H8 - Note on Covariance and Correlation William L. Silber...

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Note on Covariance and Correlation William L. Silber and Jessica Wachter 1. Theoretical Definitions The covariance of two random variables, R 1 and R 2 , is defined as: [ ] ) )( ( ) , ( 2 2 1 1 2 1 m m - - = R R E R R Cov The covariance can be calculated as follows, where the ij p are the elements of a joint probability distribution: ( 29 ( 29( 29 ij j i j i p R R R R Cov 2 2 1 1 2 1 , m m - - = In words, the covariance is calculated by summing the product of the paired deviations of each observation from its respective mean, with each pair multiplied by its probability. Correlation, r , is defined as: 2 1 ) , ( 2 1 R R R R Cov s s r = 2. Numerical Examples Suppose one of three things will happen next year. There will either be a recession, a boom, or things will continue as normal. Each of these scenarios (or observations) is equally likely. That is, each happens with probability 1/3 (these are the ij p ). (a) First, let’s compare a fund that invests in U.S. stocks to a fund that invests in U.S. bonds. Based on Table I, and using the 1/3 probability associated with each
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scenario, we calculate a mean return on the stock fund of .11.
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This note was uploaded on 02/04/2010 for the course ECON 106v taught by Professor Miyakawa during the Spring '08 term at UCLA.

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H8 - Note on Covariance and Correlation William L. Silber...

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