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
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Efficient frontier
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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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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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Example of Portfolio
Combinations and Correlation
Asset
Expected
Return
Standard
Deviation
Correlation
Coefficient
A 5.0% 15.0% 0.5
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% 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%
Portfolio Components Portfolio Characteristics
Positive
Correlation
weak
diversification
potential
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Example of Portfolio
Combinations and Correlation
Asset
Expected
Return
Standard
Deviation
Correlation
Coefficient
A 5.0% 15.0% 0
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% 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%
Portfolio Components Portfolio Characteristics
No
Correlation
some
diversification
potential
Lower
risk than
asset A
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Example of Portfolio
Combinations and Correlation
Asset
Expected
Return
Standard
Deviation
Correlation
Coefficient
A 5.0% 15.0% -0.5
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% 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%
Portfolio Components Portfolio Characteristics
Negative
Correlation
greater
diversification
potential
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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% 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%
Portfolio Components Portfolio Characteristics
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
= +
= +
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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
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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.
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