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MeanVarianceDetails

# MeanVarianceDetails - Outline Types of Securities Return...

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Mean-Variance Optimization

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Outline Types of Securities Return Statistics and their Estimation Formulation of the Mean-Variance Problem Solution of the Mean-Variance Problem Inclusion of Risk-Free Borrowing/Lending The Mutual Fund Principle (Separation Theorem) and the Capital Market Line
Types of Securities Treasury Bills (T-bills) Bonds Government Bonds Corporate Bonds Stocks: large, medium, small caps, index (S&P 500, TSX 300) Risk-Free vs. Risky assets Risk-Free rate: r f Rate of return on T-Bills Returns on other assets are often expressed in terms of their spread above the risk-free rate: ) ( asset asset f f r r r r - + =

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Return Statistics: Expected Return  (Reward) For a discrete random variable with P[R=x i ]=p i For a continuous distribution with probability density function p(x): i i i R x p R E = = μ ] [ = = dx x xp R E R ) ( ] [ μ
Return Statistics: Variance and Standard  Deviation (Risk) - = - = i R i i R R x p R E 2 2 2 ) ( ] ) [( μ μ σ For a discrete random variable with P[R=x i ]=p i For a continuous distribution with probability density function p(x): The standard deviation σ R is the square root of the variance. It measures the variability of the returns. dx x p x R E R R R ) ( ) ( ] ) [( 2 2 2 - = - = μ μ σ

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Historical Estimates {r i , i=1,…,N} observed (historical) rate of return in period i. Empirical Estimates: = = - - = = N i R i R N i i R r N r N 1 2 2 1 ) ˆ ( 1 1 ˆ 1 ˆ μ σ μ
Covariance For two assets A and B with discrete returns distributions, with scenarios indexed by i, P[R A =x i and R B =y i ]=p i : If the returns are continuous random variables, with joint distribution f(x,y) - - = - - = = i B i A i i B B A A B A AB y x p R R E R R ) )( ( )] ( ) [( ) , cov( μ μ μ μ σ ∫ ∫ - - = - - = = dxdy y x f y x R R E R R B A B B A A B A AB ) , ( ) )( ( )] ( ) [( ) , cov( μ μ μ μ σ

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Correlation The correlation between R A and R B is: The correlation must satisfy: ρ AB measures the degree of linear dependence. ρ=1 implies that R A and R B satisfy a linear relationship. ρ=0 (uncorrelated returns) does not imply independence. b A AB AB σ σ σ ρ = 1 1 - AB ρ
Historical Estimation {r Ai , r Bi ,i=1,…,N} observed (historical) rates of return in period i. ) ˆ )( ˆ ( 1 1 ˆ 1 B Bi N i A Ai AB r r N μ μ σ - - - = =

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Return Statistics for Portfolios Now consider S securities with returns given by the random variables R j . Suppose that we invest the fraction w j of wealth in security j. The portfolio return is: = = S j j w 1 1 = = S j j j R w R 1
Return Statistics for the Portfolios Suppose that E[R j ]=μ j . Then the mean return of the portfolio with weights w j is: Let σ ij =cov(R i, ,R j ), and let  Σ be the SxS matrix with entries σ ij (called the variance-covariance matrix).

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MeanVarianceDetails - Outline Types of Securities Return...

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