optimal%20portfolios%20DQ

Optimal20portfolios20dq

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Unformatted text preview: Session 7: A. Locating the ORP & OBP A portfolio of risky assets (denoted as P) contains two or more risky assets (denoted as i or j) with uncertain returns. A balanced portfolio (denoted as B) contains a risk free asset (denoted as rf) with certain return and a portfolio of risky assets. The expected return and risk of a portfolio of risky assets, E(RP) and σP, are expressed as: ⎞2 ⎛n n n n ⎟ ⎜ 22 E (RP ) = ∑ wi E (ri ) , σ P = ⎜ ∑ wi σ i + ∑∑ wi w j σ ij ⎟ i =1 j =1 i =1 ⎟ ⎜ i =1 i≠ j ⎠ ⎝ where wi and wj are the weights allocated to the respective risky assets, the total weight is one, E(ri) the expected return on risky asset i, n the total number of risky assets in the portfolio, σi the standard deviation of returns of risky asset i, and σij the covariance of returns between risky assets i and j. 1 For a balanced portfolio, the expected return and risk, E(RB) expressed as: and σB, are E (R B ) = y P E (R P ) + (1 − y P )r f , where 2 2 σ B = y P σ P + (1 − y P )2 σ r2 + 2 y P (1 − y P )σ P , r = y P σ P f f yP is the weight allocated to the portfolio of risky assets, B. When risk free lending and borrowing are not available, investors choose the portfolio of risky assets offering the highest utility. Among all the portfolio of risky assets lying on the efficient frontier, investors choose the one that touches their indifference curves tangentially. IC’’’ IC’’ IC’ ORPMRA P Expected return ORPLRA efficient frontier Optimal risky portfolio for a less risk averse investor with a smaller A and gently sloped indifference curve Optimal risky portfolio for a more risk averse investor with a larger A and steeper indifference curves Standard deviation of returns 1 Expected return CALORP CALORP(LRA) CALORP(MRA) B1 rf B2 efficient frontier ORPLRA - Optimal risky portfolio for the less risk averse investor in the absence of a risk-free asset ORP – Optimal risky portfolio for any investor in the presence of a risky free asset ORPMRA - Optimal risky portfolio for the more risk averse investor in the absence of a risk-free asset Standard deviation of returns When risk free lending and borrowing are available, investors...
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