Lecture 13-14

# Lecture 13-14 - From MVE Frontier to The Capital Asset...

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Lily Qiu, Assistant Professor Economics Department, Brown University EC1710, Lecture 13-14, Spring 2010, page 1 From MVE Frontier to The Capital Asset Pricing Model (CAPM) 1. Review of the MVE frontier and the mean-variance analysis (a) When we have one risky asset (S&P index) and one risk free asset: (b) When we have n risky assets only: The more assets we add, the more the MVE frontier extends in the north-westerly direction -> more diversification

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Lily Qiu, Assistant Professor Economics Department, Brown University EC1710, Lecture 13-14, Spring 2010, page 2 (c) one risk-free asset and multiple risky assets: The MVE frontier is the tangent line. Method 1:Using the Separation property --- the overall portfolio choice problem can be separated into two independent tasks: an investor first chooses an optimal risky portfolio consisting of the risky assets only, then his/her overally asset allocation problem is the same as in part (a), i.e., allocation of his/her wealth among one risk free asset and one risky asset (the optimal risky portfolio). Method 2: Solve the maximization problem directly: Max U = ) ~ ( p r E - A σ p 2 Let’s ssume that in the overall portfolio, portfolio weights for the two risky assets are ω 1 and ω 2 . ) ~ ( p r E = (1- ω 1 - ω 2 ) r f + ω 1 ) ~ ( 1 r E + ω 2 ) ~ ( 2 r E σ p 2 = ω 1 2 σ 1 2 + ω 2 2 σ 2 2 + 2 ω 1 ω 2 Cov 1,2 U = (1- ω 1 - ω 2 ) r f + ω 1 ) ~ ( 1 r E + ω 2 ) ~ ( 2 r E - A (ω 1 2 σ 1 2 + ω 2 2 σ 2 2 + 2 ω 1 ω 2 Cov 1,2 ) dU/d ω 1 = 0, dU/d ω 2 = 0 => ) ~ ( 1 r E - r f - A (2ω 1 σ 1 2 + 2 ω 2 Cov 1,2 ) = 0 ) ~ ( 2 r E - r f - A (2ω 2 σ 2 2 + 2 ω 1 Cov 1,2 ) = 0
Lily Qiu, Assistant Professor Economics Department, Brown University EC1710, Lecture 13-14, Spring 2010, page 3 Solve this system of equations => we’ll find ω 1 and ω 2. We have: Cov 1,p = n s p p s r E s r r E s r 1 1 1 )] ~ ( ) ( )][ ~ ( ) ( [ Pr = n s f f s r E r E r s r s r r r E s r 1 2 2 1 1 2 1 2 2 1 1 2 1 1 1 )]} ~ ( ) ~ ( ) 1 [( )] ( ) ( ) 1 )]{[( ~ ( ) ( [ Pr = n s s r E r E s r s r r E s r 1 2 2 1 1 2 2 1 1 1 1 )]} ~ ( ) ~ ( [ )] ( ) ( )]{[ ~ ( ) ( [ Pr = n s s r E s r r E s r r E s r 1 2 2 2 1 1 1 1 1 )]} ~ ( ) ( [ )] ~ ( ) ( [ )]{ ~ ( ) ( [ Pr = n s n s s s r E s r r E s r r E s r r E s r 1 1 2 2 1 1 2 1 1 1 1 1 } )] ~ ( ) ( )][ ~ ( ) ( [ Pr )] ~ ( ) ( )][ ~ ( ) ( [ Pr = ω 1 σ 1 2 + ω 2 Cov 1,2 Thus, the two equations are equivalent to: ) ~ ( 1 r E - r f = 2A Cov 1,p => A Cov r r E p f 2 ) ~ ( , 1 1 ) ~ ( 2 r E - r f = 2A Cov 2,p => A Cov r r E p f 2 ) ~ ( , 2 2 When we have n risky securities to choose from (in addition to one risk-free asset), we get n such equations. A

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## This note was uploaded on 07/21/2010 for the course ECON 1710 taught by Professor Qiu during the Spring '10 term at Brown.

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Lecture 13-14 - From MVE Frontier to The Capital Asset...

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