Fi8000 Optimal Risky Portfolios

1. Fi8000OptimalRisky Portfolios Milind Shrikhande

2. Investment Strategies • Lending vs. Borrowing (risk-free asset) • Lending: a positive proportion is invested in the risk-free asset (cash outflow in the present: CF0 < 0, and cash inflow in the future: CF1 > 0) • Borrowing: a negative proportion is invested in the risk-free asset (cash inflow in the present: CF0 > 0, and cash outflow in the future: CF1 < 0)

3. Lending vs. Borrowing A A Lend B Borrow C rf rf

4. Investment Strategies • A Long vs. Short position in the risky asset • Long: A positive proportion is invested in the risky asset (cash outflow in the present: CF0 < 0, and cash inflow in the future: CF1 > 0) • Short: A negative proportion is invested in the risky asset (cash inflow in the present: CF0 > 0, and cash outflow in the future: CF1 < 0)

5. Long vs. Short E(R) Long A and Short B Long A and Long B A Short A and Long B B STD(R)

6. Investment Strategies • Passive risk reduction: The risk of the portfolio is reduced if we invest a larger proportion in the risk-free asset relative to the risky one • The perfect hedge: The risk of asset A is offset (can be reduced to zero) by forming a portfolio with a risky asset B, such that ρAB=(-1) • Diversification: The risk is reduced if we form a portfolio of at least two risky assets A and B, such that ρAB<(+1) The risk is reduced if we add more risky assets to our portfolio, such that ρij<(+1)

7. One Risky Fund and one Risk-free Asset: Passive Risk Reduction A A Reduction in portfolio risk B Increase of portfolio Risk C rf rf

8. Two Risky Assets with ρAB=(-1):The Perfect Hedge E(R) A Minimum Variance is zero Pmin B STD(R)

9. The Perfect Hedge – an Example What is the minimum variance portfolio if we assume that μA=10%; μB=5%; σA=12%; σB=6% andρAB=(-1)?

10. The Perfect Hedge – Continued What is the expected return μmin and the standard deviation of the return σmin of that portfolio?

11. Diversification: the Correlation Coefficient and the Frontier E(R) A ρAB=(-1) -1<ρAB<1 ρAB=+1 B STD(R)

12. Diversification: the Number of Risky assets and the Frontier E(R) A C B STD(R)

13. Diversification: the Number of Risky assets and the Frontier E(R) A C B STD(R)

14. Capital Allocation:n Risky Assets State all the possible investments – how many possible investments are there? Assuming you can use the Mean-Variance (M-V) rule, which investments are M-V efficient? Present your results in the μ-σ (mean – standard-deviation) plane.

15. The Expected Return and the Variance of the Return of the Portfolio wi = the proportion invested in the risky asset i (i=1,…n) p = the portfolio of n risky assets (wiinvested in asset i) Rp = the return of portfolio p μp= the expected return of portfolio p σ2p= the variance of the return of portfolio p

16. The Set of Possible Portfoliosin the μ-σ Plane E(R) The Frontier i STD(R)

17. The Set of Efficient Portfoliosin the μ-σ Plane E(R) The Efficient Frontier i STD(R)

18. Capital Allocation:n Risky Assets The investment opportunity set: {all the portfolios {w1, … wn} where Σwi=1} The Mean-Variance (M-V or μ-σ ) efficient investment set: {only portfolios on the efficient frontier}

19. The case of n Risky Assets:Finding a Portfolio on the Frontier Optimization: Find the minimum variance portfolio for a given expected return Constraints: A given expected return; The budget constraint.

20. The case of n Risky Assets:Finding a Portfolio on the Frontier

21. Capital Allocation: n Risky Assets and a Risk-free Asset State all the possible investments – how many possible investments are there? Assuming you can use the Mean-Variance (M-V) rule, which investments are M-V efficient? Present your results in the μ-σ (mean – standard-deviation) plane.

22. The Expected Return and the Variance of the Return of the Possible Portfolios wi = the proportion invested in the risky asset i (i=1,…n) p = the portfolio of n risky assets (wiinvested in asset i) Rp = the return of portfolio p μp= the expected return of portfolio p σ2p= the variance of the return of portfolio p

23. The Set of Possible Portfoliosin the μ-σ Plane (only n risky assets) E(R) The Frontier i STD(R)

24. The Set of Possible Portfoliosin the μ-σ Plane(risk free asset included) E(R) The Frontier i rf STD(R)

25. n Risky Assets and a Risk-free Asset: The Separation Theorem The process of finding the set of Mean-Variance efficient portfolios can be separated into two stages: 1. Find the Mean Variance efficient frontier for the risky assets 2. Find the Capital Allocation Line with the highest reward to risk ratio (slope) - CML

26. The Set of Efficient Portfoliosin the μ-σ Plane The Capital Market Line: μp= rf + [(μm-rf)/ σm]·σp μ m i rf σ

27. The Separation Theorem: Consequences The asset allocation process of the risk-averse investors can be separated into two stages: 1.Decide on the optimal portfolio of risky assets m (the stage of risky security selection is identical for all the investors) 2.Decide on the optimal allocation of funds between the risky portfolio m and the risk-free asset rf – choice of portfolio on the CML (the asset allocation stage is personal, and it depends on the risk preferences of the investor)

28. Capital Allocation: n Risky Assets and a Risk-free Asset The investment opportunity set: {all the portfolios {w0, w1, … wn} where Σwi=1} The Mean-Variance (M-V or μ-σ ) efficient investment set: {all the portfolios on the Capital Market Line - CML}

29. n Risky Assets and One Risk-free Asset: Finding a Portfolio on the Frontier Optimization: Find the minimum variance portfolio for a given expected return Constraints: A given expected return; The budget constraint.

30. n Risky Assets and One Risk-free Asset: Finding the Market Portfolio

31. n Risky Assets and One Risk-free Asset: Finding the Market Portfolio

32. A Numeric Example Find the market portfolio if there are only two risky assets, A and B, and a risk-free asset rf. μA=10%; μB=5%; σA=12%; σB=6%; ρAB=(-0.5) and rf=4%

33. Example Continued

34. Example Continued

35. Practice Problems BKM Ch. 8: 1-7, 11-14 Mathematics of Portfolio Theory: Read and practice parts 11-13.