Decision Making for Risky Alternatives Lect. 14
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Watch an episode of “Deal or No Deal”
Read Chapter 10
Read Chapter 16 Section 11.0
Read Richardson and Outlaw article
Lecture 14 CEs.xls
Lecture 14 Elicit Utility.xls
Lecture 14 Ranking Scenarios.xls
Lecture 14 Ranking Scenarios Whole Farm.xls
Lecture 14 Utility Function.xls
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 – consistent with simulation
– Normative economics – role of economist is to
make recommendations – consistent with LP
Ranking Risky Alternatives
• Simulation results can be presented many
different ways to help the decision makers (DM)
make the best decision for themselves
–
–
–
–
–
Tables
PDFs and CDFs
StopLight charts
Fan graphs
SERF and SDRF
• Purpose of this lecture is to present the best
methods for ranking risky alternatives so the DM
can make the best decision
Decision Making for 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
Easy to Use Ranking Procedures
• Mean only – Pick scenario with the highest mean – ignores all risk
• Minimize Risk – Pick the scenario with lowest Std Dev – this ranking
strategy ignores the level of returns (mean and relative risk)
• Mean Variance – Always select the scenario in lower right quadrant
often difficult to read and often results in tied rankings, does not
work well for non-normal distributions.
- From diagram below A is preferred to C; E is preferred to B
- Indifferent between A and E
2
8
(risk)
B
6
E
C
4
2
A
D
0 1
2
(income)
3
4
X
Easy to Use Ranking Procedures
• Worst case – Bases decisions on scenario with highest minimum, but
it was observed with only a 1% chance. Worst case had a 1 out of 500
chance of being observed -- has merit in that it avoids catastrophic losses,
but ignores the level of returns and ignores upside risk.
• Best case –Looks at only one iteration, the best, which had < 1%
chance. Best case had a 1 out of 500 chance of being observed -ignores the overall risk and downside potential risk.
0
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 and does not
work when the mean is small.
CV = (Std Dev / Mean) * 100.0
Simetar Simulation Results for 100 Iterations. 10:54:17 AM 4/6/2002 (0.41 sec.). © 2002 Texas Agricultural E
Variable Sheet1!B25
Sheet1!C25
Sheet1!D25
Sheet1!E25
Sheet1!F25
Mean
5948.63
4894.94
3277.08
1642.44
-92.55
StDev
13012.63
14130.17
13615.85
13697.44
14419.89
CV
218.75
288.67
415.49
833.97
-15579.92
Min
-33653.55
-38159.43
-38331.16
-40112.24
-43798.94
Max
39572.48
41537.27
38627.14
37145.60
37376.81
Iteration
Option 1
Option 2
Option 3
Option 4
Option 5
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.
StopLight Chart for Probabilities Less Than 0.000 and
Greater Than 10,000.000
100%
90%
80%
0.40
0.36
0.32
0.25
0.20
70%
60%
50%
0.33
0.28
0.28
0.36
0.29
40%
30%
20%
0.36
0.39
0.42
0.44
0.32
Option 1
Option 2
Option 3
Option 4
Option 5
10%
0%
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 highest return at each
risk level or pick the distribution with the lowest risk for each
level of returns, i.e., the distribution furthest to the right.
Comparison of 5 CDF Series
1
0.9
0.8
0.7
Prob
0.6
0.5
0.4
0.3
0.2
0.1
0
-80000
-60000
-40000
-20000
Alt 1
Alt 2
0
Alt 3
Alt 4
20000
Alt 5
40000
60000
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, with a probability of 50%
– Risk neutral DM will sell the ticket for $5
– Risk averse DM will sell ticket for a “certain (nonrisky)” 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 sell the ticket
is DM’s “Certainty Equivalent” or CE
• Risk premium (RP) is the difference between the
CE and the expected value
– RP = E(Value) – CE
– RP = 5 – 4
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
– Offered a “certain payment” (say, $65,000) to 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
– Their CE is based on their risk aversion level
Utility Based Risk Ranking Procedures
Utility
Utility Function for
Risk Averse Person
$0
E($) =$5
$10
CE($) Risk Averse DM
Income
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 always prefer the risky alternative with
the greater 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 coefficient, and then calculate
the DM’s utility for a risky choice
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 DM’s risk aversion coefficient (RAC) as
it is the parameter for the Utility Function
Suggestions on Setting the RACs
• Anderson and Dillon (1992) proposed a
relative risk aversion (RRAC) definition of
0.0
0.5
1.0
2.0
3.0
4.0
risk neutral
hardly risk averse
normal or somewhat risk averse
moderately risk averse
very risk averse
extremely risk averse (4.01 is a maximum)
• Rule for setting RRAC and ARAC range is:
Utility Function
Lower RAC
Upper RAC
Neg Exponential Utility
ARAC
0
4/Wealth
Power Utility
RRAC
0
4.000001
Assuming a Utility Function for the DM
• Power utility function
– Use this function when assuming the DM
exhibits relative risk aversion RRAC
• DM willing to take on more risk as wealth increases
– Or when ranking risky scenarios with a KOV
that is calculated over multiple years, as:
• Net Present Value (NPV)
• Present Value of Ending Net Worth (PVENW)
Assuming a Utility Function for the DM
• Negative Exponential utility function
– Use this function when assuming DM exhibits
constant absolute risk aversion ARAC
• DM will not take on more risk as wealth increases
– Or when ranking risky scenarios using KOVs
for single year, such as:
• Annual net cash income or return on investments
• You get the same rankings if you use
correct the RACs
Estimate the DM’s RAC
• Calculate
RAC
• Enter
values in
the cells
that are
Yellow
• Lecture
14 Elicit
Utility
.xls
1. Stochastic Dominance
• Stochastic Dominance assumes
– Decision maker is an expected value maximizer
– Risky alternative distributions (F and G) are mutually
exclusive – These are two scenarios we simulated
– 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 stochastic dominance when CDFs
do not cross
– In this case we can say, “All decision makers prefer
distribution whose CDF is furthest to the right.”
• However, we are not always lucky enough to
have distributions that do not cross.
Stochastic Dominance wrt a Function
(SDRF) or Generalized Stoch. Dominance
• 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 DM’s ARAC.
1.0
-----------------------------------------------------------------------------------------------------------------------------------------
P(x)
F(x) is blue CDF
G(x) is red CDF
0.0
A
B
NPV for F and G
•
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))
Ranking Scenarios with SDRF in Simetar
• Interpretation of a sample Stochastic Dominance result
• For all decision makers with a RAC between -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
2. Rank Risky Scenarios Using CE
3. 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 for the DM’s 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
Ranking Scenarios with Stochastic Efficiency (SERF)
• SERF requires an assumption about the
decision makers’ utility function and like
SDRF uses a range of RAC’s
• SERF ranks risky strategies based on
expected utility which is expressed as CE
at the DM’s RAC level
• Simetar includes SERF and calculates a
table of CE’s over a range of RAC values
from the LRAC to the URAC and develops a
chart for ranking alternatives
Ranking Scenarios with SERF
• SERF results point 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
• SERF 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
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
Ranking Risky Annual NCIs with SERF
CDF of Annual Net Incomes for Five Risky Alternatives
1
0.9
0.8
0.7
Prob
0.6
0.5
0.4
0.3
0.2
0.1
0
-150000
-100000
-50000
Alt 1
0
Alt 2
50000
Alt 3
100000
Alt 4
150000
Alt 5
200000
Ranking Risky Annual NCIs with SERF
Ranking Risky Alternative NPVs with SERF
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 and find a BRAC
– 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
RAC
RAC
RAC
=
=
=
=
0
1
2
4
is for risk neutral DM’s
or 1/W is for normal slightly risk aversion DM’s
or 2/W is for moderately risk averse DM’s
or 4/W is for extremely risk averse DM’s
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 Chapter 10
Ranking Risky Alternatives
• X=random income simulated for Alter 1
• Y=random 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(X=x) * U(x) for all levels x
where P(X=x) is probability income equals x
Ranking Risky Alternatives
• Each risky alternative has a unique CE once we
assume a utility function or U(CE) = E(U(X))
• Constant absolute risk aversion (CARA) 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
• CARA is a reasonable assumption and it allows us
to demonstrate risk ranking
Ranking Risky Alternatives
• A CARA 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 maker’s RAC
by asking a series of questions regarding
gambles
Ranking Risky Alternatives
• Calculate Utility for a random return or income
given a RAC
• U(x) = A – EXP(- (x+scalar) * r)
– Let A = 1000 to scale all utility values to positive
– Can try different RAC values such as 0.001
Lecture 15
Alternative RACs
Lecture 15
Add or Subtract a Constant $ Amount
Lecture 15
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(-x*r) and repeat this step for Y
– 2nd calculate the expected value of U(x) as
E(U(X)) = ∑ P(X=x) * [A-EXP(-x*r)]
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)
Ranking Risky Alternatives
Lecture 15