[Pivotal Concepts of CAPM]
Introduction
No matter the attractiveness of an investment, an investor must always be ready for risks arising. In which case, it is always impossible to do away with entire risk. The only an investor can manage the impact of this risks is by coming up with an investment that yields a rate of return that can cover the risks. This would help in covering the investors whenever they take on risk. As the result, the need for a capital asset pricing model (CAPM) sets in as an important tool for the investors. The CAPM helps in calculation of risk associated with investment and the return that should be expected. This presents the platform for investors to compare between risk and rate of return; thereby, make a crucial decision regarding the best investment. Investors will always go for the investment options that assures them of higher return but with lower risks. In exploring the relationship between risk and return, two types of investors comes out; the risk averse and those that fear risks. The risk averse always opts for the investments with higher risks while the other group go for investment with lower risks. The CAPM, about the intuition that a positive risk-return tradeoff, points out that the expected return from any asset usually come as a positive function on a single variable: the market beta of the asset (Chou, 2012, 360). In which case, beta is defined as the covariance obtained for market return and return of the asset. However, it comes out that an investment with higher risks normally has a higher return than that with the lower risks. The relationship therein, between return on risk and risk is well articulated in the CAPM as pivotal concepts.
Intuitively, this report explores the way in which the ideas, frameworks and theories surrounding the relationship between the returns on a security and its risks are reflected within the capital pricing asset pricing model. This will present a platform for determining how the capital asset pricing model works for the investors; regardless of whether they are risk averse or not. In order to achieve this, the report reviews the fundamental features of the CAPM followed by the reviewing journal articles that are related to the tool.
Capital Asset Pricing Model (CAPM) is one of the financial models founded by Sharpe (1970, 46). This model becomes very necessary in financial world because it is impossible for investors to curb all the risk be investing. In simple terms, we can say that CAPM is a model that shows how risk can influence the expected returns during investments. Once this relationship between risks and expected returns has been analysed then it applies in pricing of risk securities. The formulae used for CAPM is;
Expected security return= Risk return + Beta (Expected market risk premium)
r= Rf + Beta (mm-Rf)
Where r is the rate on a security
Rf is the rate of risk of a risk free investment such as cash
Rm is the return rate of the appropriateness asset class
There are fundamental features of this model necessary to understand for proper application.
Security market line is also a very important component of the CAPM. The word characteristic line means the same thing. This is a linear relationship between the risks and the returns of an investor. Investors will always want high-expected returns for high risks.
The other paramount feature in this model is the risk premium. A risk premium is a form of reward or compensation for the investing in a high-risk project (George, 2008, 89). For example if an someone invests in a company which has just started and does not have a lot of public approval then that person should be given more returns as compared to the one who invested in a well established company.
Asset pricing is another also core in CAPM. The value of an item relates directly to the risks and delay of its payments. When the price is low then the anticipated rate of return is high. The price of an item equals the expected returns. It is important to calculate the price of each item before making investments to avoid impending losses and optimize on the profit.
Efficient frontier or tool showing a set of optimal portfolio (collection of financial assets) that maximizes on the returns for a given risk is also a feature of CAPM. Investors will always go for portfolios that lie above the efficient frontier since they offer bigger rewards for risks. Usually optimal portfolios that lie above the efficient frontier are highly diversified.
In conclusion, CAPM is a model that any investor would like to use before giving out there money. The main component of the model includes risk (systematic or unsystematic), security market line, efficient frontier, expected return and the risk premium. The advantages of this model are that it is easy to use and that it provides diversified portfolios to mention but a few.
Literature Review
The model of CAPM originated from the establishment of “theory of risk-return portfolio” provided by Markowitz (1959) and Tobin (1958). The theory of risk-return portfolio acts on the basis of utility model formulated by von Neuman and Morgenstern (1953).
The mean-variance efficiency determined for the market forms the basis for the CAPM. In which case, when the market portfolio is determined to be efficient, then this implies that there is existence of a positive linear correlation between market betas and expected returns; not any other variable can be effectively used to provided information about the expected return of the portfolio other than betas.
CAPM model is a representation of a great advancement in financial theory; thereby significant amount of research, both negative and positive critics, have surfaced regarding the same. Authors, Chen and Aug (2007), carried out a research involving x-ray of CAPM evolution over period, documenting on the chances of adapting the model to the emerging needs of the contemporary economy and to cut down on investment risks. Together with Chen et al (2008, 45), the researchers came up with a study that explores the relationship between the return expected on investments and evolution of risk in the market.
The most serious negative assertion regarding the validity of the model is best seen in Roll’s critique. Roll’s asserts that there is not one correct test of the concept that has been seen in the literature; besides, there is no future chances that a test like that can come to existence. His research indicates that the theory articulates that the market portfolio is always mean-variance efficient. In which case, CAPM is not in any way tested when portfolios containing subset of assets are utilized as reflection of the true market portfolio.
Many attempts have been made to utilize historical return rates of securities and market index in testing for the aspects related to CAPM. According to Chen (2007) most of these studies have been successful used in ASE for empirical testing of the model. These studies include Lintner (1965), Miller and Scholes (1972).
The assumptions used when utilizing CAPM mainly arise from the major concept that the investors would always want a strategy that allows them to maximize profit they get from their investments. Besides the risk aversion aspect, all investors have the same goal about the return they get from their securities. The returns from the securities are normally characterized by a normal distribution, which follows the aspect of homoscedasticity. In addition, the risk free rate of return is also available for the investors and it presents them with the opportunity to borrow funds or lend funds at the given rate of return with no risk involved. Another significant assumption is that there is absence of taxes or other forms of restrictions, which can result to market imperfections.
In their view, Sharpe (1964) and Lintner (1965) while making several assumptions, augmented Markowitz’s framework of mean-variance by coming up with a relation meant for excess of expected returns; the risk free rate less the returns. These returns is supposed to be equal to security return coupling the multiplication of excess market portfolio and the coefficient beta. During the analysis, the coefficient of beta is taken as the measure of risk. a large fraction of tests performed on CAPM have been done through estimation of the cross-sectional relation existing between the mean return realized on assets and the betas per se. This is usually done over a given period interval followed by making a comparison of the CAPM’s estimated relation.
Whenever there is riskless asset, Black (1972) recommended that the utilization of zero beta portfolio, RZ (which is represented by cov(RZ, Rm) = 0 ) would much significant. In this case, the zero beta portfolio would be considered as a representation of a proxy for riskless asset (Black, 1972, 34).
The model, zero-beta, gives specification on the expected return equilibrium on asset as a basis of market factor as defined with the market portfolio return Rm , besides a beta factor that is also represented by portfolio of zero beta. The portfolio, of zero-beta, is considered to be of minimum variance nature and it has not correlation to the market portfolio. The zero-beta can also be seen as playing the same function as that played by the risk-free rate of return depicted in the Sharpe-Lintner model. In a case where the intercept term is found to be zero, the CAPM will be consider to apply. The process of carrying the test besides examination of CAPM used traditionally paves way for zero-beta model verification in the ASE.
The initial process of testing CAPM occurred in two stages. BJS (1972) performed estimation of betas through the help of monthly returns obtained for the stocks found in NYSE. This was done besides an equally weighted portfolio of entire stocks listed NYSE.
However, in their research Roll and Ross indicate that a market proxy can almost achieve the state of being MV efficient although the gradient associated with OLS regression run for the expected returns on the portfolio betas is zero. Countering this argument, Kandel and Stambaugh stipulate that an almost perfect OLS for the average and betas achieved from calculation relative to a proxy, which is highly inefficient. Even though this literature use multiple evidences, neither of the two teams was able provide verification regarding if the frontier of the minimum-variance exhibits a weighted portfolio that is positive. As the result, they do not convince us beyond doubts that the results achieved are accurate when there is trueness of the CAPM; that is when weighted market portfolio is mean variance efficient. In addition, reliable proxy portfolio should be composed of positive weights. In their case, Roll and Ross were able to come up with an instance featuring a proxy portfolio that has positive weight, but the proxies found by Kandel and Stambaugh misses positive weights.
Results
The following is the table showing the calculated inputs from the data downloaded:
The table shows that the extent of correlation between two portfolios, both making up the portfolio, was 0.345 while the expected return of portfolio was 11.67 and the standard deviation 6.59724.
The 0.345 calculated for correlation shows the type of correlation between the was positive but not perfectly correlated.
The figures obtained above helped in building the efficient frontier curve, after finding the SD portfolio and Expected Return of Portfolio (%) as shown in the excel spreadsheet attached. The following is the efficient frontier curve:
Observations made from the curve are as follows:
The frontier shows that portfolio marked with a black star (risk of 7 and return of 15) dominates all the other efficient portfolio depicted in the efficient set with optimal risks and optimal returns
The rest of the points along the curve (representing other efficient portfolio) indicate some other risky assets. Lower than the black starred point the investor experiences low returns with low risk and on the upper side, the investor experiences high risk with high returns.
Discussion
The result depict a better way of efficiently diversifying the two portfolios. Of all the two stocks, the first portfolio with beta of greater than 1 has the highest monthly return percentage of 2.45% against that registered by the portfolio, 0.785%. This called for the need of building efficient frontier curve, which helped in determining a better combination of the two stocks that would help in optimizing the returns while maintaining lower risks. Intuitively, this helped in efficient diversification of the stock through plotting the efficient frontier, for expected return against standard deviation of the minimum-variance portfolio. Even though the correlation is positive, the results shows that the two are not perfectly correlated thereby resulting to the efficient frontier curve that curves to the left. The line shown on the graph shows the tangent line representing the optimal risky portfolio. The black starred point on the graph assures the investor of better risk premium for given lower risk amount as compared to any other point on the graph (Bacon,2013, 57). . The result shows that at a lower risk of 7, the investor is assured of an optimal return of 15. Further, the efficient frontier curve also proves useful by showing the investor the risks and corresponding return associated with other points, above and below the optima risky portfolio.
Justification and Discussion
In this part of the report, the main objective was to investigate how treynor ratio, appraisal ratio and sharpe ratio contributes to security analysis of mutual funds. Investing in mutual funds in the market normally requires the investor to evaluate the risks associated with the investment. This occurs because the returns from the mutual funds rely on the quality and quantity of the associated risks. In order to evaluate the risks effectively, an investor requires to keep in track of the three ratios associated. These ratios include appraisal ratio, treynor ratio and sharpe ratio. Determining the appraisal ratio help the investor in accessing the picking ability of the investment done on the mutual fund. The ratio gives an investor clue on the performance of the mutual fund on the basis of picking ability of the fund and the associated risks. The process of calculating appraisal ratio involves comparing the mutual fund’s alpha to the residual standard deviation. Further, Treynor ratio helps in measuring the excess earned on the return on a riskless investment per each unit of market risk (McMillan, 2011, 68). The investor can use this ratio as a measure of the adjusted risk on the return, with reference to the systematic risks. The following is the formula for calculating Treynor ratio:
(Average Return of the Portfolio - Average Return of the Risk-Free Rate) / Beta of the Portfolio
In this ratio, beta of the portfolio is used to represent the unit of market risks. Sharpe ratios represents another risk –adjusted measure for the excess return received as a result of experiencing an extra volatility in a riskier market. This is the most critical ratio, among the other three, since it controls the compensation for any additional risks experienced with a risk free-asset. The following is the formula for calculating sharpe ratio:
S(x)= rx – Rf)/StdDev(x)
(whereby x is the investment,
rx is the average rate of return of the investment,
Rf is the best available rate of return of the risk-free security,
stdDev(x) is the standard deviation of rx.
The report uses the ratio to analyse the security associated with investing on Vanguard High-Yield Corporate Inv (VWEHX), a US-based mutual fund, for a period of 10 years. The corresponding fama-French (FF) factors are also used to help in calculating the ratios Fama & (French, 2006, 2170).
Results
CAPM
The following is a table showing calculation of the CAPM, Beta and alfa
Appraisal ratio = alpha/standard error = 0.20/0.18 = 1.11
Fama French
The following is a table showing calculation when the Fama French was used as the benchmark model
Sharpe ratio
The following is a Sharpe ratio table showing the results obtained for the sharpe ratio
The mutual fund witnessed a sharpe ratio of 0.172 after resulting to an average excess return of 0.457 and standard deviation of 2.660.
Treynor ratio
The treynor ratio = Average Return of the Portfolio - Average Return of the Risk-Free Rate) / Beta of the Portfolio
(0.585 – 0.128)/0.455
= 1.005
Consequently, the mutual fund experienced a ratio of 1.005 over the period of 10 years.
The following is a table showing the calculation of Treynor ratio as obtained from excel:
Conclusion
Using the three rations, treynor ratio, sharpe ratio and appraisal ratio led to the achievement of the objectives, which was to analyze the security associated with investment in Vanguard High-Yield Corporate Inv (VWEHX). The results show that the mutual fund guarantees its investors of enough security even in the midst of risks. The calculation of the sharpe ratio, resulting to 0.172 shows that the mutual fund not good when it comes to generating returns on the basis of risk adjusted (Smith & Shawky, 2012, 96). The ratio shows that the mutual fund generated returns with reference to risk-adjusted. However, the use of Treynor ratio upholds good image of the mutual fund based on the generated returns. The mutual fund experienced a treynor ratio of 1.005 shows that it was able to generate higher return on the risk-adjusted basis. Determining the ability of the investment on the funds to pick involved the use of the appraisal ratio. The ratios show that the mutual fund assures the investor of security of their funds as shown in the resulting value of 1.11. 1.11 indicates a better performance from the manager as it shows that the mutual fund had higher chances for picking even in the presence of the unsystematic risks (Cernauskas & Tarantino, 2011, 145). Consequently, considering the ratios for the 10-year period, an investor should consider choosing this mutual fund because the security analysis favors its operations.
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