than a stake. The probability of ruin is 0.9084.
A share of 50% (a stake of 10 million dollars): $m = 3$ , $\psi_3 = 0.9084^3 = 0.7496$ . A share of 25%: $m = 6$ , $\psi_6 = 0.5619$ . A share of 12.5%: $m = 12$ , $\psi_{12} = 0.3157$ . A share of 5%: $m = 30$ , $\psi_{30} = 0.0560$ .
For comparison, the probability that the first $m$ wells are all dry is $0.88^m$ : 0.88, 0.6815, 0.4644, 0.2157, 0.0216.
The EMV of the first $m$ stakes: for a whole well it is 16 million dollars. For any share from 50 to 5% it is $m$ shares in wells, each with an EMV of $0.12\cdot300 - 20 = 16$ million dollars, that is $3\cdot0.5\cdot16 = 30\cdot0.05\cdot16 = 24$ million dollars.

Answer: With a stake of a whole well, the company is ruined with a probability of 90.8%, and with shares of 5% with a probability of 5.6%, although the EMV per dollar invested is the same.

What it means. Bets with the same return lead, on average, to ruin almost surely or almost never. Everything depends on how many stakes the capital can bear. The probability of ruin is higher than the probability that all the first wells fail: after a discovery the company goes on betting and can lose later. Prospects with a small chance of success and a large prize are a game for those whose capital is measured in tens of such wells. A small company does well to take small shares in them. How much capital we need to drill such wells whole is shown in the advanced section. The data in this example are assumed for illustration.

CalculatedThe probability of ruin against the number of stakes that the capital can bear, for the three types of prospects of the examples of the lesson: A (

P_g = 0.4

V/C = 5

), B (

P_g = 0.3

V/C = 7

) and D (

P_g = 0.12

V/C = 15

). Their EMV per stake is close (1.0, 1.1 and 0.8 stakes), but the risk of ruin is different. The points are the options of the example with a capital of 30 million dollars.

Diversification: independent prospects and the common risk of a play

Instead of one whole well, a company can take small shares in many. The average result does not change by this: EMVs add up with any link between the projects. What changes is the spread, and how much it falls depends on the link between the outcomes.

The average result and the spread of a portfolio

\mathrm{EMV}_\Sigma = \sum_i w_i\,\mathrm{EMV}_i, \qquad \sigma_\Sigma^2 = \sum_i w_i^2\sigma_i^2 + 2\sum_{i<j} w_i w_j\,\rho_{ij}\,\sigma_i\sigma_j

where:

$w_i$ working interest in the $i$ -th project, fraction
$\mathrm{EMV}_i,\ \sigma_i$ the EMV and the standard deviation of the NPV of the project at a share of 100%; for an exploration well with two outcomes $\sigma_i = V_i\sqrt{P_g(1 - P_g)}$ , US$ million
$\rho_{ij}$ the correlation coefficient of the outcomes of projects $i$ and $j$ ; for independent ones it is zero, —
$\mathrm{EMV}_\Sigma,\ \sigma_\Sigma$ the EMV and the standard deviation of the portfolio, US$ million

Units: US$ million at the date of the decision

Holds when

the NPVs of all the projects are at one date and at one rate
the standard deviation is a convenient measure of the spread, but the outcomes of exploration wells are strongly skewed (lopsided), so next to it we look at the probability of complete failure and at the certainty equivalent

For $n$ equal independent shares of $1/n$ , the EMV is the same as for one whole project, and the standard deviation is $\sqrt n$ times smaller. If all the pairs are linked by the same correlation $\rho$ , then as $n$ grows, the ratio of the spread to the mean does not fall below $\sqrt\rho$ of its value for one project: the linked part of the risk cannot be removed by diversification.

Where does the link between exploration prospects come from? The prospects of one play segment have a common multiplier, the probability that the play works, $P_\text{play}$ : if it does not work, all the traps will be dry (dependent and independent prospects). Hence the correlation of their outcomes.

The correlation of the outcomes of two prospects of one play

\rho = \frac{P_\text{loc}\,(1 - P_\text{play})}{1 - P_\text{play}\,P_\text{loc}}

where:

$P_\text{play}$ the probability that the play (the segment) works, fraction
$P_\text{loc}$ the local chance of success of each of the two prospects, the same for both, fraction
$\rho$ the correlation of the outcomes: “success” is taken as one, “failure” as zero, —

Units: dimensionless

Holds when

the prospects are linked only by the common risk of the play, and their local elements are independent
the local probabilities are the same; the derivation and the general case are in the advanced section

The less reliable the play and the more reliable the traps themselves, the higher the correlation: then the play almost entirely decides the outcome. In a proven play ( $P_\text{play} = 1$ ) the prospects are independent.

Worked exampleThe same 10 million dollars: one well, ten independent ones, or ten of one play

Given (assumed data)

Prospects: like block B below: a well of 10 million dollars, a discovery of 70 million dollars at a share of 100%, $P_g = 0.30$ ; in one segment $P_\text{play} = 0.5$ and $P_\text{loc} = 0.6$
Options: one prospect whole; 10% in each of ten prospects of different plays; 10% in each of ten prospects of one segment
Company: small, risk tolerance $R = 30$ million dollars, as in the lesson on expected monetary value

Solution

The EMV is the same in all three options: $0.30\cdot70 - 10 = 11$ million dollars for a whole well and $10\cdot0.1\cdot11 = 11$ million dollars for ten shares.
The standard deviation of one well: $70\sqrt{0.3\cdot0.7} = 32.1$ million dollars. Of ten independent shares: $0.1\cdot70\sqrt{10\cdot0.21} = 10.1$ million dollars, which is $\sqrt{10}$ times smaller.
The correlation of the prospects of one segment: $\rho = 0.6\cdot0.5/(1 - 0.3) = 0.429$ . Ten shares of one segment: $0.1\cdot70\sqrt{10\cdot0.21\cdot(1 + 9\cdot0.429)}$ $= 22.4$ million dollars.
The probability that there are no discoveries at all: 0.7 for one well; $0.7^{10} = 0.0282$ for ten independent ones; $0.5 + 0.5\cdot0.4^{10} = 0.5001$ for ten of one segment.
The certainty equivalent at $R = 30$ million dollars: one well, −0.52 million dollars; ten independent shares, 9.34; ten shares of one segment, 3.78 million dollars.

Answer: With the same EMV of 11 million dollars, ten independent shares cut the spread to a third and almost rule out a complete failure (2.8% against 70%). Ten shares of one segment leave it with a probability of 50%.

What it means. For a small company a whole well does not pay (the certainty equivalent is −0.52 million dollars). The same money spread over ten independent prospects is worth 9.34 million dollars to it, which is most of the EMV. Ten prospects of one segment are in essence one large bet on the play: if it does not work, and this happens with a probability of 0.5, all of them will be dry. So we diversify between plays, basins and countries, not only between the traps of one field. The data in this example are assumed for illustration.

CalculatedWhat the number of prospects with

P_g = 0.3

gives: at the top, the probability that there will be no discoveries at all, and at the bottom, the spread of the number of discoveries relative to its mean. The solid lines are prospects of different plays, and the dashed lines are prospects of one segment (

P_\text{play} = 0.5

P_\text{loc} = 0.6

, correlation 0.43). For the prospects of one segment, both quantities run into a limit.

Partnership: working interest, farm-out, promote and carry

Most exploration projects are run by several companies. Each owns a working interest: a share in the rights and duties under the contract. In the same share it pays the costs and receives the production. One of the participants is the operator. It carries out the work on behalf of all under a joint operating agreement (JOA), and the participants approve the budgets of the wells by the rules of this agreement (the well cost estimate).

Interests change hands in deals. In a farm-out, the owner of an area gives part of its interest to a partner, and the partner pays for part of the work in return, usually more than the share received would require. For the partner the same deal is a farm-in (an entry into the project). The excess of the partner's share of costs over its working interest is called the promote: it is the price of entry into the project. If the partner pays all of the owner's costs at the stage of work (for example the whole exploration well), the owner's interest is carried by the partner (a carried interest, or “carry”): the owner takes part in the result and pays nothing at this stage. Under some deals, the costs paid for the owner are later repaid to the partner out of the owner's share of production.

Such deals are especially needed when the contract obliges the company to carry out a work programme on time. Take the exploration areas of the electronic auction of the Ministry of Energy of Kazakhstan on 25 December 2026. The minimum exploration programme includes a 2D seismic survey within three years. If the terms of the area require it, the programme also includes drilling one or two wells, at least one within four years (details are in the section of the site on the auction). It is often easier for a small company that has won an area to drill the well with a partner. But in Kazakhstan, a share in the right under a contract for exploration and production of hydrocarbons cannot be transferred in the first three years of the contract. The same applies to shares in a company for which this contract is the only one. Later, such a transfer is possible only with the permission of the competent authority (transfer of the right and of the share).

A farm-out: the EMV of the sides

\mathrm{EMV}_\text{owner} = (1 - w)\,P_g V - (1 - s)\,C, \qquad \mathrm{EMV}_\text{partner} = w\,P_g V - s\,C

where:

$w$ the working interest that the partner gets, fraction
$s$ the share of the cost of the well that the partner pays; $s - w$ is the promote, and at $s = 1$ the owner's share is carried by the partner (a carry), fraction
$P_g,\ V,\ C$ the chance of success, the value of the discovery at the date of the decision, and the cost of the well at a share of 100%, —, US$ million
$\mathrm{EMV}_\text{owner},\ \mathrm{EMV}_\text{partner}$ the EMV of the owner and of the partner after the deal, US$ million

Units: US$ million at the date of the decision

Holds when

after a discovery, the sides carry the development costs in proportion to their working interests, and these costs are already counted in $V$
both sides assess $P_g$ and $V$ in the same way
the repayment of the paid costs out of production is not counted

The sum of the EMVs of the sides is always equal to the EMV of the project, $P_g V - C$ : the deal only divides it. A risk-neutral partner will accept the terms while its EMV is not negative, that is $s \le w P_g V/C$ . The better the prospect, the larger the promote that is paid for it.

Worked exampleBlock D: to keep it, to give half with a promote, or to stay with a carried share

Given (assumed data)

Block D: 100% owned by a small company; the contract requires a well of 20 million dollars to be drilled; $P_g = 0.12$ , the discovery is worth 300 million dollars at a share of 100%
Company: capital for exploration 30 million dollars, risk tolerance $R = 30$ million dollars
Partner: a large company, $R = 1000$ million dollars
Terms: Terms 1: the partner gets 50% of the area and pays 70% of the well. Terms 2: it gets 85% and pays for the whole well

Solution

Keep it: EMV $= 0.12\cdot300 - 20 = 16$ million dollars; the certainty equivalent is −16.2 million dollars. Two thirds of the capital are staked on an outcome that takes them away with a probability of 0.88.
Terms 1 (a promote of $0.70 - 0.50 = 0.20$ ): the company gets $0.5\cdot0.12\cdot300 - 0.3\cdot20 = 12$ million dollars, CE −2.19. The partner gets $18 - 14 = 4$ million dollars, CE 2.86.
Terms 2 (a carry): the company pays nothing and gets 15% of the discovery, so its EMV is $0.12\cdot0.15\cdot300 = 5.4$ million dollars, CE 2.94. The partner gets $0.85\cdot36 - 20 = 10.6$ million dollars, CE 7.38.
The bargaining range for a risk-neutral partner: for half of the area it would pay for up to $0.5\cdot36/20 = 0.90$ of the well. To pay for the whole well, it needs only $20/36 = 0.556$ of the area.

Answer: By EMV, it is best to keep the area (16 million dollars), but for a small company this is a certainty equivalent of −16.2 million dollars. Terms 1 still give it a negative CE (−2.19), and terms 2 give a positive one (2.94). The partner's CE is positive too (7.38).

What it means. The sum of the EMVs of the sides is 16 million dollars in every option: the deal neither creates nor destroys average value. It moves the risk from a company for which 20 million dollars is two thirds of its capital to a company for which this is a small stake. So both sides gain by the certainty equivalent. A carried share of 15% is worth more to the small company than a share bought with its own money: even with the best share of about 6.5% and payment of its own part of the well, its CE would be only 0.47 million dollars (what share to take). Inside the bargaining range, from 55.6% of the area for the whole well to 90% of the well for half of the area, the terms are decided by negotiation, the assessments of the chances and the demand for the area. The data in this example are assumed for illustration.

A portfolio of areas: what share to take in each

Let us put it all together. The small company of this lesson has 15 million dollars for exploration wells this year and a risk tolerance of 30 million dollars, and it is offered interests in four areas.

The areas of the portfolio at a share of 100%

Area: A
$P_g$: 0.40
Well / discovery, US$ million: 6 / 30
EMV, US$ million: 6
σ, US$ million: 14.7

Area: B
$P_g$: 0.30
Well / discovery, US$ million: 10 / 70
EMV, US$ million: 11
σ, US$ million: 32.1

Area: C
$P_g$: 0.30
Well / discovery, US$ million: 8 / 50
EMV, US$ million: 7
σ, US$ million: 22.9

Area: D
$P_g$: 0.12
Well / discovery, US$ million: 20 / 300
EMV, US$ million: 16
σ, US$ million: 97.5

Area	$P_g$	Well / discovery, US$ million	EMV, US$ million	σ, US$ million
A	0.40	6 / 30	6	14.7
B	0.30	10 / 70	11	32.1
C	0.30	8 / 50	7	22.9
D	0.12	20 / 300	16	97.5

A is a proven play next to a producing field; B and C are two prospects of one untested segment ( $P_\text{play} = 0.5$ , $P_\text{loc} = 0.6$ ); D is a large deep prospect of a new region. The discovery is the value of the development at the date of the decision, without the cost of the well; σ is the standard deviation of NPV. The data in this example are assumed for illustration.

Worked exampleFour portfolios from the same areas

Given (assumed data)

Budget and attitude to risk: 15 million dollars for wells; $R = 30$ million dollars
Links: B and C are linked by the common risk of the play; A and D are independent of them and of each other

Solution

Three quarters of D (the whole budget): EMV 12 million dollars, σ 73.1; no discoveries with a probability of 0.88; the certainty equivalent is −11.2 million dollars.
The highest EMV within the budget: the money goes where the EMV per dollar invested is larger, into B (1.1), then into A (1.0). All of B and 5/6 of A: EMV 16 million dollars, σ 34.3, no discoveries with a probability of 0.42; CE 2.17.
Equal shares in all: $15/44 \approx 34$ %; EMV 13.6 million dollars, σ 37.2, no discoveries with a probability of 0.306; CE 1.09.
The highest certainty equivalent (the shares found by calculation): A 98%, B 31%, C 32%, D 6.5%; 12.9 million dollars go to wells; EMV 12.6, σ 21.6, no discoveries with a probability of 0.306; CE 5.80 million dollars.

Answer: The best portfolio for this company is almost all of A, about a third of B and C, and 6.5% of D. Its certainty equivalent is 5.80 million dollars, against 2.17 for the portfolio with the highest EMV and −11.2 for the bet on D.

What it means. The best portfolio gives up 3.4 million dollars of EMV compared with the highest, but its spread is smaller by more than a third. It also does not spend the whole budget: 2.1 million dollars remain. The company is limited not by money but by risk: each next share adds more to the risk than to the value. It is sensible to put the free money into shares of new, independent areas. That is further diversification. The shares of B and C are smaller than they would be for independent prospects (40 and 49%): the common risk of the play turns the pair into one larger bet. The probability of a complete failure, 0.306, is the same as for equal shares: it is set by the set of areas, not by the sizes of the shares. The data in this example are assumed for illustration.

Transfer of risk for a fee: insurance, contracts, financing, hedging

To transfer risk means to agree that in a bad outcome part of the loss will be carried by someone else. That party agrees for a fee: a premium, a mark-up, interest. On average the fee is larger than the expected loss (otherwise the deal does not pay the other side), so the EMV of the one who transfers falls. The deal is justified when the possible loss costs more to the one who transfers than to the one who receives. The one who transfers has a smaller risk tolerance, or the consequences of the loss are bigger for it.

Insurance. The insurer takes over random losses: a fire at a plant, a blowout and regaining control of the well, a spill, liability to third parties. Some kinds of insurance are required by law. For the insurer this is the same diversification: thousands of policies, with losses that are rare and independent of each other, add up to a predictable portfolio. So insurers are glad to insure what can be assessed from statistics and is limited in size, and they are unwilling to insure events that will hit many clients at once. Small frequent losses are usually kept by the company. This is self-insurance: the premium for them is almost equal to the losses themselves plus the insurer's expenses.

Worked exampleWhether to insure a well against a blowout

Given (assumed data)

Risk: a blowout with a loss of 50 million dollars (regaining control, redrilling, liability); a probability of 0.4% per well
Insurance: a premium of 0.35 million dollars
Companies: small ( $R = 30$ million dollars) and large ( $R = 1000$ million dollars)

Solution

The expected loss: $0.004\cdot50 = 0.20$ million dollars. This is less than the premium, so by EMV insurance does not pay.
A small company: the certainty equivalent of self-insurance is $-30\ln(0.996 + 0.004\,\e^{50/30})$ $= -0.511$ million dollars. Insurance costs −0.35. This is better, so it pays to insure.
A large company: $-1000\ln(0.996 + 0.004\,\e^{50/1000})$ $= -0.205$ million dollars, almost the expected loss. Insurance is worse, and it pays to carry the risk itself.

Answer: Insurance pays for the small company: by the certainty equivalent it is ready to pay up to 0.51 million dollars for it. For the large company it does not pay: the risk is worth 0.21 million dollars to it, which is less than the premium.

What it means. This is the same logic as in a farm-out: the risk goes to the one for whom it is small. For the insurer, thousands of independent policies are a diversified portfolio: its own risk is small, and the premium covers the expected payments, the expenses and the profit. Large companies with many facilities are themselves like insurers. They often insure only catastrophic risks, and pass part of the rest to their own insurance companies (captive insurers). The data in this example are assumed for illustration.

Contracts. A fixed-price contractor takes on the risk of timing and of complications. Under a turnkey contract (one agreed sum for the whole job) it takes almost all of it. Under a footage contract (paid per metre drilled) it takes mainly the risk of drilling speed. The customer pays a mark-up for this (types of drilling contracts). What distribution of overrun it protects against is shown in the lesson on the time and cost of drilling. Penalties for missed deadlines put on the contractor part of the losses from a late start of production. The lease of a facility (for example a treatment plant or a storage facility), with a payment for each barrel produced, puts part of the risk of the reservoir on the lessor. If production is smaller, the payment is smaller too.

Project finance: banks lend against the cash flow of the project itself. If the project fails, the claims of the banks are limited to the property and income of the project. They have no right to claim against the rest of the company's property, or only a limited right. Part of the risk of the project falls on the lenders. So they check the reserves, the model and the cost estimate themselves, the rate is higher than usual, and the agreement restricts the borrower. The loan is repaid only from the money of the project. So this is how the development of an already appraised field is financed, but not exploration: an exploration well has no cash flow against which to lend.

Hedging of the price moves the price risk to market participants. A forward is a contract to sell in the future at a price fixed today (the forward sale in the payoff matrix). A futures contract is the same contract, but standard and exchange-traded, with daily settlement by margin. An option is a right, but not a duty, to sell (a put option) or to buy (a call option) at an agreed exercise price (the strike) for a premium that is paid. A swap is an exchange of a floating price for a fixed one in cash settlement, with no delivery of oil. Where the uncertainty of prices and exchange rates comes from is the subject of the lesson Market, financial and regulatory uncertainty.

Worked exampleA swap or a put option for a year of production

Given (assumed data)

Volume: 1 million bbl of oil in a year
Price at delivery: 50, 70 or 90 US$/bbl, with probabilities of 0.25, 0.5 and 0.25 (an assumption of the example)
Instruments: a swap at $68/bbl; a put with a strike of $60/bbl for a premium of $3/bbl

Solution

Without a hedge: the revenue is 50, 70 or 90 million dollars. The mean is 70, the standard deviation is $\sqrt{0.25\cdot20^2 + 0.25\cdot20^2} = 14.14$ million dollars, and the worst case is 50.
Swap: 68 million dollars at any price. The mean is lower by 2 million dollars, and there is no spread.
Put: at a price of 50 the company sells at 60 and gets $60 - 3 = 57$ million dollars. At 70 and 90 the option is not needed: $70 - 3 = 67$ and $90 - 3 = 87$ . The mean is 69.5, the standard deviation is 10.9, and the worst case is 57 million dollars.
The fair premium of a put is its average payout: $0.25\cdot(60 - 50) = 2.5$ US$/bbl. The seller takes 3 for risk and expenses.

Answer: A swap gives up 2 million dollars of the mean for full certainty. A put gives up 0.5 million dollars for protection against revenue below 57 million dollars, and it keeps the gain from a rise of the price.

What it means. Hedging does not raise income: on average it lowers it by the fee paid to the other side. It changes the shape of the distribution by removing the worst outcomes. This is worth the money when a low price threatens obligations: payments on a loan, the compulsory work programme. We hedge only the volume that the company will surely produce. Otherwise, if production fails, the obligation is left without oil. The price of the company's grade can differ from the benchmark price of the contract (basis risk). With futures, when the price rises, a margin has to be paid long before the revenue is received. The data in this example are assumed for illustration.

CalculatedRevenue per barrel against the price at delivery: without a hedge, with a swap at $68/bbl, and with a put option with a strike of $60/bbl for a premium of $3/bbl. The dotted verticals are the prices of the example, $50, $70 and $90/bbl.

Diversification of a company, and scenarios

The same ideas work at the level of a whole company, not only of one portfolio of areas.

Regions and countries: Production and exploration in different basins and countries weaken the dependence on one geological result, one government and one export route. The price is the cost of being present and the complexity of management.
Links of the chain (vertical integration): A company that owns production, refining and sales is partly protected from swings of the oil price: a low price hits production but makes the raw material of refining cheaper. The protection is incomplete: the income of refining depends on the difference between the prices of oil and of oil products, and this is linked with the oil price only loosely.
Business outside the industry: Investment in unrelated industries reduces the dependence on oil, but a shareholder can diversify alone by buying the shares of different companies. Without advantages in a new industry, a company risks losing to those who know it better.
Scenario planning: We draw several consistent pictures of the future (prices, technology, regulation), without probabilities, and test how the strategy survives each of them (scenarios). This does not reduce risk, but it answers in advance the question of what to do in each development of events.

Terms and symbols

Working interest: A share in the rights and duties under the contract. In the same share the participant pays the costs and receives the production.
Operator: The participant that carries out the work on behalf of all under a joint operating agreement.
Farm-out and farm-in: A deal in which a partner gets a share in an area by paying for part of the work.
Promote: The excess of the partner's share of the costs over the working interest it gets: the price of entry into the project.
Carried interest (carry): A share whose costs at the stage of work are paid by the partner.
Gambler's ruin: The loss of all the capital through a series of failures. With equal stakes its probability is $\psi_1^{\,m}$ .
Stake: In the model of ruin, the cost to the company of one well. Capital is measured by the number of stakes.
Diversification: Spreading money over projects with unlinked outcomes. It reduces the spread without changing the mean.
Common risk of a play: The probability that the whole play does not work. It links the outcomes of the prospects of one segment and is not removed by diversification within the play.
Self-insurance: The decision to carry the risk oneself when the premium is above the value of the protection for the company.
Project finance: A loan against the cash flow of a project, with claims limited to its property and income.
Forward, futures, option, swap: Instruments that move price risk: a contract at a fixed price, its exchange-traded version, the right to buy or sell at the exercise price, and an exchange of a floating price for a fixed one.
Basis risk: The difference between the price at which the company sells and the benchmark price of the hedging instrument.

Common mistakes and misconceptions

Wrong

“The EMV of a prospect is positive, so with enough attempts we will certainly win.”

In fact

The money may run out sooner. A company with capital for one well of type D is ruined with a probability of 0.91, although the EMV of such a well is 16 million dollars.

Wrong

“We have shares in ten prospects, so the portfolio is diversified.”

In fact

If all ten are in one untested play, the probability of being left with no discoveries is not less than $1 - P_\text{play}$ . In this lesson's example it is 0.50, against 0.028 for ten independent ones.

Wrong

“By giving away half of the area for the payment of the well, we lost half of its value.”

In fact

The company gave away part of the EMV, but also most of the risk. For a small company the certainty equivalent after such a deal can be higher than with the whole area. The sum of the EMVs of the sides does not change.

Wrong

“Hedging raises income.”

In fact

On average it lowers income by the fee paid to the other side; it removes the worst outcomes. Its point is to protect obligations, not to earn.

Wrong

“We must insure everything that can be insured.”

In fact

Insurance pays when the possible loss is large for the company. A large company with many independent facilities sensibly carries the small risks itself.

In depth: advanced detail

The probability of ruin and risk tolerance

Why does the equation $\psi_1 = q + p\,\psi_1^{\,g+1}$ have no more than one root between 0 and 1? The function $q + p\,x^{g+1} - x$ is convex (its curve bends upwards), equals $q > 0$ at $x = 0$ and zero at $x = 1$ , and its slope at $x = 1$ is $p(g + 1) - 1$ . If the EMV per stake, $p(g + 1) - 1$ , is positive, the curve comes to zero at $x = 1$ from below. So earlier it crossed the axis, and a second root exists. If the EMV is not positive, the curve stays above the axis up to $x = 1$ , and the only root is one.

Let us write $\psi_1 = \e^{-c/R_0}$ , where $c$ is the stake in money and $R_0$ is a so far unknown parameter in money. The equation for $\psi_1$ turns into $q\,\e^{c/R_0} + p\,\e^{-g c/R_0} = 1$ . This is the condition $\sum_j p_j\,\e^{-x_j/R_0} = 1$ for the two outcomes of one stake, $x = -c$ and $x = g c$ . It says that the certainty equivalent of the stake at a risk tolerance of $R_0$ is zero (the exponential utility function). From this comes a simple formula.

The probability of ruin through the risk tolerance

\psi = \e^{-K/R_0}, \qquad R_0 = \frac{c}{-\ln \psi_1}

where:

$K$ capital expressed in money: $K = m c$ , US$ million
$c$ the stake: the cost to the company of one well, US$ million
$R_0$ the risk tolerance at which one such stake has a zero certainty equivalent, US$ million
$\psi_1,\ \psi$ the probability of ruin with capital for one stake and with capital $K$ , fraction
$\ln$ natural logarithm, —

Units: US$ million

Holds when

the same assumptions as for the formula $\psi_1^{\,m}$ : equal independent stakes, a discovery is turned into money at once, a game with no limit on the number of wells
when the costs of the wells are random and the capital can fall by several steps at once, there is no exact formula. But if the stakes are independent and have the same distribution, and $R_0$ is found from the distribution of one stake (its certainty equivalent is zero), it is proved in insurance mathematics that $\e^{-K/R_0}$ is an upper limit of the probability of ruin; $K$ here is the reserve whose exhaustion means ruin (the Lundberg inequality)

The probability of ruin is set by the ratio of the capital to the tolerance at which the company is indifferent to one stake. For the probability of ruin not to exceed $\psi$ , we need a capital of $K = R_0 \ln(1/\psi)$ : at 5%, about $3R_0$ . A stake that is twice as small halves $R_0$ , and the same capital gives the square of the probability of ruin.

Worked exampleHow much capital we need to drill wells whole

Given (assumed data)

Types of prospects: A, B and D of the lesson at a share of 100%: $\psi_1$ = 0.6445, 0.7346 and 0.9084
Aim: a probability of ruin of no more than 5%

Solution

Type D (a stake of 20 million dollars): $R_0 = 20/(-\ln 0.9084)$ $= 20/0.09607 = 208.2$ million dollars; the capital is $208.2\cdot\ln 20 = 208.2\cdot2.996 = 624$ million dollars.
Type B (10 million dollars): $R_0 = 10/0.30843 = 32.4$ million dollars, the capital is 97 million dollars. Type A (6 million dollars): $R_0 = 6/0.43928 = 13.66$ million dollars, the capital is 41 million dollars.
A check: the small company of the lesson has $R = 30$ million dollars, which is less than $R_0 = 32.4$ for a whole well B. And indeed its certainty equivalent for such a well is negative (−0.52 million dollars, the example of the ten prospects).

Answer: To drill prospects of type D whole with a risk of ruin of no more than 5%, we need about 624 million dollars, for type B 97, and for type A 41 million dollars.

What it means. Even per stake, the difference is large: the capital must bear about 31 stakes for prospects of type D, 9.7 for B and 6.8 for A, although their EMV per stake is close. Everything is decided by the shape of the outcome: frequent small losses, or a rare large gain. So the rule is: the lower the chance of success and the larger the prize, the smaller the share in such prospects that a company with a given capital can afford. The data in this example are assumed for illustration.

The correlation of the prospects of one play: the derivation

Let us write the outcome of a prospect as a number: 1 is a success, 0 is a failure. Its mean is $P_g$ , and its variance is $P_g(1 - P_g)$ . Both prospects succeed only if the play works and both traps work: $P_\text{play}\,P_{\text{loc},1}P_{\text{loc},2}$ . The covariance is this probability minus the product of the means: $P_\text{play}P_{\text{loc},1}P_{\text{loc},2}$ $- P_\text{play}^2P_{\text{loc},1}P_{\text{loc},2}$ $= P_\text{play}(1 - P_\text{play})\,P_{\text{loc},1}P_{\text{loc},2}$ .

The correlation of two prospects of one play: the general case

\rho_{12} = \frac{P_\text{play}(1 - P_\text{play})\,P_{\text{loc},1}\,P_{\text{loc},2}}{\sqrt{P_{g,1}(1 - P_{g,1})\,P_{g,2}(1 - P_{g,2})}}, \qquad P_{g,i} = P_\text{play}\,P_{\text{loc},i}

where:

$P_{\text{loc},i},\ P_{g,i}$ the local probability and the chance of success of the $i$ -th prospect, fraction
$P_\text{play}$ the probability that the common play works, fraction

Units: dimensionless

Holds when

the local elements of the prospects are independent; the link is only through the play.

With equal local probabilities the formula reduces to $P_\text{loc}(1 - P_\text{play})/(1 - P_\text{play}P_\text{loc})$ . For the number of discoveries among $n$ equal prospects of one segment, the variance is $nP_g(1 - P_g)\,[1 + (n - 1)\rho]$ . This gives the spread in the example of the ten prospects, and a direct enumeration of the outcomes gives the same number.

Why the certainty equivalents of independent parts add up

For an exponential utility function, the certainty equivalent of the sum of two independent results equals the sum of their equivalents. The mean of $\e^{-(X+Y)/R}$ for independent $X$ and $Y$ equals the product of the means of $\e^{-X/R}$ and $\e^{-Y/R}$ , and the logarithm of a product is the sum of the logarithms. So the shares in independent areas can be chosen separately: A and D each by itself, and the linked B and C together, by their common outcomes.

The property rests on two conditions. If the areas are linked, by a common play or a common oil price, we calculate them together. If the budget sets a limit, the shares are chosen jointly: each next share competes for the same money. In this lesson's example the budget did not set the limit, so a separate choice was enough.

Connections to other topics

Expected monetary value and preference theory — EMV, the utility function, the certainty equivalent and the choice of one working interest.
Geological risk and chance of success — the chance of geological success, the common risk of a play and dependent prospects.
Decision trees and the value of information — when information is worth buying before a decision.
Geological and technical uncertainty of a project — where the uncertainty of a project comes from and how it is allowed for in the economics.
Drilling time and cost — overrun of the cost estimate of a well and the reserve for risks.
Sensitivity analysis and scenarios — the forward sale in the payoff matrix, and scenarios.
Licences and contracts for subsoil use — subsoil use contracts, work obligations and the transfer of rights.

Check yourself

1
A small company can drill one exploration well whole, or take 20% in each of five identical independent prospects. What is true?
1. A.The EMV in the second case is five times smaller
2. B.The second option is better by EMV
3. C.The probability of ruin is the same, since the money at risk is the same
4. D.The EMV is the same, and the probability of being left with no discoveries is much smaller in the second case
Show the answer
Correct answer: D. The EMV is the same, and the probability of being left with no discoveries is much smaller in the second case
Five shares of 20% have the same cost and the same EMV: $5\cdot0.2 = 1$ well. But the company is left with no discoveries only if all five are dry: at $P_g = 0.3$ this is $0.7^5 \approx 0.17$ , against 0.7 for one well.
2
Prospects with $P_g = 0.4$ and $V/C = 4$ . Check that $\psi_1 \approx 0.69$ satisfies the ruin equation, and find the probability of ruin of a company whose capital is enough for five stakes. Compare it with the probability that the first five wells are all dry.
Show the answer
Here $g = 3$ : $\psi_1 = 0.6 + 0.4\,\psi_1^4$ . At $\psi_1 = 0.69$ : $0.6 + 0.4\cdot0.227 \approx 0.691$ , which agrees. The probability of ruin is $0.69^5 \approx 0.16$ . The probability of five dry wells in a row is $0.6^5 \approx 0.08$ . A company can be ruined even after the first discoveries, so the first number is larger than the second.
3
Two prospects of one segment: $P_\text{play} = 0.8$ , and each has $P_\text{loc} = 0.5$ . Find the chance of success of each, the probability that both are dry, and the correlation of their outcomes.
Show the answer
$P_g = 0.8\cdot0.5 = 0.4$ . Both are dry if the play does not work (0.2), or if it works but both traps fail ( $0.8\cdot0.5^2 = 0.2$ ): 0.4 in all, against $0.6^2 = 0.36$ for independent ones. The correlation is $0.5\cdot0.2/(1 - 0.4) \approx 0.17$ . The play is reliable, so the link is weak.
4
A partner gets 40% of an area and pays 60% of the cost of an exploration well. What is the promote, and what share of the well does the owner pay?
1. A.There is no promote: the shares of costs and of the interest are equal
2. B.The promote is 60%; the owner pays nothing
3. C.The promote is 40%; the owner pays 60% of the well
4. D.The promote is 20% of the cost of the well; the owner pays 40% of the well and keeps 60% of the area
Show the answer
Correct answer: D. The promote is 20% of the cost of the well; the owner pays 40% of the well and keeps 60% of the area
The promote is the excess of the partner's share of costs over its working interest: $0.6 - 0.4 = 0.2$ , that is, the partner pays 20% of the well on the owner's behalf. The owner pays the remaining 40% and keeps 60% of the area. For its own share it would have had to pay 60%.
5
A company with a risk tolerance of $R = 20$ million dollars can insure a possible loss of 10 million dollars, with a probability of 2%, for a premium of 0.3 million dollars. Is it worth it?
Show the answer
The expected loss is $0.02\cdot10 = 0.2$ million dollars. The certainty equivalent of self-insurance is $-20\ln(0.98 + 0.02\,\e^{10/20})$ $= -20\ln 1.0130 \approx -0.26$ million dollars. The premium of 0.3 is more than this 0.26, so it is better to carry the risk itself: for such a company a loss of 10 million dollars is not so large.
6
A company bought a put option with a strike of $55/bbl for a premium of $2/bbl. By delivery the price has risen to $70/bbl. What is the revenue per barrel, with the premium taken into account?
1. A.68 dollars
2. B.72 dollars
3. C.55 dollars
4. D.70 dollars
Show the answer
Correct answer: A. 68 dollars
The price is above the strike, so the right to sell at 55 is not used, and the oil is sold at the market price, 70 dollars. With the premium paid, the revenue is $70 - 2 = 68$ $/bbl. Protection against a fall of the price cost $2/bbl.

Updated: 17 October 2026