factor model and arbitrage pricing theory1 [compatibility mode]

8/8/2019 Factor Model and Arbitrage Pricing Theory1 [Compatibility Mode]

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Welcome to UnitedworldFACTOR MODEL ANDARBITRAGE PRICING THEORYWelcome to UnitedworldFACTOR MODEL ANDARBITRAGE PRICING THEORY


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RETURN FROM SHARES ARE NOTBASED ON SINGLE FACTOR S RossS Ross propounded the concept that security returns arenot based on one single factor (index) of the market;instead, each security is linked with multiple factors andreturns get influenced by these factors. The theory given

by S Ross is called arbitrage pricing theory and multiplefactor theory. APT believes that the return generatingf ti l it / tf li d t tprocess of a particular security/ portfolio does not getinfluenced by its association with the market portfolio(index). Rather it is influenced by several logical factors Gross Domestic product (GDP), National Income,Personal Disposable Income, Inflation Interest Rate,Consumer Price Index, Wholesale Price Index, Bank Rate,Foreign Exchange Rate etc. therefore, one must considerseveral factors instead of only the index of market toarrive at investment decision. Similarly, disequilibrium inshare prices can lead to the process of arbitrage.

RETURN FROM SHARES ARE NOTBASED ON SINGLE FACTOR S RossS Ross propounded the concept that security returns arenot based on one single factor (index) of the market;instead, each security is linked with multiple factors andreturns get influenced by these factors. The theory givenby S Ross is called arbitrage pricing theory and multiplefactor theory. APT believes that the return generatingf ti l it / tf li d t tprocess of a particular security/ portfolio does not getinfluenced by its association with the market portfolio(index). Rather it is influenced by several logical factors Gross Domestic product (GDP), National Income,Personal Disposable Income, Inflation Interest Rate,

Consumer Price Index, Wholesale Price Index, Bank Rate,Foreign Exchange Rate etc. therefore, one must considerseveral factors instead of only the index of market toarrive at investment decision. Similarly, disequilibrium inshare prices can lead to the process of arbitrage.


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CONCEPT OF ARBITRAGE PRICINGTHEORY (APT)This model has been developed by S Ross, using a notion thatsecurity returns are not based on one single factor (index) of themarket; instead each security is linked with multiple factors andreturns get influenced by these factors. APT believes that the return

generating process of a particular security/portfolio does not getinfluenced by its association with the market portfolio; rather it isinfluenced by several logical factors gross domestic product(GDP), national income, personal disposable income, inflation,interest rate, consumer price index, wholesale price index, bankrate, foreign exchange rate, etc. APT also emphasizes thefundamental of one single price which implies that marketremains in equilibrium and two identical securities, having samedegree of risk will command same price, i.e. will have same return inthe market in the long run. However, due to the short termdisequilibrium, there may be difference in the return of two securitieswith the same risk level, which will attract the process of arbitrage

and investors will be able to generate safe returns.CONCEPT OF ARBITRAGE PRICINGTHEORY (APT)This model has been developed by S Ross, using a notion thatsecurity returns are not based on one single factor (index) of themarket; instead each security is linked with multiple factors andreturns get influenced by these factors. APT believes that the returngenerating process of a particular security/portfolio does not getinfluenced by its association with the market portfolio; rather it isinfluenced by several logical factors gross domestic product(GDP), national income, personal disposable income, inflation,interest rate, consumer price index, wholesale price index, bankrate, foreign exchange rate, etc. APT also emphasizes the

fundamental of one single price which implies that marketremains in equilibrium and two identical securities, having samedegree of risk will command same price, i.e. will have same return inthe market in the long run. However, due to the short termdisequilibrium, there may be difference in the return of two securitieswith the same risk level, which will attract the process of arbitrageand investors will be able to generate safe returns.


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CONCEPT OF ARBITRAGE PRICINGTHEORY (APT)APT is one of the modern portfolio theories, which is basedon the rational behavior of investors. Theory is of one priceopinion , which implies that two identical securities withsame level of risk must have same return; if due to certain

imperfections in the market these offer different returns, theninvestors would sell the security with low return and buy theith hi h tone with high return.Arbitrage pricing theory has two basic outcomes it helps inassigning price to securities by identifying thesecurities/portfolios asa) Underpricedb) Overpricedc) Efficiently pricedCONCEPT OF ARBITRAGE PRICINGTHEORY (APT)APT is one of the modern portfolio theories, which is based

on the rational behavior of investors. Theory is of one priceopinion , which implies that two identical securities withsame level of risk must have same return; if due to certainimperfections in the market these offer different returns, theninvestors would sell the security with low return and buy theith hi h tone with high return.Arbitrage pricing theory has two basic outcomes it helps inassigning price to securities by identifying thesecurities/portfolios asa) Underpricedb) Overpricedc) Efficiently priced


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CONCEPT OF ARBITRAGE PRICINGTHEORY (APT)It also focuses on the fact that returns for a particular share isderived from its association will multiple factors, i.e. more thanone factor affecting the performance of the company. Thisassociation of an individual share/portfolios with different

factors is established with the help of a sensitivity factor calledbeta. In single index model and CAPM, each share has onlyb t h i th l ti hi f t f th h ith one beta showing the relationship of returns of the share withthe index of the market or market portfolio. But in APT, thereare more than one beta value for a particular share; return ofa share with each of the factors are shown with the help of aseparate beta value. For example, a share is affected byseven factors then it will have seven beta values, eachshowing relationship of the returns of the share with the factorseparately. Due to the relationship with many factors, it iscalled MULTIPLE FACTOR MODEL.

CONCEPT OF ARBITRAGE PRICINGTHEORY (APT)It also focuses on the fact that returns for a particular share isderived from its association will multiple factors, i.e. more thanone factor affecting the performance of the company. Thisassociation of an individual share/portfolios with differentfactors is established with the help of a sensitivity factor calledbeta. In single index model and CAPM, each share has onlyb t h i th l ti hi f t f th h ith one beta showing the relationship of returns of the share withthe index of the market or market portfolio. But in APT, thereare more than one beta value for a particular share; return ofa share with each of the factors are shown with the help of a

separate beta value. For example, a share is affected byseven factors then it will have seven beta values, eachshowing relationship of the returns of the share with the factorseparately. Due to the relationship with many factors, it iscalled MULTIPLE FACTOR MODEL.


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ASSUMPTIONS OF APTArbitrage pricing theory is based upon certain assumptions ifthese assumptions are found valid then the findings and logicof the model are applicable. These assumptions are asfollows:1. Investors are risk averse and utility maximisers

2. Investors have homogenous beliefs3. Market are perfect4. Multi factor affect5. Dominance principle6. Equilibrium of the market7. Efficient frontierASSUMPTIONS OF APTArbitrage pricing theory is based upon certain assumptions ifthese assumptions are found valid then the findings and logicof the model are applicable. These assumptions are asfollows:1. Investors are risk averse and utility maximisers

2. Investors have homogenous beliefs3. Market are perfect4. Multi factor affect5. Dominance principle6. Equilibrium of the market7. Efficient frontier


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INVESTORS ARE RISK AVERSE ANDUTILITY MAXIMISERS1. APT is based on the assumptions that each investors is riskaverse i.e. has the tendency to either avoid the risk orminimize it. It is believed that every investor prefers highreturn as compared to low return for a given level of risk.

Investors need to be compensated suitably for the riskassumed by them. Utility maximization implies that investorsf t i i t t i l lth f th i t t bprefer to maximize to terminal wealth of the investment byselecting such avenues, which offer maximum return for therisk preference of the investors. The combined effect of riskaversion and utility maximization is that, investors select thesecurities/ portfolio using the dominance principle.INVESTORS ARE RISK AVERSE ANDUTILITY MAXIMISERS1. APT is based on the assumptions that each investors is riskaverse i.e. has the tendency to either avoid the risk or

minimize it. It is believed that every investor prefers highreturn as compared to low return for a given level of risk.Investors need to be compensated suitably for the riskassumed by them. Utility maximization implies that investorsf t i i t t i l lth f th i t t bprefer to maximize to terminal wealth of the investment byselecting such avenues, which offer maximum return for therisk preference of the investors. The combined effect of riskaversion and utility maximization is that, investors select thesecurities/ portfolio using the dominance principle.


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INVESTORS HAVE HOMOGENOUSBELIEFS2. Homogenous belief implies that each investors has similarkind of risk return estimate about different investmentavenues. This assumption has direct impact on theidentification of securities into different categories. Arbitrage

pricing theory has an objective to assign value to differentsecurities in the sense that it helps in identifying these intoth diff t d ti t i Thi t i ti three different and executive categories. This categorizationis done by comparing expected returns with the realizedreturns from a security. It is done as follows.I. Undervalued securities having expected return < realized returnII. Efficiently priced securities having expected return = realizedreturnIII. Overvalued securities having expected return > realized returnAll this identification is similar in manner by almost all investorsas each one has homogenous estimate about the expected

return and risk about securities.INVESTORS HAVE HOMOGENOUSBELIEFS2. Homogenous belief implies that each investors has similarkind of risk return estimate about different investmentavenues. This assumption has direct impact on theidentification of securities into different categories. Arbitragepricing theory has an objective to assign value to differentsecurities in the sense that it helps in identifying these intoth diff t d ti t i Thi t i ti three different and executive categories. This categorizationis done by comparing expected returns with the realizedreturns from a security. It is done as follows.

I. Undervalued securities having expected return < realized returnII. Efficiently priced securities having expected return = realizedreturnIII. Overvalued securities having expected return > realized returnAll this identification is similar in manner by almost all investorsas each one has homogenous estimate about the expectedreturn and risk about securities.


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MARKETS ARE PERFECT3. Perfect market condition is the supporting basefor arbitrage pricing theory. It implies that thereare large number of investors as soon as suchinformation gets generated. This happens onlywhen different segments of the market work on

the principle of full disclosure andtransparency. The result of this is that none ofthe investors is in a position to create any kindof extra impact in the market by undulyinfluencing the share prices. The ultimate resultof perfect market condition is that,securities/portfolios are priced efficiently in thelong run.MARKETS ARE PERFECT3. Perfect market condition is the supporting basefor arbitrage pricing theory. It implies that thereare large number of investors as soon as such

information gets generated. This happens onlywhen different segments of the market work onthe principle of full disclosure andtransparency. The result of this is that none ofthe investors is in a position to create any kindof extra impact in the market by undulyinfluencing the share prices. The ultimate resultof perfect market condition is that,securities/portfolios are priced efficiently in thelong run.


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MULTI FACTOR EFFECT4. Single index model of sharpe and CAPM are based on theassumption that each security/portfolio has an association with onlyone factor of the market and this factor is either the market portfolioor index of the market. Contrary to this, APT believes that eachsecurity has an association with more than one factor. The theory

does not talk about the type and number of factors. Accordingly,return and risk of an individual security or portfolio will always beinfluenced by more than one factor, these may be such factorshaving direct effect on the performance of the company, like marketshare of the company, foreign exchange rate if the company hasimport export business, inflation rate, interest rate, monetary policyparameters in case of a bank or financial institution, gross domesticproduct and national income affecting the demand for the productof the company, etc. This association of each individual factoraffecting with the returns of the share is represented through betaof the securitys return with the individual factor.MULTI FACTOR EFFECT

4. Single index model of sharpe and CAPM are based on theassumption that each security/portfolio has an association with onlyone factor of the market and this factor is either the market portfolioor index of the market. Contrary to this, APT believes that eachsecurity has an association with more than one factor. The theorydoes not talk about the type and number of factors. Accordingly,return and risk of an individual security or portfolio will always beinfluenced by more than one factor, these may be such factorshaving direct effect on the performance of the company, like marketshare of the company, foreign exchange rate if the company hasimport export business, inflation rate, interest rate, monetary policyparameters in case of a bank or financial institution, gross domesticproduct and national income affecting the demand for the product

of the company, etc. This association of each individual factoraffecting with the returns of the share is represented through betaof the securitys return with the individual factor.


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MULTI FACTOR EFFECT contdBeta represents the volatility of the returns fromthe share for every one percent change in thereturns of the associated factor. For exampleshare X has beta 0.50 with factor A and withfactor B its beta is 0.35; now when factor A

shows an upward movement by 1 percent then shows an upward movement by 1 percent, thenthe return of this share X on account of this willincrease by 0.50 percent and similarly, whenfactor B shows and upward movement by 1percent then also the return of this share willincrease by 0.35 percent. Similarly, when thesefactors show a decline, the returns from thisshare will decrease correspondingly.MULTI FACTOR EFFECT contdBeta represents the volatility of the returns fromthe share for every one percent change in the

returns of the associated factor. For exampleshare X has beta 0.50 with factor A and withfactor B its beta is 0.35; now when factor Ashows an upward movement by 1 percent then shows an upward movement by 1 percent, thenthe return of this share X on account of this willincrease by 0.50 percent and similarly, whenfactor B shows and upward movement by 1percent then also the return of this share willincrease by 0.35 percent. Similarly, when thesefactors show a decline, the returns from thisshare will decrease correspondingly.


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DOMINANCE PRINCIPLE5. Dominance principle implies that one security outperforms theother for the same price. Similarly, it may be that one portfoliooutperforms the other for the same given price. By price here,we mean the level of risk. It is dominance principle whichhelps in identifying efficient portfolios and generation of

efficient frontier. The fundamental of dominance principle canbe understood with the help of the following graph.be understood with the help of the following graph.Here for the risk level 1 percent standard deviation we havetwo corresponding securities one is having 20 percent returnand the other is having 15 percent, one with 20 percent returndominates the other as it is giving more return for the samelevel of risk. Similarly, if we look from the return, side for 15percent return, there are two securities one having risk 1percent and another having risk 1.5 percent. Certainly, thesecurity with 1 percent risk measurement dominates theanother.

DOMINANCE PRINCIPLE5. Dominance principle implies that one security outperforms theother for the same price. Similarly, it may be that one portfoliooutperforms the other for the same given price. By price here,we mean the level of risk. It is dominance principle whichhelps in identifying efficient portfolios and generation ofefficient frontier. The fundamental of dominance principle canbe understood with the help of the following graph.be understood with the help of the following graph.Here for the risk level 1 percent standard deviation we havetwo corresponding securities one is having 20 percent returnand the other is having 15 percent, one with 20 percent returndominates the other as it is giving more return for the same

level of risk. Similarly, if we look from the return, side for 15percent return, there are two securities one having risk 1percent and another having risk 1.5 percent. Certainly, thesecurity with 1 percent risk measurement dominates theanother.


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Equilibrium of the market :6. Equilibrium of the market : APT relies on thefact that securities markets function inequilibrium, which implies that the price of twoidentical securities having the same degree ofrisk must be the same. It clearly helps inrisk must be the same. It clearly help

s inmaking an interpretation that two securitieshaving same level of risk must generate samereturn in the long run. Thus, equilibriumtheorem creates homogenous beliefs of theinvestors and helps in identifying mispricing ofthe securities. This is the basis of bringing inthe concept of efficient portfolio.Equilibrium of the market :6. Equilibrium of the market : APT relies on thefact that securities markets function inequilibrium, which implies that the price of two

identical securities having the same degree ofrisk must be the same. It clearly helps inrisk must be the same. It clearly helps inmaking an interpretation that two securitieshaving same level of risk must generate samereturn in the long run. Thus, equilibriumtheorem creates homogenous beliefs of theinvestors and helps in identifying mispricing ofthe securities. This is the basis of bringing inthe concept of efficient portfolio.


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Equilibrium of the marketcontdHowever, due to short-term disequilibrium,two securities with identical risk levelmight have different returns which willattract the process of arbitrage. As a result

of the arbitrage process, securities will bepriced efficiently in the long run and find aplace on the efficient frontier. Theimplication of equilibrium of market resultsin the law of one price, i.e. securities withidentical risk must generate same return.Equilibrium of the marketcontdHowever, due to short-term disequilibrium,two securities with identical risk levelmight have different returns which willattract the process of arbitrage. As a result

of the arbitrage process, securities will bepriced efficiently in the long run and find aplace on the efficient frontier. Theimplication of equilibrium of market resultsin the law of one price, i.e. securities withidentical risk must generate same return.


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Efficient frontier7. Efficient frontier: An efficient frontier is theline on risk-return graph, joining all cornerportfolios. A corner portfolio is the one, which iseither efficient in itself or is created bycombining certain efficient portfolios, these combining certain efficient portfo

lios, theseefficient portfolios offer optimum returns for thegiven level of risk. It is believed that all theportfolios, which are priced according to thefundamentals of market equilibrium, willgenerate optimum return for the given level ofrisk and will be priced efficiently; consequentlyget plotted on the efficient frontier.Efficient frontier7. Efficient frontier: An efficient frontier is theline on risk-return graph, joining all cornerportfolios. A corner portfolio is the one, which is

either efficient in itself or is created bycombining certain efficient portfolios, these combining certain efficient portfolios, theseefficient portfolios offer optimum returns for thegiven level of risk. It is believed that all theportfolios, which are priced according to thefundamentals of market equilibrium, willgenerate optimum return for the given level ofrisk and will be priced efficiently; consequentlyget plotted on the efficient frontier.


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Efficient frontier contdAs market is considered to be efficient,each security and portfolio should fall onthe efficient frontier in the long-run. Thispresence and knowledge of efficientfrontier helps in the identification of

different securities and portfolios as(a) underpriced,(b) priced efficiently, and(c) overpriced.Efficient frontier contdAs market is considered to be efficient,each security and portfolio should fall onthe efficient frontier in the long-run. Thispresence and knowledge of efficientfrontier helps in the identification ofdifferent securities and portfolios as(a) underpriced,

(b) priced efficiently, and(c) overpriced.


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Efficient frontier contdSecurities or portfolios, which get plottedabove the efficient frontier are underpricedand securities or portfolios below theefficient frontier are overpriced. Thisidentification will lead to' arbitrage process

and ultimately, each security and portfoliowill be on the efficient frontier in the long-run.Efficient frontier contdSecurities or portfolios, which get plottedabove the efficient frontier are underpricedand securities or portfolios below theefficient frontier are overpriced. Thisidentification will lead to' arbitrage processand ultimately, each security and portfoliowill be on the efficient frontier in the long-run.


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Expected Return & Risk UnderAPTExpected returns : By expected return, we meanthe most likely level of the return, which should bemaintained as per the performance of the companyand association of individual share with different

factors. An estimate about the most likely return factors. An estimate about themost likely returncan be made once forecast about the differentfactors and beta value is available. Total expectedreturn can be bifurcated into two -return onaccount of company's performance called Alphaand return on account of security's association withthe multiple factors called Systematic Componentof Return.Expected Return & Risk UnderAPTExpected returns : By expected return, we mean

the most likely level of the return, which should bemaintained as per the performance of the companyand association of individual share with differentfactors. An estimate about the most likely return factors. An estimate about themost likely returncan be made once forecast about the differentfactors and beta value is available. Total expectedreturn can be bifurcated into two -return onaccount of company's performance called Alphaand return on account of security's association withthe multiple factors called Systematic Componentof Return.


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Expected Return & Risk UnderAPT contd.Alpha : Alpha component of the return is onaccount of performance of the company. It can betermed as the return, which can be expected if allother factors have zero-value or the security has

zero-beta with all the factors. Alpha component of zero beta with all the factors. Alpha component ofreturn is also termed as zero beta return.Systematic component return :This is such partof the total expected return, which is on account ofthe association of the security with differentsystematic factors, which affect returns from theshare. This is represented with the help of beta.Expected Return & Risk UnderAPT contd.Alpha : Alpha component of the return is onaccount of performance of the company. It can be

termed as the return, which can be expected if allother factors have zero-value or the security haszero-beta with all the factors. Alpha component of zero beta with all the factors. Alpha component ofreturn is also termed as zero beta return.Systematic component return :This is such partof the total expected return, which is on account ofthe association of the security with differentsystematic factors, which affect returns from theshare. This is represented with the help of beta.


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Expected Return & Risk UnderAPT contd.Random error term This represents extraordinaryreturn from a security only when some extraordinaryevent like merger, acquisition, takeover, bonusdeclaration, etc., takes place. Generally, it is

considered as zero.considered as zero.The expectation of return from a particular securityis based on the association of the security with thefactors affecting the return. As per APT theorem, thefollowing equation represents the expected return ofa security:Expected Return & Risk UnderAPT contd.Random error term This represents extraordinaryreturn from a security only when some extraordinaryevent like merger, acquisition, takeover, bonusdeclaration, etc., takes place. Generally, it is

considered as zero.considered as zero.The expectation of return from a particular securityis based on the association of the security with thefactors affecting the return. As per APT theorem, thefollowing equation represents the expected return ofa security:


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Expected Return & Risk UnderAPT contd.Expected Return =iijnijijijiejnjjj++++++bbbba...... 321 321Expected Return & Risk UnderAPT contd.

Expected Return =iijnijijijiejnjjj++++++bbbba...... 321 321


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Expected Return & Risk UnderAPT contd.return beta zero or companytheof eperformanc the of accounton return i.e.return,systematic-Non =iap yj1' 'as named factor first

ith the security w of beta 1 =ijbExpected Return & Risk UnderAPT contd.return beta zero or companytheof eperformanc the of accounton return i.e.return,systematic-Non =iap yj1' 'as named factor firstith the security w of beta 1 =ijb


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Expected Return & Risk UnderAPT contd.j2' 'as named factor second theith security wthe of beta 2 =ijbj1= value of the first factor named as j1j1 value of the first factor named as j1j2 = value of the second factor named as j2

error term random =ieExpected Return & Risk UnderAPT contd.j2' 'as named factor second theith security wthe of beta 2 =ijbj1= value of the first factor named as j1j1 value of the first factor named as j1j2 = value of the second factor named as j2error term random =ie


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RISKExpected return from a security is subject to somekind of fluctuation. This fluctuation may be positiveor negative. Fluctuation in the expected return isrepresented with the help of variance in the return.As expected return is divided into two -alpha

component and systematic component; similarly,risk of a share can also be divided into two -systematic risk on account of system-wide factorsand non-systematic risk on account of company-wide factors. Risk of a share is represented withthe help of the following equation. Risk is representas variance of the expected return which is asfollowsRISKExpected return from a security is subject to somekind of fluctuation. This fluctuation may be positiveor negative. Fluctuation in the expected return is

represented with the help of variance in the return.As expected return is divided into two -alphacomponent and systematic component; similarly,risk of a share can also be divided into two -systematic risk on account of system-wide factorsand non-systematic risk on account of company-wide factors. Risk of a share is represented withthe help of the following equation. Risk is representas variance of the expected return which is asfollows


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RISK CONTD.)var( )var( .... )2var( )1var( )var( 22212 ejnjjiijnijij+++=bbb

Var (i) = Variance of expected return of i' securityVar (j1) = Variance of factor 'j1'2'as namedfactor secondith thesecurity wthe of beta'1'as namedfactor first ith thesecurity wthe of beta21jjijij==bbVar (j2) = Variance of factor 'j2'

Var (e) = Company-specific risk/non-systematic risk on account ofcompany-wide factorsRISK CONTD.)var( )var( .... )2var( )1var( )var( 22212 ejnjjiijnijij+++=bbbVar (i) = Variance of expected return of i' securityVar (j1) = Variance of factor 'j1'2'as namedfactor secondith thesecurity wthe of beta'1'as namedfactor first ith thesecurity wthe of beta2

1jjijij==bbVar (j2) = Variance of factor 'j2'Var (e) = Company-specific risk/non-systematic risk on account ofcompany-wide factors


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Beta: Beta of a security is the sensitivitymeasurement, representing volatility of thereturn for a given change in the factor to whichthis beta value associates. Beta is calculated byconsidering considering covariance of the security s returncovariance of the security's return

and value of the related factor. For example, ashare has beta of 1.20 with factor' Y', then itmeans for every one per cent change in thefactor' Y', the returns from this share will changeby 1.20% only due to this factor. It is calculatedas follows:


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BETA121,1covariance

jjiijsb='1'factor with the ''security the of beta 1 jiij=b'1'factor of variancereturn affecting factors the of one '1'security individual ''12 jjij===s

Using this formula, beta of the security with each factor can be calculated separatelyBETA121,1covariancejjiijsb='1'factor with the ''security the of beta 1 jiij=b'1'factor of variance

return affecting factors the of one '1'security individual ''12 jjij===sUsing this formula, beta of the security with each factor can be calculated separately


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Calculation of expected return using APTequationIllustration 1, pg no. 454Illustration 1, pg no. 454Calculation of expected return using APTequationIllustration 1, pg no. 454Illustration 1, pg no. 454


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The Arbitrage ProcessA combined effect of investor's utilitymaximization theorem, homogenous beliefs ofinvestors and perfect markets is that,securities are priced efficiently in the long-run.Two identical securities with the same level of

risk (OJ) must have the same price, i.e. meanreturn for each of the security. However, dueto market imperfections and short-term,disequilibrium in the market, two identicalsecurities might be priced differently; but thiswill last only in the short run.The Arbitrage ProcessA combined effect of investor's utilitymaximization theorem, homogenous beliefs ofinvestors and perfect markets is that,securities are priced efficiently in the long-run.Two identical securities with the same level of

risk (OJ) must have the same price, i.e. meanreturn for each of the security. However, dueto market imperfections and short-term,disequilibrium in the market, two identicalsecurities might be priced differently; but thiswill last only in the short run.


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The Arbitrage ProcessFor example, two securities have the samelevel of risk, i.e. a standard deviation of 3%but these are having a return of 15 %(lowyielding share) and 17% (high yieldingh ) thi diff i th t ill share), this difference in the returns will

attract the process of arbitrage. This willimply that investors. who have security,giving 15% return will sell and buy thesecurity with 17% return so that they canhave more return by having the same levelof risk.The Arbitrage ProcessFor example, two securities have the samelevel of risk, i.e. a standard deviation of 3%but these are having a return of 15 %(lowyielding share) and 17% (high yieldingh ) thi diff i th t ill share), this difference in the returns will

attract the process of arbitrage. This willimply that investors. who have security,giving 15% return will sell and buy thesecurity with 17% return so that they canhave more return by having the same levelof risk.


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The Arbitrage ProcessThis process will create a supply for lowyielding share and demand for high yielding;consequently prices of the earlier willdecline, leading to an increase in the yieldand prices of the later will rise, leading to

decline in the yield. The change in the yieldwill ultimately bring both the securities atequilibrium and both will lie on the efficientfrontier.The Arbitrage ProcessThis process will create a supply for lowyielding share and demand for high yielding;consequently prices of the earlier willdecline, leading to an increase in the yieldand prices of the later will rise, leading todecline in the yield. The change in the yieldwill ultimately bring both the securities at

equilibrium and both will lie on the efficientfrontier.


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The Arbitrage ProcessBy arbitrage in portfolio context, we mean selling thesecurities or portfolios, which generate low or less return fora particular level of risk and buying the other security orportfolio, having same level of risk but generating high ormore return. This happens because of the risk-aversion

behaviour and utility maximization by the investors The behaviour and utility maximization by the investors. Theeffect of this process is that, the price of the security/portfoliowith low return starts declining on account of increasedsupply and price of the security with high return increases onaccount of increased demand. This arbitrage process willcontinue till the time both the securities do not find a placeon the efficient frontier. This process is explained with thehelp of following example:The Arbitrage ProcessBy arbitrage in portfolio context, we mean selling thesecurities or portfolios, which generate low or less return for

a particular level of risk and buying the other security orportfolio, having same level of risk but generating high ormore return. This happens because of the risk-aversionbehaviour and utility maximization by the investors The behaviour and utility maximization by the investors. Theeffect of this process is that, the price of the security/portfoliowith low return starts declining on account of increasedsupply and price of the security with high return increases onaccount of increased demand. This arbitrage process willcontinue till the time both the securities do not find a placeon the efficient frontier. This process is explained with thehelp of following example:


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The Arbitrage ProcessPortfolio Mean return in % RiskA 10 1B 15 1.5%sC 20 2

U 17 1.25O 9 1.25O2 13 1.50The Arbitrage ProcessPortfolio Mean return in % RiskA 10 1B 15 1.5%sC 20 2U 17 1.25O 9 1.25O2 13 1.50


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The Arbitrage ProcessBy reading the data given in the table andalso plotted on the graph above, a portfoliomanager can identify that the portfolios A, U& C are on efficient frontier, whereasportfolios B, 0 and 02 do not find a place on portfolios B, 0 and 02 do not find

a place onthe efficient frontier. Here, portfolio Uhaving return 17 per cent with a risk of 1.25per cent is under priced and anotherportfolio with the same level of risk (1.25per cent) has the return of only 9 per cent isover-priced. This identification will lead toarbitrage process.The Arbitrage ProcessBy reading the data given in the table andalso plotted on the graph above, a portfoliomanager can identify that the portfolios A, U

& C are on efficient frontier, whereasportfolios B, 0 and 02 do not find a place on portfolios B, 0 and 02 do not finda place onthe efficient frontier. Here, portfolio Uhaving return 17 per cent with a risk of 1.25per cent is under priced and anotherportfolio with the same level of risk (1.25per cent) has the return of only 9 per cent isover-priced. This identification will lead toarbitrage process.


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Generating profit with zero initialoutflowNow, an investor can generate extra return with zeroinitial outflow. Let us assume that an investor short sells(selling the share or portfolio without having it with theunderstanding of buying it back in the future or having

arbitrage profit) portfolio '0' of the value Rs 10,000 andbuys the portfolio U' by using these Rs 10 000 After buys the portfolio U by using these Rs 10,000. Afterdoing this and waiting for a period of one year, he has topay 9 per cent for the short sale of the portfolio '0' but hewill receive 17 per cent from the portfolio U'. Thus withthe help of arbitrage, he can earn 8 per cent return withzero initial outflow, now he might reverse the deal, i.e.selling portfolio U' and cover the short sale of '0' bybuying it.Generating profit with zero initialoutflow

Now, an investor can generate extra return with zeroinitial outflow. Let us assume that an investor short sells(selling the share or portfolio without having it with theunderstanding of buying it back in the future or havingarbitrage profit) portfolio '0' of the value Rs 10,000 andbuys the portfolio U' by using these Rs 10 000 After buys the portfolio U by using these Rs 10,000. Afterdoing this and waiting for a period of one year, he has topay 9 per cent for the short sale of the portfolio '0' but hewill receive 17 per cent from the portfolio U'. Thus withthe help of arbitrage, he can earn 8 per cent return withzero initial outflow, now he might reverse the deal, i.e.selling portfolio U' and cover the short sale of '0' by

buying it.


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Table showing arbitrage profitthrough zero initial investmentPresent cashflowReturn after oneyear

Cash flow afterone yearShort sell O +10000 -900 -10000Buy U -10000 + 1700 + 10000Net cash flow 0 +800 0Thus, by making such arbitrage, one can have Rs 800 withzero initial investment.Likewise, portfolio B has risk 1.5 per cent and return 15 percent; at the same time, we can observe that the portfolio '02'also has a risk of 1.5 per cent but its' return is only 13 percent. Here portfolio '02' is identified as overvalued and 'B' asefficient as it is on the efficient frontier.

Table showing arbitrage profitthrough zero initial investmentPresent cashflowReturn after oneyearCash flow afterone yearShort sell O +10000 -900 -10000Buy U -10000 + 1700 + 10000Net cash flow 0 +800 0Thus, by making such arbitrage, one can have Rs 800 withzero initial investment.

Likewise, portfolio B has risk 1.5 per cent and return 15 percent; at the same time, we can observe that the portfolio '02'also has a risk of 1.5 per cent but its' return is only 13 percent. Here portfolio '02' is identified as overvalued and 'B' asefficient as it is on the efficient frontier.


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Limitation of APTArbitrage Pricing Theory has practicalimplication, as it considers association of thesecurity and portfolio with more than one factoraffecting return and risk. It is an improvementover Sharpe's Single Index model and CAPM,

t it h li it ti :yet it has limitations:It does not specify the type and number of factorsaffecting return and riskIt is difficult to identify the factors affecting return andriskDifferent investors might identify different factors forthe same security/portfolio; this will violate theassumption of homogenous beliefsLimitation of APTArbitrage Pricing Theory has practicalimplication, as it considers association of thesecurity and portfolio with more than one factor

affecting return and risk. It is an improvementover Sharpe's Single Index model and CAPM,t it h li it ti :yet it has limitations:It does not specify the type and number of factorsaffecting return and riskIt is difficult to identify the factors affecting return andriskDifferent investors might identify different factors forthe same security/portfolio; this will violate theassumption of homogenous beliefs


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Limitation of APT contdDifficulty in calculating different beta valuesEffect of one factor on the return and risk cannot be assessed preciselyIf the list of factors affecting the return andrisk is infinite, then this theory does not find

the practical implicationWith passage of time, type and number offactors for one security/portfolio mightchange, leading to inconsistency incomparison.Limitation of APT contdDifficulty in calculating different beta valuesEffect of one factor on the return and risk cannot be assessed preciselyIf the list of factors affecting the return andrisk is infinite, then this theory does not findthe practical implication

With passage of time, type and number offactors for one security/portfolio mightchange, leading to inconsistency incomparison.


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CONCLUSIONArbitrage Pricing Theory is an improvement over thesingle index model like that of Sharpe or CAPM.This theory believes that each security and portfoliohas relationship with more than one factor and thisrelationship affects the return of it. APT does not

specify the type and number of factors affectingreturns from a security/portfolio, but it discusses thatone single factor cannot influence returns and risk ofa security/portfolio. Instead, return and risk areinfluenced by a set of multiple factors. Due to this, itis also called as multi-factor model. Theory restsupon the basic assumption of risk-averse and utilitymaximization behaviour of the investors.CONCLUSIONArbitrage Pricing Theory is an improvement over thesingle index model like that of Sharpe or CAPM.This theory believes that each security and portfolio

has relationship with more than one factor and thisrelationship affects the return of it. APT does notspecify the type and number of factors affectingreturns from a security/portfolio, but it discusses thatone single factor cannot influence returns and risk ofa security/portfolio. Instead, return and risk areinfluenced by a set of multiple factors. Due to this, itis also called as multi-factor model. Theory restsupon the basic assumption of risk-averse and utilitymaximization behaviour of the investors.


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CONCLUSION CONTD.It also assumes that markets are perfect and eachsecurity and portfolio is in equilibrium in the long run.Due to this equilibrium, two identical securities musthave the same return. However, certain short-termdisequilibrium in the market can create distortion in the

return of these securities, which will attract the process return of these securities, which will attract the processof arbitrage. Accordingly, if two securities have samelevel of risk, but are giving different return, then peoplewill sell low yielding security and buy high yieldingsecurity, this process is called arbitrage. This arbitragewill finally put all the securities/portfolio on the efficientfrontier. An efficient frontier is the one, which is createdby joining the entire comer portfolio on the risk-returngraph.CONCLUSION CONTD.It also assumes that markets are perfect and each

security and portfolio is in equilibrium in the long run.Due to this equilibrium, two identical securities musthave the same return. However, certain short-termdisequilibrium in the market can create distortion in thereturn of these securities, which will attract the process return of these securities, which will attract the processof arbitrage. Accordingly, if two securities have samelevel of risk, but are giving different return, then peoplewill sell low yielding security and buy high yieldingsecurity, this process is called arbitrage. This arbitragewill finally put all the securities/portfolio on the efficientfrontier. An efficient frontier is the one, which is createdby joining the entire comer portfolio on the risk-return

graph.


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THANK YOU THANK YOU


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factor model and arbitrage pricing theory1 [compatibility mode]

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