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Arbitrage Pricing Theory and Uk Stock Exchange Finance Essay

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Contents

To estimate empirically the Arbitrage Pricing Theory (APT) model we focus our attention to the UK's stock exchange market. Our study employs monthly time series data spanning the period 2000:9 to 2010:9 (121 observations). The sample of our analysis is dictated solely by the demands of the coursework. The variables involved are: the closing share prices for 25 UK companies listed on the London Stock Exchange, the FTSE 100 stock index, the UK Libor as proxy for the short-run risk-free rate, the 20-year government bond yield as proxy for the long-run risk-free rate, the exchange rate series between the British Pound and the US Dollar and finally the Brent crude oil prices. [1] The abbreviated notation of the above variables is as follows: Sharei = Si with i= 1 to 25 FTSE 100 stock index = indext UK Libor= free_s_ratet 20-year government bond yield = free_l_ratet Exchange rate series = fxt Brent crude oil prices = brentt Given the availability of the Si series it is trivial to calculate the return series for each of the 25 selected shares (r_ Si) by taking the first logarithmic differences of the share prices (growth rate). To do so in E-views the relevant command is the one described below: For !i=1 to 25 series r_S!i = dlog(S!i) Next To construct the equally weighed portfolio return series (portfoliot) we merely estimate the average return of the 25 share returns for each time period of the sample. The commands applied are described as follows: series sum_stock_returns = r_S1+ r_S2+ r_S3+ r_s4+ r_S5+ r_S6+ r_S7+ r_S8+ r_S9+r_S10+r_S11 +r_S12+ r_S13+ r_S14+ r_S15+ r_S16+ r_S17+ r_S18+ r_S19+ r_S20+ r_S21+ r_s22+ r_S23+ r_S24+ r_S25 series portofolio = sum_stock_returns / 25 Figure 1 below presents the five main variables used in this study as well as the equally weighed portfolio return series constructed by the returns of the 25 involved shares. Figure 1. The variables of the study Figure 1. The variables of the study (continued)

Question 1

The E-views command to construct the portfolio excess returns is presented beneath: series exs_r_portofolio = portofolio - (free_s_rate/100) Figure 4. Portfolio excess returns and the associated descriptive statistics Similarly, figure 5 presents the market excess returns over the short-run risk free proxy (exs_r_index) along with a set of descriptive statistics (see footnote 3). The E-views command to construct initially the market returns and afterwards the market excess returns are presented below: series r_index = dlog(index) series exs_r_index = r_index - (free_s_rate/100) Figure 5. Market excess returns and the associated descriptive statistics Before we comment on the distributional properties of these two variables, it is worth to mention that both variables appear to be stationary with constant mean and constant variance. [4] Again the applied ADF tests revealed that indeed both variables are clearly stationary. [5] What we know for the case of the symmetric distributions is that the mean and the median statistics are equal between them.

Differences between these two measures occur with skewed distributions. In our case, the portfolio excess returns series appear to have almost identical values for the mean and the median statistics, that is -0.036483 and -0.036315 respectively, which is an indication for a symmetric behaviour. The maximum value is 0.14, the minimum value is -0.24 and finally the standard deviation receive the value of 0.057. Skewness measures the distribution's asymmetry around the mean. A symmetric distribution like the normal has skewness equal to 0, while positive skewness means that the distribution has a long right tail and negative skewness implies that the distribution has a long left tail. The value of -0.300 for the skewness implies that the distribution of the portfolio excess returns presents a slightly long left tail, which is a result of the negative returns during the crisis. Kurtosis measures the fatness of the distribution of the series. The kurtosis of the normal distribution is 3, while in cases where the kurtosis exceeds the said value, the distribution is leptokurtic relative to the normal; and if the kurtosis is less than 3, the distribution is platykurtic relative to the normal. The value of 4.67 for the Kurtosis implies that the distribution of the portfolio excess returns series is pretty leptokurtic. This is common for stock returns. Finally, we tested for normality by making use of the Jarque-Bera statistic. The null hypothesis in the test is that the distribution is normal. The estimated Jarque-Bera statistic along with the associated p-value, for the portfolio excess returns series, are 15.76 and 0.000, respectively. Consequently, judging by the reported p-value we clearly reject the null hypothesis of normality.

Therefore, the portfolio excess returns series is not distributed normally. Provided that the Anderson-Darling normality test presents better small sample properties than the Jarque-Bera test, we implemented it also. The Anderson-Darling normality test has the same null hypothesis as the Jarque-Bera test. The Anderson-Darling test statistic receives the value of 24.13 with p-value 0.000, and as a result we reject the null hypothesis of normality. Overall, both tests affirm that the portfolio excess returns series is not distributed as a normal variable. The non-normality is a characteristic of small samples and in our case we only had 121 observations. If the sample size increases the distribution will be closer to the normal distribution. Furthermore, it is a well known fact that asset returns are not normally distributed. Turning now to the market excess returns series, we may say that the mean and the median of the series are quite similar, that is -0.042 and -0.039 respectively, indicating symmetric behaviour. The maximum value is 0.07, the minimum value of the series is -0.20 and finally the standard deviation receive the value of 0.05. The skewness is -0.31 and kurtosis is 3.66, indicating quite normal behaviour given that these two values are quite close to the benchmark values of the normal distribution (0 and 3, respectively). The p-value of the estimated Jarque-Bera statistic is 0.12, indicating that we fail to reject the null hypothesis of normality at the conventional 0.05 level of significance. Finally, the p-value of the estimated Anderson-Darling statistic is 0.06, indicating again that we fail to reject the null hypothesis of normality at the conventional 0.05 level of significance. In general the market excess returns appear to distribute like a normal variable. What has been revealed from the above analysis is that a) both series are stationary, b) the market excess return series is less volatile than the portfolio excess returns series and c) the market excess return series is distributed normally, while this not true for the portfolio market excess return series.

Question 2

In this section we estimate the APT model having as a dependent variable the excess portfolio returns and as independent variables 1) the excess market returns 2) the percentage change of the term spread 3) the percentage change of the exchange rate series and finally 4) the percentage change of the Brent crude oil prices. The specification of the above described APT model is provided by equation (1): (1) where, is the excess portfolio returns series at time t , c is the constant term, is the excess market returns series at time t, is the percentage change of the term spread at time t, is the percentage change of the exchange rate series at time t, is the percentage change of the Brent crude oil prices at time t, are parameters to be estimated and finally, is the error term assuming the usual properties. Parameter estimates for equation (1), by means of the OLS estimation technique, along with their associated standard errors, t-statistics and p-values, are analytically illustrated in Table 1. The E-views command for the estimation of the above mentioned model is as follows: equation model1.ls exs_r_portofolio c exs_r_index g_term_spread g_fx g_brent Table 1. Estimation output for equation 1 Variable Coefficient Std.

error t-Statistic p-value constant 0.008164 0.002893 2.822046 0.0056 [ Rm-Rf ]t 1.047768 0.043587 24.03840 0.0000 GTSt 0.001993 0.003167 0.629342 0.5304 GFXt 0.039145 0.086038 0.454976 0.6500 GBPt 0.005340 0.024816 0.215195 0.8300 Regression Diagnostic Statistics R-squared 0.844807 Mean dependent var -0.036483 Adjusted R-squared 0.839409 S.D. dependent var 0.057761 S.E. of regression 0.023147 Akaike info Criterion -4.653130 Log likelihood 284.1878 Schwarz criterion -4.536984 F-statistic 156.5033 Hannan-Quinn criter. -4.605962 Prob(F-statistic) 0.000000 Durbin-Watson stat. 1.927105 White hetero. Test 0.926993 LM test ser. cor. (2 lags) 0.203657 White's test p-value 0.532800 LM test ser. cor. p-value 0.816000 Anderson-Darling nor. test 0.858200 LM test ser. cor. (8 lags) 0.521658 Anderson-Darling p-value 0.440800 LM test ser. cor. p-value 0.837900 In general, the estimated sign for every single parameter is theoretically meaningful, only two of the parameters appear to be statistically significant at the conventional level of 0.05 and finally, the model seems to fit the data pretty well. In more detail and in relation to the expected signs we stress the following: the excess return of a well diversified portfolio is expected to experience similar co-movements with the excess returns of the market. Therefore, the expected sign is positive as it happens in our case. As it is well known, the term spread variable is widely used by economists and the practitioners in order to predict the real economic activity. Given that the stock market generally follows real economic activity, then it comes that the expected sign for the term-spread variable would be positive.

The estimated sign in our model for the term spread variable is also positive. For the exchange rate variable we know that a company may be affected by the changes in the exchange rates directly if its orientation has to do with the foreign trade or indirectly if its inputs or outputs are affected by the exchange rate. In the literature there is no consensus with respect to the expected sign. Some studies have shown that devaluation of the currency has a strong positive effect in the long-run for the stock prices and a negative effect in the short-run. In general, we have no reasons to expect a particular sign for the exchange rate variable. Our results suggest the effect of the exchange rate changes is positive. Finally, we know how crucial the price of oil is for the operation of all firms but again we do not expect an a priori sign for the returns of a portfolio. The sign might be positive if most of the firms in the portfolio experience profits by such an increase and negative if the opposite is true. Furthermore, oil prices may be seen as the expectation for the future inflation. The estimated sign in our model is again positive, which of course does not contradict the theoretical underpinnings of the APT model.

Overall, all the estimated signs of the coefficients, which are presented in the second column of Table 1, are theoretically meaningful and this fact indicates that our model is well specified. Among the independent variables used only the constant and the excess market returns appear to be statistically significant even at the 0.01 significance level, while all the remaining variables are statistically insignificant. The significance for a coefficient can be affirmed by the corresponding t-statistic or alternatively by the associated p-value. The t-statistic is calculated by the ratio of the estimated coefficient (column two) to the associated standard error (column three). If the absolute value of the t-statistic is greater than 2, then we may say that the coefficient is significant at the 0.05 significance level (but this is a rule of thumb). More accurate information with respect to the significance can be derived from the p-value. Hence, if the p-value is lower than the selected level of significance (e.g. 0.01), then the coefficient is considered significant at that particular level of significance. Provided that we only have the constant and one variable that are statistically significant, we continue by interpreting only the two respective coefficients.

The constant can be interpreted as follows: if all the independent variables are simultaneously equal to zero then portfolio's excess return is equal to 0.008. For the second coefficient we can say that if the excess market returns increase by 1 unit, then the portfolio excess return will be increased by 1.047 units, provided that all the other variables remain constant. Additionally, as it can be inferred by the value of the adjusted R-square (corrected with the degrees of freedom), which is 0.839, included into the model independent variables explain more that the 4/5 of the portfolio's excess returns variability. At last, the value of the F-statistic for testing the joint significance of all the independent variables included in the model is pretty high (156.50) with the associated p-value to be in practice equal to zero. Therefore, we reject the null hypothesis () that all the coefficients are jointly insignificant at 0.05 significance level and we can support that there is at least one coefficient which is significantly different from zero. The F-statistic provides further evidence for the validity of the estimated APT model.

Question 3

In this section we conduct diagnostic testing in order to assess our models statistical strength. For this reason we investigate by testing analogously if there is presence of multicollinearity, heteroskedasticity, serial correlation and finally non-normality in the residuals. Before the diagnostic testing, it is important to stress that all the regressors used in equation 1 are stationary and therefore we exclude the possibility of estimating a spurious regression. It is well known that the estimated results in cases where the regression is characterized as spurious, are meaningless and the statistical inference is worthless. Clearly, this is not the case for our estimated model presented in Table 1. Turning now to the diagnostic testing procedure, our first concern is to ensure that there is no presence of multicollinearity. The problem with multicollinearity is that it inflates the standard errors and therefore it is hard to assess the significance of the regressors used in the model. Furthermore, we know that multicollinearity does not affect the efficiency of the estimated parameters. Provided that there is no availability of an official testing procedure for the detection of multicollinearity we make use of a practical solution. According to this approach evidence for multicollinearity would be a high correlation among the regressors. High value for the correlation coefficient is considered a value of above 0.8. For this reason we estimate the correlation coefficients for all the regressors involved in the estimation of equation 1. The correlation coefficients are illustrated in Table 2. Undoubtedly, the results in Table 2 reveal that all the correlation coefficients are well below the threshold value of 0.8 and as a consequence we may say that there is no evidence of multicollinearity for that particular set of regressors.

Question 4

As is clearly shown in question 3, the diagnostic testing performed for the statistical validity of the estimated model revealed the following a) the regressors are stationary, b) multicollinearity is not considered a threat, c) there is no serial correlation in the residuals, d) the residuals are homoskedastic and finally, e) the residuals are distributed normally. Therefore, we came to the conclusion that all the basic assumptions of the classical linear regression model hold and no further actions are required.

Question 5

In this part of the coursework we augment equation (1) with the squares of the factor changes. The new specification is given by equation (2): (2) Parameter estimates for equation (2) along with their associated standard errors, t-statistics and p-values, are analytically illustrated in Table 3. The E-views command for the estimation of the above mentioned model is as follows: equation model2.ls exs_r_portofolio c exs_r_index g_term_spread g_fx g_brent (g_term_spread)^2 (g_fx)^2 (g_brent)^2 Examining the results in Table 3, we can say that from the three additionally included variables only one proves to be statistically significant (the GBPt2) at the 0.1 significance level (not 0.05 or 0.01). The significance and the magnitude for the non-squared regressors do not alter in any important way with respect to the corresponding results presented in Table 1. Additionally, the adjusted R-squared improved marginally from 0.839 to 0.848, implying that the additional regressors have contributed less than 1% in explaining the variability of the dependent variable. The diagnostic testing for equation (2), which is presented at the lower part of Table 3, reveals that the new model is well specified. In more detail, we realize that all the main assumptions of the classical linear regression model are adequately satisfied. There is no serial correlation, the residuals are homoskedastic and finally the residuals are distributed normally. Table 3. Estimation output for the augmented specification (equation 2) Variable Coefficient Std. error t-Statistic p-value constant 0.011998 0.003107 3.861951 0.0002 [ Rm-Rf ]t 1.037345 0.042858 24.20433 0.0000 GTSt 0.002770 0.003517 0.787418 0.4327 GFXt 0.039821 0.083831 0.475017 0.6357 GBPt -0.004681 0.024362 -0.192148 0.8480 (GTSt)2 -0.001131 0.000921 -1.229082 0.2216 (GFXt)2 -0.831541 1.931252 -0.430571 0.6676 (GBPt)2 -0.340632 0.184292 -1.848331 0.0672 Regression Diagnostic Statistics R-squared 0.857205 Mean dependent var -0.036483 Adjusted R-squared 0.848280 S.D. dependent var 0.057761 S.E. of regression 0.022499 Akaike info Criterion -4.686387 Log likelihood 289.1832 Schwarz criterion -4.500554 F-statistic 96.04850 Hannan-Quinn criter. -4.610919 Prob(F-statistic) 0.000000 Durbin-Watson stat. 1.843856 White hetero. Test 0.840402 LM test ser. cor. (2 lags) 0.546468 White's test p-value 0.704900 LM test ser. cor. p-value 0.580600 Anderson-Darling nor. test 0.737227 LM test ser. cor. (8 lags) 0.615434 Anderson-Darling p-value 0.528500 LM test ser. cor. p-value 0.763100 In order to test whether the additionally included variables provide a better specification, we perform the Wald test for coefficient restrictions. The Wald test is applied to all the possible combinations that may arise among the three variables.

The results of the Wald testing procedure along with their associated p-values are illustrated in Table 4. Table 4. Wald testing results Null hypothesis F-Statistic (p-value) 3.24 (0.02) 0.81 (0.44) 4.65 (0.00) 4.03 (0.02) When we tested the null of it was realised that we reject the null at the 0.05 level, suggesting that the three additional variables contribute significantly in explaining the dependent variable. In the case where the following is tested: we fail to reject the null at the 0.05 level, signifying therefore that the square of the oil inflation ( coefficient) is quite crucial. Moreover, in testing the restriction the null is rejected at the 0.01 level. Finally, when the restriction is tested we reject the null at the 0.05 level. Overall, the Wald testing procedure suggests that the preferred specification would be the one that excludes the square of the exchange rate percentage change. Therefore, the new adopted specification after the Wald testing procedure receives the form presented in equation (3): (3) Equation (3) is estimated with OLS and the results are illustrated in Table 5. The E-views command for the estimation of the model is the following: equation model3.ls exs_r_portofolio c exs_r_index g_term_spread g_fx g_brent (g_term_spread) (g_brent)^2 Table 5. Estimation output for the specification of equation 3 Variable Coefficient Std. error t-Statistic p-value constant 0.011796 0.003060 3.854984 0.0002 [ Rm-Rf ]t 1.035900 0.042572 24.33292 0.0000 GTSt 0.002638 0.003491 0.755653 0.4514 GFXt 0.042275 0.083335 0.507297 0.6129 GBPt -0.004001 0.024223 -0.165184 0.8691 (GTSt)2 -0.001100 0.000914 -1.202750 0.2316 (GBPt)2 -0.392334 0.139300 -2.816473 0.0057 Regression Diagnostic Statistics R-squared 0.856968 Mean dependent var -0.036483 Adjusted R-squared 0.849374 S.D. dependent var 0.057761 S.E. of regression 0.022417 Akaike info Criterion -4.701399 Log likelihood 289.0840 Schwarz criterion -4.538796 F-statistic 112.8391 Hannan-Quinn criter. -4.635365 Prob(F-statistic) 0.000000 Durbin-Watson stat. 1.831355 White hetero. Test 1.066227 LM test ser. cor. (2 lags) 0.552305 White's test p-value 0.396100 LM test ser. cor. p-value 0.577200 Anderson-Darling nor. test 0.690392 LM test ser. cor. (8 lags) 0.557522 Anderson-Darling p-value 0.566900 LM test ser. cor. p-value 0.810300 The econometric inference for equation (3), which is presented at the lower part in Table 5, reveals that again the selected model is well specified. There is no serial correlation, the residuals are homoskedastic and finally the residuals are distributed normally.

The rationale for the inclusion of the initial variables squared lies in our intension to assess the presence of a non-linear impact that the independent variables may have on the dependent variable. The intuition in other words is that we actually generate a quadratic term. Consequently, if we have for example a positive coefficient for a variable and a negative coefficient for the square of the same variable, then it is implied that as the variable receives higher values the effect increases with a decreasing rate. Therefore, the interpretation of the coefficient for the (GTSt)2 variable is as follows: as the GTSt increases then the effect on the dependent variable decreases with the rate of 0.0011 (Table 5). The interpretation for the rest squared coefficients is quite similar.

Question 6

The Chow breakpoint test is implemented for equation (3). The Chow breakpoint test is used to assess the stability of the estimated coefficients over a pre-specified breakpoint. The test depends heavily on the correct selection of the breakpoint. After the selection of the breakpoint the test is carried out by separating the initial sample into two sub-samples, with the first sample to be from the beginning of the sample up to the breakpoint and the second sample from the breakpoint up to the end. The main intuition of the test is based on the similarity of the sum of squared residuals resulting from the whole sample with the respective sum of squared residuals resulting from the equations that are fitted to each sub-sample. If there is a significant difference then this is indicative of a structural change in the coefficients derived from the whole sample regression. At this point we need to be very careful of the selection of the breakpoint. Based on the results presented above, and especially in question 1, we have realized that all the variables illustrated graphically show systematically a spike (extreme value) which takes place during the last quarter of 2008. The period indicated by the data coincides with the beginning of the global economic crisis. The beginning of the crisis is chronologically oriented by the collapse of the investment bank Lehman rothers on September of 2008 (2008m09). Consequently, the choice of the 2008m09 as a break date for our application seems to be theoretically and empirically fully justified. The results of the Chow breakpoint test are presented in Table 6. The test is implemented for a break to all the estimated coefficients of the regression. As can be realized from the p-values of the three illustrated statistics we clearly reject in all cases the null hypothesis of no breaks at the 0.01 significance level. Table 6. Chow Breakpoint Test (equation 3) Null Hypothesis: No breaks at specified breakpoints Breakpoint: 2008:m9 Varying regressors: All equation variables Equation Sample: 2000:m10 2010:m09 F-statistic 3.226107 Prob. F(7,106) 0.0039 Log likelihood ratio 23.17603 Prob. Chi-Square(7) 0.0016 Wald Statistic 22.58275 Prob. Chi-Square(7) 0.0020 Clearly the confirmation of the structural change in the coefficients of the estimated regression reveals that our specification needs to be revised analogously in order to take into account the break. Such a re-specification may be the inclusion of a dummy variable for the period after the break date or otherwise cross products between the dummy and the independent variables in order to determine the magnitude of change for the initially estimated slopes.

Question 7

In this section we will compare the three alternative specifications which have been presented (equations 1, 2 and 3) and estimated (Tables 1, 3 and 5) in the previous sections. For this reason we will make use of four different Statistics which are considered appropriate for the task at hand. These Statistics are the Adjusted R-square, the Akaike information criterion, The Schwartz criterion and finally the Hannan-Quinn criterion. For the adjusted R-square, this receives values between 0 and 1, the higher the value the better for the corresponding model. High values imply that high percentage of the dependent's variable variability is explained by the regressors. For the three remaining Statistics, the lower the values they receive the better the model is. Table 7 below presents all these Statistics in order to select the final model.

Table 7. Model selection criteria

Statistic Model 1 Model 2 Model 3 Adjusted R-square 0.839409 0.848280 0.849374 Akaike -4.653130 -4.686387 -4.701399 Schwartz -4.536984 -4.500554 -4.538796 Hannan-Quinn criterion -4.605962 -4.610919 -4.635365 Based on the reported results in Table 7 it is immediately realized that model 2 is preferred in comparison to model 1 (higher adjusted R-square and lower values for the rest of the statistics) and model 3 is preferred in comparison to model 2 (also there is higher adjusted R-square and lower values for the rest of the statistics). Model's 3 fit to the data is considered more than satisfactory as almost 85% of the variability that the dependent variable has is explained by the selected regressors.

Question 8

Based on the model presented in Table 5 we will assess the results from a financial perspective. This task is mainly focused on the interpretation and the significance of the estimated coefficients. We have already proved that model 3 is well specified and as a result we may proceed to the analysis of the results. Regarding the expected theoretical sign of the regressors it can be stressed that the estimated signs do not deviate from those expected. The justification for the sign of each variable has been analytically presented in question 2 and the same rationale applies also to the finally selected specification. The most notable fact is that the market index, FTSE 100, excess returns was found to be significant at 99% confidence level. This implies that this factor is the single-most important factor in explaining our portfolios excess returns. Alternatively, the estimated coefficient of 1.035900 can be seen as a measure of risk for the portfolio constructed since it is inferred that if the market's excess returns increase by one unit then the portfolio's excess returns will increase also by the value of the coefficient. Immediately we realise that our constructed portfolio is riskier than the market. This is because our portfolio is only a subset of the market portfolio and the market portfolio is more diversified and contains less individual risk. The constant can be interpreted as follows: if all the independent variables are simultaneously equal to zero then portfolio's excess return is equal to 0.011796. The fact that the constant term is statistically different from zero suggests that our choice to use the APT model is correct provided that the CAPM model is a special case of the APT model. A non-significant constant term would favour the use of the CAPM model. Finally, the three included factors in the specification remain statistically insignificant implying that these factors do not contribute considerably in the explanation of our portfolios excess returns. There are probably other factors that may play a significant role in explaining our portfolios excess returns. Such factors, among others, may be the industrial production, money supply, inflation and market's capitalization. Their effect remains under further investigation.

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