CHAPTER 6 PORTFOLIO SELECTION

1
CHAPTER 6PORTFOLIO SELECTION
2
Portfolio management

• 3-step process
• Capital allocation
• How much in risk-free vs risky assets
• Asset allocation
• How much in each asset class
• Security selection
• Which securities to hold

3
Chapter Summary

• ObjectiveTo present the basics of modern
  portfolio selection process
• Capital allocation decision
• Derive the set of all possible investment
  opportunities
• Determine where the investor will choose to
  invest in this set
• Two-security portfolios and extensions
• The Markowitz portfolio selection model

4
Allocating Capital Between Risky Risk Free
Assets

• OPTIMAL PORTFOLIO portfolio which consists of
  risk-free asset and risky asset
• Risk free asset proxy is T-bills
• Risky asset stock (or a portfolio)

b
5
Possible Combinations(Capital Allocation Line
CAL)
BORROW
CAL
B
6
Different lending and borrowing rates
7
Chapter Summary

• ObjectiveTo present the basics of modern
  portfolio selection process
• Capital allocation decision
• Derive the set of all possible investment
  opportunities
• Determine where the investor will choose to
  invest in this set
• Two-security portfolios and extensions
• The Markowitz portfolio selection model

8
Indifference Curves

• U E ( r ) .5 A s²

9
Indifference Curves and Risk Aversion

• Certainty equivalent of portfolio Ps expected
  return for two different investors

Flatter curve means lower risk aversion
P
E(rp)15
10
Question

• The investor has the same utility anywhere on the
  curve. How does he choose WHERE to lie on the
  curve?
• Answer superimpose the utility curves on top of
  the investment opportunities available (CAL)

11
CAL with Risk Preferences
12
Risk Aversion and Allocation

• Greater levels of risk aversion lead to larger
  proportions of the risk free rate
• Lower levels of risk aversion lead to larger
  proportions of the portfolio of risky assets

b
13
Reducing risk through Diversification
14
Summary Reminder

• ObjectiveTo present the basics of modern
  portfolio selection process
• Capital allocation decision
• Two-security portfolios and extensions
• The Markowitz portfolio selection model

15
PORTFOLIOS

• We saw that we will combine a risk-free asset
  with a risky porfolio
• QUESTION How do you construct the risky
  portfolio?

16
Two-Security Portfolio Return
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
17
Two-Security Portfolio Risk
?12 variance of Security 1 ?22 variance of
Security 2 Cov(r1,r2) covariance of returns for
Security 1 and Security 2
18
Covariance
?1,2 Correlation coefficient of returns ?1
Standard deviation of returns for Security 1 ?2
Standard deviation of returns for Security 2

19
Correlation Coefficients Possible Values
Range of values for ?1,2
1.0 gt ????gt ?-1.0
If ?? 1.0, the securities would be perfectly
positively correlated If ?? - 1.0, the
securities would be perfectly negatively
correlated
20
Returning to the Two-Security Portfolio
and
, or
Question What happens if we use various
securities combinations, i.e. if we vary r?
21
Three-Security Portfolio
22
Generally, for an n-Security Portfolio
23
Two-Security Portfolios with Different
Correlations
24
Portfolio of Two Securities Correlation Effects

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

25
Minimum-Variance Combination

• Suppose our investment universe comprises the
  following two securities

• What are the weights of each security in the
  minimum-variance portfolio?

26
Minimum-Variance Combination ? .2

• Solving the minimization problem we get

• Numerically

27
Minimum -Variance Return and Risk with ? .2

• Using the weights wA and wB we determine
  minimum-variance portfolios characteristics

28
Minimum -Variance Combination ? -.3

• Using the same mathematics we obtain
• wA 0.6087
• wB 0.3913
• While the corresponding minimum-variance
  portfolios characteristics are
• rP 11.57 and
• sP 10.09

29
Summary Reminder

• ObjectiveTo present the basics of modern
  portfolio selection process
• Capital allocation decision
• Two-security portfolios and extensions
• The Markowitz portfolio selection model

Addresses the question of where will the investor
invest
30
Extending Concepts to All Securities

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

b
31
The Minimum-Variance Frontier of Risky Assets
32
The Minimum-Variance Frontier of Risky Assets
33
Extending to Include A Riskless Asset

• The set of opportunities again described by the
  CAL
• The choice of the optimal portfolio depends on
  the clients risk aversion
• A single combination of risky and riskless assets
  will dominate

34
Alternative CALs
35
CAL with Risk Preferences