Modern Porfolio Theory by p.rai87@Gmail

8/8/2019 Modern Porfolio Theory by p.rai87@Gmail

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Modern Portfolio Theory

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PORTFOLIO

Do not put all your eggs in one basket.

The term portfolio is usually applied to define

combination of securities..

It is done to reduce risk of investor without sacrificing

returns..

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Markowitz (1952) Portfolio

selectionReturn of portfolio

Characteristics of a portfolio:

1. Expected return

2. Risk : Variance/Standard deviation

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Portfolio ReturnsSimply the Weighted Average of Expected Returns

RelativeWeight

ExpectedReturn

WeightedReturn

Stock X 0.400 8.0% 0.03

Stock Y 0.350 15.0% 0.05

Stock Z 0.250 25.0% 0.06

Expected Portfolio Return = 14.70%

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Standard Deviation or risk for

individual securityThe formula for the standard deviation

when analyzing sample data (realized

returns) is:

1

)(1

2

=

=

n

kkn

i

ii

Where k is a realized return on the stock and n is the

number of returns used in the calculation of the mean.

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Standard Deviation or risk for

individual securityThe formula for the standard deviation whenanalyzing forecast data (ex ante returns) is:

it is the square root of the sum of the squared

deviations away from the expected value.

=

=n

i

iii Pkk1

2)(

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Expected Risk For Portfolios

Standard Deviation of a Two-Asset Portfolio using Covariance

))()((2)()()()( ,2222 BABABBAAp COVwwww ++=

Risk of Asset A

adjusted for weightin the portfolio

Risk of Asset B

adjusted for weightin the portfolio

Factor to take into

account comovement ofreturns. This factor

can be negative.

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Expected Risk For Portfolios

Standard Deviation of a Two-Asset Portfolio using Correlation

Coefficient

))()()()((2)()()()( ,2222 BABABABBAAp wwww ++=

Factor that takes into

account the degree of

comovement of returns.

It can have a negative

value if correlation is

negative.

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Grouping Individual Assets into

PortfoliosThe riskiness of a portfolio that is made of different risky assets is afunction of three different factors:

the riskiness of the individual assets that make up the portfolio

the relative weights of the assets in the portfolio

the degree of comovement of returns of the assets making up theportfolio

The standard deviation of a two-asset portfolio may be measured

using the Markowitz model:

BABABABBAAp wwww ,2222 2++=

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Correlation

The degree to which the returns of two stocks co-move is

measured by the correlation coefficient ().

The correlation coefficient () between the returns on two

securities will lie in the range of +1 through - 1.+1 is perfect positive correlation

-1 is perfect negative correlation

BA

ABAB

COV

=[8-13]

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Covariance and Correlation

CoefficientSolving for covariance given the correlation

coefficient and standard deviation of the

two assets:

BAABABCOV =[8-14]

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Expected Portfolio ReturnImpact of the Correlation Coefficient

15

10

5

0

Standa

rdDeviation(%)of

Portfo

lioReturns

Correlation Coefficient ()

-1 -0.5 0 0.5 1

8 - 15


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Efficient frontier

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8 - 17

Efficient FrontierThe Two-Asset Portfolio Combinations

A is not attainable

B,E lie on theefficient frontier and

are attainable

E is the minimumvariance portfolio

(lowest risk

combination)

C, D are attainablebut are dominated by

superior portfolios

that line on the line

above E

Expected

Return

% Standard Deviation (%)

A

E

B

C

D


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8 - 18

Efficient FrontierThe Two-Asset Portfolio Combinations

Expected

Return

% Standard Deviation (%)

A

E

B

C

D

Rational, risk

averse

investors will

only want tohold portfolios

such as B.

The actual

choice will

depend on

her/his risk

preferences.


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Diversification Potential

The potential of an asset to diversify a portfolio is

dependent upon the degree of co-movement of returns of

the asset with those other assets that make up the

portfolio.

In a simple, two-asset case, if the returns of the two assets

are perfectly negatively correlated it is possible

(depending on the relative weighting) to eliminate allportfolio risk.


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Example of Portfolio

Combinations and Correlation

Asset

Expected

Return

Standard

Deviation

Correlation

Coefficient

A 5.0% 15.0% 1

B 14.0% 40.0%

Weight of A Weight of B

Expected

Return

Standard

Deviation

100.00% 0.00% 5.00% 15.0%

90.00% 10.00% 5.90% 17.5%

80.00% 20.00% 6.80% 20.0%

70.00% 30.00% 7.70% 22.5%

60.00% 40.00% 8.60% 25.0%50.00% 50.00% 9.50% 27.5%

40.00% 60.00% 10.40% 30.0%

30.00% 70.00% 11.30% 32.5%

20.00% 80.00% 12.20% 35.0%

10.00% 90.00% 13.10% 37.5%

0.00% 100.00% 14.00% 40.0%

Portfolio Components Portfolio Characteristics

Perfect

Positive

Correlation

no

diversification

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Asset

Expected

Return

Standard

Deviation

Correlation

Coefficient

A 5.0% 15.0% 0.5

B 14.0% 40.0%


Expected

Return

Standard

Deviation

100.00% 0.00% 5.00% 15.0%

90.00% 10.00% 5.90% 15.9%

80.00% 20.00% 6.80% 17.4%

70.00% 30.00% 7.70% 19.5%

60.00% 40.00% 8.60% 21.9%50.00% 50.00% 9.50% 24.6%

40.00% 60.00% 10.40% 27.5%

30.00% 70.00% 11.30% 30.5%

20.00% 80.00% 12.20% 33.6%

10.00% 90.00% 13.10% 36.8%

0.00% 100.00% 14.00% 40.0%


Positive

Correlation

weak

diversification

potential

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Asset

Expected

Return

Standard

Deviation

Correlation

Coefficient

A 5.0% 15.0% 0

B 14.0% 40.0%


Expected

Return

Standard

Deviation

100.00% 0.00% 5.00% 15.0%

90.00% 10.00% 5.90% 14.1%

80.00% 20.00% 6.80% 14.4%

70.00% 30.00% 7.70% 15.9%

60.00% 40.00% 8.60% 18.4%50.00% 50.00% 9.50% 21.4%

40.00% 60.00% 10.40% 24.7%

30.00% 70.00% 11.30% 28.4%

20.00% 80.00% 12.20% 32.1%

10.00% 90.00% 13.10% 36.0%

0.00% 100.00% 14.00% 40.0%


No

Correlation

some

diversification

potential

Lower

risk than

asset A

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Asset

Expected

Return

Standard

Deviation

Correlation

Coefficient

A 5.0% 15.0% -0.5

B 14.0% 40.0%


Expected

Return

Standard

Deviation

100.00% 0.00% 5.00% 15.0%

90.00% 10.00% 5.90% 12.0%

80.00% 20.00% 6.80% 10.6%

70.00% 30.00% 7.70% 11.3%

60.00% 40.00% 8.60% 13.9%50.00% 50.00% 9.50% 17.5%

40.00% 60.00% 10.40% 21.6%

30.00% 70.00% 11.30% 26.0%

20.00% 80.00% 12.20% 30.6%

10.00% 90.00% 13.10% 35.3%

0.00% 100.00% 14.00% 40.0%


Negative

Correlation

greater

diversification

potential

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Asset

Expected

Return

Standard

Deviation

Correlation

Coefficient

A 5.0% 15.0% -1

B 14.0% 40.0%


Expected

Return

Standard

Deviation

100.00% 0.00% 5.00% 15.0%

90.00% 10.00% 5.90% 9.5%

80.00% 20.00% 6.80% 4.0%

70.00% 30.00% 7.70% 1.5%

60.00% 40.00% 8.60% 7.0%50.00% 50.00% 9.50% 12.5%

40.00% 60.00% 10.40% 18.0%

30.00% 70.00% 11.30% 23.5%

20.00% 80.00% 12.20% 29.0%

10.00% 90.00% 13.10% 34.5%

0.00% 100.00% 14.00% 40.0%


Perfect

Negative

Correlation

greatest

diversification

potential

Risk of the

portfolio is

almost

eliminated at

70% asset A

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Diversification of a Two Asset Portfolio Demonstrated


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The Effect of Correlation on Portfolio Risk:

The Two-Asset Case

Expected

Return

StandardDeviation

0%0% 10

%

4%

8%

20% 30% 40%

12%

B

AB =+1

A

AB =0

AB =-0.5AB =

-1

Diversification of a Two Asset Portfolio Demonstrated

Graphically

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The Single Index Model

It is given by William sharpe.

It Introduces Market index,which is a

tangent to efficient frontier.

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According to Single Index Model:

Where, Ri= return on security i

Ai= constant return

Bi= measure of thesensitivity of the security Isreturn to the return on the marketindex

Rm= return on the marketindex

Ei= error term

E(Ri) = Ai+biE(Rm)

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Assumptions for SingleIndex Model

The Error term(ei) has anexpected value of Zero and a

finite variance.

Cov (Ei, Rm)=0

Cov (Ei

,Ej

) = 0

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Calculation of the SingleIndex Model

E(Ri) = Ai+biE(Rm)+ei

If there are n securities, in this

model we need 3n+2estimates,

By contrast the Markowitz

model requires n(n+3)/f2estimates,

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Multi-IndexModel

The general form of amulti-index model:

1 1 2 2 ...

where constant

return on the market index

return on an industry index

Security 's beta for industry index

Security 's market beta

retur

i i im m i i in n

i

m

j

ij

im

i

R a I I I I

a

I

I

i j

i

R

= + + + + +

=

=

=

=

=

=

% % % % %

%

%

% n on Security i

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AssumptionforCAPM

Individuals are risk averse.They seek to maximize the expected utilitof their portfolio.

There is a one-period time horizonThey have homogenous expectations.They can borrow and lend freely at a risk

free interest rate.The market is perfect.The quantity of risky securities in the

market is given.

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CAPM & Market Efficiency

CAPM can test Efficient Market Hypothesis.

Market is efficient if only risk-free assetsgive risk-free rates of return (e.g., Treasurybills).

Deviations may indicate opportunities.

Modeling predictions can suggestimprovements to market functioning.

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L di & B i U d


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Lending & BorrowingUndertheCaPM

Assumption of unlimited lending and borrowing atrisk-free rate.

Lending if portion of portfolio held in risk-free

assets.Borrowing (leverage) if more than 100% of

portfolio is invested in risky assets.Superior returns made possible with lending and

borrowing; creates spectrum of risk preference for

different investors.

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CAPITAL ASSETPRICING MODEL

Capital Market Line: Linear risk-returntrade-off for all investment portfolios.

Standard Deviation (total portfolio risk)

E(R)

M

Rf

= market

z

K

L

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The Capital Market Line

(CML)The equation for CML is:

( )( )

( )( )

( )( )

( )[ ]

E R RE R R

SD RSD R

R SD RSD R

E R R

P F

M F

M

P

FP

M

M F

= +

= +

40

Security Market Line


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Security Market Line(SML)

The equation for SML is,

E(Ri)= Rf+E(Rm)-Rf im

Expected return on security i=

Risk free return + Market risk premium* Beta of security

i= im / 2M

2

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Inputs required for applying

CAPMRisk-free rate

Market Risk Premium

Beta

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What drives the Market Risk

Premium

Variance in the underlying economy.Political risk

Market structure

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The BETA Factor

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Results

The relation appears to be linear.

In general 0 is greater than the risk-free rate

and 1 is less than Rm - Rf.

In addition to beta, some other factors such as

standard deviation of returns and company size,

too have a bearing on return.

Beta does not explain a very high percentage of

the variance in return among securities.

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KEY TERMSCapital Asset Pricing

investment portfolio

portfolio theory

expected return

risk

probability distribution

utility

disutility

risk averse

zero-risk portfolio

efficient portfolio

efficient frontier

optimal portfolio

market portfolio

capital asset pricing model

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Efficient Market Hypothesis

Market efficiency is defined in relation to

information that is related in security prices.

Three levels of market efficiency:1. Weak- form efficiency

2. Semi-strong form efficiency

3. Strong form efficiency

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Empirical evidences on weak- form

efficient market hypothesis

1.Serial correlation test-

check the auto-correlations.if such auto-correalations are negligible, the

price changes are considered to be seriallyindependent.

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2. Run tests-: Given a series of stock price changes.

+ represents an increase in price. represents a decrease in price.

++-++--+

A run occurs when there is no difference

between the sign of two changes.

++ - ++ -- +

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3.Filter Rules Test

An x% filter rule may be defined as follows:

if the price of a stock increases by at least x%, buy and

hold it until its price decrease by at least x% from a

subsequent high. When the price decreases by at least x% or more, sell it.

If the behavior of stock price changes is random, filter

rules should not perform a simple buy-hold strategy.

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E i

i l E id i


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Empirical Evidence on semi-Empirical Evidence on semi-

strong form efficient marketstrong form efficient market

1. Discount rate change:-

The average security price change a littlebefore the announcement of discount rate

changes.

Such a change is not enough to yield a

trading profit

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2. Stock Spilt:-

If there is a stock split and the accompanying

information with respect to change in dividend

policy.it give favorable finding to conclude that

market was efficient.

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i i l idE i

i l E id


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Empirical Evidence onEmpirical Evidence on

Strong FormStrong Form

To test the strong form efficient market

hypothesis, researchers analyzed the returns

earned by certain groups(like corporateinsider, specialist on stock exchange) who

have access to information which is not

publicly available and ostensibly possessgreater recourses and the abilities to

intensively analyze information which is in the

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Different degree of


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Different degree ofmarket price efficency

Tim

e

1

23

Rs50

1- weak-EMH, 2- semi strong-EMH 3- strong-EMH

price

t+0 t+1 t+2 t+3 t+4 t+5 t+6

t+7 t+8 t+9

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Portfolio Evaluation

Portfolio evaluating refers to the evaluation

of the performance of the portfolio.

It is essentially the process of comparingthe return earned on a portfolio with the

return earned on one or more other portfolio

or on a benchmark portfolio.

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M h d P f liM h d P f li


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Methods to measure PortfolioMethods to measure Portfolio

PerformancePerformance

Sharpes Ratio- SR = (Rp Rf)/p

Where,

SR = Sharpes Ratio

Rp = Average return on portfolio

Rf= Risk free returnp = Standard deviation of the portfolio return.

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Treynors Measure

Tn = (Rp Rf)/p

Where, Tn= Treynors measure of performance

Rp = Return on the portfolio

Rf = Risk free rate of return

p = Beta of the portfolio ( A measure of

systematic risk)

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Jensen Measure

Ri = Rf+ (RMI Rf) x

Where,

Ri = Return on portfolioRMI = Return on market index

Rf= Risk free rate of return

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Various plans for PortfolioVarious pl

ans for Portfolio


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Various plans for PortfolioVarious plans for Portfolio

RevisionRevision

Portfolio revision can be studied under the

fallowing plane:-

Rupee Cost Averaging

Constant Rupee Plan

Variable Ratio Plan

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Problem-2

The following information is available:

Expected return for the market=14%

Standard deviation of market return=20%

Risk-free return=6%Correlation coefficient b/w stock A and market= 0.7%

Correlation coefficient b/w stock B and market= 0.8%

Standard deviation for stock A=24%

Standard deviation for stock B=32%

a). Calculate the beta for stock A and B

b).Calculate the required return for each stock.

Modern Porfolio Theory by p.rai87@Gmail

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