Arbitrage+Pricing+Theory

7/31/2019 Arbitrage+Pricing+Theory

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ARBITRAGE PRICING THEORY

NMIMS/PTMBA FIN. DIV A & B


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Arbitrage Pricing TheoryWhy & What is Need?

1. CAPM establishes relationship between risk and expected return for

securities. (Where investors base their decision on comparision ofreturn distribution of alternative portfolios.)

2. Equilibrium predictions of CAPM asserts that investors form portfolio

consisting of riskless assets and market porttfolio; combination vary

from investor to investor depending upon individual preference forexpected risk and return.

3. Equilibrium expected return should be linear function of return on

risk free assets (rf) plus market portfolio risk premium [E(rm) rf]

which is proportional to the beta of a security.

Arbitrage Pricing Theory


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What was observed?

1. Small cap stocks outperformed large cap stocks on risk adjusted basis

(finding by Mr. Banz)

e.g.

Tata Sponge Jindal Saw SBT LIC Housing

Equity 15 45 18 87P/E 2.40 6.0 3.80 9.30

Market Price 190 488 465 602

(27.08.09)

2. Low P/E stocks outperformed large P/E stocks; after adjustment for risk

(finding by Mr. Banz)P/E Market Price

TATA Steel 3.1 440

BPCL 7.5 504

SBI 9.1 1750

Grasim 9.4 2625



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3. Value Stocks Tata Chem, HLL (i.e. stocks with higher

book to market value) generated larger returns than

Growth Stocks JP Associates, GMR, RPL (stocks with

low book-to-market value) on risk adjusted basis. (finding

by Mr. Fama & French)

Value Stock Growth Stock

Low Earnings per share growth High Earnings per share growth

Low P/E Ratio High P/E Ratio

Low price to book-value ratio High price to book-value ratio

High Div. yield Low Div. yield

Betas tend to be less than one Beta tends to be >1Out of Favour Popular

TATA Chem/HLL/Shipping Corp JP/GMR/RPL


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What does this mean?

1. Are markets not efficient for long period of time?

2. Markets efficient but CAPM (a single factor model) does not capture

risk adequately.

Look for Alternative Risk Return Model beyond CAPM

Hence

ARBITRAGE PRICING THEORY(developed by Mr. Stephen Ross)


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Arbitrage Pricing ModelSay, there are 2 securities having same risk but Different Expected

Return.

Investors to arbitrage / eliminate these differences.

Trading to buy Low Priced and sell High Priced to continue till both

are at same expected return

Hence Ross developed a Model explaining

i) Assets Risk

ii) Expected Return (required return) in terms ofsensitivity to each ofbasic economic factors.



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Arbitrage - arises if an investor can construct a

zero investment portfolio with a sure profit

Since no investment is required, an investor

can create large positions to secure large

levels of profit In efficient markets, profitable arbitrage

opportunities will quickly disappear



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Arbitrage Pricing Theory - APT

Three major assumptions:1. Capital markets are perfectly competitive and

no transaction cost. (This assumption makespossible the arbitraging of mispriced securities,thus forcing an equilibrium price)

2. Investors always prefer more wealth to less wealth

3. Various factors give rise to returns on securitiesand the relation between the security return and

that these factor is LINEAR. The stochastic processgenerating asset returns can be expressed as alinear function of a set ofKrisk factors or indexes



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CAPMs STRONG Assumption are not required

1. Single Period Investment horizon.2. No Taxes.

3. Investors can freely borrow or lend at rf.

4. Investors select portfolio based on theexpectedmean and variance of return.

ARBITRAGE PRICING THEORY - APT


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The return on any asset i, ri can be expressed as a linear Function of set of M

factors/indexes

Ri = 10 + i1 I1 + i2 I2 + i3 I3 + . + iM IM + Ei

Ri = Return on asset I during a particular time period I = 1, 2 n

10= Expected Return on asset i, if all risk factors have zero charges (zero returns)

ij = Sensitivity of asset is return to the jth index i.e. Reaction in asset is returns to

movement in a common factor. (Various systematic risk like interest rate risk etc.)

Ij = Value of Jth index j = 1,2, 3 . M i.e. a common factor with zero mean that

influences the returns of all assets i.e. Deviation of a systematic risk factor j from its

expected value.

Ei = Random error term for asset i, with mean of zero and variance E i.e. a unique

effect on asset is return that by assumption is completely diversifiable in large portfolios

and has a mean of zero

N = Nos. of assets

Note : 1] APT does not specify the risk factors I1, I2, . Ik that have bearing on assetreturn.

2] Error Term can be reduced to zero through appropriate diversification.



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LINEAR RELATIONSHIP

1] Depicts similar to Single Index Model except that under APT there are several indexes/

factors that may have influence on particularsecuritys index (Factor)

2] These indexes represent systematic influence on securitys return

e.g. Growth in GP

Changes in Inflation, Interest Rate Risk, Forex Rate Risk etc.

ONLY SYSTEMATIC FACTORS ARE IMPORTANT IN PRICING OF SECURITIES.

3] ij element in equation is analogus to i in Single Index Model. They represent the

impact that a particular factor has on a particular securitys return e.g. a ij value of 3.0

means that for every 1% change in factor j, security is return is expected to change 3%or thrice the change in the value of Ij.


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Arbitrage Pricing Theory (APT)

* ij determines how each asset reacts to the common factor

* Each asset may be affected by (though effects are not identical)

- Growth in GP

- Changes in expected inflation

- Changes in the market risk premium

- Changes in oil prices etc.

* Systematic risks (i.e. Non-diversifiable) factors are important in the

pricing of securities.



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APT APT assumes that, in equilibrium, the

return on a zero-investment, zero-

systematic-risk portfolio is zero when the

unique effects are diversified away

The expected return on any well-

diversified portfolio i (Ei) can be expressed

as:



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APT

E(ri) = 0 + 1i1 + 2i2 + 3i3 + . + kik

Where

E(ri) = Expected Return on asset I i.e. on well diversified portfolio

0 = Expected Return on asset with zero systematic risk

1 = The risk premium related to each of the common factor e.g. the risk premium

related to interest rate risk.

ij = the pricing relationship between the risk premium and asset i i.e. how responsive

asset i is to this common factor j i.e. sensitivity or beta coefficient for security i

that is associated with index j



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TWO FACTOR MODEL

Assume, There are two factors which

generate returns on assets I. APT Model

becomes

E (ri) = 0 + 1bi1 +2bi2


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ONE FACTOR MODEL

Assume, there is only one factor which generates returns on asset i, APT Model boils down

to

E(ri) = io + ij1

io = Risk Free Return or Zero Beta Security

Slope of arbitrage price line is and intercept is io. The arbitrage price line shows the

equilibrium relation between an assets systematic risk and expected return.


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Risk Premium

Factor % is Response to Portfolio Response to Portfolio

Rate of Inflation 1 1 = 0.01 bx1 0.50 by1 2

Growth in GNP 2 2 = 0.02 bx2 1.50 by2 1.75

Zero Systematic 3 3 = 0.03

Risk AssetE1 = 0 + 1bi1 + 2bi2

= 0.03 + (0.01) bi1 + (0.02) bi2

Ex = 0.03 + (0.01) (0.50) + (0.02) (1.50)

= 0.065 = 6.5%

Ey = 0.03 + (0.01) bi1 + (0.02) bi2= 0.03 + (0.01) (2) + (0.02) (1.75)

= 0.085 = 8.5%


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Example of Two Stocksand a Two-Factor Model= changes in the rate of inflation. The risk premium

related to this factor is 1 percent for every 1percent change in the rate

1

)01.0( 1

= percent growth in real GNP. The average risk

premium related to this factor is 2 percent for

every 1 percent change in the rate

= the rate of return on a zero-systematic-risk asset

(zero beta: boj=0) is 3 percent

2

)02.0( 2

)03.0( 0 0



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Example of Two Stocksand a Two-Factor Model

= the response of portfolio Xto changes in the rate

of inflation is 0.50

1xb

)50.( 1

xb= the response of portfolio Yto changes in the rate

of inflation is 2.00 )00.2( 1 yb1yb

= the response of portfolio Xto changes in the

growth rate of real GNP is 1.50

= the response of portfolio Yto changes in the

growth rate of real GNP is 1.75

2xb

2yb

)50.1( 2 xb

)75.1(2

y

b



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Example of Two portfolios

and a Two-Factor Model

= 0.03 + (0.01)bi1 + (0.02)bi2

Ex= 0.03 + (0.01)(0.50) + (0.02)(1.50)= 0.065 = 6.5%

Ey= 0.03 + (0.01)(2.00) + (0.02)(1.75)

= 0.085 = 8.5%

22110 iii bbE



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For same risks

Asset U has higher return than Asset O

Asset U is underpriced and assets O is overpriced.

Sell asset O or go short on O

Buy asset U or go long on U

Investor makes Riskless profit

Impact

Demand on asset U goes up and supply of O also goes up

Price of U increases and price of O decreases

Thus, Arbitrage goes on till prices are traded at same level.

Arbitrage Mechanism



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ARBITRAGEStatus as on 07-08-09

Cash Derivative Lot Size What is Needed

RIL 1995 (buy)

Rs. 598500

2000 (sale)

Rs. 600000

300 Funds

BHEL 2180 (buy)

Rs. 654000

2200 (sale)

Rs. 660000

300 Funds

EDUCOMP 3905(sale)

Rs. 292875

3891 (buy)

Rs. 291825

75 Shares

DLF 368 (buy)

Rs. 588800

372 (sale)

Rs. 595200

1600 Funds

L & T 1465 (buy)

Rs. 586000

1470 (sale)

Rs. 588000

400 Funds

SBI 1742 (buy)

Rs. 459988

1750 (sale)

Rs. 462000

264 Funds



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Arbitrage Pricing Theory (APT)Examples of multiple factors expected to

have an impact on all assets:

Inflation Growth in GNP

Major political upheavals

Changes in interest rates And many more.

Contrast with CAPM insistence that only beta

is relevant



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APT and CAPM Compared APT applies to well diversified portfolios and not necessarily to individual

stocks

With APT it is possible for some individual stocks to be mispriced in

equilibrium - not lie on the SML APT is more general in that it gets to an expected return and beta

relationship without the assumption of the market portfolio

APT can be extended to multifactor models

CAPM APT

Nature of relation Linear Linear

Number of risk factors 1 k

Factor risk premium [E (RM) Rf] j

Factor risk sensitivity i bij

Zero-beta return Rf 0



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IMPLICATIONS

1. While the CAPM is still probably the best available estimate of risk for most corporate

investment decision, managers must recognise that their stock prices may fluctuate more

than what the standard theory suggests.

2. The market is usually smarter than the individual. Hence managers should weight the

evidence of the market over the evidence of experts.

3. Markets function well when participants pursue diverse decision rules and their errors are

independent. Markets, however, can become very fragile when participants display herd-

like behaviour, imitating one another.

4. It may be futile to identify the cause of a crash or boom because in a non-linear system

small things can cause large scale changes.

5. The discounted cash flow model provides an excellent framework for valuation. Indeed, it

is the best model for figuring out the expectations embedded in stock prices.



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APT SUMMARY

Compared to the CAPM, the APT is much more general. It assumes that the

return on any stock is linearly related to a set of systematic factors.

Ri = ai + bi1 I1 + bi2 I2 + .. + bij Ij + ei

The equilibrium relationship according to the APT is

E(Ri) = 0 + i1 1 + bi2 2+ . bij I

Some studies suggest that, in comparison to the CAPM, the APT explains

stock returns better; other studies hardly find any difference between the two.



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EXAMPLESArbitrage Pricing Theory

1] Three portfolio with risk return characteristics as under

Portfolio E(ri) Risk ( i)A 10.6 0.80

B 13.4 1.20

U 15.0 1.00

Portfolio U offers returns disproportionate to risk Portfolios A & B returns arecommensurate with the risk

Opportunity to profit

Riskless arbitrage conversion. How?

Construct Portfolio F with same risk level as U

This can be done by investing equal amounts in Portfolio A & B

of Portfolio F is 0.50 X beta of A + 0.50 X beta of B

= 0.50 X 0.80 + 0.50 X 1.20

= 0.40 + 0.60 = 1.00

Return on Portfolio F is = 0.50 X 10.6 + 0.50 X 13.4

= 12%

This return is in line with the Portfolios risk.


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EXAMPLESArbitrage Pricing Theory

2] How arbitrage Portfolio can be constructed to benefit from differential return given byportfolio F & U having same level of risk.

Ans. Strategy is short sell FGo Long on U

Portfolio Investment(Rs.) Return (Rs. ) Risk ( i)F 10,000 - 1200 - 1

U - 10,000 + 1500 + 1

Arbitrage Portfolio 0 + 300 0

Arbitrage Portfolio O

i) Short sell F Realise Rs. 10,000 i.e.ve value of investmentii) Invest in U Invest Rs. 10,000

iii) Expected Return on F & U @ 12% and 15%

iv) Net Return on Arbitrage Rs. 300

v) Since eta of F & U are equal but opposite in value, Beta of arbitrage portfolio is zero.

vi) Arbitrage Portfolio offers profit without any Net Investment or Risk.


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EXAMPLES (two factor Arbitraging Process)

Returns generated by two indices or factors as per following pricing relation

E(ri) = 5 + 7i1 + 4 i2

AnsE(ri) Risk ( i1) Risk ( i2)

A 15.4 0.80 1.20

B 16.6 1.20 0.80

O 14.0 1.00 1.00

A & B are correctly priced. Return O is not commensurate with risk.

Scope of arbitrate

Construct Portfolio F (same risk as O)

E(rf) = (0.50 X 15.4) + (0.50 X 16.6) = 16

Bf1 = (0.50 x 0.80) + (0.5 X 1.20) = 1.0

Bf2 = (0.50 X 1.20) + (0.5 X 0.80) = 1.0

Portfolio F has the same systematic risk as portfolio O

Arbitrage portfolio can be created by selling overprice i.e. O and going long on F.



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ExampleThree portfolios as under:-

Portfolio E(ri) Bi1 Risk Bi2 Risk

A 15.4 0.80 1.20

B 16.6 1.20 0.80

O 14.0 1.00 1.00

A & B are correctly priced.

Return on O is not commensurate with underlying risk measured by Bil & Biz

We can construct a Portfolio F which also has same risk characteristic as

portfolio O.

This portfolio f can be constructed by combining portfolios A & B in equal

proportion.

i.e. E(rf) = (0.50)(15.4) + (0.50)(16.6)= 7.7 + 8.3 = 16

F1 = (0.50)(0.80) + (0.50)(1.20) = 0.4 + 0.6 = 1

F2 = (0.50)(1.20) + (0.50)(0.80) = 0.6 + 0.4 = 1



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Portfolio F has same systematic risk as portfolio O differs on

higher return (i.e. 16% & not 14%). This offers an arbitrageopportunity.

Portfolio Investment Return Risk Bi1 Risk Bi2

O -10,000 -1400 -1 -1

F 10,000 +1600 +1 +1________ ________ ____ ______

0 200 0 0



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Examples

1. Consider the following data relating to beta and expected return of two

portfolios:-

Portfolio Return % Beta

A 14 1.0

B 16 1.0

Given one factor arbitrage pricing model is E(ri) = 6 + 8 i1 show the

arbitrage opportunity, if any, can be exploited.

Ans.: E (rA) = 6 + 8 i1 = 6 + 8(1) = 14

E (rB) = 6 + 8 i1 = 6 + 8(1) = 14

The return from B is higher @16% than 14% expected. Hence, we should go

long on B & short sell A suppose we have Rs. 1,00,000/-

Portfolio Investment Return Risk (Beta)

A -1,00,000 -14,000 1

B +1,00,000 +16,000 1

+2,000

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