The Relevance of Portfolio Theory and the Capital Asset Pricing Model - Coursework Example

1. Introduction One of the most important decisions of economic agents is how to allocate their wealth against competing invesments. The economic agent is concerned with how his/her wealth can be invested among competiting investments. Portfofolio theory helps to determine how best the economic agent can allocate his/her investment. Secondly, the agent is concerned with the required rate of return on his or her investment. This paper provides a discussion of teh relevance of portfolio theory and the Capital Asset Pricing Model (CAPM) to investors. Section 2 looks and portfolio theory and section 3 looks at the CAPM while section 4 provides some conclusions. 2. Portfolio Theory Portfolio selection involves making decisions under uncertainty. The founder of modern portfolio theory is Hary Markowitz who for the first time in 1952 formulated and solved the problem of portfolio selection (Constantinides and Malliaris, 1995). Harry Markowitz focused on the notion of ”not putting all your eggs in one basket”. Modern portfolio theory therefore suggests that the best portfolio to invest in is a well diversified portfolio. (Constantinides and Malliaris, 1995). Each asset or security has an expected return and variance. Variance or standard deviation measures the risk inherent in the asset. According to portfolio theory, when an investor invests in two or more risky assets, the risk of the portfolio is reduced. The investor is interested in portfolios that provide the highest level of return at a lower level of risk (Bodie et al., 2006). 3. Capital Asset Pricing Model (CAPM) The capital asset pricing model (CAPM) is a model that establishes the equilibrium relationship between the risk and return on a risky asset. (Bodie et al., 2005). The model has often been used by corporations during capital budgeting decisions to establish the appropriate discount rate for evaluating investments. Understanding how to price risky assets or otherwise determining the cost of capital for risky investments has been a prominent problem in finance. (Furman and Zitikis, 2008). This has led to the development of a number of capital asset pricing models. (CAPMs) (Furman and Zitikis, 2008). The most frequently used of these CAPMs is that of Sharpe (1964), Lintner (1965) and Black (1972). The CAPM relates the expected return on an asset to the expected return on the market portfolio of all assets in the market. The CAPM states that “the expected excess return of any asset is linear in its covariance with the expected return on the market portfolio”. (Hogson et al., 2001: p. 2). If we assume that Ri and Rm are random variables that represent the return on security i and the return on the market portfolio m, respectively, then under specific assumptions such as 1. the investor’s utility functions are either quadratic or logarithmic, or 2. the pair (Ri, Rm) posseses the bivariate normal or elliptical distributions, then the CAPM implies that the expected return on the risky asset is the risk-free rate of return rf plus a risk premium that is proportional to the difference between the expected return on the market portfolio and rf., that is, we have the equation (Hodgson et al., 2001; Furman and Zitikis, 2008; Ross et al., 2008): : (1) Where (2) The equilibrium return on risky assets can be determined using the capital asset pricing model. If we consider capital markets to be continuously in equilibrium, with price changes over time reflecting the instantaneous adjustment to new equilibrium, and focus on the market portfolio, that is the portfolio of all existing assets traded in the markets, then in equilibrium, all investors should individually hold different portfolios of risky assets such that a combination of all the individual portfolios should sum up to the market portfolio. (Bodie et al., 2005; Ross et al., 2008). In other words if we assume that all investors have the same estimate of systematic and unsystematic risk, then each investor will find it optimal to hold the market portfolio. Assume also that all investors are risk adverse but with slight variations. All investors will investors will consider the market portfolio to be the optimal portfolio. All investors will therefore hold a combination of risk-free assets Rf and the market portfolio of risky assets. The proportion that each investor chooses to invest in each of the portfolios will depend on the risk aversion of the individual investor. Those with relatively low degrees of risk aversion will borrow (that is, hold a negative quantity of the risk-free asset) and invest everything in the risky portfolio. Such an investor will invest a negative proportion in the risk-free asset and a greater than one proportion in the risky asset. Such an investor will invest in security A as shown in the security market line (SML) in figure 1 below. Figure 1. The Security Market Line Source: (Ross et al., 2008) As can be seen, the less risk adverse investor bears more systematic, which is measured by the standard deviation (WM Stdev (RM)) of the portfolio. As we can see, this security has a higher return (RA). The variance of this portfolio is calculated by squaring the standard deviation, that is, WM2Var(RM) which is higher than the variance on the market portfolio. (Var2(RM)) since WM2 is greater than one. The expected return on portfolio A is greater than the expected return on portfolio M because the systematic risk component in portfolio A is higher than that for portfolio M. As earlier mentioned, this risk component is measured by beta and is calculated as shown in equation (2) above. Beta in equation 2 above measures the systematic risk of asset i relative to the overall risk in the market, given by the variance of the market portfolio. An asset that approximates the market portfolio is therefore going to have a beta of 1 since the covariance of an asset with itself is equal to its variance. (Ross et al., 2008). The equilibrium relationship between a security and the expected return can be stated as shown in equation 1. Equation (1) is otherwise referred to as the security market line. (Ross et al., 2008). The security market line indicates that the higher the beta, the higher will be the return that the asset must pay, thus reflecting investor’s reward for systematic risk. (Ross et al., 2008). A beta of 1 results in an expected return equal to the return on the market portfolio. When beta is smaller than 1, the expected return is lower than the return on the market portfolio and if beta is greater than 1, the expected return is greater than the return on the market portfolio. (Ross et al., 2008). From figure 1 above, we can confirm that the beta of security A is greater than 1 since the proportion invested in the security is greater than one. Consequently, the expected return on the security is higher than the expected return on the market portfolio. In effect, we can see that the SML is actually the CAPM relationship. For example, the return on security A can be stated in terms of the risk-free rate and the market return as follows: (3). Where (4) The CAPM can be used to estimate the cost of capital for risky investments. The decision rule for evaluating risky investments is often based on an estimation of the NPV of the project. Since there are always uncertainties, it is difficult to be a hundred percent sure that the estimated cash flows that are being discounted to present value to arrive at the NPV will actually be realized. The CAPM enables companies to compute the NPV using an appropriate measure of the cost of capital of the project. (Ross et al., 2008). If the firm is an all-equity financed firm, the cost of equity capital can be estimated from the security market line. If the firm is financed by both debt and equity, the discount rate is the overall cost of capital often referred to as the weighted average cost of capital. (Ross et al., 2008). 2.1 Tests of the CAPM. Earlier tests of the CAPM involved determining a sample period of say 5 years and estimating the security characteristic line in a first pass regression equation. Beta coefficients from the first pass regression are then used to estimate the security market line. The security characteristic line can be estimated using the following equation: As earlier mentioned, the SCL is estimated by running the regression (first pass) equation for each of the securities. Equation one can be rewritten as follows: (5) After estimating the SCL from equation 2, the SML is estimated using the following equation: (6) Where represents the sample averages of the excess returns on each of the securities; represents the sample estimate of the beta coefficients of each of the securities; represents the sample average of the excess return on the market portfolio; and represents the estimate of the variance of the residuals for each of the 100 stocks. It is an estimate of the diversifiable risk (unsystematic risk) of each of the 100 stocks. (Bodie et al., 2005). Equation (6) indicates that if the CAPM is true, then we should have the following hypotheses 1. That =0; 2. That = ; 3. That = 0. Using first pass and second pass to test the CAPM as shown above have been criticized for a number of reasons. To begin with, the CAPM assumes that all investors can borrow at the risk free rate, secondly, it assumes that the market index included in the model is the true market portfolio and thirdly, the estimation of beta based on averaging returns has been blamed for a number of reasons. In addition, the Betas used from the first pass regression as inputs to the second pass regression are assumed to be constant through time. (Bodie et al., 2005). However, all these are casting doubts on using the CAPM to as a means of determining the discount rate to be used for evaluating investments. According to Furman and Zitikis (2008) equation (2) above does not hold in general and its validity has only been established under certain assumptions on the investor’s utility function and or the distribution of the random pair (R­­i, RM). on the contrary, it is not possible to observe the utility function and this restricts the applicability of the CAPM. (Furman and Zitikis, 2008). The CAPM also assumes that the utility function of investors is quadratic. On the contrary, Hodgson et al. (2001) the CAPM is valid for a much broader class of utility functions thereby making the CAPM to be comforting to its proponents. Hodgson et al. (2001) further states that the assumption of normality is not appropriate for asset returns. Therefore it may be difficult for the CAPM to hold unless the assumption of normality in asset returns is true. If it is true that the asset returns are non-normal, then this may have serious implications for the econometric implications of the CAPM. (Hodgson et al., 2001). Hodgson et al. (2001) also notes that the standard estimator of the CAPM is ordinary least squares (OLS) regression, which is only fully efficient under nomarlity but is likely to fail should normality fail. For example, Hodgson et al. (2001) cites Zhou (1993) who considers implementation of OLS under possible non-normality deriving a procedure to mitigate the problems associated with sample size assuming that the returns distribution is elliptical. and rather than normally distributed. Based on the assumption that returns have an elliptical symmetry rather than normally distributed as assumed in earlier tests of the CAPM, Hodgson et al. (2001) developed a test of the CAPM. The results retain betas that are lower than the OLS estimates and parameter estimates that are less consistent with the CAPM restrictions than corresponding OLS estimates. 4. Conclusions Both portfolio theory and the CAPM are relevant for making investment decisions. Portfolio theory on its part enables investors to invest in a diversified portfolio. The CAPM on its part enables the investor to make inferences about the required return on his/her investment. However, both portfolio theory and the CAPM are based on underlying assumptions that may flaw their relevance to investors. For example, the CAPM assumes that each investor holds the market portfolio, which may not be the case in real life. BIBLIOGRAPHY Bodie Z., Kane A., Marcus A. J. (2006). Investments. Sixth Edition. McGraw-Hill. Constantinides, G.M., Malliaris A. G. (1995), “Portfolio Theory”, R. Jarrow et al., Eds., Handbooks" in OR & MS, VoL 9 Furman, Edward and Zitikis, Ricardas (2008). "General Stein-Type Decompositions of Covariances: Revisiting the Capital Asset Pricing Model". Available at SSRN: http://ssrn.com/abstract=1103333 Hodgson, Douglas J., Linton, Oliver B. and Vorkink, Keith (2001). "Testing the Capital Asset Pricing Model Efficiently Under Elliptical Symmetry: A Semiparametric Approach". Available at SSRN: http://ssrn.com/abstract=283364 or DOI: 10.2139/ssrn.283364 Myers S. C. Brealey R. (2002). Principles of Corporate Finance. Seventh Edition. McGraw-Hill Irwin. Ross, Westerfield, Jaffe & Jordan. (2008). Modern Financial Management", (authors), McGraw-Hill International Edition, 8th Edition,   ISBN: 978-0-07-128652-7 Read More