The Capital Asset Pricing Theory and the Arbitrage Pricing Theory - Assignment Example

CRITICAL ANALYSIS OF CAPM AND APT Introduction In the field of finance, as with all economic activity, there is always an element of risk that creates the potential for growth and development. Generally, investors are averse to unreasonable or indeterminable risk (Block & Hirt, 2006). However, if no risk exists, then prices have no variation and no margin is created by which profit can be made. The idea of investing in the financial market is to purchase the asset while the price is low, and to sell when the price appreciates. The seeming arbitrary movement of prices of assets, such as stocks, has been at the center of most research, in an attempt by investors to try to forecast if prices will be going up or down. Various mathematical models have been developed in order to determine the true value of financial assets. Two of these models, the Capital Asset Pricing Theory and the Arbitrage Pricing Theory, have emerged as two of the most widely accepted methods by which returns may be determined and valuation models developed from these returns (Malevergene & Sornette, 2007). However, as much as they have had loyal followers, there have also been criticisms forwarded by researchers on the validity of these models (MacKinlay, 1995). The models and the criticisms about them shall will be examined in this study. Emphasis of the Models The models are based on the theory that the basis for choosing assets for investments relies on whether or not they interact with one another, whether they balance each other’s risks, instead of looking at their individual performance when taken individually or in isolation (Scott, 2003). The portfolio theory provides the basis of the CAPM and the APT, and it states that an optimal portfolio is the combination of assets that provides the investor the highest return for the least possible risk level for a specified return (Constantinides & Malliaris, 1995). History of the CAPM and the Arbitrage Pricing Theory The Arbitrage Pricing Theory and the Capital Asset Pricing Model do not owe its existence to a single person or a single effort, but was developed over the years by researchers’ independent inquiry. In the late 40s, Harry Markowitz, a graduate student in economics with a penchant for mathematical processes, was developing his dissertation on the stock market. Originally, he was fashioning his paper after the present value model of John Burr Williams. Halfway through, however, Markowitz realized that Williams failed to account for the impact of risk in his model. His insight into this vital role played by risk analysis in the valuation of portfolio assets led to Markowitz’s now famous theory on portfolio allocation under uncertainty, which was published in the Journal of Finance in 1952 (Fabozzi, & Markowitz, 2002). The Markowitz Efficient Portfolio is an investment portfolio the further diversification of which could not result in a lower risk, given a particular rate of return (Downes & Goodman, 1995). The Markowitz Efficient Frontier, depicted in the diagram on the following page, is the locus or set or all portfolio combinations that will yield the highest possible return for a given degree of risk. It will be seen from the graph that the efficient frontier is tangent to the risk-free asset’s risk-return tradeoff, but does not surpass it. Before the point of tangency, the further assumption of risk tends to increase returns, but beyond the point of tangency further assumption of risk will tend to decrease returns. The point of tangency, where the risk-return tradeoff approximates the risk-free asset, is Markowitz’s efficient portfolio (Reilly & Brown, 2006). Source: http://en.wikipedia.org/wiki/Modern_portfolio_theory On the basis of Markowitz’s study of portfolio risk, several researchers began to build on Markowitz’s modern portfolio theory by studying the relationship of diversification and risk. Independently, Jack Treynor (1961), William Sharpe (1964), John Lintner (1965), and Jan Mossin (1966) contributed to what would, however, eventually be known as the Sharpe-Lintner Capital Asset Pricing Model, or the CAPM (Elsas, El-Shaer & Theissen, 2003). Eventually, the CAPM was adopted by most stock analysts and practitioners, investors, and portfolio managers, citing its ease of use and the clear relationships it draws among risk, return of the stock, return of the market, and risk-free rate (Omran, 2007). On the other hand, the Arbitrage Pricing Theory or APT Model was developed in 1976 by Steven Ross, in his article in the Journal of Economic Theory, “The Arbitrage Theory of Capital Asset Pricing.” In the main the APT is built on the assumption that each stock’s return to the investor is influenced by a number of independent factors. The APT is actually a theory that took off from the fundamental precepts laid down by the CAPM. It states that the risk premium of an asset or a stock is dependent on the risk premium of the factors identified in the tenet (factor 1 to factor n), and the asset’s own sensitivity to each of the factors (an adaptation of the beta concept in CAPM) (Jagannathan & McGrattan, 1995). Tenets of the CAPM and the APT The CAPM is a model in finance which aims to determine the required rate of return of an asset, taking into consideration the non-diversifiable risk termed the beta (ß), the expected return of the market of all assets, and the theoretical risk-free rate of return (Reilly & Brown, 2006). Expressed in mathematical form, the model is described by the equation: where: E(Ri) = the rate of return that may be expected of a stock Rf = the risk-free rate of return βi = the non-diversifiable risk of the stock E(Rm) = the market rate of return In the equation given, the coefficient βi refers to the degree of sensitivity of the asset to the change in the rate of return of the market. The beta βi, or an approximation thereof, may be found by regressing the historical data that correlates the market return E(Rm) and the risk free rate, with the return of the stock E(Ri). The beta is said to be a measure of risk because it defines to what degree the stock’s return will outperform or underperform the market. When a stock’s price moves in tandem with the market index, it is said to have a beta of 1. A stock that moves opposite to the market but to the same degree has a beta of -1. If the stock’s gain over the risk free rate is only a fraction of the market’s premium over the risk-free rate, then the stock is said to have a beta of between 0 and 1. If the resultant gain of the stock is more than that of the market over the risk-free rate, then it is said to have a beta of greater than 1 (Litterman, 2003). The Arbitrage Pricing Theory, on the other hand, is built on the law of one price; that is, two items identical to each other should sell for the same price, otherwise a riskless profit materializes in the form of arbitrage. Arbitrage is a technique which takes advantage of a price discrepancy, wherein identical items are transacted in two separate markets under two different equilibrium prices. When a person buys the item in the market with the lower price, and simultaneously sells the same item in the second item where the price is higher, he incurs a profit without risk. This same principle applies to the market for financial assets and securities. E(rp) = rf + p1 [E(r1) – rf] + p2 [E(r2) – rf] On its face, the formula for APT closely resembles the formula for CAPM, except that the APT provides for several factors while the CAPM relies only on one factor (Armstrong & Brodie, 1994). Relationship between CAPM and BETA The CAPM is a model that describes the relationship among the expected rate of return, the rate of return of the market, and the rate of return of a risk-free instrument. The model is based on the principle that the return on a risky instrument is at a premium above the risk-free rate, and that this premium is a proportion of the premium of the market to the risk-free rate. In effect, the CAPM expresses a relationship between the risk of the stock (denoted by beta) and the return to be realized on it. In the CAPM, therefore, the concept of the beta is the innovation introduced into the field of portfolio theory. While the modern portfolio theory of Markowitz introduced the necessity of factoring in the risk of the asset, that is, the variability of an asset’s return, as an explanatory variable in determining the asset’s gain, it is the beta coefficient as a direct proportion between market premium and asset premium over the risk free instrument that is the distinguishing concept of the CAPM. Stated otherwise, the beta is the central innovation that distinguishes the CAPM (Kane, 2004). Benefits and Limitations of CAPM The most apparent benefit of the CAPM is its relative simplicity and straightforward application. Practitioners appreciate the ease of use of the CAPM on which to base their investment decisions, and the relative dispatch with which the model arrives at a figure hastens the decision-making process that leads to a buy or sell decision in fast-moving markets (Low & Nayak, 2009). On the other hand, CAPM’s simplicity is misleading, according to several researchers, because the model makes too many assumptions which are understood to be impossible at any time. For instance, the model assumes that the market is efficient at all times, whereas in truth not all investors are aware of, nor have they discounted, all the available information affecting the market. Another questionable assumption is that the market portfolio is composed of all possible assets, including real assets; however, at any one time, the return of other assets is not ascertainable for lack of a single exchange or a standard index in the same manner as the stock market. One other assumption stipulates that all investors can borrow or lend at the risk-free rate at all times, something is that known to be impossible as investors have different credit risk profile (Pastor & Stambaugh, 2000). Benefits and Limitations of APT The APT has far fewer restrictions than the CAPM, which has eight assumptions which should be held true if CAPM is to be accurate. The APT, on the other hand, allows the investor to create and manipulate his own model, depending on the nature of the stock. By specifying many factors, the model can be fine-tuned so that any movement in the level of the factors would influence the stock variably. This makes the model more sensitive to important variations which a more generalized model such as the CAPM would not be responsive to (Simpson & Ramchander, 2008). In the APT, while the equation is relatively straightforward to apply, the identification of the factors may present a challenge. The investor is faced with the task of identifying each of the factors affecting a particular stock, the expected returns for each of these factors, and the sensitivity of the stock to each of these factors. Some of the macroeconomic factors which have been identified by Ross and other researchers as having an important influence on the return of the stock are: the inflation rate, the gross national product, investor confidence, and the shift in the yield curve. The factors are not limited to these, though, but may be added to serve the purposes of the analyst or the investor. Similarities, Differences, Weaknesses, application in practical life. Similarities between CAPM and APT – The Capital Asset Pricing Model and the Arbitrage Pricing Theory were both arrived at in an effort to find a reliable means of computing for the expected return of an investible asset. They both relate rate of return to factors that are currently developing in the investment environment. They both utilize coefficients that describe the sensitivity of the rate of return to the factor being related, and they both allow for the consideration of risk in portfolio development. The models are similar, such that if the factors of APT were reduced to only one, and this factor was specified to be the index gain, then the APT reduces to the form of the CAPM. Differences between CAPM and APT – The CAPM was motivated by a desire to improve Markowitz’s efficient portfolio theory, and the APT in turn was motivated by the failure of CAPM to be decisively and convincingly proven through empirical data. The most apparent difference between the two models is that CAPM is a single-factor model, that factor being the premium of the market return over the risk-free rate. The coefficient relating this single factor to the return of the asset is the beta, representing the systematic or non-diversifiable risk of the asset. The APT, on the other hand, admits the effect of several factors and not just one. As the foregoing stated, it requires at least to include the inflation rate, the gross national product, investor confidence, and shifts in the yield curve, as well as other factors pertaining to the nature of the investment and the investor’s profile. The APT seeks to accommodate that which the CAPM does not factor in, the unsystematic risk that affects all assets in the market, as well as the idiosyncratic risk, that is, that portion of the risk that is not explained by any of the factors (van & Rias, 2009). The APT provides the impetus for the development of multiple factor models that seeks to minimize dependence on the theoretical market portfolio proxy as the basis for determination, and allows for predictive power for excess return that are due to other factors (Rindisbacher, 2002). Weaknesses of the two models – It is apparent that certain weaknesses affect both models. They both depend upon the estimation of difficult-to-estimate factors, as well as more difficult-to-estimate coefficients (betas). Another common weakness is their necessary assumption that the betas relating to the factors may be estimated by a regression model that is assumed to be linear, whereas in fact these factors move randomly. The weakness of the CAPM lies in its many assumptions and in its reliance on the theoretical efficient market portfolio (Choi, 1995). The weakness of the APT lies in its many factors and betas which are non-specified; furthermore, some factors that would be introduced in the model may be correlated, thus introducing an estimation error in the regression equation (Garleanu, 2009). Example - An example will be given here using the APT and the CAPM in a practical application. We take the case of a theoretical Investor A, retired and advanced in years, who would want to avoid excess risk because he is no longer employed and is investing his life’s savings, his “nest egg” so to speak. His son, Investor B, is a young and upcoming professional who has a high paying job and is willing to lose some money in order to make more money. Assuming that one was an investment advisor asked to design their respective portfolios, then one is constrained to use an asset pricing model. This is because each of these investors has a different risk profile, and is prone to different risk tolerances (Scott, 2003). For Investor A, the kinds of stocks that would be preferable for his portfolio are relatively safe and reliable stocks that have a strong dividend pay-out history. This is to enable the elderly Investor A to realize a periodic income on his investment, inasmuch as he is no longer gainfully employed. The investment advisor should thus allocate for Investor A a choice of blue chip stocks, which, with reference to CAPM, normally have a beta of close to 1, or even slightly lower than 1. Thus, if the index were to move, the stock would have a relatively predictable movement that mirrors the market. More defensive stocks that have a beta that is less than 1 may be added to the portfolio, so that the stock’s cumulative risk becomes lower. This means that if the index goes up the stock or portfolio may not move up as fast as the stock, but then again if the index moves down, the stock will not lose too much compared to the rest of the market (Koo & Olson, 2004). For Investor B, the investment advisor may allocate for him stocks with a higher beta, such as the high growth counters and speculative stocks that typically have a beta with magnitude higher than 1 (thus, either greater than 1 or less than -1). He is capable of greater risk, so that in case he incurs losses on the market he is not financially impaired unduly. It is foolhardy, however, for an investor, even a young and moneyed one, to not guard against extreme market movements that may cause substantial losses from which it will take time for him to recover. Therefore, the wise investment advisor will usually diversify the stocks so that there would be stocks with positive betas and those with negative betas in the portfolio (Pastor & Stambaugh, 2000). Such a combination of stocks are collectively described as “negatively correlated”, meaning that when some stocks will be falling, there are those that would maintain their value or even rise because the have opposite beta signs (Koo & Olson, 2004). The risks would offset each other to an extent, so that in case of unexpected market volatility, such as the market crash at the start of the recent subprime financial crisis, certain stocks would be defensive and maintain some of the value of the principal investment. As for the use of the APT, the portfolio of each investor may have differently specified factors. For both portfolios, there would be the common macroeconomic factors recommended by Ross, such as the GNP, inflation rate, investor confidence, and the shift in the yield curve. Beyond that, the other factors may be tailored to the purposes of the individual investor (Bernstein & Damodaran, 1998). A practical problem shall be given to illustrate both models. Investor A above would wish to invest in a blue chip stock, which we shall call ALFA. The historical returns on the stock and historical returns on the market for the past 50 weeks are given in the table following, where stock returns are designated as Y and market returns as X. From historical price and index readings, the beta is estimated by regression as follows: Computation of Beta BETA is computed through the regression of the return of the stock and the return of the market, which in the case of stocks is represented by the index. Beta is computed as the slope of the best fit linear regression line (m) that describes the degree of dependence of the two variables. The slope of the linear regression equation is given by statistical theory to be: Where x is the return of the market (index) and y is the return of the stock. The sample returns may be daily, weekly or monthly (Björck, 1996). In this example, beta is m, which is 0.901. It is computed as follows, with data picked up from the table on the next page. 50 (3,835) – (72) (114) m (Beta) = ------------------------------------ = 0.90058 50 (4,336) – (114)2 Computation of expected asset returns through CAPM From the preceding beta, determined to be 0.901 for theoretical stock ALFA, the expected return on the stock could be computed. Assume the one-year treasury bond benchmark rate to be 8% and the stockmarket index, taken as a proxy for the efficient market, shows a gain of 15%. The application of CAPM would be as follows: RE = 8% + 0.901 x (15% - 8%) RE = 14.3 % Thus the stock is expected to yield a return of 16.4% when the market gains by 15%. To show its importance in the valuation of the asset, assuming that the stock regularly declares an annual cash dividend of $1.00, without further growth, then the stock may be fairly priced at: Div $1.00 P = ---------- = ----------- RE 0.143 P = $ 6.99 Thus, Investor A knows that he may transact his stock (buy or sell) reasonably at a price of $6.99. Buying at below this price or selling at above this price would be advantageous for Investor A. Computation of expected assets returns through APT Investor B, on the other hand, is contemplating theoretical stock CHI that has and inflation coefficient of 1.5, a GNP coefficient of 1.2, and an investor coefficient of 2.5. During a bull market, the annual inflation rate registered 4%, the GNP growth rate showed 6%, and the rise in investor confidence was determined by the government economic research bureau to be at 20%. For the same risk-free rate of 8% as in the case of Investor A, the rate of return expected on stock CHI would be: RE = 8% + 1.5 (4% - 8%) + 1.2 (6% - 8%) + 2.5 (15% - 8%) RE = 8% + 1.5 (4% - 8%) + 1.2 (6% - 8%) + 2.5 (20% - 8%) RE = 8% - 6% - 2.4% + 17.5% RE = 17.1% This return shall be used as the discount rate for valuing the stock. Assuming that the dividend pay-out for stock CHI at the end of the current year is expected $1.00, and the company CHI grows at the rate of 5% per annum, the Investor B arrives at the price: Div $1.00 P = ---------- = ----------------- RE - g (0.171 – 0.05) P = $ 8.26 Therefore, Investor B could reasonably transact the stock at $8.26. If he buys the stock at below $8.26 or sells above it, he would be doing so to his advantage. These illustrate the use of the two asset pricing models (Pastor & Stambaugh, 2000). Conclusion There are many studies and commentaries that highlight the weaknesses of either or both the CAPM and the APT as methods for estimating the expected rates of return for the assets that make up the portfolio. Despite their weaknesses, however, a means of estimation, though deficient, is better than none, and at least provides a basis for making decisions. The CAPM and APT have proven useful to practitioners for many years, and will continue to be used well into the future. REFERENCES Armstong, J S & Brodie, R J 1994 Effects of portfolio planning methods on decision making: Experimental results. International Journal of Research in Marketing, vol. 11, pp. 73-84 Bernstein, P L & Damodaran, A 1998 Investment Management. John Wiley & Sons, Inc. 1998. Björck, A 1996 Numerical methods for least squares problems. Philadelphia: SIAM Block, S B & Hirt, G A 2006 Foundations of Financial Management, eleventh edition. McGraw-Hill Irwin. 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