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3. Portfolio optimization with risk-free asset: Now assume that in addition to the risky assets described above with expected return vector u and positive definite

3. Portfolio optimization with risk-free asset: Now assume that in addition to the risky assets described above with expected return vector u and positive definite variance-covariance matrix E, there is a risk-free asset with deterministic return R >0. We can think of this asset as a one-year treasury bond, or alternatively as a certificate of deposit (CD) in a bank. Consider a portfolio of h ERN in the risky assets, and the remaining 1-h71 in the risk-free asset. You can verify that the expected return on this portfolio is R+h(u Ri), (8) and that the variance is (as before) hTh. (9) The case when h71> 1 corresponds to a situation when money is borrowed from the bank to invest in the market. The case when h1 < 0 corresponds to short-selling assets in the market and depositing the money in the bank. Thus, h E RN is completely unconstrained in this case with a risk-free asset. The risk averse investor's optimization problem when the risky asset is present is then U(h) = max R+h" ( R1) - h?sh, (10) 2 with no restrictions on h ERN. (a) Show that there is so-called one-fund separation in the market in this case, in that all investors, regardless of their 7, will hold the same stock market portfolio w 1 2-'(u - Ri), (11) B - AR and only differ in the amount of money they deposit in the bank. This is another classical result! (b) What is the relationship between an investor's 7 and the expected return (8) of the chosen portfolio (including the bond) in this case?

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