Portfolio Analyis, Risk & Return Pws Dmb Ipb

8/12/2019 Portfolio Analyis, Risk & Return Pws Dmb Ipb

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Portfolio AnalysisThe Characteristics of theOpportunity Set Under Risk

Dr. Perdana Wahyu Santosa

1


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PWS

-60

-40

-20

0

20

40

60

2 6 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0 8 5 9 0 9 5 2 0

0 0

Common StocksLong T-BondsT-Bills

P e r c e n t a g e

R e t u r n

Year

Rates of Return 1926-2000

Source: Ibbotson Associates


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Risk and Return for Single Asset

Rate of Return on Investment and Wealth

1 1 0

0

1

1 1 1 0

Rate of Return is defined, R:

(1)

the contribution of the stock investment to the investor's wealth

one year hence, W is defined as follows:

(1 ),

or s

P d P R

P

W P d P R

1 10

tated in terms of present wealth (single period),

(1 ) P d

W R

PWS DMB Finance IPB 4


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Expected Rate of Return

The Expected Rate of Return (Expected Return)is simply the expected value or average of therandom or probabilistic rate of return.

An expected value or average is easy to compute.For notational convenience, we will use:


( )i i E R R


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Measures and Sources of Risk



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Measure of Risk

The dispersion in return of investment can bemeasured using the variance which is the expectedvalue of the squared deviation from the expectedreturn.

22

1

i

2

1

Using probability ( ) to designate the variance in returns,

(4) )

We calculate the standard deviation, as follows:

(5)

M

i ij ij i j

N

i i ij ii

R R

R R



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Measure of Risk


2i

2

1

2

1

The formula for measure of risk or variance (dispersion)

of the return on the- th asset (symbolized ),

(6)

and standard deviation:

(7)

M ij i

i j

M ij i

i j

i

R R

M

R R

M


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Calculating the Expected Rate of Return


3

1 13

1

12 9 6( ) 9

3 3

( ) 0.333(12) 0.333(9) 0.333(6) 9

M ij ij

i

j j

i ij ij j

Solution

R Ri R

M

ii R R

Event Probability Returns

1 0.333 12

2 0.333 9

3 0.333 6


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Calculating the Expected Rate ofReturn and Variance

Probability End ofYear Stock

Price

CashDividen

Rate ofReturn

0.1 $7.00 $ 1.00 -0.20

0.2

9.00

1.00

0.00

0.4 10.00 1.00 0.10

0.2 11.00 1.00 0.10

0.1 12.00 1.00 0.30

2

Calculate:

a. the expected rate of return of one-year investment

b. the variance of return of one-year investment

i R



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Calculating the expected rate of returnand variance

1

a.

0.1( 0.20) 0.2(0.00) 0.4(0.10) 0.20(0.20) 0.10(0.30)

0.09 or 9.0%

Thus expected value of one-year investment return is 9.0%

b. We can also calcul

M

i ij ij j

R R

2

2 2 2 2

2 2

2

ate the ( ) as follows:

0.10( 0.20 0.9) 0.20(0.00 0.9) 0.40(0.10 0.9)

+0.20(0.20-0.9) 0.10(0.30 0.9) 0.00841 0.00162 0.00004 0.00242 0.00441

R

0.0169 1.69% and 0.0169 0.13 13%



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Portfolio Expected Return

Portfolio is merely a collection of assets. ThereforeExpected Return on a portfolio is the weighted average ofexpected rates of return of assets its contains.

1 1 2 2

1

For a two-asset portfolio containing securities 1 and 2, theexpected portfolio return is defined as follows:

is the portfolio return

is the proportion (weight) of the

P

p

R w R w R

R

w

2 1

portfolio invested in asset 1and (1 ) is the proportion placed in asset 2.w w



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Portfolio Expected Return


1

1

The expected return is also a weighted average expected returns

on the individual assets. For the N-asset case:

( )

( )

This is a

N

P P i ii

N

P i ii

R E R E w R

R E w R

1

perfectly general formula of expected retrun (N- assets):

(8) N

P i ii

R w R


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Calculating Portfolio Expected Return


Stock InvestmentProportions

ExpectedReturns

A 0.5 0.20

B 0.5 0.14

1 1 2 2 3 3

If we adds a third stock to this portfolio with investment equal to that

stock A and B and an expected return of 21%, what will be the expected

portfolio return?

Solution P R w R w R w R

1 1 1 (0.20) (0.14) (0.21)

3 3 3 0.1833 18.33%

P R

1 1 2 2

The expected portfolio return is

0.5(0.20) 0.5(0.14)

0.17 17%

P R w R w R

1

N

P i ii

R w R


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Variance of Portfolio (1): Risk

The variance of a portfolio is the varianceof its rate of return. The variance on a

portfolio is a little more difficult todetermine than the expected return.


2 2( ) P P P E R R


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22 21 1 2 2 1 1 2 2

2

1 1 1 2 2 2

2 2 2

2 21

( ) ( )

( ) ( )

Recall that ( ) 2

Applying this to the previous expression:

P P P j j

j j

X Y

P

E R R E w R w R w R w R

E w R R w R R

X Y X XY Y

E w

2 2121 2

2 2 21 1 1 2 1 1 2 2 2 2 2

2 2 2 21 1 1 1 2 1 1 2 2 2 2 2

1 1 2 2

( ) 2 ( )( ) ( )

( ) 2 ( )( ) ( )

where ( )( ) is called

j j j j

j j j j

Cov

j j

R R w w R R R R w R R

w E R R w w E R R R R w E R R

E R R R R

12the covariance .

1

N

P i ii

R w R


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Variance of Portfolio(3): Risk


2 2 2 2 21 1 2 2 1 2 12

We define the variance of a two asset portfolio:

(9) 2 P w w w w

2 2 2

1 1 1

This formula can be extended to any number of assets:

(10) ( ) ( ) j k

N N N

P j j j k jk

j j k

w w w


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Variance of Portfolio (4): 3 Assets


32 2 2 2 2 2 2 2

1 1 2 2 3 31

3 3

1 2 12 1 3 13 2 3 23 1 2 211 1

1 3 31 2 3 32

securities case:

( ) ( )

( ) ( )

2 2

(

j k

N

j j j

N N

j k jk j k

jk kj

jk

Three

i w w w w

ii w w w w w w w w w w

w w w w

Since

w

3 3

1 2 12 1 3 13 2 3 231 1

) 2 2 2 j k

k jk j

w w w w w w w

2 2 2 2 2 2 21 1 2 2 3 3 1 2 12 1 3 13 2 3 23

and ( ) ( ) thus:

2 2 2 P

i ii

w w w w w w w w w


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2 2

1

The first part of expression for variance of portfolio is the sum

of variances on individual assets times square of the proportion

invested in each, as follows:

( ) ( )

The second s

N i i

i

i w

1 1

et of term in the expression for variance of

a portfolio is covariance terms:

( ) ( ) j k

N N

j k jk j k

ii w w


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Covariance and the Benefits ofDiversification


The covariance between two random variables, X and Y can be

defined:

(13) ( , ) ( ) ( ) ,

where E denotes the expectation of the term in braces. Thus,intuitively covariance betw

XY Cov X Y E X E X Y E Y

2P

een two random variables is measure of

the degree to which the variables move together, or covary.

A positive covariance, other things equal, makes (risk) larger,

while a negative covariance reduc

2Pes .


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Securitis and Predetermined



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Calculating the Expected Rate of Returnand Variance of Portfolio (1)


A,B [0.1125 + 0,09 ( A,B)] 1/2 p

+1,0 [0,1125 + (0,09) (1,0)] 1/2 45,0%

+0,5 [0,1125 + (0,09) (0,5)] 1/2 39,8%

+0,2 [0,1125 + (0,09) (0,2)] 1/2 36,1%

0 [0,1125 + (0,09) (0,0)] 1/2 33,5%

-0,2 [0,1125 + (0,09) (-0,2)] 1/2 30,7%

-0,5 [0,1125 + (0,09) (-0,5)] 1/2 25,9%

-1,0 [0,1125 + (0,09) (-1,0)] 1/2 15%

2 2 2 2 2P 2 A A B B A B AB A Bw w w w

2 2 2 2 2AB

AB

12

AB

[(0,5) (0,3) + (0,5) (0,6) + 2 (0,5)(0,5)( )(0,3)(0,6)]

[0,0225 + 0,09 + (0,09) ( )]

[0,1125 + 0,09( )]

P

P


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Example

Suppose you invest 65% of your portfolio in Coca-Cola (CC)and 35% in Reebok. Assume a correlation coefficient of 1 .

Solution:

The expected dollar return on your CC is 10% x 65% = 6.5%and on Reebok it is 20% x 35% = 7.0%. The expected return

on your portfolio is 6.5 + 7.0 = 13.50%.


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PWS


2 2 2 2 2P [(.65) x(31.5) ]+[(.35) x(58.5) ] 2(.65x.35x1x31.5x58.5)

1,006.1Standard Deviation ( ) 1,006.1 31.7 % P

222222211221

2112212221

21

)5.58()35(.x5.585.311

35.65.xxReebok

5.585.311

35.65.xx)5.31()65(.xCola-Coca

Reebok Cola-Coca


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Markowitz Portfolio Theory

Coca Cola

Reebok

Standard Deviation

Expected Return (%)

35% in Reebok

Expected Returns and Standard Deviations varygiven different weighted combinations of the stocks


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Efficient Frontier

Standard Deviation

E x p e c

t e d R e t u r n

( % )

Each half egg shell represents the possible weightedcombinations for two stocks. The composite of all stock sets constitutes the efficient frontier


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Combination of US Stocks andInternational Stocks



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Diversified-N assets


ASSET 1 ASSET 2 ASSET 3 ASSET N

ASSET 1 W 1W 1 1 1 W 1W 2 12 W 1W 3 13 W 1W N 1N

ASSET 2 W 2W 1 12 W 2W 2 2 2 W 2W 3 23 W 2W N 2N

ASSET 3 W 3W 1 13 W 2W 3 23 W 3W 3 3 3 W 3W N 3N

ASSET N W N W 1 N1 W N W 2 N2 W N W 3 N3 W N W N N N

N variances dan [N(N-1)]/2 covariances


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Risk in Equally-Weighted Portfolios:In General


2 2 2P

1

In this case 0 and the formula of variance becomes:

( ) 0

Furhermore, assume equal amount are invested in each asset. With N assets

the proportion invested in each asset i

jk

N

j j j

w

22 2 2P

1 1

2 2 2P

s 1/N. Applying our formula yields.

(14) (1 ) 1

The term in the bracket is expression for an average. Thus the formulareduces to 1 , where represents th

N N j

j j j

j j

N N N

N

e average variance of the

stock in the portfolio. As N gets larger, the variance of the portfolio

gets smaller.

case 0 jk


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Risk in Equally-Weighted Portfolios


2 2 2

1 1 1

In the most market the correlation coefficient and the covariance

between assets is positive. The variance of portfolio of assets is:

( ) ( )

Consider equal inve j k

N N N

P i i j k jk i j k

w w w

2 2 2

1 1 1

stment in N assets.

With equal investment, proportion 1 :

(1 ) 1 1 j k

j

N N N

P j jk j j k

w N

N N N


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2

2

1 1 1

2

Factoring out 1 from first summation and -1 from

the second yields:

1(15) 1( 1)

Both of the terms in the bracket are averages.

(16)

j k

N N N j jk

P j j k

P

N N N

N N N N N N

21 1 j jk

N N N


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2 2

average variance average covariance

1 1

This expression is much more realistic representation of whatoccurs when we invest in a portfolio of assets. The contribution:

P j jk

N N N

variance of the individual asset or securities goes to zero as N gets

very large. However, the contribution of the covariance terms

approaches the average covariance as N gets large.


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2 2

/ / / /

1 1 P j jk

Diversified Undiversified Unsystematic Risk Systematic Risk Unique Risk Market Risk

N N N

Total Risk in Diversifiable Risk Non-Diversifiable Riska Securities (Non market Risk) (Market Risk)


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What happens when N goes to infinity?

Variance of Portfolio Return = average covariance of

returnRisk of Portfolio=Undiversified Risk (market risk)


2 2

0 / /

average covariance1 1

P j jk

Undiversified Systematic Risk Market Risk

N N N

jk


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Risk Diversification



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Table Elton & GruberRisk Diversification


No. ofSecurities

PortfolioVariance

1 46.61

2 26.83

4 16.94

6 13.658 12.03

10 11.01

20 9.036

50 7.849

100 7.453

150 7.321

200 7.255

500 7.137

1000 7.097

Infinity 7.058

As can be seen from above the maximumrisk dispersion can be attained when thenumber of securities increased from 1 to20. After that the Law of DiminishingReturns prevail where any further additionof securities will not reduce risk muchmore significantly.

2 2Market Risk(Undiversified) Non-Market Risk

(Diversified)

1(17) P j jk jk N


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The Effect of Number of Securities onRisk of the Portfolio (US)



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The Effect of Number of Securities onRisk of the Portfolio (UK)



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Diversified Risk (Non market risk)



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References

Elton, E.J, Gruber, M.J, et al (2003) Modern PortfolioTheory and Investment Analysis , 6th Ed, WilleyInternational.Brealy & Myers (2008) Principle of Corporate Finance ,7th Ed, McGraw-Hill, USAMartin, John.D et al (1988) Theory of Finance: Evidence& Applications , The Dryden Press, NY.


Portfolio Analyis, Risk & Return Pws Dmb Ipb

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