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Optimal risky portfolio

Optimal risky portfolio. Chapter 6. Optimal Risky Portfolio. The capital allocation decision revisited: portfolio of two risky assets Extension to the N-asset case Applications and practical issues. Idea of Diversification. Notations. P: risky portfolio F: risk-free asset

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Optimal risky portfolio

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  1. Optimal risky portfolio Chapter 6

  2. Optimal Risky Portfolio • The capital allocation decision revisited: portfolio of two risky assets • Extension to the N-asset case • Applications and practical issues

  3. Idea of Diversification

  4. Notations • P: risky portfolio • F: risk-free asset • y: weight allocated to P • 1-y: weight allocated to F • C: combined portfolio (P and F together) • wi: weight allocated to risky asset i in P • All the wi’s must sum up to one

  5. Portfolio of Two Risky Assets Portfolio expected return Portfolio Variance 12 = variance of asset 1’s returns 22 = variance of asset 2’s returns Cov(r1,r2) = covariance of returns of asset 1 and asset 2

  6. Covariance and Correlation Coefficient • Relationshipbetween the two: = Correlation coefficient of returns between assets 1 and 2 = Standard deviation of asset 1’s returns = Standard deviation of asset 2’s returns

  7. Correlation Coefficient • Range of values for : • If = +1, the asset returns are perfectly positively correlated • If= -1, the asset returns are perfectly negatively correlated

  8. Re-writing the variance • Re-write as: • Question: what if  1,2 = 1.0?

  9. Example • Consider two mutual funds: • What does the line that links all the combinations of D and E look like?  Depends on D,E

  10. Combinations of D and E

  11. Portfolio of Two Risky Assets: Correlation Effects • The smaller the correlation, the greater the potential in risk reduction • If= +1.0, there is no reduction in risk, in the sense the standard deviation of the portfolio returns (P) is just a linear combination of D and E, i.e., no diversification benefit • Risk reduction: horizontal move toward the y axis

  12. Minimum Variance Portfolios

  13. Vary the Weights to Generate Portfolios

  14. Minimum Variance Portfolio • Solving the minimization problem we get: • Using our example, with  = 0.3: • wD= 0.82, and therefore wE = 0.18

  15. Risk and Return of the Min. Var. Portfolio • Using the weights wD and wE, we can determine the risk-return characteristics of the minimum variance portfolio:

  16. Extending to include a risk-free asset • Now bring back the risk-free asset, and include it in the choice of assets • What happens when you combine a different risky portfolio with the risk-free asset?

  17. Alternative CALs

  18. Optimal Risky Portfolio

  19. When the risk-free asset is included… • The opportunity set is again described by the (linear) CAL • The optimal risky portfolio is obtained when the slope of the CAL is maximized (i.e., when the Sharpe Ratio is maximized) • The choice of the optimal combined portfolio depends on the client’s risk attitude

  20. Now bring in the Client D

  21. Asset Allocation/Mix

  22. The Markowitz Portfolio Selection model • Extending the concepts to n risky assets • Many possible combinations/portfolios of risky assets • Focus on portfolios that have the highest expected return for a given level of risk • Portfolios that satisfy this optimal trade-off lie on the efficient frontier • These portfolios are dominant in a mean-variance sense

  23. Extending to n Risky Securities

  24. And if there is a Risk-free Security

  25. Main difference between 2 risky assets and n risky assets • Two risky assets • The optimal risky portfolio is a portfolio on the curve linking all possible combinations of the two assets • n risky assets • Different possible combinations form an area, rather than a curve • Focus on the efficient frontier to determine the optimal risky portfolio

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