A World of Arbitrageurs and Vendors of Information - Essay Example

Summary This paper is an analysis of the Arbitrage Pricing Theory (APT) of Ross (1976). The paper includes the determination of the problems that exist during the selection of the factors in asset pricing. Arbitrage Pricing Theory was developed by Ross in 1976, the start of a great change in the market industry. The previous method which is the Capital Asset Pricing Model (CAPM) of Roll was further developed by Ross thru APT. In this paper the two postulates done by Ross, Roll & Burmeister (2003) was shown. The macroeconomic risk affecting the analysis of APT was also discussed. To be able to show the problems, I make use of the study done by Lehman and Modest (1985), which come up into three conclusions. The analysis of Lehman and Modest was able to show that one of the problems in determining the factor for asset pricing is the proper or the correct use of procedure. Lehman and Modest opposed Fama-Macbeth in using the maximum likelihood analysis in determining the factors. Another study included in this paper is the one done by Enrico Altay (2003) using the Germany and Turkish stock exchange. In his study he uses the Fama-Macbeth maximum likelihood analysis. This causes the difference in the result. Therefore, in analysing the stock exchange one should be aware of the models and theory being used. The arbitrage pricing theory may encounter several problems especially in analyzing the factors. The macroeconomic factors may affect the outcome in pricing the asset. The analysis in which the best portfolio perform best remains. The arguments are presented in the later part of the paper. Arbitrage Pricing Theory of Asset Pricing Arbitrage Pricing Theory (APT), is a general theory of asset pricing. It holds the expected return of a financial asset that can be modelled as a linear function of various macro-economic factors or theoretical market indices, where sensitivity to changes in each factor is represented by a factor specific beta coefficient. This theory was initiated by the economist Stephen Ross in 1976. The Arbitrage Pricing Theory establishes an equilibrium pricing relation between each asset's expected return and all others. In analyzing the theory, we must first discuss the covariance matrix and the portfolio risk. There are several method in estimating covariance matrix one can be by using the simple volatility estimator (Garman-Klass, 1980) or with GARCH estimators (Engle,1982; Bollerslev, 1986). Another method is that, assuming the drivers of volatility is known the historical data can be estimated if the variables are picking and the relation between each driver and each stock are identified. Portfolio return is the weighted average of the individual asset returns, using the portfolio holdings as weighs. The portfolio risk is the weighted sum of the individual asset variances and covariances with all other assets, using as weights the squared portfolio weights. Portfolio risk is its weights the squared portfolio weights. The original return units can be used to compute variance. The portfolio risk is usually reported as the square root of the variance, the volatility of portfolio returns. To compute for the portfolio risk we must collect the return variances and covariances in a table - the "covariance matrix". We must identify the portfolio holdings weights and apply the portfolio risk formula - a function dubbed a quadratic form by mathematics. The portfolio analysis is important in correlating with the common factors in showing the validity of the APT. The APT based on Ross (1976) takes the view that there is no single way to measure systematic risk. The risks arise from the unanticipated changes in the following fundamental economic variables: 1. Investor confidence 2. Interest rate 3. Inflation 4. Real Business Activity 5. A market index Each stock and portfolio has exposures or betas with respect to each of these systematic risks. Risk Exposure Profile is the pattern of economic betas for a stock or portfolio. The profile indicated how a stock or portfolio will perform under different economic conditions. The equations of APT based on Ross (1976) are as follows: The APT model follows from two basic postulates: The first postulate is that, in every time period, the difference between the actual (realized) return and the expected return for any asset is equal to the sum, over all risk factors, of the risk exposure (the beta for that risk factor) multiplied by the realization (the actual end-of-period value) for that risk factor, plus an asset-specific (idiosyncratic) error term. This postulate the equation: Where r i (t) = the total return on asset i (capital gains plus dividends) realized at the end of period t, E[ri(t)] = the expected return, at the beginning of period t, = the risk exposure or beta of asset i to risk factor j for j = 1, .., K, f j(t) = the value of the end-of-period realization for the j-th risk factor, j = 1, K, and = the value of the end-of-period asset-specific (idiosyncratic) shock. The following are the assumptions of Ross, Burmeister and Roll (2003) on the above Postulate: It is assumed that at the beginning of the period, for all the factor realizations and for asset - specific shock are zero, i.e. Another assumption is that the asset-specific shock is uncorrelated with the factor realization i.e. It is assumed that all the factor realizations and the asset-specific shocks are uncorrelated across time, i.e. The second postulate of the APT is that pure arbitrage profits are impossible. This was assumed because of competition in financial markets, it is impossible for investor to earn a positive expected rate of return on any combination of assets without undertaking some risk and without making some net investment of funds. The APT Risk The structure of the covariance matrix enforces the arbitrage pricing. The APT shows that the assets expected return beyond the risk-free rate will simply be the sum of its exposure to some shared sources of risk, weighted by the prices the market assigns to these risks - the risk premia. The theory is a straight mathematical result relating each stock performance to uncorrelated portfolios of all stocks, each portfolio 'mimicking" their independent contribution due to shared features. The following are the risk in APT: Confidence risk, Time horizon risk, Inflation risk, Business Cycle risk, and Market timing risk. Confidence risk is the unpredicted changes in investors' willingness to undertake relatively risky investment. Time horizon risk is the unanticipated changes in investor's desired time to payouts. Inflation risk is a combination of the unexpected components of short and long rum inflation rates. Business cycle risk represents unanticipated changes in the level of real business activity. And the Market timing risk is computed as the part of the S&P 500 total return that is not explained by the first four macroeconomic risks and an intercept term. Empirical Study and Problems in Determining the Factors for Asset Pricing According to Lehman and Modest (1985) one of the main lines in empirical research in asset pricing is the Arbitrage Pricing Theory of Ross (1976). The implementation of the APT requires an implicit measurement of the common factors underlying security returns. The APT empirical studies have typically constructed basis or reference portfolios to mimic the factors. In the analysis of Lehman and Modest, the theory requires portfolio that are supposed to be highly correlated with the common factors affecting security returns and to be relatively free of unsystematic risk. A basis portfolio which is poorly correlated with the common factors can lead to incorrect inferences about the validity of the APT or the interpretation of its application to capital budgeting, performance evaluation, and macroeconomic activity. The problem is which basis portfolio construction procedures do a good job of mimicking the factors and which is not. Fama-Macbeth type portfolios with maximum likelihood factor analysis of thirty to sixty securities in mimicking the factors are being opposed several times. Chen (1983) uses an instrumental variable procedure. Lehman and Modest (1985) did the empirical study to be able to evaluate the best procedure in mimicking the factors to be able to implement the APT accurately. The Arbitrage Pricing Theory (APT) of Ross (1976) in the empirical study done by Lehman and Modest (1985) begins with the assumption that K common factors are the dominant sources of co variation among security returns and that other sources of risk impinging on security returns can be removed in large well-diversified portfolios. (Lehmann, B., Modest, D., 1985). Based on the study done by Lehmann and Modest (1985), Ross made an assumption that the common factor affect the security returns in a linear fashion and that the security returns are generated by the model: Where: = Return on security i between time t-1 and time t for i = 1,.,N = Expected return on security i = Realization of the kth common factor ( i.e. source of systematic risk ) between time t-1 and t. = Sensitivity of the return of security i to the kth common factor (called the factor loading) and = The idiosyncratic or residual risk of the return on the tth security between time t-1 and time t. These residual risks are assumed to have zero mean, finite variance and to be sufficiently independent across securities for a law of large numbers to apply. Lehmann and Modest (1985), came up with a question during their analysis of the theory. The question is "how should expected returns be determined in a well-functioning capital market, if security returns satisfy these assumptions and there are no taxes, transaction cost or constraints on short sales "In the analysis of Lehmann and Modest (1985), Ross argued that if the portfolios bear the systematic risk inherent in the K common factors then the investor should be compensated. Since in large portfolios the idiosyncratic risk are eliminated.. In their analysis, they have found out that as the number of securities grows large, the portfolios will contain no risk at all and will earn zero profits to prevent the occurrence of risk less arbitrage opportunities. The equation below as assumed by Lehman and Modest shows the approximate satisfaction if the arbitrage portfolio tends towards infinity. This was proven right by Ross and others who work on the same study. Where: = the intercept in the pricing relation = the risk premium on the kth common factor, k = 1,. K. Lehman and Modest in their study assumed that it is important to distinguish between riskless rate and zero beta of the APT. The study done by Lehman and Modest (1985) has come up into three conclusions: First, to improve the basis portfolio performance, it is best to increase the number of securities included in the analysis. Second, the instrumental variable and the principal components analysis is less efficient than the maximum likelihood and restricted maximum likelihood factor analysis. Third, the procedure proposed by Lehman and Modest outperformed the study done by Fama-Macbeth which is the minimum idiosyncratic risk portfolio formation procedure. This has proven by Lehman and Modest to be as effective with the expensive well-diversified quadratic programming procedure. Enrico Altay in 2003 made his own empirical study of the APT theory. Unlike Lehman and Modest, Altay did not oppose the Fama-Macbeth procedure in analysing the factors that affects the asset pricing. He makes use of the maximum likelihood factor analysis, which uses only 30 to 60 securities in mimicking the factors. The APT Model of Altay (2003), starts with the same assumption as Lehman and Modest (1985) that k is the linear factor wherein N represents the assets under no arbitrage condition and can be shown as follows: Where Rt is a (Nx1) vector of asset returns, is an (Nx1) vector of expected asset returns, B is a (NxK) matrix of factor beta coefficients (factor loadings), is a (Kx1) vector of common factor realisations and is a (Nx1) vector of idiosyncratic return.(Altay, 2003). In the study done by Altay (2003) using the German and Turkish stock market, Ross (1976) shows the approximate relationship between expected returns and factor betas by: Where t is a (Nx1) vector of ones, is a scalar of zero beta parameter and is a (Kx1) vector factor risk premia. The above is an approximate relation between exact relations with additional assumptions. The exact pricing relation base on the assumption done by Altay (2003) can be shown as follows: Altay (2003), specified that the APT does not specify neither the number nor the contents of the common risk. And is therefore concluded by Altay (2003), that the first step in the APT analysis should be the determination of potential systematic risk factors. Altay (2003) uses the time series regression in estimating the factor beta coefficients of each portfolio. And then estimate the relation between factor betas and average asset returns by cross sectional regression process. Altay (2003) concluded in his analysis that the asset prices react to macroeconomic factors and is expected to be rewarded in stock markets. He found out that the empirical result can be altered using a large number of macroeconomics variables in the factor analysis methods. Relationship of Arbitrage Pricing Theory (APT) with Capital Asset Pricing Model (CAPM) The APT along with the Capital Asset Pricing Model (CAPM) is one of two influential theories on asset pricing. The APT differs from the CAPM in that it is less restrictive in its assumptions. It allows for an explanatory (as opposed to statistical) model of asset returns. It assumes that each investor will hold a unique portfolio with its own particular array of betas, as opposed to the identical "market portfolio". In some ways, the CAPM can be considered a "special case" of the APT in that the securities market line represents a single-factor model of the asset price, where Beta is exposure to changes in value of the Market. The APT is considered as the supply side model, since its beta coefficients reflect the sensitivity of the underlying asset to economic factors. And the Capital Asset Pricing Model is the demand side models its result arise form a maximization problem of each investors utility function, and from the resulting market equilibrium. Arguments in Arbitrage Pricing We assume that the following are the arguments in the Arbitrage pricing: There exist some important systematic risks driving security returns in a linear fashion The sensitivity of the security and the risks can be perceived by investors. There are some investor who want to take the risk in economy By undertaking risk arbitrage the investor can exploit the differences in expected return. Expected returns will be determined such that the expected returns of securities in the economy plot on or close to the security market line in as many dimensions of risk as there are factors. Conclusion Arbitrage Pricing Theory has a number of benefits. Although it is sometimes confusing because of its generality. First, it is not as a restrictive as the CAPM in its requirement about individual portfolios. It is also less restrictive with respect to the information structure it allows. The APT is a world of arbitrageurs and vendors of information. It also allows multiple sources of risk, indeed these provide an explanation of what moves stock returns. The benefits also come with drawbacks. The APT demands that investors perceive the risk sources, and that they can reasonably estimate factor sensitivities. In fact, even professionals and academics can't agree on the identity of the risk factors, and the more betas you have to estimate, the more statistical noise you must live with. Proper use of the portfolio in determining the factors in asset pricing is important. Lehman and Modest was able to prove that the maximum likelihood analysis for factor analysis is somewhat less effective because of its limited number of securities. As in the study done by Altay (2003), in the argument of systematic risk it is for him the first step in APT. Determining first the systematic risk factors. These systematic risk factors will have a great effect in asset pricing. There are investors who are brave enough to take the challenge in using several methods in analyzing the asset pricing. Improper use of factors in asset pricing, as well as the wrong implementation of the APT would result in disaster. By exploiting the risk arbitrage the investors will able to see the difference in expected arguments. Problems in determining the factors in asset pricing can be resolve by adopting the APT in a proper way. Risk analysis and portfolios are needed in analyzing the APT. References: 1. Burmeister, E., Roll, R., Ross, S. , Using Macroeconomics Factors to Control Portfolio Risk, March 9, 2003. http://www.birr.com/documen2.htm 2. Lehman, B.N., Modest, D. (1985) The Empirical Foundations of the Arbitrage Pricing Theory, Journal of Financial Economics, 21, 213-254. http://www.nber.org/papers/w1726.pdf 3. Altay, E. 2003, The effect of Macroeconomics Factors on Asset Returns: A comparative Analysis of the German and the Turkish Stock Markets in an APT Framework, www.econturk.org/Turkisheconomy/0307006.pdf 4. . Read More