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Warm-up Problems

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]?

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Warm-up Problems

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  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 • List alternatives • For each alternative • Describe cashflow stream • Calculate NPV • Choose alternative with largest NPV

  3. New Approach • List alternatives • For each alternative • Describe average cashflow stream • Calculate average NPV • Choose alternative with largest average NPV

  4. New Approach • List alternatives • For each alternative • List possible scenarios and their probabilities • Describe cashflow stream • Calculate NPV • Calculate E[NPV] • Choose alternative with largest E[NPV]

  5. alternative 1 alternative 2 alternative 3 scenario A pa pb scenario B pc scenario C NPV= x Decision Trees • Decision nodes(we choose) • Chance nodes(stuff happens) • Outcome nodes

  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. scenario A pa pb scenario B pc scenario C alternative 1 alternative 2 alternative 3 Solving Decision Trees • Calculate value V at each node • At outcome node: do NPV calculation • At chance node: take expectation of value of scenariosV(node) = pa V(a) + pb V(b) + pc V(c) • At decision node: • Pick value of largest alternativeV(node) = max { V(1), V(2), V(3) } • Prune sub-optimal branches (rejected alternatives)

  8. 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? Oil Example Cont.

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