The Capital Asset Pricing Model Test Using Regression - Assignment Example

CAPM Test Using Regression By Word Count 2740 CAPM: Concept analysis The CAPM model combines two concepts; asset pricing and portfolio selection (Perold 2004). The model is based on the asset-pricing theory (Fama & French 2004) and Markowitz’s portfolio selection concept (Marling & Emanuelson 2012). Markowitz assumed that all investors were risk averse, and they chose mean-variance efficient portfolios (Washington.edu n.d.). That is why; his approach is called mean variance efficient portfolio model. To understand the concept of mean-variance efficient portfolio, first we need to understand the concept of capital market line (CML). Figure 1 describes the CML equation on σ – E (r) plane. Figure 1. Capital Market Line (Fama & French 2004). On the σ – E ( r) plane, the curve is drawn (See Figure 1); it is called minimum variance frontier (Fama & French 2004). The tangent to the curve is called capital market line (CML); it is the mean-variance frontier with the riskless assets (Fama & French 2004).The CML is expressed through E (r i) = r f + [{E(rM) – rf } / σM] x σ. The intercept rf denotes risk-free return, and the slope [{E(rM – rf) / σM] denotes the market price of risk or market premium. The point, where CML touches the curve is called the market portfolio. This point on the σ – E (r) plane is denoted by M (The Capital Asset Pricing Model n.d.). The CML depicts that the risk is measured in terms of standard deviation on the portfolio (Fama & French 2004). The risk-return relationship of the efficient portfolio satisfies the equation ri – rf = ß x [E(rM-rf)]. It tells that excess return, ri-rf is a function of β and the dependency is linear. In finance, it is presented in the form E(ri) = rf + [E(rM) – rf] x ß. It is called CAPM equation. The parameter ß is expressed as Cov (ri, rM)/ σ2M ((Modigliani, Pogue, & Solnik 1973). It expresses the relation of covariance of the return on asset and market to the variance of market. It measures systematic risk in terms of volatility of an asset in the portfolio (Allen, Kumar Singh, & Powel 2009). When the value of ß is 1, asset moves with the market; when ß > 1, it indicates the asset is more volatile than market and opposite for ß < 1. If rewritten the CAPM equation in the form E(ri) - rf = [E(rM) – rf] x ß, then it implies that excess return on an asset is a linear function of ß. The assignment tests the above-described concept through time-series regression analysis using observational data. The result and interpretation of the test are given below. Bivariate Regression Analysis The dataset consists of 607 monthly observations acquired for a period from January 1963 to July 2013. It represents asset pricing factors; market premium, size premium, and asset premium, risk-free rates and returns on portfolios. These portfolios represent food industry, small size low B/M, small size medium B/M, small size high B/M, big size low B/M, big size medium B/M, and big size high B/M companies. The regression model is expressed as (rt –rft) = α + ß (rmkt, t – rft). The goal is to verify the validity of CAPM equation (Modigliani, Pogue, & Solnik 1973). The food industry and six other portfolios data are continuous; sample period of data is one month. The mean of the food industry data is 1.01, the median is 1.18, and the standard deviation is 4.36. Multivariate Regression Analysis The CAPM model states that the rate of returns are compensation for bearing risks (Goyal 2011). However, the question why different assets earn a different rate of returns remains as the challenge of modern finance (Bhatnagar, C.S., Ramlogan, R., n.d). Researchers suggest using more factors other than market premium. This assignment uses Fama-French three-factor model (Lozano 2006) in order to study if the excess return can be explained by factors other than market premium. The regression model becomes (rt – rft) = α + β1 (rmkt,t – rft) + β2 SMBt + β3 HML + et. It considers that market premium, size premium, and asset premium influence the excess return (Multifactor Explanation n.d). Diagnostic Test of Regression Models. The F-test and coefficient of determination provide overall diagnosis of regression models. The F-test verifies if parameters of independent variables are statistically significant. It defines the probability for which the regression output is merely a chance of random occurrence. The coefficient of determination, R2, and adjusted R2 define how good a regression line fits the dataset. It defines what percentage of data that fall on the regression line. A bivariate regression analysis uses R2, and multivariate uses adjusted R2. Adjusted R2 considers the number of independent variables used in the analysis. Table 1 illustrates F-test results of CAPM model of seven portfolios and Table 2 shows results of Fama-Frencnh model of the same portfolios. Table 1. The F-Test of CAPM Model Portfolio F-Statistics: CAPM F-Critical; df1= 1 , df2=605, Level = 0.05 Test (Two-tailed) Significance at 0.05 Result Food 715 3.94 Fstat>Fcrt 0.000 Reject Ho Small and Low B/M 1971 3.94 Fstat>Fcritical 0.000 Reject Ho Small and Med B/M 2071 3.94 Fstat>Fcritical 0.000 Reject Ho Small and High B/M 1501 3.94 Fstat>Fcritical 0.000 Reject Ho Big and Low B/M 9542 3.94 Fstat>Fcritical 0.000 Reject Ho Big and Med B/M 3981 3.94 Fstat>Fcritical 0.000 Reject Ho Big and High B/M 1957 3.94 Fstat>Fcritical 0.000 Reject Ho The parameter of independent variable (mkt – rf) is statistically significant; at the significance level 0.05, F = 0.000 states that there is only a 0% chance that the regression output is merely a chance of random occurrence Table 2. The F-Test of Fama-French Model Portfolio F-Statistics; Fama-French F-Critical Test Significance at 0.05 Result Food industry 262 2.70 F-stat. > F-crit. 0.000 Reject Ho Small and Low B/M 1062 2.70 F-stat. > F-crit. 0.000 Reject Ho Small and Med B/M 9390 2.70 F-stat. > F-crit. 0.000 Reject Ho Small and High B/M 19370 2.70 F-stat. > F-crit. 0.000 Reject Ho Big and Low B/M 7985 2.70 F-stat. > F-crit. 0.000 Reject Ho Big and Med B/M 2676 2.70 F-stat. > F-crit. 0.000 Reject Ho Big and High B/M 4316 2.70 F-stat. > F-crit. 0.000 Reject Ho The test results show that parameters of independent variables (mkt-rf, SMB, HML) are statistically significant. At the significance level 0.05, F = 0.000 states that there is only a 0% chance that the regression output is merely a chance of random occurrence. Table 3. Coefficient of Determination: CAPM and Fama – French models Portfolio R2 CAPM Adj. R2 Fama-French Conclusion Food Industry 54.2% 56.3% No changes Food Industry (bias corrected) 68% N/A N/A Small and Low B/M 76.5% 98% Better fit in Fama-French model Small and Med B/M 77.4% 97.9% Better fit in Fama-French model Small and High B/M 71.3% 99% Better fit in Fama-French model Big and Low B/M 94% 97.5% Better fit in Fama-French model Big and Med B/M 86.8% 93% Better fit in Fama-French model Big and High B/M 76.4% 95.5% Better fit in Fama-French model Note: Table 3 shows that Fama-French model explains data better than CAPM model. Hypothesis Test for Coefficients Linear regression analysis returns values of the intercept and variables. These coefficients give the least squares estimate. Hypothesis test for coefficient determines significance of the regression coefficient in the model. For CAPM test, we expect ∞ to be zero but β cannot be zero. For Fama-French model, we expect ∞ to be zero; depending on the test on β1, β2, and β3, we can predict which pricing factor influence on the excess return. Table 4. Coefficient Hypothesis Test: CAPM Model Portfolio t-statistics for α Ho:α=0 Ha:α≠0 Two-tailed Ho: α=0 Ha: α>0 Right-tailed Ho: α=0 Ha: α0 Right-tailed Food Industry 2.66 Reject Ho Reject Ho Reject Ho 26.73 Reject Ho Reject Ho Small and Low B/M -1.20 Do not reject Ho Do not reject Ho Do not reject Ho 44.40 Reject Ho Reject Ho Small and Med B/M 3.39 Reject Ho Reject Ho Reject Ho 45.51 Reject Ho Reject Ho Small and High B/M 4.35 Reject Ho Reject Ho Reject Ho 38.75 Reject Ho Reject Ho Big and Low B/M -0.77 Do not reject Ho Do not reject Ho Do not reject Ho 97.78 Reject Ho Reject Ho Big and Med B/M 1.22 Do not reject Ho Do not reject Ho Do not reject Ho 63.0 Reject Ho Reject Ho Big and High B/M 2.66 Reject Ho Reject Ho Reject Ho 44.24 Reject Ho Reject Ho t-critical, two-tailed at significance level 0.05, df = 605 is 1.972 t-critical, right-tailed at significance level 0.05, df = 605 is 1.652 t-critical, left-tailed at significance level 0.05, df = 605 is 1.652 Figure 3. Principal concept of CAPM: Dependency of E(ri) from β (Fu-berlin-de n.d) The regression analysis tested the main concept of CAPM, E(ri) = rf + [E(rM) – rf] x ß, or E(ri) - rf = [E(rM) – rf] x ß. For CAPM to be valid, the regression model for the food industry should be rt-rft= ß (rmkt,t-rft). However, hypothesis test of coefficients returns rt-rft = 0.32+ 0.72 (rmkt,t-rft). It implies that the time-series regression does not confirm the validity of CAPM for this portfolio. We conducted bias study through standardization method of residuals. Observations with absolute values of standardized error greater than 2 were deleted from the dataset. The dataset reduced to 570 from 607 observations. The new dataset produced regression equation (rt-rft) = 0.27 + 0.73 (rmkt, t – rft). The coefficient α remained statistically significant. The bias removal failed to confirm the validity of CAPM for this portfolio. However, hypothesis tests on α for other six different portfolios demonstrate that the intercept is not statistically significant for small and low B/M, big and low B/M and big and medium B/M portfolios. The regression produces the following equations: (rt – rft) = 1.34 (rmkt, t – rf) – small and low B/M ; (rt – rft) = 1.009 (rmkt, t – rf) – big and low B/M ; (rt – rft) = 0.9 (rmkt, t – rf) – big and medium B/M. Table 5. Hypothesis Test: Fama-French Model Portfolio t-statistics for α Ho:α=0 Ha:α≠0 Two-tailed t-statistics for β1 Ho: β1=0 Ha: β1≠0 Two-tailed t-statistics for β2 Ho: β2=0 Ha:β2≠0 Two-tailed t-statistics for β3 Ho: β3=0 Ha:β3≠0 Two-tailed Food Industry 2.26 Reject Ho 27.5 Reject Ho -4.0 Reject Ho 3.46 Reject Ho Small and Low B/M -4.62 Reject Ho 113.32 Reject Ho 75.89 Reject Ho -17.91 Reject Ho Small and Med B/M 1.95 Do not Reject Ho 123.85 Reject Ho 74.36 Reject Ho 30.10 Reject Ho Small and High B/M 2.80 Reject Ho 179.40 Reject Ho 109.01 Reject Ho 81.59 Reject Ho Big and Low B/M 4.35 Reject Ho 136.83 Reject Ho -15.29 Reject Ho 27.03 Reject Ho Big and Med B/M -1.42 Do not reject Ho 88.26 Reject Ho -8.90 Reject Ho 19.81 Reject Ho Big and High B/M -2.89 Reject Ho 108.49 Reject Ho -0.035 Do not Reject Ho 50.39 Reject Ho t-critical, two-tailed at significance level 0.05, df = 603 is 1.972 Conclusion The assignment tested the CAPM claim that the excess return linearly increases with risk. The CAPM uses β to define risk. In addition to the set forth discussion, regression results for seven different portfolios presented in Figure 3 do not confirm this claim. Regression analysis of the same portfolio with Fama-Franch model does not confirm that market premium is the only factor that produces an excess return on an asset. However, using the results of the hypothesis test of intercepts and coefficients and subsequent removal of factors with negative coefficients we evaluated the following equations showing dependency of excess returns on parameters for a particular sector: (rt-rft) = 0.96 (rmkt, t- rft) + 0.82 SMBt – For small and medium B/M; (rt-rft) = 0.99 (rmkt, t- rft) + 0.34 HMLt – For big and medium B/M (BM); (rt-rft) = 0.12 + 1.06 (rmkt, t- rft) + 0.74 HMLt –For big and high B/M (BH). Audit Fee In an audit operation, there are two parties; auditor and auditee. The auditee pays an amount of fees for the audit process conducted by the auditor. In recent years, there is a growing concern about the issue how auditors determine an audit fee (Clatworthy & Peel 2007). Does it depend on the attributes of the audit company or client’s company? This assignment accepts the assumption that it depends on the attributes of the client’s company. A client’s company can be categorized by various factors, such as size, complexity, risk and financial parameters (Gammal-El 2012). Larger client of the same industry needs more auditing hours than smaller. The client’s company with subsidiaries locally or internationally adds complexity and an audit firm consumes more time. Audit risk characterizes the odds of an auditor issuing improper judgment on misstated financial statements (Chan, Ezzamel, & William 1993). Companies that report high profits undergo stringent audit testing, which incurs higher audit fees. Audit fee in this study becomes dependent variable and rest of the factors described above become independent variables. A best-fit relationship between dependent and independent variables will be obtained by conducting linear regression analysis with multiple variables. We used dataset created by random sampling of 5,000 UK firms. Some of the variables in the dataset are log transformed. Our goal is not to study the numerical effect of variables on the model but to analyze statistical significance of the hypothesized model. Search for a Regression Model The purpose is to find a model that can predict behavior of the dependent variable for the selected independent variables. It is achieved by conducting a hypothesis test of coefficients of independent variables. Coefficients are tested for Ho=0 and Ha ≠ 0. We use both critical and p-value methods to conduct the test. If a coefficient satisfies null, its corresponding variable is deleted from the model. This procedure is continued until the model is reached where Ha is satisfied instead of Ho for all coefficients. Model 1: Y - audit fee (log fee); X1 - sales (log sales); X2 - total assets (log assets); X3 - number of subsidiaries (nsubs); X4 - exports divided by sales (expsales); X5 - total liabilities divided by total assets (gearing) (tlta); X6 - return on total assets (retta); X7 - current assets divided by current liabilities (cacl); Y = ∫ (X1, X2, X3, X4, X5, X6, X7) Log (Y) = b0 + b1*Log(sales) +b2*Log(assets) + b3* nsubs + b4 *expsales + b5 *tlata + b6* retta + b7 * cacl Table 6. Regression results: Model 1 Variables Coefficients p-value at α=0.05 t-statistics t-critical Ho=0 Ha≠0 Intercept 1.66 0.000 27.74 1.972 Reject Ho X1 0.29 0.000 34.33 1.972 Reject Ho X2 0.188 0.000 23.00 1.972 Reject Ho X3 0.02 0.000 16.14 1.972 Reject Ho X4 0.60 0.000 8.71 1.972 Reject Ho X5 0.024 0.000 3.88 1.972 Reject Ho X6 -0.021 0.054 -1.92 1.972 Do not reject Ho X7 -.00014 0.17 -1.36 1.972 Do not reject Ho Adjusted R2 = 0.775 confirms that 77.5% observation data fall on the regression line. F – test states that independent variables used in the model are statistically significant at significance level 0.05; F 0.000 states that there is only a 0% chance that the regression output is merely a chance of random occurrence. The value of t-critical for tow-tailed at significance level α=0.05 for df = 4992 is 1.972. We used the criteria to reject Ho if t-statistics > t-critical and p-value > 0.05. Based on the above hypothesis test of Model 1, variables “retta” and “cacl” are deleted from the model 1. Model 2 Y = ∫ (X1, X2, X3, X4, X5) Log (Y) = b0 + b1*Log(sales) +b2*Log(assets) + b3* nsubs + b4 *expsales + b5 *tlata Table 7: Regression output: Model 2 Variables Coefficients p-value at α=0.05 t-statistics t-critical Ho=0 Ha≠0 Intercept 1.62 0.000 28.573 1.972 Reject Ho X1 0.29 0.000 34.985 1.972 Reject Ho X2 0.19 0.000 24.346 1.972 Reject Ho X3 0.02 0.000 16.04 1.972 Reject Ho X4 0.60 0.000 8.682 1.972 Reject Ho X5 0.03 0.000 4.075 1.972 Reject Ho Adjusted R2 for the model is = 0.775 confirms that 77.5% observation data fall on the regression line. F – test states that independent variables used in the model are statistically significant at significance level 0.05; F 0.000 states that there is only a 0% chance that the Regression output is merely a chance of random occurrence. The value of t-critical for tow-tailed at significance level α=0.05 for df = 4994 is 1.972. Conclusion The above regression analysis demonstrates that a regression equation Log (logafee) = 1.62 + 0.29 *Log(sales) + 0.19 *Log(assets) + 0.02 * nsubs + 0.60 *expsales + 0.03 *tlata can model audit fee. Audit fee is mostly influenced by the size of the company. 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