Materials for Lecture 14

1
Materials for Lecture 14

• Watch an episode of Deal or No Deal
• Read Chapter 10
• Read Chapter 16 Section 11.0
• Lecture 15 CEs.xls
• Lecture 15 Elicit Utility.xls
• Lecture 15 Ranking Scenarios.xls
• Lecture 15 Ranking Scenarios Whole Farm.xls
• Lecture 15 Utility Function.xls

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Ranking Risky Alternatives

• After simulating multiple scenarios your job is
  to help the decision maker pick the best
  alternative
• Two ways to approach this problem
• Positive economics role of economist is to
  present consequences and not make recommendations
• Normative economics role of economist is to
  make recommendations
• Simulation results can be presented many
  different ways to help the decision maker (DM)
  make the best decision for him/herself
• Purpose of this lecture is to present the best
  methods for ranking risky alternatives so the DM
  can make the best decision

3
Ranking Risky Alternatives

• Decision makers rank risky alternatives based on
  their utility for income and risk
• Several of the ranking procedures ignore utility,
  but they are easy to use
• The more complex procedures incorporate utility
  but can be complicated to use

4
Easy to Use Ranking Procedures

• Mean only Pick scenario with the highest mean
  -- throw away the risk

• Minimize Risk Pick the scenario with lowest
  Std Dev this ranking strategy ignores the
  level of returns (mean and relative risk)

 

• Mean Variance often difficult to read, does
  not work well for non-normal distributions.
  Always select the scenario in lower right
  quadrant.

- From diagram below A is preferred to C E is
preferred to B
(income)
5
Easy to Use Ranking Procedures

• Worst case ignores the level of returns but
  has merit in that it avoids catastrophic
  losses, ignores upside risk. Bases decisions on
  scenario with highest minimum, but it observed
  with a 1 chance. Worst case had a 1 out of
  500 chance of being observed.

• Best case ignores the overall risk and
  downside potential risk. Looks at only one
  iteration, the best, with a 1 chance. Worst case
  had a 1 out of 500 chance of being observed.

0
6
Easy to Use Ranking Procedures

• Relative Risk Coefficient of Variation (CV),
  pick the scenario that has lowest absolute CV.
  Easy to use, considers risk relative to the level
  of returns but ignores the decision makers risk
  aversion.

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Easy to Use Ranking Procedures

• Probabilities of Target Values Calculate and
  report the probability of achieving a preferred
  target and probability of failing to achieve a
  minimum target, i.e., the StopLight chart. This
  method is easy to use and easy to present to
  decision makers who do not understand risk.

8
Easy to Use to Rank Procedures

• Rank Scenarios Based on Complete Distribution
  Graph the distributions as CDFs and compare the
  relative risk of the returns for each
  distribution at alternative levels of return.
  Pick the distribution with the most return for
  each risk level or pick the distribution with the
  lowest risk for each level of returns, i.e., the
  distribution further to the right.

9
Utility Based Risk Ranking Procedures

• Utility and risk are often stated as a lottery
• Assume you own a lottery ticket that will pay you
  10 or 0, each with a probability of 50
• Risk neutral DM will sell the ticket for 5
• Risk averse DM will sell ticket for a certain
  (non-risky) payment less than 5, say 4
• Risk loving DM will sell if paid a certain amount
  greater than 5, say 7
• Amount of the certain payment to part with the
  ticket is DMs Certainty Equivalent or CE
• Risk premium (RP) is the difference between the
  CE and the expected value
• RP E(value) CE
• RP 5 4

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Utility Based Risk Ranking Procedures

• CE is used everyday when we make risky decisions
• We implicitly calculate a CE for each risky
  alternative
• Deal or No Deal game show is a good example
• Player has 4 unopened boxes with amounts of
• 5, 50,000, 250,000 and 0
• Host offers a certain payment (say, 65,000) NOT
  to open another box and exit the game, the
  certain payment is always less than the expected
  value (E(x) 75,001.25 in this example)
• If a contestant takes the Deal, then the Certain
  Payment offer exceeded their CE for that
  particular gamble

11
Ranking Risky Alternatives Using Utility

• With a simple assumption, the DM prefers more to
  less, then we can rank risky alternatives with
  CE
• DM will prefer the risky alternative with the
  greatest CE
• To calculate a CE, all we have to do is assume
  a utility function and that the DM is rational
  and consistent, calculate their risk aversion
  function, and then calculate the DMs utility for
  a risky choice

12
Ranking Risky Alternatives Using Utility

• Utility based risk ranking tools in Simetar
• Stochastic dominance with respect to a function
  (SDRF)
• Certainty equivalents (CE)
• Stochastic efficiency with respect to a function
  (SERF)
• Risk Premiums (RP)
• All of these procedures require estimating the
  DMs risk aversion coefficient (RAC) as it is the
  parameter for the Utility Function

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Suggestions on Setting the RACs

• Anderson and Dillon (1992) proposed a RAC
  definition of
• 0.0 risk neutral
• 0.5 hardly risk averse
• 1.0 normal or somewhat risk averse
• 2.0 moderately risk averse
• 3.0 very risk averse
• 4.0 extremely risk averse (4.01 is a maximum)
• Rule for setting RRAC and ARAC range is
• Relative RACs of 0 to 4.001 for the Power
  utility function
• Absolute RACs of 0 to 4/Wealth for the
  Negative Exponential utility function

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Assuming a Utility Function for the DM

• Power utility function
• Use this function when assuming the DM exhibits
  evidence of relative risk aversion
• DM willing to take on more risk as wealth
  increases
• Or when ranking risky scenarios with a KOV that
  is summarized over multiple years, as
• Net Present Value (NPV)
• Present Value of Ending Net Worth (PVENW)
• Negative Exponential utility function
• Use this function when assuming DM exhibits
  constant absolute relative risk aversion
• DM does not take on more risk as income increases
• Or when ranking risky scenarios using KOVs for
  single years, such as
• Annual net cash income or return on investments

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Estimate the DMs RAC

• Calculate RAC
• Enter values in the cells that are Yellow
• Lecture 15 Elicit Utility .xls

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Stochastic Dominance

• Stochastic Dominance assumes
• Decision maker is an expected value maximizer
• Alternative distributions (F and G) are mutually
  exclusive
• Distributions F and G are based on population
  probability distributions. In simulation, these
  are 500 iterations for alternative scenarios of a
  KOV, e.g. NPV
• First degree dominance when CDFs do not cross.
• In this case we can say, All decision makers
  prefer distribution furthest to the right.
• However, we are not always lucky enough to have
  distributions that do not cross.

17
Stochastic Dominance wrt a Function (SDRF)

• Generalized SDRF measures the difference between
  two risky distributions, F and G, at each value
  on the Y axis, and weights differences by a
  utility function using the DMs ARAC.

1.0

• F(x) dominates G(x) for NPV values from zero to A
  and G(x) dominates from A to B, F(x) dominates
  for NPV values B
• At each probability, calculate F(x) minus G(x)
  (the horizontal bars between F and G) and weight
  the difference by a utility function for the
  upper and lower RACs
• Sum the differences and keep score of U(F(x))
  U(G(x))

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Ranking Scenarios with SDRF in Simetar

• Interpretation of a sample Stochastic Dominance
  result

• For all decision makers with a RAC of -0.01 to
  0.1
• The preferred scenarios are Options 1 and 2 the
  efficient set
• If Options 1 and 2 are not available, then Option
  3 is preferred
• Options 4 and 5 are the least preferred
• Note that Stochastic Dominance resulted in a
  split decision
• The Efficient Set has more than one alternative

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Rank Risky Scenarios Using CE
Lecture 15
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Ranking Scenarios with Stochastic Efficiency
(SERF)

• Stochastic Efficiency with Respect to a Function
  (SERF) calculates the certainty equivalent for
  risky alternatives at 25 different RAC levels
• Compare CE of all risky alternatives at each RAC
  level
• Scenario with the highest CE at the DMs RAC is
  the preferred scenario
• Summarize the CE results for possible RACs in a
  chart
• Identify the efficient set based on the highest
  CE within a range of RACs
• Efficient Set
• This is utility shorthand for saying the risky
  alternative(s) that is (are) the most preferred

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Ranking Scenarios with Stochastic Efficiency
(SERF)

• SERF requires an assumption about the decision
  makers utility function and like SDRF uses a
  range of RACs
• SERF ranks risky strategies based on expected
  utility which is expressed as CE
• Simetar includes SERF and calculates a table of
  CEs over a range of RAC values from the LRAC to
  the URAC and develops a chart for ranking
  alternatives

22
SERF Table of CE Values

• In this example the RAC range of 0.0005 to
  0.0005 is specified and 25 separate RACs are
  calculated and used over this range
• The RACs can be changed dynamically and the CE
  values are re-calculated by Simetar
• The CEs for each of the 5 scenarios are
  calculated for each of the 25 RAC values, given
  the Utility function
• Read across the table to find which scenario is
  best at each RAC it has the highest CE at each
  RAC level

23
Ranking Scenarios with SERF

• The SERF Table points out the reason that SDRF
  produces inconsistent rankings
• SDRF only uses the minimum and maximum RACs
• The efficient set (ranking) can differ from
  minimum the RAC to the maximum RAC
• Changing the RACs and re-running SDRF can be slow
• The SERF Table can show the actual RAC where the
  decision maker is indifferent between scenarios
  (this is the BRAC or breakeven risk aversion
  coefficient)
• The SERF Table is best understood as a chart
  developed by Simetar

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Ranking Scenarios with SERF

• Two examples are presented next
• The first is for ranking an annual decision using
  annual net cash income
• Uses negative exponential utility function
• Lower ARAC zero
• Upper ARAC 4.0/Wealth
• The second example is for ranking a multiple year
  decision using NPV variable
• Uses Power Utility function
• Lower RRAC zero
• Upper RRAC 4.001

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Ranking Risky Annual NCIs with SERF
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Ranking Risky Annual NCIs with SERF
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Ranking Risky Alternative NPVs with SERF
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Ranking Risky Alternatives with SERF

• Interpret the SERF chart as follows
• The risky alternative that has the highest CE at
  a particular RAC is the preferred strategy
• Within a range of RACs the risky alternative
  which has the highest CE line is preferred
• If the CE lines cross at that point the DM is
  indifferent between the two risky alternatives
• If the CE line goes negative, the DM would rather
  earn nothing than to invest in that alternative
• Interpret the rankings within risk aversion
  intervals
• RAC 0 is for risk neutral DMs
• RAC 1 or 1/W is for normal slightly risk
  aversion DMs
• RAC 2 or 2/W is for moderately risk averse DMs
• RAC 4 or 4/W is for extremely risk averse DMs

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Ranking Risky Alternatives

• Advanced materials provided as an appendix
• The following overheads are to good to trash but
  make the lecture to long
• They complement the data in Chapter 10

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Ranking Risky Alternatives

• Xrandom income simulated for Alter 1
• Yrandom income simulated for Alter 2
• Level of income realized for either is x or y
• If risk neutral, prefer Alter 1 if E(X) E(Y)
• In terms of utility theory, prefer Alter 1 iff
• E(U(X)) E(U(Y))
• Given that expected utility is calculated as
• E(U(X)) ? P(Xx) U(x) for all levels x
• where P(Xx) is probability income equals x

31
Ranking Risky Alternatives

• Each risky alternative has a unique CE once we
  have assumed a utility function or U(CE)
  E(U(X))
• Constant risk aversion (CRA) means that if we add
  1 to each outcome we do not change the ranking
• If a bet pays 10 or 0 with probability of 50
  it may have a CE of 4
• Then if a bet pays 11 or 0 with Probability of
  50 the CE is greater than 4
• CRA is a reasonable assumption and it allows us
  to demonstrate risk ranking

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Ranking Risky Alternatives

• A CRA simple utility function is the negative
  exponential function
• U(x) A - EXP(-x r)
• A is a constant to convert income to positives
• r is the ARAC or absolute risk aversion
  coefficient
• x is the realized income for the alternative
• EXP is the exponent function in Excel
• We can estimate the decision makers RAC by
  asking a series of questions regarding gambles

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Ranking Risky Alternatives

• Calculate Utility for a random return or income
  given a RAC
• U(x) A EXP(- (xscalar) r)
• Let A 1000 to scale all utility values to
  positive
• Can try different RAC values such as 0.001

Lecture 15
34
Alternative RACs
Lecture 15
35
Add or Subtract a Constant Amount
Lecture 15
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Ranking Risky Alternatives

• Three steps in Utility Analysis
• 1st convert the monetary payoffs to utility
  values using a utility function as U(X)
  A-EXP(-xr) and repeat this step for Y
• 2nd calculate the expected value of U(x) as
• E(U(X)) ? P(Xx) A-EXP(-xr)
• Repeat this step for Y
• 3rd convert the E(U(X)) and the E(U(Y))to a CE
• CE(X) CE(Y) means we prefer X to Y based on
  the DM ARAC of r and the utility function and the
  simulated Y and X values
• A short cut is to calculate CE directly for a
  decision makers RAC
• Simetar includes a function for calculating
• CE CERTEQ(risky income, RAC)

37
Ranking Risky Alternatives
Lecture 15