Capital Asset Pricing Model in the Stock Exchange - Research Proposal Example

APPLIED STATISTICS FOR FINANCE & ECONOMICS By of the of the School This paper examines the CAPM in the stock exchange using Black, Jensen and Scholes-BJS approach based on weekly stock returns from 257 companies of the Standard & Poor 500 index (sap) listed on the stock exchange market from February 2002 to January 2007. The tests were conducted to examine the linearity or nonlinearity of the relationship between expected returns and betas reinforce the presumed hypothesis that the expected return-beta relationship is linear. The results show that there is indeed a positive linear relation between the expected portfolio returns () and the risk (). The risk free rate () of the CAPM was found to be approximately zero, implying that the companies are subject to less market risk. Finally the Beta () which is given as the slope of the equation and in this case it was found to be 1.031406, implying that the movement the asset is in the same direction as though more than the movement of the benchmark. Keywords: CAPM, portfolio returns, beta, risk free rate, systematic risk Introduction Capital Asset Pricing Model (CAPM) For many years, the Capital Asset Pricing Model (CAPM) has been used as one of the best tools for analyzing the risk-return trade-off of investors and is considered one of the main contributions of academic research to financial managers: The only way an investor can get a much higher return for his investment is by taking a greater risk. This particular intuition/belief is summarized in the CAPM of Treynor (1961) and Sharpe (1964) and was extended further by Black (1972), Mossin (1966) and Lintner (1965).The CAPM aims at quantifying the association between the beta (coefficient) of an asset and its corresponding expected return. The CAPM model makes a number of assumptions, most of which relates the investor behavior and the traces of a single common risk factor. The very first assumption is that investors mostly care only about volatility and the expected returns. As rational consumers, they will therefore always attempt to maximize the expected return for any particular level of expected volatility. Secondly, it argues that all investors have homogeneous sort of beliefs about the reward or risk tradeoffs in the market. The next assumption is that only one risk factor is common among the broad-based market portfolio. This risk factor is the systematic market risk which drives non-diversifiable volatility. Investors are assumed to hold diversified portfolios, as the market does not reward investors for the bearing of diversifiable risk. Based on these, the CAPM states that if a security’s beta (coefficient) is known, then it is possible to compute the corresponding expected return. Risk and Return Generally, investors believe that it is prudent to take a high risk so as to receive a high return. Conversely, investors are always willing and ready to sacrifice return (receive less than their current return) with an aim of minimizing the risk. An asset exhibits two main types of risk that is, systematic and unsystematic risk. Unsystematic risk affects only an individual portfolio or security and that it does not affect in any way the market as a whole. Unsystematic risk is considered as "random noise" in a given portfolio, example is in volatility of returns. By adding more securities the random noise component eventually achieves a zero mean. Also as the number of securities in a given portfolio increases so does the standard deviation of the returns. If there are enough assets in any given portfolio in such a way that adding more securities does not affect the performance, then the volatility of the portfolios returns automatically matches that of the entire markets returns. Risk that can be never be palliated through diversification (adding more securities) is referred to as systematic risk. The portion of the volatility of returns that is considered systematic is actually measured by the degree to which its (systematic risk) returns vary relative to those of the entire market. The Simple Return/Risk Relationship CAPM is used to analyze the performance of portfolios and other mutual funds. The technique compares the historical risk adjusted returns of the fund against those of an estimated index, and then use least-squares regression (OLS)  to fit a straight line through the data points In arriving CAPM regression model a number of derivations has to be done. The general equation is given as; Where; , , and The slope of the line is the Beta () while alpha () is the vertical intercept and it generally tells us how much better or worse the fund progressed than the CAPM predicted (positive tells us how much better it faired on while negative tells us how worse it faired on). R-Squared (R2) is used to tell the quality of the fit. The values of R2 ranges from 0 to 1 and a value of 1 would mean that model and the data fit perfectly. The expected return and standard deviation of a given portfolio can be written as follows: The of in the above equations represents excess demands for a risky asset. This can be written as follows; Where; Empirical Results In this section we present the results of descriptive techniques (means and medians etc.) and graphs (time series plots, kernel density plots and box plots etc.) aimed at describing the main properties of the two log return series and the regression analysis. We also report the interpretation and detailed discussion of the empirical findings in this section. Finally, a possible explanation, based on the financial theory earlier on reviewed above, is presented so as to explicate the empirical findings. Summary Statistics Descriptive Statistics We began by testing whether the variables and are normally distributed: We can reject the hypothesis that is normally distributed, but we cannot reject the hypothesis that is normally distributed, at least at the 5.14% level. The kurtosis for is 2.47292, as can be verified by issuing the command . summarize ly, detail and the p-value of 0.0293 shown in the table above indicates that it is significantly different from the kurtosis of a normal distribution at the 5% significance level. However, on the basis of skewness alone, we cannot reject the hypothesis that is normally distributed. In both cases the normal curves are skewed to the left with outliers. Their histograms and box plot diagrams below therefore show similar conclusion as made above. The table below gives the summary statistics of and , the mean for is given as 0.0013793 while that of is given as 0.0007498. Their respective standard deviations are 0.041417 and 0.0311088. We present a scatter plot of against The scatter plot shows that there exists a relationship between and and based on the shape of the graph we can deduce that it is a positive relationship. The slope of this line is the Beta () and Alpha () is the vertical intercept, and it indicates how much better the fund did than the CAPM predicted. We expect the regression line to pass through the origin if is zero. Looking at the graph below, the regression line is approaching the origin implying that we expect to be zero or close to zero. The graph below shows a plot of the log of the end-of-week share price for a particular stock (denoted ) and the log of the end-of-week value for the Standard & Poor 500 index () Time series regression This is the final empirical analysis that focuses on the estimation of a simple Capital Asset Pricing Model (CAPM). This involves the Ordinary Least Squares Estimation (OLS) of the following regression model: Which can be written as follows; Where; However, in our case we have; Such that, , while CAPM model describes the relationship between risk and expected return and that is used in the pricing of risky securities. From the table below the p-value=0.0000 Read More