Optimal Risky Portfolios

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, s10.12r2 0.13, s20.20 w1 w2
  0.5 (assumption)
• rp w1r1 w2r2 0.5(0.08) 0.5(0.13)
  0.105s2pw21s21 w22s22 2w1w2cov12
  0.25(0.0144)0.25(0.04)
  2(0.5)(0.5)cs1s2
• case (1) Assume c1.0 s2p 0.0256 sp 0.16

5
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
Case 2 c 0.3 s2p 0.017187 sp 0.1311, rp
0.105
7
Portfolio Return/Risk
Return
c-1
c0.3
c-1
c1
Stand. Dev.
8
Capital Allocation for Two Risky Assets
Return
2
1
rf
Sp
Max (rp -rf)/sp ww 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-rfr2-rf)cov(1,2)
w2 1-w1
Given r10.1, s10.2 r20.3, s20.6 c(coeff. of
corr)-0.2 Then cov-0.24 w10.68 w21-w10.32
10
Lending v.s Borrowing
Return
U
2
p
1
rf
Lending
Sp
Assume two portfolios (p, rf), weightin
portfolio, y, will bey (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