Fin 2802: Investments
Spring, 2010
Dragon Tang
Lecture 18
Optimal Investment Portfolio
March 30, 2010
Readings: Chapter 7
Practice Problem Sets: 1-15, 17-21
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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!
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Diversification and Portfolio Risk
• Market risk or beta risk
– Systematic or Nondiversifiable
• Firm-specific risk
– Diversifiable or nonsystematic
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Portfolio Risk as a
Function of the Number of Stocks
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Portfolio Risk as a
Function of Number of Securities
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Two Asset Portfolio Return – Stock and Bond
r w r w r
r  Portfolio Return
w  Bond Weight
r  Bond Return
w  Stock Weig ht
r  Stock Return
p
B
B
S
S
p
B
B
S
S
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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
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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
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Two Asset Portfolio St Dev – Stock and Bond
s  w s  w s  2w w s s r
s  Portfolio Variance
s  Portfolio Standard Deviation
2
2
2
2
2
p
B
B
S
S
B
S
S
B
B,S
2
p
2
p
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In General, For an n-Security Portfolio:
rp = Weighted average of the
n securities
sp2 = (Consider all pair-wise
covariance measures)
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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
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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
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Investment Opportunity Set for Stock and Bonds
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Investment Opportunity Set for Stock
and Bonds with Various Correlations
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Table 7.1 Descriptive Statistics for Two Mutual Funds
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Table 7.3 Expected Return and Standard Deviation
with Various Correlation Coefficients
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Figure 7.3 Portfolio Expected Return
as a Function of Investment Proportions
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Figure 7.4 Portfolio Standard Deviation
as a Function of Investment Proportions
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Figure 7.5 Portfolio Expected Return
as a function of Standard Deviation
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Table 7.4 Risk Reduction of Equally Weighted Portfolios
in Correlated and Uncorrelated Universes
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Portfolio Selection
• Asset allocation
• Security selection
• These two are separable!
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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
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Optimal Portfolio Construction
Step 1: Using available risky securities (stocks) to
construct efficient frontier.
Step 2: Find the optimal risky portfolio using riskfree asset
Step 3: Now We have a risk-return tradeoff, choose
your most favorable asset allocation
Step 4: Calculate optimal portfolio weights
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Portfolios Constructed from Three Stocks A, B and C
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The Efficient Frontier of
Risky Assets and Individual Assets
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Optimal Capital Allocation Line for
Bonds, Stocks and T-Bills
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The Complete Portfolio
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The Complete Portfolio – Solution
to the Asset Allocation Problem
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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.
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Investor Personalities
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•
•
•
Measured investors: Rich and greedy
Reluctant investors: Rich and humble
Competitive investors: Like to trade, which is hazardous
Unprepared investors: Poor, greedy, and ignorant
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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
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Summary
•
•
•
•
Diversification
Optimal risky portfolio and efficient frontier
Allocation among risky and risk-free assets
Next Class: Practical Portfolio Management
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