Capital Asset Pricing Model & Arbitrage Pricing Theory Model - Assignment Example

Capital Asset Pricing Model & Arbitrage Pricing Theory Model Introduction During the roaring twenties, American affluence fuelled the rise of the stock market, and individual investors played the counters like high rollers in a casino; the focus then was on choosing the lucky stock that would meteorically rise and bring a windfall of profits. The selection was then a trial-and-miss affair, and this unregulated trading brought the Wall Street crash that triggered the Great Depression of the thirties. In the forties, the approach to stock investing became more studied and academic pursuits sought to develop a logical and organized way of choosing stocks. Markowitz’s theory of portfolio investment drew attention to the determination of not single stocks, but groups of assets that are designed to minimize risk while maximizing returns. It was in this background that the Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT) came to be. 1.0 CAPM and APT: When and how these theories started Capital Asset Pricing Model and the Arbitrage Pricing Theory both trace their beginning to the Modern Pricing Theory of Harry M. Markowitz. Markowitz, in the late 1940s, was a graduate student in the University of Chicago. Faced with a need to find a topic for his dissertation, a casual conversation gave him the idea to develop a mathematical method for stock investing. In going through the reading list given to him by Professor Marshall Ketchum, his interest was piqued when he read John Burr Williams’ Theory of Investment Value. According to Williams, the value of a stock should equal the present value of its future dividends, which is presently known as the dividend discount model of stock valuation. Markowitz noted that future values are uncertain, and if one were to follow Williams’ rationale then investors would only be investing in one stock of which dividends they know. Markowitz rationalized that this is not the way investors behaved, because they tended to diversify their holdings, invest in several stock at a time, in order to minimize the risk. Because of this, Markowitz developed his now famour modern pricing theory, wherein he inputted the risk factor as a determinant of the expected returns on a portfolio. His article was published in 1952 “Portfolio Section”. For his body of work which introduced the concept of risk in portfolio selection, he was awarded the Nobel Prize in economics in 1990 (Markowitz, 1990). By the 1960s, several financial economists were intrigued by the idea of a systematic risk which might mathematically be tied in with Markowitz’s portfolio theory. The result, the Capital Asset Pricing Model or CAPM, was developed separately by William F. Sharpe, John Lintner, and Jan Mossin in the mid 1960s, who just happened to come up with the same idea at about the same time. The CAPM quickly gained popularity among investment advisers and finance practitioners and professionals because of its simplicity and the clear theoretical appeal of a single risk coefficient (beta). Theorists, however, were quick to point out that the CAPM could not explain reliably empirical data, and adjudged the CAPM insufficient as a predictive model. In an attempt to improve the idea of the CAPM, several multiple-variable models were developed, one of which is the Arbitrage Pricing Theory, or APT, which was developed by Stephen A. Ross in 1976. Ross’s model was similar to the beta, only it had multiple “betas” corresponding to the several variables resorted to in order to fine-tune the model. (Le Compte, 2008). 2.0 The Emphasis of the Models (CAPM & APT) Modern managerial finance theory operates on the assumption that the primary goal of the firm is to maximise the wealth of its stockholders and that quite automatically gets translated into maximisation of the price of the firm’s common stock. It must be noted that the traditional objective of a commercial firm, as stated by economists, of maximising profits is no longer sufficient for any business to survive in the modern world of extreme competition. The concept can be fully appreciated when we realise that profit maximisation is basically a single-period or a short term goal. A firm may attempt to maximise its profits within a given time period at the expense of its long term profitability and still manage to reach its short term goal. As against this concept, wealth maximisation of stockholders is essentially a long term goal because the stockholders are essentially interested in both immediate and future profits. Maximisation of stockholders’ wealth is quite obviously the preferred option of majority of stockholders because it considers: Wealth for the long term Risk and uncertainty inherent in any long-term projection The timing of returns – This is of special significance to firms as they would be able to plan their future investment and expansion plans in a better way if there is some sort of surety about the future cash inflows Stockholders’ returns – This assumes substantial importance as stockholders are essentially interested to know how the projected cash flow would be timed since such a forecasted schedule helps them to manage their personal expenditures in a more planned manner (Shim and Siegel 2007) Maximisation of shareholders’ wealth brings us to the all important and most crucial issue of the trade off that exists between risk and return. It is common economic phenomenon that investments that yield higher returns carry higher levels of risk too. The higher return is in fact the added incentive to an entrepreneur to bear the associated risk, or, as some would like to place it, the price for undertaking a higher level of risk. Thus, a finance manager is perpetually faced with the dilemma of counterbalancing the two diametrically opposed forces of risk and return in an attempt to maximise stockholders’ wealth. 3.0 The Tenets of the Modules 3.1 Capital Asset Pricing Model The basic assumption of capital asset pricing model (CAPM) is that investors demand additional incentive in the form of returns if they are asked to bear additional risk. CAPM is an economic model that attempts to value financial assets as stocks, securities and even derivatives or real assets as plant and machinery by correlating associated risks and expected returns. Thus asset pricing models seek to establish the determinants of financial assets expected rates of return. Classic asset pricing models, such as that theorised by several famous economists (Sharpe 1964), (Lintner 1965 ) and (Black 1972) predict that an assets expected return should be positively related to its systematic market risk. The CAPM states that the expected returns that the investors will demand is the sum of return on risk free security plus a risk premium. Any security is accompanied by two types of risk – diversifiable risk and non-diversifiable risk. Diversifiable risk is also referred to as controllable or unsystematic risk and represents that risk which can be controlled or reduced through diversification. This type of risk, quite naturally, is specific to a particular type of security. Examples of such types of risk are business risks, liquidity risks and default risks. Non-diversifiable risks, also known as non-controllable or systematic risks, arise from forces beyond the control of a particular firm and are therefore not specific to a particular security. Purchasing power risk, interest rate risk and market risk fall under this category. Non-diversifiable risk is estimated relative to market portfolio, i.e., the risk associated with a diversified portfolio of securities and is measured by ‘beta’ coefficient. CAPM associates the risk estimated through ‘beta’ to the level of expected rate of return on a specific security. This method, also known as Security Line Model (SLM), is deduced as: rj = rf + b (rm – rf) where, rj = required rate of return on security j rf = return on a risk–free security as T-bill rm = expected return on market portfolio (as Dow Jones 30 Industrial) b = beta, an index of non-diversifiable risk Beta is a measure of the volatility of the selected scrip with respect to an average security. b = 1 implies the security is of average risk. Thus if a security has b = 0.25 it implies that the said security is only 25% volatile as compared to the average security while b = 3 implies that the said security is three times as volatile as the average security. b (rm – rf) is the additional return that is necessary to motivate investors to undertake the additional risk. So, the expected return on a specific security (rj) is equal to rate of return on securities that have no risk as T-bills (rf) plus a risk premium for investors who are interested in such risky securities. It is quite obvious that the higher the value of beta, i.e. systematic risk, the higher will be the premium demanded by investors. If rf = 10% and rm = 15% then for various levels of beta, the expected rates of returns would be; For b = 0 (risk free security) rj = 10% + 0(15% - 10%) = 10% For b = 1/2 (half volatile as average security) rj = 10% + 1/2 (15% - 10%) = 12.5% For b = 1.0 (market portfolio) rj = 10% + 1(15% - 10%) = 15% For b = 2.0 (twice as volatile) rj = 10% + 2(15% - 10%) = 20% Based on Mertons ICAPM (Merton 1973), and researches done by Fama and French in 1992 and 1993 (Fama and French 1992), (Fama and French, Common Risk Factors in the Returns on Stocks and Bonds 1993), it can be safely concluded that their findings of higher returns for high book-to-market stocks and low capitalization stocks reflect compensation for risk. That is, those stocks are expected to earn higher rates of return because they are riskier. It is prudent at this stage to highlight the assumptions that underlie CAPM model. These are: Investors as a class are averse to risk taking. All investors have same set of information at any particular point of time. Returns on assets follow normal distribution. There always exists a risk-free asset and investors can lend or borrow unlimited amounts at a constant risk-free rate. The number and volume of assets are fixed for a particular period. These assets are perfectly divisible and they are traded under perfectly competitive conditions. Since perfectly competitive conditions exist in asset market, there is no friction in such a market. Moreover, information is not only available free of cost but is also equally available to all participants in the market. The economic implication of such an assumption is equality between borrowing rate and lending rate. Taxes, regulations and restrictions on short selling do not exist in an asset market. It is rather difficult, if not completely impossible for all these conditions to be simultaneously fulfilled. However, CAPM remains to be one of the most popular models of pricing capital assets. 3.2 Arbitrage Pricing Model CAPM assumes that required rates of return are dependent on a single factor – the stock’s beta. As it can be safely be stated at this juncture, this is definitely not the case in real life. Arbitrage Pricing Model (APM) refutes this incorporate numerous risk factors and the expected return on a stock or a given portfolio is determined as: r = rf + b1RP1 + b2RP2 + b3RP3 + ... + bnRPn where; r = expected return of a given stock or portfolio rf = risk-free rate bi = the sensitivity of the returns on stocks to unexpected changes in economic forces represented by ‘i’ where i ranges from 1 ... n RPi = the premium demanded by investors for unexpected changes in the ‘i’th economic force n = number of relevant economic forces For the APT, it is theoretically possible to have as many explanatory values as are needed to suit the particular circumstances of the investor. However, Roll and Ross have listed five relevant (usually necessary) economic factors the indicators of which should included in the model, and they are: Expected changes in inflationary rate Unexpected changes in inflationary rate Unforeseen changes in the level of economic production Unanticipated changes in default risk premium Unexpected changes in the structure and condition of long-term interest rates (Roll and Ross 1980) 4.0 The Relationship between CAPM AND BETA The CAPM is a model developed around the idea that the sum of all systematic risks may be described by a single factor coefficient. That factor coefficient is the beta of the stock, which is taken as the proxy for the risk of the stock. The beta relates the premium of the stock over the riskless instrument to the premium of the market return over the same riskless instrument. The CAPM is thus built upon the concept of the beta and revolves around the concept of risk. Without the beta, the CAPM will lose its significance. (Singh 2008) 5.0 Benefits of CAPM and its limitation and benefits of APT AND its limitation The benefit of the Capital Asset Pricing Model is its ease of use and simplicity of approach, a factor that appealed greatly to most stock market analysts and professionals working in the field. Stock market move rather quickly especially during trending markets, and so stock analysts and investment advisers as well as investors themselves preferred a quick way by which to gauge if particular stock has already been overpriced by the market, or is still undervalued. As the weakness of the dividend discount model shows, future dividends are difficult to estimate, and besides it was easier to gauge the performance of the stock against the general market index through the use of benchmark indicators such as the beta of the stock as indicator of its relative performance. CAPM is, however, limited because it merely presents a single factor, that of the market in general. Use of the single factor generally is less sensitive to the other forces influencing the market (Copeland et al., 2005). Where the CAPM is limited, the APT is capable. The APT is essentially a multifactor model, with several “betas” or factor coefficients linking each indicator to the overall required rate of return. Since several factors are each described by its relative movement to the result, the APT becomes a more sensitive model than the CAPM and is thus a great improvement over the latter as a predictive model. However, APT has it limitations also, in that its accuracy is only as reliable as the accuracy of its many betas or factor coefficients. Where these estimates are wrong, the result is rendered wrong, and since there are several factors it becomes more difficult to ensure the correctness and reliability of the resulting expected rate of return (Grinblatt and Titman, 2002). 6.0 Similarities, weaknesses and differences of CAPM and APT The similarities between CAPM and APT are that they are both asset pricing models that employ the concept of risk in the process of determining the rate of return of the stock based on the prevailing indicators. Both these models are refinements of the Markowitz efficient portfolio. Both these models involve the same mathematical form; in fact, if the APT were reduced to only one explanatory variable and one factor coefficient (Harvey, 1991). The difference between CAPM and APT, on the other hand, is their comparative complexity of resolution, considering CAPM has only one factor and the APT has several factors. On the other hand, APT having several factors increases its reliability as a forecasting model, making it a more accurate predictor than CAPM. The weakness of CAPM and APT lie in the fact that they both require estimation of the betas or factor coefficients. There is no constant relationship among the required return being computed and the different indicators, since economic variables are dynamic and these relationships are constantly being redefined. Therefore the accuracy can only be approximated depending upon the quality of the estimation of the factor indicators (Dimson et al., 2002). There are some weaknesses that affect econometric models of financial markets in general. In the practical application of both the CAPM and APT, the ideal variables are not easily obtainable and proxies have to be resorted to. Although asset pricing models aim at explaining cross-sectional variation in expected return, researchers have been forced to use realized return as a proxy for expected return in tests of these models. The use of realized returns implies that such tests are conditioned on the joint hypothesis of rational expectations, in the sense that the average realization is a good proxy for expectation and that the null asset pricing model describes the relation between expectations and firm attributes. However, realized return may not be a perfect proxy for expected return. First, noise in realized returns is likely to be large as pointed out by several researchers (Blume and Friend 1973), (Sharpe, New Evidence on the Capital Asset Pricing Model: Discussion 1978). Second, realized returns may be poor estimates of expected returns if information surprises do not cancel out over the period of study (Froot and Frankel 1989), (Elton 1999). Third, realized returns may also be noisy and biased estimates of expected returns due to complex learning effects. More recently, some economists have argued that rational learning can generate price paths that, ex post, yield "value-" and "momentum-" type effects in stock returns (Lewellen and Shanken 2002), (Brav and Heaton 2002). Furthermore, the CAPM model does not make much economic sense in investing in individual securities since it is quite possible to accurately replicate the reward-risk equilibrium of any security through an appropriate mixture of cash and prudently selected asset class. This could be one of the main reasons as to why those who follow scrupulously follow the tenets of CAPM rarely invest in securities, and instead opt for low cost index funds. 7.0 Practical Situation In order to illustrate the use of the CAPM and the APT in a practical situation, a hypothetical portfolio shall be assumed as belong to a hypothetical investor. (Because of confidentiality reasons, actual investors and portfolios could not be employed in this academic exercise). It will be assumed that the investor decides to invest in a stock for his portfolio, and the results are the values X and Y in the second and third columns incorporated in the table. 7.1 Application of the CAPM The computation of the stock’s beta is computed by the statistical linear regression formula, specifically the slope of the regression equation. The slope’s formula is given by the equation: . In the foregoing formula, m is the slope (the beta), n is the number of samples, x is the independent factor (in this case, the stock market return), and y is the dependent factor (here, it is the individual stock’s return) (Draper & Smith, 1998). In the following table, beta as denoted by the slope was computed as: 30 (2,942) – (92) (142) m (Beta) = ------------------------------------ 30 (2,466) – (92) 2 m (Beta) = 1.1672 or approximately 1.17 Table for computation of beta coefficient using linear regression slope Assuming that the risk-free rate (given by the rate of return of the benchmark government security) is equal to 4% and the general market return is given as -5% (meaning that the market index is expected to fall by 5%), then the stock is expected to have a rate of return of: rj = 4% + 1.17 (-5% – 4%) rj = -6.53% Thus in this simple example, the stock has a beta of higher than 1 and would tend to outperform the market when the market is bullish, but where the market is expected to incur losses or negative returns, the stock is expected to incur a higher rate of loss than the market. It also shows that the market beta is computed by linear regression and the slope of the best fit regression line is taken as the calculation of the beta. A diagram of the regression line of best fit is shown below. source: http://upload.wikimedia.org/wikipedia/en/1/13/ Linear_regression.png 7.2 Practical Application of the APT In drawing an example of the APT, it will be necessary to assume certain macroeconomic growth rates to show the workings of a multifactor model. Assuming the same stock as presented in the CAPM problem above had a rate of return that was determined to be sensitive to the GNP, inflation rate, and interest rate of fixed income instruments. It is to be assessed in an economic environment where the premium demanded by investors over the GNP growth rate is 8%; that over the prevailing inflation rate is 3%; and that over the interest rate is pegged at 6%. It was also determined, through a regression method similar to that of determining the beta illustrated in the preceding page, that the factor coefficient of the stock against the GNP growth rate is 1.2, that of the stock against the inflation rate is negative 0.8, and that of the stock against the interest rate is 0.3. The APT model would thus be resolved as: rj = rf + b1RP1 + b2 RP2 + b3RP3 + ... + bnRPn rj = 4% + 1.2 (8%) + (-0.8) (3%) + 0.3 (6%) rj = 13% Immediately evident in this simple exercise is the relative complexity of the APT compared to the CAPM, a major consideration why CAPM continues to enjoy a strong following among investors and practitioners. Conclusion This brief discussion on the CAPM and the APT draw attention to the inadequacy of econometric models in capturing the nuances of market volatility. However, this is also true of most business models that pertain to economic markets, and while they are not perfect, they provide some basis for decision-making. 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