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Topic 3: Capital Asset Pricing Model and Arbitrage Pricing Theory BUS 442 Investment Theory and Portfolio Management The three portfolios we looked at in Topic 2 helped down the foundation for many of the asset pricing models commonly used in the financial industry today. Two of such models are the capital asset pricing model (CAPM ) and the arbitrage pricing theory (APT). We will focus first on the capital asset pricing model fo r two reasons: (i) many of you have seen the CAPM in an introductory finance co urse, and (ii) the approach the CAPM takes to estimate the risk of a portfolio is very similar to the approach of the third portfolio we analy zed in Topic 2. 1. Assumpti ons of the Capital Asset Pricing Model Before we look at how the CAPM can be used to price a portfolio (or an investment), it is important for you to understand that it is after all a theoretical model, which means that it is based on an idealistic investment environment different fro m the real world. Despite its “simplistic” assumptions abou t the investment environment, the CAPM still serves as a valuable tool in understanding the relationship between the risk and return. The following are the assumptions of the CAPM. Briefly explain what each assumption means. (a) Investors are price takers (b) Investors have identical single-period holding horizons (c) Investors have access to all investments and have access to unlimited borrowing and lending opportunities at the risk-free rate (d) The financial markets are frictionless (e) Investors are rational mean-variance optimizer (f) Investors have homogenous expectations Investment Theory -1- Topic 3 Handout 2. Relati onshi p Between the CAPM and the CML In Topic 2, we know that when you have access to all the different investments available in the financial market, the “best place” you can be is on the capital market line (CM L). Portfolios that are located on this line will provide you the best (or optimal) comb ination of risk and return. As a result, the CML is a good measure for the relationship between risk and return. Just in case you forgot, the CML is represented by the following formu la: E (rm ) rrf E (r p ) rrf p rm What is the similarity and difference between the CAPM and the CM L in measuring the relat ionship between risk and return? We need to first re-arrange the formu la (which is presented below) for the CM L before we will address the question. p E (r p ) rrf m E(r m ) rrf If you remembered what you learned in your Introductory Finance course, no doubt you will recognize the equation for the CAPM : E (rp ) rrf p E (rm ) rrf where is the beta of the portfolio. Do you begin to see the resemblance between the CML and the CAPM? According to the two formulae, the return of the portfolio can be broken down into two components: (i) the guaranteed risk-free rate and (ii) the compensation for taking on risk. In addition, the co mpensation is determined by two things: (i) a relative measurement of the portfolio’s risk and (ii) the market risk premiu m [i.e. E(rm ) rrf ]. What about the differences between the CML and the CAPM? Can you tell wh at are the t wo differences between the CM L and CAPM? (a) Difference 1 (b) Difference 2 3. The CAPM, Beta (i.e. , and S ML Now that we know more about the similarities and differences between the CML and the CAPM, we need to go back and look at some of the details related to the CAPM. Even though the formu la presented earlier for the CAPM is for a portfolio, the formu la can easily be modified to determine the return of a single investment as follo ws: E (ri ) rrf i E (rm ) rrf Investment Theory -2- Topic 3 Handout Refer to In-class Example 1 Since the risk-free rate and the market return should be the same fo r every investment in the financial market, the only thing that is different fro m investment to investment is the beta of the investment. As a result, we can claim that the only driving force behind the determination of an investment’s return is its beta. What is the beta? It represents an investment’s non-diversifiable risk (and not its total risk) relative to the market risk. In other words, the beta of an investment measures the co -movement of the investment’s expected return with the market’s expected return. The fo rmula of an investment’s beta is as follows: im i i m where im = correlat ion between investment i’s return and the market ’s return i = investment i’s non-diversifiable risk m = market risk Refer to In-class Example 2 We know the CAPM can be easily mod ified to determine the expected return of either a port folio or an individual investment. The only difference between the two is the beta: beta of a portfolio and beta of an individual investment. What is the relationship between the two? The beta of a portfolio is simply the weighted average of the betas of the investments included in the portfolio. The formula for the beta of a portfolio is as follows: p wi i Refer to In-class Example 3 Just as in the case with the capital allocation line, we can also represent the CAPM in a graphical manner. The straight line that represents the relationship between risk and return (according to the CAPM) is known as the security market line (SM L). E(ri) A SML E(rm) B rrf i 1.0 The security market line will help you determine if an investment is co rrectly price. In other words, help you determine if the investment is offering a return that is appropriate for its level of risk (as measured by the beta). If an Investment Theory -3- Topic 3 Handout investment’s return falls on the SML, the investment is considered to be correctly price bec ause the expected return of the investment matches the one according to the CAPM (based on for its beta). However, if the expected return of the investment differs fro m the one as “predicted” by the CAPM, the investment is considered to be either underpriced or overpriced. The difference between the investment’s actual expected return and its fair return (as dictated by the CAPM) is known as the investment’s alpha (i.e. ). Let’s analy ze the two investments A and B as depicted in the graph above. Based on your analysis, what can you say about the two investments? (a) Investment A (b) Investment B Refer to In-class Example 4 4. Es timati ng the Beta of an Investment Using the Index Model Since the driving force behind the CAPM in determining the return of an investment is its beta, it is important that you know the process commonly adopted to estimate the beta of an investment. Before we can proceed with the discussion on how to estimate beta, you need to first understand that we cannot implement the CAPM in the real world as it is because of two main issues. First, the CAPM assumes that the market portfolio (wh ich includes all investments in the financial market) is available to all investors. Second, it focuses on the expected return of an investment. To apply the CAPM in the real world, we need to use the index model, which addresses the above two issues as follows: (a) The index model uses a proxy such as a market index (e.g. S&P 500) to represents a more relevant market portfolio (and the market risk). (b) The index model uses realized returns (rather than expected returns, which are not easily observable). If we are to estimate the beta of an investment using CAPM, we will need to establish the following regression model, which is based on the realized excess returns of the investment in relation to the realized excess returns of the market : E(ri ) rrf i i [ E(rm ) rrf ] However, since we are using the index model (i.e. using realized returns), the regression model will look as follows: ri rrf i i [rm rrf ] To estimate the beta of an investment, we need first to determine the holding period returns of the investment and the chosen market index. Once we have identified the proxy for the risk-free rate, we can determine the excess Investment Theory -4- Topic 3 Handout returns of the investment and the excess returns of the market index. If you plot the excess returns of the investment and the market index as follows, you will have a very good idea whether the beta will be positive or negative. Excess Return of Investment (%) 20 16 12 8 4 0 0 4 8 12 16 20 Excess Return of a Market Portfolio (%) Based on the graph above, can you tell if the beta will be positive or negative? One thing that is crucial to remember is that because of the setup of the regression model, the excess re turns of the investment have to be on the y-axis and the excess returns of the market index have to be on the x-axis. Once you have the excess returns of the investment and the market index plotted as above, you want to find a straight line that “best fit” the data as presented in the graph below: Excess Return of Investment (%) 20 16 12 8 4 0 0 4 8 12 16 20 Excess Return of a Market Portfolio (%) What does it mean to have a straight line that “best fit” the data points? The straight line, wh ich “best fit” the data points, is known as the security characteristic line (SC L). Once again, a straight line is determined by its y-intercept and its slope. How do you determine the y-intercept and the slope of the SCL? You can do so by performing a regression analysis using any statistical packages or Microsoft Excel. Refer to In-class Example 5 Investment Theory -5- Topic 3 Handout 5. Problems of the Capi tal Asset Pricing Model Although the Capital Asset Pricing Model is the most popular tool among many of the investors and investment analysts, it does have its problems. We know the CAPM uses 3 variables to determine the expected return of an asset: the risk-free rate, the expected return of the market portfolio, and the beta of the asset. An error in the estimation of any of these variables might lead to a wrong reco mmendation or investment decision. The following are some of the sources of error in estimating the 3 variab les for the CAPM : (a) Although a 1-year T-b ill is co mmonly used as a representation for a risk-free asset, it might not be the most appropriate choice in certain situations. Some analysts have suggest ed that a 30-year T-bond might be a more appropriate choice because its time horizon closely matches the investment horizons of most investors. In this case, the choice of the representation of a risk-free asset might lead to a wrong investment decision because the return of a 1-year T-bill can differ significantly fro m the return of a 30-year T-bond. (b) We know there are many representations (or proxies) for the market, wh ich means that there are many choices to represent the market portfolio : the Dow Jones Industrial Average, the S&P 500 index, the NYSE Co mposite index, etc. Each of these choices will p rovide a different estimate for the market return. Just as in the case of the risk-free asset, the choice of the representation for the market portfolio will affect an investor’s investment decisions. (c) It has been proven empirically that the beta of an investment is unstable over time. In other words, the value of the beta of an investment changes over time. This could be due to changes in the company’s management, its financing policy, etc. In addition, the estimates for the beta of a particular investment vary among analysts and publications for several reasons: (i) The proxy fo r the market can be different among analysts and publications. For examp le, one analyst might be using the Value Line index (which contains 1700 stocks), while another analyst might be using the S&P 500 index. (ii) The time period used in estimating the beta of a stock can be different among analysts and publications. For examp le, the beta of an investment estimated using 5 years of return will differ fro m the one estimated using 10 years of return. (iii) The intervals of the measurement of the returns will also affect the estimates of the betas. For example, a beta estimated with weekly returns will differ fro m the one estimated with monthly returns. Refer to In-class Example 6 6. Capi tal Asset Pricing Model and Arbi trage Theory The major crit icism of the CAPM is that it uses only a single factor in determin ing the return of a portfolio, namely the beta of the portfolio. In other words, the non-diversifiable risk of the portfolio (in relation to the market risk) is the sole determinant of its return. No other factors will have any effect on the portfolio’s return. To address this criticis m of the CAPM, a new model has been developed based on the arbitrage pricing theory (APT). Similar to the CAPM, the APT assumes that there is a relationship between the risk and return of a portfolio. However, co mpared to the CAPM, the APT has fewer assumptions. The following assumptions are required for the CAPM but not for the APT: (a) A single-period investment horizon (b) Borro wing and lending at the risk-free rate (c) Investors are mean-variance optimizer Investment Theory -6- Topic 3 Handout The APT is based on the concept of arbitrage (or law of one price), wh ich states that any two identical investments cannot be sold at a different price. In other words, the theory states that market forces will adjust to eliminate any arbitrage opportunities, where a zero investment portfolio can be created to yield a ris k-free profit. The key thing you need to understand is that, unlike the CAPM, the APT does not assume that the market risk is the only factor that influences the return of a portfolio. The APT recognizes that several other factors (or risks) can influence the return of a portfolio . The APT preserves the linear relat ionship between risk and return of the CAPM but abandons the single measure of risk by the beta of the portfolio. The APT model is a mu ltip le factor model, wh ich uses factors such as the inflat ion rate, the growth rate of the economy, the slope of the yield curve, etc. in addition to the beta of the portfolio in determining the return of the portfolio. Keep in mind that just as in the case with the CAPM, the APT can also be modified to determine the return of an individual investment. The formu la of the APT can be presented as follows: E (ri ) rrf 1 E (r1 ) rrf 2 E (r2 ) rrf ... n E (rn ) rrf where 1, 2, …, n represent the different factors that have impact over an investment’s return. The problem with the APT is that the factors are not well-specified ex-ante. So me research had been conducted to determine the appropriate factors that should be included in the model. However, there is no consensus on what the factors should be. One study suggested that the factors that should be included are changes in expected inflation, unanticipated changes in inflation, unanticipated changes in industrial production, unanticipated changes in the default risk-premiu m, and unanticipated changes in the term structure of interest rates. On the other hand, another study suggested that the factors should be default risk, the term structure of interest rates, inflation or deflation, the long-run expected growth rate of profits for the economy, and residual market risk. Refer to In-class Example 7 What does this all means to an investor like you? Should you use the CAPM or the APT? The key thing you need to remember is that neither of the theories dominates the other one. The APT is more general because it does not require as many assumptions as the CAPM. However, the CAPM is more general because it applies to all individual investments without reservation (whereas the APT works better with well -diversified portfolio). Investment Theory -7- Topic 3 Handout