let’s start with a review of what we did in the class!!!
TRANSCRIPT
Let’s start with
a review of
what we did
in the class!!!
In the last class…
We discussed Markowitz Model
… a model helping us to create an optimum portfolio
Markowitz Model of Portfolio
•First, we discussed ----- First, we discussed ----- calculation of calculation of returns and risk for each portfolio returns and risk for each portfolio as they have to be evaluated in as they have to be evaluated in this two parametric framework.this two parametric framework.
Markowitz Model of Portfolio
•We were trying to find an optimum portfolio!!!!!We were trying to find an optimum portfolio!!!!!
•One step towards that is ----- One step towards that is ----- Reduce Reduce choice set by using Mean-choice set by using Mean-Variance Dominance Principle Variance Dominance Principle and thus, obtain EFFICIENT and thus, obtain EFFICIENT FRONTIER.FRONTIER.
Efficient Frontier
OPTIMUM SELECTION OF A PORTFOLIO DEPENDS UPON RISK - RETURN TRADE - OFF!!!
Standard Deviation
Exp
ecte
d R
etu
rn
F
E
OPTIMUM PORTFOLIO
P
What are the most important contribution of Markowitz model?
????????!!!!!!!!!!
What are the most important contributions of Markowitz
model?
It has two important contributions:
FIRSTFIRST, it has provided tools of
‘quantification of ‘Risk and Return ’!!!
What are the most important contributions of Markowitz
model?
SecondSecond is the concept of
‘Efficient Portfolio’!!!
Is there
anything in the
Markowitz
Model at which
you would like
to ‘ATTACK’?
FIRST...
Are you comfortable with Two-Parametric model to
evaluate a security/portfolio?
Are Mean and Variance
sufficient to evaluate a
security or a portfolio?
Return (%) Probability Return (%) Probability2 0.05 1 0.029 0.29 4 0.08
12 0.24 7 0.1016 0.17 9 0.1319 0.12 12 0.1623 0.07 16 0.1828 0.03 21 0.3130 0.04 30 0.02
1.00 1.00Expected
Return14.10%
Expected Return
14.10%
Standard Deviation
6.40Standard Deviation
6.40
Skewness 0.80 Skewness -1.07
SHARE - BSHARE - A
Look at the following two shares…
Now, look at their distribution …PROBABILITY DISTRIBUTION OF RETURNS
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 5 10 15 20 25 30 35
RETURN
PRO
BA
BIL
ITY
SHARE A SHARE B
What do you think in which shares should
you invest?
Return (%) Probability Return (%) Probability2 0.05 1 0.029 0.29 4 0.08
12 0.24 7 0.1016 0.17 9 0.1319 0.12 12 0.1623 0.07 16 0.1828 0.03 21 0.3130 0.04 30 0.02
1.00 1.00Expected
Return14.10%
Expected Return
14.10%
Standard Deviation
6.40Standard Deviation
6.40
Skewness 0.80 Skewness -1.07
SHARE - BSHARE - A
Now, look at again the following two shares…
It shows that Skewnwss
is also that may matter
in making a choice!!!!
It shows that Skewnwss
is also that may matter
in making a choice!!!!
SECOND...
Why is Markowitz Model not working?
I want to invest in RISK-FREE ASSET and the Markowtiz model
does not allow is this!!!!!!
I do not know
how to help
her!!!!
THIRD...
Large Volume of
data required.
I would become I would become mad!!! I really do mad!!! I really do
not know how not know how many pieces of many pieces of
input data I need input data I need to generate my to generate my best portfolio?best portfolio?
Too much information required!!!
• This model requirement of information is huge and it increases exponentially with increase in the number of securities.
• Markowitz model requires (n (n+3))/2 pieces of input data.
FOURTH...
Have you ever wondered why returns of shares of companies from various industries are correlated?
Scatter Diagram
R = 0.2814
-20
-10
0
10
20
30
40
-10 -5 0 5 10 15
ACC (Return%)
ON
GC
(R
etu
rn%
)
SCATTER DIAGRAM OF RETURNS
-3
-2
-1
0
1
2
3
4
-3 -2 -1 0 1 2 3 4
INFOSYS TECHNOLOGIES LTD.(%)
RA
NB
AX
Y L
AB
OR
AT
OR
IES
LT
D.(
%)
R = 0.2674
Scatter Diagram
R = 0.289
-6
-4
-2
0
2
4
6
-10 -5 0 5 10 15
ACC (Return%)
RIL
(R
etu
rn%
)
SCATTER DIAGRAM OF RETURNS
-4
-2
0
2
4
6
8
10
-3 -2 -1 0 1 2 3 4
RANBAXY LABORATORIES LTD.(%)
ST
AT
E B
AN
K O
F IN
DIA
(%)
R = 0.3027
What makes shares’ return to have correlation across the companies from the different industries?
THINK!!!
Is there some
underlying
FACTOR which
makes these
correlations to
exist?
If that factor exists, then your
data requirement will also be
considerably reduced!!!!
If that factor exists, then your
data requirement will also be
considerably reduced!!!!
But, are we in a position to identify that factor?
Yes!!!! We can identify that factor...
And, this takes us to ...
And, now…
?????????????????????????………
RmRi
SHARPE’S SINGLE FACTOR/INDEX MODELSHARPE’S SINGLE FACTOR/INDEX MODEL
• It is ex-post relationship.
• It shows how a factor leads to generation of returns in a security.
• Its intercept represents unique return of a security which is independent of Market Index.
• The slope of the Single Index Model represents which is a measure of SYSTEMATIC RISK.
RmRi
It is a linear relation between the return of a security and the underlying factor which is the MARKET INDEX.
Systematic Risk Vs. Unsystematic
Risk • Systematic Risk: Return on an asset is systemically
influenced by 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 is independent 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; andit is diversifiable risk.
• Systematic Risk: Return on an asset is systemically influenced by 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 is independent 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; andit is diversifiable risk.
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
y = 0.4619x - 0.2251
R2 = 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 SECURITY
• Total Risk of a security is determined by the variance of the returns.
• It is equal to Unsystematic Risk and Systematic Risk. That is---
TOTAL RISK = UNSYSTEMATIC RISK + TOTAL RISK = UNSYSTEMATIC RISK + SYSTEMATIC RISK.SYSTEMATIC RISK.
– Where
Total Risk of ith security = i
Systematic Risk = i2 m
; and
Unsystematic Risk = Total Risk - Systematic Risk = i
i2 m
Is there any statistical measure that can tell us - out of total variation, how much per cent variation is due to systematic part and how much is due to unsystematic part?
• YES!!!
• It is R2. It represents proportion of total risk which is SYSTEMATIC.
• In what way, the information of R2 is useful for an investment manager?
What’s the difference between …
• Total Systematic Risk?
• β?
• R2?
ESTIMATION OF • The estimation of of a security needs the
following steps:
– 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.
ESTIMATION [EXCEL output]SUMMARY OUTPUT
Regression StatisticsMultiple R 0.423823119R Square 0.179626036Adjusted R Square 0.178161082Standard Error 6.876151354Observations 562
ANOVAdf SS MS F Significance F
Regression 1 5797.440487 5797.440487 122.6155199 6.61196E-26Residual 560 26477.61617 47.28145745Total 561 32275.05666
Coefficients Standard Error t Stat P-valueIntercept 0.841961864 0.290330094 2.900015814 0.003877893X Variable 1 0.753268111 0.068026302 11.07318924 6.61196E-26
Dr. Reddy'S Laboratories Ltd.
ESTIMATION [EXCEL output]SUMMARY OUTPUT
Regression StatisticsMultiple R 0.339940172R Square 0.11555932Adjusted R Square 0.113237954Standard Error 7.369805289Observations 383
ANOVAdf SS MS F Significance F
Regression 1 2703.791967 2703.791967 49.78072823 8.16384E-12Residual 381 20693.64543 54.31403Total 382 23397.4374
Coefficients Standard Error t Stat P-valueIntercept 0.275169466 0.376636098 0.730597698 0.465473937X Variable 1 0.696256196 0.098682115 7.05554592 8.16384E-12
Oil & Natural Gas Corpn. Ltd.
ESTIMATION [EXCEL output]SUMMARY OUTPUT
Regression StatisticsMultiple R 0.714636907R Square 0.510705909Adjusted R Square 0.509833727Standard Error 5.35907977Observations 563
ANOVAdf SS MS F Significance F
Regression 1 16816.83319 16816.83319 585.5497139 3.95203E-89Residual 561 16111.77189 28.71973599
Total 562 32928.60508
Coefficients Standard Error t Stat P-valueIntercept 0.214380529 0.226065807 0.94831028 0.343379833X Variable 1 1.282653728 0.053006306 24.19813451 3.95203E-89
Reliance Industries Ltd.
Any comment??
Source: BSE Site
Beta of a Portfolio …
• Beta of a portfolio is the weighted average of individual securities betas.
∑1=
=n
iiiP X ββ
What next…?
And, now…something exciting…
Markow
itz’s Idea of
Efficient F
rontier Ris
k F
ree
Ass
et
Con
cept of B
eta from
Sin
gle Ind
ex Mod
el
What a cocktail!!!!
All these take us to …All these take us to …
?????Markowitz Markowitz Efficient Efficient FrontierFrontier
Risk-Free Asset
Concept of Systematic Risk - βeta
Dr. C. P. Gupta
WHAT’S THE
WORTH OF A
CAPITAL ASSETS???
CAPITAL ASSET PRICING MODELCAPITAL 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 is LINEARITY and SIMPLICITY.
Assumptions of Capital Assets Assumptions of Capital Assets Pricing ModelPricing 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 Assumptions of Capital Assets Pricing ModelPricing Model(continued…)(continued…)
Capital assets are perfectly divisible.
Investors can have unlimited borrowing and lending at
risk free rate.
All investors have homogenous probability
distributions and expected 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.
WHAT HAPPENS TO EFFICIENT WHAT HAPPENS TO EFFICIENT
FRONTIER WHEN A RISK FREE ASSET FRONTIER WHEN A RISK FREE ASSET
IS INTRODUCED INTO CAPITAL IS INTRODUCED INTO CAPITAL
MARKET???MARKET???
Will it be a non-linear or
linear ???
EFFICIENT FRONTIER EFFICIENT FRONTIER
becomes a straight line becomes a straight line
that is tangent to that is tangent to
Markowitz Efficient Markowitz Efficient
Frontier and it is calledFrontier and it is called
CAPITAL MARKET LINE.CAPITAL MARKET LINE.
Capital Market Line (CML)Capital Market Line (CML)
CML is a line rising from the risk free rate, Rf, on the vertical axis and tangential to the Markowitz Efficient Frontier at M, which is market portfolio.
It consists of efficient portfolios constructed by combing risk free security and market portfolio.
It represents equilibrium in the capital market.
M
Rf
Lending
Borrowing
Risk
Exp
ecte
d R
etur
n
Capital Market Line (CML) Capital Market Line (CML) (continued…)(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 portfolio and 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
FMFP
RRERRE
)(
)(
The slope of CML represents the price per unit of risk.
It does not show how the expected rate of return of an asset relates to its individual risk.
Therefore, in an equilibrium situation, the market will price only systematic risk and eta measures the systematic risk. This is known as the ‘SYSTEMATIC RISK PRINCIPLE’ which states that the expected return on an asset depends only on its systematic risk.
Capital Market Line (CML) Capital Market Line (CML) (continued…)(continued…)
ONE - FUND THEOREMONE - 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 WHICH PORTFOLIO FROM CML
SHOULD BE SELECTED BY AN SHOULD BE SELECTED BY AN
INVESTOR…???INVESTOR…??? Depending upon an investor’s return - risk trade-off which is reflected in his indifference map, he selects an optimum portfolio for himself.
M
Rf
Risk
A
B
Exp
ecte
d R
etur
n
Does the idea of Capital Market
Line ensure better risk-
return trade-off for me???
Yes!!! It will improve risk-return Yes!!! It will improve risk-return trade-off for our Topiwalla.trade-off for our Topiwalla.
M
Rf
Risk
Exp
ecte
d R
etur
n
Do you see this?
AARE Investment Decisions andRE Investment Decisions and
Financing Decisions Financing Decisions
independentindependent ??????
TOBIN’S SEPARATION THEOREMTOBIN’S 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 ]
TOBIN’S SEPARATION THEOREMTOBIN’S SEPARATION THEOREM (continued…)(continued…)
* Investment Decision is same for all investors as every one selects the market portfolio of 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 independentinvestment decisions are same for all; andfinancing decisions are different and independent of
investment decisions.
SML is a line drawn in E(R) and space.
It shows a linear relation between a security’s expected return and its .
Security lying above SML is under-priced while security below SML is over-priced.
Security lying to the right of = 1 is aggressive while security on the left of = 1 is defensive.
The equation of SML is:
E(Ri) = RF + ( E(RM) - RF ) i
SECURITY MARKET LINE SECURITY MARKET LINE (SML)(SML)
The equation of SML is called CAPITAL ASSET PRICING MODEL.
E(RM) - RF is called risk premium per unit of systematic risk.
SECURITY MARKET LINE SECURITY MARKET LINE (SML)(SML)
Rf
Exp
ecte
d R
etur
n
SML
M
Aggressive Security
Defensive Security
WWhat should be hat should be the pricethe price of a of a security in an equilibrium capital security in an equilibrium capital
market …???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
P
PPRE
11
0
0
01
0
01
CAPM CAPM andandits IMPLICATIONSits IMPLICATIONS
CAPM makes investment decision simple. Just 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:
Jensen’s Index
Sharpe’s Index
Treynor’s 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
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-post relationship while CAPM represents ex-ante relationship.
SFM - shows how a factor leads to generation of returns in a security, i.e. it shows return generating process 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 return of a security when the return on the factor is zero while the intercept of CAPM represents risk free rate.
The slope of SFM represents while the slope of CAPM represents the risk premium.
CAPM CAPM vs.vs.
SINGLE FACTOR MODELSINGLE FACTOR MODEL
What’s Next…???
?????
Dr. C. P. Gupta
MEASURING PORTFOLIO PERFORMANCE …???
Portfolio performance MEASUREMENT AND EVALUATION is the last step in the process of portfolio management.
The basic objective of measuring performance is - to judge the return of a portfolio vis-à-vis with the risk involved in it.
That is to say, ASSOCIATE A MEASURE OF RISK WITH THE RETURN and then, determine whether the portfolio manager is able to generate more returns than expected.
Portfolio Evaluation is concerned with the evaluation of the PORTFOLIO AS A WHOLE without examining the performance of individual securities in the portfolio.
Before, we proceed further...
We should also evaluate to what extent a portfolio is diversified.
For that we must use - R2.
WHY?
Measures of Portfolio Evaluation
THE SHARPE INDEX
THE TREYNOR’S INDEX
THE JENSEN INDEX (ALSO KNOWN AS THE JENSEN’S )
})({ PFMtFPt RRRRJ
Pt
FPt RRS
Pt
FPt RRT
Measures of Portfolio Evaluation(continued…)
APPRAISAL RATIO - P/(eP): It divides the alpha of the
portfolio by the non-systematic risk of the portfolio. It measures
abnormal return per unit of risk that in principle could be
diversified away by holding a market index portfolio.
Measures of Portfolio Evaluation(continued…)
The M2 Measure of Performance: This measure wad made popular by Leah Modigliani,
grand daughter of Franco Modigiliani.
To compute M2, an imaginary portfolio is constructed by mixing the managed portfolio(say, P*) with a position in risk free assets in such a manner that the variance of such a portfolio matched with the variance of the market portfolio. Then,
M2 = RP* - RM
Why a portfolio
manager is able to
perform - better or worse?
Manager
Looking for the exact source of
success/failure!!
FAMA’S DECOMPOSITION OF TOTAL RETURN ...
E. Fama has provided an analytical framework that allows a detailed breakdown of a fund’s performance into the source or components of performance.
Such a decomposition of total return is useful in identifying the different skills in portfolio management and to what extent the portfolio manager is capable of managing each one of them.
This may suggest the areas of strength and those of weakness in the ability of a portfolio manager.
FAMA SUGGESTED THE FOLLOWING DECOMPOSITION OF THE TOTAL
RETURN FROM A PORTFOLIO...
TOTAL RETURN
RISK FREE
RETURN
EXCESS RETURN
RISK PREMIUM
RETURN FROM SHARE SELECTION
DUE TO
SYSTEMATIC RISK
DUE TO UNSYSTEMATIC
RISK
FAMA’S DECOMPOSITION...
Using the decomposition scheme discussed, Fama suggested the following:
RP = RF + R1 + R2 + R3
where RP = Return on the managed portfolio;
RF = Return on a risk free asset;
R1 = Return from SYSTEMATIC RISK and is equal to (RM - RF)P;
R2 = Return from UNSYSTEMATIC RISK & is equal to (RM - RF)(P/M - P); and
R3 = Residual Return and Fama named as NET SELECTIVITY MEASURE.
PORTFOLIOS RETURNSTANDARD DEVIATION
BETA
A 12% 18% 0.7Z 19% 25% 1.3M (MARKET INDEX) 15% 20% 1.0Risk - Free Return = 7%
A 0.28Z 0.48M 0.40
A 7.14Z 9.23M 8.00
A -0.60Z 1.60M 0.00
Unsystematic RiskAPPRAISAL
RATIOSA 11.31% -5.30Z 25.00% 6.40M 20.00% 0.00
AZM
Risk - Free RateDue to
Systematic Risk
Due to Unsystematic
Risk
Net Selectivity Measure
Total
A 7% 5.60% 1.60% -2.20% 12%Z 7% 10.40% -0.40% 2.00% 19%M 7% 8.00% 0.00% 0.00% 15%
0.000
PORTFOLIO PERFORMANCE EVALUTION - AN ILLUSTRATION
Consider the following information about Portfolio A, Portfolio Z and the Market Portfolio -M:
Proportion of Investment in Fund1.11
M-SQUARE MEASURE -0.017
FAMA'S DECOMPOISTION
JENSEN RATIOS
SHARPE RATIOS
TREYNOR RATIOS
APPRAISAL RATIOS
M - MEASURE
Risk Premium
0.801.00
0.002
2
Other Measures...
Expense Ratio: It is a ratio of the total
expenses of a fund to the average net assets of a fund.
Portfolio Turnover Ratio: It is defined as
minimum of assets bought or assets sold during a year divided by average assets of a fund.
Other Measures…(continued)
Tracking Error: It is defined as the standard deviation of the difference in returns between the portfolio under consideration and a specified benchmark or target; that is to say, STANDARD DEVIATION OF (Rp-Rb) where Rp is the return on the portfolio under consideration while Rb is the return on the benchmark portfolio.
Portfolio Evaluation completes the cycle
of activities comprising portfolio management. And,
thus, we come to an end of the course.
But, before that - the last words
At, the end of the Course, I feel that we have enough light about Investment
Management !!!!!