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Optimal Risky Portfolios

Optimal Risky Portfolios. Portfolio Diversification Portfolios of Two Risky Assets Asset Allocation Markowitz Portfolio Model. Portfolio Diversification. Return. Stock y. x + y. Stock x. Time. Implications : combination of stocks can reduce overall risk (variance).

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Optimal Risky Portfolios

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  1. Optimal Risky Portfolios • Portfolio Diversification • Portfolios of Two Risky Assets • Asset Allocation • Markowitz Portfolio Model

  2. Portfolio Diversification Return Stock y x + y Stock x Time Implications: combination of stocks canreduce overall risk (variance).

  3. Portfolio Risk behavior Variance Unsystematic risk systematic risk # Assets After a certain number of securities, portfolio variance can no longer be reduced

  4. Portfolios of Two Risky Assets • Givenr1 =0.08, s1=0.12r2 =0.13, s2=0.20 w1 =w2 = 0.5 (assumption) • rp = w1r1 + w2r2 =0.5(0.08) + 0.5(0.13) = 0.105s2p=w21s21 + w22s22 + 2w1w2cov12 =0.25(0.0144)+0.25(0.04) + 2(0.5)(0.5)cs1s2 • case (1): Assume c=1.0 s2p = 0.0256 sp = 0.16

  5. return 0.08 0.2 0.12 0.16 stand. dev Portfolio Return/Risk 2 0.13 0.105 1 If more weight is invested in security 1, thetradeoff line will move downward. Otherwise,it will move upward.

  6. Return 2 . 0.105 0.08 1 0.2 0.12 0.13 stand. dev Case 2: c =0.3s2p= 0.017187sp = 0.1311, rp = 0.105

  7. Portfolio Return/Risk Return c=-1 c=0.3 c=-1 c=1 Stand. Dev.

  8. Capital Allocation for Two Risky Assets Return 2 1 rf Sp Max (rp -rf)/sp {w}w* =f(r1, r2, s1, s2, cov(1,2))then, we get: rp, sp

  9. Example of optimal portfolio The optimal weight in the less risky asset will be: (r1-rf)s22-(r2-rf)cov(1,2) w1= (r1-rf)s22+(r2-rf)s21-(r1-rf+r2-rf)cov(1,2) w2 =1-w1 Given: r1=0.1, s1=0.2 r2=0.3, s2=0.6 c(coeff. of corr)=-0.2 Then: cov=-0.24 w1=0.68 w2=1-w1=0.32

  10. Lending v.s Borrowing Return U 2 p 1 rf Lending Sp Assume two portfolios (p, rf), weightin portfolio, y, will be:y = (rp -rf)/0.01As2p

  11. Markowitz Portfolio Selection • Three assets casereturn and variance formula for the portfolio • N-assets caseReturn and variance formula for the portfolio

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