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```									                             Fin 2802: Investments
Spring, 2010
Dragon Tang

Lecture 18
Optimal Investment Portfolio
March 30, 2010

Practice Problem Sets: 1-15, 17-21

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                  1
Optimal Portfolio Choice
Objectives:
• Show how covariance and correlation affect the
power of diversification
• Construct efficient portfolio
• Calculate the composition of the optimal risky
portfolio
• Take risk wisely!

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                 2
Diversification and Portfolio Risk

• Market risk or beta risk
– Systematic or Nondiversifiable
• Firm-specific risk
– Diversifiable or nonsystematic

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             3
Portfolio Risk as a
Function of the Number of Stocks

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                   4
Portfolio Risk as a
Function of Number of Securities

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                   5
Two Asset Portfolio Return – Stock and Bond

r w r w r
p                B      B            S        S

r  Portfolio Return
p

w  Bond Weight
B

r  Bond Return
B

w  Stock Weig ht
S

r  Stock Return
S

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                      6
Covariance
Cov(r1r2) = r1,2s1s2
r1,2 = Correlation coefficient of
returns
s1 = Standard deviation of
returns for Security 1
s2 = Standard deviation of
returns for Security 2
Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             7
Correlation Coefficients: Possible Values

Range of values for r 1,2
-1.0 < r < 1.0
If r = 1.0, the securities would be
perfectly positively correlated
If r = - 1.0, the securities would be
perfectly negatively correlated

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                   8
Two Asset Portfolio St Dev – Stock and Bond

s  w s  w s  2w w s s r
2        2   2            2        2
p        B   B            S        S               B       S   S   B   B,S

s  Portfolio Variance
2
p

s  Portfolio Standard Deviation
2
p

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                                       9
In General, For an n-Security Portfolio:

rp = Weighted average of the
n securities
sp2 = (Consider all pair-wise
covariance measures)

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                 10
Numerical Example: Bond and Stock
Returns
Bond = 6%        Stock = 10%
Standard Deviation
Bond = 12%       Stock = 25%
Weights
Bond = .5 Stock = .5
Correlation Coefficient
(Bonds and Stock) = 0

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                              11
Return and Risk for Example

Return = 8%
.5(6) + .5 (10)

Standard Deviation = 13.87%
[(.5)2 (12)2 + (.5)2 (25)2 + …
2 (.5) (.5) (12) (25) (0)] ½
[192.25] ½ = 13.87

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                              12
Investment Opportunity Set for Stock and Bonds

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             13
Investment Opportunity Set for Stock
and Bonds with Various Correlations

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                14
Table 7.1 Descriptive Statistics for Two Mutual Funds

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             15
Table 7.3 Expected Return and Standard Deviation
with Various Correlation Coefficients

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             16
Figure 7.3 Portfolio Expected Return
as a Function of Investment Proportions

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                              17
Figure 7.4 Portfolio Standard Deviation
as a Function of Investment Proportions

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                              18
Figure 7.5 Portfolio Expected Return
as a function of Standard Deviation

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             19
Table 7.4 Risk Reduction of Equally Weighted Portfolios
in Correlated and Uncorrelated Universes

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             20
Portfolio Selection

• Asset allocation
• Security selection
• These two are separable!

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                               21
Asset Allocation
• John Bogle: “Asset allocation accounts for 94% of
the differences in pension fund performance”
• Identify investment opportunities (risk-return
combinations)
• Choose the optimal combination according to
investor’s risk attitude

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                              22
Optimal Portfolio Construction

Step 1: Using available risky securities (stocks) to
construct efficient frontier.
Step 2: Find the optimal risky portfolio using risk-
free asset
Step 3: Now We have a risk-return tradeoff, choose
Step 4: Calculate optimal portfolio weights

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             23
Portfolios Constructed from Three Stocks A, B and C

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             24
The Efficient Frontier of
Risky Assets and Individual Assets

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                  25
Optimal Capital Allocation Line for
Bonds, Stocks and T-Bills

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                    26
The Complete Portfolio

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                27
The Complete Portfolio – Solution
to the Asset Allocation Problem

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                  28
Discussion: Practical Portfolio Rules

• Rule #1: do not be a amateur stock trader (don’t do it or do it
full time!), choose to be a trader or investor first!
• Investment philosophy: define value! Be cost cautious!
• Investment psychology: do not chicken out!
– Don’t get sentimental, history doesn’t matter
– Stop loss and let your winner run
–…
•   Research, research, research!
•   Sector rotation, familiarity, estimation risk
•   Offense wins game, defense wins championship
•   Amateurs practice until they get it right, pros practice until
they can’t get it wrong.

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                       29
Investor Personalities

•   Measured investors: Rich and greedy
•   Reluctant investors: Rich and humble
•   Competitive investors: Like to trade, which is hazardous
•   Unprepared investors: Poor, greedy, and ignorant

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                30
Mistakes Investors Make
•   Overconfident, underestimate market force
•   Short-sighted, resulting in unnecessary transactions
•   Mental accounting, do not see the big picture
•   Can’t see “everyone is unique, just like everyone else”
•   Disposition: holding on losers too long and selling winner
too fast
•   Averaging down in price rather than up in buying
•   Buying on tips and rumors
•   Speculating too heavily in options or futures wanting to get
rich quick
•   No investment strategy, or having one without persistence

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                                    31
Summary

•   Diversification
•   Optimal risky portfolio and efficient frontier
•   Allocation among risky and risk-free assets
•   Next Class: Practical Portfolio Management

Chapter 7: Optimal Investment Portfolio
FIN 2802, Spring 10 - Tang                                             32

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