1. Warm-up Problems N(2,4) is a normal random variable. What is E[3+N(2,4)]? Random variable X equals 0 with probability 0.4, 3 with probability 0.5, and -10 with probability 0.1. –What is E[X]? –What is E[X | X ≤ 1]? Let Z be the return of a stock. Then with 90% probability, log Z is normally distributed with mean 0.1 and standard deviation 0.15. However, 10% of the time, log Z is normally distributed with mean -0.2 and standard deviation 0.4. What is E[log Z]?

2. Previous Approach 1.List alternatives 2.For each alternative a)Describe cashflow stream b)Calculate NPV 3.Choose alternative with largest NPV

3. New Approach 1.List alternatives 2.For each alternative a)Describe average cashflow stream b)Calculate average NPV 3.Choose alternative with largest average NPV

4. New Approach 1.List alternatives 2.For each alternative a)List possible scenarios and their probabilities I.Describe cashflow stream II.Calculate NPV b)Calculate E[NPV] 3.Choose alternative with largest E[NPV]

5. Decision nodes (we choose) Chance nodes (stuff happens) Outcome nodes Decision Trees alternative 1 alternative 2 alternative 3 NPV= x scenario A scenario B scenario C papa pbpb pcpc

6. Oil Well Example An oil field has a 50% probability of being rich, in which case it will produce cashflows of $5 million per year for 15 years, starting one year after an oil well is drilled. The field has a 50% probability of being poor, in which case it will produce cashflows of $1 million per year for 15 years, starting one year after an oil well is drilled. Drilling a well costs $15 million. The discount rate is 10%. What should you do?

7. Solving Decision Trees Calculate value V at each node At outcome node: do NPV calculation At chance node: take expectation of value of scenarios V(node) = p a V(a) + p b V(b) + p c V(c) At decision node: –Pick value of largest alternative V(node) = max { V(1), V(2), V(3) } –Prune sub-optimal branches (rejected alternatives) alternative 1 alternative 2 alternative 3 scenario A scenario B scenario C papa pbpb pcpc

8. Oil Example Cont. Old Problem An oil field has a 50% probability of being rich, in which case it will produce cashflows of $5 million per year for 15 years, starting one year after an oil well is drilled. The field has a 50% probability of being poor, in which case it will produce cashflows of $1 million per year for 15 years, starting one year after an oil well is drilled. Drilling a well costs $15 million. The discount rate is 10%. What should you do? Extension If you spend $1 million testing the oil field, then after 1 year you will learn whether the oil field is rich or poor, and you can decide then whether or not to drill. What should you do?