PETROLEUM ENGINEERING 689 Special Topics in Unconventional Resource Reserves Lecture 4 Expected Valu

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PETROLEUM ENGINEERING 689Special Topics
inUnconventional Resource ReservesLecture
4Expected Value Decision TreesTexas AM
University - Spring 2007
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Expected Value and Decision Trees

• Expected Value

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Learning Objectives

• You will be able to
• Calculate the expected value and standard
  deviation of a random variable
• Calculate expected monetary value (EMV) using a
  payoff table
• Determine sensitivity of EMV analysis results to
  probabilities used

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Learning Objectives

• You will be able to
• Define expected profitability index, EPI
• Use EMV and EPI to screen investments
• Use EMV and EPI to rank investment alternatives
• Define and explain performance index, I
• Define and use expected opportunity loss, EOL, to
  select investments

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Learning Objectives

• You will be able to
• Apply mean-standard deviation screening method to
  select investments
• Define and explain use of deterministic dominance
  in investment selection
• Define and explain use of stochastic dominance in
  investment selection
• Interpret what expected value represents in
  applications

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Expected Value of Random Variable

• where
• EX expectation operator, read expectation
  of
• P(xi) P(Xxi), unconditional probability
  associated with variable x
• E(x) often referred to as mean of X

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Standard Deviation of Random Variable

• Variance of discrete random variable given by
• where
• S2X variance of X

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Sums of Expected Values, Variances

• When sum of expected values or variances has to
  be determined, process straightforward when
  variables are both independent and random
• EX Y EX EY
• and
• s2X Y s2X s2Y

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Example Calculation of Expected Value

• Expected results from drilling prospect
• 30 chance of 20 MSTB
• 50 chance of 60 MSTB
• 20 chance of 95 MSTB
• What are mean, variance, and standard deviation
  of expected reserves?

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Example Calculation of Expected Value
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Example Calculation of Expected Value

• Mean, or expected value, of reserves 55.0 MSTB
• Variance 700.0 MSTB2
• Standard deviation 26.5 MSTB (v700)
• Interpretation
• Over large number of similar trials, expect to
  recover 55 MSTB with 67 confidence result will
  lie between 28.5 (55-26.5) and 81.5 (5526.5) MSTB

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Expected Monetary Value

• When random variable in expected value is
  monetary value, calculated expected value called
  expected monetary value, EMV
• EMV is weighted average of possible monetary
  values (usually NPVs), weighted by respective
  probabilities
• Monetary values can be undiscounted or
  undiscounted
• EMV of NPVs called expected present value profit

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Expected Monetary Value

• Best choice of mutually exclusive investment
  alternatives is one with largest EMV
• For screening investment alternatives, all
  alternatives with EMVgt0 are acceptable
• When monetary values represent opportunity
  losses, alternative with smallest expected value
  is optimal choice

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Structural Elements in EMV Calculations

• Acts or strategies, Aj alternatives for action
  available
• Outcome states, Si different results that may
  occur
• Consequences or payoffs, Cij gains, rewards,
  losses, etc. associated with jth act that results
  in ith outcome state

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Structural Elements in EMV Calculations

• Outcome state probabilities, P(Si) probabilities
  assigned to outcome states
• Criterion Basis decision makers use for most
  appropriate course of action from among the
  alternatives
• Two ways to display structural elements
• Payoff table (tabular)
• Decision tree (graphical)

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Generalized Payoff Table
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Example Payoff Table Application

• Outcomes likely for drilling prospect
• Dry hole probability 65, loss 250,000
• Successful well probability 35, NPV of future
  net revenues 500,000
• Can farm out prospect, remove exposure to
  drilling expenditure, retain overriding royalty
  interest, NPV 50,000
• Determine whether to drill or farm out

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Example Payoff Table Application
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Example Payoff Table Application

• Since EMV of farm-out (17.5M)gtEMV of drilling,
  we should farm out
• Result highly sensitive to probability of
  producer
• If increased from 35 to 36, drillling better
  option
• Sensitivity analysis useful if unsure about
  probabilities
• Variance of drill option much greater than
  variance of farm-out option (drilling much more
  risky)

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Example Payoff Table Application

• A company with 100 acres leased wants to drill a
  well on 160-acre prospect area
• We can join unit by leasing remaining 60 acres in
  unit
• Evaluation assumes we acquire acreage
• Gross well cost (with equipment) 110M
• Gross dry hole cost 80M

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Example Payoff Table Application

• We have identified 3 options and determined NPVs
  for several outcomes
• Participate in drilling with 37.5 non-operating
  WI (60/160x100 37.5)
• Farm out acreage and retain 1/8-th of 7/8-ths
  royalty interest on 60 acres
• Be carried with back-in privilege (37.5 WI)
  after investing parties have recovered 150 of
  investment

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Example Payoff Table Application
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Example Payoff Table Application

• Answer questions
• Should we lease adjacent land (mineral rights)?
• If so, what maximum amount should we pay?
• If we lease adjacent land, which option will be
  most valuable to us?

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Example Payoff Table Application
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Example Payoff Table Application

• Back-in has largest EMV and is best option
• Maximum value of additional acreage is 25,375/60
  423 per acre
• If acreage acquired for exactly 25,375, rate of
  return will be 10 (discount rate used to
  determine NPVs)
• Rate of return increases as we pay less to lease
  land

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Sensitivity Analysis on Probabilities

• Probabilities used in EMV analysis usually most
  uncertain parameters
• We need to determine influence of changes in
  probabilities on apparent optimal decision to
  improve our decision making
• Consider earlier example with two acts (drill or
  farm out) and two events (dry hole or producer)

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Example Sensitivity Analysis on Probabilities
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Example Sensitivity Analysis on Probabilities

• Let p probability of dry hole
• (1-p) probability of producer
• Then
• EVdrill p(-250) (1-p)(500)
• -750p 500
• EVfarmout p(0) (1-p)(50)
• -50p 50

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Example Sensitivity Analysis on Probabilities

• Decision maker indifferent if EV of two
  alternatives equal
• Probability at point of indifference given by
• EMVDrill EMV Farmout
• -750p 50 -50p 50
• p 0.6429 or 64.29
• Farm-out optimal for pgt0.6429, but results highly
  sensitive to change in probability
• We must do our best to ensure probability correct

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Example Sensitivity Analysis on Probabilities
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Sensitivity Analysis on Probabilities

• For previous example with three alternatives
  (drill, farm out, back-in), graphical method
  easier to implement

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Sensitivity Analysis on Probabilities
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Expected Profitability Index

• Expected profitability index, EPI, is ratio of
  EPV of net operating revenue to EPV capital
  investment
• For non-mutually exclusive investments, we will
  maximize corporate NPV if we maximize EPI
  provides capital budgeting strategy when we dont
  have enough funds to invest in all economically
  viable (EMVgt0 or EPIgt1) alternatives
• To develop budget, rank investments in descending
  order of EPI, start at top and select investments
  until investment budget fully allocated
• Method fails for farm-out and royalty strategies
  because investment is zero (and EPI is infinity)

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Expected Profitability Index Example

• Information for 3 drilling prospects given in
  following table
• Calculate EPI for each prospect
• Select economic optimum prospect, using decision
  rule of maximizing EMV
• Assume prospects mutually exclusive
• Select optimal prospects if we have only 150M to
  invest and prospects are non-mutually exclusive

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Expected Profitability Index Example
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Expected Profitability Index Example

• Mutually exclusive prospects, no capital
  constraint
• Prefer prospect B largest EMV

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Expected Profitability Index Example

• Non-mutually exclusive prospects, capital
  constraint of 150M
• Cannot select prospect B, ECAPEX 198.75gt150
• Choose A and C, total ECAPEX 145,250
• If both projects successful, budget overspent by
  15M (120M 45M 165M completed cost)
• A success, C dry, budget matched exactly

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Performance Index

• Definition of performance index, I
• I EMV/s expected monetary
  value/standard deviation
• Maximum I maximizes economic return at given
  level of risk
• Minimum I sometimes used as threshold for
  screening investments
• Unlike EMV criterion, takes into account risk
  preferences of investor

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Expected Opportunity Loss

• Definition EOL is difference between actual
  profit or loss and profit or loss that would have
  resulted if decision maker had had perfect
  information at time decision made
• Example choose to drill well, turns out to be
  dry hole, lose 30M
• Farm-out would have had zero loss
• EOL 30M 0 30M

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Expected Opportunity Loss

• EOL minimization rule can be used in place of EMV
  maximization rule as basis for decision making
• Result same with either rule
• EMV easier to work with in complex situations

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Example Expected Opportunity Loss

• Analyze data in following table using EOL
  criterion (for drill, farm-out, back-in
  alternatives)

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Example Expected Opportunity Loss
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Example Expected Opportunity Loss

• Construct opportunity loss table
• Identify maximum value entry in each row in
  previous table
• Subtract each entry in same row from maximum
  value
• Compute expected values by multiplying
  probabilities of outcomes by conditional
  opportunity losses
• Results in following table

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Example Expected Opportunity Loss
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Example Expected Opportunity Loss

• Best choice is back-in alternative, which has
  minimum EOL
• Same decision as with maximum EMV criterion

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Summary of Decision Criteria

• Choose alternative with largest EMV when profit
  is payoff variable and alternatives are mutually
  exclusive
• Choose alternative with smallest EOL when cost is
  payoff variable and alternatives are mutually
  exclusive

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Summary of Decision Criteria

• Rank alternatives from largest to smallest EPI
  when profit is payoff variable and alternatives
  are non-mutually exclusive
• When capital is limited, select alternatives with
  largest EPI for budget and stop when expected
  capital expenditures equal or exceed capital
  budget

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Mean-Variance Criterion

• Risk-averse decision maker may use EMV and
  standard deviation to screen or rank alternatives
• Mean-variance approach favored by some
  risk-averse decision makers seeks to choose
  alternative that yields highest expected return
  with lowest variance

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Mean-Standard Deviation Screening Method
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Mean-Standard Deviation Screening Method

• Investments 1, 7, 8, 9, and 10 provide greater
  EMV for given level of risk than other investment
  opportunities
• Determines efficient frontier
• Choice between investments depends on how risk
  averse decision maker is
• Investment 10 offers greatest reward, but is
  highest risk

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Mean-Standard Deviation Screening Method

• Approach more appropriate when probability
  distribution of each alternative can be
  represented by mean and standard deviation
• Normal distribution good example of appropriate
  distribution
• When distribution not described completely by
  mean and standard deviation, best to compare
  distributions themselves

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Dominance

• When we can compare pdf and cdf of alternatives,
  we can use dominance rules to choose between
  alternatives
• Situations include
• Deterministic dominance pdfs, cdfs, dont
  intersect, one alternative always better
• First degree stochastic dominance pdfs
  intersect, cdfs dont, one alternative still
  clear choice over other
• Second degree stochastic dominance less clear

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Deterministic Dominance
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First Degree Stochastic Dominance
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More Complex Situation
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Second-Degree Stochastic Dominance

• Compare areas before and after cross-over of
  cdfs
• Alternative with larger area dominates other
  alternative
• Called second-degree stochastic dominance

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Application of Dominance Rules
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Application of Dominance Rules

• Assume normal distributions for each case
• Allows use of EMV, standard deviation to generate
  cdf

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Application of Dominance Rules
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Application of Dominance Rules

• Back-in option dominates drill option for
  0ltEMVlt27M
• Drill option dominates back-in option for
  EMVgt27M
• Area dominated by back-in option slightly larger
  than area dominated by drill option
• Back-in option has second-degree stochastic
  dominance over drill option

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Interpretation of Expected Value

• Expected value is average value per decision
  realized when the alternative is repeated over
  many trials
• What EV is not
• Not the most probable outcome of selecting an
  alternative
• Not the number which we expect to equal or exceed
  50 of the time

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Accomplishments

• You can now
• Calculate the expected value and standard
  deviation of a random variable
• Calculate expected monetary value (EMV) using a
  payoff table
• Determine sensitivity of EMV analysis results to
  probabilities used

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Accomplishments

• You can now
• Define expected profitability index, EPI
• Use EMV and EPI to screen investments
• Use EMV and EPI to rank investment alternatives
• Define and explain performance index, I
• Define and use expected opportunity loss, EOL, to
  select investments

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Accomplishments

• You can now
• Apply mean-standard deviation screening method to
  select investments
• Define and explain use of deterministic dominance
  in investment selection
• Define and explain use of stochastic dominance in
  investment selection
• Interpret what expected value represents in
  applications

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Expected Value and Decision Trees

• Expected Value

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Expected Value and Decision Trees

• Decision Trees

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Learning Objectives

• You will be able to
• Construct a decision tree
• Solve a decision tree to determine optimal
  decision and payoff
• Calculate expected value, variance, and standard
  deviation for optimal outcomes from decision tree
• Use Excel to simplify arithmetic in value,
  variance, standard deviation calculations
• Use PrecisionTree to create and solve trees

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Decision Trees Described

• Decision Tree is diagrammatic representation of
  decision situation
• Value of decision trees
• Help decision maker develop clear understanding
  of structure of problem
• Make it easier to determine possible scenarios
  that can result if particular course of action
  chosen
• Help decision maker judge nature of information
  needed for solving given problem
• Help decision maker identify alternatives that
  maximize EMV
• Serve as excellent communication medium

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Example Decision Tree Drill or Dont Drill

• From left to right
• Typically start with a decision to be made
• Proceed to other decisions or chance events in
• chronological order

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Conventions on Decision Tree

• Decision Nodes
• Represented by square ?
• Point at which we have control
• and must make a choice
• Assigned sequential numbers (D1 here)
• May be followed by another decision node
• or chance node
• Branches emanating from square called
• decision forks, correspond to choices
• available

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Conventions on Decision Tree

• Chance Nodes
• Represented by circle o, numbered
• sequentially (C1 in example)
• Point at which we have no control,
• chance determines outcome
• Chance event probabilistic
• May be followed by series of decision
• nodes or chance nodes
• Branches emanating from circle called
• chance forks, represent possible outcomes

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Conventions on Decision Tree

• Probability or chance
• Likelihood of possible outcomes happening
• End, terminal, or payoff node
• Payoff deterministic financial outcome of
  decision
• Node represented by triangle (not on example)
• Has no branches following, returns payoff and
  probability for associated path

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Guidelines for Designing Trees

• Tree construction is iterative we can change
  our minds as we learn more
• We should keep trees as simple as possible
• Define decision nodes so we can choose only one
  option (but we should describe every option)

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Guidelines for Designing Trees

• We should design chance nodes so they are
  mutually exclusive and collectively exhaustive
• Tree should proceed chronologically from left to
  right
• Sum of probabilities should equal one at each
  chance node
• Remember that often we can draw a tree in number
  of different ways that look different but that
  are structurally equivalent

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Solving Decision Trees

• Decision analysis on tree can produce expected
  value of model, standard deviation, and risk
  profile of optimum strategy
• Method of calculating optimum path called folding
  back or rolling back tree
• Solve from right to left consider later
  decisions first

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Solving Decision Trees

• Chance node reduction
• Calculate expected value of rightmost chance
  nodes and reduce to single event
• Decision node reduction
• Choose optimal path of rightmost decision nodes
  and reduce to single event (choose maximum ECi
  at decision node)
• Repeat
• Repeat procedure until you arrive at final,
  leftmost, decision node

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Example Decision Tree

• Aggie Oil Company wants to decide whether to
  drill new prospect
• Geologists and engineers expect
• Probability of dry hole 60, NPV -65M
• Probability of 60M STB 30, NPV 120M
• Probability of 90M STB 10, NPV 180M

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Example Decision Tree
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Example Decision Tree
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Another Example Decision Tree

• Aggie Oil Company plans to drill a well, wants to
  determine EMV of drilling
• 35 chance of dry hole, NPV -65M
• 65 chance of producer if successful
• 60 chance of 30M STB, NPV 60M
• 30 chance of 60M STB, NPV 120M
• 10 chance of 90M STB, NPV 180M

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Another Example Decision Tree
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Alternative Approach Collapse Tree
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Alternative Approach Collapse Tree
Too much collapsing obscures important detail
not recommended
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Constructing Risk Profiles

• Risk profile is distribution function describing
  chance associated with every possible outcome of
  decision model
• Steps to generate risk profile
• Reduce chance nodes (collapse tree)
• Reduce decision nodes consider only optimal
  branches

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Steps in Constructing Risk Profiles

• Repeat steps 1 and 2 until tree is reduced to
  single chance node with set of values and
  corresponding probabilities
• Generate risk profile
• Final set of payoff and probability pairs defines
  discrete probability distribution used to
  generate risk profile
• Can graph risk profile as discrete cumulative
  density distribution or scatter diagram

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Steps in Constructing Risk Profile

• 5. Calculate expected value, variance, and
  standard deviation, as in example

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Spreadsheet Applications

• Excel built-in functions simplify calculation of
  EMV, variance, standard deviation
• Palisades PrecisionTree assists us in
  constructing and solving decision trees

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Excel SUMPRODUCT Function
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PrecisionTree

• Part of Palisades suite
• Add-in to Microsoft Excel
• Allows us to create and solve decision trees in
  Excel
• Also capable of performing sensitivity analysis,
  displaying results as spider graphs and tornado
  charts

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Running Precision Tree

• Pages 212 to 226 of Mian, Vol. II, serve as a
  tutorial for using PrecisionTree to create and
  solve decision trees
• Be sure that you can reproduce Examples 3-10,
  3-11, and 3-12

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Learning Objectives

• You can now
• Construct a decision tree
• Solve a decision tree to determine optimal
  decision and payoff
• Calculate expected value, variance, and standard
  deviation for optimal outcomes from decision tree
• After study and practice, you will be able to
• Use Excel to simplify arithmetic in value,
  variance, standard deviation calculations
• Use PrecisionTree to create and solve trees

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Expected Value and Decision Trees

• Decision Trees