Expected Value

Expected Value The Expected Value of X equals the sum of each possible outcome of X multiplied by its probability The Xs must be: Mutually exclusive – you can’t get both heads and tails at the same time. Collectively exhaustive – you must account for all possible outcomes. Statistically independent – one X cannot affect the probability of another X The sum of all the probabilities must equal 1

Expected Value: What is the expected value of a single roll of a fair (unweighted) 6-sided die? 1+2+3+4+5+6 = 21/6= 3.5 Test yourself: What is the dies isn’t fair and is weighted to have a higher probability of landing on certain values? Make up weights and figure out the expected values. REMEMBER: weights need to sum to 1!

You make an initial investment of $10,000 Expected Value: Example You make an initial investment of $10,000 A: The market goes up, gain $1,000 = 20% B: The market stays the same, gain $0 = 50% C: The market goes down, loose $5,000 = 30% What is the Expected Value of your position? ($11,000 x .2) + ($10,000 x .5) + ($5,000 x .3) = $8,700 Test yourself: Is this a good investment? How much would you be willing to pay for this investment? What would the probabilities and/or pay-offs have to be to justify a $10,000 investment?

Key Tool: Decision Trees Nodes .2 .5 .3 =1 Branches $11,000 $10,000 $5,000 +$1,000 $0 -$5,000 ($11,000 x .2) + ($10,000 x .5) + ($5,000 x .3) = $8,700 Be careful to use values that answer the question asked! What is the expected gain or loss of your position? ($1,000 x .2) + ($0 x .5) + (-$5,000 x .3) = -$1,300

Expected Utility!