CAPM and APT Model Development, Evaluation and Interpretation on ABC’s Share Price

Group – 02 FNB 501, QTF

Abstract: Over the last decades spontaneous anddevelopment which come under the label ofarbitrage pricing model, following the developmentof CAPM model (Sharpe, 1964 Lintner, 1965), are todevelop an accurate estimation model for expectedreturn. The aim of this academic coursework is totest the CAPM model and APT model on givensecondary data and experience with hypothesistesting which makes selected models whether fitsor not. Using samples of monthly data (2006-2012),regression model has developed to review the CAPMand APT model. The findings shows that there areslight differences between CAPM, APT and ExtensionAPT. As the assumptions of arbitrage pricing modelholds two basic assumptions of APT – efficientmarkets information and diversified investment.When portfolios investment are diversified thismeans Market Risk Exposure includes most of micro-macroeconomics factors’ effects. But using APT andExtension of APT have better explanatory power (R2

and Adjusted R2) then CAPM which makes APT modelmore attractive to the scholars to suggest use ofAPT in estimation of expected return. And FTSEMarket Excess Return makes the most importantfactors apart from other small affecting factorsin most researches. But scholars find thataccepting APT is better than CAPM but it is noterror-free to estimate expected return in allcondition. As assumptions makes it difficult topredict actually in the markets. However, FF3Model (Fama and French, 1992) has developedfurther supportive model but in the end like allof the scholars, it is expected that missinginformation (Aggelidis and Mandotinos, 2007) willfurther improve through development andmodification of arbitrage pricing model for thedemand of the time.

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Keywords: CAPM, APT, Asset Pricing, Regression Analysis.

Introduction:The Capital Asset Pricing Model (CAPM) is widely accepted asan appropriate technique for evaluating financial assets. Itis used to construct portfolios, measure the performance ofinvestment managers, develop project screening rates forcapital budgeting, and value companies. The ArbitragePricing Theory (APT) which offers an alternative explanationof the relationship between risk and return including Macroand Micro-economic factors, while CAPM has the limitationespecially in the emerging markets where the perfect marketassumption do not apply, are only superficially mentioned.

CAPM, APT and FF3 (modified theory of CAPM by usingadditional two Fama and French Factors) models are appliedfor constructing portfolios, measuring the performance ofinvestment managers, developing project screening rates forcapital budgeting, valuation of companies, determining costof capital and so forth (Dhankar and Esq, 2005). Therefore,asset pricing model has definitely got a wide range ofsignificant and practical implications as a research basedcurrent issue especially in the domain of finance andinvestment.

Objectives: This academic assignment is done with the aim to

represent an easy and transparent method of testing CAPMand APT model utilize widely used equation on ABC shareprice.

The first and foremost objective of this academic work isto realize the CAPM and APT model in the investmentmanagement.

Use of CAPM and APT model to estimate expected returnswhich depends on various dependent factors includingMarket Index, Inflation and Retail Sales.

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Comparison between CAPM and APT Model is also included inthe objective to realize how these two model differ togive most predictable solution of return.

Literatures Review:After development of CAPM as a model of predicting asset’sreturn from the market by Sharpe (1964) and Lintner (1965),in the introductory period this model was best in the worldto predict returns on investment. But for the time-beingdifferences limitations occurred for which new Asset PricingTheory, which is developed by Ross (1976) is known as APTModel (or Arbitrage Pricing Theory). And also CAPM model isexpanded by FF3 model by Fama and French (1992, 1993, and1996) by adding two new factors with the exiting marketindex for more fitted model in predicting the returns.

CAPM theoretically suggests that a security could be addedto a portfolio, based only on its systematic risk or beta.Fact is that the beta is only priced by the market becauseall non-systematic risk is eliminated by diversification.CAPM basic equation is:

E (ri)=rf+βi [E (rm )−rf ]Where subscript i refers to individual price; E(Ri) isexpected return on ith security; rf is the return on risk-free asset; E(Rm) is expected return on market portfolio andβi is the measure of risk or definition of marketsensitivity.

Thus given that investors are risk averse, it seems sensiblethat high risk (high beta) stock should have higher expectedreturn than low risk (low beta) stocks and if zerosystematic risk (β=0) then expected return just equal tothat on the riskless asset Rf.

CAPM/SML concepts are based upon expectations, but betas arecalculated using historical data. A company‘s historicaldata may not reflect investors’ expectations about futureriskiness. Therefore, because of several limitations thatCAPM has got, many researchers were not completely satisfiedwith the CAPM. Moreover, theoretically the CAPM suggests

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that returns on an asset will follow the market returnsexcept for differences in risk. Also CAPM is unable topredict return in all situation and in all share market. Aresearch was done by Gonzalez F. (2001) to validate the CAPMmodel in Caracas Stock Exchange (CSE) from 1992 to 1998.After different hypothesis testing significant evidenceshows that CAPM should not be used to predict stock returnsin the CSE. As the researchers found that the model islinear and significant evidence on the existence of otherfactors different from Beta that are important to predictreturns. This research was based regression model

Where, Rm represents the CSE index return ad proxy for themarket portfolio. When the researchers also include varianceequation to assess the risk of security by:

In the end the result shows that CAPM cannot be used topredict assets returns in Venezuela, though it was likesurprise but the strong assumptions of CAPM do not apply inVenezuela. Also researchers include with that there areother factors that influence stock returns and these factorsshould be included in the pricing model.

Like this research several other researches shows that CAPMhas poor overall explanatory power, whereas APT model whichallows multiple sources of systematic risks to be taken intoaccount, performs better than the CAPM (Aggelidis, V. andMaditinos, D. 2007). Aggelidis and Maditinos also suggestedlike Gonzalez F. (2001) that shares and portfolios in theASE seem to be significantly influenced by a number ofsystematic forces and their behavior can be explained onlythrough the combined explanatory power of several factorsbut CAPM cannot.

Also the parameters of the two-factor CAPM were linearlyrelated to other variables and changed over time in responseto changes in these variables (Cho, Elton and Gruber, 1984).In order to develop CAPM more realistic and significantmeasurement for portfolio performance Jensen measure ofportfolio was tested but was proved invalid (Peasnell,Skerratt and Taylor, 1979). A similar test also used in

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Ross’ APT model but became again invalid (Morris and Pope,1980).

In 1986, single factor CAPM model was used in UK, and foundthat tests of the single factor CAPM model were verydisappointing and CAPM was always rejected in favor of APT(Beenstock and Chan, 1986). A cross-section regression wasused in the case of CAPM:

Where, Ri (T1T2) is the mean return for security i over theperiod T1T2 while Rf (T1T2) is the mean value of the Tresury Billrate over the same period. CAPM implies 1 = 0 sincesecurity returns have been expressed as a premium over therisk free rate.

Similarly like these limitations of CAPM models in modernera, it was same after the development of basic CAPM in1964/ The CAPM of Sharpe-Lintner-Black (SLB) have beensubjected to extensive empirical testing in the past 30years (Black, Jensen and Scholes, 1972; Blume and Friend,1973; Fama and MacBeth, 1973; Basu, 1977; Reiganum, 1981;Banz, 1981; Gibbons, 1982; Stambaugh, 1982 and Shanken,1985). In general, the empirical results have offered verylittle support of the CAPM, although most of them suggestedthe existence of a significant linear positive relationbetween realised return and systematic risk.

As a consequence APT, founded upon the work of Ross (1976),aims to analyse the equilibrium relationship between assets’risk and expected return just as the CAPM does. The two keyCAPM assumptions of perfectly competitive and efficientmarkets and homogeneous expectations are maintained.Moreover, in line with the CAPM, the APT assumes thatportfolios are sufficiently diversified, so that thecontribution to the total portfolio risk of assets’ unique(unsystematic) risk is approximately zero (Aggelidis andManditinos, 2007). Therefore, the arbitrage pricing theory(APT) has been proposed as an alternative to the capitalasset pricing model (Dankar and Esq, 2005). The appeal ofthe APT probably comes from its implication thatcompensation for bearing risk may be comprised of severalrisk premiums (macroeconomic factors), rather than just one

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risk premium as in the CAPM (Reinganum, 1981). The generalequation for Arbitrage Pricing Model is:

Y=α+β1x1+β2x2+…+βnxn+μ

Where, Y = returns on the individual asset; X1, X3… Xn arethe general macroeconomic factors; is the random error;and b1, b2, b3 … show the effects of each individual Xvariable on Y when the other X variables are held constant.

However, the room for debate and subjective argument aboutwhich macroeconomic factors have the most important effectson asset and market returns. In addition, some researcherscontinue to include market returns as one of the X variableswhile others do not (Javed 2003; Benninga 2006, Pp. 443). Itshould be possible to test these issues in a regression. TheAPT has recently attracted considerable attention as atestable alternative to capital asset pricing model ofSharpe-Lintner and Black.

APT model (Ross, 1976) became highly accepted both in US andUK market in compare CAPM model, as a relatively highproportion of the variance of estimated expected returns inthe market for 220 UK securities can be explained in termsof the APT (Beenstock and Chan, 1984). Also explanatorypower of a 20 factor APT model was significantly greaterthan a four factor model by using cross-sectional regressionAPT model where 2 = 0:

Roll and Ross have written what has quickly become theclassic article on testing the Arbitrage Pricing Theory(APT) originally proposed by Ross, it became inconclusivebecause APT needs efficient markets and information systemto perform (Solnik, B. 1983). It also became inconclusive innational and international cases where common factors aredifferent. That is why another research conduct Solnik urgesfor new theory which can be attractive in both national andinternational markets, moreover can access some commonfactors for the portfolio which engaged in national andinternational market simultaneously.

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Though Gonzalez F. (2001) found CAPM is insignificant inVenezuela and suggestion was to use models like APT for morefactors which affects returns. But in 2007, Aggelidis andManditinos found that APT is better than CAPM in AthensStock Exchange but APT does not explain the overall varianceproperly, maybe there is a missing information and that iswhy APT fails to explain fully the returns covariance andmeans returns. There can be several possible explanations(Cheng, 1995). First, risk and expected return may not bestationary as it is assumed not to change during the period;secondly, APT prcing relationship could hold only in somemonths of the year, and there is evidence of a “Januaryeffect” on the capability of the APT to explain the return-risk-relationship (Gultekin and Gultekin, 1987); andthirdly, there is a possibility of non-linear pricingrelationships. In the end, it is recommended that higher-order factor models would provide more accurate predictions.

Similar problems arise in (Huberman and Wang, 2005;Aggelidis and Manditinos, 2007) in Burmeister and McElory(1988) article where January effects also found, rejectionof CAPM in favor of the APT; however, it cannot reject theAPT restrictions on the linear factor model.

But there are also articles which shows APT is comfortablyconsistence with CAPM in predicting returns like forestryindustry in China (Sun and Zhang, 2001) and in case of 3selected companies APT performed robustly when using basicand n-factor model where E (Ri) is used instead of alpha.

Apart from success and limitations of APT in differentregions and markets, what about South-Asian Market which ismore nearest to our Bangladeshi markets. In Indian stockmarket, use of APT model shows that APT may lead to betterestimates of expected rate of return than CAPM (Dhankar,2005). But APT results may vary depending on the sample,time period and estimation methods used. But it is suggestedthat investor should give due consideration to multifactormodels like APT and not rely soley on beta and CAPM. Theregression model which is used as follows.

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Where E(Ri) is expected return on asset i, R0 is return on arisk-free asset because all its b are zero (0). Thesensibility to factors are estimated as:

Beside APT to cover the limitations of CAPM model morerecently Fama and French (FF) (1992) work confirms theinadequacy of two-moment CAPM. Their empirical workindicates that book-to-market ratio and marketcapitalization (firm size) explain the cross-sectionvariation of average returns more appropriately than beta.FF (1993) propose a model wherein the excess return versionof CAPM is modified by adding two more items namely SMB(small minus big; the difference between the return on aportfolio of small stocks and the return on a portfolio oflarge stocks) and HML (high minus low; the differencebetween the return on a portfolio of low book to marketstocks). And scholars found FF three-factor modeloutperforms the other models (Lawrence, Geppert and Prakash,2007) when comparing among traditional CAPM, three-momentCAPM and the FF three-factor model.

In addition to this, a recently performed test of FF3 model(1992, 1993, 1996) which also suggests that the test havesupported to accept the FF3 pricing model and they revealstrong evidence of significantly positive risk premiums,particularly in the case of SMB and HML factors by using FF3three-factor model and extension of the model by adding Bankrate (Shawkat, M. H. 2014). But like the author, one vitalissue of the FF model that potentially makes it lessappealing (than the CAPM) is the difficulty surrounding thenature and construction of factors. FF3 model will bedifficult to implement in markets like Bangladesh. Where itis arguably CAPM is most preferable but APT could be betterafter experimental test. Also this article used the basicregression model which is used APT and CAPM model asfollows:

Y = a + b1X1 + b2X2 + b3X3+b4X4+u

Author actually tested it for the support of FF3 foracademic purpose and it is noticeable that adding Bank ratein Extension of FF3 basic model doesn’t change beta for

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Market Excess Return, SMB and HML from previous model buthas changed the value of R2 = 83.2% to 81.2% and Adjusted R2

= 73.2% to 76.8%, in which R2 decreased but Adjusted R2

increased. Thus, Bank rate is created confusion in affectingoverall Quality of the model. Actually we find it difficultto keep the same beta value for FTSE Market Excess returnwhen including or excluding new variables.

After all, the author has expressed same feeling we havethat there is still the models gap, room for extension,development and modification of arbitrage pricing model forthe demand of the time.

Methodology and Data Analysis:The main focus of this assignment is to assess the role andpractical implementation of CAPM and APT model as themechanism for investment strategy. To serve the purpose thework has mainly emphasised on quantitative analysis. Intesting the models, the time series (historical) data havebeen considered where the values of one or more variables areobserved at different points in time. Using SPSS 21 versionData from Excel has analysed through regression line for CAPMand APT model which state below sequentially.

The Capital Asset Pricing Model (CAPM):The Asset Pricing Model is based on CAPM (Sharpe 1964,Linter 1965). CAPM states that it is only degree ofsystematic risk as expressed by beta (Bp) that willdetermine the degree of returns for diversified portfolio.When we should no longer rely on /SD at portfolio (as perMarketing 1954, Tofin 1959), CAPM does not considerMacro/Micro Factor, CAPM consider the market index is theonly risk as single risk factor. Basic CAPM equation asfollows:

E (Rp)=Rf+(Rm−Rf)×βp

In the basic model of CAPM βp can be measured by theformula:

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βp=σx×r (x,y )

σy

Also it can be measured by the regression line and here wecalculate the Beta by regression line between Given ABCshare Excess Return and FTSE Market Index.

Analysis and interpretations:

Annual Return & Excess Return Data:Calculated Annual Return and Excess Return for ABC share andFTSE market index table is given below based on these two formulas:

Annual Rate of Return = (Pt – Pt – 1) / Pt – 1 *100Where,

Pt = Current PricePt – 1 = Beginning Price

Excess Rate of Return = Annual Return (%) – Risk-free Return (%)

= Rp - Rf (Treasury Bills Rate)

Regression model (Excel):As in the early we stated that basic model of CAPM:E (Rp)=Rf+(Rm−Rf)×βp where the βp can be measured by Regressionmodel between ABC Share Excess Return and FTSE Market IndexExcess Return (Data from Annual & Excess Return Table – 01).Regression Model will be:

yi=α+β1x1+μ

[Where, = alpha or intercept between two data set, β1 is the coefficient betweenABC Excess Return and FTSE Market Index Excess Return; and error terms]

As per Regression model for CAPM, it has been estimated fromthe Excel Output:

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ABC excess return = - 1.009 + 1.406*FTSE excess return +.

o R = 0.944;o R Square (R2) = 0.891;o Adjusted R Square (R2) = 0.890

Sample Interpretations:From output on dependent variable ABC excess return, we caninterpret the result as:

If FTSE excess return goes up by $1 ABC share will alsogo up by $1.406359.

Alpha abc:From MS Excel output we have got = - 1.008869 which is theconstant or intercept value between ABC Share Excess Returnand FTSE Market Index Excess Return. But this is inclusivebecause of P-value and t-value of the result, because:

As P-value of Alpha is .31 which is higher than 0.05(at 95% interval) which mean it is insignificant.

t-value of Alpha is -1.02 which is lower than 2. As t <2 then it is inclusive with ABC Share Excess Return.

B abc (Beta):There is only one beta result as CAPM model only considerthe market index as market risk, no other Macro or Microfactors. Beta represent degree of systematic risk. Beta forABC share is:

β = 1.406359 P-value of β is 0.00 which lower than .05 so value is

significant. t-value of β is 23.97 which higher than 2 (t > 2) which

indicates there is relation between ABC Excess returnand FTSE Market Index Excess return.

R Squared:R Squared (R2) is an measure of –

Overall goodness of fit of the model.

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Explanatory power of the model indicating how closelythe ABC excess return is associated with Market,Inflation and Retail Sales.

It represents whether the model fits the real data set R2 = 1 represents linearly fits the data set

Here, R square (R2) = .891 which indicates that 89.1% of thevariability in ABC returns is explained by explanatoryvariable Market Index (FTSE). R2 shows very good explanatorypower and linearly fits data set.

Correlation coefficient:Correlation coefficient or Multiple Regression which isdenoted by R refers to the degree of relationship whosevalue range is 0 to +1. Near to 1 means there is a highdegree of relationship between variables. From the SPSSoutput, we get

R = 0.944

This result shows high degree of relationship between FTSEmarket Index and ABC Share excess return variables.

The Arbitrage Pricing Theory (APT):Arbitrage Pricing theory is proposed alternative inpredicting require rate of return in spite of CAPM in stockmarket. It is also the expansion of CAPM beta where CAPMconsider only one beta (Stock to Market Index), however APTconsider multi-factor beta to determine the required rate ofreturn such as Market Index, Inflation, Retail Sales,Central Bank Rate etc. APT or Arbitrage Pricing theory worksthrough regression line where dependent variable expressedas of independent variables beta. The regression equation ofAPT by most scholars is:

Y=α+β1x1+β2x2+β3x3+…+βnxn+μWhere,

Y = required rate of return for individual stock. = incept between dependent and independent variables β1, β2, β2 … βn are the beta for different factors regarding

returns

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X1, X2 … Xn is the individual stock which affect by betafrom independent variables.

is the independent error terms.

SPSS Output:As per multi-factor asset pricing model using ArbitragePricing Theory (APT), it has been estimated from the SPSSOutput: ABC excess return = - 4.250 + 1.363*FTSE excess return

- .084*Inflation Rate + 1.015*RetailsSales Rate + .

R = .947; R Square (R2) = .897; Adjusted RSquare (R2) = .892

Sample Interpretations:From the SPSS output on dependent variable ABC excessreturn, we can interpret the result as:

If FTSE excess return goes up by $1 ABC share will alsogo up by $1.363 by holding Inflation Rate and RetailsSales Rate constant.

Holding FTSE excess return and Retails Sales Rateconstant, if Inflation rate goes up by $1 ABC sharewill go down by 8 pence.

Holding FTSE excess return and Inflation rateunchanged, if retail sales rate goes by $1 ABC shareprice will all go up by $1.015

Hypothesis Testing:Hypothesis Testing is important for the reasons below:

Significance testing for accepting a result of a modelbased on a sample.

To infer the result of an operation based on sample. Procedure used to accept/reject the null hypothesis or

alternative hypothesis Whether the variable support the model scientifically

or not.

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In the model γi = + β1x1 + β2x2 + β3x2 + - the collectivename for , β1, β2, β3 are the Coefficients (or parameters).Testing of the results from these parameters is included inHypothesis testing. In every test, we have two hypothesis

H0 = null Hypothesis and H1 = alternative hypothesisWhere it is expected to have the null hypothesis rejectedand alternative hypothesis is accepted through differenthypothesis testing.

t-statistic (or t-ratio):In order to carry out the hypothesis test we need toconsider the estimated coefficient in relation to the sizeof its associated standard error.In t-statistic (or t-ration) hypothesis ratio if the resultis t 2 (equal or higher than 2) than it is assumed thatthere is a relation but on the contrary there is no relationbetween dependent and independent variables.

Hypothesis: H0: a = 0 (or t < 2, then there is norelation)

H1: a 0 (or t 2, then there is relationbetween variables)

Parameters Alpha FTSEExcessReturn

InflationRate

RetailSales Rate

Hypothesis H0: true = 0H1: 0

H0: trueβ1 = 0H1: β1 0

H0: true β2

= 0H1: β2 0

H0: true β3

= 0H1: β2 0

SPSS Result -1.836 <2

21.771 >2

- .145 < 2 1.806 < 2

Accept & Reject

H1

rejected& H0

accepted

H0

rejected& H1

accepted

H1

rejected &H0

accepted

H1 rejected& H0

accepted

Interpretation

As interceptor (alpha) is not

As FTSE Excess Return issignificant from

As Inflation (%) is notsignificant from 0,

As Retail Sales (%) is not significantfrom 0, so

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significant from 0, so it is not meaningful

0, so market index hasrelation with ABC

so Inflation has no relation with ABC

it has no relation with ABC

P-value:For 95% interval if P > .05 then reject null hypothesis (H0)and P < .05 then accept the alternative hypothesis (H1). So,hypothesis are:

H0: a = 0 (P > .05; no relation and reject)H1: a 0 (P < .05; there is relation and accept)

Parameters Alpha FTSEExcessReturn

InflationRate

RetailSales Rate

Hypothesis H0: true = 0H1: 0

H0: true β1

= 0H1: β1 0

H0: trueβ2 = 0H1: β2 0

H0: true β3

= 0H1: β2 0

SPSS Result .071> .05

.00 < .05 .885> .05

.075 > .05

Accept & Reject

H1

rejected& H0

accepted

H0

rejected &H1

accepted

H1

rejected& H0

accepted

H1 rejected& H0

accepted

Interpretation

Interceptor alpha value is not meaningful

FTSE Excess Return hasrelation with ABC share excess return

Inflation(%) has no relation with ABC

Retail Sales (%) has no relation with ABC

Testing R-square (R2) & Adjusted R-square (R2):In the model summary of SPSS output, the R2 variable givesthe proportion of variance that can be predicted by theregression model using the data provided (Rahman, 2006). So,R2 is an measure of –

Overall goodness of fit of the model.

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Explanatory power of the model indicating how closelythe ABC excess return is associated with Market,Inflation and Retail Sales.

It represents whether the model fits the real data set R2 = 1 represents linearly fits the data set

On the other hand Adjusted R2 is the modified R2 that adjuststhe degree of explanatory variables. The relation between R2

and adjusted R2 is that if R2 increase, Adjusted R2 should beincreased but never exceeds R2.Here, R square (R2) = .897 which indicates that 89.7% of thevariability in ABC returns is explained by variations inexplanatory variables like Market Index (FTSE), Inflationand Retail Sales. R2 result shows very good explanatory powerand linearly fits data set.

The F-test:F-test is used to test the R2 which justify the reliabilityof overall model (but t-value is for individual variable).The hypothesis for F-test are:

H0: the true R2 = 0 (no relationship) H1: the true R2 0 (there is a relationship)

Here, significance F less than < .05 or Sig. FChange .00<.05 which means alternative hypothesis H1 isaccepted and there is much evidence of overall relationshipin the model.

Durbin-Watson d Test:Auto-correlation is a concept if the variables have thetendency to follow negative or positive. Normally errorterms mean = 0 and no auto-correlation is good subject tobiased negative or positive tendency.

Auto-correlation complicates statistical analysis byaltering the variance of variables, changing theprobabilities that statisticians commonly attach to makingincorrect statistical decisions (Griffith 1987). Themechanism which has used for testing auto-correlation is DW

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d test which is calculated from the residuals. So, Durbin-Watson (D.W) d test is:

To test the auto-correlation we have to test D.W d test Auto-correlation complicates the statistical decisions

by changing the values of variances. Successive errors terms have the same sign error.

The Hypothesis for D.W d test are H0: = 0 (means there is no auto-correlation) H1: > 0 (means there is positive auto-correlation) H1: < 0 (means there is negative auto-correlation)

Also it is important to see the result from Durbin-Watsontable from which we can formulate, if

du > DW value, then H0: accepted. dL < DW value, then H1: accepted. dL > DW value, the test is inclusive.

In the model summary, the Durbin-Watson value is 1.05 whichis higher than 0, so we could have positive auto-correlationindicating that successive error terms have the same sign.Here, the DW value = 1.05 less than dL = 1.52 (using DWtable). Therefore, we would reject the null hypothesis andconclude that there is positive auto-correlation.

Testing value of tolerance (multicollinearity):

The value of tolerance close to 0 indicates that avariable is almost a liner combination of the otherindependent variables and such data are called multi-collinear (Norusis, 2005).

If the tolerance statistic is below .2,multicollinearity can be biasing the results (Menard,1995). Also there is no-correlation between variables.

Strong relationship between two, then if one increasesthen other will increase but dependent may notincrease. So it may be biased.

For testing value of tolerance (multicollinearity) thehypothesis are:

H0: a = 0 (or H0 < .02 multicollinearity can bias theresult)

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H1: a 0 (or H1 > .02 multicollinearity cannot biasthe result)

Since tolerance value of X variables (.861, .908, .787)all are close to 1, so H1 Hypothesis is accepted whichindicates X variables are independent and there is noroom for multicollinearity.

Redesigning APT model:Analysis and interpretation of APT model shows there is highdegree of relationship between independent variables anddependent variables; shows only one variables (FTSE MarketIndex) is significant proved by t-value and p-value. Othermeasurements like Durbin-Watson d test or F test also showsthe overall model is significant and good fit to the APTmodel.

But what would happen if we add a new variable and excludeexisting variable to redesign the existed APT model. Itcould worse in quality or better in quality for adding thenew variable. For this test or redesign task we includedBank Rate (%) as a new variable and excluded inflation rate

Therefore, the new Regression Model is:

Y=α+β1x1+β2x2+β3x3+β4x4+μ

Where, Y = ABC Excess Return = alpha or intercept or constant value or fixed

costΒ1, 2, 3, 4 = Beta coefficient for independent

variablesX1, 2, 3, 4 = Independent Variables = error terms

SPSS Output:ABC Excess -2.885 + 1.373 * FTSE Excess Return +

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Return = 1.117 * Retail Sales - 0.478 * BankRate

Determinants Value Comments Multiple R = .947 High degree of correlation R Square (R2) = .897 89.7% of independent

variables can be explained bydependent variables

Adjusted RSquare

= .893

Durbin-Watson

= 1.066 Free from Auto-correlation problem

Sig. FChange

= .000 Overall of fitness of model is good

Sample Interpretation:From the SPSS output of regression line we can interpretthat,

If FTSE Excess Return increase 1, ABC Excess Returnwill be increased by 1.373.

If Bank Rate increase 1%, ABC Excess Return will bedecreased by -.478%

And so on for other two independent variables

t-statistic (or t-ratio):In order to carry out the hypothesis test we need toconsider the estimated coefficient in relation to the sizeof its associated standard error. Value of t can be derivedfrom:

t−value=βi

Std.Error

In t-statistic (or t-ration) hypothesis ratio if the resultis t 2 (equal or higher than 2) than it is assumed that

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there is a relation but on the contrary there is no relationbetween dependent and independent variables.

Hypothesis: H0: a = 0 (or t < 2, then there is norelation)

H1: a 0 (or t 2, then there is relationbetween variables)

Alpha FTSE ExcessReturn

Retail SalesRate

Bank Rate

H0: true = 0H1: 0

H0: true β1 = 0H1: β1 0

H0: true β3 = 0H1: β3 0

H0: true β4 =0H1: β4 0

-.978 < 2 21.538 > 2 1.936 < 2 -.720 < 2H1

rejected& H0

accepted

H0 rejected & H1

acceptedH1 rejected &H0 accepted

H1 rejected &H0 accepted

As interceptor (alpha) is not significant from 0, so it is not meaningful

As FTSE Excess Return is significant from0, so market index has relation with ABC

As Retail Sales is insignificant from 0, so Bank rate has no relation with ABC

As Bank Rate is significant from 0, so Bank rate hasno relation with ABC

P-value:For 95% interval if P > .05 then reject null hypothesis (H0)and P < .05 then accept the alternative hypothesis (H1). So,hypothesis are:

H0: a = 0 (P > .05; no relation and reject)H1: a 0 (P < .05; there is relation and accept)

Alpha FTSE Excess Return Retail SalesRate

Bank Rate

H0: true H0: true β1 = 0 H0: true β3 = H0: true β4 =

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= 0H1: 0

H1: β1 0 0H1: β3 0

0H1: β4 0

.331 > .05 .000 < .05 .051 > .05 .474 > .05H1 rejected& H0

rejected

H0 rejected & H1

acceptedH1 rejected& H0

accepted

H1 rejected& H0

acceptedIntercept or alpha value is not meaningful

FTSE Excess Returnhas relation with ABC share excess return

Retail Sales(%) has no relation with ABC

Bank Rate (%) has no relation with ABC

Comparison: CAPM, APT and Redesigned APT

Particulars

CAPM APT Redesigned APT

Alpha -1.009 -4.250 -2.885Beta β1 (FTSE) =

1.406β1 (FTSE) =

1.363β2 (Inflation)

= -.084β3 (R. Sales) =

1.015

β1 (FTSE) =1.373

β2 (R. Sales) =1.117

β3 (Bank R.) = -0.478

R, R Square & Adjusted R Square

R = .944R Square

= .891Adj. R2 = .890

.947

.897

.892

.947

.897

.893

t-value = -1.019β1 (FTSE) =

23.97

= -1.836β1 (FTSE) =

21.771β2 (Inflation)

= -.145β3 (R. Sales) =

= -.978β1 (FTSE) =

21.538β2 (R. Sales) =

1.986 β3 (Bank R.) =

Page | 21


1.806 -.720p-value = .321

β1 (FTSE)= .000

= .071β1 (FTSE)

= .000β2 (Inflation)

= .885β3 (R. Sales) =

.075

= .331β1 (FTSE) = .000

β2 (R. Sales)= .051

β3 (Bank R.)= .474

F-test .000 .000 .000DW d-test .987 1.050 1.140Tolerance FTSE = 1.000 FTSE = .861

Inflation= .908

Retail Sales =.787

FTSE = .824Retail Sales

= .779Bank Rate = .805

Comparison in Details:When to compare the results we get from three asset pricingmodel (CAPM, APT, and APT redesigned) under to distinctivebasic model (CAPM & APT). Actually there is no suchdifference in predicting the rate of return by using CAPMand APT. Because APT is the extended version of CAPM. Aswhen we count Market Index beta it’s almost cover the macroand micro economics factor but not in specific figure.That’s different hypothesis test and fitness test are samewith near result for specific variables and determinants.Comparison are done sequentially –

Alpha (): it is the constant or intercept or fixedcost which is insignificant, inclusive, and avoidablein any situation whether it is CAPM, APT or RedesignedAPT by t-value and p-value.

Variables Beta (βi): Beta shows how close is CAPM andAPT. As FTSE Market Index Beta and Retail Sales isalways same or nearest in value always where FTSE isthe main determinant for predicting rate of return.

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R, R2, and Adjusted R2: Another measurement which showsCAPM and APT is just similar by showing least change invalue. Actually the overall degree of relationship (R),explanatory power (R2) improved by adding/excludingvariables in APT and Redesigned APT model.

F-test: .000 is the result of F-test in all cases whichmeans overall fitness.

Tolerance: Tolerance remain almost same for FTSE andRetail Sales when Bank rate has also good tolerancevalue variables.

Actually FTSE and Retails Sales are more insignificant in atleast two cases when others not which clear that CAPM andAPT has no such difference but APT gives better and specificprediction for return rate.

Limitation of APT:In the major limitation of APT lies in the major APT’sAssumptions which are Capital markets are perfectlycompetitive; Investors always prefer more to less wealth andPrice-generating process is a K factor model. So, thescenario is that:

If the assumptions did not work in a market, then APTfails to estimates the accurate expected return.Moreover in the world there are few countries who havesufficient efficiency in the market where informationcan be gathered easily.

Also the APT demands that investor perceive the risksources, and that they can reasonably estimate factorsensitive. In fact, even professional and academiciancan’t agree on the identity of the risk factors and themore betas investor have to estimate the morestatistical noise the investor must live with.

Strictness in linear relationship mostly createslimitation for APT model in some parts of the worldlike ASE (Athens Stock Exchange).

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Conclusion:The findings shows that there are slight differences betweenCAPM, APT and Extension APT. As the assumptions of arbitragepricing model holds two basic assumptions of APT – efficientmarkets information and diversified investment. Whenportfolios investment are diversified this means Market RiskExposure includes most of micro-macroeconomics factors’effects. But using APT and Extension of APT have betterexplanatory power (R2 and Adjusted R2) then CAPM which makesAPT model more attractive to the scholars to suggest use ofAPT in estimation of expected return. But scholars find thataccepting APT is better than CAPM but it is not error-freeto estimate expected return in all condition. As assumptionsmakes it difficult to predict actually in the markets.However, FF3 Model (Fama and French, 1992) has developedfurther supportive model but in the end like all of thescholars, it is expected that missing information (Aggelidisand Mandotinos, 2007) will further improve throughdevelopment and modification of arbitrage pricing model forthe demand of the time.

References:

Adair, T. (2004). Excel Applications for corporate finance, London:McGraw-Hill/Irwin.

Aggelidis, T. N. V. and Maditinos. D. (2007). Testing therelation between risk and returns using CAPM and APT: TheCase of Athens Stock Exchange (ASE).

Benninga, S. (2006). Principles of finance with Excel. New York:Oxford University Press.

Blume, M. and Friend, I. (1973). A new look at the capitalasset pricing model. Journal of Finance.

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Bower, D. H., Bower, R. S. and Logue, D. E. (1984).Arbitrage pricing and utility stock returns. Of Finance, 39,Pp. 1041-1054.

Brealey, R. A. Myers, S. C. and Allen, F. (2006). Corporatefinance. 8th ed., Boston: McGraw-Hill/Irwin.

Burmeister, E. & McElroy M. B. (1988). Joint Estimation ofFactor Sensitivities and Risk Premia for the ArbitragePricing Theory. The Journal of Finance, Vol. XLIII, No. 3.

Chan, Kam-Fai & Beenstock, M. (1986). Testing the ArbitragePricing Theory in the United Kingdom.

Chen, N., Roll, R. and Ross, S. (1986). Economic forces andthe stock market. Journal of Business, 59, Pp. 383-403.

Cho, D. C., Elton, E. J. and Gruber, M. J. (1984). On theRobustness of the Roll and Ross Arbitrage Pricing Theory.Journal of Financial and Quantitative Analysis, Vol. 19, No. 1.

Connor, G. and Korajczyk, R. (1988). Risk and Return in anequilibrium APT: application of a new test methodology.Journal of Financial Economics, 21.

Damodaran, A. (2001). Corporate finance: theory and practice. 2 ed.,Chichester: Wiley.

Dhankar, R. S. (1988). A new look at the criteria ofperformance measurement for business enterprises in India: astudy of public sector undertakings. Finance India.

Dhankar, R. S. (1996). An empirical testing of capital assetpricing model in the Indian context. Journal of FinancialManagement and Analysis.

Dhankar, R. S. and Esq, R.S. (2005). Arbitrage pricingtheory and the capital asset pricing model from the Indianstock market. Journal of Financial Management and Analysis, 18(1),pp.14-27.

Huberman, G. and Wang, Z. (2005). Arbitrage Pricing Theory.

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Gonzalez, M. F. (2001). CAPM performance in the CaracasStock Exchange from 1992 to 1998. International Review of FinancialAnalysis, No. 2, Pp. 333-341.

Lawrence, E. R., Geppert, J. and Prakash, A. J. (2007).Asset Pricing Models: a comparison. Applied Financial Economics,Vol. 17, Pp. 933-940.

Morris, R. C. and Pope, P. F. (1980). The Jensen Measure ofPortfolio Performance in an Arbitrage Pricing TheoryContext.

Ross, S. A. (1976). The Arbitrage Theory of Capital AssetPricing. Journal of Economic Theory, 13, Pp. 341-360.

Solnik, B. (1983). International Arbitrage Pricing Theory.The Journal of Finance, Vol. XXXVIII, No. 2.

Sun, C. and Zhang, D. (2001). Assessing the FinancialPerformance of Forestry-related investment Vehicles: CapitalAsset Pricing Model vs. Arbitrage Pricing Theory.

http://www.investopedia.com/terms/a/aar.asp

http://www.investopedia.com/articles/08/performance-measure.asp

http://en.wikipedia.org/wiki/Arbitrage_pricing_theory

http://en.wikipedia.org/wiki/Correlation_coefficient

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http://en.wikipedia.org/wiki/Correlation_coefficient



http://www.investopedia.com/terms/a/aar.asp


AppendixTable Annual Return and Excess Return

Date ABC AnnualReturn

FTSE AnnualReturn

ABCExcessReturn

FTSEExcessReturn

1-Jan-07 30.4901608 26.98864296 26.66 23.161-Feb-07 42.51641 27.53314232 38.62 23.631-Mar-07 50 26.57398659 46.02 22.591-Apr-07 25.2287009 18.28390167 21.13 14.181-May-07 19.6624668 11.83342391 15.47 7.641-Jun-07 35.237484 13.05814555 30.90 8.721-Jul-07 8.52834891 7.156054785 3.95 2.581-Aug-07 5.88235294 7.238199483 1.24 2.601-Sep-07 17.2380559 12.03075375 12.52 7.311-Oct-07 12.7187616 8.106353247 8.03 3.421-Nov-07 25.1610932 9.246198852 20.48 4.571-Dec-07 28.0985232 9.213184862 23.44 4.551-Jan-08 23.9028429 11.61903891 19.22 6.941-Feb-08 14.89352 11.23512867 10.23 6.581-Mar-08 23.1477422 11.86907422 18.46 7.181-Apr-08 9.4569048 7.13838755 4.69 2.371-May-08 18.452956 12.78675272 13.75 8.09

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1-Jun-08 11.9330124 14.8743421 7.27 10.211-Jul-08 19.5834386 20.642545 14.96 16.021-Aug-08 19.9627046 20.0985462 15.50 15.641-Sep-08 18.2454492 20.87098918 13.84 16.461-Oct-08 27.2817367 15.96145644 22.88 11.561-Nov-08 23.2030713 16.87865905 18.80 12.481-Dec-08 16.7502816 18.09685782 12.33 13.681-Jan-09 18.0036048 19.96296934 13.57 15.531-Feb-09 26.6302838 18.45992322 22.24 14.071-Mar-09 30.663344 24.01524984 26.28 19.641-Apr-09 41.1330732 28.25180952 36.73 23.851-May-09 25.0076454 17.45625868 20.59 13.041-Jun-09 27.7222525 15.91339638 23.22 11.411-Jul-09 13.9356697 13.59410152 9.40 9.051-Aug-09 24.9560917 13.09787493 20.43 8.571-Sep-09 30.4471327 11.09516751 25.70 6.351-Oct-09 20.4997961 17.86781264 15.66 13.031-Nov-09 18.9646788 13.81952172 14.02 8.881-Dec-09 20.9190564 13.15059255 15.91 8.141-Jan-10 18.1726132 9.673013358 13.09 4.591-Feb-10 7.51774828 8.191819006 2.22 2.891-Mar-10 -2.5474525 7.718277143 -7.89 2.381-Apr-10 2.72850884 9.151470598 -2.60 3.821-May-10 11.9481367 17.89087543 6.52 12.461-Jun-10 1.10027349 14.71097662 -4.45 9.161-Jul-10 0.17697027 9.48114024 -5.49 3.811-Aug-10 -1.5412702 8.411277103 -7.31 2.641-Sep-10 -1.8684046 8.734805471 -7.66 2.941-Oct-10 1.77230769 9.98735858 -3.92 4.301-Nov-10 -9.5869671 5.161145568 -15.20 -0.451-Dec-10 -14.525253 2.025504281 -20.03 -3.471-Jan-11 -20.588748 -

6.592482814-25.89 -11.89

1-Feb-11 -21.955806 -5.792488463

-27.08 -10.91

1-Mar-11 -13.137008 -10.84792018

-18.16 -15.87

1-Apr-11 -16.153891 -7.61890571 -21.03 -12.501-May-11 -16.809196 -

10.36554512-21.64 -15.20

1-Jun-11 -29.051022 - -34.00 -21.06

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16.111264521-Jul-11 -23.127008 -

16.41502894-28.24 -21.53

1-Aug-11 -23.540029 -12.01632888

-28.62 -17.10

1-Sep-11 -32.542983 -25.12051952

-37.49 -30.07

1-Oct-11 -49.617547 -36.78013503

-54.36 -41.52

1-Nov-11 -39.297676 -34.95658164

-42.98 -38.64

1-Dec-11 -40.060752 -32.7802913 -42.05 -34.771-Jan-12 -48.323915 -30.7049765 -49.61 -31.991-Feb-12 -51.115926 -

35.95296414-52.01 -36.84

1-Mar-12 -62.158248 -32.2126373 -62.88 -32.931-Apr-12 -45.84298 -

29.89993355-46.44 -30.50

1-May-12 -42.672014 -26.91596426

-43.30 -27.55

1-Jun-12 -26.34328 -23.93852274

-26.87 -24.47

1-Jul-12 -24.873607 -14.39468065

-25.37 -14.89

1-Aug-12 -14.687641 -12.13201845

-15.13 -12.57

1-Sep-12 3.07784321 6.084544243 2.69 5.691-Oct-12 2.99138888 7.774913106 2.49 7.271-Nov-12 9.66647356 18.11957882 9.23 17.681-Dec-12 26.8650378 19.25958113 26.48 18.87

Page | 29


SPSS Output: CAPM Model

Descriptive StatisticsMean Std.

DeviationN

ABC Excess Return -.7061 25.30773 72

FTSE Excess Return .2153 16.98976 72

CorrelationsABC Excess

ReturnFTSE Excess

ReturnPearson Correlation

ABC Excess Return 1.000 .944FTSE Excess Return

.944 1.000

Sig. (1-tailed) ABC Excess Return . .000FTSE Excess Return

.000 .

N ABC Excess Return 72 72FTSE Excess Return

72 72

Model Summaryb

Model

R RSquare

Adjusted RSquare

Std.Errorof theEstimate

Change Statistics Durbin-

Watson

RSquar

eChang

e

FChange

df1

df2

Sig.F

Change

1 .944a

.891 .890 8.40035

.891 574.421

1 70 .000 .987

a. Predictors: (Constant), FTSE Excess Returnb. Dependent Variable: ABC Excess Return

ANOVAa

Model Sum ofSquares

df MeanSquare

F Sig.

Page | 30


1 Regression

40534.535 1 40534.535 574.421

.000b

Residual 4939.615 70 70.566Total 45474.149 71

a. Dependent Variable: ABC Excess Returnb. Predictors: (Constant), FTSE Excess Return

Coefficientsa

Model UnstandardizedCoefficients

Standardized

Coefficients

t Sig.

CollinearityStatistics

B Std.Error

Beta Tolerance

VIF

1 (Constant) -1.009

.990 -1.019

.312

FTSE Excess Return

1.406 .059 .944 23.967

.000

1.000 1.000

a. Dependent Variable: ABC Excess Return

Coefficient Correlationsa

Model FTSE ExcessReturn

1 Correlations

FTSE Excess Return

1.000

Covariances

FTSE Excess Return

.003


SPSS Output: APT Model


DeviationN

ABC Excess Return -.7061 25.30773 72FTSE Excess Return

.2153 16.98976 72

Inflation Rate 2.7624 1.80250 72Retail Sales 3.4290 1.98014 72

CorrelationsABC

ExcessReturn

FTSEExcessReturn

Inflation Rate

RetailSales

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Pearson Correlation

ABC Excess Return 1.000 .944 .075 .416FTSE Excess Return .944 1.000 .063 .370Inflation Rate .075 .063 1.000 .299Retail Sales .416 .370 .299 1.000

Sig. (1-tailed)

ABC Excess Return . .000 .265 .000FTSE Excess Return .000 . .300 .001Inflation Rate .265 .300 . .005Retail Sales .000 .001 .005 .

N

ABC Excess Return 72 72 72 72FTSE Excess Return 72 72 72 72Inflation Rate 72 72 72 72Retail Sales 72 72 72 72

Model Summaryb

Model R RSquare

Adjusted R

Square

Std.Error of

theEstimate

Change Statistics Durbin-

WatsonR SquareChange

FChange

df1 df2 Sig. FChange

1 .947a .897 .892 8.31571 .897 196.535

3 68 .000 1.050

a. Predictors: (Constant), Retail Sales, Inflation Rate, FTSE Excess Returnb. Dependent Variable: ABC Excess Return

ANOVAa

Model Sum ofSquares

df Mean Square F Sig.

1

Regression 40771.877 3 13590.626 196.535 .000b

Residual 4702.273 68 69.151

Total 45474.149 71

a. Dependent Variable: ABC Excess Returnb. Predictors: (Constant), Retail Sales, Inflation Rate, FTSE Excess Return

Coefficientsa

Model Unstandardized

Coefficients

Standard.Coefficie

nts

t Sig.

Correlations CollinearityStatistics

B Std.Error

Beta Zero-order

Partial

Part Tolerance

VIF

1

(Constant) -4.250 2.314

-1.836

.071

FTSE Excess Return

1.363 .063 .915 21.771

.000

.944 .935 .849 .861 1.162

Inflation Rate -.084 .575 -.006 -.145 .885

.075 -.018 -.006

.908 1.102

Retail Sales 1.015 .562 .079 1.806 .075

.416 .214 .070 .787 1.271


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Model RetailSales

InflationRate

FTSE ExcessReturn

1

Correlations

Retail Sales 1.000 -.298 -.369

Inflation Rate -.298 1.000 .054

FTSE Excess Return

-.369 .054 1.000

Covariances

Retail Sales .316 -.096 -.013

Inflation Rate -.096 .330 .002

FTSE Excess Return

-.013 .002 .004


SPSS Output: APT Model (Redesigned)


DeviationN

ABC Excess Return -.7061 25.30773 72FTSE Excess Return

.2153 16.98976 72

Retail Sales 3.4290 1.98014 72Bank Rate 4.0757 1.65189 72

CorrelationsABC ExcessReturn

FTSE ExcessReturn

RetailSales

Bank Rate

Pearson Correlation

ABC Excess Return 1.000 .944 .416 .307FTSE Excess Return

.944 1.000 .370 .329

Retail Sales .416 .370 1.000 .396Bank Rate .307 .329 .396 1.000

Sig. (1-tailed) ABC Excess Return . .000 .000 .004FTSE Excess Return

.000 . .001 .002

Retail Sales .000 .001 . .000

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Bank Rate .004 .002 .000 .

N

ABC Excess Return 72 72 72 72FTSE Excess Return

72 72 72 72

Retail Sales 72 72 72 72Bank Rate 72 72 72 72

Model Summaryb

Model R R Square AdjustedR Square

Std. Errorof the

Estimate

Change Statistics Durbin-WatsonR Square

ChangeF

Changedf1

df2

Sig. FChange

1 .947a .897 .893 8.28545 .897 198.139

3 68 .000 1.066

a. Predictors: (Constant), Bank Rate, FTSE Excess Return, Retail Salesb. Dependent Variable: ABC Excess Return

ANOVAa

Model Sum ofSquares

df Mean Square F Sig.

1

Regression 40806.034 3 13602.011 198.139 .000b

Residual 4668.115 68 68.649

Total 45474.149 71

a. Dependent Variable: ABC Excess Returnb. Predictors: (Constant), Bank Rate, FTSE Excess Return, Retail Sales

Coefficientsa

Model UnstandardizedCoefficients

Standardized

Coefficients

t Sig.

CollinearityStatistics

B Std.Error

Beta Tolerance

VIF

1

(Constant) -2.885 2.949 -.978 .331

FTSE Excess Return

1.373 .064 .922 21.538

.000

.824 1.214

Retail Sales 1.117 .563 .087 1.986 .051

.779 1.284

Bank Rate -.478 .664 -.031 -.720 .474

.805 1.243



Model Bank Rate FTSE ExcessReturn

RetailSales

Page | 34


1

Correlations

Bank Rate 1.000 -.214 -.312FTSE Excess Return

-.214 1.000 -.276

Retail Sales -.312 -.276 1.000

Covariances

Bank Rate .440 -.009 -.117FTSE Excess Return

-.009 .004 -.010

Retail Sales -.117 -.010 .317a. Dependent Variable: ABC Excess Return

Page | 35

CAPM and APT Model Development, Evaluation and Interpretation on ABC’s Share Price

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