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 CEs.xlsx• Lecture 21 Elicit Utility.xlsx• Lecture 21 Ranking Scenarios.xlsx• Lecture 21 Utility Function.xlsx• Lecture 21 Ranking Scenarios Whole

Farm.xlsx

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

(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 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 implicit CE for that particular gamble

– Their CE is based on their risk aversion level

Utility Based Risk Ranking Procedures

Utility

IncomeE($) =$5

Utility Function for Risk Averse Person

$10$0

CE($) Risk Averse DM

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

• Anderson and Dillon (1992) proposed a relative risk aversion (RRAC) definition of0.0 risk neutral0.5 hardly risk averse1.0 normal or somewhat risk averse2.0 moderately risk averse3.0 very risk averse4.0 extremely risk averse (4.01 is a

maximum)• Rule for setting RRAC and ARAC

range is:

Suggestions on Setting the RACs

Utility Function Lower RAC Upper RAC

Neg Exponential UtilityARAC

0 4/Wealth

Power UtilityRRAC

0 4.000001

• 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

Assuming a Utility Function for the DM

Estimate the DM’s RAC

• Calculate RAC

• Enter values in the cells that are Yellow

• Lecture 14 Elicit Utility .xls

• 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.

1. Stochastic Dominance

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)

0.0NPV for F and G

-----------------------------------------------------------------------------------------------------------------------------------------

F(x) is blue CDF

G(x) is red CDF

BA

• 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))

• Interpretation of a sample Stochastic Dominance result

Ranking Scenarios with SDRF in Simetar

• 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

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-150000 -100000 -50000 0 50000 100000 150000 200000

Pro

b

Alt 1 Alt 2 Alt 3 Alt 4 Alt 5

Ranking Risky Annual NCIs with SERF

Power Utility Function in 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 = 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

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)

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

Ranking Using Risk Premiums

• 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 Base 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.

Based on the Risk Premium, decision maker would pay to move from Base to Alt 4

Risk Premium decision maker must be paid to accept an inferior scenario

Neg. Exponential Utility Weighted Risk Premiums Relative to Alt 1

Alt 1

Alt 2

Alt 3

Alt 4

Alt 5

(20,000)

(15,000)

(10,000)

(5,000)

-

5,000

10,000

0.000000 0.000050 0.000100 0.000150 0.000200 0.000250 0.000300 0.000350

ARAC

Alt 1 Alt 2 Alt 3 Alt 4 Alt 5

• 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 “Stop Light chart”

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-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

Roy’s Safety First Rule

Roy’s Safety First RuleRoy's Probability

0.335556 =SUM(M9:U108)/900=IF(AND(B9<0,C9<0),1,0)

Iteration Roys 1-2 Roys 2-3 Roys 3-4 Roys 4-5 Roys 5-6 Roys 6-7 Roys 7-8 Roys 8-9 Roys 9-101 0.000 0.000 0.000 1.000 1.000 0.000 0.000 0.000 0.0002 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.0003 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.0004 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.0005 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.0006 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.0007 0.000 1.000 1.000 1.000 1.000 0.000 0.000 0.000 0.0008 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.0009 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

10 0.000 0.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000

• 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

Roy’s Safety First RuleStopLight Chart for Probabilities Less Than 0.000 and Greater Than 1.000

0.860.80

0.70 0.670.56 0.55 0.60 0.63 0.61

0.000.00

0.000.00

0.00 0.000.00

0.00 0.00

0.140.20

0.30 0.330.44 0.45 0.40 0.37 0.39

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Roys 1-2 Roys 2-3 Roys 3-4 Roys 4-5 Roys 5-6 Roys 6-7 Roys 7-8 Roys 8-9 Roys 9-10

• 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 not marginal probabilities and thus no yellow in the Stop Light

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!!!

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) asE(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 CECE(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