1. Expected Value

2. 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

3. Expected Value:
What is the expected value of a single roll of a fair (unweighted) 6-sided die?
= 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!

4. 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?

5. 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

6. Expected Utility!