Optimal Risky Portfolios Portfolio Diversification Portfolios of Two

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).

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

Portfolios of Two Risky Assets • Given r 1 =0. 08, s 1=0. 12 r 2 =0. 13, s 2=0. 20 w 1 =w 2 = 0. 5 (assumption) • rp = w 1 r 1 + w 2 r 2 =0. 5(0. 08) + 0. 5(0. 13) = 0. 105 s 2 p=w 21 s 21 + w 22 s 22 + 2 w 1 w 2 cov 12 =0. 25(0. 0144)+0. 25(0. 04) + 2(0. 5)cs 1 s 2 • case (1): Assume c=1. 0 s 2 p = 0. 0256 sp = 0. 16

Portfolio Return/Risk return 2 0. 13 0. 105 0. 08 1 0. 12 0. 16 0. 2 stand. dev If more weight is invested in security 1, the tradeoff line will move downward. Otherwise, it will move upward.

Case 2: c =0. 3 s 2 p= 0. 017187 sp = 0. 1311, rp = 0. 105 Return 2 . 0. 105 0. 08 1 0. 12 0. 13 0. 2 stand. dev

Portfolio Return/Risk Return c=-1 c=0. 3 c=1 Stand. Dev.

Capital Allocation for Two Risky Assets Return 2 rf 1 Sp Max (rp -rf)/sp {w} w* =f(r 1, r 2, s 1, s 2, cov(1, 2)) then, we get: rp, sp

Example of optimal portfolio The optimal weight in the less risky asset will be: w 1= (r 1 -rf)s 22 -(r 2 -rf)cov(1, 2) (r 1 -rf)s 22+(r 2 -rf)s 21 -(r 1 -rf+r 2 -rf)cov(1, 2) w 2 =1 -w 1 Given: r 1=0. 1, s 1=0. 2 r 2=0. 3, s 2=0. 6 c(coeff. of corr)=-0. 2 Then: cov=-0. 24 w 1=0. 68 w 2=1 -w 1=0. 32

Lending v. s Borrowing Return U p rf 2 1 Lending Sp Assume two portfolios (p, rf), weight in portfolio, y, will be: y = (rp -rf)/0. 01 As 2 p

Markowitz Portfolio Selection • Three assets case return and variance formula for the portfolio • N-assets case Return and variance formula for the portfolio