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lec6v1_1up - Stat 104 Quantitative Methods for Economists...

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Stat 104: Quantitative Methods for Economists Class 6: Portfolios (cont) 1

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Review: Correlation Summary • Scatter diagrams show relationships between variables •The covariance gives you the direction of a linear relationship between the two variables • The correlation coefficient measures the strength of a linear relationship 2 • Correlation ranges between -1 and 1 •Covariance can be any number •Both covariance and correlation measure association, not causation •They can be misleading if there are outliers or a nonlinear association
Review: Combining Data Sets s Covariance (correlation) shows up when we combine two data sets together. s That is, suppose X and Y are two data sets, oth of size both of size n . s Create Z = aX+bY for constants a and b. 3 ( ) Mean Z Z aX bY = = + 2 2 2 2 2 2 2 ( ) ( ) ( ) 2 ( , ) 2( ) Z X Y XY Var Z s a Var X b Var Y abCov X Y a s b s ab s = = + + = + +

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For the curious…………. s Suppose you have two data sets, X and Y (each of sample size n ), and want to create a new data set Z = aX+bY 4 1 1 1 1 1 1 ( ) n n n n i i i i i i i i i a b Z Z aX bY X Y aX bY n n n n = = = = = = + = + = +
For the curious (cont)…… s Suppose you have two data sets, x and y (each of sample size n ), and want to create a new data set Z = aX+bY 2 2 1 1 2 1 1 ( ) ( ) [( ) ( )] 1 1 1 ) ( )] n n i i i i i n i Var Z Z Z aX bY aX bY n n X aX bY bY = = = - = + - + - - - + - 5 1 2 2 1 2 2 2 2 1 1 1 [( 1 1 ( ) ( ) 2( )( ) 1 2 ( ) ( ) ( )( ) 1 1 1 i i n i i i i i n n n i i i i i i i aX n aX aX bY bY aX aX bY bY n a b ab X X Y Y X X Y Y n n n a = = = = = = - = - + - + - - - = - + - + - - - - - = 2 2 ( ) ( ) 2 ( , ) Var X b Var Y abCov X Y + +

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Numerical Example 6
Example (cont) s Create Z = .3X+.7Y 37.42 .3*39.8 .7*36.4 Z = = + 2 2 2 ) .3 (29.34 ) .7 (28.25 ) 2(.3)(.7)( 118.467) ar Z + + - 7 2 ( ) Var Z = 2 20.46 =

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Modern Portfolio Theory (MPT) s Finance professor Harry Markowitz began a revolution (in 1954!) by suggesting that the value of a security (stock) to an investor might best be evaluated by its mean, its standard deviation, and its correlation to other ecurities in the portfolio. 8 securities in the portfolio. s This audacious suggestion amounted to ignoring a lot of information about the firm -- its earnings, its dividend policy, its capital structure, its market, its competitors -- and calculating a few simple statistics. In this section, we give a basic introduction to what the finance guys refer to as modern portfolio theory.
The standard deviation is often used by investors to measure the risk of a stock or a stock portfolio. The basic idea is that the standard deviation is a measure of volatility: the more a stock's returns vary from the stock's average return, the more volatile the stock. Consider the following two stocks and their respective returns (in per

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lec6v1_1up - Stat 104 Quantitative Methods for Economists...

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