Efficient Diversification and optimal Portfolio

1
Chapter 6

• Efficient Diversification and optimal Portfolio

2
Outline

• Risk and Return of a two-security portfolio
• Risk and return combination of two-security
  portfolio
• Extending to Include riskless Asset

3
I Risk and Return of a two-security portfolio
4
Two-Security Portfolio Return
rp W1r1 W2r2 W1 Proportion of funds in
Security 1 W2 Proportion of funds in Security
2 r1 Expected return on Security 1 r2
Expected return on Security 2
5
Two-Security Portfolio Risk

• How risk might be reduced?
• How to measure the relationship between two
  stocks? Are they Moving together or not?
• The idea of covariance and correlation of
  coefficient.

6
Returns distribution for two perfectly negatively
correlated stocks (? -1.0)
40
40
15
15
-10
7
Returns distribution for two perfectly positively
correlated stocks (? 1.0)
8
Correlation Coefficients Possible Values
Range of values for r 1,2 -1.0 lt r lt 1.0
If r 1.0, the securities would be perfectly
positively correlated If r - 1.0, the
securities would be perfectly negatively
correlated
9
How to find Correlation
r1,2 Cov(r1r2) /(s1s2) Cov(r1r2) (r1,2
)(s1s2)
r1,2 Correlation coefficient of
returns
s1 Standard deviation of returns for Security
1 s2 Standard deviation of returns for
Security 2
10
Two-Security Portfolio Risk

or
or


sp w12s12 w22s22 2W1W2 r1,2s1s21/2
Or
11
II Risk and return combination of two-security
portfolio

• the investment opportunity set of two-security
  portfolio

12
INVESTMENT OPPORTUNITY SETS FOR TWO-SECURITY
PORTFOLIOS WITH DIFFERENT CORRELATIONS
r 0
13
Portfolio Risk/Return Two Securities Correlation
Effects

• Relationship depends on correlation coefficient
• -1.0 lt r lt 1.0
• The smaller the correlation, the greater the risk
  reduction potential
• If r 1.0, no risk reduction is possible

14
Magic of combining two risky assets

• Put yourself in the two-dimensional world risk
  and return
• Risk /return changes in a non-linear pattern when
  increasing the weight in asset with higher
  risk/return
• Offers more choice more risk/return combination
• Risk can be reduced while return increases while
  you move up and left on the investment
  opportunity set
• You only invest on efficient frontier Graph
  representing a set of portfolios that maximizes
  expected return at each level of portfolio risk

15
Extending Concepts to All Securities

• The optimal combinations result in lowest level
  of risk for a given return, or highest return for
  a given level of risk
• The optimal trade-off is on the efficient
  frontier
• These portfolios are dominant

16
The minimum-variance frontier of risky assets
E(r)
Efficient frontier
Individual assets
Global minimum variance portfolio
Minimum variance frontier
St. Dev.
17
III. Extending to Include Riskless Asset

• Portfolio choice can be separated into two
  independent steps 1) determination of the risky
  portfolio and 2) the capital allocation between
  risky portfolio and risk-free asset
• Optimal risky portfolio lies on efficient
  frontier
• CAL with steepest slope dominates all other CALs.
  The optimal CAL will be tangent to the efficient
  frontier
• There is only one optimal risky portfolio, the
  tangent portfolio between CAL and efficient
  frontier

18
ALTERNATIVE CALS
CAL (P)
CAL (A)
E(r)
M
M
P
P
CAL (Global minimum variance)
A
A
G
F
s
P
PF
AF
M
19
Dominant CAL with a Risk-Free Investment (F)

• CAL(P) dominates other lines -- it has the best
  risk/return or the largest slope
• Slope (E(R) - Rf) / s Sharp ratio
• E(RP) - Rf) / s P gt E(RA) - Rf) / sA
• Regardless of risk preferences combinations of P
  F dominate