chapter 8 portfolio theory

8/8/2019 Chapter 8 Portfolio Theory

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Chapter 8

PORTFOLIO THEORY

The Benefits of Diversification


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PORTFOLIO EXPECTED RETURN

n

E(RP) = w

i E(Ri)i=1

whereE(RP) = expected portfolio return

wi= weight assigned to security i

E(Ri) = expected return on security in = number of securities in the portfolio

Example A portfolio consists of four securities with expected returns

of 12%, 15%, 18%, and 20% respectively. The proportions of

portfolio value invested in these securities are 0.2, 0.3, 0.3, and 0.20respectively.

The expected return on the portfolio is:

E(RP) = 0.2(12%) + 0.3(15%) + 0.3(18%) + 0.2(20%)

= 16.3%


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PORTFOLIO RISK

The risk of a portfolio is measured by the variance (orstandard deviation) of its return. Although the expected

return on a portfolio is the weighted average of the

expected returns on the individual securities in the

portfolio, portfolio risk is not the weighted average of the

risks of the individual securities in the portfolio (except

when the returns from the securities are uncorrelated).


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MEASUREMENT OF COMOVEMENTS

IN SECURITY RETURNS

To develop the equation for calculating portfolio risk we

need information on weighted individual security risks

and weighted comovements between the returns of

securities included in the portfolio.

Comovements between the returns of securities are

measured by covariance (an absolute measure) and

coefficient of correlation (a relative measure).


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COVARIANCE

COV (Ri,Rj) =p1 [Ri1E(Ri)] [Rj1E(Rj)]

+p2 [Ri2E(Rj)] [Rj2E(Rj)]

+

+pn [RinE(Ri)] [RjnE(Rj)]


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ILLUSTRATION

T

S R R

-

9

4

7

T

R - 7

TR 9 4

T


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State of

nature

(1)

Probability

(2)

Return on

asset 1

(3)

Deviation of

the return on

asset 1 from its

mean

(4)

Return on

asset 2

(5)

Deviation of

the return on

asset 2 from

its mean

(6)

Product of the

deviations

times

probability

(2) x (4) x (6)

1 0.10 -10% -26% 5% -9% 23.4

2 0.30 15% -1% 12% -2% 0.6

3 0.30 18% 2% 19% 5% 3.0

4 0.20 22% 6% 15% 1% 1.2

5 0.10 27% 11% 12% -2% -2.2

Sum = 26.0

Thus the covariance between the returns on the two assets is 26.0.


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CO EFFIENT OF CORRELATION

Cov (Ri , Rj)

Cor (Ri , Rj) orVij=W(Ri ,Rj)

Wij

WiW

j

Wij= Vij. Wi . Wj

where Vij = correlation coefficient between the returns on

securities iandj

Wij = covariance between the returns on securities

iand j

Wi , Wj = standard deviation of the returns on securities

iandj

=


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PORTFOLIO RISK : 2 SECURITY CASE

Wp = [w12 W12 + w22 W22 + 2w1w2V12 W1 W2]

Example : w1 = 0.6 , w2 = 0. ,

W1 = 10%, W2 = 16%

V12 = 0.5

Wp = [0.62 x 102 + 0. 2 x 162 +2 x 0.6 x 0. x 0.5 x 10 x 16]

= 10. %


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PORTFOLIO RISK : n SECURITY CASE

Wp = [ wiwjVijWiWj]

Example : w1 = 0.5 , w2 = 0.3, and w3 = 0.2

W1 = 10%, W2 = 15%, W3 = 20%

V12 = 0.3, V13 = 0.5, V23 = 0.6

Wp = [w12 W1

2 + w22 W2

2 + w32 W3

2 + 2 w1 w2 V12 W1 W2

+ 2w2 w3 V13 W1 W3 + 2w2 w3 V23W2 W3]

= [0.52 x 102 + 0.32 x 152 + 0.22 x 202

+ 2 x 0.5 x 0.3 x 0.3 x 10 x 15

+ 2 x 0.5 x 0.2 x 05 x 10 x 20

+ 2 x 0.3 x 0.2 x 0.6 x 15 x 20]

= 10. %


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DOMINANCE OF COVARIANCE

As the number of securities included in a portfolio

increases, the importance of the risk of each individual

security decreases whereas the significance of the

covariance relationship increases


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PORTFOLIO OPTIONS

AND THE EFFICIENT FRONTIER

12%

20%

20% 0%

Risk, Wp

Expected

return , E(Rp)

1 (A)

6 (B)

23


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EFFICIENT FRONTIER FOR THE

n SECURITY CASE

Standard deviation, Wp

Expected

return ,E(Rp)

A O

N

M

X

F

B

D

Z


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OPTIMAL PORTFOLIO

P*

Q*

A

O

N

M

X


Expected

return ,E(Rp)

IQ3IQ2

IQ1

IP3 IP2 IP1


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A

O

N

M

X


Expected

return ,E(Rp)

Rf

u

D

E

VG

B

CF Y

I

II

S

RISKLESS LENDING AND

BORROWING OPPORTUNITY

Thus, with the opportunity of lending and borrowing, the efficient frontier changes. It is no

longerAFX. Rather, it becomes RfSG as it domniates AFX.


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SEPARATION THEOREM

Since RfSG dominates AFX, every investor would do well to

choose some combination ofRf and S. A conservative investor

may choose a point like U, whereas an aggressive investor may

choose a point like V.

Thus, the task of portfolio selection can be separated into twosteps:

1. Identification of S, the optimal portfolio of risky securities.

2. Choice of a combination ofRf

and S, depending on ones

risk attitude.

This is the import of the celebrated separation theorem


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SINGLE INDEX MODEL

INFORMATION INTENSITY OF THE MARKOWITZ MODEL

n SECURITIES

n VARIANCE TERMS & n(n 1) /2

COVARIANCE TERMS

SHARPES MODEL

Rit = ai + biRMt + eitE(Ri) = ai + biE(RM)

VAR (Ri) = bi2 [VAR (RM)] + VAR (ei)

COV (Ri,Rj) = bibjVAR (RM)

MARKOWITZ MODEL SINGLE INDEX MODEL

n (n + 3)/2 3n + 2

E(Ri) & VAR (Ri) FOR EACH bi, bjVAR (ei) FOR

SECURITY n( n 1)/2 EACH SECURITY &

COVARIANCE TERMS E(RM) & VAR (RM)


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SUMMING UP

Portfolio theory, originally proposed by Markowitz, is the

first formal attempt to quantify the risk of a portfolio anddevelop a methodology for determining the optimal portfolio.

The expected return on a portfolio ofn securities is :

E(Rp) = wiE(Ri)

The variance and standard deviation of the return on an

n-security portfolio are:

Wp2 = wiwjVijWiWj

Wp = wiwjVijWiWj

A portfolio is efficient if (and only if) there is an alternative

with (i) the same E(Rp) and a lower Wp or (ii) the same Wp anda higher E(Rp), or (iii) a higher E(Rp) and a lower Wp


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chapter 8 portfolio theory

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