Asset Pricing Lbsim

7/28/2019 Asset Pricing Lbsim

1/72

ASSET PRICING


2/72

The basic issue is

how to determine

the price of a financial

asset?

REAL

ASSETS

FINANCIAL

ASSETS


3/72

There are broadly two approaches for

asset pricing in Theory of Finance One approach is EQUILIBRIUM

APPROACH; and

Another approach is ARBITRAGEAPPROACH.


4/72

The Story of Asset Pricing started

with

MarkowitzPortfolio Theory


5/72

Therefore, we start with

HARRY MARKOWITZ MODEL


6/72

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.


7/72


8/72


9/72

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


10/72

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.


11/72

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


12/72

How to estimate RISK from thepast data?

The Standard Deviation of the returns

can be taken as a proxy for Risk of a

security.


13/72

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.


14/72

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


15/72

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?


16/72

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.


17/72

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


18/72

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.


19/72

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


20/72

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


21/72

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


22/72

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


23/72

Can we have a zero risk portfolio???

Yes! We can have it if we can find

two securities having between them

perfect negative correlation.


24/72

And now, we start with

EFFICIENT FRONTIER


25/72

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


26/72

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


27/72

OPTIMUM SELECTION OF A PORTFOLIO

DEPENDS UPON RISK - RETURN TRADE - OFF!!!

Standard Deviation

Expec

tedReturn

F

E

OPTIMUM

PORTFOLIO

P

h h


28/72

What is the most importantcontribution of Markowitz model?

It is the concept of

Efficient Portfolio!!!


29/72

Is there

anything in the

Markowitz

Model at which

you would like to

ATTACK?


30/72

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


31/72

One of the serious limitation of

the Markowitz Model is

Huge data requirement!!!!

Can we overcome thisproblem?


32/72

Have you ever

wondered whyreturns of shares

of companies from

various industriesare correlated?


33/72

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


34/72

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


35/72

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


36/72

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


37/72

What makes shares

return to havecorrelation across

the companies fromthe different

industries?THINK!!!


38/72

Is there some

underlyingFACTOR

which makes

these

correlations

to exist?


39/72

If that factor exists, then your

data requirement will also be

considerably reduced!!!!

But, are we in a position

to identify that factor?


40/72

Yes!!!! We can identify thatfactor...

And, this takes us to ...


41/72

RmRi

SHARPES SINGLE


42/72

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


43/72

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


44/72

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


45/72

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


46/72

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.


47/72

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


48/72

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.


49/72

What next?


50/72

Dr. Vibha Jain

WHATS THEWORTH OF A

CAPITALASSETS???


51/72

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


52/72


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.



53/72


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.


54/72

WHAT HAPPENS TO EFFICIENT

FRONTIER WHEN A RISK FREE ASSET IS

INTRODUCED INTO CAPITAL MARKET???

Will it be a non-linearor

linear ???


55/72

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)


56/72

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


57/72

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)


58/72

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)


59/72

Why is the portfolioat which the

new efficient frontier is tangent to

the Markowitz Efficient Frontier

called MARKET PORTFOLIO?


60/72

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


61/72

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


62/72

ARE Investment Decisions and

Financing Decisions

independent???

TOBINS SEPARATION THEOREM


63/72


* 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 ]



64/72

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)


65/72

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




66/72

The equation of SML is called CAPITAL ASSET

PRICING MODEL.

E(RM)- RF is called risk premium per unit of systematicrisk.


Rf

SML

M

1

Aggressive Security

Defensive Security

W

hat should be

the price

of a


67/72

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


68/72

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


69/72

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


70/72

What next?

References


71/72

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


72/72

Do visit

www.projectbaba.com

Documents

Asset Pricing Lbsim