Lecture 13-14

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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

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

Page1 / 11

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online