Efficient Diversification I

1
Efficient Diversification I

• Covariance and Portfolio Risk
• Mean-variance Frontier
• Efficient Portfolio Frontier

2
Some Empirical Evidence

• In 2000, 40 of stocks in Russell 3000 had
  returns of -20 or worse.
• Meanwhile, less than 12 of U.S. stock mutual
  funds had returns of -20 or below.
• Of the 2,397 U.S. stocks in existence throughout
  1990s, 22 had negative returns.
• In contrast, 0.4 of U.S. equity mutual funds had
  negative returns.

3
Diversification and Portfolio Risk

• Dont put all your eggs in one basket
• Effect of portfolio diversification

?
Diversifiable risk, non-systematic risk,
firm-specific risk, idiosyncratic risk
Non-diversifiable risk, systematic risk, market
risk
5
15
10
20
of securities in the portfolio
4
Covariance and Correlation

• Covariance and correlation
• Degree of co-movement of two stocks
• Covariance non-standardized measure
• Correlation coefficient standardized measure

r2
r2
r2
r1
r1
r1
0lt?12 lt1
-1lt?12 lt0
?12 0
5
Covariance and Correlation

• Example Two risky assets
• Calculating the covariance

Means
Std. Dev.
Cov.
Corr.
6
Diversification and Portfolio Risk

• A portfolio of two risky assets
• w1 invested in bond
• w2 invested in stock
• Expected return
• Variance

7
Diversification and Portfolio Risk

• Example Portfolio of two risky securities
• w in security 1, (1 w) in security 2
• Expected return (Mean)
• Variance
• What happens when w changes?
• Expected return decreases with increasing w
• How about variance ?..

8
Mean-Variance Frontier

• w from 0 to 1

GMVP Global Minimum Variance Portfolio
Mean-variance frontier
Security 2
Security 1
GMVP
9
Mean-Variance Frontier

• Global Minimum Variance Port. (GMVP)
• A unique w
• Associated characteristics

10
Efficient Portfolio Frontier

• 67 in Security 1 and 33 in Security 2, whats
  so special?
• Efficient portfolio has lt 67 in 1, and gt 33 in
  2

w10
P
Efficient Frontier
w1 .6733
GMVP
Inefficient Frontier
w11
11
Efficient Portfolio Frontier

• Portfolio P dominates Security 1
• The same standard deviation
• The higher expected return
• How to find it?
• Since the portfolio has the same standard
  deviation as Security 1
• Solve the quadratic equation
• w 1 (Security 1) or w .3465 (Portfolio P)

12
Efficient Portfolio Frontier

• The effect of correlation
• Lower correlation means greater risk reduction
• If?r? 1.0, no risk reduction is possible

13
Efficient Portfolio Frontier

• Efficient Portfolio of Many securities
• Erp Weighted average of n securities
• ?p2 Combination of all pair-wise covariance
  measures
• Construction of the efficient frontier is
  complicated
• Analytical solution without short-sale
  constraints
• Numerical solution with short-sale constraints
• General Features
• Optimal combination results in lowest risk for
  given return
• Efficient frontier describes optimal trade-off
• Portfolios on efficient frontier are dominant

14
Efficient Frontier
Er
Efficient frontier
Individual assets
Global minimum variance portfolio
Minimum variance frontier
St. Dev.
15
Wrap-up

• How to estimate portfolio return and risk?
• What is the mean-variance frontier?
• What is the efficient portfolio frontier?
• Why do portfolios on efficient frontier dominate
  other combinations?