Unit III Saim

8/3/2019 Unit III Saim

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UNIT III:

PORTFOLIO ANALYSISAND

SELECTION


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Portfolio Theory

Investment Portfolio: collectionof securities that together

provide an investor with an

attractive trade-off betweenrisk and return

Portfolio Theory: concept of

making security choices basedon portfolio expected returns

and risks


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PORTFOLIO THEORY: Basic Assumptions

Expected Return: anticipated profit oversome relevant holding period

Risk: return dispersion, usually measured

by standard deviation of returns Probability Distribution: apportionment of

likely occurrences

Utility: positive benefit Disutility: psychic loss

Risk Averse: desire to avoid risk


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PORTFOLIO THEORY: Three Fundamental Assertions

Investors seek to maximize utility.

Investors are risk averse: Utility rises with

expected return and falls with an increasein volatility.

The optimal portfolio has the highest

expected return for a given level of risk, orthe lowest level of risk for a given expected

return.


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Portfolio Risk and Return

An investment portfolio is any collection or combinationof financial assets.

If we assume all investors are rational and therefore riskaverse, that investor will ALWAYS choose to invest in

portfolios rather than in single assets.

Investors will hold portfolios because he or she willdiversify away a portion of the risk that is inherent inputting all your eggs in one basket.

If an investor holds a single asset, he or she will fullysuffer the consequences of poor performance.

This is not the case for an investor who owns a diversified

portfolio of assets.


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Risk of a Portfolio

Diversification is enhanced depending upon the extent to

which the returns on assets move together.

This movement is typically measured by a statistic known

as correlation as shown in the figure below.


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Risk of a Portfolio (cont.)

Even if two assets are not perfectly negatively

correlated, an investor can still realize diversification

benefits from combining them in a portfolio as shown in

the figure below.


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Portfolio Expected Return

For a portfolio, the expected returncalculation is straightforward. It is simply a

weighted average of the expected returns of

the individual securities:

Where wi is the proportion (weight) ofsecurity i in the portfolio.

N

i

iiP RERE1

w


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Stock fund Bond Fund Rate of Squared Rate of Squared

Scenario Return Deviation Return DeviationRecession -7% 3.24% 17% 1.00%

Normal 12% 0.01% 7% 0.00%

Boom 28% 2.89% -3% 1.00%Expected return 11.00% 7.00%

Variance 0.0205 0.0067

Standard Deviation 14.3% 8.2%

The Return and Risk for Portfolios

Note that stocks have a higher expected return than bonds andhigher risk. Let us turn now to the risk-return tradeoff of a portfolio

that is 50% invested in bonds and 50% invested in stocks.


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Rate of Return

Scenario Stock fund Bond fund Portfolio squared deviation

Recession -7% 17% 5.0% 0.160%

Normal 12% 7% 9.5% 0.003%

Boom 28% -3% 12.5% 0.123%

Expected return 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010

Standard Deviation 14.31% 8.16% 3.08%

The rate of return on the portfolio is a weighted average of the

returns on the stocks and bonds in the portfolio:

SSBBP rwrwr +%)17(%50%)7(%50%5 +-



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Rate of ReturnScenario Stock fund Bond fund Portfolio squared deviation

Recession -7% 17% 5.0% 0.160%

Normal 12% 7% 9.5% 0.003%

Boom 28% -3% 12.5% 0.123%





%)7(%50%)12(%50%5.9 SSBBP rwrwr +



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Recession -7% 17% 5.0% 0.160%

Normal 12% 7% 9.5% 0.003%

Boom 28% -3% 12.5% 0.123%

Expected return 11.00% 7.00% 9.0%

Variance 0.0205 0.0067 0.0010




%)3(%50%)28(%50%5.12 -SSBBP rwrwr +


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Recession -7% 17% 5.0% 0.160%

Normal 12% 7% 9.5% 0.003%

Boom 28% -3% 12.5% 0.123%



The expectedrate of return on the portfolio is a weighted average

of the expectedreturns on the securities in the portfolio.

%)7(%50%)11(%50%9 )()()( SSBBP rEwrEwrE +


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Recession -7% 17% 5.0% 0.160%

Normal 12% 7% 9.5% 0.003%

Boom 28% -3% 12.5% 0.123%



The variance of the rate of return on the two risky assets portfolio is

BSSSBB

2

SS

2

BB

2

P ))(w2(w)(w)(w +where BS is the correlation coefficient between the returns on thestock and bond funds.


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Rate of Return

Scenario Stock fund Bond fund Portfolio squared deviation

Recession -7% 17% 5.0% 0.160%

Normal 12% 7% 9.5% 0.003%

Boom 28% -3% 12.5% 0.123%

Expected return 11.00% 7.00% 9.0%

Variance 0.0205 0.0067 0.0010


Observe the decrease in risk that diversification offers.

An equally weighted portfolio (50% in stocks and 50% in bonds)

has less risk than stocks or bonds held in isolation.


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A measure of the degree to which two variables movetogether relative to their individual mean values over time

The Covariance between the returns on two stocks can becalculated as follows:

N

Cov(RA,RB) = sA,B = S pi(RAi - E[RA])(RBi - E[RB])i=1

Where: sA,B = the covariance between the returns on stocks A and B

N = the number of states pi = the probability of state i

RAi = the return on stock A in state i

E[RA] = the expected return on stock A

RBi = the return on stock B in state i

E[RB] = the expected return on stock B

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The correlation coefficient is obtained by standardizing

(dividing) the covariance by the product of the individualstandard deviations

The Correlation Coefficient between the returns on two

stocks can be calculated as follows:sA,B Cov(RA,RB)

Corr(RA,RB) = A,B = sAsB = SD(RA)SD(RB)

Where:

A,B=the correlation coefficient between the returns on stocks A and B

sA,B=the covariance between the returns on stocks A and B,

sA=the standard deviation on stock A, and

sB=the standard deviation on stock B

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Two-Security Portfolios with Various Correlations

100%

bonds

return

s

100%

stocks

= 0.2 = 1.0

= -1.0

Relationship depends on correlation coefficient-1.0 < < +1.0

If = +1.0, no risk reduction is possible

If =1.0, complete risk reduction is possible

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Diversification Potential

The potential of an asset to diversify a portfolio isdependent 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 all portfolio risk.

This is demonstrated through the following chart.

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E l f P tf li C bi ti d


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Example of Portfolio Combinations andCorrelation

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

PerfectPositive

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%


PositiveCorrelation

weakdiversification 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%


NoCorrelation

somediversification potential

Lowerriskthanasset 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%


NegativeCorrelation 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%


PerfectNegative

Correlation greatestdiversification potential

Risk of theportfolio isalmosteliminated at70% asset A


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Di ifi ti f T A t P tf li


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The Effect of Correlation on Portfolio Risk:The Two-Asset Case

Expected Return

Standard Deviation

0%

0% 10%

4%

8%

20% 30% 40%

12%

B

AB= +1

A

AB = 0

AB = -0.5AB = -1

Diversification of a Two Asset PortfolioDemonstrated Graphically

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An Exercise using T-bills, Stocks and Bonds

Base Data: Stocks T-bills BondsExpected Return 12.73383 6.151702 7.007872

Standard Deviation 0.168 0.042 0.102

Correlation Coefficient Matrix:

Stocks 1 -0.216 0.048

T-bills -0.216 1.000 0.380

Bonds 0.048 0.380 1.000

Portfolio Combinations:

Combination Stocks T-bills Bonds

Expected

Return Variance

Standard

Deviation

1 100.0% 0.0% 0.0% 12.7 0.0283 16.8%

2 90.0% 10.0% 0.0% 12.1 0.0226 15.0%

3 80.0% 20.0% 0.0% 11.4 0.0177 13.3%

4 70.0% 30.0% 0.0% 10.8 0.0134 11.6%5 60.0% 40.0% 0.0% 10.1 0.0097 9.9%

6 50.0% 50.0% 0.0% 9.4 0.0067 8.2%

7 40.0% 60.0% 0.0% 8.8 0.0044 6.6%

8 30.0% 70.0% 0.0% 8.1 0.0028 5.3%

9 20.0% 80.0% 0.0% 7.5 0.0018 4.2%

10 10.0% 90.0% 0.0% 6.8 0.0014 3.8%

11 0.0% 100.0% 0.0% 6.2 0.0017 4.2%

Weights Portfolio

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Results Using only Three Asset Classes

Attainable Portfolio Combinationsand Efficient Set of Portfolio Combinations

0.0

2.04.0

6.0

8.0

10.0

12.0

14.0

0.0 5.0 10.0 15.0 20.0

Standard Deviation of the Portfolio (%)

Portfo

lioExpectedReturn(%) Efficient Set

Minimum

Variance

Portfolio

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Plotting Achievable Portfolio Combinations

Expected Return onthe Portfolio

Standard Deviation of thePortfolio

0%

0% 10%

4%

8%

20% 30% 40%

12%

Plotting achievable Portfolio Combinations

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The Efficient Frontier



0%

0% 10%

4%

8%

20% 30% 40%

12%



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ExpectedPortfolioReturn, r

p

Risk, sp

Efficient Set

Feasible Set

Feasible and Efficient Portfolios

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

An efficient portfolio is one that offers: the most return for a given amount of risk, or

the least risk for a give amount of return.

The collection of efficient portfolios iscalled the efficient set or efficient frontier.


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The efficient frontier consists of the set portfolios

that has the maximum expected return for a given

risk level.

Optimal portfolio: the portfolio that lies at thepoint of tangency between the efficient frontier

and his/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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The Capital Market Line



0%

0% 10%

4%

8%

20% 30% 40%

12%

Risk-free

rate

CapitalMarket

Line

Capital Market Line

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The Capital Market Line and Iso Utility Curves



0%

0% 10%

4%

8%

20% 30% 40%

12%

Risk-free

rate

CapitalMarket

Line

HighlyRisk

AverseInvesto

r

A risk-taker

Capital Market Line

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The Capital Market Line and Iso Utility Curves



0%

0% 10%

4%

8%

20% 30% 40%

12%

Risk-free

rate

CapitalMarket

Line

A risk-takersutility curve

The risk-takersoptimalportfolio

combination

Capital Market Line

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Capital Market Line

The capital market (securities markets) is the

market for securities

The capital market includes the stock market and

the bond market.

CML is used to illustrate all of the efficient

portfolio combinations available to investors.

The CML is derived by drawing a tangent

line from the intercept point on the efficient

frontier to the point where the expected return

equals the risk-free rate of return.

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Portfolio E(R) Beta

Risk-free asset Rf 0

Market portfolio E(Rm) 1

Consider a portfolio composed of the following two assets: An asset that pays a risk-free return Rf, , and

A market portfolio that contains some of every risky asset inthe market.

The Security Market Line

Security market line: the line connecting the risk-free

asset and the market portfolio

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The Security Market Line shows how an investor can construct a

portfolio of T-bills and the market portfolio to achieve the desired

level of risk and return.


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In equilibrium, all assets lie on this line.

If individual stock or portfolio lies above the line: Expected return is too highstock is undervalued.

Investors bid up price until expected return falls.

If individual stock or portfolio lies below SML: Expected return is too lowstock is overvalued.

Investors sell stock driving down price until

expected return rises.

Plots relationship between expected return and betas.


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i

E(RP)

RF

SML

Slope = (y2-y1) / (x2-x1)= [E(RM) RF] / (M-0)= [E(RM) RF] / (1-0)= E(RM) RF= Market Risk Premium

A - Undervalued

RM

M =1.0

B

A

B - Overvalued


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Capital Market Line v/s


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Capital Market Line v/sSecurity Market Line

The capital market line (CML) is a line used in the capital

asset pricing model to illustrate the rates of return forefficient while the security market line (SML) is a line that

graphs the systematic, or market, risk versus return of the

whole market at a certain time and shows all risky

marketable securities. The CML is derived by drawing a tangent line from the

intercept point on the efficient frontier to the point where

the expected return equals the risk-free rate of return while

the SML essentially graphs the results from the capitalasset pricing model (CAPM) formula. The x-axis

represents the risk (beta), and the y-axis represents the

expected return. The market risk premium is determined

from the slope of the SML.

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Capital Market Line v/s


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Capital Market Line v/sSecurity Market Line

What is plotted CML plots efficient portfolios, i.e.

combinations of the risky portfolio and the risk-

free asset (it is not valid for individual assets)

SML plots individual assets and portfolios

Measure of risk

for CML standard deviation (because welldiversified portfolios)

for SMLbeta (because individual assets)

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Portfolio Risk as a Function of the Number of


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Portfolio Risk as a Function of the Number ofStocks in the Portfolio

Nondiversifiable risk;

Systematic Risk;Market Risk

Diversifiable Risk;

Nonsystematic Risk;

Firm Specific Risk;Unique Risk

n

s In a large portfolio the variance terms are effectivelydiversified away, but the covariance terms are not.

Thus diversification can eliminate some, but not all of the risk of

individual securities.

Portfolio risk

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Market Risk / Systematic Risk

As more and more assets are added to aportfolio, risk measured by s decreases.

However, we could put every conceivableasset in the world into our portfolio and still

have risk remaining.

This remaining risk is called Market Risk/

Systematic Riskand is measured by Beta.

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A measure of systematic risk: Beta

The tendency of a stock to move up ordown with the market is reflected in its

beta coefficient.

Indicates how risky a stock is if the stock

is held in a well-diversified portfolio.

1-46Definition of Risk When Investors Hold


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Definition of Risk When Investors Holdthe Market Portfolio

Researchers have shown that the bestmeasure of the risk of a security in a large

portfolio is the beta ()of the security.

Beta measures the responsiveness of a

security to movements in the market

portfolio.

)()(

2

,

M

Mi

iRRRCovs

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h


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Computed as the slope of a regression linebetween periodic rates of return on the

Market Portfolio and periodic rates of return

for security j.

The slope of the regression line (sometimes

called the securitys characteristic line) isdefined as the beta coefficient for the

security.

Estimating with regression

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i i i h i


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Estimating with regression

Se

curityReturn

s

Return onmarket %

Ri = ai + iRm + ei

Slope = i

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C b


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Comments on beta

Beta(b) measures how the return of an individual

asset (or even a portfolio) varies with the market. b = 1.0 : same risk as the market

b < 1.0 : less risky than the market

b > 1.0 : more risky than the market

Most stocks have betas in the range of 0.5 to 1.5.

Can the beta of a security be negative?

Yes, if the correlation between Stock i and the market isnegative (i.e., i,m < 0).

If the correlation is negative, the regression line would slope

downward, and the beta would be negative.

However, a negative beta is highly unlikely.

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Calculating Portfolio Beta

The beta of a portfolio of stocks is equal tothe weighted average of their individualbetas:

bp = S wibi

Example: What is the portfolio beta for aportfolio consisting of 25% Home Depot withb = 1.0, 40% Hewlett-Packard with b = 1.35,

and 35% Disney with b = 1.25?


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SELECTION OFPORTFOLIO:

MARKOWITZs THEORY

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Markowitzs Portfolio Theory

Modern portfolio theory was initiated byUniversity of Chicago graduate student,

Harry Markowitz in 1952.

Markowitz showed how the risk of a

portfolio is NOT just the weighted average

sum of the risks of the individual

securitiesbut rather, also a function of the

degree of comovement of the returns of

those individual assets.

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Ri k d R t MPT


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Risk and Return - MPT

Prior to the establishment of Modern

Portfolio Theory, most people only focused

upon investment returnsthey ignored risk.

With MPT, investors had a tool that they

could use to dramatically reduce the risk of

the portfolio without a significant reduction

in the expected return of the portfolio.

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k f l h


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Markowitz Portfolio Theory

Markowitzs portfolio theory is based upon two

principles: To maximize the expected return of a portfolio

To minimize the risk of portfolio

Portfolios variance is a function of not only thevariance of returns on the individual investmentsin the portfolio, but also of the covariancebetween returns of these individual investments.

In a large portfolio, the covariance are much moreimportant determinants of the total portfoliovariance than the variances of individual

investments.

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M k i P f li Th


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Markowitz Portfolio Theory

Combining stocks into portfolios can reduce

standard deviation, below the level obtained

from a simple weighted average calculation.

Correlation coefficients make this possible.

The various weighted combinations ofstocks that create this standard deviations

constitute the set ofefficient portfolios.

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M k it A ti


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Markowitzs Assumptions

Investors consider investments as the probability

distribution of expected returns over a holding

period.

Investors seek to maximize expected utility

Investors measure portfolio risk on the basis ofexpected return variability

Investors make decisions only on the basis of

expected return and risk

For a given level of risk, investors prefer higher

return to lower returns.

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M k it A ti


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Markowitzs Assumptions

Under these five assumptions, a singleasset or portfolio of assets is efficient if

no other asset or portfolio of assets

offers higher expected return with thesame (or lower) risk, or lower risk with

the same (or higher) expected return.

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D b k f M k it M d l


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Drawbacks of Markowitz Model

Estimates of input parameters includingexpected asset returns and covariance matrix

could be fairly unstable and inaccurate.

The optimized portfolio of Markowitz Method

is in fact not the optimal one as it is only an

estimate of the best portfolio based on theestimated input parameters.


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SELECTION OFPORTFOLIO:

SINGLE INDEX MODEL

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Introduction: Inputs to Mean-


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pVariance analysis

In Markowitz Moedel, we need to estimate both

expected returns and the variance-covariance

matrix

N expected returns

N variances N(N-1)/2 covariances

This results in 1325 estimates for 50 stocks

And 501500 estimates for 1000 stocks

**problem - too many inputs

It would be nice:

To cut down on number of estimates

Put more structure on the estimates

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Single Index Model


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Single Index Model

The CAPM is a theory about expected returns

The application of the CAPM, i.e., the

empirical version, is ex-post, or after the fact

The empirical version is often referred to as

the Single Index Model

One step removed from the theoretical

CAPM and all of its assumptions

Understanding of single-index model shedslight on APT (Arbitrage Pricing Theory or

multiple factor model)

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Si l I d M d l


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Single Index Model

A broad stock market index is assumed to

be the single, common factor for all stocks

ai = expected return of stock i ifmarkets excess return is

zero

i(rmt - rft) = component of return due to market movements

eit = component of return due to unexpected firm-specific

events

ifmiifi errrr +-+- )()( a

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Single Index Model


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Examples of variable I: a stock index,

inflation, GDP growth, interest rates

The SIM also assumes:

Means that the correlation between asset i

and j is assumed to be only due tomovements in the common factor I.

Single Index Model

0),(

jtitCov

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Single Index Model


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Single Index Model

According to the model,

All asset returns derive only from the commonfactor, RM

ei is firm-specific, and hence uncorrelated acrossassets

Therefore,

Cov(Ri, Rj) = Cov(iRM, jRM ) = ijs2M

This setup allows security analysts to specialize Provides rationale for why analysts do not have to

research other sectors

Model says only the common factor (the market)

matters; there is no relationship otherwise

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Properties of SIM


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Properties of SIM

Assume we use an observable stock index as factor:

Then:

Looks like CAPM but here we can use any stock market index(or any factor), rather than the unobservable true market

portfolio

The variance of asset i

Hence the explained variance is:

In words, the R2 = the percentage of total risk of asset i that

can be explained by its systematic risk

itmtiiit RR a ++

)(

),(

m

mii

RVar

RRCov

)()()(

2

imii VarRVarRVar +

)(

)(22

i

mi

RVar

RVarR

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Properties of SIM


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Properties of SIM

Use the NSE as index I:

Then:

We now only require:

N estimates of expected returns

N estimates of i N estimates of firm-specific risk

1 estimate of

In total (3N+1) estimates: For 1000 stocks, we now need only

3001 estimates vs. 501500 before

)(),(

)()()(

)()(

2

mjiji

imii

miii

RVarRRCov

VarRVarRVar

RERE

a

+

+

)(i

Var

)( mRVar

itmtiiit RR a ++

1-67

Estimating the Single Index Model


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Estimating the Single Index Model

Regression analysis

Typically, use monthly returns over the past5 years (i.e., 60 observations) to estimate Y: excess return on individual security (or

individual portfolio)

X: excess return on market index

Intercept is ai, slope is i

itmtiiit eRR ++ a

1-68

Security Characteristic Line


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Security Characteristic Line

1-69

Interpreting the Results


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Interpreting the Results

alpha

statistical significancebeta


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CAPITAL THEORY:CAPITAL ASSET PRICING

MODEL (CAPM)

1-71

Introduction


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Introduction

The Capital Asset Pricing Model (CAPM) is a

model developed in an attempt to explain variationin yield rates on various types of investments

CAPM is based on the idea that investors demandadditional expected return (called the risk

premium) if they are asked to accept additional

risk

The CAPM model says that this expected return

that these investors would demand is equal to the

rate on a risk-free security plus a risk premium

1-72

Contd


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The model was the work of financial economist (and,

later, Nobel laureate in economics) William Sharpe,set out in his 1970 book "Portfolio Theory And

Capital Markets"

His model starts with the idea that individualinvestment contains two types of risk:

Systematic Risk (orMarket risk)

Unsystematic Risk (or Specific risk)

CAPM considers only systematic risk and assumes

that unsystematic risk can be eliminated by

diversification

Contd.

1-73

C it l A t P i i M d l (CAPM) A ti


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Capital Asset Pricing Model (CAPM): Assumptions

Market efficiency: The Capital Market efficiency implies

that share prices reflect all available information. Also,individual are not able to effect the prices of securities.

This means that there are large number of investors

holding small amount of wealth.

Risk aversion and mean-variance optimization: Investors

are risk-averse. They evaluate a securitys return and risk

in terms of expected return and variance or standard

deviation respectively. They prefer the highest expectedreturns for a given level of risks. This implies that

investors are mean-variance and they form efficient

portfolios.

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CAPM Assumptions contd


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CAPM Assumptions contd.

Homogeneous Expectations: All investors have

the same expectations about the expected returnsand risk of securities.

Single Time period: All investors decisions arebased on single time period.

Risk-free rate: All investors can lend and borrowat a risk-free rate of interest. They form portfolios

from publicly traded securities like shares and

bonds.

1-75Relationship between Risk and ExpectedR t (CAPM)


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Return (CAPM)

Expected Return on the Market:

Expected return on an individual security:

PremiumRiskMarket+F

M RR

)( FMiFi RRRR -+

Market Risk Premium

This applies to individual securities held within well-diversified portfolios.

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Expected Return on an Individual Security


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Expected Return on an Individual Security

This formula is called the Capital Asset PricingModel (CAPM)

)( FMiFi RRRR -

Assume i = 0, then the expected return isRF. Assume i = 1, then Mi RR

Expected return

on a

security

=Risk-

free rate+

Beta of

the

security

Market risk

premium

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Relationship Between Risk & Expected Return


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Relationship Between Risk & Expected Return

Expecte

dreturn

)( FMiFi RRRR -

FR

1.0

MR

1-78

Example: CAPM


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Example: CAPM

1-79

Example (cont'd)


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Example (cont d)

1-80

Example: CAPM


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Example: CAPM

Assume the risk-free return is 5% and the

market portfolio has an expected return of12% and a standard deviation of 44%.

ATP Oil and Gas has a standard deviation of

68% and a correlation with the market of0.91.

What is ATPs beta with the market?

Under the CAPM assumptions, what is itsexpected return?

1-81

Solution of Example


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Solution of Example

Solution

i

( ) ( , ) (.68)(.91)1.41

( ) .44

i i Mkt

Mkt

SD R Corr R R

SD R

[ ] ( [ ] ) 5% 1.41(12% 5%) 14.87% + - + - Mkti f i Mkt f E R r E R r

1-82

Limitations of CAPM


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Limitations of CAPM

In real world, assumptions of CAPM will

not hold good.

In practice, it is difficult to estimate the

risk-free return, market rate of return andrisk premium.

CAPM is a single period model while mostprojects are often available only as large

indivisible projects. It is, therefore, more

difficult to adjust.


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CAPITAL THEORY:ARBITRAGE PRICING

THEORY(APT)

1-84

Arbitrage Price Theory(APT)


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Arbitrage Price Theory(APT)

The Arbitrage Pricing Theory (APT) is a theory of

expected asset returns due to Ross (1976). The APT

explicitly accounts for multiple factors.

The APT requires three assumptions:

Returns can be described by a factor model

There are no arbitrage opportunitiesThere are large numbers of securities that permit the

formation of portfolios that diversify the firm-specific

risk of individual stocks

1-85

Arbitrage Pricing Theory (APT)


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Based on the law of one price. Two items

that are the same cannot sell at different

prices

If they sell at a different price, arbitrage

will take place in which arbitrageurs buy

the good which is cheap and sell the one

which is higher priced till all prices for thegoods are equal

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In APT, the assumption of investors utilizing a

mean-variance framework is replaced by anassumption of the process of generating securityreturns.

APT requires that the returns on any stock belinearly related to a set of indices.

In APT, multiple factors have an impact on thereturns of an asset in contrast with CAPM modelthat suggests that return is related to only onefactor, i.e., systematic risk


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Factors that have an impact the returns of

all assets may include inflation, growth inGNP, major political upheavals, or changesin interest rates

ri = ai + bi1F1 + bi2F2 + +bikFk+ ei

Given these common factors, the bik

termsdetermine how each asset reacts to thiscommon factor.


1-88



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In order to implement the APT we need to know

what the factors are! Here the theory gives no

guidance. There is some evidence that the following

macroeconomic variables may be risk factors:

Changes in monthly GDP

Changes in the default risk premium

The slope of the yield curve

Unexpected changes in the price level

Changes in expected inflation


1-89

The Capital Asset Pricing Model andthe Arbitrage Pricing Theory


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APT applies to well diversified portfolios and not

necessarily to individual stocks.

With APT it is possible for some individual stocks

to be mispriced - not lie on the SML.

APT is more general in that it gets to an expected

return and beta relationship without the assumption

of the market portfolio.

APT can be extended to multifactor models.

the Arbitrage Pricing Theory

1-90

The APT Model


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The APT Model

MULTIPLE-FACTOR MODELS FORMULA

ri = rj + bi1 F1 + bi2 F2 +. . .

+ biKFK+ ei

where ri is the return on security irj risk free rate of retrun

biK is the sensitivity of the asset to the factor

FK is the macro-economic factor

ei is the error term

1-91

Required Return for Stock iunder the APT


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ri = rRF + (r1 - rRF)b1 + (r2 - rRF)b2+ ... + (rj - rRF)bj.

bj = sensitivity of Stock i to economic

Factor j.

rj = required rate of return on a portfolio

sensitive only to economic Factor j.

under the APT

1-92

An Illustration of APT


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For example, suppose we have identified three

systematic risks on which we want to focus:1. Inflation

2. GDP growth

3. The dollar-euro spot exchange rate, S($,)

Our model is:

riskicunsystemattheis

betarateexchangespottheis

betaGDPtheis

betainflationtheis

FFFRR

mRR

S

GDP

I

SSGDPGDPII ++++

++

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Suppose we have made the following estimates:

1. I = -2.30

2. GDP = 1.503. S = 0.50.

Finally, the firm was able to attract a superstar CEO and

this unanticipated development contributes 1% to the

return.

FFFRR SSGDPGDPII ++++

%1

%150.050.130.2 +++- SGDPI FFFRR


1-94



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We must decide what surprises took place in the systematic

factors.

If it was the case that the inflation rate was expected to be by3%, but in fact was 8% during the time period, then

FI = Surprise in the inflation rate

= actualexpected

= 8% - 3%= 5%

%150.050.130.2 +++- SGDPI FFFRR

%150.050.1%530.2 +++- SGDP FFRR

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If it was the case that the rate ofGDP growth was

expected to be 4%, but in fact was 1%, then

FGDP = Surprise in the rate ofGDP growth= actualexpected

= 1% - 4%

= -3%

%150.050.1%530.2 +++- SGDP FFRR

%150.0%)3(50.1%530.2 ++-+- SFRR

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If it was the case that dollar-euro spot exchange rate, S($,),

was expected to increase by 10%, but in fact remained

stable during the time period, then

FS = Surprise in the exchange rate

= actualexpected

= 0% - 10%= -10%

%150.0%)3(50.1%530.2 ++-+- SFRR

%1%)10(50.0%)3(50.1%530.2 +-+-+- RR

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Finally, if it was the case that the expected return on

the stock was 8%, then

%150.0%)3(50.1%530.2 ++-+- SFRR

%12

%1%)10(50.0%)3(50.1%530.2%8

-

+-+-+-

R

R

%8R

1-98

What is the status of the APT?


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The APT is being used for some realworld applications.

Its acceptance has been slow because themodel does not specify what factorsinfluence stock returns.

More research on risk and return modelsis needed to find a model that istheoretically sound, empirically verified,

and easy to use 1-99


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THANK YOU

Documents

Unit III Saim