## Capital Asset Pricing Model(CAPM)

The core idea of the Capital Asset Pricing Model (CAPM) is that on the assumption of homogeneous expectation on financial markets, a security’s return is related linearly with beta, which is sensitivity of the capital asset to market return of equilibrium.The CAPM is developed by Sharpe W F., Lintner J., Mossin J. respectively. Sharpe published his upgraded dissertation in the Journal of Finance in 1964. Later Sharpe won a Nobel Prize in Economics for the contribution of financial economics in 1990.The CAPM is grounded in diversification and the Modern Portfolio Theory (MPT) by Harry Markowitz. So, it’s needed to check the theory of MPT first.I’ll just glimpse on 5 basic concepts of individual securities, because it’s quite common.1.    Expected Return is usually calculated by historical rate of return data, believing history is the best source for future expectation. There are many other ways to get expected return, for example, CAPM, building block method using risk premium, etc.2.    Variance and Standard Deviation (STD).Variance is the most standard way to assess the volatility of the securities.We can use the formula like this,=      (Equation 1)* N : the number of observationR : the rate of return: average of return.In other expression, Var(R) is expected value ofAnd Standard Deviation is the square root of the Variance.3.    Covariance and Correlation are the relationship of the securities. It can be positive of negative. These factors are really important when to analyze the portfolio of many securities.The formula for covariance is as follows :(Equation 2)Covariance can be changed to more standardized form of correlation.The formula for correlation can be written as :(Equation 3)If  is equal to 1 then the return of security A move exactly the same direction with the return of security B. If  is equal to 0 then the return of two securities have no relationship with each other. If is equal to -1 then the return of security A move opposite direction to the return of security B.The diversification effect of portfolio can be changed depending on correlation of the return of the securities within portfolio.When it comes to a portfolio, the return and risk are different from that of individual securities.The expected return of a portfolio is a little simple because it’s just weighted average of the return of individual securities. I’ll skip expressing the equation or example of the expected return of portfolio.Variance and standard deviation of a portfolio are a little complicated at first glimpse.Given 2 securities A and B, the variance of the portfolio is;Var(P)=          (Equation 4)=     (Equation 5)* XA : weight of security AXB : weight of security BStandard deviation of the portfolio’s return is squared root of the variance. So, it’s simple.Standard deviation of the portfolio is less than the weighted average of the standard deviation of the individual securities in the portfolio, unless the correlation is equal to 1.With the expected return and standard deviation of the two securities, we can draw the efficient set as follows;Figure 1* XA : the weight of security A.We can notice that the diversification effect occurs when  is below 1. The MVP means Minimum Variance Portfolio. The MVP~A line is inefficient because the more risk, the less  return.The MVP~B line is so called efficient frontier, which is the most efficient sets of the portfolio with 2 securities.When it comes to more than 2 securities, it’s quite similar that of 2 securities case. And the efficient frontier is the same as multi securities case, because the reasonable investors will select upper west side of possible sets of portfolios. It’s the line between MVP and B.When the number of securities of a portfolio is big, practical calculation is overwhelming. To calculate the variance of a portfolio, we can use matrix method as follows;Table 1.Stock    1    2    3    …    N123…NWe can get the variance of portfolio by sum of the terms of all boxes. It’s quite a workload to calculate this. But, by the help of computer development, the cost of doing this is not a much time consuming job.One point I don’t want to miss is that the more securities are included in a portfolio, the less dependant the risk of portfolio on individual security’s standard deviations. The portion of correlation is the matter in a portfolio with many securities.When there is risk-free asset in a portfolio, efficient frontier is different from that of risky asset only portfolio. In this case, efficient frontier is the line from the point of risk-free asset through the point of market portfolio, assuming the rate of borrowing and lending is same.The detailed discussion of portfolio theory could have been easily condensed into not more than a page. The key idea is to show how one moves from portfolio theory to the CAPM, rather than a detailed development of portfolio selection principles.Figure 2The efficient frontier of risk asset and riskless asset is also the most upper west feasible sets of portfolios.Here is another important concept which is so called separation theorem. The first step of investing which is to find market portfolio among feasible portfolios is separated from investor’s risk preference. And investor’s decision to invest how much in market portfolio and how much in risk-free asset by his risk tolerance is next step of investing which has no relationship with the first step.Under the assumption of homogeneous expectations on the market, beta can be a best measure of the risk of a security in a large portfolio. Beta is the sensitivity of the security to the market portfolio.The definition of beta is ;(Equation 6)In practice, beta can be assessed by the regression analysis of a security return and market return.Expected return of the individual security is described as follows, based on capital asset pricing model.(Equation 7)This formula implies that the expected return on a security is linearly and positively related to its beta.Using CAPM, we can easily calculate the return on a security, or appropriate price of a security. It’s simple and intuitive. Intuitiveness of derivation is one of advantages of CAPM.When a firm decides to invest in a new project, it depends on the cost of equity capital, which can be calculated using CAPM.To calculate the cost of equity capital, we should know 3 valuables.- The risk-free rate, RF- The market risk premium, RM-RF- The company beta,These are the unknown valuables to calculate the expected return of a company using equation of CAPM.In practice, we can accept expected rate of return on T-Bond or T-Bill as the risk-free rate. When it comes to E(RM), we can take it from the historical data of the S&P500, for example.But the company beta should be estimated with some skills, and it is a little matter.One way to estimate beta is using formula by definition as above equation 6.For example ;When beta is 0.39, which is Yahoo’s recent beta according to ‘finance.yahoo.com’, and assuming RM is 0.12, RF is 0.05 then the cost of equity capital can be calculated as follows;E(Ri) = 0.05 + 0.39 * (0.12 – 0.05)= 0.0773%So, Yahoo must accept the project if only its expected return is larger than 7.73%.In other words, Yahoo can use the cost of equity capital as discount rate of return on a prospective  project to calculate Net Present ValueIn real world, we can calculate the company beta using regression analysis technique. But few observations will be suspected, and too many observations will be out dated. ??Changes in industry, in market, or regulation affect a firm’s beta, but generally speaking, betas are stable remaining in the same industry.Industry beta can be used to reduce estimation error, when a firm’s operation is similar to other firms in the same industry. But the firm should use its own beta, when its operation is different from others.If the project is new one, which has quite different risk profile, then the project’s own beta should be used instead of that of the company’s. In practice, adjustment is made to a firm’s beta by the experience that is ad hoc.When the project is financed by both debt and equity, an adjustment should be made to the calculation I mentioned above. Because by far assumed I the project is financed by only equity.The cost of debt is the firm’s borrowing rate, RB.Considering tax deduction, after tax cost of debt is RB(1-tC).* tC : corporate tax rateSo, average cost of capital is weighted average of the cost of equity and the cost of debt. It’s usually referred to as the weighted average cost of capital, WACC.The equation is as follows ;WACC =      (Equation 8)* S = the market value of equity capital of the firmB = the market value of debt capital of the firmRS = the cost of equity capitalRB = the cost of debt capitaltC = the corporate tax rateUnder the theory of CAPM, there are many assumptions. These are unrealistic, but make the theory simple and easy to apply. The concern is that when the assumptions lifted, how much will the theory be affected and still be meaningful or not?The assumptions are like these. (Edwin J. Elton et al. 2003)1.    Unlimited short sales are allowed2.    Unlimited lending and borrowing at the riskless rate3.    The absence of personal income tax4.    All assets are marketable5.    All investors have identical expectations with respect to the necessary inputs to the portfolio decision and all investors are concerned with the mean and variance of returns and define the relevant period in exactly the same manner. It’s so called heterogeneous expectations.6.    Investors are expected to make decisions solely in terms of expected values and standard deviations of the returns on their portfolios7.    Investors are risk-averse and rational8.    An individual cannot affect the price of a stock by his buying or selling action9.    There are no transaction costs10.    Assets are infinitely divisibleAll these assumptions are limitations of CAPM, and also there are a few more limitations.First, estimation of beta is difficult. The beta estimation depends on history data, so the method and period is matter.Second, CAPM is not testable Roll (1977) critique. Because the market portfolio is theoretical but not observable in a real world, so we cannot test the relation between the return on the asset and the market portfolio.  But note that there have been tests of the empirical implications of the model. These empirical tests should have been summarised and reviewed to determine the extent to which the empirical results depend on the assumptions of the model as required by the question.Third, CAPM dose not explain the return on security that differ over time, on dividend yield, and on market value.The CAPM assumes that the return of assets is normally distributed random variable. But in real world, sometimes the return of assets is not normal. For example, thick tail distributions occur.After the release of the CAPM by Sharpe in 1964, there were many studies to easing the assumptions and to test the theory.The assumption of short sales make CAPM simplified for its mathematical derivation, but it’s not necessary. In market equilibrium, there are no short sales. Thus, CAPM can be derived without short sale assumption.The assumption of riskless lending and borrowing is not realistic. In real, we can assume that riskless lending is possible, but riskless borrowing is impossible. Because, we can buy government bond at any time but investors can not borrow or issue bonds in financial market.Starting with the case of ‘No riskless lending or borrowing’, we can get the slightly changed model, but this model still holds the core concept of CAPM and the second most widely used general equilibrium model.In equilibrium, efficient portfolio is a straight line with two securities with the same systematic risk in the space of beta and expected return. The market portfolio is one of portfolio on the straight line composed of average weighted two assets.The equation is as follows ;.     (Equation 9)※ : expected return of the security i: Expected return of the security with zero beta: Beta of the security iThis is so called zero beta capital asset pricing model or a two factor model of equilibrium. And its efficient frontier is the line MC in the figure as follows ;Figure 3In real, even individuals can lend without risk using government bond, so assuming riskless lending is more realistic.In this case, T is not the same as M in the Figure 4. Since, zero beta portfolio Z has variability, but RF has no risk. Thus, is above RF and M is upper right of T.The efficient frontier is given by the straight line segment RFT and curve TMC.When it comes to security market line, the only difference of this model is the intercept and slope.Figure 4When there are other lending and borrowing assumptions, SML and its equation are identical to previous case, but it’s portfolio of only risky assets.Considering personal taxes, expected return equation of the security should include dividend yield as a valuable. Thus, the equation changed to 3-dimentinal. And the relation between beta and expected return of the security is no longer linear.If the assumption of marketable assets were change and nonmarketable assets, like human capital, social security payment, house, etc., were included in portfolio, the equilibrium model should be changed but had the same form as the standard CAPM.In real, heterogeneous expectations occurs. In this case, equilibrium can still be expressed using expected return and variance and covariance, but these valuables will have various estimates values and should be calculated as weighted average value. The calculation of weighted average may be vary complicated.But Lintner and Gonedes developed ways to make equilibrium equation respectively?? meaning. And CAPM still holds.The assumption of price taking behavior is unrealistic too. Institutional investors or pension investors can affect market price. Without the assumption of price taker, the CAPM still holds according to the study of Lindenberg (year).Many changes of assumption can be hold in standard of slightly changed CAPM model, but when the changes are applied simultaneously the effect is serious.Lastly, according to famous study by Roll(1977), the CAPM can not be testable. Since CAPM is based on market portfolio and genuine market portfolio is possible in theory but it can not be observed. Thus, test for relation between beta and expected return can not be tested.A number of papers cited in the essay are not in the references. Why is it so?The structure of arguments for the second part of the question could have been improved as I point out in comments above. Nevertheless, there is some understanding of the basic requirements of the question. The first part of the essay is better than the second part even though there is no clear separation of the two parts of the essay.The essay deserves an overall mark of 63%.ReferencesEdwin J. Elton, Martin J. Gruber, Stephen J. Brown, William N. Goetzmann (2003) Modern Portfolio Theory and Investment Analysis, Sixth Edition, John Willey & Sons, Inc.Jang, Kwang Young (2002) Portfolio Management using Execl, Park Young Sa.Ross SA, RW Westerfield, J Jaffe and B Jordan (2008) Modern Financial Management, Eight Edition, New York : McGraw-Hill/Irwin.Sharpe,W.F. (1964) Capital Asset Prices : A Theory of Market Equilibrium Under Conditions of Risk, Journal of Finance