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ByPriya Kansal

Assistant Professor

Jaipuria Institute of Management

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A portfolio is a combination of two or more

securities.

Combining securities into a portfolioreduces risk.

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Expected Return Risk

The Expected

Returns

of the

Securities

The

Portfolio

Weights

The Risk

of the

Securities

The

Portfolio

Weights

The

Correlation

Coefficients

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The Expected Return on a Portfolio is computed as theweighted average of the expected returns on the stocks whichcomprise the portfolio.

The weights reflect the proportion of the portfolio invested in thestocks.

This can be expressed as follows:N

E[Rp] = 7 wiE[Ri]i=1

Where: E[Rp] = the expected return on the portfolio

N = the number of stocks in the portfolio

wi = the proportion of the portfolio invested in stock i

E[Ri] = th

e expected return on stock i 4

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The variance/standard deviation of a portfolio reflects not onlythe variance/standard deviation of the stocks that make up theportfolio but also how the returns on the stocks which comprisethe portfolio vary together.

Two measures ofhow the returns on a pair of stocks varytogether are the covariance and the correlation coefficient.

Covariance is a measure that combines the variance of a stocks returnswith the tendency of those returns to move up or down at the same timeother stocks move up or down.

Since it is difficult to interpret the magnitude of the covariance terms, arelated statistic, the correlation coefficient, is often used to measure thedegree of co-movement between two variables. The correlation coefficientsimply standardizes the covariance.

6

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ijjij

n

i

n

j

ipVWW[[W

! !

!

1 1

2

For a two asset portfolio,

ABBABABBAAp VVV[[W[W[W 222222!

For a three asset portfolio,

CAACACBCCBCBABBABACCBBAAp VVV[[VVV[[VVV[[W[W[W[W 2222222222

!

For a n asset portfolio,

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Traditional Portfolio Analysis

Modern Portfolio Analysis

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Emphasizes balancing the portfolio using awide variety of stocks and/or bonds

Uses a broad range of industries to diversifythe portfolio

Tends to focus on well-known companies Perceived as less risky Stocks are more liquid and available Familiarity provides higher comfort levels

for investors

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10

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Emphasizes statistical measures to develop aportfolio plan

Focus is on: Expected returns Standard deviation of returns Correlation between returns

Combines securities that have negative (or low-positive) correlations between each others ratesof return

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12

Any asset or portfolio can be described by twocharacteristics:

1. The expected return

2. The risk measure (variance) Portfolios variance is a function of not only the variance

of returns on the individual investments in the portfolio,but also of the covariance between returns of these

individual investments. In a large portfolio, the covariances are much moreimportant determinants of the total portfolio variancethan the variances of individual investments.

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13

V1,2= Correlation coefficient ofreturns

V1,2= Correlation coefficient ofreturns

Cov(r1r2) =VW1W2Cov(r1r2) =VW1W2

W1 = Standard deviation of

returns for Security 1W2 = Standard deviation of

returns for Security 2

W1 = Standard deviation of

returns for Security 1W2 = Standard deviation of

returns for Security 2

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14

Investors consider investments as the probabilitydistribution of expected returns over a holding

period. Investors seek to maximize expected utility Investors measure portfolio risk on the basis of

expected return variability

Investors make decisions only on th

e basis ofexpected return and risk For a given level of risk, investors prefer higher

return to lower returns.

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15

Utility = Expected Return- Risk PenaltyWhere risk penalty depends upon portfolios risk &

investors risk tolerance.

Risk Penalty = Variance/ Risk ToleranceRiskTolerance varies from 0 to 100

Risk penalty is less as tolerance increases

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16

Indifference curves represent differentcombinations of risk and return, which provide thesame level of utility to the investor.

An investor is indifferent between any twoportfolios that lie on the same indifference curve.

Flat indifference curves indicate that an individualhas a higher tolerance for risk. Very steep

indifference curves belong to highly risk-averseinvestors. The optimal portfolio offers the greatest amount of

utility to the individual investor.

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return

risk

Highly risk averse

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return

risk

Highly risk tolerant

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Efficient Frontier

The efficient frontier consists of the set portfolios that

has the maximum expected return for a given risk level. The leftmost boundary of the feasible set of portfolios that

include all efficient portfolios: those providing the bestattainable tradeoff between risk and return

Portfolios that fall to the right of the efficient frontier are notdesirable because their risk return tradeoffsare inferior

Portfolios that fall to the left of the efficient frontier are notavailable for investments

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In a real world investment universe with all of the investmentalternatives (stocks, bonds, money market securities, hybrid

instruments, gold real estate, etc.) it is possible to constructmany different alternative portfolios out of risky securities.

Each portfolio will have its own unique expected return andrisk.

Whenever you construct a portfolio, you can measure two

fundamental characteristics of the portfolio: Portfolio expected return (ERp)

Portfolio risk (p)

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You could start by randomly assembling tenrisky portfolios.

The results (in terms of ER p and p )mightlook like the graph on the following page:

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Portfolio Risk (p)

10Achievable RiskyPortfolio

Combinations

ERp

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You could continue randomly assembling

more portfolios. Thirty risky portfolios might look like the

graph on the following slide:

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Portfolio Risk (p)

30 is rtf li

i ti s

ERp

Thirty Combinations Naively Created

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When you construct many hundreds of

different portfolios naively varying the weightof the individual assets and the number oftypes of assets themselves, you get a set of

achievable portfolio combinations asindicated on the following slide:

All Securities Many Hundreds of Different Combinations

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Portfolio Risk (p)

ERp

E

E is the

minimumvariance

portfolio Achievable Set of Risky

Portfolio Combinations

The highlighted

portfolios are

efficient in that

they offer the

highest rate of return

for a given level of

risk. Rationale

investors will choose

only from this

efficient set.

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Portfolio Risk (p)

Achievable Set of Risky

Portfolio Combinations

ERp

E

Efficient

frontier is the

set ofachievable

portfoliocombinationsthat offer the

highest rate of

return for a

given level of

risk.

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28

Optimal portfolio: the portfolio that lies at thepoint of tangency between the efficient frontier andhis/her utility (indifference) curve.

An investors optimal portfolio is the efficientportfolio that yields the highest utility.

A risk averse investor has steep utility curves.

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Optimal Portfolio (O)

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Complex model, involving tough calculations.

For n securities, n returns, n variances & n(n-1)/2

co variances. Exact estimation of investors risk tolerance of is

not an easy task

Revision is not easy

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Developed byWilliam F. Sharpe in 1963

Indicates the allocation of the investments in

the portfolio b/w individual equity shares. Substantially reduced the number of

required inputs when estimating portfoliorisk. Instead of estimating the correlation

between every pair of securities, simplycorrelate each security with an index of all

of the securities included in the analysis

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32

The single-index modelcompares allsecurities to a single benchmark

An alternative to comparing a security to eachof the others

By observing how two independent securities

behave relative to a third value, we learnsomething about how the securities are likelyto behave relative to each other

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33

Beta of a portfolio:

Variance of a portfolio:

1

n

p i i

i

xF F!

!

2 2 2 2

2 2

p p m ep

p m

W F W WF W

! }

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34

Variance of a portfolio component:

Covariance of two portfolio components:

2 2 2 2

i i m eiW F W W!

2 AB A B mW F F W !

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The return of the securities are related onlythrough common relationship with some

basic underlying factors This factor may be the level of stock marketas a whole, the GNP, price index or any otherfactor thought to be most important

The only reason s

hares vary toget

hersystematically is because of a common co

movement with the market & there are noeffects beyond the market

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Relationship b/w the stock return & market

return is given by,

imiii eRR ! FE

Where,

iR

iE

iF

R

ie

= Expected return on security I

= Market Return

= return free from market

= slope of the line

= residual term/risk not relatedto the market risk i.e.

unsystematic risk

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2

2

( , )

where return on the market index

variance of the market returns

return on Security

i mi

m

m

m

i

COV R R

R

R i

FW

W

!

!

!

!

% %

%

%

M

sSM

i

M

MsSM

i

r

r

W

WF

W

WWF

!

! 2

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securitytodevotedportfoliotheofproportionthe,

)(

!

! !

i

N

i

Miiip

where

RR

[

FE[

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Systematic Risk

Unsystematic Risk

Systematic Risk

Unsystematic Risk

Total Risk

22

2IndexofVariance

Mi

i

WF

F

v!

v!

2

ie!

222

iMieWF

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][

])[(

N

1i

1

!

!

!

ii

M

N

i

iip

e[

WF[W

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Calculate the excess return to beta ration

for each security under review i.e.

Rank from highest to lowest

The optimal portfolio consists of investment

in all securities for which excess return tobeta ratio is greater then a particular cut off

point,

i

fi RRF

*C

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!

!

!N

ie

iM

N

i e

ifi

M

i

i

RR

1

2

2

2

12

2

*

1

)(

W

FW

W

F

W

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Once known which securities are to be

included in the optimum portfolio, the %age

invested in each security is

Where ,

!

!N

i

i

i

i

Z

Z

1

[

)( *CRR

Zi

fi

e

ii

i

!FW

F

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It also include lots of calculations.

Estimation of Beta in real life situation is notvery easy.

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The CAPM is an equilibrium modelthat specifies the relationshipbetween risk and required rate of

return for assets held in well-diversified portfolios.

It is based on the premise that only

one factor affects risk.

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Investors all think in terms ofa single

holding period.

All investors have identical expectations.

Investors can borrow or lend unlimited

amounts at the risk-free rate.

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All assets are perfectly divisible.

There are no taxes and no transactions costs.

All investors are price takers, that is,investors buying and selling wont influence

stock prices.

Quantities of all assets are given and fixed.

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ExpectedPortfolio

Return, rp

Risk, Wp

Efficient Set

Feasible Set

Feasible and Efficient Portfolios

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The feasible set of portfolios represents allportfolios that can be constructed from a givenset of stocks.

An efficient portfolio is one that offers:

the most return for a given amount of risk, or

th

e least risk for a give amount of return. The collection of efficient portfolios is called

the efficient set or efficient frontier.

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IB2 IB1

IA2IA1

Optimal Portfolio

Investor A

Optimal Portfolio

Investor B

Risk Wp

Expected

Return, rp

Optimal Portfolios

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Indifference curves reflect an investorsattitude toward risk as reflected in his or

her risk/return tradeoff function. Theydiffer among investors because ofdifferences in risk aversion.

An investors optimal portfolio is definedby the tangency point between theefficient set and the investors

indifference curve.

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When a risk-free asset is added to thefeasible set, investors can create portfolios

that combine this asset with a portfolio ofrisky assets.

The straight line connecting rRF with M, the

tangency point between th

e line and th

eold efficient set, becomes the new efficientfrontier.

What impact does Risk free asset have on

the efficient frontier?

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Risk

ER

RF

A

Ap WW w![9-2]

Equation 9 2

illustrates

what you canseeportfolio

risk increases

in direct

proportion to

the amount

invested in the

risky asset.

RF-)E(R

RFERA

A

PP WW

-

!

Rearranging 9

-2 where w=

p/ A andsubstituting in

Equation 1 we

get an

equation for a

straight line

with a

constant

slope.

This means

you can

achieve any

portfolio

combinationalong the blue

coloured line

simply by

changing the

relative weight

ofRFandA inthe two asset

portfolio.

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Which risky

portfolio

would a

rational risk-

averseinvestor

choose in the

presence of a

RF

investment?

PortfolioA?

Tangent

Portfolio T?Risk

ER

RF

A

T

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Risk

ER

RF

A

T

Clearly RFwith

T(the tangent

portfolio) offers

a series of

portfolio

combinationsthat dominate

those produced

by RFandA.

Further, they

dominate all but

one portfolio on

the efficient

frontier!

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Risk

ER

RF

A

T

Portfolios

between RF

and Tare

lending

portfolios,

because theyare achieved by

investing in the

Tangent

Portfolio and

lending funds to

the government(purchasing a

T-bill, the RF).

Lending Portfolios

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Risk

ER

RF

A

T

The line can be

extended to risk

levels beyond

T by

borrowing at RF

and investing itin T. This is a

levered

investment that

increases both

risk and

expected returnof the portfolio.

Lending Portfolios Borrowing Portfolios

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ER

RF

A2

T

A

B

B2

Capital Market LineThe optimal

risky portfolio

(the market

portfolio M)

Clearly RF with

T (the market

portfolio) offers

a series of

portfoliocombinations

that dominate

those produced

by RF and A.

Further, they

dominate all butone portfolio on

the efficient

frontier!

This is now

called the new

(or super)

efficient frontier

of risky

portfolios.

Investors can

achieve any

one of these

portfolio

combinations

by borrowing orinvesting in RF

in combination

with the market

portfolio.

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The Capital Market Line (CML) is all linearcombinations of the risk-free asset andPortfolio M.

Represent the relationship b/w risk andreturn of efficient portfolio

Portfolios below the CML are inferior. The CML defines the new efficient set.

All investors will choose a portfolio on the CML.

What is the Capital Market Line?

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rp = rRF +

SlopeIntercept

Wp.

The CML Equation

rM

- rRF

WM

Risk

measure

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The Security Market Line (SML)

)(fmsfsrrrr ! F )( fmsfs rrrr

! F

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What does the SML tell us

The required rate of return on a security

depends on:

the risk free rate

t

he beta of t

he security, and

the market price of risk.

The required return is a linear function of the

beta coefficient.

All else being the same, higher the beta coefficient, higher is the

required return on the security.

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Graphical Representation of the SML

=0.5 =1.0 =1.5

Rf

Rm

Rate of

Return

Defensive

Security

Aggressive

Security

Conservative

Investment

Aggressive

Investment

F

A

M

P

Q

Underpriced

Overpriced

SML

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It is based on highly restrictive assumptions

Market factor is not th

e sole factorinfluencing stock return.

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A pricing model that uses multiple factors to relateexpected returns to risk by assuming that asset returns arelinearly related to a set of indexes, which proxy risk factorsthat influence security returns.

It is based on the no-arbitrage principle which is the rulethat two otherwise identical assets cannot sell at differentprices.

Underlying factors represent broad economic forces whichare inherently unpredictable.

...11110 niniii

FbFbFbaER ![9-10]

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Where: ERi= the expected return on security i a

0= the expected return on a security with zero

systematic risk bi = the sensitivity of security i to a given risk

factor Fi = the risk premium for a given risk factor

The model demonstrates that a securitys risk isbased on its sensitivity to broad economic forces.

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Ross and Roll identify five systematic factors:1. Changes in expected inflation

2. Unanticipated changes in inflation

3. Unanticipated changes in industrial production4. Unanticipated changes in the default-risk

premium

5. Unanticipated changes in the term structure ofinterest rates

Clearly, something that isnt forecast, cant beused to price securities todaythey can only beused to explain prices after the fact.