# Investments

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8/8/2012

Portfolio Theory
A. The capital allocation decision: portfolio of one
risk-free asset and one risky asset
B. Extension to the case of two risky assets
C. The Markowitz portfolio selection model (N
assets)
D. Excel application and practical issues

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Idea of Diversification

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Notations
 P: risky portfolio
 F: risk-free asset
 y: weight allocated to P
 1-y: weight allocated to F
 C: combined portfolio (P and F together)
 wi: weight allocated to risky asset i in P
 All the wi’s must sum up to one

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B. Extension to Case of Two Risky Assets

p  w1 1  w2 2  2w1w2Cov(r1,r2)
2   2   2   2   2

12 = variance of asset 1’s returns
22 = variance of asset 2’s returns
Cov(r1,r2) = covariance of returns of asset 1 and asset 2

Slide 6-4
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Covariance and Correlation Coefficient

Cov(r1,r2)  1,2  1  2

1,2 = Correlation coefficient of returns between
assets 1 and 2
1 = Standard deviation of returns for asset 1
2 = Standard deviation of returns for asset 2

Slide 6-5
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Correlation Coefficients: Possible Values

Range of values for 1,2

+ 1.0 >  1,2 > -1.0

If  1,2 = 1.0, the assets are perfectly positively
correlated

If  1,2 = - 1.0, the assets are perfectly negatively
correlated
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Expected return and variance of a
portfolio (2 assets)

E(rp )  w1E(r1 )  w2E(r2 )   and

p  w1 1  w2 2  2w1w2Cov(r1,r2)
2    2   2      2   2

Re-write as:
p  w1 1  w2 2  2w1w21,212
2    2   2      2   2

Question: How does 1,2affect the variance of a portfolio?

Slide 6-7
Numerical Example
 Consider two mutual funds
Debt fund       Equity fund
(D)              (E)
E(r)      8%              13%
        12%              20%

 What does the CAL look like?
 Depends on D,E

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Two Risky Assets: Different
Correlations

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Portfolio of two assets: Correlation
Effects
 The smaller the correlation, the greater the
potential in risk reduction
 If= +1.0, there is no reduction in risk, in the sense
the variance of the portfolio is just a linear
combination of D and E, i.e., no diversification
benefit
 Risk reduction: horizontal move toward the y axis

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Minimum-Variance Portfolio
• Solving the minimization problem we get:

 E 2  Cov(rD , rE )
wD 
 D   E  2Cov(rD , rE )
2       2

• Numerical example ( = 0.2):
(20)2  (0.2)( 20)(12)
wD                                 0.7857
(20)  (12)  2(0.2)( 20)(12)
2       2

wE  1 wD  0.2143
7-11
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Minimum-Variance: Return and Risk

 Using the weights wD and wE we can determine the
risk-return characteristics of the minimum-variance
portfolio:
E(rP )  (0.7857)(8%)  (0.2143)(13%)  9.07%

P2  (0.7857)2 (12)2  (0.2143)2 (20)2 
 2(0.7857)(0.2143)(12)(20)(0.2)  123.43
P  123.43  11.11%

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Extending to include a risk-free asset

 Now bring back the risk-free asset, and include it
in the choice of assets
 What happens when you combine a different risky
portfolio with the risk-free asset?

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Alternative CALs

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Optimal Risky Portfolio

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When the risk-free asset is included…

 The opportunity set is again described by the
(linear) CAL
 The choice of the optimal combined portfolio
depends on the client’s risk attitude
 As before, a single combination of risky and risk-
free assets will dominate (i.e., the tangency
portfolio that maximizes utility)

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Bring in the Client

Slide 6-17
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Asset Allocation/Mix

Slide 6-18
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C. The Markowitz Portfolio Selection model
 Extending the concepts to n risky assets
 Many possible combinations/portfolios of risky assets
 Focus on portfolios that have the highest expected
return for a given level of risk
 Portfolios that satisfy this optimal trade-off lie on the
efficient frontier
 These portfolios are dominant in a mean-variance sense

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Extending to n risky securities

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GMV

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An Application of M-V Portfolios

 TDAM Emerald Low Volatility Canadian Equity Pooled
Fund, and Global Equity Pooled Fund
 New Product – Launched June 2009
 Marketed as “Minimum Variance Portfolios”

Slide 6-22

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