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ASSET PRICING
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The basic issue is
how to determine
the price of a financial
asset?
REAL
ASSETS
FINANCIAL
ASSETS
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There are broadly two approaches for
asset pricing in Theory of Finance One approach is EQUILIBRIUM
APPROACH; and
Another approach is ARBITRAGEAPPROACH.
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The Story of Asset Pricing started
with
MarkowitzPortfolio Theory
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Therefore, we start with
HARRY MARKOWITZ MODEL
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MARKOWITZ MODEL Markowitz is called father of Modern Portfolio Theory.
He gave for the very first time a quantitative
measurement of risk and return of a security as well asof a portfolio.
He also suggested a methodology for constructing anoptimum portfolio.
He changed the whole character and the nature of thetheory of Finance.
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HOW TO CALCULATE RETURN ??
Return on an equity share consists ofdividend income; and
capital gain/loss
Under uncertainty situation, it is measured in termsofEXPECTED RETURN.
And, expected return is defined as -
j
n
1jji pr)R(E
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How to estimate RETURN from thepast data?
Return may be defined as
(P1- P0)/P0 assuming that the compounding is discrete (one may
include any other cash flow that may arise during the period.).
Ln(P1/P0) assuming that the compounding is continuous (onemay include any other cash flow that may arise during the
period.).
The mean of the return can be taken as a proxy for
the Expected Return.
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HOW TO CALCULATE RISK ??
Risk on equity is understood in the sense of variation;higher the variation higher the risk; lower thevariation lower the risk.
Under certainty situation, there is no risk.
Under uncertainty situation, it is measured in termsof STANDARD DEVIATION/VARIANCE/ COEFFICIENT OFVARIATION.
And, variance is defined as -
2ii
2
j
n
1j
2iji
)]R(E[)R(E
p)]R(Er[()R(Var
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How to estimate RISK from thepast data?
The Standard Deviation of the returns
can be taken as a proxy for Risk of a
security.
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Markowitz said that since a security is
evaluated in terms of Risk and Return
parameters, a portfolio should also be
evaluated in terms ofRisk and Return.
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Return and Risk of a Portfolio
ji
),()(
)()(
n
i
n
i
n
j
jijiiiP
n
iiiP
RRCovR
RERE
1 1 1
22
1
Where
E(RP) = Expected Return of the Portfolio
s(RP) = Standard Deviation of return on a portfolio
ai = Proportion of ith security in the portfoliosi
2 = Variance of ith security
E(Ri) = Return on ith security
Cov(Ri,Rj) = Covariance between the return of ith security and the return of the jth
security
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Portfolio DIVERSIFY AWAYRisk ????!!!!!
Markowitz questioned NAVE
DIVERSIFICATION - a diversification that isobtained by just adding a number of different
securities into a portfolio. Can we really conclude that adding too many
securities simply into a portfolio reduce risk?
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Markowitz said - not Necessarily.It may be or may not be.
Then, what determines whether risk in a
portfolio can be reduced?
Markowitz said - it is the nature and the
degree of covariances existing among
securities that determine whether risk in aportfolio could be reduced.
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Diversification pays when the securities arehaving less degree of correlation and negative
correlation.
Standard Deviation
ExpectedR
eturn
r = 1
r = -1
M k it M d l f P tf li
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Markowitz Model of Portfolio(Journal of Finance : 1952)
Assumptions: The investor is rational.
The investor is risk averter.
Securities and portfolios can be evaluated only in terms oftwoparameters - Mean and Variance.
Security Market is perfectly competitive.
Securities are perfectly divisible.
Investors have complete information about Mean,
Variance and Correlation of all securities. Investors have one period as holding period.
Investors are not E(R) maximiser but E(U) maximiser andU = f(Risk and Return)
Either Utility Function is quadratic or the returns arefollowin normal robabilit distribution.
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Are you searching for an OPTIMUM
PORTFOLIO ??? If YES!? Then, first, look for an efficient set of portfolios.
A set of portfolios is called an efficient set if all the portfolios in it
are non-dominated portfolios in the sense of mean-variance
dominance principle.
MEAN - VARIANCE DOMINANCE PRINCIPLE says that a
portfolio is a dominating over the other portfolio if
for the same or more expected return a portfolio is havingsame or less risk.
for the same or less risk a portfolio is having more expected
return.
MINIMUM VARIANCE SET OF
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MINIMUM VARIANCE SET OF
PORTFOLIOS?
It is a set of those portfolios which have minimum variance for agiven expected return on a portfolio.
It is usually referred as a BULLET because of its shape.
Standard DeviationO
If two or more portfolios
from minimum variance
set are combined, then the
resultant portfolio also has
minimum variance.
MINIMUM VARIANCE PORTFOLIOS
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MINIMUM VARIANCE PORTFOLIOS -
SOME THEOREMS
A portfolio with two shares will have minimum variance
if the weight of one of the shares in the portfolio will
be
A portfolio of a group of shares that minimises thereturn variance is the portfolio that has an equal
covariance with every share return.
2122
12
212
2
2x
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CHART SHOWING HOW TO FIND MINIMUM VARIANCE
PORTFOLIO
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
1
0.
95
0.
9
0.
85
0.
8
0.
75
0.
7
0.
65
0.
6
0.
55
0.
5
0.
45
0.
4
0.
35
0.
3
0.
25
0.
2
0.
15
0.
1
0.
05 0
PROPORTION OF X AND (1-X) PROPORTION OF Y
VA
RIANCEAND
COVARIANCES
COV(P,X)
COV(P,Y)
PORTFOLIO
VARIANCE
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Can we have a zero risk portfolio???
Yes! We can have it if we can find
two securities having between them
perfect negative correlation.
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And now, we start with
EFFICIENT FRONTIER
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EFFICIENT FRONTIER ???
A curve that shows non-dominatedportfolios in terms of mean-variancedominance is called EFFICIENTFRONTIER.
No portfolio on it is dominated byany one.
It always have positive slope.
It steepnees depends upon thedegree of correlation that exists
between portfolios.
It is concave with respect to riskand convex with respect toexpected return.
Standard Deviation
Expec
tedReturn
F
E
A B C
X
Y
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TWO - FUND SEPARATION
THEOREM
This theorem says that-
all portfolios on the mean - variance efficient
frontier can be formed as a weighted average of
any two portfolios(or funds) on the efficient
frontier.
OPTIMUM SELECTION OF A PORTFOLIO
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OPTIMUM SELECTION OF A PORTFOLIO
DEPENDS UPON RISK - RETURN TRADE - OFF!!!
Standard Deviation
Expec
tedReturn
F
E
OPTIMUM
PORTFOLIO
P
h h
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What is the most importantcontribution of Markowitz model?
It is the concept of
Efficient Portfolio!!!
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Is there
anything in the
Markowitz
Model at which
you would like to
ATTACK?
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Limitations of Markowitz Model
Large Volume of data requiredNo consideration of risk free rate
It does not explain how securities are priced in market.
It is a normative model. It does not explain the market behaviour.
It is not a multi-period model.
It suggest that different persons can have differentportfolios of risk assets which is unlikely a case in a
perfectly competitive market.
It shows that risk of a portfolio can be reduced to zero
that is again unrealistic conclusion in the real world.
O f h l f
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One of the serious limitation of
the Markowitz Model is
Huge data requirement!!!!
Can we overcome thisproblem?
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Have you ever
wondered whyreturns of shares
of companies from
various industriesare correlated?
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Scatter Diagram
R = 0.289
-6
-4
-2
0
2
4
6
-10 -5 0 5 10 15
ACC (Return%)
RIL(Return%)
SCATTER DIAGRAM OF RETURNS
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SCATTER DIAGRAM OF RETURNS
-4
-2
0
2
4
6
8
10
-3 -2 -1 0 1 2 3 4
RANBAXY LABORATORIES LTD.(%)
STATEBANKOFI
NDIA(%)
R = 0.3027
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SBI AND SAIL
R2
= 0.208
-15
-10
-5
0
5
10
15
20
-10 -8 -6 -4 -2 0 2 4 6 8 10
RETURN OF SBI
RETURN
OF
SAI
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ONGC & Ranbaxy Laboratories Ltd.
R2
= 0.1468
-10
-5
0
5
10
15
-8 -6 -4 -2 0 2 4 6 8
RETURN OF ONGC
RETURN
OFRANBAX
What makes shares
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What makes shares
return to havecorrelation across
the companies fromthe different
industries?THINK!!!
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Is there some
underlyingFACTOR
which makes
these
correlations
to exist?
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If that factor exists, then your
data requirement will also be
considerably reduced!!!!
But, are we in a position
to identify that factor?
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Yes!!!! We can identify thatfactor...
And, this takes us to ...
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RmRi
SHARPES SINGLE
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SHARPE S SINGLE
FACTOR/INDEX MODEL
It is ex-postrelationship.
It shows how a factor leads to generation of returns in
a security.
Its intercept represents unique returnof a security
which is independent of Market Index.
The slopeof the Single Index Model represents whichis a measure ofSYSTEMATIC RISK.
RmRi
It is a linear relation between the return of a securityand the underlying factor which is the MARKET
INDEX.
Systematic Risk
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Systematic Risk
Vs. Unsystematic Risk
Systematic Risk: Return on an asset is systemically influencedby return on market portfolio; hence if any variation in the return of an
asset is explained by the variation in the market return, then such a
variation is called SYSTEMATIC RISK.
Such a risk is caused mainly by the macro factors; and
it is non-diversifiable risk.
Unsystematic Risk:Any variation in the return of an asset that
is not explained by the variation in the market return and isindependent of the market risk, or that resides within the asset itself
is called UNSYSTEMATIC RISK.
Such a risk is caused mainly by the micro factors; and
it is diversifiable risk.
CHARACTERISTIC LINE
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CHARACTERISTIC LINE
A regression line fitted to the scatter plot of returns from the
market portfolio and a security is called CHARACTERISTIC
LINE.
This is also a line that gives us the estimates of the parameters of
the Single Factor Model.
The slope of the characteristic line is called that represents
SYSTEMATIC RISK.
It is called a characteristic line as its slope showing the risk
characteristics of a security which is different for different
securities.
CHARACTERISTICS LINE
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CHARACTERISTICS LINE
y = 0.4619x - 0.2251
R
2
= 0.1813
-3
-2
-1
0
1
2
3
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3
COMPONENTS OF TOTAL RISK OF A
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COMPONENTS OF TOTAL RISK OF ASECURITY
Total Risk of a security is determined by the variance of thereturns.
It is equal to Unsystematic Risk and Systematic Risk. That is---
TOTAL RISK = UNSYSTEMATIC RISK + SYSTEMATIC
RISK.
Where
Total Risk of ith security = i2;Systematic Risk = i2 m2 ; andUnsystematic Risk = Total Risk - Systematic Risk = i2 i2m2.
R2 represents proportion of total risk which is SYSTEMATIC.
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Is there any statistical measure that can tell us -out of total variation, how much per cent
variation is due to systematic part and howmuch is due to unsystematic part?
YES!!!
It is R2. It represents proportion of total risk which isSYSTEMATIC.
In what way, the information of R2 is useful for aninvestment manager?
E
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ESTIMATION OF The estimation of of a security needs the followingsteps:
First, identify a suitable MARKET INDEX.
Collect information about the prices of the security and
the Index.
Fit the regression equation on the returns of the security
and the Index where the security return will be taken as a
dependent variable and the return on the Index will be
taken as an independent variable.
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What next?
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Dr. Vibha Jain
WHATS THEWORTH OF A
CAPITALASSETS???
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CAPITAL ASSET PRICING MODEL
It is a model that tries to answer the following questions:
What is the relevant CHOICE SET OF SECURITIES/PORTFOLIOS given
the risk free asset and risky assets?
How investors select the final OPTIMAL PORTFOLIO?
What risk is considered by the market in pricing a security?
What should be the equilibrium return and price?
It makes use of the foundations built by the Markowitz
Model and the Single Factor Model of Sharpe.
Its main contribution isLINEARITYandSIMPLICITY.
Assumptions of Capital Assets Pricing
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Assumptions of Capital Assets Pricing
Model
Investments are judged on the basis of risk and return
associated with them.
Returns are visualized in stochastic manner by investors.
Investors maximise their expected utility function which is
determined by return and risk.
Investors are rational investors.
Investors are risk averse.
Market is perfectly competitive.
Market is frictionless i.e. it has no transaction cost and
information is also cost free.
Assumptions of Capital Assets Pricing
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Assumptions of Capital Assets Pricing
Model(continued)
Capital assets are perfectly divisible.
Investors can have unlimited borrowing and lending at risk
free rate.
All investors have homogenous probability distributions andexpected returns for future returns.
All investors have same one holding period time horizon.
All investors are Markowitz efficient.
None is expecting any unanticipated inflation.
All assets are available in fixed quantities.
Capital market is in equilibrium.
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WHAT HAPPENS TO EFFICIENT
FRONTIER WHEN A RISK FREE ASSET IS
INTRODUCED INTO CAPITAL MARKET???
Will it be a non-linearor
linear ???
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EFFICIENT FRONTIER
becomes a straight line that is
tangent to Markowitz Efficient
Frontier and it is called
CAPITAL MARKET LINE.
C it l M k t Li (CML)
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Capital Market Line (CML)
CML is a line rising from the risk free rate, Rf, on the vertical axis andtangential to the Markowitz Efficient Frontier at M, which is market
portfolio.
It consists of efficient portfolios constructed by combing risk free securityand market portfolio.
It represents equilibrium in the capital market.
M
Rf
Lending
Borrowing
RiskM
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Capital Market Line (CML)(continued)
All risky assets are included in the market portfolio to extent of their
supply.
All portfolios on CML are perfectly correlated with the market portfolioand it implies that they are completely diversified and hence, possesses no
unsystematic risk.
CML relates the expected rate of return of an efficient portfolio to its
standard deviation.
The equation of CML is -
P
M
FM
FP
RRERRE
)()(
Capital Market Line (CML)
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The slope of CML represents the price per unit of risk.
It does not show how the expected rate of returnof an asset relates toits individual risk.
Every portfolio on the efficient frontier has perfectly positive correlation
with the market portfolio and hence, all variations in it are explained bythe market portfolio. Thus, they have no diversifiable risk.
Therefore, in an equilibrium situation, the market will price onlysystematic riskand eta measures the systematic risk. This is known as
theSYSTEMATIC RISK PRINCIPLE
which states that the expectedreturn on an asset depends only on its systematic risk.
Capital Market Line (CML)(continued)
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Why is the portfolioat which the
new efficient frontier is tangent to
the Markowitz Efficient Frontier
called MARKET PORTFOLIO?
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ONE - FUND THEOREM
It says that
one can generate an Efficient Portfolio by taking
only ONE FUND and that is, the Market Portfolio
and combine it with a risk free asset.
WHICH PORTFOLIO FROM CML SHOULD
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WHICH PORTFOLIO FROM CML SHOULD
BE SELECTED BY AN INVESTOR???
Depending upon an investors return - risk trade-offwhich isreflected in his indifference map, he selects an optimumportfolio for himself.
M
Rf
Risk
A
B
M
A
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ARE Investment Decisions and
Financing Decisions
independent???
TOBINS SEPARATION THEOREM
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TOBINS SEPARATION THEOREM
* Decision to invest in a capital asset has two
stages:
How to find the proportion of optimal portfolio of
risky assets? [Investment Decision ] and
How to finance the portfolio of risky assets?
[Financing Decision ]
TOBINS SEPARATION THEOREM
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TOBIN S SEPARATION THEOREM(continued)
* Investment Decision is same for all investors as every one
selects the market portfolioof risky assets.* Financing Decision is left for the individual investor. He/she
can decide how much to borrow or to lend at risk free rate
depending upon his/her degree of risk averseness.
* Thus, investment decision and financing decision of each
investor are totally independent
investment decisions are same for all; and
financing decisions are different and independent of investment decisions.
SECURITY MARKET LINE (SML)
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SML is a line drawn in E(R) and space.
It shows a linear relation between a securitys expectedreturn and its .
Security lying above SML is under-priced while securitybelow SML is over-priced.
Security lying to the right of = 1 is aggressive whilesecurity on the left of = 1 is defensive.
The equation of SML is:
E(Ri) = RF + ( E(RM) - RF ) i
SECURITY MARKET LINE (SML)
SECURITY MARKET LINE (SML)
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The equation of SML is called CAPITAL ASSET
PRICING MODEL.
E(RM)- RF is called risk premium per unit of systematicrisk.
SECURITY MARKET LINE (SML)
Rf
SML
M
1
Aggressive Security
Defensive Security
W
hat should be
the price
of a
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What should be the priceof asecurity in an equilibrium capital
market ??? CAPM directly does not provide price of a security. However,
indirectly through expected return, it provides price as return
and price are inversely related. Let P1 and P0 represent price of a security at time 1 and time 0
respectively. Also, if P1 is the expected price, then by definition,the expected return, E(R), would be:
}))(({
))((
)(
FMF
FMF
RRER
PP
P
PPRRER
PPPRE
1
1
0
0
01
0
01
CAPM and
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CAPM andits IMPLICATIONS
CAPM makes investment decision simple.J ust buy market portfolio.
CAPM helps in identifying over - and under - priced securities.
CAPM helps in the performance evaluation of an investment portfolio.A
number of measures are developed to evaluate a portfolio. They are: Jensens Index
Sharpes Index
Treynors Index
CAPM says Simplified diversification works.
CAPM is very useful in capital budgeting decisions. It helps in finding:
Certainty Equivalent; and
Risk Adjusted Discount Rate
CAPM
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SFM - a linear relation between the return of a security and the underlying factor.
CAPM - a linear relation between the return of a security and its .
SFM - represents ex-postrelationship while CAPM represents ex-anterelationship.
SFM - shows how a factor leads to generation of returns in a security, i.e. it shows return generatingprocess while CAPM shows how the market price a security and how much risk premium, the
market is willing to pay for one unit of systematic risk.
SFM - its intercept represents unique returnof a security when the return on the factor is zero whilethe intercept ofCAPM represents risk free rate.
The slopeof SFM represents while the slope ofCAPM represents the risk premium.
CAPMvs.
SINGLE FACTOR MODEL
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What next?
References
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References ...1. Investments Analysis and Management Charles P. Jones; John Wiley & Sons, Inc. (CPJ)
2. Investments: Analysis and Management - Jack Clark Francis; McGraw Hill International Editions,
McGraw-Hill, Inc. New York. (CF)
3. Security Analysis and Portfolio Management - Donald E. Fischer and Ronald J. Jordan; Prentice- Hall of India Private Limited; New Delhi. (FJ)
4. Modern Portfolio Theory and Investment Analysis - Edwin J Elton and Martin J Gruber; JohnWiley, New York. (EG)
5. Portfolio Construction, Management and Protection - Robert A. Strong; South-Western CollegePublishing, Thomson Learning, USA. (RS)
6. Introduction to Investments - Haim Levy; South-Western College Publishing, Thomson Learning,USA. (HL)
7. Investments-An introduction - Herbert B. Mayo; The Dryden Press, Harcourt College Publishers,USA. (MA)
8. Quantitative Analysis for Investment Management - Robert A. Taggart; Prentice-Hall, NewJersey, USA. (RT)
9. Investment Science - D. G. Luenberger; Oxford University Press, 1998
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