8. Uncertainty. Reading: BGVW, Chapter 7

Transcription

1 8. Uncertainty Reading: BGVW, Chapter 7

2 1. Introduction Uncertainties abound future: incomes/prices/populations analysis: dose-response/valuation/climate/effects of regulation on environmental quality/longevity of a piece of capital equipment/markets Incorporate uncertainty into the analysis using contingencies probabilities

3 Outline Expected Value Analysis Decision Analysis Sensitivity Analyses Partial sensitivity Best Case - Worst Case Monte Carlo Analyses Value of Information

4 2. Expected Value Analysis Asteroid Defense Example EV (Forward Base) = ($5, ) + ($1, ) + ($60.995) = 69 billion EV (Near-Earth Base) = ($10, ) + ($2, ) + ($60.995) = 38 billion ** EV (No Base) = ($30, ) + ($6, ) + ($0.995) = 54 billion

5 Weighted average over outcomes General rule over n contingencies or n possible states of the world: EV ( NB) = p ( B C ) p ( B C ) EV ( NB) = p ( NB ) p ( NB ) 1 1 n n n n n Stockpile of oil ~ major disruption, minor disruption, moderate / slack mkt, tight mkt.

6 Risk preferences Consider two choices A and B A: Certain payoff of $200 B: 50% chance of winning $1000, 50% chance of loosing $500 à Uncertain outcome w/ expected payoff of $250 If risks do not matter then B is preferred to A since $250 < $200 If risk matters a person might rightly prefer A to B if they do not like the risk Especially true at extremes. Think about B : 99% chance of winning $1000 and 1% chance of loosing $30,000 à expected payout $690

7 Risk preferences defined Risk Neutral Preferences à Use EV calculations, B is preferred Risk Adverse Preferences à Risky outcomes get discounted so A might be preferred to B Risk Loving Preferences à Risky outcomes get a premium so B might be preferred to A in case where its expected cost is the same or higher (say certain outcome is $300)

8 Certainty equivalents Consider the following lottery.1 $10, $0 Expected value is $1000 What is the certain outcome that a risk adverse person would accept in exchange for this lottery? $999? $950? 750? Whatever that value is is said to be the certainty equivalent (CE) and it embodies risk preferences. What about a risk loving person? >$1000. The CE in principle can be substituted for expected values in a decision tree

9 Risks and Government Projects Consider two projects A and B A: Certain cost of $800 million B: 50% chance of $1000 million, 50% chance of $500 million à Uncertain outcome w/ expected cost of $750 million If population is risk adverse might want to go with A If, however, costs are spread over 300 million people, implications per person are quite small. Spreading risks Generally ignore risks unless Concentrated Massive in scope

10 3. Decision Analysis More flexible framework for working with EVA Lets NB or probabilities in some periods depend on what happen in past periods, allows for many decisions overtime Simple Decision Tree (Fund raising event indoors or outdoors)

11 Define Decision nodes Chance nodes Probabilities Payoffs Gates

12 Vaccination Program

13 Vaccination Program E( C ) = C + C + PC + (1 P ) P C /(1 + d) v a s 1 e v 1 2 e v E( C ) = PC + (1 P ) P C /(1 + d) nv 1 e nv 1 2 e nv E( C ) vs E( C ) v nv E( NB) = E( C ) - E( C ) nv Choose smallest one. v

14 Read Tree Nodes/Probabilities/Payoffs Time periods Solve Tree: Folding back Calculate expected values at each probability node Choose best out come at each decision node Do all of this working backward Look at Payoff boxes in Fig 7.2 Express it E( C ) = C + C + PC + (1 P ) P C /(1 + d) v a s 1 e v 1 2 e v E( C ) = PC + (1 P ) P C /(1 + d) nv 1 e nv 1 2 e nv E( C ) vs E( C ) v nv E( NB) = E( C ) - E( C ) nv Choose smallest one. v

15 4. Sensitivity Analysis Example with Numbers

16

17 Partial Sensitivity Analysis Sensitivity is over probability of an outbreak

18 Best Case/Worst Case Base case take most plausible outcomes Best case: set all probabilities and parameters to favor no vaccination (If outcome still favors vaccine, you re on strong footing to proceed) Worst case: set all probabilities and parameters to favor vaccination (If outcome still favors no vaccine, youre on strong footing to take no action) Ranges as well

19 Monte Carlo Simulate different possible outcomes and report a distribution of Net Benefits Two Steps Specify probabilities for the all uncertain quantitative assumptions Take random draws from these distributions, can be in the hundreds or thousands

20 5. Value of Information Sometimes delaying a decision to gather information may make sense or at least is a option worth considering Metropole example

21 Suppose there is a test that can be done to learn about the system. Cost of the test is T. New decision tree: Value of the test (information) is = 1.2

22 6. Quasi Option Value NOT COVERED The expected value of information gained by delaying an irreversible decision. Development example Exogenous learning Endogenous learning

23

24