portfolio study

7/29/2019 portfolio study

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PORTFOLIO ANALYSIS

DR SUNANDA MITRA GHOSH


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Portfolio Construction


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Objectives

To know the concept of portfolio construction

To determine the objectives in the traditional

approach

To select the securities to be included in the

portfolio

To learn the basics of modern approach


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Portfolio

Portfolio is a combination of securities such as

stocks, bonds and money market instruments.

The process of blending together the assetclasses

so as to obtain optimum return with minimum

risk is called portfolio construction.

Diversification of investments helps to spread

risk over many assets.


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Approaches in

Portfolio Construction

Traditional approach evaluates the entire

financial plan of the individual.

In the modern approach, portfolios are

constructed to maximise the expected return for

a given level of risk.


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Traditional Approach

The traditional approach basically deals with two

major decisions:

Determining the objectives of the portfolio

Selection of securities to be included in the portfolio


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Steps in Traditional ApproachAnalysis of Constraints

Determination of Objectives

Selection of Protfolio

Assessment of risk and return

Diversification

BondBond and Common stock Common stock


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Analysis of Constraints

Strong lyefficientmar ketAllinf ormationisreflect edonprices.

Weaklyeffici entmarketAllhistorical informationisreflectedon security

Sem istrongefficien tmarketAll publicinformat ionisref lectedonsecurit yprices







Income needs

Need for current income

Need for constant income

Liquidity

Safety of the principal

Time horizon

Tax consideration

Temperament


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Determination of Objectives

The common objectives are stated below:

Current income

Growth in income

Capital appreciation

Preservation of capital


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Selection of Portfolio

Objectives and asset mix

Growth of income and asset mix

Capital appreciation and asset mix

Safety of principal and asset mix

Risk and return analysis


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Diversification

According to the investors need for income and

risk tolerance level portfolio is diversified.

In the bond portfolio, the investor has to strike a

balance between the short term and long term

bonds.


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Stock Portfolio

Selection of Industries

Selection of Companies in the Industry

Determining the size of participation


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EFFECTS OF COMBINING THESECURITIES

Modern portfoliotheory (MPT


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Modern portfoliotheory (MPT)

Modern portfolio theory (MPT) is a theoryof investment

which attempts to maximize

portfolio expected return

for a given amount of portfolio risk,

by carefully choosing the proportions of

various assets.


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Modern Approach

Modern approach gives more attention to theprocess of selecting the portfolio.

The selection is based on the risk and return

analysis. Return includes the market return and dividend.


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several of its creators won aNobel prize for the theory

MPT is widely used in practice in the financialindustry


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MPT is a mathematical formulation of the

concept of diversification in investing,

Aim of selecting a collection of investmentassets that has collectively lower risk thanany individual asset.


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Different types of assets often change in value

in opposite ways.

For example, when prices in the stock

market fall, prices in the bond market oftenincrease, and vice versa

A collection of both types of assets can

therefore have lower overall risk than eitherindividually.

diversification lowers risk.


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The fundamental concept

behind MPT is that the assets in an

investment portfolio should not be selected

individually,

each on their own merits.

Rather, it is important to consider

how each asset changes in price relative tohow every other asset in the portfolio changes

in price.


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Investing is a trade-off between risk andexpected return.

In general, assets with higher expected returns areriskier.

For a given amount of risk,

MPT describes how to select a portfolio with thehighest possible expected return.

Or, for a given expected return,

MPT explains how to select a portfolio with the lowestpossible risk

(the targeted expected return cannot be more thanthe highest-returning available security, of course,

unless negative holdings of assets are possible.)


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MPT is therefore a form of diversification.

Under certain assumptions and for

specific quantitative definitions of risk and

return, MPT explains

how to find the best possible diversification

strategy.


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EFFECTS OF COMBINING THESECURITIES

As per Markowitz ,

Given the return , risk can be reduced bydiversifying of investment in to number of

scripsTwo scrips A & B, A is more riskierA B

Expected return 40% 30%

RISK 15% 10%

INVESTMENT IN A = 60% (.6) , B= 40%(.4)

RETURN ON PORTFOLIO=( 40 *.6) + (30 * .4) = 36%

RISK ON PORTFOLIO = =( 15 *.6) + (10 * .4) = 13%


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MATHEMATICAL

MODEL

Risk and expected return


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If given two portfolios

that offer the same expected return,

investors will prefer the less risky one.

Thus, an investor will take on increased risk

only if compensated by higher expectedreturns.


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an investor wants higher expected returns

must accept more risk.

The exact trade-off will be the same for all

investors,

but different investors will evaluate

the trade-off differently

based on individual risk aversioncharacteristics.


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The implication is that a rational investor will

not invest in a portfolio if a second portfolio

exists with a more favourable

risk-expected return profile

i.e., if for that level of risk an alternative

portfolio exists which has better expected

returns. The theory uses standard deviation of return

as a proxy for risk.


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Portfolio return is the

proportion-weighted combination of the assets'returns.


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Expected return:

E( Rp) = Wi E(Ri)

Rp = return on portfolio

Ri =return on assets I

Wi = weighting of component assets (share of the asset in portfolio)


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Portfolio risk

the risk involved in individual securities can bemeasured by standard deviation or variance

When two securities are combined we need to

consider their- Interactive risk or covariance

If rates of return of two securities move together

-Interactive risk or covariance is positive

-If rates are independent- covariance is zero

-Inverse movement - covariance is negative

M h i ll i i


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Mathematically covariance isdefined as -

N

Covxy = 1/N (RxRx ) (RyRy )

Covxy = Covariance between X & Y

Rx= Return on security X

Rx = expected return to security X

Ry = Return on security Y

Ry =expected return to security Y

N = number of observations


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RETURN EXPECTED DIFFERNCES


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RETURN EXPECTEDRETURN

DIFFERNCES

STOCK X 7 9 -2

STOCK Y 13 9 4

PRODUCT - 8

STOCK X 11 9 2

STOCK Y 5 9 -4

PRODUCT - 8COV= (7-9)(13-9)+(11-9)(5-9)

=1/2(-8)+(-8)=-16/2 =8


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Modern portfolio theory


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The fundamental concept behind MPTis that the assets in aninvestment portfolio should not be

selected individually, each on their ownmerits. Rather, it is important toconsider how each asset changes in

price relative to how every other assetin the portfolio changes in price.

Investing is a trade off between risk andexpected return.


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In general, assets with higher expected

returns are riskier.


MPT describes how to select a portfolio

with the highest possible expected return.

Harry Markowitz introduced MPT in a1952 articleand a 1959 book


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MPT assumes that investors are risk averse,meaning that given two portfolios that offer thesame expected return, investors will prefer theless risky one.

Thus, an investor will take on increased risk only ifcompensated by higher expected returns.

Conversely, an investor who wants higherexpected returns must accept more risk.

The exact trade-off will be the same for allinvestors, but different investors will evaluate thetrade-off differently based on individual riskaversion characteristics.


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The implication is that a rational investor will not investin a portfolio if a second portfolio exists with a morefavorable risk-expected return profile i.e., if for thatlevel of risk an alternative portfolio exists which hasbetter expected returns.

Theory uses standard deviation of return as a proxyfor risk.

Under the model:

Portfolio return is the proportion-weighted

combination of the constituent assets' returns. Portfolio volatility is a function of the correlationsij of

the component assets, for all asset pairs (i,j).


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Portfolio return is the proportion-weighted

combination of the constituent assets' returns.

Portfolio volatility is a function of

the correlationsij of the component assets,

for all asset pairs (i,j).


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Expected return:

E( Rp) = Wi E(Ri)

Rp = return on portfolio

Ri =return on assets I

Wi = weighting of component assets (share of the asset in portfolio)


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p2 =Wi2i2 +WiWj ij

ij=Correlation coefficientbetween

the returns on assets i and j(Between assets A & B & C..N)

Portfolio return variance --p2


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p2 =wi2i2 +wiwj ij

ij = 1 for i=j


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For Two assets portfolio

Portfolio return:

E( Rp) = wAE(RA)+ WBE(RB)= WA E(RA)+ (1- WA)E(RB)

SIMPLY = wAE(RA)+ WBE(RB)

Portfolio variance: p

2 = wA

2

A

2 + wB

2

B

2 +2 wA

wB

A B AB


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For THREE assets portfolio

Portfolio return:

E( Rp) = wAE(RA)+ WBE(RB) + WCE(RC)

Portfolio variance:p

2 =w

A

2

A

2 + wB

2

B

2 +2 wA

wB

A B rAB+2 wAwC A C rAC +2 wBwC BC rBC

Diversification


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Diversification

An investor can reduce portfolio risk simply by

holding combinations of instruments which are

not perfectly positively correlated

correlation coefficient -1 rij1


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In other words, investors can reduce theirexposure to individual asset risk by holdinga diversified portfolio of assets.

Diversification may allow for the same portfolio

expected return with reduced risk. If all the asset pairs have correlations of 0they

are perfectly uncorrelatedthe portfolio's returnvariance is the sum over all assets of the squareof the fraction held in the asset times the asset's

return variance (and the portfolio standarddeviation is the square root of this sum).


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Portfolio return volatility

(STANDARD DEVIATION)

pp2


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Risk Return

Individual securities has risk return

characteristics

PORTFOLIO is combination of securities

May or may not take aggregate risk returncharacteristics of Individual securities

Consists various blend of risk return

characteristics of Individual securities


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Traditional portfolio analysis

Traditional portfolio analysis recognizes

Key importance of Risk Return to investors

Each security ends up

with some rough measurement of

likely return and

potential downside risk for the future


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Portfolio or combination of securities

Helpful spreading the risk over many securities

However the interrelationship between

securities may be specified.

Example Auto stocks are Risk interrelated with

Tire stocks

Utility stock display defensive movementrelative to steel etc.


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Why portfolio

Expected return from individual security will

carry

Some degree of risk

Risk is defined as standard deviation aroundthe expected return (risk surrogate not

synonyms for risk)

We equated securitys risk with variability of itsreturn


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Do not put all the eggs in one basket


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Since security carries different degree of

expected risk

Investors hold more than one security

Attempt to spread risk


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Diversification

Effort to spread risk takes the form of

Diversification

Most traditional type holding a number of

security type Utility, mining, banking ,pharma, manufacturing

group

To inherent differences in bonds and equity


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Effects of combining securities

Although holding two types of securities is

better than holding only one type

Is it possible to reduce the risk of portfolio

by incorporating in to it a security whoserisk is greater than that of any of theinvestment initially?


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Lets take two securities X & Y

Y is more riskier than X

A portfolio consist of some of X & some of Y

Is better than

Holding exclusively X or Y


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STOCK X STOCK Y

Return% 7 or 11 13 or 5

probability .5 each return .5 each return

Expected return 9 9

Variance % 4 16

Standard deviation 2 4


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Expected Return

X= (.5)(7) +(.5)(11)=9

Y= (.5)(13)+(.5)(5)=9

Expected Return

X= (.5)(7) +(.5)(11)=9

Y= (.5)(13)+(.5)(5)=9


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Since X & Y have similar return

Y is riskier (SD is 4 in Y)

Is a portfolio of some of X and some of Y is

superior than holding exclusively X ?


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Lets take 2/3 X as well as 1/3 Y

N

Rp = Xi Rii=1

/ /


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Lets take 2/3 X as well as 1/3 Y

Rp = EXPECTED RETURN TO PORTFOLIO

Rp = (2/3)(9)+(1/3)+(9)=9

In a period when X is better as an

investment we have

(2/3)(11)+(1/3)( 5)=9

When y is remunerative

We have

(2/3)(7)+(1/3)( 13)=9


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Thus by putting part of money in riskier stock

like Y

We are able to reduce risk also


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The crucial question is how to achieve the

proper proportion of X and y

Finding two securities each of which tends to

perform whenever other does poorly Makes more certain and reasonable return of

the portfolio as a whole

Even if one of its component happens to bequite risky.

Cl l k t tf li i k


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Closer look at portfolio risk

the risk involved in individual securities can bemeasured by standard deviation or variance

When two securities are combined we need toconsider their

- Interactive risk or covariance

If rates of return of two securities move together

-Interactive risk or covariance is positive

-If rates are independent- covariance is zero

-Inverse movement - covariance is negative

Mathematically covariance is


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Mathematically covariance isdefined as -

N

Covxy = 1/N (RxRx ) (RyRy )

Covxy = Covariance between X & Y

Rx= Return on security X

Rx = expected return to security X

Ry = Return on security Y

Ry =expected return to security Y

N = number of observations


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RETURN EXPECTEDRETURN

DIFFERNCES


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RETURN

STOCK X 7 9 -2

STOCK Y 13 9 4

PRODUCT - 8

STOCK X 11 9 2

STOCK Y 5 9 -4

PRODUCT - 8COV= (7-9)(13-9)+(11-9)(5-9)

=1/2(-8)+(-8)=-16/2 =8


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MEASUREMENT OFRISK

M t f i k


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Measurement of risk

Sensitivity or Range analysis

Probability distribution

Standard deviation

Coefficient of variation



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Particulars Assets A Assets B

INITIAL CASH OUTLAY 100 lakhs 100 lakhs

Rate of return %

Pessimistic 10 6

Most likely 12 12

Optimistic 14 18

Range 4 12

Probability distribution


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y

Probability of an event represents the changesof its occurrences

If change of occurrences = 3 out of 5 the

Probability of the event = 60% or .6


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n

E(R)= PiRii=1

E(R)= expected return

R = rate of return for the ith possible out come

P= possibility associated with the ith possible outcomesN = number of possible outcomes


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Possibleoutcome

Probability

Pi

Rate ofreturn

Ri

expectedreturnE(R)= PiRi

Rateofreturn

expectedreturnE(R)= PiRi

1 .25 10 ? 6 ?2 .50 12 ? 12 ?

3 .25 14 ? 18 ?

Standard deviation


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Most common quantitative measure of risk

It considered every possible event and weight

equal to its Probability is assigned to each

event


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The greater the SD of return of an assets ,

the greater is the risk of the assets

Investor prefers higher rate of return

with lower ?????????.

Coefficient of variation


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Coefficient of variation = SD

mean

Cv = E(R)


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For Two assets portfolio


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p

Portfolio return:

E( Rp) = wAE(RA)+ WBE(RB)= WA E(RA)+ (1- WA)E(RB)

SIMPLY = wAE(RA)+ WBE(RB)

Portfolio variance: p

2 = wA2 A

2 + wB2 B

2 +2 wAwB A B AB


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Modern portfolioh MPT


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theory (MPT)

Modern portfolio theory (MPT) is a theoryof investment

which attempts to maximize

portfolio expected return

for a given amount of portfolio risk,

or equivalently minimize risk for a given levelof expected return,

by carefully choosing the proportions ofvarious assets.


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several of its creators won aNobel prize for the theory

MPT is widely used in practice in the financialindustry

MPT is a mathematical formulation of the

t f di ifi ti i i ti


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concept of diversification in investing,

with the aim of selecting a collection of

investment assets that has collectively lower

risk than any individual asset.

That this is possible can be seen intuitivelybecause different types of assets often change

in value in opposite ways.

For example, when prices in the stockmarket fall, prices in the bond market often

increase, and vice versa


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A collection of both types of assets cantherefore have lower overall risk than either

individually.

But diversification lowers risk.


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The fundamental concept

behind MPT is that the assets in an

investment portfolio should not be selected

individually, each on their own merits.

Rather, it is important to consider

how each asset changes in price relative tohow every other asset in the portfolio changes

in price.

Investing is a trade-off between risk andexpected return.


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p

In general, assets with higher expected returnsare riskier.


MPT describes how to select a portfolio with thehighest possible expected return.

Or, for a given expected return, MPT explains how to select a portfolio with the

lowest possible risk

(the targeted expected return cannot be more

than the highest-returning available security,of course, unless negative holdings of assetsare possible.)


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MPT is therefore a form of diversification.

Under certain assumptions and for

specific quantitative definitions of risk and

return, MPT explains

how to find the best possible diversification

strategy.


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Mathematical model



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p

MPT assumes that investors are risk averse,meaning that given two portfolios that offer the

same expected return, investors will prefer the

less risky one.

Thus, an investor will take on increased risk

only if compensated by higher expected

returns.

Conversely, an investor who wants higherexpected returns must accept more risk.


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The exact trade-off will be the same for allinvestors,

but different investors will evaluate the trade-off

differently based on individual risk aversion

characteristics.

The implication is that a rational investor will not

invest in a portfolio if a second portfolio exists with

a more favourable risk-expected return profile i.e., if for that level

of risk an alternative portfolio exists which has

better expected returns.


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Note that the theory uses standard deviation ofreturn as a proxy for risk. There are problems

with this, however;

Under the model: Portfolio return is the proportion-weighted

combination of the constituent assets'returns.

Portfolio volatility is a function of

the correlationsij of the component assets,for all asset pairs (i,j).


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Expected return:


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Markowitz Diversification


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Combining assets that are less than perfectlypositively correlated in order to reduce portfoliorisk without sacrificing portfolio returns.

It is more analytical than simple diversificationand considers assets correlations.

The lower the correlation among assets, themore will be risk reduction through Markowitzdiversification

Portfolio Expected Return


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A weighted average of the expected returns of individualsecurities in the portfolio.

The weights are the proportions of total investment in

each security

n

E(Rp) = wi x E(Ri)

i=1

Where n is the number of securities in the portfolio

Portfolio Risk


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Measured by portfolio standard deviation Not a simple weighted average of the standard

deviations of individual securities in the

portfolio. Why? How to compute portfolio standard deviation?


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Markowitz Diversification


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Combining assets that are less than perfectlypositively correlated in order to reduce portfoliorisk without sacrificing portfolio returns.

It is more analytical than simple diversificationand considers assets correlations. The lowerthe correlation among assets, the more will berisk reduction through Markowitz diversification

Example ofMarkotwitzs Diversification

The emphasis in Markowitzs Diversification ison portfolio expected return and portfolio risk

Portfolio Expected Return


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A weighted average of the expected returns of individualsecurities in the portfolio.

The weights are the proportions of total investment in

each security

n E(Rp) = wi x E(Ri)

i=1

Where n is the number of securities in the portfolio

Example:

Portfolio Risk


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Measured by portfolio standard deviation Not a simple weighted average of the standard

deviations of individual securities in the

portfolio. Why? How to compute portfolio standard deviation?

Significance of Covariance


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An absolute measure of the degree ofassociation between the returns for a pair of

securities.

The extent to which and the direction in whichtwo variables co-vary over time

Example:

Why Correlation?


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What is correlation? Perfect positive correlation

The returns have a perfect direct linear relationship

Knowing what the return on one security will do allows an

investor to forecast perfectly what the other will do

Perfect negative correlation Perfect inverse linear relationship

Zero correlation No relationship between the returns on two securities


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Combining securities with perfect positivecorrelation or high positive correlation does not

reduce risk in the portfolio

Combining two securities with zero correlation

reduces the risk of the portfolio. However,

portfolio risk cannot be eliminated

Combining two securities with perfect negative

correlation could eliminate risk altogether


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