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Today Portfolios Return on a Portfolio

Lecture 2.2: Characteristics of PortfoliosInvestment Analysis

Fall, 2012

Anisha Ghosh

Tepper School of BusinessCarnegie Mellon University

November 8, 2012



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Readings and Assignments

Chapter 3 of the course textbook (EGBG) covers relatedmaterial.

Homework 2 is available on the Courses Wall for Week 2.



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Characteristics of Portfolios

Most investors hold portfolios of a large number of assets rather than

individual assets.

The risk on a portfolio is more complex than a simple average of the risk

of individual assets - it depends on whether the returns on individualassets tend to move together or whether some assets give good returns

when others give bad returns.

Benefits of Diversification

There is risk reduction from holding a portfolio of assets if the assets donot move in perfect unison.



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individual assets.








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individual assets.






T d P f li R P f li



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Return on a portfolio

Definition

The return on a portfolio of assets is a weighted average of the return on the

individual assets where the weight applied to each return is the fraction of the

portfolio invested in the asset

R pj =N i =1

X i R ij , whereN i =1

X i = 1

X i > 0⇒ long position in asset i

X i < 0⇒ short position in (short sale of) asset i

T d P tf li R t P tf li



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Definition




R pj =N i =1


X i = 1






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Expected Return on a portfolioProperties of Expected Return:

1 The expected value of the sum of two returns is equal to the sum

of the expected value of each return:

E (R 1 + R 2) = E (R 1) + E (R 2)

2 The expected value of a constant C times a return is the constant

times the expected return:

E (CR i ) = CE (R i )

Expected Return on a Portfolio

The expected return on a portfolio is:

E (R p ) = E N

i =1

X i R ij

=N i =1

E (X i R ij )

=

N =

X i E (R i )




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E (R 1 + R 2) = E (R 1) + E (R 2)



E (CR i ) = CE (R i )



E (R p ) = E N

i =1

X i R ij

=N i =1

E (X i R ij )

=

N =

X i E (R i )




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y

Variance of Return on a portfolio


The variance of a portfolio is the expected value of the squared

deviations of the returns on the portfolio from its mean return:

σ2p = E (R p − E (R p ))2

= E (X 1R 1 j + X 2R 2 j −

[X 1E (R 1) + X 2E (R 2)])2

= E (X 1 [R 1 j − E (R 1)] + X 2 [R 2 j − E (R 2)])2

= E

X 21 [R 1 j − E (R 1)]2 + X 22 [R 2 j − E (R 2)]2

+2X 1X 2 [R 1 j − E (R 1)] [R 2 j − E (R 2)]

= X 21 E [R 1 j − E (R 1)]2 + X 22 E [R 2 j − E (R 2)]2

+2X 1X 2E [R 1 j − E (R 1)] [R 2 j − E (R 2)]

= X 21 σ

21 + X

22 σ

22 + 2X 1X 2E [R 1 j − E (R 1)] [R 2 j − E (R 2)]




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y



The variance of a portfolio is the expected value of the squared

deviations of the returns on the portfolio from its mean return:

σ2p = E (R p − E (R p ))2

= E (X 1R 1 j + X 2R 2 j −

[X 1E (R 1) + X 2E (R 2)])2

= E (X 1 [R 1 j − E (R 1)] + X 2 [R 2 j − E (R 2)])2

= E

X 21 [R 1 j − E (R 1)]2 + X 22 [R 2 j − E (R 2)]2

+2X 1X 2 [R 1 j − E (R 1)] [R 2 j − E (R 2)]

= X 21 E [R 1 j − E (R 1)]2 + X 22 E [R 2 j − E (R 2)]2

+2X 1X 2E [R 1 j − E (R 1)] [R 2 j − E (R 2)]

= X 21 σ

21 + X

22 σ

22 + 2X 1X 2E [R 1 j − E (R 1)] [R 2 j − E (R 2)]




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Covariance and Correlation

Covariance: The covariance of two assets is a measure of how returns

on the assets move together:

σ12 = E [R 1 j − E (R 1)] [R 2 j − E (R 2)]

Correlation: The correlation has the same properties as the covariance

but with a but with a range of −1 to 1:

ρ12 =

σ12

σ1σ2




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The variance of a portfolio of N assets is

σ2p =

N

i =1

X 2i σ2i +

N

i =1

N

k =1

i =k

X i X k σik

In the case where all the assets are independent, the covariance

between them is zero, and the formula for the variance becomes

σ

2

p =

N i =1

X

2

i σ

2

i


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