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Lecture 11-12

# Lecture 11-12 - Asset Allocation and the Mean-Variance...

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Lily Qiu, Assistant Professor Economics Department, Brown University EC1710, Lecture 11, Spring 2010, page 1 Asset Allocation and the Mean-Variance Efficient Frontier 1. a basic asset allocation decision Two investment choices available: 1) riskless one-year t-bill, which offers a certain annual rate of return of 2.5%; 2) the S&P 500 index, a risky asset, priced at 1100. It will go to 1250 in a year with 0.6 probability, and go to 1000 in a year with 0.4 probability. Mean(E) Standard Deviation(σ) T-bill % S&P500 % (0.6*(0.136-0.045) 2 + 0.4*(-0.091-0.045) 2 ) 0.5 = 0.111 Q1: Find the equation representing all risk-return combinations of all possible portfolios. Let’s assume the portfolio weight of the risky asset (S&P 500 index) is ω. ) ~ ( p r E = (1- ω) r f + ω ) ~ ( & p s r E σ p 2 = (1- ω) 2 σ f 2 + ω 2 σ s&p 2 + 2(1- ω) ω Cov(r f , r s&p ) = ω 2 σ s&p 2 => σ p = ω σ s&p Because: σ f 2 = σ f = 0; Cov(r f , r s&p ) = 0 so: ) ~ ( p r E = σ p 2 = To generalize: ) ~ ( p r E = (1- ω) r f + ω ) ~ ( & p s r E σ p = ω σ s&p These two equations give us a linear relationship: p p s f p s f p r r E r r E & & ) ~ ( ) ~ (

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Lily Qiu, Assistant Professor Economics Department, Brown University EC1710, Lecture 11, Spring 2010, page 2 Graphically: The straight line is the investment opportunity set , which is the set of all feasible expected return and standard deviation pairs of all porfolios resulting from different values of ω , the portfolio weight of the risky asset (S&P 500 index).
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Lecture 11-12 - Asset Allocation and the Mean-Variance...

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