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Fi8000 Valuation of Financial Assets Spring Semester 2010 Dr. Isabel Tkatch Assistant Professor of Finance

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Today ☺ Portfolio Theory ☺ The Mean-Variance Criterion ☺ Capital Allocation ☺ The Mathematics of Portfolio Theory

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Nation’s Financial Industry Gripped by Fear NY Time, September 15, 2008 By BEN WHITE and JENNY ANDERSON ‘Fear and greed are the stuff that Wall Street is made of.’

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The Mean-Variance Criterion (M-V or μ-σ criterion) STD(R) – “fear” E(R) - “g reed ” ☺ ☺

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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 returns of risky assets is normal with the following parameters AssetABC Expected Return 5.6%4.2%1.7% Standard Deviation of the Return 2.5%5.0%2.1%

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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.

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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 { μ A ≥ μ B and σ A < σ B } or if { μ A > μ B and σ A ≤ σ B } Note that these rules apply only when we assume that the distribution of returns is normal.

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

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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).

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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.

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The Expected Return and STD of Return of the Portfolio α = the proportion invested in the risky asset A p = the portfolio with α invested in the risky asset A and (1- α) invested in the risk-free asset rf and (1- α) invested in the risk-free asset rf R p = the return of portfolio p μ p = the expected return of portfolio p σ p = the standard deviation of return of portfolio p R p = α· R A + (1- α)·rf μ p = E[ α· R A + (1- α)·rf ] = α·μ A + (1- α)·rf σ 2 p = V[ α· R A + (1- α)·rf ] = (α·σ A ) 2 Or σ p = α·σ A

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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 }

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The Capital Allocation Line

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

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The Capital Allocation Line (CAL): Four Basic Investment Strategies rf C A B A P1P1 P2P2

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Portfolios on the CAL Portfolioα E(R p ) = μ p Std(R p ) = σ p rf01.00%0.00% P1P1P1P10.252.15%0.625% A15.60%2.50% P2P2P2P21.57.90%3.75%

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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.

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

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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?)}

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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.

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The Expected Return and STD of Return of the Portfolio w A = the proportion invested in the risky asset A w B = (1- w A ) = the proportion invested in the risky asset B p = the portfolio with w A invested in the risky asset A and (1- w A ) invested in the risky asset B (1- w A ) invested in the risky asset B R p = the return of portfolio p μ p = the expected return of portfolio p σ p = the standard deviation of the return of portfolio p R p = w A ·R A + (1-w A )·R B μ p = E[ w A ·R A + (1-w A )·R B ] σ 2 p = V[ w A ·R A + (1-w A )·R B ]

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Two Risky Assets: The Investment Opportunity Set STD(R p ) E(R p ) B A

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

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

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

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

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

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

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Two Risky Assets: The M-V Efficient Set (Frontier) STD(R) E(R) B A P1P1 P2P2 P3P3 P min

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Portfolios on the Efficient Frontier w A = the proportion invested in the risky asset A w B = (1- w A ) = the proportion invested in the risky asset B What is the value of w A for each one of the portfolios indicated on the graph? - Assume that μ A =10%; μ B =5%; σ A =12%; σ B =6%; ρ AB =(-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?

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Portfolios on the Frontier Portfolio wAwAwAwA E(R p ) = μ p Std(R p ) = σ p P1P1P1P11.311.50%16.57% A110.00%12.00% P2P2P2P20.356.75%4.06% P min ??? B05.00%6.00% P3P3P3P3-0.52.50%13.08%

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The Minimum Variance Portfolio

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Practice Problems BKM 7th Ed. Ch. 6: 15-18, 20-21, 25, 32, 34-35; BKM 8th Ed. Ch. 6: 15-18, 26-27, 21, CFA: 6, 8-9; Mathematics of Portfolio Theory: Read and practice parts 6-10.

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