Decision Making for Risky Alternatives Lect. 14 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 maker (DM) make the best decision for him/herself –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 (income) D A C E B (risk)

Easy to Use Ranking Procedures Worst case – 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 -- 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, with a 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

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.

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 further to the right.

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 –Host offers 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 CE for that particular gamble –Their CE is based on their risk aversion level

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 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: Suggestions on Setting the RACs Utility FunctionLower RACUpper RAC Neg Exponential Utility ARAC 04/Wealth Power Utility RRAC

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 does not take on more risk as wealth increases –Or when ranking risky scenarios using KOVs for single years, such as: Annual net cash income or return on investments Same results if you use correct 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 furthest to the right.” However, we are not always lucky enough to have distributions that do not cross. 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) 0.0 NPV 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 of 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

Rank Risky Scenarios Using CE

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 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

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 = 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

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 Lecture 15

Alternative RACs Lecture 15

Add or Subtract a Constant $ Amount Lecture 15

Ranking Risky Alternatives Three steps in Utility Analysis –1 st convert the monetary payoffs to utility values using a utility function as U(X) =A-EXP(-x*r) and repeat this step for Y –2 nd calculate the expected value of U(x) as E(U(X)) = ∑ P(X=x) * [A-EXP(-x*r)] Repeat this step for Y –3 rd 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