Optimal Risky Portfolios

1
Optimal Risky Portfolios

• Chapter 7

2
Risk Reduction with Diversification
St. Deviation
Unique Risk
Market Risk
Number of Securities
3
Risk Reduction with Diversification
Empirical support Statman (1987)
4
Two-Security PortfolioReturn and Risk
5
Covariance
?D,E Correlation coefficient of
returns
?D Standard deviation of returns for
Security D ?E Standard deviation of
returns for Security E
6
Correlation Coefficients Possible Values
Range of values for ?D,E
1.0 gt r gt -1.0
If r 1.0, the securities would be perfectly
positively correlated If r -1.0, the securities
would be perfectly negatively correlated
7
In General, For An N-Security Portfolio
8
Two-Security PortfolioExample

• Allocation between Debt and Equity
• rD8, rE13, sD12 , sE20
• rD, E -1, 0, .3, 1
• Different risk/return tradeoff with different
  weights

9
Two-Security PortfolioExample

• Excel skill
• Formula
• Graphs
• X-Y scatter plot

10
(No Transcript)
11
(No Transcript)
12
Correlation Effects

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

13
Minimum-Variance Combination

• When is it achieved?
• Again, think about the first and second order
  derivatives.

14
Minimum-Variance Combination

• Special situation
• Correlation coefficient between D and E is -1.

15
Minimum-Variance Combination

• In class exercise
• sD12 , sE20
• ? .2
• wD
• wE
• ? -.3
• wD
• wE

16
Minimum-Variance Combination

• In excel, one could use solver to find the weight
  to minimize the portfolio variance.
• Target cell
• Changing cells
• Constraints
• Do the weights from mathematical formula agree
  with Excel solutions?

17
Asset Allocation with Stocks, Bonds and Bills
Figure 7.6 The Opportunity Set of the Debt and
Equity Funds and Two Feasible CALs
18
Asset Allocation with Stocks, Bonds and Bills
Figure 7.7 The Opportunity Set of the Debt and
Equity Funds with the Optimal CAL and the Optimal
Risky Portfolio
19
Asset Allocation with Stocks, Bonds and Bills

• To maximize utility, one needs to find the
  weights wD , wE that result in the highest slope
  of the CAL (that is, the weights that result in
  the risky portfolio with the highest
  reward-to-variability ratio)
• Thus our objective function is the slope that we
  have called Sp

20
Asset Allocation with Stocks, Bonds and Bills
21
Asset Allocation with Stocks, Bonds and Bills

• Figure 7.8 Determination of the Optimal Overall
  Portfolio

22
Asset Allocation with Stocks, Bonds and Bills

• Therefore we solve a mathematical problem
  formally written as
• Subject to ? wi 1
• In the case of two risky assets D and E, the
  solution for the weights of the optimal risky
  portfolio, P, can be shown to be as follows

 
23
Asset Allocation with Stocks, Bonds and Bills

• In our example, where
• rD8, rE13, sD12 , sE20, and rD, E .3

24
Asset Allocation with Stocks, Bonds and Bills

• If A4, how much do we put in the optimal risky
  portfolio of D and E, and how much do we put in
  the risk free asset?

25
Figure 7.9 The Proportions of the Optimal Overall
Portfolio
26
Extending Concepts to All Securities Markowitz
Model

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

27
Figure 7.10 The Minimum-Variance Frontier of
Risky Assets
28
Figure 7.12 The Efficient Portfolio Set
29
Extending to Include Riskless Asset

• The optimal combination becomes linear.
• A single combination of risky and riskless assets
  will dominate.

30
Figure 7.13 Capital Allocation Lines with Various
Portfolios from the Efficient Set
31
Portfolio Selection Risk Aversion
U
U
U
E(r)
Efficient frontier of risky assets
S
P
Q
Less risk-averse investor
More risk-averse investor
St. Dev
32
Efficient Frontier with Lending Borrowing
CAL
E(r)
B
Q
P
A
rf
F
St. Dev
33
The Separation Theorem

• A portfolio manager only need the same risky
  portfolio, P, to all clients regardless of their
  degree of risk aversion
• The portfolio choice problem may be separated
  into two independent tasks
• First determine the optimal risky portfolio
• Then choose the allocation of the complete
  portfolio to risk-free assets

34
The Separation Theorem

• Simplifying assumptions
• No market frictions (tax, transaction costs,
  market segmentation)
• No heterogeneity in investors (wealth level,
  posses of information, etc.)
• Static expected return and variance
• Violation of the assumptions
• Individuals may not hold identical risky
  portfolios.

35
Assignments

• Chapter 7
• Problems 1-7, 17-21, 23-26, 29