CAPM and Three Factor Model in Cost of Equity Measurement

CAPM and Three Factor Model in Cost of Equity Measurement 1.0 INTRODUCTION AND OBJECTIVES Central to many financial decisions such as those relating to investment, capital budgeting, portfolio management and performance evaluation is the estimation of the cost of equity or expected return. There exist several models for the valuation of equity returns, prominent among which are the dividend growth model, residual income model and its extension, free cash flow model, the capital asset pricing model, the Fama and French three factor model, the four factor model etc. Over the past few decades, two of the most common asset pricing models that have been used for this purpose are the Capital Asset Pricing Model (a single factor model by Sharpe 1964, Lintner 1965) and the three factor model suggested by Fama and French (1993). These two models have been very appealing to both practitioners and academicians due to their structural simplicity and are very easy to interpret. There have however been lots of debates and articles as to which of these two models should be used when est imating the cost of equity or expected returns. The question as to which of these two models is better in terms of their ability to explain variation in returns and forecast future returns is still an open one. While most practitioners favour a one factor model (CAPM) when estimating the cost of equity or expected return for a single stock or portfolio, academics however recommend the Fama and French three factor model (see eg. Bruner et al, 1998). The CAPM depicts a linear relationship between the expected return on a stock or portfolio to the excess return on a market portfolio. It characterizes the degree to which an assets return is correlated to the market, and indirectly how risky the asset is, as captured by beta. The three-factor model on the other hand is an extension of the CAPM with the introduction of two additional factors, which takes into account firm size (SMB) and book-to-market equity (HML). The question therefore is why practitioners prefer to use the single factor model (CAPM) when there exist some evidence in academics in favour of the Fama and French three factor model. Considering the number of years most academic concepts are adopted practically, can we conclude that the Fama and French three factor model is experiencing this so-called natural resistance or is it the case that the Fama and French model does not perform significantly better than the CAPM and so therefore not worth the time and cost? The few questions I have posed above form the basis for this study. It is worth noting that while the huge academic studies on these models produce interesting results and new findings, the validity of the underlying models have not been rigorously verified. In this paper, while I aim to ascertain which of the two models better estimates the cost of equity for capital budgeting purposes using regression analysis, I also will like to test whether the data used satisfy the assumptions of the method most academicians adopt, i.e. the Ordinary Least Squares (OLS) method. I will in particular be testing for the existence or otherwise of heteroscedasticity, multicollinearity, normality of errors serial correlation and unit roots, which may result in inefficient coefficient estimates, wrong standard errors, and hence inflated adjusted R2 if present in the data. I will then correct these if they exist by adopting the Generalised Least Squares (GLS) approach instead of the widely used Ordinary Least Squares (OLS) before drawing any inference from the results obtained. My conclusion as to which of the models is superior to the other will be based on which provides the best possible estimate for expected return or cost of equity for capital budgeting decision making. Since the cost of capital for capital budgeting is not observed, the objective here, therefore, is to find the model that is most effective in capturing the variations in stock returns as well as providing the best estimates for future returns. By running a cross sectional regression using stock or portfolio returns as the dependent variable and estimated factor(s) based on past returns as regressors, R2 measures how much of the differences in returns is explained by the estimation procedure. The model that produces the highest adjusted R2 will therefore be deemed the best. The Fama-French (1993, 1996) claimed superiority of their model over CAPM in explaining variations in returns from regressions of 25 portfolios sorted by size and book-to-market value. Their conclusion was based on the fact that their model produced a lower mean absolute value of alpha which is much closer to the theoretical value of zero. Fama and French (2004, working paper) stated that if asset pricing theory holds either in the case of the CAPM (page 10), or the Fama and French three-factor model (page 21), then the value of their alphas should be zero, depicting that the asset pricing model and its factor or factors explain the variations in portfolio returns. Larger values of alpha in this case are not desirable, since this will imply that the model was poor in explaining variation in returns. In line with this postulation, the model that yields the lowest Mean Absolute Value of Alpha (MAVA) will therefore be considered the best. But since alpha is a random variable, I will pro ceed to test the null hypothesis H0: ÃŽ ±i = 0 for all i, by employing the GRS F-statistic postulated by Gibbons, Ross and Shanken (1989). My third and fourth testing measures are based on postulates by econometricians that, the statistical adequacy of a model in terms of its violations of the classical linear regression model assumptions is hugely irrelevant if the models predictive power is poor and that the accuracy of forecasts according to traditional statistical criteria such as the MSE may give little guide to the potential profitability of employing those forecasts in a market trading strategy or for capital budgeting purposes. I will therefore test the predictive power of the two models by observing the percentage of forecast signs predicted correctly and their Mean Square Errors (MSE). One other motivation for this study is also to ascertain whether the results of prior studies are sample specific, that is, whether it is dependent on the period of study or the portfolio grouping used. Theoretically, the effectiveness of an asset pricing model in explaining variation in returns should not be influenced by how the data is grouped. Fama and French (1996) claimed superiority of their model over the CAPM using the July 1963 to December 1993 time period with data groupings based on size and book-to-market equity. I will be replicating this test on the same data grouping but covering a much longer period (from July 1926 to June 2006) and then on a different data grouping based on industry characteristics. Testing the models using the second grouping of industry portfolios will afford me the opportunity to ascertain whether the effectiveness of an asset pricing model is sample specific. I will also carry out the test by employing a much shorter period (5 years) and compari ng it to the longer period and then using the one with the better estimate in terms of alpha and R2 to carry out out-of-sample forecasts. The rest of this paper is structured as follows. Chapter 2 will review the various models available for the estimation of equity cost with particular emphasis on the two asset-pricing models and analysing some existing literature. Chapter 3 will give a description of the data, its source and transformations required, with Chapter 4 describing the methodology. Chapter 5 will involve the time series tests of hypothesis on the data and Chapter 6 will involve an empirical analysis of the results for the tests of the CAPM and the Fama and French three-factor model. Finally, Chapter 7 contains a summary of the major findings of my work and my recommendation as well as some limitations, if any, of the study and recommended areas for further studies. 2.0 RELEVANT LITERATURE The estimation of the cost of equity for an industry involves estimation of what investors expect in return for their investment in that industry. That is, the cost of equity to an industry is equal to the expected return on investors equity holdings in that industry. There are however a host of models available for the estimation of expected returns on an industrys equity capital including but not limited to estimates from fundamentals (dividends and earnings) and those from asset pricing models. 2.1 Estimations from Fundamentals Estimation of expected returns or cost of equity in this case from fundamentals involves the use of dividends and earnings. Fama and French (2002) used this approach to estimate expected stock returns. They stated that, the expected return estimates from fundamentals help to judge whether the realised average return is high or low relative to the expected value (pp 1). The reasoning behind this approach lies in the fact that, the average stock return is the average dividend yield plus the average rate of capital gain: A(Rt) = A(Dt/Pt-1) + A(GPt) (1) where Dt is the dividend for year t, Pt-1 is the price at the end of year t 1, GPt = (Pt Pt-1)/Pt-1 is the rate of capital gain, and A( ) indicates an average value. Given in this situation that the dividend-price ratio, Dt/Pt , is stationary (mean reverting), an alternative estimate of the stock return from fundamentals is: A(RDt) = A(Dt/Pt-1) + A(GDt) (2) Where GDt = (Dt Dt-1)/Dt-1is the growth rate of dividends and (2) is known as the dividend growth model which can be viewed as the expected stock return estimate of the Gordon (1962) model. Equation (2) in theory will only apply to variables that are cointegrated with the stock price and may not hold if the dividend-price ratio is non-stationary, which may be caused by firms decision to return earnings to stockholders by moving away from dividends to share repurchases (Fama and French 2002). But assuming that the ratio of earnings to price, (Yt/Pt), is stationary, then an alternative estimate of the expected rate of capital gain will be the average growth rate of earnings, A(GYt) = A((Yt Yt-1)/Yt-1). In this case, the average dividend yield can be combined with the A(GYt) to produce a third method of estimating expected stock return, the earnings growth model given as: A(RYt) = A(Dt/Pt-1) + A(GYt) (3) It stands to reason from the model in Lettau and Ludvigson (2001) that the average growth rate of consumption can be an alternative mean of estimating the expected rate of capital gain if the ratio of consumption to stock market wealth is assumed stationary. Fama and French (2002) in their analysis concluded that the dividend growth model has an advantage over the earnings growth model and the average stock return if the goal is to estimate the long-term expected growth of wealth. However, it is a more generally known fact that, dividends are a policy variable and so subject to changes in management policy, which raises problems when using the dividend growth model to estimate the expected stock returns. But this may not be a problem in the long run if there is stability in dividend policies and dividend-price ratio resumes its mean-reversion (although the reversion may be at a new mean level). Bagwell and Shoven (1989) and Dunsby (1995) have observed that share repurchases after 1983 has been on the ascendancy, while Fama and French (2001) have also observed that the proportion of firms who do not pay dividends have been increasing steadily since 1978. The Fama and French (2001) observation implies that in transition periods where firms who do not pay dividends increases steadily, the market dividend-price ratio may be non-stationary; overtime, it is likely to decrease, in which case the expected return will likely be underestimated when the dividend growth model is used. The earnings growth model, although not superior to the dividend growth model (Fama and French (2002)), is not affected by possible changes in dividend policies over time. The earnings growth model however may also be affected by non-stationarity in earnings-price ratio since it ability to accurately estimate average expected return is based on the assumption that there are permanent shifts in the expected value of the earnings-price ratio. 2.2 Estimations from Asset-Pricing Models One of the most fundamental concepts in the area of asset-pricing is that of risk versus reward. The pioneering work that addressed the risk and reward trade-off was done by Sharpe (1964)-Lintner (1965), in their introduction of the Capital Asset Pricing Model (CAPM). The Capital Asset Pricing Model postulates that the cross-sectional variation in expected stock or portfolio returns is captured only by the market beta. However, evidence from past literature (Fama and French (1992), Carhart (1997), Strong and Xu (1997), Jagannathan and Wang (1996), Lettau and Ludvigson (2001), and others) stipulates that the cross-section of stock returns is not fully captured by the one factor market beta. Past and present literature including studies by Banz (1981), Rosenberg et al (1985), Basu (1983) and Lakonishok et al (1994) have established that, in addition to the market beta, average returns on stocks are influenced by size, book-to-market equity, earnings/price and past sales growth respecti vely. Past studies have also revealed that stock returns tend to display short-term momentum (Jegadeesh and Titman (1993)) and long-term reversals (DeBondt and Thaler (1985)). Growing research in this area by scholars to address these anomalies has led to the development of alternative models that better explain variations in stock returns. This led to the categorisation of asset pricing models into three: (1) multifactor models that add some factors to the market return, such as the Fama and French three factor model; (2) the arbitrage pricing theory postulated by Ross (1977) and (3) the nonparametric models that heavily criticized the linearity of the CAPM and therefore added moments, as evidenced in the work of Harvey and Siddique (2000) and Dittmar (2002). From this categorization, most of the asset-pricing models can be described as special cases of the four-factor model proposed by Carhart (1997). The four-factor model is given as: E(Ri) Rf = ÃŽ ±i + [ E(RM) Rf ]bi + si E(SMB) + hi E(HML) + wiE(WML) + ÃŽ µi (4) where SMB, HML and WML are proxies for size, book-to-market equity and momentum respectively. There exist other variants of these models such as the three-moment CAPM and the four-moment CAPM (Dittmar, 2002) which add skewness and kurtosis to investor preferences, however the focus of this paper is to compare and test the effectiveness of the CAPM and the Fama and French three-factor model, the two premier asset-pricing models widely acknowledged among both practitioners and academicians. 2.3 Theoretical Background: CAPM and Fama French Three-Factor Model There exist quite a substantial amount of studies in the field of finance relating to these two prominent asset pricing models. The Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965) has been the first most widely recognized theoretical explanation for the estimation of expected stock returns or cost of equity in this case. It is a single factor model that is widely used by Financial Economists and in industry. The CAPM being the first theoretical asset pricing model to address the risk and return concept and due to its simplicity and ease of interpretation, was quickly embraced when it was first introduced. The models attractiveness also lies in the fact that, it addressed difficult problems related to asset pricing using readily available time series data. The CAPM is based on the idea of the relationship that exists between the risk of an asset and the expected return with beta being the sole risk pricing factor. The Sharpe-Lintner CAPM equation which describes individual asset return is given as: E(Ri) = Rf + [ E(RM) Rf ]ÃŽ ²iM i = 1,,N (5) where E(Ri) is the expected return on any asset i, Rf is the risk-free interest rate, E(RM) is the expected return on the value-weighted market portfolio, and ÃŽ ²iM is the assets market beta which measures the sensitivity of the assets return to variations in the market returns and it is equivalent to Cov(Ri, RM)/Var(RM). The equation for the time series regression can be written as: E(Ri) Rf = ÃŽ ±i + [ E(RM) Rf ]ÃŽ ²iM + ÃŽ µi i = 1,,N; (6) showing that the excess return on portfolio i is dependent on excess market return with ÃŽ µi as the error term. The excess market return is also referred to as the market premium. The model is based on several key assumptions, portraying a simplified world where: (1) there are no taxes or transaction costs or problems with indivisibilities of assets; (2) all investors have identical investment horizons; (3) all investors have identical opinions about expected returns, volatilities and correlations of available investments; (4) all assets have limited liability; (5) there exist sufficiently large number of investors with comparable wealth levels so that each investor believes that he/she can purchase and sell any amount of an asset as he or she deems fit in the market; (6) the capital market is in equilibrium; and (7) Trading in assets takes place continually over time. The merits of these assumptions have been discussed extensively in literature. It is evident that most of these assumptions are the standard assumptions of a perfect market which does not exist in reality. It is a known fact that, in reality, indivisibilities and transaction costs do exist and one of the reasons assigned to the assumption of continual trading models is to implicitly give recognition to these costs. It is imperative to note however that, trading intervals are stochastic and of non-constant length and so making it unsatisfactory to assume no trading cost. As mentioned earlier, the assumptions made the model very simple to estimate (given a proxy for the market factor) and interpret, thus making it very attractive and this explains why it was easily embraced. The CAPM stipulates that, investors are only rewarded for the systematic or non-diversifiable risk (represented by beta) they bear in holding a portfolio of assets. Notwithstanding the models simplicity in estimation and interpretation, it has been criticized heavily over the past few decades . Due to its many unrealistic assumptions and simple nature, academicians almost immediately began testing the implications of the CAPM. Studies by Black, Jensen and Scholes (1972) and Fama and MacBeth (1973) gave the first strong empirical support to the use of the model for determining the cost of capital. Black et al. (1972) in combining all the NYSE stocks into portfolio and using data between the periods of 1931 to 1965 found that the data are consistent with the predictions of the Capital Asset Pricing Model (CAPM). Using return data for NYSE stocks for the period between 1926 to 1968, Fama and MacBeth (1973) in examining whether other stock characteristics such as beta squared and idiosyncratic volatility of returns in addition to their betas would help in explaining the cross section of stock returns better found that knowledge of beta was sufficient. There have however been several academic challenges to the validity of the model in relation to its practical application. Banz (1981) revealed the first major challenge to the model when he provided empirical evidence to show that stocks of smaller firms earned better returns than predicted by the CAPM. Banzs finding was not deemed economically important by most academicians in the light that, it is unreasonable to expect an abstract model such as the CAPM to hold exactly and that the proportion of small firms to total market capital is insignificant (under 5%). Other early empirical works by Blume and friend (1973), Basu (1977), Reinganum (1981), Gibbons (1982), Stambaugh (1982) and shanken (1985) could not offer any significant evidence in support of the CAPM. In their paper, Fama and French (2004) noted that in regressing a cross section of average portfolio returns on portfolio beta estimates, the CAPM would predict an intercept which is equal to the risk free rate (Rf) and a beta coefficient equal to the market risk premium (E(Rm) Rf). However, Black, Jensen and Scholes (1972), Blume and Friend (1973), Fama and MacBeth (1973) and Fama and French (1992) after running series of cross-sectional regressions found that the average risk-free rate, which is proxied by the one month T-bill, was always less that the realised intercept. Theory stipulates that, the three main components of the model (the risk free, beta and the market risk premium) must be forward-looking estimates. That is they must be estimates of their true future values. Empirical studies and survey results however show substantial disagreements as to how these components can be estimated. While most empirical researches use the one month T-bill rate as a proxy to the risk-fr ee rate, interviews depicts that practitioners prefer to use either the 90-day T-bill or a 10-year T-bond (normally characterised by a flat yield curve). Survey results have revealed that practitioners have a strong preference for long-term bond yields with over 70% of financial advisors and corporations using Treasury-bond yields with maturities of ten 10 or more years. However, many corporations reveal that they match the tenor of the investment to the term of the risk free rate. Finance theory postulates that the estimated beta should be forward looking, so as to reflect investors uncertainty about future cash flows to equity. Practitioners are forced to use various kinds of proxies since forward-looking betas are unobservable. It is therefore a common practice to use beta estimates derived from historical data which are normally retrieved from Bloomberg, Standard Poors and Value Line. However, the lack of consensus as to which of these three to use results in different betas for the same company. These differences in beta estimates could result in significantly different expected future returns or cost of equity for the company in question thereby yielding conflicting financial decisions especially in capital budgeting. In the work of Bruner et al. (1998), they found significant differences in beta estimates for a small sample of stocks, with Bloomberg providing a figure of 1.03 while Value Line beta was 1.24. The use of historical data however requires th at one makes some practical compromises, each of which can adversely affect the quality of the results. Forinstance, the statistically reliability of the estimate may improve greatly by employing longer time series periods but this may include information that are stale or irrelevant. Empirical research over the years has shown that the precision of the beta estimates using the CAPM is greatly improved when working with well diversified portfolios compared to individual securities. In relation to the equity risk premium, finance theory postulates that, the market premium should be equal to the difference between investors expected returns on the market portfolio and the risk-free rate. Most practitioners have to grapple with the problem of how to measure the market risk premium. Survey results have revealed that the equity market premium prompted the greatest diversity of responses among survey respondents. Since future expected returns are unobservable, most of the survey participants extrapolated historical returns in the future on the assumption that future expectations are heavily influenced by past experience. The survey participants however differed in their estimation of the average historical equity returns as well as their choice of proxy for the riskless asset. Some respondents preferred the geometric average historical equity returns to the arithmetic one while some also prefer the T-bonds to the T-bill as a proxy for the riskless asset. Despite the numerous academic literatures which discuss how the CAPM should be implemented, there is no consensus in relation to the time frame and the data frequency that should be used for estimation. Bartholdy Peare (2005) in their paper concluded that, for estimation of beta, five years of monthly data is the appropriate time period and data frequency. They also found that an equal weighted index, as opposed to the commonly recommended value-weighted index provides a better estimate. Their findings also revealed that it does not really matter whether dividends are included in the index or not or whether raw returns or excess returns are used in the regression equation. The CAPM has over the years been said to have failed greatly in explaining accurate expected returns and this some researchers have attributed to its many unrealistic assumptions. One other major assumption of the CAPM is that there exists complete knowledge of the true market portfolios composition or index to be used. This assumed index is to consist of all the assets in the world. However since only a small fraction of all assets in the world are traded on stock exchanges, it is impossible to construct such an index leading to the use of proxies such as the SP500, resulting in ambiguities in tests. The greatest challenge to the CAPM aside that of Banz (1981) came from Fama and French (1992). Using similar procedures as Fama and MacBeth (1973) and ten size classes and ten beta classes, Fama and French (1992) found that the cross section of average returns on stocks for the periods spanning 1960s to 1990 for US stocks is not fully explained by the CAPM beta and that stock risks are multidimensional. Their regression analysis suggest that company size and book-to-market equity ratio do perform better than beta in capturing cross-sectional variation in the cost of equity capital across firms. Their work was however preceded by Stattman (1980) who was the first to document a positive relation between book-to-market ratios and US stock returns. The findings of Fama and French could however not be dismissed as being economically insignificant as in the case of Banz. Fama and French therefore in 1993 identified a model with three common risk factors in the stock return- an overall market factor, factors related to firm size (SMB) and those related to book-to-market equity (HML), as an alternative to the CAPM. The SMB factor is computed as the average return on three small portfolios (small cap portfolios) less the average return on three big portfolios (large cap portfolios). The HML factor on the other hand is computed as the average return on two value portfolios less the average return on two growth portfolios. The growth portfolio represents stocks with low Book Equity to Market Equity ratio (BE/ME) while the value portfolios represent stocks with high BE/ME ratio. Their three-factor model equation is described as follows: E(Ri) Rf = ÃŽ ±i + [ E(RM) Rf ]bi + si E(SMB) + hi E(HML) + ÃŽ µi (7) Where E(RM) Rf, , E(SMB) and E(HML) are the factor risk premiums and bi , si and hi are the factor sensitivities. It is however believed that the introduction of these two additional factors was motivated by the works of Stattman (1980) and Banz (1981). The effectiveness of these two models in capturing variations in stock returns may be judged by the intercept (alpha) in equations (6) and (7) above. Theory postulates that if these models hold, then the value of the intercept or alpha must equal zero for all assets or portfolio of assets. Fama and French (1997) tested the ability of both the CAPM and their own three-factor model in estimating industry costs of equity. Their test considered 48 industries in which they found that their model outperformed the CAPM across all the industries considered. They however could not conclude that their model was better since their estimates of industry cost of equities were observed to be imprecise. Another disturbing outcome of their study is that both models displayed very large standard errors in the order of 3.0% per annum across all industries. Connor and Senghal (2001) tested the effectiveness of these two models in predicting portfolio returns in indias stock market. They tested the models using 6 portfolio groupings formed from the intersection of two size and three book-to-market equity by examining and testing their intercepts. Connor and Senghal in this paper examined the values of the intercepts and their corresponding t-statistics and then tested the intercepts simultaneously by using the GRS statistic first introduced by Gibbons, Ross and Shanken (1989). Based on the evidence provided by the intercepts and the GRS tests, Connor and Senghal concluded generally that the three-factor model of Fama and French was superior to the CAPM. There have been other several empirical papers ever since, to ascertain which of these models is better in the estimation of expected return or cost of equity, most producing contrasting results. Howard Qi (2004) concluded in his work that on the aggregate level, the two models behave fairly well in their predictive power but the CAPM appeared to be slightly better. Bartholdy and Peare (2002) in their work came to the conclusion that both models performed poorly with the CAPM being the poorest. 3.0 DATA SOURCES T CAPM and Three Factor Model in Cost of Equity Measurement CAPM and Three Factor Model in Cost of Equity Measurement 1.0 INTRODUCTION AND OBJECTIVES Central to many financial decisions such as those relating to investment, capital budgeting, portfolio management and performance evaluation is the estimation of the cost of equity or expected return. There exist several models for the valuation of equity returns, prominent among which are the dividend growth model, residual income model and its extension, free cash flow model, the capital asset pricing model, the Fama and French three factor model, the four factor model etc. Over the past few decades, two of the most common asset pricing models that have been used for this purpose are the Capital Asset Pricing Model (a single factor model by Sharpe 1964, Lintner 1965) and the three factor model suggested by Fama and French (1993). These two models have been very appealing to both practitioners and academicians due to their structural simplicity and are very easy to interpret. There have however been lots of debates and articles as to which of these two models should be used when est imating the cost of equity or expected returns. The question as to which of these two models is better in terms of their ability to explain variation in returns and forecast future returns is still an open one. While most practitioners favour a one factor model (CAPM) when estimating the cost of equity or expected return for a single stock or portfolio, academics however recommend the Fama and French three factor model (see eg. Bruner et al, 1998). The CAPM depicts a linear relationship between the expected return on a stock or portfolio to the excess return on a market portfolio. It characterizes the degree to which an assets return is correlated to the market, and indirectly how risky the asset is, as captured by beta. The three-factor model on the other hand is an extension of the CAPM with the introduction of two additional factors, which takes into account firm size (SMB) and book-to-market equity (HML). The question therefore is why practitioners prefer to use the single factor model (CAPM) when there exist some evidence in academics in favour of the Fama and French three factor model. Considering the number of years most academic concepts are adopted practically, can we conclude that the Fama and French three factor model is experiencing this so-called natural resistance or is it the case that the Fama and French model does not perform significantly better than the CAPM and so therefore not worth the time and cost? The few questions I have posed above form the basis for this study. It is worth noting that while the huge academic studies on these models produce interesting results and new findings, the validity of the underlying models have not been rigorously verified. In this paper, while I aim to ascertain which of the two models better estimates the cost of equity for capital budgeting purposes using regression analysis, I also will like to test whether the data used satisfy the assumptions of the method most academicians adopt, i.e. the Ordinary Least Squares (OLS) method. I will in particular be testing for the existence or otherwise of heteroscedasticity, multicollinearity, normality of errors serial correlation and unit roots, which may result in inefficient coefficient estimates, wrong standard errors, and hence inflated adjusted R2 if present in the data. I will then correct these if they exist by adopting the Generalised Least Squares (GLS) approach instead of the widely used Ordinary Least Squares (OLS) before drawing any inference from the results obtained. My conclusion as to which of the models is superior to the other will be based on which provides the best possible estimate for expected return or cost of equity for capital budgeting decision making. Since the cost of capital for capital budgeting is not observed, the objective here, therefore, is to find the model that is most effective in capturing the variations in stock returns as well as providing the best estimates for future returns. By running a cross sectional regression using stock or portfolio returns as the dependent variable and estimated factor(s) based on past returns as regressors, R2 measures how much of the differences in returns is explained by the estimation procedure. The model that produces the highest adjusted R2 will therefore be deemed the best. The Fama-French (1993, 1996) claimed superiority of their model over CAPM in explaining variations in returns from regressions of 25 portfolios sorted by size and book-to-market value. Their conclusion was based on the fact that their model produced a lower mean absolute value of alpha which is much closer to the theoretical value of zero. Fama and French (2004, working paper) stated that if asset pricing theory holds either in the case of the CAPM (page 10), or the Fama and French three-factor model (page 21), then the value of their alphas should be zero, depicting that the asset pricing model and its factor or factors explain the variations in portfolio returns. Larger values of alpha in this case are not desirable, since this will imply that the model was poor in explaining variation in returns. In line with this postulation, the model that yields the lowest Mean Absolute Value of Alpha (MAVA) will therefore be considered the best. But since alpha is a random variable, I will pro ceed to test the null hypothesis H0: ÃŽ ±i = 0 for all i, by employing the GRS F-statistic postulated by Gibbons, Ross and Shanken (1989). My third and fourth testing measures are based on postulates by econometricians that, the statistical adequacy of a model in terms of its violations of the classical linear regression model assumptions is hugely irrelevant if the models predictive power is poor and that the accuracy of forecasts according to traditional statistical criteria such as the MSE may give little guide to the potential profitability of employing those forecasts in a market trading strategy or for capital budgeting purposes. I will therefore test the predictive power of the two models by observing the percentage of forecast signs predicted correctly and their Mean Square Errors (MSE). One other motivation for this study is also to ascertain whether the results of prior studies are sample specific, that is, whether it is dependent on the period of study or the portfolio grouping used. Theoretically, the effectiveness of an asset pricing model in explaining variation in returns should not be influenced by how the data is grouped. Fama and French (1996) claimed superiority of their model over the CAPM using the July 1963 to December 1993 time period with data groupings based on size and book-to-market equity. I will be replicating this test on the same data grouping but covering a much longer period (from July 1926 to June 2006) and then on a different data grouping based on industry characteristics. Testing the models using the second grouping of industry portfolios will afford me the opportunity to ascertain whether the effectiveness of an asset pricing model is sample specific. I will also carry out the test by employing a much shorter period (5 years) and compari ng it to the longer period and then using the one with the better estimate in terms of alpha and R2 to carry out out-of-sample forecasts. The rest of this paper is structured as follows. Chapter 2 will review the various models available for the estimation of equity cost with particular emphasis on the two asset-pricing models and analysing some existing literature. Chapter 3 will give a description of the data, its source and transformations required, with Chapter 4 describing the methodology. Chapter 5 will involve the time series tests of hypothesis on the data and Chapter 6 will involve an empirical analysis of the results for the tests of the CAPM and the Fama and French three-factor model. Finally, Chapter 7 contains a summary of the major findings of my work and my recommendation as well as some limitations, if any, of the study and recommended areas for further studies. 2.0 RELEVANT LITERATURE The estimation of the cost of equity for an industry involves estimation of what investors expect in return for their investment in that industry. That is, the cost of equity to an industry is equal to the expected return on investors equity holdings in that industry. There are however a host of models available for the estimation of expected returns on an industrys equity capital including but not limited to estimates from fundamentals (dividends and earnings) and those from asset pricing models. 2.1 Estimations from Fundamentals Estimation of expected returns or cost of equity in this case from fundamentals involves the use of dividends and earnings. Fama and French (2002) used this approach to estimate expected stock returns. They stated that, the expected return estimates from fundamentals help to judge whether the realised average return is high or low relative to the expected value (pp 1). The reasoning behind this approach lies in the fact that, the average stock return is the average dividend yield plus the average rate of capital gain: A(Rt) = A(Dt/Pt-1) + A(GPt) (1) where Dt is the dividend for year t, Pt-1 is the price at the end of year t 1, GPt = (Pt Pt-1)/Pt-1 is the rate of capital gain, and A( ) indicates an average value. Given in this situation that the dividend-price ratio, Dt/Pt , is stationary (mean reverting), an alternative estimate of the stock return from fundamentals is: A(RDt) = A(Dt/Pt-1) + A(GDt) (2) Where GDt = (Dt Dt-1)/Dt-1is the growth rate of dividends and (2) is known as the dividend growth model which can be viewed as the expected stock return estimate of the Gordon (1962) model. Equation (2) in theory will only apply to variables that are cointegrated with the stock price and may not hold if the dividend-price ratio is non-stationary, which may be caused by firms decision to return earnings to stockholders by moving away from dividends to share repurchases (Fama and French 2002). But assuming that the ratio of earnings to price, (Yt/Pt), is stationary, then an alternative estimate of the expected rate of capital gain will be the average growth rate of earnings, A(GYt) = A((Yt Yt-1)/Yt-1). In this case, the average dividend yield can be combined with the A(GYt) to produce a third method of estimating expected stock return, the earnings growth model given as: A(RYt) = A(Dt/Pt-1) + A(GYt) (3) It stands to reason from the model in Lettau and Ludvigson (2001) that the average growth rate of consumption can be an alternative mean of estimating the expected rate of capital gain if the ratio of consumption to stock market wealth is assumed stationary. Fama and French (2002) in their analysis concluded that the dividend growth model has an advantage over the earnings growth model and the average stock return if the goal is to estimate the long-term expected growth of wealth. However, it is a more generally known fact that, dividends are a policy variable and so subject to changes in management policy, which raises problems when using the dividend growth model to estimate the expected stock returns. But this may not be a problem in the long run if there is stability in dividend policies and dividend-price ratio resumes its mean-reversion (although the reversion may be at a new mean level). Bagwell and Shoven (1989) and Dunsby (1995) have observed that share repurchases after 1983 has been on the ascendancy, while Fama and French (2001) have also observed that the proportion of firms who do not pay dividends have been increasing steadily since 1978. The Fama and French (2001) observation implies that in transition periods where firms who do not pay dividends increases steadily, the market dividend-price ratio may be non-stationary; overtime, it is likely to decrease, in which case the expected return will likely be underestimated when the dividend growth model is used. The earnings growth model, although not superior to the dividend growth model (Fama and French (2002)), is not affected by possible changes in dividend policies over time. The earnings growth model however may also be affected by non-stationarity in earnings-price ratio since it ability to accurately estimate average expected return is based on the assumption that there are permanent shifts in the expected value of the earnings-price ratio. 2.2 Estimations from Asset-Pricing Models One of the most fundamental concepts in the area of asset-pricing is that of risk versus reward. The pioneering work that addressed the risk and reward trade-off was done by Sharpe (1964)-Lintner (1965), in their introduction of the Capital Asset Pricing Model (CAPM). The Capital Asset Pricing Model postulates that the cross-sectional variation in expected stock or portfolio returns is captured only by the market beta. However, evidence from past literature (Fama and French (1992), Carhart (1997), Strong and Xu (1997), Jagannathan and Wang (1996), Lettau and Ludvigson (2001), and others) stipulates that the cross-section of stock returns is not fully captured by the one factor market beta. Past and present literature including studies by Banz (1981), Rosenberg et al (1985), Basu (1983) and Lakonishok et al (1994) have established that, in addition to the market beta, average returns on stocks are influenced by size, book-to-market equity, earnings/price and past sales growth respecti vely. Past studies have also revealed that stock returns tend to display short-term momentum (Jegadeesh and Titman (1993)) and long-term reversals (DeBondt and Thaler (1985)). Growing research in this area by scholars to address these anomalies has led to the development of alternative models that better explain variations in stock returns. This led to the categorisation of asset pricing models into three: (1) multifactor models that add some factors to the market return, such as the Fama and French three factor model; (2) the arbitrage pricing theory postulated by Ross (1977) and (3) the nonparametric models that heavily criticized the linearity of the CAPM and therefore added moments, as evidenced in the work of Harvey and Siddique (2000) and Dittmar (2002). From this categorization, most of the asset-pricing models can be described as special cases of the four-factor model proposed by Carhart (1997). The four-factor model is given as: E(Ri) Rf = ÃŽ ±i + [ E(RM) Rf ]bi + si E(SMB) + hi E(HML) + wiE(WML) + ÃŽ µi (4) where SMB, HML and WML are proxies for size, book-to-market equity and momentum respectively. There exist other variants of these models such as the three-moment CAPM and the four-moment CAPM (Dittmar, 2002) which add skewness and kurtosis to investor preferences, however the focus of this paper is to compare and test the effectiveness of the CAPM and the Fama and French three-factor model, the two premier asset-pricing models widely acknowledged among both practitioners and academicians. 2.3 Theoretical Background: CAPM and Fama French Three-Factor Model There exist quite a substantial amount of studies in the field of finance relating to these two prominent asset pricing models. The Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965) has been the first most widely recognized theoretical explanation for the estimation of expected stock returns or cost of equity in this case. It is a single factor model that is widely used by Financial Economists and in industry. The CAPM being the first theoretical asset pricing model to address the risk and return concept and due to its simplicity and ease of interpretation, was quickly embraced when it was first introduced. The models attractiveness also lies in the fact that, it addressed difficult problems related to asset pricing using readily available time series data. The CAPM is based on the idea of the relationship that exists between the risk of an asset and the expected return with beta being the sole risk pricing factor. The Sharpe-Lintner CAPM equation which describes individual asset return is given as: E(Ri) = Rf + [ E(RM) Rf ]ÃŽ ²iM i = 1,,N (5) where E(Ri) is the expected return on any asset i, Rf is the risk-free interest rate, E(RM) is the expected return on the value-weighted market portfolio, and ÃŽ ²iM is the assets market beta which measures the sensitivity of the assets return to variations in the market returns and it is equivalent to Cov(Ri, RM)/Var(RM). The equation for the time series regression can be written as: E(Ri) Rf = ÃŽ ±i + [ E(RM) Rf ]ÃŽ ²iM + ÃŽ µi i = 1,,N; (6) showing that the excess return on portfolio i is dependent on excess market return with ÃŽ µi as the error term. The excess market return is also referred to as the market premium. The model is based on several key assumptions, portraying a simplified world where: (1) there are no taxes or transaction costs or problems with indivisibilities of assets; (2) all investors have identical investment horizons; (3) all investors have identical opinions about expected returns, volatilities and correlations of available investments; (4) all assets have limited liability; (5) there exist sufficiently large number of investors with comparable wealth levels so that each investor believes that he/she can purchase and sell any amount of an asset as he or she deems fit in the market; (6) the capital market is in equilibrium; and (7) Trading in assets takes place continually over time. The merits of these assumptions have been discussed extensively in literature. It is evident that most of these assumptions are the standard assumptions of a perfect market which does not exist in reality. It is a known fact that, in reality, indivisibilities and transaction costs do exist and one of the reasons assigned to the assumption of continual trading models is to implicitly give recognition to these costs. It is imperative to note however that, trading intervals are stochastic and of non-constant length and so making it unsatisfactory to assume no trading cost. As mentioned earlier, the assumptions made the model very simple to estimate (given a proxy for the market factor) and interpret, thus making it very attractive and this explains why it was easily embraced. The CAPM stipulates that, investors are only rewarded for the systematic or non-diversifiable risk (represented by beta) they bear in holding a portfolio of assets. Notwithstanding the models simplicity in estimation and interpretation, it has been criticized heavily over the past few decades . Due to its many unrealistic assumptions and simple nature, academicians almost immediately began testing the implications of the CAPM. Studies by Black, Jensen and Scholes (1972) and Fama and MacBeth (1973) gave the first strong empirical support to the use of the model for determining the cost of capital. Black et al. (1972) in combining all the NYSE stocks into portfolio and using data between the periods of 1931 to 1965 found that the data are consistent with the predictions of the Capital Asset Pricing Model (CAPM). Using return data for NYSE stocks for the period between 1926 to 1968, Fama and MacBeth (1973) in examining whether other stock characteristics such as beta squared and idiosyncratic volatility of returns in addition to their betas would help in explaining the cross section of stock returns better found that knowledge of beta was sufficient. There have however been several academic challenges to the validity of the model in relation to its practical application. Banz (1981) revealed the first major challenge to the model when he provided empirical evidence to show that stocks of smaller firms earned better returns than predicted by the CAPM. Banzs finding was not deemed economically important by most academicians in the light that, it is unreasonable to expect an abstract model such as the CAPM to hold exactly and that the proportion of small firms to total market capital is insignificant (under 5%). Other early empirical works by Blume and friend (1973), Basu (1977), Reinganum (1981), Gibbons (1982), Stambaugh (1982) and shanken (1985) could not offer any significant evidence in support of the CAPM. In their paper, Fama and French (2004) noted that in regressing a cross section of average portfolio returns on portfolio beta estimates, the CAPM would predict an intercept which is equal to the risk free rate (Rf) and a beta coefficient equal to the market risk premium (E(Rm) Rf). However, Black, Jensen and Scholes (1972), Blume and Friend (1973), Fama and MacBeth (1973) and Fama and French (1992) after running series of cross-sectional regressions found that the average risk-free rate, which is proxied by the one month T-bill, was always less that the realised intercept. Theory stipulates that, the three main components of the model (the risk free, beta and the market risk premium) must be forward-looking estimates. That is they must be estimates of their true future values. Empirical studies and survey results however show substantial disagreements as to how these components can be estimated. While most empirical researches use the one month T-bill rate as a proxy to the risk-fr ee rate, interviews depicts that practitioners prefer to use either the 90-day T-bill or a 10-year T-bond (normally characterised by a flat yield curve). Survey results have revealed that practitioners have a strong preference for long-term bond yields with over 70% of financial advisors and corporations using Treasury-bond yields with maturities of ten 10 or more years. However, many corporations reveal that they match the tenor of the investment to the term of the risk free rate. Finance theory postulates that the estimated beta should be forward looking, so as to reflect investors uncertainty about future cash flows to equity. Practitioners are forced to use various kinds of proxies since forward-looking betas are unobservable. It is therefore a common practice to use beta estimates derived from historical data which are normally retrieved from Bloomberg, Standard Poors and Value Line. However, the lack of consensus as to which of these three to use results in different betas for the same company. These differences in beta estimates could result in significantly different expected future returns or cost of equity for the company in question thereby yielding conflicting financial decisions especially in capital budgeting. In the work of Bruner et al. (1998), they found significant differences in beta estimates for a small sample of stocks, with Bloomberg providing a figure of 1.03 while Value Line beta was 1.24. The use of historical data however requires th at one makes some practical compromises, each of which can adversely affect the quality of the results. Forinstance, the statistically reliability of the estimate may improve greatly by employing longer time series periods but this may include information that are stale or irrelevant. Empirical research over the years has shown that the precision of the beta estimates using the CAPM is greatly improved when working with well diversified portfolios compared to individual securities. In relation to the equity risk premium, finance theory postulates that, the market premium should be equal to the difference between investors expected returns on the market portfolio and the risk-free rate. Most practitioners have to grapple with the problem of how to measure the market risk premium. Survey results have revealed that the equity market premium prompted the greatest diversity of responses among survey respondents. Since future expected returns are unobservable, most of the survey participants extrapolated historical returns in the future on the assumption that future expectations are heavily influenced by past experience. The survey participants however differed in their estimation of the average historical equity returns as well as their choice of proxy for the riskless asset. Some respondents preferred the geometric average historical equity returns to the arithmetic one while some also prefer the T-bonds to the T-bill as a proxy for the riskless asset. Despite the numerous academic literatures which discuss how the CAPM should be implemented, there is no consensus in relation to the time frame and the data frequency that should be used for estimation. Bartholdy Peare (2005) in their paper concluded that, for estimation of beta, five years of monthly data is the appropriate time period and data frequency. They also found that an equal weighted index, as opposed to the commonly recommended value-weighted index provides a better estimate. Their findings also revealed that it does not really matter whether dividends are included in the index or not or whether raw returns or excess returns are used in the regression equation. The CAPM has over the years been said to have failed greatly in explaining accurate expected returns and this some researchers have attributed to its many unrealistic assumptions. One other major assumption of the CAPM is that there exists complete knowledge of the true market portfolios composition or index to be used. This assumed index is to consist of all the assets in the world. However since only a small fraction of all assets in the world are traded on stock exchanges, it is impossible to construct such an index leading to the use of proxies such as the SP500, resulting in ambiguities in tests. The greatest challenge to the CAPM aside that of Banz (1981) came from Fama and French (1992). Using similar procedures as Fama and MacBeth (1973) and ten size classes and ten beta classes, Fama and French (1992) found that the cross section of average returns on stocks for the periods spanning 1960s to 1990 for US stocks is not fully explained by the CAPM beta and that stock risks are multidimensional. Their regression analysis suggest that company size and book-to-market equity ratio do perform better than beta in capturing cross-sectional variation in the cost of equity capital across firms. Their work was however preceded by Stattman (1980) who was the first to document a positive relation between book-to-market ratios and US stock returns. The findings of Fama and French could however not be dismissed as being economically insignificant as in the case of Banz. Fama and French therefore in 1993 identified a model with three common risk factors in the stock return- an overall market factor, factors related to firm size (SMB) and those related to book-to-market equity (HML), as an alternative to the CAPM. The SMB factor is computed as the average return on three small portfolios (small cap portfolios) less the average return on three big portfolios (large cap portfolios). The HML factor on the other hand is computed as the average return on two value portfolios less the average return on two growth portfolios. The growth portfolio represents stocks with low Book Equity to Market Equity ratio (BE/ME) while the value portfolios represent stocks with high BE/ME ratio. Their three-factor model equation is described as follows: E(Ri) Rf = ÃŽ ±i + [ E(RM) Rf ]bi + si E(SMB) + hi E(HML) + ÃŽ µi (7) Where E(RM) Rf, , E(SMB) and E(HML) are the factor risk premiums and bi , si and hi are the factor sensitivities. It is however believed that the introduction of these two additional factors was motivated by the works of Stattman (1980) and Banz (1981). The effectiveness of these two models in capturing variations in stock returns may be judged by the intercept (alpha) in equations (6) and (7) above. Theory postulates that if these models hold, then the value of the intercept or alpha must equal zero for all assets or portfolio of assets. Fama and French (1997) tested the ability of both the CAPM and their own three-factor model in estimating industry costs of equity. Their test considered 48 industries in which they found that their model outperformed the CAPM across all the industries considered. They however could not conclude that their model was better since their estimates of industry cost of equities were observed to be imprecise. Another disturbing outcome of their study is that both models displayed very large standard errors in the order of 3.0% per annum across all industries. Connor and Senghal (2001) tested the effectiveness of these two models in predicting portfolio returns in indias stock market. They tested the models using 6 portfolio groupings formed from the intersection of two size and three book-to-market equity by examining and testing their intercepts. Connor and Senghal in this paper examined the values of the intercepts and their corresponding t-statistics and then tested the intercepts simultaneously by using the GRS statistic first introduced by Gibbons, Ross and Shanken (1989). Based on the evidence provided by the intercepts and the GRS tests, Connor and Senghal concluded generally that the three-factor model of Fama and French was superior to the CAPM. There have been other several empirical papers ever since, to ascertain which of these models is better in the estimation of expected return or cost of equity, most producing contrasting results. Howard Qi (2004) concluded in his work that on the aggregate level, the two models behave fairly well in their predictive power but the CAPM appeared to be slightly better. Bartholdy and Peare (2002) in their work came to the conclusion that both models performed poorly with the CAPM being the poorest. 3.0 DATA SOURCES T

King Arthur Essay

While significant attention is given to the potential historical origins of King Arthur, the Arthur who lingers in pop consciousness is the product of the romantic imaginings of literature and poetry. It is frequently asserted that he is a product of a medieval zeitgeist, which needed a heroic figure to represent the values of its culture. Arthur is a rather complex hero in the wake of the Greek heroes that preceded him, a medieval modern who stands in stark contrast to his classically antiquated predecessors. The tales identify Arthur as the bastard child of the lady Igraine. Merlin used his skills to help Uther Pendragon come to her under the guise of Igraine’s husband the Duke of Cornwall, and Arthur was the fruit of that adulterous union. Merlin takes Arthur and raises him, and when he reached manhood he becomes King of Britain. Arthur’s ascension is detailed as having begun with an embrace with his destiny by obtaining the sword Excalibur. Depending on how the tale is told, he either obtains it by pulling it from a stone or anvil, or it is given to him by the Lady of the Lake. Either way, this dramatically proves his worth despite numerous challengers to the throne. As King, Arthur takes a wife and assembles a gallery of knights who convene around a round table and go on quests in search of the legendary Holy Grail, as well as unifying the various geographical and political factions of Britain as one kingdom, governed under the utopia capital of Camelot. However, despite his desire for peace and prosperity for Britain, the foibles and flaws of his peers continue to disappoint him. Eventually he faces his own mortality, when he is challenged by his own son Mordred for the right to rule, despite his desire for a peaceable settlement. Mordred is defeated but Arthur is fatally wounded and dies. Arthur exemplifies a hero who struggles with duality. Despite the mythic stature accorded to him, his ability to command respect, his strength of courage and his fair-minded sense of justice makes him an exemplar of pure Christianity: liberal in disposition and democratic in temperament, but he is also rather arrogant and inflexible. His crusades also place him in a position in which he frequently neglects his responsibilities as a leader, despite having fought for the right to rule among petty bickering would-be monarchs. The Queen he romanced is left neglected during his adventures, and she betrays him through adultery. His cunning and courage is remarkable, but is witness only abroad never to be put to good use within the kingdom. The Greek heroes are significantly darker, which sets them apart from the rather romantic and idealist tone of Arthur’s mythical self-journey. Consider for example, Achilles: Like Arthur, he was born of an arranged union. Zeus was afraid of a prophecy which told that a child of Thetis would overthrow him, and thus provided for her to conceive with a half-mortal child. Whereas Arthur’s parentage was largely a design of Merlin’s to obtain a new protà ©gà © to raise, while Achilles’ parentage was a political one, insofar as the machinations of gods can be political. Additionally, Achilles’ alienation from humanity is more pronounced, because he is literally ‘less than human’ in his partial divinity, though this divinity makes him cruel rather than virtuous, a far cry from Arthur. The superhuman Beowulf is also comparable to Arthur. While he is able to defeat monstrous opponents in single combat, he parallels Arthur as a monarch whose grim demeanor softens as he ages, and concludes his life with a confrontation with his own mortality. He emerges victorious but fatally wounded, ready to embrace death with the same kind of acceptance that Arthur does.

Analysis of Aluminium †Analytical Chemistry Essay

Introduction Atomic Absorption relies on the principle that each atom absorbs light of a particular wavelength, and hence at that wavelength the quantity of that absorption is proportional to the elements concentration. The first technique used in this analysis was Electrothermal Atomisation, known as Graphite Furnace-AAS. The components of a Graphite Furnace AAS are similar to that of Flame AAS, they more or less use the same components, with the exception of the component used to heat the sample. For a Graphite Furnace Atomic Absorption Spectrometer the schematic is: Hollow Cathode Lamp → Graphite Tube → Monochromator → Detector The Hollow Cathode Lamp (HCL) is a selective/specific light source made from the element of interest. The HCL contains an inert gas which is usually Argon or Neon. Ionization of the inert gas takes place, which causes acceleration of the gas into the Cathode. This causes the metal atoms (Al) of the cathode to sputter into the gas phase. The collision of sputtered atoms with the Argon atoms or electrons, excite the metal to higher energy levels. It is the process of decay that occurs which causes excited electrons to emit light in their return down to lower energy levels which is shone through the sample in the Graphite tube. Ar + e- → Ar+ = 2e- M(s) + Ar+ → M(g) + Ar M(g) + Ar+ → M*(g) + Ar M*(g) → M(g) + hv (light) The monochromator isolates the absorption line obtained from the sample, giving an accurate depiction to the detector. While the detector measures the intensity of the light and produces an electrical signal equivalent to the intensity. The sample is injected directly into the graphite tube, which is then heated. The tube provides thermal energy enough to break bonds within the sample and produce free atoms of the analyte. The sample undergoes three stages of heating, which are all imperative to the analysis of the sample: Step 1: drying of sample (which is the removal of water and the solvent) happens at approximately 125 degrees celsius. Step 2: ‘ashing’ of organic matter (removal of organic and inorganic material, essentially any matrix) happens at approximately1300 degrees celsius. Step 3: Vaporization of analyte atoms (free analyte atoms in light path) – happens at approximately 2400 degrees celsius Step 2 is important for preventing Spectral Interference in the analysis. Due to the fact, there is no flame used in this particular technique there is no combustion products, hence there improved sensitivity of between 10 and 10^3. Method 1. From stock standard of Aluminium provided, prepare concentrations of 10ppb, 20ppb and 50ppb. 2. Add 2ml of 10% HNO3 to each standard and make up to volume in polycarbonate flasks using Ultra High Purity water (UHP water). 3. Prepare a blank: 2ml of 10% HNO3 diluted with UHP water. 4. Take 50ml samples of tap water from: a laboratory sink, a tap in C block, a sink from alternative lab. 5. Add 1ml of 10% HNO3 to each sample as a preservative. 6. Ensure all solutions are mixed well before loading into carousel. 7. Load carousel in order of: blank, then set of standards filling all inner positions. 8. Load samples into outer carousel using small cups. 9. Run analysis. Optimising the GF-AAS is important to ensure maximum efficiency of detection. This is done via running the standards over a range of pre-treatment and atomisation temperatures to determine the conditions for maximum response. Running the blank is important in establishing the base-line reading – this acts as a correction factor, as you can determine how much absorption you find in your samples as a result of the solvent.

Graduation Speech Avoiding College Debt - 895 Words

Avoiding College Debt From elementary school career days to college visits in high school, the importance of continuing our education is drilled into our heads from day one. What the adults who so graciously give up their time to speak at career day fail to mention is enormous cost of college. So, what is a parent to do, dash their child’s dream of becoming a doctor, or find a way to make it happen? We all know a college education comes with a high price tag, but with all the advantages like high income potential and job stability, the investment is well worth it. With time and effort a college education does not automatically come with tremendous debt. Advantages of a College Degree Meeting new people, taking interesting course and the added bonus leaving away from mom and dad, just some of the things a new college thinks are the advantages of going to college. True is while those are all great life experiences, there are true advantages to earning a Bachelor’s Degree or higher. First of all is that college graduates have higher annual incomes and greater lifetime earning potential than non-graduates. According to the United States Census Bureau â€Å"median annual salaries for college and advanced degrees : PhD’s, $100,000 or more; master’s, $63,000; bachelor’s, $55,700; associate’s, $42,000; high school diploma, $32,500.† (collegeatlas.org, 2015). The potential lifetime earnings of college graduates are,† bachelor’s degree holders earn about $2.3 million over theirShow MoreRelatedCase Studies13817 Words   |  56 Pagesregisters members for $30, who are then allowed to bid on exceptional deals. In an effo rt to stand out from the crowded field of online bargain sites, Fashionforward! reached out to the local community in search of help marketing their company to college students. Part of this effort included hiring a student intern, Carly LeBlanc. At that point, Fashionforward! had no formal marketing strategy for targeting consumers. 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