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Fi8000 Capital Allocation and Efficient Portfolios

Fi8000 Capital Allocation and Efficient Portfolios. Milind Shrikhande. Today. Portfolio Theory The Mean-Variance Criterion The Normal Distribution Capital Allocation The Mathematics of Portfolio Theory. The Mean-Variance Criterion (M-V or μ - σ criterion).

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Fi8000 Capital Allocation and Efficient Portfolios

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  1. Fi8000Capital Allocation andEfficient Portfolios Milind Shrikhande

  2. Today • Portfolio Theory • The Mean-Variance Criterion • The Normal Distribution • Capital Allocation • The Mathematics of Portfolio Theory

  3. The Mean-Variance Criterion(M-V or μ-σ criterion) Let A and B be two (risky) assets. All risk-averse investors prefer asset A to B if { μA ≥ μB and σA < σB } or if { μA > μB and σA ≤ σB } * Note that these rules apply only when we assume that the distribution of returns is normal.

  4. The Mean-Variance Criterion(M-V or μ-σ criterion) ☺ E(R) = μR ☺ STD(R) = σR

  5. The Normal Distribution of Returns Pr(R) 68% 95% μ- 2σ μ μ+σ μ+2σ μ- σ R

  6. The Normal Distribution of Returns Pr(Return) σR: Risk μR: Reward 0 R=Return

  7. The Normal DistributionHigher Reward (Expected Return) Pr(Return) μB μA < R=Return

  8. The Normal DistributionLower Risk (Standard Deviation) Pr(Return) A σA < σB B μA= μB R=Return

  9. Capital Allocation - Outline • n mutually exclusive assets (i.e. one can only invest in one asset but not in a portfolio) • One risky asset and one risk-free asset • n risky assets and one risk-free asset (the risky investments are mutually exclusive) • Two risky assets • n risky assets • n risky assets and one risk-free asset

  10. Capital Allocation - Data There are three (risky) assets and one risk-free asset in the market. The risk-free rate is rf = 1%, and the distribution of the returns of the risky assets is normal with the following parameters

  11. Capital Allocation: n mutually exclusive assets State all the possible investments. Assuming you can use the Mean-Variance (M-V) rule, which investments are M-V efficient (i.e. which assets can not be thrown out of the set of desirable investments by a risk-averse investor who uses the M-V rule)? Present your results on the μ-σ (mean – standard-deviation) plane.

  12. The Expected Return andthe STD of the Return (μ-σ plane) A B C rf

  13. The Mean-Variance Criterion(M-V or μ-σ criterion) ☺ E(R) ☺ STD(R)

  14. Capital Allocation: n mutually exclusive assets The investment opportunity set: {rf, A, B, C} The Mean-Variance (M-V or μ-σ ) efficient investment set: {rf, A, C} Note that investment B is not in the efficient set since investment A dominates it (one dominant investment is enough).

  15. Capital Allocation:One Risky Asset (A) and One 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 on the μ-σ (mean – standard-deviation) plane.

  16. The Expected Return and STD of the Return of the Portfolio α = the proportion invested in the risky asset A p = the portfolio with αinvested in the risky asset A and (1- α) invested in the risk-free asset rf Rp = the return of portfolio p μp= the expected return of portfolio p σ p= the standard deviation of the return of portfolio p Rp = α·RA + (1-α)·rf μp = E[ α·RA + (1-α)·rf ] = α·μA + (1-α)·rf σ2p = V[ α·RA + (1-α)·rf ] = (α·σA)2 Or σp = α·σA

  17. Capital Allocation:One Risky Asset and One Risk-free Asset The investment opportunity set: {all portfolios with proportion α invested in A and (1-α) invested in the risk-free asset rf} The Mean-Variance (M-V or μ-σ ) efficient investment set: {all the portfolios in the opportunity set}

  18. The Capital Allocation Line

  19. The Expected Return andthe STD of the Return (μ-σ plane) A A B C rf rf

  20. The Capital Allocation Line (CAL):Four Basic Investment Strategies A P2 B A P1 C rf rf

  21. Portfolios on the CAL

  22. Capital Allocation: n Mutually Exclusive Risky Asset and One 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 on the μ-σ (mean – standard-deviation) plane.

  23. The Expected Return andthe STD of the Return (μ-σ plane) A B C rf

  24. Capital Allocation:One Risky Asset and One Risk-free Asset The investment opportunity set: {all the portfolios with proportion α invested in the risky asset j and (1-α) invested in the risk-free asset, (j = A or B or C)} The Mean-Variance (M-V or μ-σ ) efficient investment set: {all the portfolios with proportion α invested in the risky asset A and (1-α) invested in the risk-free asset – (why A?)}

  25. Capital Allocation:Two 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 on the μ-σ (mean – standard-deviation) plane.

  26. The Expected Return and STD of the Return of the Portfolio wA = the proportion invested in the risky asset A wB = (1-wA) = the proportion invested in the risky asset B p = the portfolio with wAinvested in the risky asset A and (1-wA) invested in the risky asset B Rp = the return of portfolio p μp= the expected return of portfolio p σ p= the standard deviation of the return of portfolio p Rp = wA·RA + (1-wA)·RB μp = E[ wA·RA + (1-wA)·RB ] σ2p= V[ wA·RA + (1-wA)·RB ]

  27. Two Risky Assets:The Investment OpportunitySet E(Rp) A B STD(Rp)

  28. Two Risky Assets:The M-V Efficient Set (Frontier) E(Rp) A B STD(Rp)

  29. Two Mutually Exclusive Risky Assets: The M-V Efficient Set E(R) A B STD(R)

  30. Two Risky Assets:The M-V Efficient Set (Frontier) E(R) A P B STD(R)

  31. Capital Allocation:Two Risky Assets The investment opportunity set: {all the portfolios on the frontier: with proportion wA invested in the risky asset A and (1-wA)invested in the risky asset B} The Mean-Variance (M-V or μ-σ ) efficient investment set: {all the portfolios on the efficient frontier}

  32. Two Risky Assets:The M-V Efficient Set (Frontier) E(R) P1 A P2 Pmin B P3 STD(R)

  33. Portfolios on the Efficient Frontier wA = the proportion invested in the risky asset A wB = (1-wA) = the proportion invested in the risky asset B What is the value of wA for each on of the portfolios indicated on the graph? - Assume that μA=10%; μB=5%; σA=12%; σ B=6%; ρAB=(-0.5). What is the investment strategy that each portfolio represents? How can you find the minimum variance portfolio? What is the expected return and the std of return of that portfolio?

  34. Portfolios on the Frontier

  35. The Minimum Variance Portfolio

  36. The Minimum Variance Portfolio

  37. Investment Strategies • Lending vs. Borrowing (bonds) • Long vs. Short position (stocks) • Passive risk reduction • Diversification • The number of risky assets in the portfolio • The correlation between the returns of the assets • A perfect hedge

  38. Practice Problems BKM Ch. 7: 1-6, 8, 9, 13, 20, 22, 23 BKM Ch. 8: 1-7 Mathematics of Portfolio Theory: Read and practice parts 4-10.

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