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Fi8000 Valuation of Financial Assets

Fi8000 Valuation of Financial Assets. Spring Semester 2010 Dr. Isabel Tkatch Assistant Professor of Finance. Investment Strategies. Lending vs. Borrowing (risk-free asset)

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Fi8000 Valuation of Financial Assets

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  1. Fi8000Valuation ofFinancial Assets Spring Semester 2010 Dr. Isabel Tkatch Assistant Professor of Finance

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

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

  16. 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.

  17. 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

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

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

  20. 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}

  21. 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.

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

  23. 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.

  24. 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

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

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

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

  28. The Separation Theorem The asset allocation process of the risk-averse investors can be separated into two stages: 1.Choose the optimal portfolio of risky assets m (The allocation between risky securities is identical for all the investors) 2.Choose the optimal allocation of funds between the risky portfolio m and the risk-free asset rf – choose a portfolio on the CML (The allocation between the risky portfolio and the risk free asset is personal and depends on the risk preferences of each investor)

  29. 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}

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

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

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

  33. 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%

  34. Example Continued

  35. Example Continued

  36. Practice Problems BKM 7th Ed. Ch. 7: 1-13, 17-22, 25-26 BKM 8th Ed. Ch. 7: 4-19 CFA: 4-6, 10-11 Mathematics of Portfolio Theory: Read and practice parts 11-13

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