Suppose investors can invest in two risky assets deriving returns R, R2 and in a riskless...

  

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Suppose investors can invest in two risky assets deriving returns R₁, R2 and in a riskless asset deriving a return Rf. Consider a portfolio P of risky assets with return Rp=wR₁+(1-w) R₂ where w is a real number. (a) Express E(Rp) and Var(Rp) in terms of w, μ₁ = E(R₁), 2= E(R₂), of = Var (R₁), 02= Var (R₂) and 12 = Cov(R₁, R₂). (b) Explain why one has to maximize the function g(w): = (₁-R₁)w+(₂-R₁)(1-w) √ow² +0(1-w)² +20₁2w(1 − w) - in order to obtain the efficient frontier of investment strategies when short selling is allowed and the risk-free lending and borrowing rates are both equal to Rf. (c) Let c = ₁ R, for i=1,2. Show that when g'(w) = w=w* = C102-C₂012 c₁0+0₂07-C1012 - 02012 and when c₁0+ c₂07-C1012-C2012 > 0, the portfolio (w*, 1-w*) of risky assets is on the efficient frontier of investment strategies. [Hint: You may use the fact that the derivative of g(w) with respect to w is without proof.] (-₁0-₂0+9₁012 + C₂012)w + (c₁02 - C₂012) [ow² +0(1-w)² + 2012w(1-w)1³/2 Suppose investors can invest in two risky assets deriving returns R₁, R2 and in a riskless asset deriving a return Rf. Consider a portfolio P of risky assets with return Rp=wR₁+(1-w) R₂ where w is a real number. (a) Express E(Rp) and Var(Rp) in terms of w, μ₁ = E(R₁), 2= E(R₂), of = Var (R₁), 02= Var (R₂) and 12 = Cov(R₁, R₂). (b) Explain why one has to maximize the function g(w): = (₁-R₁)w+(₂-R₁)(1-w) √ow² +0(1-w)² +20₁2w(1 − w) - in order to obtain the efficient frontier of investment strategies when short selling is allowed and the risk-free lending and borrowing rates are both equal to Rf. (c) Let c = ₁ R, for i=1,2. Show that when g'(w) = w=w* = C102-C₂012 c₁0+0₂07-C1012 - 02012 and when c₁0+ c₂07-C1012-C2012 > 0, the portfolio (w*, 1-w*) of risky assets is on the efficient frontier of investment strategies. [Hint: You may use the fact that the derivative of g(w) with respect to w is without proof.] (-₁0-₂0+9₁012 + C₂012)w + (c₁02 - C₂012) [ow² +0(1-w)² + 2012w(1-w)1³/2

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a Expressions for ERp and VarRp in terms of w ER1 ER2 VarR1 VarR2 and CovR1 R2 Expected return of portfolio ... View the full answer