Programa Ejecutivo de Capacitación en el Sector Energía Risk Analysis Michèle Breton, November...

3Risk Analysis - Michèle Breton

Course Outline

Risk and the risk analysis processDeterministic models and sensitivity analysisBasic probabilities, expectation and decision criteriaProbabilistic models and Monte-Carlo simulationDecision trees, utility theory and real options

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Risk Analysis - Michèle Breton 4

INTRODUCTION

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Geologists tell you that the chances of finding oil are 90% for a given prospect. You find a dry wellEconomists tell you that there is a 90% chance that the demand for electricity will exceed your capacity in the next 25 years. You build a power plant, but its capacity is under-used

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What Is a Good Decision?

Did you take the right decision?

How can you tell if your decision was right?

Were your experts wrong?

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Capabilities

Information processingComputationConsistency

StructureIntuitionAlternative generation

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Decision Problems

Unique

Complex

Uncertain

Subjective

Conceptual framework

Mathematical model

Probabilities

Analysis

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Risk Analysis

Risk evaluation• Identify, quantify, characterize unwanted events

Risk management• Decision making, acceptable risks, communication,

risk mitigation and hedging

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What Is Risk/Uncertainty?

Technical EconomicEnvironmentalPolitical…

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Unpredictable events• Account for the risk of hazardous material spillage

Probability• Quantify the risk of a nuclear accident

Consequence• Evaluate the risk of a negligence

Menace• Consider the political risk

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What Is Risk/Uncertainty?

Risk Definition

There is no precise definition – depends on the context and applicationMultidimensional concept

• Damages, consequences (values)• Uncertainty (probability)• Impact (decision maker’s circumstances)

There is no universally accepted measure

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10 000 $

-10 000 $

10 000 Iranian Rials

-10 000 Iranian Rials

10 000 $

-10 000 $

10 000 $

50 000 $

Perception vs. Attitude

Different decision• Perception of risk• Risk preference

Examples10 000 $

-10 000 $

For Carlos Slim …

1 MXN = 1886.0989 IRR

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Global Risk

Is a combination of many elementsIs related both to uncertainty and adverse consequencesDecreases with time, with increasing informationIs estimated subjectively – no single measure is universally accepted

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The Risk Analysis Process

Structuring the problem• Alternatives, uncertainties, variables and parameters,

values, relations and dependencies

Deterministic modeling• Computation of results, preference measures,

sensitivity analysis

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The Risk Analysis Process

Probabilistic modeling• Probability distributions, parameter estimation,

scenario forecasting, dependencies

Analysis• Risk profile, optimal strategy, value of information,

value of flexibility, sensitivity analysis

Results

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Applications

Equipment replacement decisionsDetermination and revision of the probability of a discoveryProject evaluation and selectionDrilling, testing and development decisionsFarming-out, joint-ventures and abandonment decisionsEvaluation of reserves

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Structuring Decision Problems

Parameters, futuresVariables, alternatives, plansAvailable information and uncertaintyDependenciesObjective and risk mitigation

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Examples: deciding about

Size of a production facilityExploration for oilEnergy resource plan

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DETERMINISTIC MODELSEvaluation of a situation

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Investment Opportunities

Most decisions under risk may be considered as investment opportunities

• Exploration for hydrocarbons in new regions• Expansion schedule for power plant's capacities• Development of new capacity, new energy sources• Reliability, quality, process, product improvement

programs• Long-term contracts

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Modeling and Economic Analysis

The determination of the economic value of these investment opportunities must take into account many elements, some of which are uncertain:

• Development and operation costs• Demand, revenues• Price of energy

While others result from decisions:• Development plan• Production method• Pricing• Capacity

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Deterministic Model:a Basic Tool

Choose the “best” optionEvaluate projectsAnalyze risk

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Scenario Evaluation

Scenario: a given situation, for a given set of decisions and of values for the uncertain parametersDeterministic model: evaluation tool, generally in the form of a spreadsheet cash-flow model

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Inputs

Model parameters are based on information gathered from various departments or experts:

• Investments & development planning• Operation costs• Demand• Oil price

The precision of this information varies. Usually, it increases during the life of the project

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Base Scenario

The most probable values for all parameters make up what is called the base scenario

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Outputs

Evolution of some interesting variables:• Revenues• Cash flows• Cumulated cash flows

Corresponding economic indicators:• Total discounted flows• Payback time• Maximum exposure• Surplus• Internal rate of return

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Setting Up a Cash Flow

The following table presents the investment schedule for a nuclear plant project of 2000 MW capacity. Operation costs are expected to be $15 million per year plus $17 per megawatt produced, and capacity factor to be 90%. Operation costs are expected to increase by 6% per year after 20 years of operation. Abandonment cost is $500M.

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Suppose that there is no inflation and that electricity price, presently at $34/MWh, will increase by 0.2% during the first 25 years, increase by 1.9% per year for the next 10 years, and by 0.6% per year afterwards.

Year 0 1 2 3 4 5 6 7 8 9 10 Investment ($ million)

10 10 15 15 15 95 600 1200 750 300 150

nucleardet

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Cash-ins• Revenues (sales) • Salvage value of assets• Refunds of past expenses• Sales of services

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Inputs of Cash Flow Models

Cash-outs• Technical costs (separated according to fiscal

treatment): • Capital expenditures• Operating costs (fixed or variable)

• Government take

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Inputs of Cash Flow Models

Economic indicators “summarizing” the value of a project

• Time evolution of cash flows• Cumulated values• Net present value, return• Decomposition of cash flows

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Outputs of cash-flow models

nucleardet

The cash flows generated by a project are divided between a Company (or a group of companies) and the State (possibly via a State Company)

• Royalties• Taxes (fiscal depreciation)• Advances and refunds

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Dividing the Cash Flows

nucleardet

Investing in a Risky Project

Investing in a project irrevocably commits resources in the expectation of receiving future gainsTwo important factors have to be accounted for when evaluating an investment project:

• Return on investment in alternative investment opportunities (cost of capital)

• Risk associated with the project

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Valuation of a Risk-free Project

Investing in a project prevents the use of capital in other opportunities, and particularly in "no risk" ventures with guaranteed rate of returnIn order for a project to be interesting, the return on investment should be at least as high as the best "no risk" alternative

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Discounting

Discounting allows comparison of non risky amounts received at different time and of different streams of amounts

• Present value for a given rate is the amount which is equivalent to the stream of cash flows generated by the project

• Internal rate of return is a rate such that the NPV of a stream of cash flows is 0

nucleardet

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Discount Rate

Discount rates for projects must be chiefly based on rates of return* of alternative investment opportunities. These may vary in time, but may also vary according to the available capital The long term cost of capital is generally a lower bound when evaluating investment projects

* May include anticipated inflation

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Use of Index Numbers

The computations in a cash-flow model must be done in money of the day (e.g. taxes, agreements, depreciation, refunds)When setting up the model, parameters are estimates in estimate date money

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Use of Index Numbers

Cost and price data is obtained in estimate date moneyThis data is adjusted using an acceleration index - adjustment rates may differ (nominal or money of the day data)If desired, the cash-flows can be deflated to obtain constant money data Discounting is applied to account for time value. If it is applied on constant money data, if must not contain anticipated inflation PESCE 2013

Risk Premium

The previous argument is based on the assumption that cash flows are risk-free amounts. The investor will prefer a "risk-free" investment to an uncertain project with a null net present valueThe risk premium is the amount the investor will require of a risky project in excess of the present value of a risk-free project

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Risk Preference

This risk premium depends on the investor (risk aversion) and on the magnitude of the risked amounts. It is usually difficult to ascertainExamples

• Insurance• Higher return from speculative stock• Higher price for reputable product• Price of flexibility

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Accounting for the risk premium by increasing the discount rate is a common practice – but cannot be used in the context of risk analysis. Using a “risky” discount rate supposes that there exist comparable risky investment for which comparable amounts are required.The alternative is the use of risk analysis, sensitivity analysis or Monte-Carlo simulation in order to evaluate the risk premium directly

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Estimating the Risk Premium

Examples

Valuation of capital investment (nuclear plant Texas Gulf Coast)Development of an oil fieldIntegrated resource plan

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SENSITIVITY ANALYSISIllustration of the effects of risk

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The Base Case

Most probable realization for the uncertain parameter and corresponding decisionsIs often used for decision making (e.g. npv analysis)However, the probability of the realization of the base case may be very small

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Steps of a Sensitivity Analysis

Identification of the various risk factorsEstimation of the range of possible values Computation of the (economic) effectAnalysis (risk management)

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Sensitivity Analysis

In a sensitivity analysis, input parameters are varied one by one around their base case value in order to judge the impact of these variations on the profitability indicatorsThis provides the answers to "what if" type questions

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Example

In the nuclear plant valuation example, the experts who estimated the data point out that it may vary. Thus:

• The forecast of long range electricity prices could be lower (no increase in real term price) or higher (environmental legislation)

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Example

• Increase in variable operating costs could go from 4% to 7%

• Plant performance could be between 80% and 92% • Construction costs could vary by 200$ per nominal

KW, more or less• Fixed operating costs could be between 13 and 16 M$

per year.

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Sensitivity TableElectricity price high 2785 815.61

probable 2223 -53.45low 1904 -362.56

Variable costs high 1759 -96.81probable 1576 -53.45low 1293 79.69

Plant efficiency high 92% -8.51probable 90% -53.45low 80% -278.12

Construction costs high 1.7 -296.52probable 1.5 -53.45low 1.3 189.63

fixed costs high 16 -60.54probable 15 -53.45low 13 -39.27

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Spider Diagram

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Tornado Diagram

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Is easy to do and easy to understandDetermines the range of probable values Illustrates extreme casesAnswers what if and what should questionsPoints out critical parametersPoints out the range of acceptable situationsIndicates trade-offs and allow risk management

nuclearsens

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Considers only independent variations of parametersDoes not take probabilities into accountDoes not account for correlations in parameter values

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Sensitivity analysis can be used to evaluate the risk of an investment projectIt can also help in choosing among many possible projectsExample: BED

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PROBABILITIESMeasure of uncertainty

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What Is a Probability

A probability is a measure of an uncertain eventThis measure is applied to the “likelihood” of the eventAn event which is more likely to happen than another has a higher probabilityBy convention, probabilities are between 0 and 1

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Three Kinds of Probability

Geometric probability

Honest dice

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Limit of frequencies

Statistical data

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Subjective probability

Personal firm belief of the decision maker

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61Risk Analysis - Michèle BretonPESCE 2013

A Probability Tree

medium

A probability tree is the representation of a state of mind

Dependent and Independent Events

Notation

: Probability that B is realized, given that A is realized: Probability that both A and B are realized: Probability that either A or B is realized

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when new informationbecomes available

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Change of Probability

A1 A2 A3 A4 A5

Example

In a region, the probability of drilling a dry well is 90%. When there is no oil, a test gives positive results 40% of the time. When there is oil, the test gives a positive result 80% of the time. Suppose that the test in some prospect is negative, what is the probability of this prospect containing oil?

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Random Variables

X: number of points on the dieProbability distribution:

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x 1 2 3 4 5 6

P(X=x)

1/6 1/6 1/6 1/6 1/6 1/6

Random Variables

X: construction costProbability density:

Construction costs ($/KW)

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Probability Distribution/Density

abxa i dxxfxXPbXaP

i)()()(

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Understanding a distribution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 190

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Definitions

ModeMedianMeanRangeStandard deviationVaRTVaR

95% VaR=7,5

TVaR95=9

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Mathematical Expectation Decision Criterion

Preventive maintenance, equipment backupAsymmetrical distributionsExploration wellsResearch and developmentExpected Net Present Value where

• p: probability of a success• V: net present value if success• C: sunk cost if failure

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Examples

reliability

Best Strategy

expnpv

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Risk Measures

Variance, standard deviation

Percentiles (VaR)Conditional expectations (TVaR)

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Risk-return Trade-off: the Efficient Frontier

Maximise return for a given level of riskMinimise risk for a given returnLagrangean interpretation, risk tolerance

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Expected Value Adjusted for Risk

where R is the risk tolerance (order of the total amount that one is willing to risk)

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MONTE-CARLO SIMULATIONRisk evaluation in a complex model

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Monte-Carlo Simulation

Allows simultaneous variations of parametersUses probability distributionsAccounts for correlation

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Steps in a Monte-Carlo Simulation

1. Stochastic modeling: obtain probability distributions for the uncertain input parameters (statistical information or analysis)

2. Select a combination of input parameters from these distributions using random numbers

3. Compute economic indicators (outputs) for this combination of parameters

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Steps in a Monte-Carlo Simulation

4. Repeat steps 2 and 3 many times5. Present results in the form of a probability

distribution

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Number of equipment failures in a week: Poisson distribution of mean 2

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Number of producing wells in100: Binomial (100,0.9)

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Net pay: triangular of minimum 20m, probable 30m and maximum 60m

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Reserves: Lognormal with mean of 150 BCF and standard deviation of 150 BCF

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Variable opex: Normal with mean of $10 per MW and standard deviation of $2 per MW

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Delay: Uniform between 0 minutes and 5 minutes

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Random Valuesrandom

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Sampling

Electricity price: highVariable costs increase: 5.65%Plant efficiency: 85%Construction costs: 1.42$/WFixed costs: 13.7M$/yrAbandonment cost: 475M$

Porosity: 0.17Water saturation: 0.30Net pay: 9mArea: 32 haRecovery factor: 30%

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A “trial” corresponds to the generation of random values for all parameters

Results

For each specific trial, a “result” is computed using the deterministic model. This result is recorded and will be used to produce the distribution of results of all trials

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Example

Compute recoverable reserves according to pessimistic and optimistic scenariosEvaluate nuclear plant project according to pessimistic and optimistic scenariosCompare with Monte-Carlo simulation.

RESERVNuclearsimul

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Distribution of Results

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Trend Chart

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Examples

Exploration for oilCorrelated reserves

Drill BitOilsim

Multizone

Oil field

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Monte-Carlo Simulation

Illustrates possible consequences of a decision or a strategy, but cannot choose best strategy (specifically in sequential problems)Often the only possible solution when stochastic model is complexCan be used along with an optimizer – the objective is then a summary statistic

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DECISION TREESSequential decision processes

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What Is a Tree?

A tree is a graph made up of nodes and arcs such that there exists a unique path from one specific node (called the root) to all the other nodes

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A Decision Tree

Construction0,20

environmental legislation 1044,56 920,691

0,00 920,69stop

permit 0,00 -123,87

-104,52 85,0435802Construction

0,800,55 no legislation -65,85 -189,72

high efficiency 21 0,00 -123,87

0,00 85,04 stop

0,00 -123,87

0,00 -19,35

feasibility study Construction0,20

-19,35 49,9137894 environmental legislation 654,23 530,361

0,00 530,36stop

permit 0,00 -123,87

-104,52 6,97737843Construction

0,800,45 no legislation -312,14 -436,01

1 low efficiency 249,91 1 0,00 -123,87

0,00 6,98 stop

0,00 -123,87

0,00 -19,35

0,00 0,00

Decision and Event Nodes

Decision nodes (represented by squares) model a choice in a limited number of possible actions

Event nodes (represented by circles) model uncertainty

Decision 1outcome 1

cost 11

Decision 2outcome 2

cost 2

prob 1Event 1

outcome 1gain 1

prob 2Event 2

outcome 2gain 2

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Drawing a Decision Tree

Identify the horizon and the evaluation dateIdentify all alternatives, present and future; do not forget “wait”, “stop” and “seek more information” options

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Identify all uncertain events that can affect the consequences of decisions, or that can give useful information for future decisions. Make sure that all uncertain events at a given node form partitions

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Draw the tree in chronological order - retard decisions as much as possibleEvaluate consequences at terminal nodes (leaves) using a deterministic modelEvaluate probability of uncertain events (subjective probabilities)

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Possible, probable, a working possibility, betting chanceVery possible, reasonably certain, without doubtNot probable, impossible, a small chance, need to be very lucky, doubtful, highly improbable, almost impossible...

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Evaluating a Subjective Probability

A way to quantify a subjective probability is to offer a choice between the event and a reference lottery

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Which event is more probable?One adjusts the proportion of black balls in the box until the decision maker feels that both events are equally probable

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Solving Simple Trees

A Decision Node Is Replaced by the Best Option

500500

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Solving Simple Trees

An Event Node Is Replaced by its Certainty Equivalent

200200

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Certainty Equivalent

The decision maker is indifferent between the lottery and the certainty equivalentFor a risk neutral decision maker, the certainty equivalent is equal to the expected value of the lotteryUtility theory provides a mechanism to find certainty equivalent for risk averse or risk seeking decision makers

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Solving Complex Trees

All trees are simple trees at the end of the horizonTo solve a complex tree, simply “roll back” the horizon

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Using a Software

Nucleartree

drilling

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Optimal Strategy and Risk Profile0.20

environmental legislation Construction0.55 1 920.69

high efficiency permit 0.00 920.69 920.691

0.00 111.037979 -104.52 111.037979 0.80no legislation Construction

1 -91.38feasibility study 0.00 -91.38 -91.38

164.2106148 -19.35 64.2106148 0.20

environmental legislation Construction0.45 1 530.36

low efficiency permit 0.00 530.36 530.361

0.00 6.97717017 -104.52 6.97717017 0.80no legislation stop

2 -123.870.00 -123.87 0.00 -123.87

nuclearstrategy

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Sensitivity

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Value of Information

The expected value of information is the maximum price that the decision maker is willing to pay to obtain additional informationIt is equal to the additional gain the decision maker expects if this information is obtained

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Expected Value of Perfect Information

Sequential decision problems are hard because the decision maker has to commit resources before knowing the realization of uncertain eventsIf uncertainty were resolved, the decision maker would have perfect information

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No Information

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Perfect Information

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Expected Value of Perfect Information

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Expected Value of Imperfect Information

Imperfect information allows the decision maker to revise probabilitiesValuable information has an impact on the decisionPrice of the information should not exceed its expected value

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Example

A consulting firm is offering to analyze the situation for environmental legislationPast performances of this consulting firm indicate a 75% predicting accuracy when legislation is passed, 50% when it is not What is the maximum amount you should pay for this analysis?

nucleartest

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Decision Tree

Clearly lays out the problem so that all options can be challenged Allows to analyze fully the possible consequences of a decisionModels adaptive strategiesHelps to choose the best strategy and evaluates the strategic value in a project (real options) Provides a framework to quantify the values of outcomes and the probabilities of achieving them

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Examples

Bidding for a contractBuy or lease decisionOil exploration

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The Real Option Approach

Useful when a predominant source of uncertainty can be monitored continuously (e.g. electricity prices)A simple decision tree with an underlying asset processVarious solution approaches, the most general is dynamic programmingReal option: flexibility value

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Stochastic optimization

Taking the best decision under uncertaintyWhen uncertainty is revealed progressively, choosing the best strategy (tree)Monte-Carlo simulation can be used to assess strategies Simulation can also be coupled with search heuristics Decision trees are stochastic dynamic programs – curse of dimensionality Oil field

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UTILITY THEORYModeling attitude to risk

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Attitudes to Risk

0.5Risk neutral

0.5100

Risk seeker

Risk averse

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Risk Premium

The risk premium is the difference between the expected value and the certainty equivalentThe amount of the risk premium indicates the attitude to risk of the decision makerAttitude to risk may differ according to the consequences

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Attitude to Risk

The certainty equivalent reflects the preferences of the decision makerThere is no “correct” attitude

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Amount at Risk

0 200 400 600 800 1000

expected value certainty equivalent

riskpremium

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Uncertainty

0102030405060708090

0 0.2 0.4 0.6 0.8 1

expected value certainty equivalent

risk premium

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Gains vs. Losses

< 32000.8

0.2Risk averse

>-32000.8

0.2Risk seeker

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Consistent Risk Attitude

Finding the certain equivalent for each event is difficult and time consumingConsistency problemDecentralization problem

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Using a Utility Function

Accept the behavioral axiomsFind the utility function describing attitude towards risk takingUse the utility function to find certain equivalents

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Behavioral Axioms

> Indicates preference of outcomesA>B: outcome A is preferred to B~ Indicates indifference of outcomesA~B: the decision maker is indifferent between A and B

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If A>B and B>C, then A>C

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Transitivity

If A>B>C, then there exists a probability p such that

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Existence

1-p~ B

Then B can be substituted to the lottery without changing preferences

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Substitution

1-p~ B

If A>B and p is greater than q, then

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Preference

Probability (No Fun in Gambling)

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Existence of a Utility Function

Acceptation of these axioms leads to the existence of a utility function that can be used for any uncertain ventureUtility function can treat any variables and multiple attributesUtility is a scale measuring preferences for outcomes. Conventionally, it takes values from 0 to 1

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Building a Utility Function

0 2000 4000 6000 8000 10000

utilit

amount

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Building a Utility Function

-2000$

+8000$0.5

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Using a Utility Function0

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Value of Information

With non-linear utility, expected utility of information cannot be computed directlyOne has to find iteratively the cost which would make the decision maker indifferent between obtaining the information or not

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The Exponential Utility Function

Within a reasonable range of values, many utility curves can be fit by an exponential function

R is the risk tolerance (in the units of x)Rules of thumb: net income or 20% market value or 15% of equity

utility

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Expected Value Adjusted for Risk

Taylor expansion of the utility function shows that the certain equivalent of a risky amount X is approximately equal to

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Market for Risk and Side Bets

Shifting the risk to less risk-averse party increases the utility of both partiesRisk transferring or risk sharing instruments

• Joint ventures• Insurance• Derivatives• …

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Allais' paradox

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Failure? of Utility Theory

EmotionsComprehension

• Probability• Consequences• Complexity

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Conclusion

Utility theory and certainty equivalent approach accounts for risk explicitly with the risk penalty or risk tolerance, and for time considerations with a risk-free discount rate to evaluate streams of payoff Monte-Carlo simulation directly illustrates the effect of risk and allows one to choose, between various alternatives, the preferred risk/return profile

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Thank you for your attention

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Programa Ejecutivo de Capacitación en el Sector Energía Risk Analysis Michèle Breton, November...

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