Download Markowitz Model of Portfolio

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 ----- calculation of
returns and risk for each portfolio
as they have to be evaluated in
this two parametric framework.
Markowitz Model of Portfolio
•We were trying to find an optimum portfolio!!!!!
•One step towards that is ----- Reduce
choice set by using Mean-Variance
Dominance Principle and thus,
obtain EFFICIENT FRONTIER.
Efficient Frontier
OPTIMUM SELECTION OF A PORTFOLIO DEPENDS
UPON RISK - RETURN TRADE - OFF!!!
Expected Return
F
P
OPTIMUM
PORTFOLIO
E
Standard Deviation
What are the most important
contribution of Markowitz model?
????????
!!!!!!!!!!
What are the most important
contributions of Markowitz model?
It has two important contributions:
FIRST, it has provided tools of
‘quantification of ‘Risk and Return ’!!!
What are the most important
contributions of Markowitz model?
Second 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 TwoParametric model to evaluate a
security/portfolio?
Are Mean and
Variance sufficient
to evaluate a
security or a
portfolio?
Look at the following two shares…
SHARE - A
Return (%)
Probability
2
0.05
9
0.29
12
0.24
16
0.17
19
0.12
23
0.07
28
0.03
30
0.04
1.00
Expected
14.10%
Return
Standard
6.40
Deviation
Skewness
0.80
SHARE - B
Return (%)
Probability
1
0.02
4
0.08
7
0.10
9
0.13
12
0.16
16
0.18
21
0.31
30
0.02
1.00
Expected
14.10%
Return
Standard
6.40
Deviation
Skewness
-1.07
Now, look at their distribution …
PROBABILITY DISTRIBUTION OF RETURNS
0.35
SHARE A
SHARE B
0.30
PROBABILITY
0.25
0.20
0.15
What do you
think in which
shares should
you invest?
0.10
0.05
0.00
0
5
10
15
20
RETURN
25
30
35
Now, look at again the following two
shares…
SHARE - A
Return (%)
Probability
2
0.05
9
0.29
12
0.24
16
0.17
19
0.12
23
0.07
28
0.03
30
0.04
1.00
Expected
14.10%
Return
Standard
6.40
Deviation
Skewness
0.80
SHARE - B
Return (%)
Probability
1
0.02
4
0.08
7
0.10
9
0.13
12
0.16
16
0.18
21
0.31
30
0.02
1.00
Expected
14.10%
Return
Standard
6.40
Deviation
Skewness
-1.07
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 mad!!! I
really do not know how many
pieces of input data I need to
generate my 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
40
R = 0.2814
30
ONGC (Return%)
20
10
0
-10
-5
0
5
-10
-20
ACC (Return%)
10
15
SCATTER DIAGRAM OF RETURNS
4
R = 0.2674
RANBAXY LABORATORIES LTD.(%)
3
2
1
0
-3
-2
-1
0
1
-1
-2
-3
INFOSYS TECHNOLOGIES LTD.(%)
2
3
4
Scatter Diagram
6
R = 0.289
4
RIL (Return%)
2
0
-10
-5
0
5
-2
-4
-6
ACC (Return%)
10
15
SCATTER DIAGRAM OF RETURNS
10
R = 0.3027
8
STATE BANK OF INDIA(%)
6
4
2
0
-3
-2
-1
0
1
-2
-4
RANBAXY LABORATORIES LTD.(%)
2
3
4
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?
But, are we in a position
to identify that factor?
If that factor exists, then your
data requirement will also be
considerably reduced!!!!
Yes!!!! We can identify that
factor...
And, this takes us to ...
And, now…
?????????????????????????………
Ri    Rm  
SHARPE’S SINGLE FACTOR/INDEX MODEL
 It is a linear relation between the return of a security
and the underlying factor which is the MARKET
INDEX.
Ri    Rm  
• It is ex-post relationship.
• It shows how a factor leads to generation of
returns in a security.
• Its intercept represents unique return of
security which is independent of Market Index.
a
• The slope of the Single Index Model represents 
which is a measure of SYSTEMATIC RISK.
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; and
it 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
3
y = 0.4619x - 0.2251
R2 = 0.1813
2
1
0
-1.5
-1
-0.5
0
-1
-2
-3
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
=
SYSTEMATIC RISK.
UNSYSTEMATIC
RISK
+
Risk
-
– Where
Total Risk of ith security = si2;
Systematic Risk = i2 sm2 ; and
Unsystematic Risk
= Total
Systematic Risk = si2 - i2 sm2.
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
Dr. Reddy'S Laboratories Ltd.
Regression Statistics
Multiple R
0.423823119
R Square
0.179626036
Adjusted R Square
0.178161082
Standard Error
6.876151354
Observations
562
ANOVA
df
Regression
Residual

Total
Intercept
X Variable 1
1
560
561
SS
MS
F
Significance F
5797.440487 5797.440487 122.6155199 6.61196E-26
26477.61617 47.28145745
32275.05666
Coefficients Standard Error
t Stat
P-value
0.841961864 0.290330094 2.900015814 0.003877893
0.753268111 0.068026302 11.07318924 6.61196E-26
 ESTIMATION [EXCEL output]
SUMMARY OUTPUT
Oil & Natural Gas Corpn. Ltd.
Regression Statistics
Multiple R
0.339940172
R Square
0.11555932
Adjusted R Square
0.113237954
Standard Error
7.369805289
Observations
383
ANOVA
df
Regression
Residual

Total
Intercept
X Variable 1
1
381
382
SS
MS
F
Significance F
2703.791967 2703.791967 49.78072823 8.16384E-12
20693.64543
54.31403
23397.4374
Coefficients Standard Error
t Stat
P-value
0.275169466 0.376636098 0.730597698 0.465473937
0.696256196 0.098682115 7.05554592 8.16384E-12
 ESTIMATION [EXCEL output]
SUMMARY OUTPUT
Reliance Industries Ltd.
Regression Statistics
Multiple R
0.714636907
R Square
0.510705909
Adjusted R Square
0.509833727
Standard Error
5.35907977
Observations
563
ANOVA
df
Regression
Residual

Total
Intercept
X Variable 1
SS
MS
F
Significance F
1 16816.83319 16816.83319 585.5497139 3.95203E-89
561 16111.77189 28.71973599
562
32928.60508
Coefficients Standard Error
t Stat
P-value
0.214380529 0.226065807 0.94831028 0.343379833
1.282653728 0.053006306 24.19813451 3.95203E-89
Any comment??
Source: BSE Site
Beta of a Portfolio …
• Beta of a portfolio is the weighted average of
individual securities betas.
n
β P = ∑X i β i
i =1
What next…?
And, now…
something exciting…
Concept of Beta from
Single Index Model
What a
cocktail!!!!
All these take us to …
Concept of
Systematic
Risk - βeta
Markowitz
Efficient
Frontier
?????
Risk-Free
Asset
WHAT’S THE
WORTH OF A
CAPITAL
ASSETS???
Dr. C. P. Gupta
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 is LINEARITY and SIMPLICITY.
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
Model(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
FRONTIER WHEN A RISK FREE ASSET IS
INTRODUCED INTO CAPITAL MARKET???
Will it be a non-linear
or
linear ???
EFFICIENT FRONTIER
becomes a straight line that is
tangent to Markowitz Efficient
Frontier and it is called
CAPITAL MARKET LINE.
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.
Borrowing
Lending
M
Rf
sM
Risk
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 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 -
E (RP )  RF 
E (RM ) - RF
sM
sP
Capital Market Line (CML)
(continued…)
 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.
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
BE SELECTED BY AN INVESTOR…???
 Depending upon an investor’s return - risk trade-off which is
reflected in his indifference map, he selects an optimum
portfolio for himself.
B
M
A
Rf
sM
Risk
Does the idea of Capital
Market Line ensure
better risk-return tradeoff for me???
Yes!!! It will improve risk-return
trade-off for our Topiwalla.
M
Rf
Do you see this?
sM
Risk
ARE Investment Decisions and
Financing Decisions
independent ???
TOBIN’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 THEOREM
(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 independent
investment decisions are same for all; and
financing decisions are different and independent of
investment decisions.
SECURITY MARKET LINE (SML)
 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 (SML)
 The equation of SML is called CAPITAL ASSET
PRICING MODEL.
 E(RM) - RF is called risk premium per unit of systematic
risk.
SML
M
Aggressive Security
Rf
Defensive Security
 1

What should be the price of a
security 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:
E (R ) 
P1 - P0
P0
 RF  ( E ( RM ) - RF ) 
 P0

1  {RF

P1 - P0
P0
P1
 ( E ( RM ) - RF )  }
CAPM and
its 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
CAPM
vs.
SINGLE FACTOR MODEL
 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.
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
S

s Pt
THE TREYNOR’S INDEX
T 

RPt - RF
RPt - RF
 Pt
THE JENSEN INDEX (ALSO KNOWN AS THE JENSEN’S )
J  RPt - {RF  ( RMt - RF ) P }
Measures of Portfolio
Evaluation
(continued…)

APPRAISAL RATIO - P/s(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
Looking for the
exact source
of
success/failure!
!
Why a
portfolio
manager is
able to
perform better or
worse?
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
EXCESS
RETURN
RISK FREE
RETURN
RISK
PREMIUM
DUE TO
SYSTEMATIC
RISK
RETURN FROM
SHARE SELECTION
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)(sP/sM - P); and
R3 = Residual Return and Fama named as NET
SELECTIVITY MEASURE.

PORTFOLIO PERFORMANCE EVALUTION - AN ILLUSTRATION
Consider the following information about Portfolio A, Portfolio Z and the Market Portfolio -M:
PORTFOLIOS
RETURN
A
Z
M (MARKET INDEX)
Risk - Free Return = 7%
12%
19%
15%
STANDARD
DEVIATION
18%
25%
20%
BETA
0.7
1.3
1.0
SHARPE RATIOS
A
Z
M
0.28
0.48
0.40
TREYNOR RATIOS
A
Z
M
7.14
9.23
8.00
JENSEN RATIOS
A
Z
M
-0.60
1.60
0.00
APPRAISAL RATIOS
Unsystematic Risk
A
Z
M
11.31%
25.00%
20.00%
APPRAISAL
RATIOS
-5.30
6.40
0.00
2
A
Z
M
M - MEASURE
Proportion of Investment in Fund
1.11
0.80
1.00
FAMA'S DECOMPOISTION
Risk - Free Rate
A
Z
M
7%
7%
7%
M-SQUARE MEASURE
-0.017
0.002
0.000
Risk Premium
Due to
Due to
Unsystematic
Systematic Risk
Risk
5.60%
1.60%
10.40%
-0.40%
8.00%
0.00%
Net
Selectivity
Measure
-2.20%
2.00%
0.00%
Total
12%
19%
15%
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
deviation of the difference in returns
portfolio under consideration and
benchmark or target; that is to say,
DEVIATION OF (Rp-Rb) where Rp is the
portfolio under consideration while Rb is
the benchmark portfolio.
the standard
between the
a specified
STANDARD
return on the
the return on
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 !!!!!