03-Indexes and Expectation

# When we dont know anything about the pdf but rather

• Notes
• shi1hong
• 25

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When we don’t know anything about the PDF, but rather, observe a sample generated by the PDF, we can estimate expected value as a simple average This works for both discrete and continuous PDFs = = n i i i r P r r E 1 ) ( ] [ = = n i i r P r g r g E 1 ) ( ) ( )] ( [

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Excel Random Number Generation Data – Data Analysis – Random Number Generation “Number of Variables” = number of samples to create “Number of Random Numbers” = sample size Choose “Discrete” for the distribution “Value and Probability Input Range” Here is where you specify the outcomes, and the probability of each outcome. Example on next slide generates 1 sample of 1000 outcomes from the PDF below, and puts them in column “D”.
Example

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Statistics Rule #1 Rule 1: If x and y are random variables and a and b are constants, then See equation 4-13 in CFA reading, p.153 bE[y] aE[x] by] E[ax + = +
Statistics Rule #1: Example Portfolio z: 30% A, 70% B Note the return for portfolio z is of the form we can therefore use stat rule #1 = = = = = ) (recession 3 and % 0 for 20 . 0 ) (expansion 5 and % 10 for 80 . 0 ) , ( % r r % r r r r P B A B A B A B A z r r r 7 . 3 . + = variable) random constant ( variable) random constant ( return × + × = % 62 . 5 046 . 7 . 08 . 3 . ] [ 7 . ] [ 3 . ] [ = × + × = × + × = B A z r E r E r E

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Statistics Rule #1: Example Let’s see why stat rule #1 works. Note that the PDF for the return on portfolio z is Expected return for portfolio z: E[r]=.8(.065)+.2(.021) = 5.62% = × + × = = × + × = = 2.1 (3%) .7 (0) .3 for .20 6.5% (5%) .7 (10%) .3 for .80 ) ( z z z r r r P
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• Fall '10
• BrianBoyer
• Variance, Probability theory, probability density function, Cumulative distribution function

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