1. Decision Making for Risky Alternatives Lect. 21
Watch an episode of “Deal or No Deal”
Read Chapter 10
Read Chapter 16 Section 11.0
Read Richardson and Outlaw article
Lecture 21 Simetar_SERF Example.xlsx
Lecture 21 Utility Based Ranking.xlsx
Lecture 21 CEs.xlsx
Lecture 21 Elicit Utility.xlsx
Lecture 21 Ranking Scenarios.xlsx
Lecture 21 Utility Function.xlsx
Lecture 21 Ranking Scenarios Whole Farm.xlsx

2. Reminder Model Design Steps
NPV
BNW, EndNW, Annual
Cash Withdrawals
Receipts, Expenses, Interest & Principal Payments, Taxes, etc.
History for Stochastic Prices and Yields
Exogenous and Control Variables: PPI, CPI, Interest Rates, Inflation Rates
Design
Build

3. Utility Based Risk Ranking Procedures
Utility and risk are often stated as a lottery
Assume you own a lottery ticket that will pay $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 (non-risky)” payment less than $5, say $4
Risk loving DM will only sell the ticket 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

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, in this example)
If a contestant takes the Deal, then the “Certain Payment” offer exceeded their implicit CE for that particular gamble
Their CE is based on their risk aversion level

5. Utility Based Risk Ranking Procedures
Black Utility Function is for Normal Risk Averse Person
Red line is for a Risk Lover
Maroon line is for a very risk averse person
Utility
E($) =$5 $0
$10
Income
CE($) Risk Averse DM

6. 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

7. 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 four procedures require estimating the DM’s risk aversion coefficient (RAC) as it is the parameter for the Utility Function

8. Suggestions on Setting the RACs
Anderson and Dillon (1992) proposed a relative risk aversion (RRAC) schedule 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:
Utility Function
Lower RAC
Upper RAC
Neg Exponential Utility
ARAC
4/Wealth
Power Utility
RRAC

9. 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
Poor person buys a $1 lottery ticket
A rich person buys 1,000 $1 lottery tickets
Both feel that same amount of risk relative to wealth
Use 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)

10. 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
Poor person buys one $1 lottery ticket
A rich person buys one $1 lottery ticket
Both feel that same amount of risk relative to wealth
Use when ranking risky scenarios using KOVs for single year, such as:
Annual net cash income or return on investments
You get the same rankings with Power and Negative Exponential utility functions, if you use correct the RACs

11. Estimate the Your Risk Aversion Coefficient (RAC)
Calculate RAC
Enter values in the cells that are Yellow
Lecture 21 Elicit Utility .xlsx

12. 1. Stochastic Dominance Stochastic Dominance assumes
Decision maker is an expected value maximizer
Risky alternative distributions (F and G) are mutually exclusive – (They are two scenarios we simulated.)
Distributions F(y) and G(y) are based on population probability distributions. In simulation, they 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.

13. First Degree Stochastic Dominance
Distribution to the right has higher income at every probability level so it is preferred by all DMs.

14. Second Degree Stochastic Dominance
SDRF measures the difference between two risky distributions, F and G, at each value on the Y axis
No utility function is used
P(x)
0.0
NPV for F and G
F(x) is blue CDF
G(x) is red CDF B A
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 keep track of the net sum of the differences
The net sum of the differences in the end determines which alternative dominates.

15. Stochastic Dominance wrt a Function (SDRF) or Generalized Stoch
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 the differences by a utility function using the DM’s ARAC. Assumes a Negative Exponential Utility function in Simetar and Jack Meyer’s original article.
P(x)
0.0
NPV for F and G
F(x) is blue CDF
G(x) is red CDF B A
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))

16. Ranking Scenarios with Stochastic Dominance in Simetar

17. First and Second Degree Stochastic Dominance
In the SDRF worksheet you will find these tables
First table is blank because CDFs cross and no clear winner
Second table indicates that Alt 1, Alt 2, and Alt 5 are second degree dominate over Alt 3; and Alt 5 is second degree dominate over Alt 1

18. Generalized Stochastic Dominance Ranking (SDRF)
Interpretation of a sample Stochastic Dominance result
For all decision makers with a RAC between to 0.1:
The preferred scenarios is Option 2 – the efficient set
Note that Stochastic Dominance resulted in a split decision for the second place rankings
The Lower RAC says Alt 1 is second most preferred while the Risk averse DM prefers Alt 5 second
This is a problem with SDRF that is remedied by SERF

19. 2. Certainty Equivalent (CE)
Compare CE of all risky alternatives at each RAC level
Simetar function for CE is
=CERTEQ(range of data, RAC)
Assumes a Negative Exponential Utility Function, but has options for alternative utility functions
Pick the scenario with the largest CE

20. Two Variable Example of CE

21. 3. Stochastic Efficiency (SERF)
Stochastic Efficiency with Respect to a Function (SERF) calculates the certainty equivalent for risky alternatives at 25 different RAC levels between the min and max RACs
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
Ranks all scenarios simultaneously
Efficient Set is utility shorthand for saying the risky alternative(s) that is (are) the most preferred

22. 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

23. Ranking Scenarios with SERF
SERF results point out why SDRF often produces inconsistent rankings
SDRF only uses the minimum and maximum RACs
The efficient set (ranking) generally differs from the minimum 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 where the CE lines cross (this is the BRAC or breakeven risk aversion coefficient)
The SERF Table is best understood as a chart

24. SERF Ranking of 5 Alternatives Used Earlier

25. 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 = 0 is for risk neutral DM’s
RAC = 1 or 1/W is for normal slightly risk aversion DM’s
RAC = 2 or 2/W is for moderately risk averse DM’s
RAC = 4 or 4/W is for extremely risk averse DM’s

26. 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 = /Wealth
The second example is for ranking a multiple year decision using NPV variable
Uses Power Utility function
Lower RRAC = zero
Upper RRAC =

27. Ranking Risky Annual NCIs with SERF

28. Power Utility Function Ranking NPV in SERF

29. 4. Ranking Using Risk Premiums
Risk Premium (RP) - calculate the risk premium between each of the scenarios and a base scenario.
Risk Premium equals difference between the CE’s for the risky scenarios:
RPG to F = CEG – CEF
Rank the risky scenarios based on the RPs
Advantage is that the full distribution (F(x) and G(x)) of values for the KOV are compared to each other, based on the decision maker’s RAC, i.e., their utility function
A wide range of RACs can be tested to allow for a wider range of decision makers given an assumed utility function
Base scenario should be the current situation or the scenario picked best by stochastic efficiency (SERF)

30. Ranking Using Risk Premiums Table
The RP Table is calculated like the SERF Table using the same range of 25 RACs
The user specifies the base scenario; Option 1 was selected for this example
Select the scenario that has highest risk premium for the RAC which best defines the decision maker

31. Ranking Using Risk Premiums
Risk Premium decision maker must be paid to accept an inferior scenario
Based on the Risk Premium, decision maker would pay to move from Base to Alt 4
Risk premiums are presented relative to a base scenario, Alt 1, above
Alt 4 is preferred for all risk averse decision makers.
Distance between Red line and Alt 1 line, $18,347; is how much a risk averse decision maker would pay to move from Alt 2 to Alt 1.
Risk averse decision makers prefer Alt 4 to Alt 1 and would pay about $8,000 to gain Alt 4 over Alt 1.

32. 5. StopLight Chart
Not utility based, but risk averse DMs focus on the Red probabilities

33. 6. Roy’s Safety First Rule
Roy (Econometrica, 1952)
Select the strategy which minimizes the chance of falling below a critical level of net cash income
Rank risky alternatives based on the scenario with the smallest probability of low net cash incomes
This is essentially a two light “StopLight chart” or a Stop and Go Light

34. Roy’s Safety First Rule
A Roy’s Safety First Rule presented as the probability of NCIi < target each year i
With Roy’s Rule, can calculate the probability of a “low” net cash income for two or more consecutive years, as:
=IF(AND(NCI1<0, NCI2<0),1,0)
=IF(AND(NCI2<0, NCI3<0),1,0)
=IF(AND(NCI3<0, NCI4<0),1,0)
=IF(AND(NCI4<0, NCI5<0),1,0)
Repeat the =IF(AND()) statement for all years 2 to T and summarize the counter variables for all iterations
Roy’s probability is sum for all of the =IF(AND()) values divided by [(No. Years – 1)*No. Iterations]
If 10 years and 500 iterations the denominator is 4,500 representing all possible sample observations that could be 1

35. Roy’s Safety First Rule
The scenario was simulated 100 iterations
Net cash income is for 10 years
Roy’s values are for 2 consecutive years with negative NCI
Roy’s Probability is the sum of the =IF(AND()) counter variables divided by 900, which is = 9 * no. iterations

36. Roy’s Safety First Rule
The Stop Light displays the probabilities of having two years of negative NCI in a row, years 1 & 2 or Years 3 & 4, etc.
Chart developed from the data in the previous overhead, over all 100 iterations
The counter variables can be 0 or 1 so no marginal probabilities and thus no yellow in the Stop Light

37. Additional Ranking Risky Considerations
Advanced materials provided as an appendix
The following overheads are to good to trash but make the lecture to long
They complement Chapter 10
READ CHAPTER 10!!!

38. 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

39. 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

40. 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

41. 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

42. Alternative RACs
Lecture 15

43. Add or Subtract a Constant $ Amount
Lecture 15

44. 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)

45. Ranking Risky Alternatives
Lecture 15