let’s start with a review of what we did in the class!!!

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 !!!!!