FINC4101 Investment Analysis

1. 1 FINC4101 Investment Analysis Instructor: Dr. Leng Ling Topic: Asset Pricing Models

2. 2 Learning objectives Distinguish between systematic and unsystematic risk. Define the single index model and identify its components. Relate a security’s beta to its systematic risk. Describe the Capital Asset Pricing Model (CAPM) and list its assumptions. Identify the implications, applications and limitations of the CAPM. Use the CAPM to compute an asset’s expected return. Define the Security Market Line (SML). Understand the concepts of fairly priced, underpriced and overpriced. Describe the implementation of the CAPM. CAPM is pronounced as ‘CAP’ + ‘M’. CAPM is pronounced as ‘CAP’ + ‘M’.

3. 3 Concept Map

4. 4 Why do we need an asset pricing model? An asset pricing model allows us to figure out the required return of an asset. Required rate of return can be used for: Valuation of assets. Identification of attractive investments. Capital budgeting. Use this slide to motivate the topic. From FI3300, we know that we can use the required rate of return as the discount rate to find the PV of financial securities like stocks and bonds. Later on, we will see how we can compare the required rate of return and the expected rate of return to find attractive investments. The required rate of return of the firm can also be used as the discount rate for capital budgeting. E.g., to compute the NPV of a project, the firm can use its required rate of return as the discount rate to compute the PV of project cash flows.Use this slide to motivate the topic. From FI3300, we know that we can use the required rate of return as the discount rate to find the PV of financial securities like stocks and bonds. Later on, we will see how we can compare the required rate of return and the expected rate of return to find attractive investments. The required rate of return of the firm can also be used as the discount rate for capital budgeting. E.g., to compute the NPV of a project, the firm can use its required rate of return as the discount rate to compute the PV of project cash flows.

5. 5 Preparation for CAPM To understand the CAPM, we need to know the following: Systematic vs. unsystematic risk Single index model To prepare students for the CAPM, we need to understanding the following concepts: 1) Systematic risk, unsystematic risk 2) Single-index model and its relation to these two types of risks. This will make it easier to explain the expected return-beta relationship of the CAPM. To prepare students for the CAPM, we need to understanding the following concepts: 1) Systematic risk, unsystematic risk 2) Single-index model and its relation to these two types of risks. This will make it easier to explain the expected return-beta relationship of the CAPM.

6. 6 Unsystematic risk Unsystematic risk: uncertainty or variability in returns that affects a specific asset without affecting other assets. Also known as unique risk, firm-specific risk, diversifiable risk. Sources of unsystematic risk: litigation, patents, R&D, management style and philosophy, financial leverage, etc Bring this in before discussion of single-index model because the risk decomposition of the single index model illustrates the dichotomy between systematic and unsystematic risk. Bring this in before discussion of single-index model because the risk decomposition of the single index model illustrates the dichotomy between systematic and unsystematic risk.

7. 7 Systematic risk Systematic risk: uncertainty or variability in returns that affects all risky assets. Also known as: market risk, nondiversifiable risk Sources of systematic risk: fluctuation in the stock market, business cycles, inflation rate, monetary policy, exchange rates, wars, political unrest, technological change etc. We can divide the risk of an asset into two parts: systematic and unsystematic. We can divide the risk of an asset into two parts: systematic and unsystematic.

8. 8 Systematic vs unsystematic risk Diversification works but has its limit. At the very most, diversification eliminates unsystematic risk. When there is no unsystematic risk, only systematic risk remains, but that is not diversifiable. It is risk that investors bear in exchange for enjoying the return from investing. After all, bearing risk is a fundamental part of investment. Investors holding well-diversified portfolios will only demand a risk premium for bearing the systematic risk.

9. 9 Graphical depiction of systematic and unsystematic risks The picture shows that market risk cannot be eliminated. How do we quantify systematic and unsystematic risk of a security or portfolio? We use a statistical model call the single-index or single-factor model. Next slide. The picture shows that market risk cannot be eliminated. How do we quantify systematic and unsystematic risk of a security or portfolio? We use a statistical model call the single-index or single-factor model. Next slide.

10. 10 Single-index model of security returns Statistical model that estimates the systematic and unsystematic risk of a security or portfolio. Looks at “excess return”, denoted by “R” What is a security’s excess return Security’s HPR in excess of the risk free rate. Excess return of security i, Ri = security i’s HPR – risk-free rate = ri – rf The following discussion focuses on excess returns. Another use of the single-index model is that it makes it easier to compute inputs for portfolio analysis. To construct the efficient frontier from a universe of 100 securities, we would need to estimate 100 expected returns, 100 variances and (100 x 99)/2 = 4950 covariances. And a universe of 100 securities is actually quite small. A universe of 1000 securities would require estimates of (1000 x999)/2=499,500 covariances, as well as 1000 expected returns and variances. The assumption that a common single factor is responsible for all the covariability of stock returns, with all other variability due to firm-specific factors, dramatically simplifies the analysis. To study the ‘anatomy’ of the single-index model, we start with the security return. In the single-index model, we look at the return of the security in excess of the risk-free rate. This is called the security’s excess return. We use capital R to denote excess return. Small “r” denotes the HPR (gross) security return. Rf is risk-free rate. Another use of the single-index model is that it makes it easier to compute inputs for portfolio analysis. To construct the efficient frontier from a universe of 100 securities, we would need to estimate 100 expected returns, 100 variances and (100 x 99)/2 = 4950 covariances. And a universe of 100 securities is actually quite small. A universe of 1000 securities would require estimates of (1000 x999)/2=499,500 covariances, as well as 1000 expected returns and variances. The assumption that a common single factor is responsible for all the covariability of stock returns, with all other variability due to firm-specific factors, dramatically simplifies the analysis. To study the ‘anatomy’ of the single-index model, we start with the security return. In the single-index model, we look at the return of the security in excess of the risk-free rate. This is called the security’s excess return. We use capital R to denote excess return. Small “r” denotes the HPR (gross) security return. Rf is risk-free rate.

11. 11 Anatomy of single-index model Uses a broad index of securities (e.g., S&P500) to represent systematic risk. This broad index is called, “market index”, “market factor”, or just “market” for short. Market excess return is denoted RM . The model says that a security’s excess return consists of 3 parts: Before showing the equation, we need one more ingredient, which is the index. The single-index model is saying that the market index summarizes the aggregate impact of all macroeconomic factors (i.e., the impact of all the possible sources of systematic risk). Before showing the equation, we need one more ingredient, which is the index. The single-index model is saying that the market index summarizes the aggregate impact of all macroeconomic factors (i.e., the impact of all the possible sources of systematic risk).

12. 12 Ri = ai + biRM + ei The equation specifies two sources of security risk: market or systematic risk, attributable to the security’s sensitivity (as measured by beta) to movements in the overall market, and firm-specific risk (ei), which is the part of uncertainty independent of the market factor. Using the equation shown on this slide, we can decompose a security’s risk in the following way. The rationale for saying that E(ei) = 0 is that ei represents the impact of unanticipated events, which by definition must average out to zero. The equation specifies two sources of security risk: market or systematic risk, attributable to the security’s sensitivity (as measured by beta) to movements in the overall market, and firm-specific risk (ei), which is the part of uncertainty independent of the market factor. Using the equation shown on this slide, we can decompose a security’s risk in the following way. The rationale for saying that E(ei) = 0 is that ei represents the impact of unanticipated events, which by definition must average out to zero.

13. 13 Using the single-index model to decompose security risk(p.168) Variance of Ri = Systematic risk + Firm-specific risk = Variance(biRM) + Variance(ei) = bi2 sM2 + s2(ei) Because the firm-specific component of the firm’s return is uncorrelated with the market return, we can write the variance of the excess return of the stock as shown on the slide. The equation says that the total variability of a security’s return depends on two components: The variance attributable to the uncertainty common to the entire market. This systematic risk is attributable to the uncertainty in RM. Notice that the systematic risk of each stock depends on both the volatility in RM (i.e., sM2 ) and the sensitivity of the stock to fluctuations in RM. That sensitivity is measured by bi. The variance attributable to firm-specific risk factors, the effects of which are measured by ei. This is the variance in the part of the stock’s return that is independent of market performance. The greater the beta, the greater the security’s systematic risk, as well as its total variance. The average security has a beta of 1. because the market is composed of all securities, the typical response to a market movement must be one for one. An “aggressive” investment will have a beta higher than 1.0; i.e., the security has above-average market risk. A “defensive” investment will have a beta less than 1.0.Because the firm-specific component of the firm’s return is uncorrelated with the market return, we can write the variance of the excess return of the stock as shown on the slide. The equation says that the total variability of a security’s return depends on two components: The variance attributable to the uncertainty common to the entire market. This systematic risk is attributable to the uncertainty in RM. Notice that the systematic risk of each stock depends on both the volatility in RM (i.e., sM2 ) and the sensitivity of the stock to fluctuations in RM. That sensitivity is measured by bi. The variance attributable to firm-specific risk factors, the effects of which are measured by ei. This is the variance in the part of the stock’s return that is independent of market performance. The greater the beta, the greater the security’s systematic risk, as well as its total variance. The average security has a beta of 1. because the market is composed of all securities, the typical response to a market movement must be one for one. An “aggressive” investment will have a beta higher than 1.0; i.e., the security has above-average market risk. A “defensive” investment will have a beta less than 1.0.

14. 14 Graphical representation of single-index model: Ri = ai + biRM + ei The equation, Ri = ai + biRM + ei , may be interpreted as a single-variable regression equation of Ri on the market excess return RM. The excess return on the security ( Ri ) is the dependent variable that is to be explained by the regression. Regression analysis lets us use the sample of historical returns on security i and the market index to estimate a relationship between the dependent variable and the explanatory variable. The estimated relationship can be graphically depicted as the line drawn in the scatter diagram. This regression line “best fits” the data in the scatter diagram. This line is called the security characteristic line. Security characteristic line: plot of a security’s excess return as a function of the excess return of the market. The slide shows the sample returns of Dell and the associated regression line. Make the following points: 1) The regression intercept corresponds to ??i . This is because any point on the vertical axis represents zero market excess return, so the intercept gives us the expected excess return when the market is neutral. This is exactly the definition of ??i. 2) The slope of the line corresponds to beta i. The slope is also called the regression coefficient or slope coefficient or just beta. For a security with a beta > 1 (“aggressive” investment), the slope will be steeper than the 45 degree line and upward sloping. For a 0< beta < 1, the slope will be less steep than the 45 degree line (“defensive” investment) and upward sloping. For a security with a beta of 1, the slope will be exactly that of the 45 degree line. 3) If beta is negative, beta < 0, the regression line will then slope downward, meaning that, for more favorable macro events (higher RM), we would expect a lower return, and vice versa. The latter means that when the economy goes bad (negative RM) and securities with positive beta are expected to have negative excess returns, the negative beta security will have positive excess returns. So, negative beta securities have negative systematic risk, they provide a hedge against systematic risk. Sometimes, gold has been suggested as an asset with a negative beta to the stock market.The equation, Ri = ai + biRM + ei , may be interpreted as a single-variable regression equation of Ri on the market excess return RM. The excess return on the security ( Ri ) is the dependent variable that is to be explained by the regression. Regression analysis lets us use the sample of historical returns on security i and the market index to estimate a relationship between the dependent variable and the explanatory variable. The estimated relationship can be graphically depicted as the line drawn in the scatter diagram. This regression line “best fits” the data in the scatter diagram. This line is called the security characteristic line. Security characteristic line: plot of a security’s excess return as a function of the excess return of the market. The slide shows the sample returns of Dell and the associated regression line. Make the following points: 1) The regression intercept corresponds to ??i . This is because any point on the vertical axis represents zero market excess return, so the intercept gives us the expected excess return when the market is neutral. This is exactly the definition of ??i. 2) The slope of the line corresponds to beta i. The slope is also called the regression coefficient or slope coefficient or just beta. For a security with a beta > 1 (“aggressive” investment), the slope will be steeper than the 45 degree line and upward sloping. For a 0< beta < 1, the slope will be less steep than the 45 degree line (“defensive” investment) and upward sloping. For a security with a beta of 1, the slope will be exactly that of the 45 degree line. 3) If beta is negative, beta < 0, the regression line will then slope downward, meaning that, for more favorable macro events (higher RM), we would expect a lower return, and vice versa. The latter means that when the economy goes bad (negative RM) and securities with positive beta are expected to have negative excess returns, the negative beta security will have positive excess returns. So, negative beta securities have negative systematic risk, they provide a hedge against systematic risk. Sometimes, gold has been suggested as an asset with a negative beta to the stock market.

15. 15 Measuring the importance of systematic risk ?2 is the ratio of systematic risk to total risk. = Systematic Risk/Total Risk = bi2 sM2 /[ bi2 sM2 + s2(ei) ] ?2 ranges from 0 to 1 As ?2 ?? 1, Systematic risk becomes a bigger part of total risk. Points on scatter diagram lie close to regression line. As ?2 ? 0, Unsystematic risk becomes a bigger part of total risk. Points on scatter diagram lie away from regression line. As ?2 approaches 1, systematic risk becomes more important than unsystematic risk, i.e., systematic risk dominates total risk and unsystematic risk is relatively unimportant. Conversely, as ?2 approaches 0, unsystematic risk becomes more important than systematic risk, i.e., unsystematic risk dominates total risk and systematic risk is relatively unimportant. When ?2 = 1 exactly, security return is fully explained by the market return, i.e., there are no firm-specific effects. All the points of the scatter diagram will lie exactly on the regression line. This is called perfect correlation; security return is perfectly predictable from the market return. As ?2 approaches 1, systematic risk becomes more important than unsystematic risk, i.e., systematic risk dominates total risk and unsystematic risk is relatively unimportant. Conversely, as ?2 approaches 0, unsystematic risk becomes more important than systematic risk, i.e., unsystematic risk dominates total risk and systematic risk is relatively unimportant. When ?2 = 1 exactly, security return is fully explained by the market return, i.e., there are no firm-specific effects. All the points of the scatter diagram will lie exactly on the regression line. This is called perfect correlation; security return is perfectly predictable from the market return.

16. 16 Different types of securities Use this slide to show how different betas, intercepts, systematic risk and unsystematic risk can be depicted graphically by the scatter diagram and regression line. (Text book, fig 6.12) 1. Securities R1-R6 have a positive beta. These securities move, on average, in the same direction as the market. R1, R2, R6 have large betas, so they are ‘aggressive’ in that they carry more systematic risk than R3,4,5, which are ‘defensive’. R7 and R8 have negative betas. These are hedge assets that carry negative systematic risk. 2. Intercept. R1,4,8 have a positive intercept, while R2,3,5,6,7 have negative intercepts. To the extent that one believes these intercepts will persist, a positive value is preferred. 3. R2,3,7 have relatively low unsystematic risk (residual variance) because the points of the scatter diagrams lie close to the regression line. This implies that excess return due to unexpected firm-specific events is a relatively smaller part of total excess return. As a result, return variability due to firm-specific events is also less important. With sufficient diversification, unsystematic risk eventually will be eliminated and hence the difference in the residual variance is of little economic significance. Total variance: R3 has a low beta and low residual variance, so its total variance will be low. R1,6 have high betas and residual variance, so their total variance will be high. But R4 has a low beta and high residual variance, while R2 has a high beta with a low residual variance. In sum, total variance often will misrepresent systematic risk, which is the part that matters.Use this slide to show how different betas, intercepts, systematic risk and unsystematic risk can be depicted graphically by the scatter diagram and regression line. (Text book, fig 6.12) 1. Securities R1-R6 have a positive beta. These securities move, on average, in the same direction as the market. R1, R2, R6 have large betas, so they are ‘aggressive’ in that they carry more systematic risk than R3,4,5, which are ‘defensive’. R7 and R8 have negative betas. These are hedge assets that carry negative systematic risk. 2. Intercept. R1,4,8 have a positive intercept, while R2,3,5,6,7 have negative intercepts. To the extent that one believes these intercepts will persist, a positive value is preferred. 3. R2,3,7 have relatively low unsystematic risk (residual variance) because the points of the scatter diagrams lie close to the regression line. This implies that excess return due to unexpected firm-specific events is a relatively smaller part of total excess return. As a result, return variability due to firm-specific events is also less important. With sufficient diversification, unsystematic risk eventually will be eliminated and hence the difference in the residual variance is of little economic significance. Total variance: R3 has a low beta and low residual variance, so its total variance will be low. R1,6 have high betas and residual variance, so their total variance will be high. But R4 has a low beta and high residual variance, while R2 has a high beta with a low residual variance. In sum, total variance often will misrepresent systematic risk, which is the part that matters.

17. 17 Relevant risk measure for diversified investors In highly diversified portfolios, unsystematic risk can be virtually eliminated and thus becomes irrelevant. Only systematic risk remains in diversified portfolios. In measuring security risk for diversified investors, we focus on the security’s systematic risk, i.e., bi2 sM2 . Equivalently, we can use bi as the measure of the security’s systematic risk. Firm-specific events (source of unsystematic risk) are independent of each other. So their risk effects are offsetting. This means that in a diversified portfolio, the unsystematic risk becomes negligible. Why can we use bi as the measure of systematic risk? Because bi is proportional to systematic risk, bi2 sM2 . So the higher the bi, the higher the systematic risk. Thus, we can equivalently use bi to measure systematic risk. This insight will be used in discussing CAPM. Beta is also the measure of the contribution of a security to the risk of a well-diversified portfolio since sM2 is the same for all stocks. Firm-specific events (source of unsystematic risk) are independent of each other. So their risk effects are offsetting. This means that in a diversified portfolio, the unsystematic risk becomes negligible. Why can we use bi as the measure of systematic risk? Because bi is proportional to systematic risk, bi2 sM2 . So the higher the bi, the higher the systematic risk. Thus, we can equivalently use bi to measure systematic risk. This insight will be used in discussing CAPM. Beta is also the measure of the contribution of a security to the risk of a well-diversified portfolio since sM2 is the same for all stocks.

18. 18 Capital Asset Pricing Model (CAPM) Equilibrium model that underlies all modern financial theory Derived using principles of diversification with simplified assumptions Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development Theoretical model based on a list of simplifying assumptions

19. 19 Assumptions of the CAPM (1) Individual investors are price takers Single-period investment horizon Investments are limited to traded financial assets No taxes, and transaction costs Information is costless and available to all investors Investors are rational mean-variance optimizers Homogeneous expectations Say right at the start that the assumptions of the CAPM are highly unrealistic. That is not surprising because all theoretical models are abstraction from reality, i.e., based on unrealistic assumptions. The way to judge a model is to see how well it’s predictions describe the real world. Thus, it’s important to test the model. A number of simplifying assumptions lead to the basic version of the CAPM. The fundamental idea is that individuals are as alike as possible, with notable exceptions of initial wealth with risk aversion. Investors cannot affect prices by their individual trades. This means that there are many investors, each with an endowment of wealth that is small compared with the total endowment of all investors. This assumption is analogous to the perfect competition assumption of microeconomics. All investors plan for one identical holding period. Investors form portfolios from a universe of publicly traded financial assets, such as stocks and bonds and have access to unlimited risk-free borrowing or lending opportunities. Investors pay neither taxes on returns nor transaction costs (commissions and service charges) on trades in securities. in such a simple world, investors will not care about the difference between returns from capital gains and those from dividends. No elaboration. All investors attempt to construct efficient frontier portfolios. All investors analyze securities in the same way and share the same economic view of the world. Hence, they all end with identical estimates of the probability distribution of future cash flows from investing in the available securities. this means that, given a set of security prices and risk-free interest rate, all investors use the same expected returns, standard deviations, and correlations to generate the efficient frontier and the unique optimal risky portfolio. Say right at the start that the assumptions of the CAPM are highly unrealistic. That is not surprising because all theoretical models are abstraction from reality, i.e., based on unrealistic assumptions. The way to judge a model is to see how well it’s predictions describe the real world. Thus, it’s important to test the model. A number of simplifying assumptions lead to the basic version of the CAPM. The fundamental idea is that individuals are as alike as possible, with notable exceptions of initial wealth with risk aversion. Investors cannot affect prices by their individual trades. This means that there are many investors, each with an endowment of wealth that is small compared with the total endowment of all investors. This assumption is analogous to the perfect competition assumption of microeconomics. All investors plan for one identical holding period. Investors form portfolios from a universe of publicly traded financial assets, such as stocks and bonds and have access to unlimited risk-free borrowing or lending opportunities. Investors pay neither taxes on returns nor transaction costs (commissions and service charges) on trades in securities. in such a simple world, investors will not care about the difference between returns from capital gains and those from dividends. No elaboration. All investors attempt to construct efficient frontier portfolios. All investors analyze securities in the same way and share the same economic view of the world. Hence, they all end with identical estimates of the probability distribution of future cash flows from investing in the available securities. this means that, given a set of security prices and risk-free interest rate, all investors use the same expected returns, standard deviations, and correlations to generate the efficient frontier and the unique optimal risky portfolio.

20. 20 Implications of the CAPM All investors will hold the same portfolio for risky assets – market portfolio (M). Market portfolio contains all securities. The proportion of each security is its market value as a percentage of total market value of all securities. Market portfolio will be on the efficient frontier and will be the tangency portfolio. Capital Allocation Line is now called the Capital Market Line (CML). These implications are the implications from the equilibrium in the CAPM world. Market value of a security = price per share times the number of shares outstanding. All investors will choose to hold the market portfolio (M), which includes all assets of the security universe. The proportion of each security in the market portfolio equals the market value of the security divided by the total market value of all stocks. The market portfolio will be on the efficient frontier. Moreover it will be the tangency portfolio. All investors hold M as their optimal risky portfolio, differing only in the amount invested in it as compared to investment in the risk-free asset. These implications are the implications from the equilibrium in the CAPM world. Market value of a security = price per share times the number of shares outstanding. All investors will choose to hold the market portfolio (M), which includes all assets of the security universe. The proportion of each security in the market portfolio equals the market value of the security divided by the total market value of all stocks. The market portfolio will be on the efficient frontier. Moreover it will be the tangency portfolio. All investors hold M as their optimal risky portfolio, differing only in the amount invested in it as compared to investment in the risk-free asset.

21. 21 Implications of the CAPM (cont’d) Risk premium on the market depends on: Average risk aversion of all market participants. Variance of the market portfolio. Risk premium on an individual security depends on: Its beta, b Market portfolio risk premium 3. The risk premium on the market portfolio, E(rM) – rf = A x ??M2 where ?M is the SD of the return on the market portfolio and A represents the degree of risk aversion of the average investor. 4. The risk premium of an individual security, E(ri ) – rf = ?i x [ E(rM) – rf ] Beta is the same beta that we looked at when we studied the single-index model. As a reminder, beta measures the extent to which returns on the stock respond to returns of the market portfolio. Formally, beta is the regression (slope) coefficient of the security return on the market portfolio return, representing the sensitivity of the stock return to fluctuations in the overall security market. 3. The risk premium on the market portfolio, E(rM) – rf = A x ??M2 where ?M is the SD of the return on the market portfolio and A represents the degree of risk aversion of the average investor. 4. The risk premium of an individual security, E(ri ) – rf = ?i x [ E(rM) – rf ] Beta is the same beta that we looked at when we studied the single-index model. As a reminder, beta measures the extent to which returns on the stock respond to returns of the market portfolio. Formally, beta is the regression (slope) coefficient of the security return on the market portfolio return, representing the sensitivity of the stock return to fluctuations in the overall security market.

22. 22 Why all investors hold the market portfolio? All investors Use identical mean-variance analysis Apply it to the same set of securities Have the same time horizon Use the same security analysis Have identical tax consequences ?Arrive at the same efficient frontier and tangency portfolio. With everyone choosing to hold the same risky portfolio, stocks will be represented in the aggregate risky portfolio in the same proportion as they are in each investor’s common risky portfolio. If GM represents 1% in each common risky portfolio, GM will be 1% of the aggregate risky portfolio. This is in fact the market portfolio since the market is no more than the aggregate of all individual portfolios. Because each investor uses the market portfolio for the optimal risk portfolio, the CAL in this case is called the capital market line or CML (see next slide). The market portfolio is the aggregate of all individual portfolios. All individual portfolios are identical and is the tangency portfolio. So the market portfolio must be the tangency portfolio. With everyone choosing to hold the same risky portfolio, stocks will be represented in the aggregate risky portfolio in the same proportion as they are in each investor’s common risky portfolio. If GM represents 1% in each common risky portfolio, GM will be 1% of the aggregate risky portfolio. This is in fact the market portfolio since the market is no more than the aggregate of all individual portfolios. Because each investor uses the market portfolio for the optimal risk portfolio, the CAL in this case is called the capital market line or CML (see next slide). The market portfolio is the aggregate of all individual portfolios. All individual portfolios are identical and is the tangency portfolio. So the market portfolio must be the tangency portfolio.

23. 23 The Efficient Frontier and the Capital Market Line

24. 24 Risk premium of market portfolio E(rM) – rf = A* x ?(?M)2 Expected return: E(rM) = rf + (A* x ??M2 ) Explanation (focuses on explaining why risk premium depends on average risk premium): p.207Explanation (focuses on explaining why risk premium depends on average risk premium): p.207

25. 25 Risk premium & expected return of single security For security i, risk premium: E(ri) – rf = ?i [ E(rM) – rf ] Security i’s expected return: E(ri) = rf + ?i [ E(rM) – rf ] Expected return-beta relationship Formulas also hold for a portfolio of securities. We know that unsystematic risk can be reduced to an arbitrarily low level through diversification; therefore investors with well-diversified portfolios do not require a risk premium as compensation for bearing unsystematic risk. They only need to be compensated for bearing systematic risk. To compute this compensation, we need an appropriate systematic risk measure. From the discussion of the single-index model, we know that the appropriate systematic risk measure of a security is its beta BECAUSE beta is proportional to the systematic risk which the security contributes to the portfolio. Therefore, it’s not surprising that the risk premium of an asset is proportional to its beta. For example, if you double a security’s systematic risk, you must double its risk premium for investors still to be willing to hold the security. If you half a security’s systematic risk, you also half its risk premium. Thus, the ratio of risk premium to beta should be the same for any two securities or portfolios. E.g., if we were to compare the ratio of risk premium to systematic risk for the market portfolio, which has a beta of 1.0 with the corresponding ratio for Dell stock, we would conclude that E(rM) – rf / 1 = E(rD) – rf / ?D Rearranging this relationship results in the CAPM’s expected return-beta relationship, E(ri) = rf + ?i [ E(rM) – rf ] This says that the expected rate of return on any asset exceeds the risk-free rate by a risk premium equal to the asset’s systematic risk measure (its beta) times the risk premium of the market portfolio. This expected return-beta relationship is the most familiar expression of the CAPM. Stress that beta of the market portfolio is 1 because the market portfolio has to respond in a 1-to-1 fashion to itself. Can also use the expected-return beta relationship to show this. If the expected return-beta relationship holds for any individual asset, it must hold for any combination of assets. The beta of a portfolio is simply the weighted average of the betas of the stocks in the portfolio, using as weights the portfolio proportions. Thus, beta also predicts the portfolio’s risk premium in accordance with what is shown on the slide. Stress this to make sure students see the BIG PICTURE. Key economic idea: the higher the (systematic) risk as measured by beta, the higher the expected return that investors want in equilibrium. So this is very intuitive because one would expect risk-averse investors to demand a higher reward (expected return) if they have to bear more risk. We know that unsystematic risk can be reduced to an arbitrarily low level through diversification; therefore investors with well-diversified portfolios do not require a risk premium as compensation for bearing unsystematic risk. They only need to be compensated for bearing systematic risk. To compute this compensation, we need an appropriate systematic risk measure. From the discussion of the single-index model, we know that the appropriate systematic risk measure of a security is its beta BECAUSE beta is proportional to the systematic risk which the security contributes to the portfolio. Therefore, it’s not surprising that the risk premium of an asset is proportional to its beta. For example, if you double a security’s systematic risk, you must double its risk premium for investors still to be willing to hold the security. If you half a security’s systematic risk, you also half its risk premium. Thus, the ratio of risk premium to beta should be the same for any two securities or portfolios. E.g., if we were to compare the ratio of risk premium to systematic risk for the market portfolio, which has a beta of 1.0 with the corresponding ratio for Dell stock, we would conclude that E(rM) – rf / 1 = E(rD) – rf / ?D Rearranging this relationship results in the CAPM’s expected return-beta relationship, E(ri) = rf + ?i [ E(rM) – rf ] This says that the expected rate of return on any asset exceeds the risk-free rate by a risk premium equal to the asset’s systematic risk measure (its beta) times the risk premium of the market portfolio. This expected return-beta relationship is the most familiar expression of the CAPM. Stress that beta of the market portfolio is 1 because the market portfolio has to respond in a 1-to-1 fashion to itself. Can also use the expected-return beta relationship to show this. If the expected return-beta relationship holds for any individual asset, it must hold for any combination of assets. The beta of a portfolio is simply the weighted average of the betas of the stocks in the portfolio, using as weights the portfolio proportions. Thus, beta also predicts the portfolio’s risk premium in accordance with what is shown on the slide. Stress this to make sure students see the BIG PICTURE. Key economic idea: the higher the (systematic) risk as measured by beta, the higher the expected return that investors want in equilibrium. So this is very intuitive because one would expect risk-averse investors to demand a higher reward (expected return) if they have to bear more risk.

26. 26 Portfolio beta, ?P The beta of a portfolio, P, Weighted average of the individual asset betas. Weights are the portfolio proportions. ?P = (w1 x ?1) + (w2 x ?2) + … + (wN x ?N) To compute risk premium and expected return of a portfolio, first we need to compute the portfolio beta. Suppose portfolio P has N assets, then portfolio beta is computed as shown. To compute risk premium and expected return of a portfolio, first we need to compute the portfolio beta. Suppose portfolio P has N assets, then portfolio beta is computed as shown.

27. 27 Problem involving portfolio beta Suppose the market risk premium is 8%. What is the risk premium of a portfolio invested 25% in Coca-Cola and 75% in BellSouth. Coca-Cola has a beta of 0.85 and BellSouth has a beta of 1.2. Verify that portfolio beta is 1.1125 Verify that risk premium is 8.9% Suppose you now invest 20% in the risk-free asset, 25% in Coca-Cola and the rest in BellSouth. What is the portfolio beta and risk premium? Using the formula from the last slide, portfolio beta = (0.25 x 0.85) + (0.75 x 1.2) = 1.1125 With a beta of 1.1125, portfolio risk premium is = 1.1125 x 8 = 8.9% For second part, re-compute the portfolio beta. Note that risk-free has beta of zero. Therefore, portfolio beta = (0.25 x 0.85) + ((1-0.25-0.2) x 1.2) = 0.2125 + 0.66 = 0.8725 Portfolio risk premium = 0.8725 x 8 = 6.98% Putting some of the portfolio into riskfree and reducing weight on Bellsouth reduce the portfolio systematic risk and hence reduce the risk premium. Using the formula from the last slide, portfolio beta = (0.25 x 0.85) + (0.75 x 1.2) = 1.1125 With a beta of 1.1125, portfolio risk premium is = 1.1125 x 8 = 8.9% For second part, re-compute the portfolio beta. Note that risk-free has beta of zero. Therefore, portfolio beta = (0.25 x 0.85) + ((1-0.25-0.2) x 1.2) = 0.2125 + 0.66 = 0.8725 Portfolio risk premium = 0.8725 x 8 = 6.98% Putting some of the portfolio into riskfree and reducing weight on Bellsouth reduce the portfolio systematic risk and hence reduce the risk premium.

28. 28 Questions Are the following statements true or false? Explain. Stocks with a beta of zero offer an expected rate of return of zero. The CAPM implies that investors require a higher return to hold highly volatile securities. You can construct a portfolio with a beta of 0.75 by investing 0.75 of the budget in T-bills and the remainder in the market portfolio. Chp 6 EOC Q6 For option (b), highly volatile securities mean high standard deviation (total risk). False. ? = 0 implies E(r) = rf , not zero. False. Investors require a risk premium for bearing systematic (i.e., market or undiversifiable) risk. C.False. You should invest 0.75 of your portfolio in the market portfolio, and the remainder in T-bills. Then: ?P = (0.75 ´ 1) + (0.25 ´ 0) = 0.75 Chp 6 EOC Q6 For option (b), highly volatile securities mean high standard deviation (total risk). False. ? = 0 implies E(r) = rf , not zero. False. Investors require a risk premium for bearing systematic (i.e., market or undiversifiable) risk. C.False. You should invest 0.75 of your portfolio in the market portfolio, and the remainder in T-bills. Then: ?P = (0.75 ´ 1) + (0.25 ´ 0) = 0.75

29. 29 More questions What is the beta of a portfolio with E(rP) = 20%, if rf = 5% and E(rM) = 15%. Assume both portfolios A and B are well diversified, that E(rA) = 14% and E(rB) = 14.8%. If the economy has only one factor, and ?A = 1.0 while ?B = 1.1, what must be the risk-free rate? 1) EOC Q3 Remind students that SML applies to both securities and assets. E(rP) = rf + b[E(rM) – rf ] 20% = 5% + b(15% – 5%) ? b = 15/10 = 1.5 2) EOC Q24 Take the factor to be the market portfolio and apply the SML. Substituting the portfolio returns and betas in the expected return-beta relationship, we obtain two equations in the unknowns, the risk-free rate (rf ) and the factor return (F): 14.0% = rf + 1 ? (F – rf ) 14.8% = rf + 1.1 ? (F – rf ) From the first equation we find that F = 14%. Substituting this value for F into the second equation, we get: 14.8% = rf + 1.1 ? (14% – rf ) 14.8 = rf + 15.4 – 1.1rf -0.6 = -0.1rf Rf = -0.6/-0.1 = 6%1) EOC Q3 Remind students that SML applies to both securities and assets. E(rP) = rf + b[E(rM) – rf ] 20% = 5% + b(15% – 5%) ? b = 15/10 = 1.5 2) EOC Q24 Take the factor to be the market portfolio and apply the SML. Substituting the portfolio returns and betas in the expected return-beta relationship, we obtain two equations in the unknowns, the risk-free rate (rf ) and the factor return (F): 14.0% = rf + 1 ? (F – rf ) 14.8% = rf + 1.1 ? (F – rf ) From the first equation we find that F = 14%. Substituting this value for F into the second equation, we get: 14.8% = rf + 1.1 ? (14% – rf ) 14.8 = rf + 15.4 – 1.1rf -0.6 = -0.1rf Rf = -0.6/-0.1 = 6%

30. 30 Security market line (SML) The SML graphically shows that the higher the risk of a security (beta), the higher the expected return. Graphically depicts the relationship between beta and expected return. When beta is 1, the asset must earn the same expected return as the market portfolio because it has the same systematic risk as the market portfolio. So, one point on the SML is beta=1, expected return = E(rM). Also, if beta = 0, the asset has no risk, so it must only earn the risk-free rate. So another point on the SML is beta = 0, expected return = rf. Using these two points, we can compute the slope using the rise over run formula. Slope = rise/run = (E(rM) – rf )/(1 – 0) = E(rM) – rf = market risk premium. CML vs SML The SML graphically shows that the higher the risk of a security (beta), the higher the expected return. Graphically depicts the relationship between beta and expected return. When beta is 1, the asset must earn the same expected return as the market portfolio because it has the same systematic risk as the market portfolio. So, one point on the SML is beta=1, expected return = E(rM). Also, if beta = 0, the asset has no risk, so it must only earn the risk-free rate. So another point on the SML is beta = 0, expected return = rf. Using these two points, we can compute the slope using the rise over run formula. Slope = rise/run = (E(rM) – rf )/(1 – 0) = E(rM) – rf = market risk premium. CML vs SML

31. 31 Security market line (SML) Graphical representation of the expected return-beta relationship. Provides a benchmark for evaluating investment performance. Given an investment’s beta, the SML tells us the return we should require or demand from this investment. “Fairly priced” assets plot exactly on the SML. “Underpriced” assets plot above the SML. “Overpriced” assets plot below the SML. Point 1 needs no explanation. Point 2: given the risk of an investment as measured by its beta, the SML provides the required rate of return that will compensate investors for the risk of that investment, as well as for the time value of money (riskfree rate). Point 3: The expected returns of fairly priced assets are commensurate with their risk. For a fairly priced asset, its expected return based on its current price and future cashflows is exactly equal to the return that the CAPM says investor should require based its risk (beta). Whenever the CAPM holds, all securities must lie on the SML in equilibrium. Underpriced assets plot above the SML: Given their risk (betas), their expected returns are greater than is indicated by the CAPM. Overpriced assets plot blow the SML: Given their risk (betas), their expected returns are lower than is indicated by the CAPM. In the next few slides, I’ll illustrate what fairly, under and overpriced mean with an example and graphs. Point 1 needs no explanation. Point 2: given the risk of an investment as measured by its beta, the SML provides the required rate of return that will compensate investors for the risk of that investment, as well as for the time value of money (riskfree rate). Point 3: The expected returns of fairly priced assets are commensurate with their risk. For a fairly priced asset, its expected return based on its current price and future cashflows is exactly equal to the return that the CAPM says investor should require based its risk (beta). Whenever the CAPM holds, all securities must lie on the SML in equilibrium. Underpriced assets plot above the SML: Given their risk (betas), their expected returns are greater than is indicated by the CAPM. Overpriced assets plot blow the SML: Given their risk (betas), their expected returns are lower than is indicated by the CAPM. In the next few slides, I’ll illustrate what fairly, under and overpriced mean with an example and graphs.

32. 32 Fairly priced security Suppose stock A is currently priced at $45. Everyone agrees that its year-end price will be $50 and a dividend of $2.02 will be paid then. Stock A’s beta is 1.2, the market portfolio’s expected return is 14% and the risk-free rate is 6%. Given the stock price and future cash flows, expected HPR = (50 + 2.02 – 45)/45 = 0.156 or 15.6% CAPM’s expected return = 6 + 1.2[14 – 6] = 15.6% ! Stock A is FAIRLY PRICED. Its current price leads to an expected return that is EXACTLY equal to the expected return indicated by the CAPM. So, CAPM says that $45 is the “correct” price for A given it’s risk. This stock plots exactly on the SML. Example to illustrate the concepts of fairly priced, underpriced and overpriced.Example to illustrate the concepts of fairly priced, underpriced and overpriced.

33. 33 Underpriced security Now supposed everything stays the same, BUT stock A is now priced at $44.46. Given the stock price and future cash flows, expected HPR = (50 + 2.02 – 44.46)/44.46 = 0.17 or 17% But CAPM says stock A should provide 15.6% return and it should be worth $45. Stock A is UNDERPRICED. It’s price of $44.46 is LESS than the CAPM price of $45. Also, A’s expected return given its price is 17% which is MORE than the CAPM expected return of 15.6% Stock A now plots ABOVE the SML.

34. 34 Overpriced security Now supposed everything stays the same, BUT stock A is now priced at $46.04. Given the stock price and future cash flows, expected HPR = (50 + 2.02 – 46.04)/46.04 = 0.13 or 13% But CAPM says stock A should provide 15.6% return and it should be worth $45. Stock A is OVERPRICED. It’s price of $46.04 is MORE than the CAPM price of $45. Also, A’s expected return given its price is 13% which is LESS than the CAPM expected return of 15.6% Stock A now plots BELOW the SML.

35. 35 All together now By comparing the actually expected return (based on current price and future cash flows) with the expected return indicated by the CAPM, we can compute what is called an asset’s Alpha. Alphas in each scenario is computed in column 5. Alpha: the abnormal rate of return on a security in excess of what would be predicted by an equilibrium model such as the CAPM. For fairly priced, alpha = 0. For underpriced, alpha is positive. For overpriced, alpha is negative. Graphically, a stock’s alpha is indicated by the stock’s position relative to the SML, given the stock’s beta. This is shown on the graph on the next slide. There is a correspondence between alpha and where the stock plots relative to the SML. Fairly priced => alpha = 0 => plots on SML Underpriced => alpha > 0 => plots above SML Overpriced => alpha < 0 => plots below SML. By comparing the actually expected return (based on current price and future cash flows) with the expected return indicated by the CAPM, we can compute what is called an asset’s Alpha. Alphas in each scenario is computed in column 5. Alpha: the abnormal rate of return on a security in excess of what would be predicted by an equilibrium model such as the CAPM. For fairly priced, alpha = 0. For underpriced, alpha is positive. For overpriced, alpha is negative. Graphically, a stock’s alpha is indicated by the stock’s position relative to the SML, given the stock’s beta. This is shown on the graph on the next slide. There is a correspondence between alpha and where the stock plots relative to the SML. Fairly priced => alpha = 0 => plots on SML Underpriced => alpha > 0 => plots above SML Overpriced => alpha < 0 => plots below SML.

36. 36 Fairly priced, underpriced, overpriced

37. 37 Capital Allocation Line (CAL)

38. 38 Capital Market Line (CML) vs. Security Market Line (SML) CML Graphs risk premium of efficient portfolios against standard deviation. Efficient portfolios SD is the valid risk measure. SML Graphs risk premium of individual asset against beta. Individual assets, portfolios Beta is valid risk measure It is useful to compare the SML to the CML. The CML graphs the risk premiums of efficient portfolios (i.e., complete portfolios made up of the risky market portfolio and the risk-free asset) as a function of portfolio SD. This is appropriate because SD is a valid measure of risk for portfolios that are candidates for an investor’s complete (overall) portfolio. The SML, in contrast, graphs individual asset risk premiums as a function of asset risk. The relevant risk measure for individual assets (which are held as parts of a well-diversified portfolio) is not the asset’s SD; it is, instead, the contribution of the asset to the portfolio SD as measured by the asset’s beta. The SML is valid for both portfolios (need not be efficient) and individual assets.It is useful to compare the SML to the CML. The CML graphs the risk premiums of efficient portfolios (i.e., complete portfolios made up of the risky market portfolio and the risk-free asset) as a function of portfolio SD. This is appropriate because SD is a valid measure of risk for portfolios that are candidates for an investor’s complete (overall) portfolio. The SML, in contrast, graphs individual asset risk premiums as a function of asset risk. The relevant risk measure for individual assets (which are held as parts of a well-diversified portfolio) is not the asset’s SD; it is, instead, the contribution of the asset to the portfolio SD as measured by the asset’s beta. The SML is valid for both portfolios (need not be efficient) and individual assets.

39. 39 Questions (1) Which of the following statements about the SML are true? The SML provides a benchmark for evaluating expected investment performance. The SML leads all investors to invest in the same portfolio of risky assets. The SML is a graphic representation of the relationship between expected return and beta. Properly valued assets plot exactly on the SML. EOC Q1. Statements A, C, D are correct. B is not true, it is the CML that leads all investors to invest in the same portfolio of risky assets.EOC Q1. Statements A, C, D are correct. B is not true, it is the CML that leads all investors to invest in the same portfolio of risky assets.

40. 40 Questions (2) The SML depicts: A security’s expected return as a function of its systematic risk. The market portfolio as the optimal portfolio of risky securities. The relationship between a security’s return and the return on an index. The complete portfolio as a combination of the market portfolio and the risk-free asset. EOC, Q32, Answer: A only.EOC, Q32, Answer: A only.

41. 41 Applications of the CAPM Identification of attractive investments. Positive alpha / underpriced assets. Valuation of assets. CAPM expected return is used as discount rate to find the value of assets. Capital budgeting. “Hurdle rate” for project under consideration. Rate-setting for utilities. Rate of return that a regulated utility should be allowed to earn. The CAPM may be used in the investment management industry. Suppose the SML is taken as the benchmark to assess the fair expected return on a risky asset. Then an analyst calculates the return he or she actually expects. If an asset (e.g., a stock) is perceived to be underpriced (based on our preceding discussion), it will provide a positive alpha, that is an expected return in excess of the fair return stipulated by the SML. Such an asset will be identified as an attractive investment, i.e., a good buy. If an asset has a negative alpha, it’s overpriced and a poor investment. Such an asset will be avoided. If the manager can short sell, he/she should short sell the asset. From FI3300, we know that we can use the required rate of return as the discount rate to find the PV of financial securities like stocks and bonds. The CAPM is also useful for capital budgeting decisions. If a firm is considering a new project, the CAPM can provide the return that project needs to yield to be acceptable to investors. Managers can use the CAPM to obtain this cutoff IRR or “hurdle rate” for the project. Another use of the CAPM is in utility rate-making cases. Here the issue is the rate of return a regulated utility should be allowed to earn on its investment in plant and equipment. Application 4 is familiar to students since they study valuation of stocks and bonds in FI3300. So will only elaborate on the first three applications in the following slides.The CAPM may be used in the investment management industry. Suppose the SML is taken as the benchmark to assess the fair expected return on a risky asset. Then an analyst calculates the return he or she actually expects. If an asset (e.g., a stock) is perceived to be underpriced (based on our preceding discussion), it will provide a positive alpha, that is an expected return in excess of the fair return stipulated by the SML. Such an asset will be identified as an attractive investment, i.e., a good buy. If an asset has a negative alpha, it’s overpriced and a poor investment. Such an asset will be avoided. If the manager can short sell, he/she should short sell the asset. From FI3300, we know that we can use the required rate of return as the discount rate to find the PV of financial securities like stocks and bonds. The CAPM is also useful for capital budgeting decisions. If a firm is considering a new project, the CAPM can provide the return that project needs to yield to be acceptable to investors. Managers can use the CAPM to obtain this cutoff IRR or “hurdle rate” for the project. Another use of the CAPM is in utility rate-making cases. Here the issue is the rate of return a regulated utility should be allowed to earn on its investment in plant and equipment. Application 4 is familiar to students since they study valuation of stocks and bonds in FI3300. So will only elaborate on the first three applications in the following slides.

42. 42 Identifying attractive investment (1) Karen Kay, a portfolio manager at Collins Asset Management, is using the CAPM for making recommendations to her clients. Her research department has developed the information shown in the following exhibit EOC Q2 EOC Q2

43. 43 Identifying attractive investment (2) Calculate expected return and alpha for each stock. Identify and justify which stock would be more appropriate for an investor who wants to: Add this stock to a well-diversified equity portfolio. Hold this stock as a single-stock portfolio. EOC Q2 a) E(rX) = 5% + 0.8(14% – 5%) = 12.2% ??X = 14% – 12.2% = 1.8% E(rY) = 5% + 1.5(14% – 5%) = 18.5% ???Y = 17% – 18.5% = –1.5% b). i. For an investor who wants to add this stock to a well-diversified equity portfolio, Kay should recommend Stock X because of its positive alpha, while Stock Y has a negative alpha. In graphical terms, Stock X’s expected return/risk profile plots above the SML, while Stock Y’s profile plots below the SML. Also, depending on the individual risk preferences of Kay’s clients, Stock X’s lower beta may have a beneficial impact on overall portfolio risk. ii. For an investor who wants to hold this stock as a single-stock portfolio, Kay should recommend Stock Y, because it has higher forecasted return and lower standard deviation than Stock X. Stock Y’s Sharpe ratio is: (0.17 – 0.05)/0.25 = 0.48 Stock X’s Sharpe ratio is only: (0.14 – 0.05)/0.36 = 0.25 The market index has an even more attractive Sharpe ratio: (0.14 – 0.05)/0.15 = 0.60 However, given the choice between Stock X and Y, Y is superior. When a stock is held in isolation, standard deviation is the relevant risk measure. For assets held in isolation, beta as a measure of risk is irrelevant. Although holding a single asset in isolation is not typically a recommended investment strategy, some investors may hold what is essentially a single-asset portfolio (e.g., the stock of their employer company). For such investors, the relevance of standard deviation versus beta is an important issue.EOC Q2 a) E(rX) = 5% + 0.8(14% – 5%) = 12.2% ??X = 14% – 12.2% = 1.8% E(rY) = 5% + 1.5(14% – 5%) = 18.5% ???Y = 17% – 18.5% = –1.5% b). i. For an investor who wants to add this stock to a well-diversified equity portfolio, Kay should recommend Stock X because of its positive alpha, while Stock Y has a negative alpha. In graphical terms, Stock X’s expected return/risk profile plots above the SML, while Stock Y’s profile plots below the SML. Also, depending on the individual risk preferences of Kay’s clients, Stock X’s lower beta may have a beneficial impact on overall portfolio risk. ii. For an investor who wants to hold this stock as a single-stock portfolio, Kay should recommend Stock Y, because it has higher forecasted return and lower standard deviation than Stock X. Stock Y’s Sharpe ratio is: (0.17 – 0.05)/0.25 = 0.48 Stock X’s Sharpe ratio is only: (0.14 – 0.05)/0.36 = 0.25 The market index has an even more attractive Sharpe ratio: (0.14 – 0.05)/0.15 = 0.60 However, given the choice between Stock X and Y, Y is superior. When a stock is held in isolation, standard deviation is the relevant risk measure. For assets held in isolation, beta as a measure of risk is irrelevant. Although holding a single asset in isolation is not typically a recommended investment strategy, some investors may hold what is essentially a single-asset portfolio (e.g., the stock of their employer company). For such investors, the relevance of standard deviation versus beta is an important issue.

44. 44 Valuation of assets (1) The market price of a security is $40. its expected rate of return is 13%. The risk-free rate is 7%, and the market risk premium is 8%. What will the market price of the security be if its beta doubles (and all other variables remain unchanged)? Assume the stock is expected to pay a constant dividend in perpetuity. EOC Q4 Assume that the security is fairly priced. If the beta of the security doubles, then so will its risk premium. The current risk premium for the stock is: (13% - 7%) = 6%, so the new risk premium would be 12%, and the new discount rate for the security would be: 12% + 7% = 19% If the stock pays a constant dividend in perpetuity, then we know from the original data that the dividend (D) must satisfy the equation for a perpetuity: Price = Dividend/Discount rate 40 = D/0.13 ? D = 40 x 0.13 = $5.20 At the new discount rate of 19%, the stock would be worth: $5.20/0.19 = $27.37 The increase in stock risk has lowered the value of the stock by 31.58%. EOC Q4 Assume that the security is fairly priced. If the beta of the security doubles, then so will its risk premium. The current risk premium for the stock is: (13% - 7%) = 6%, so the new risk premium would be 12%, and the new discount rate for the security would be: 12% + 7% = 19% If the stock pays a constant dividend in perpetuity, then we know from the original data that the dividend (D) must satisfy the equation for a perpetuity: Price = Dividend/Discount rate 40 = D/0.13 ? D = 40 x 0.13 = $5.20 At the new discount rate of 19%, the stock would be worth: $5.20/0.19 = $27.37 The increase in stock risk has lowered the value of the stock by 31.58%.

45. 45 Valuation of assets (2) The risk-free rate is 4%. Suppose that the expected return required by the market for a portfolio with a beta of 1.0 is 12%. According to the CAPM: What is the expected return on the market portfolio? What would be the expected return on a zero-beta stock? Suppose you consider buying a share of stock at a price of $40. the stock is expected to pay a dividend of $3 next year and to sell then for $41. The stock’s beta is -0.5. is the stock overpriced or underpriced? A) Since the market portfolio, by definition, has a beta of 1.0, its expected rate of return is 12%. B) beta = 0 means the stock has no systematic risk. Hence, the portfolio's expected rate of return is the risk-free rate, 4%. C) Using the SML, the fair rate of return for a stock with b= –0.5 is: E(r) = 4% + (–0.5)(12% – 4%) = 0.0% Value of the stock under the CAPM = PV of future cash using 0% as discount rate. Value of stock = (41 + 3)/1 = $44 The price is $40 < the CAPM price of $44. Therefore, stock is underpriced. Another way to check is to compute the expected return using the cash flows. The expected rate of return, using the expected price and dividend for next year: E(r) = ($44/$40) – 1 = 0.10 = 10% Because the expected return exceeds the fair return (0%), the stock must be under-priced.A) Since the market portfolio, by definition, has a beta of 1.0, its expected rate of return is 12%. B) beta = 0 means the stock has no systematic risk. Hence, the portfolio's expected rate of return is the risk-free rate, 4%. C) Using the SML, the fair rate of return for a stock with b= –0.5 is: E(r) = 4% + (–0.5)(12% – 4%) = 0.0% Value of the stock under the CAPM = PV of future cash using 0% as discount rate. Value of stock = (41 + 3)/1 = $44 The price is $40 < the CAPM price of $44. Therefore, stock is underpriced. Another way to check is to compute the expected return using the cash flows. The expected rate of return, using the expected price and dividend for next year: E(r) = ($44/$40) – 1 = 0.10 = 10% Because the expected return exceeds the fair return (0%), the stock must be under-priced.

46. 46 Capital budgeting (1) Texaco is considering a new oil rig in the Gulf of Mexico. The business plan forecasts an internal rate of return of 17% on the investment. Research shows that the beta of similar projects is a whopping 2.2! The risk-free rate is 4% and the market risk premium is 8%. Compute the hurdle rate for the project. Should the project be accepted? Based on example 7.5 CAPM expected rate of return = 4 + 2.2 x [8] = 21.6%. Since the hurdle rate is higher than the IRR, the project should not be accepted. The hurdle rate has the same role as the cost of capital in FI3300 chapter 11. Based on example 7.5 CAPM expected rate of return = 4 + 2.2 x [8] = 21.6%. Since the hurdle rate is higher than the IRR, the project should not be accepted. The hurdle rate has the same role as the cost of capital in FI3300 chapter 11.

47. 47 Capital budgeting (2) You are an analyst at Goldman Sachs. You are covering a company which is considering a project with the following net after-tax cash flows : an outlay of $20 mil right now (t= 0), $10 mil at the end of each of the first nine years and $20 mil at the end of year 10. The project is terminated after 10 years. The project’s beta is 1.7. Assuming rf = 9% and E(rM) = 19%, what is the project’s NPV? EOC Q5 The appropriate discount rate for the project is: rf + b[E(rM) – rf ] = 9% + 1.7(19% – 9%) = 26% Solve NPV using 26% as the discount rate. In TI BA II Plus, do: CF0 = -20 C1 = 10, F1 = 9 C2 = 20, F2 = 1 I = 26, CPT, NPV = $15.6396 mil To answer the second question, first find the project’s IRR, with the cashflow information still in the financial calculator, press, IRR, CPT, you get IRR = 49.5492%. Expected return must go as high as 49.5492% before NPV becomes negative. Now back out the beta. 49.5492 = 9 + b x [19 – 9] 49.5492 – 9 = b x 10 B = (49.5492 – 9)/10 = 4.05492, 4.05 (to 2 d.p.)EOC Q5 The appropriate discount rate for the project is: rf + b[E(rM) – rf ] = 9% + 1.7(19% – 9%) = 26% Solve NPV using 26% as the discount rate. In TI BA II Plus, do: CF0 = -20 C1 = 10, F1 = 9 C2 = 20, F2 = 1 I = 26, CPT, NPV = $15.6396 mil To answer the second question, first find the project’s IRR, with the cashflow information still in the financial calculator, press, IRR, CPT, you get IRR = 49.5492%. Expected return must go as high as 49.5492% before NPV becomes negative. Now back out the beta. 49.5492 = 9 + b x [19 – 9] 49.5492 – 9 = b x 10 B = (49.5492 – 9)/10 = 4.05492, 4.05 (to 2 d.p.)

48. 48 Rate-setting for utilities Example: Suppose GPower is a utility providing power to Georgia. The state of Georgia decides the rate of return that GPower can earn on its investment in plant and equipment. Suppose the firm is 100% equity-owned and assets are $100 million. The firm’s beta is 0.8. The risk-free rate is 5% and the market risk premium is 8%. If regulators use the CAPM for setting rates, they will allow GPower to set prices at a level to generate what level of profits? If the firm has debt, we want the cost of capital to compute the fair annual profit. For simplicity assume that the firm is 100% equity owned. Compute the CAPM expected rate of return for GPower. Expected return = 5 + (0.8 x 8) = 5 + 6.4 = 11.4% [CAPM determined ROE=ROA because equity=assets]. The fair annual profit will be 11.4% of $100 million. Annual profit = 0.114 x 100 = $11.4 million. So regulators will allow GPower to set rates such that it earns a profit of $11.4 million on its assets. If the firm has debt, we want the cost of capital to compute the fair annual profit. For simplicity assume that the firm is 100% equity owned. Compute the CAPM expected rate of return for GPower. Expected return = 5 + (0.8 x 8) = 5 + 6.4 = 11.4% [CAPM determined ROE=ROA because equity=assets]. The fair annual profit will be 11.4% of $100 million. Annual profit = 0.114 x 100 = $11.4 million. So regulators will allow GPower to set rates such that it earns a profit of $11.4 million on its assets.

49. 49 Implementing the CAPM (1) The CAPM can be used in several ways. To apply it, we need to calculate a security’s beta. There are two major problems in calculating beta: Market portfolio of all assets is unobservable. The CAPM gives a relationship between expected return and beta. Expected returns are unobservable. How do we get round these problems? 1. The CAPM market portfolio includes all assets (such as real estate, foreign stocks, human capital etc) and is unobservable i.e., composition is unknown. Even if one can exactly identify the market portfolio, many of the component assets are not traded. So investors would not have full access to the market portfolio even if they could exactly identify it. 2. The CAPM deals with expected as opposed to actual returns. However expectations cannot be observed. 1. The CAPM market portfolio includes all assets (such as real estate, foreign stocks, human capital etc) and is unobservable i.e., composition is unknown. Even if one can exactly identify the market portfolio, many of the component assets are not traded. So investors would not have full access to the market portfolio even if they could exactly identify it. 2. The CAPM deals with expected as opposed to actual returns. However expectations cannot be observed.

50. 50 Implementing the CAPM (2) Solutions: Use an actual portfolio, e.g., S&P 500 stock index, as a proxy for the theoretical market portfolio. The proxy represents systematic risk in the economy. Use realized (i.e., past) returns instead of expected returns to estimate beta. . The advantages of an actual portfolio or index is that the composition is known and the rate of return is easily measured and unambiguous. The advantages of an actual portfolio or index is that the composition is known and the rate of return is easily measured and unambiguous.

51. 51 Implementing the CAPM (3) Use linear regression to estimate the single-index model: ri - rf = ?i + ?i(RM-rf) + ei Remind student’s that this is just a repeat of the single-index model. For the moment, drop the t subscript. Strictly speaking, the regression model must have time subscripts to denote periods. Each period can be month, week, or day. Once we estimate the single index model using regression, we can an estimate of beta. We can then use beta in the various applications we talked about earlier. Of course, you also need the expected return on the market proxy and the risk-free rate, but we assume these things can be obtained. Remind student’s that this is just a repeat of the single-index model. For the moment, drop the t subscript. Strictly speaking, the regression model must have time subscripts to denote periods. Each period can be month, week, or day. Once we estimate the single index model using regression, we can an estimate of beta. We can then use beta in the various applications we talked about earlier. Of course, you also need the expected return on the market proxy and the risk-free rate, but we assume these things can be obtained.

52. 52 Implementing the CAPM (4) If we express the single-index model in terms of expectations, we get: E(ri) – rf = ?i + ?i [ E(rM) – rf ] Recall the CAPM equation: E(ri) – rf = ?i [ E(rM) – rf ] => The CAPM predicts that ?i = 0. Drop the t subscript and now only consider expected return for a specific period. By the definition of the single index model, E(ei,t) = 0.Drop the t subscript and now only consider expected return for a specific period. By the definition of the single index model, E(ei,t) = 0.

53. 53 Example of estimating beta Collect and process data (12/2000-12/2005) Monthly prices of Coca-Cola (KO) Monthly index values of S&P 500 (^GSPC) Get monthly risk-free rate from Prof Ken French’s website: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html Compute monthly excess returns of KO and ^GSPC. Use Excel’s Data Analysis tool to estimate the regression. Use the period 12/2000-12/2005 to collect data used to estimate the market model. We use the prices adjusted for dividends and split in order to obtain holding periods. In the class demonstration, make sure the risk-free rate is already downloaded. For any class exercise, provide all price series and risk-free rate to students. The excel file for this demonstration is “asset pricing workbook.xls”. The worksheet “Regression” already has the excess returns of KO and ^GSPC and also the regression output. When demonstrating to students, remove the output and do it right from the start.Use the period 12/2000-12/2005 to collect data used to estimate the market model. We use the prices adjusted for dividends and split in order to obtain holding periods. In the class demonstration, make sure the risk-free rate is already downloaded. For any class exercise, provide all price series and risk-free rate to students. The excel file for this demonstration is “asset pricing workbook.xls”. The worksheet “Regression” already has the excess returns of KO and ^GSPC and also the regression output. When demonstrating to students, remove the output and do it right from the start.

54. 54 Multifactor asset pricing models CAPM says that the market portfolio is the only source of systematic risk. But there can be multiple sources of systematic risk, e.g., fluctuation in interest rates, fluctuation in energy prices, uncertainty about inflation, etc. Multifactor asset pricing models try to do better than the CAPM by using more than one systematic risk factor to explain security returns. Multifactor models: Models of security returns positing that returns respond to several systematic factors. CAPM alone is not enough to explain how security returns behave. One way to improve is to consider multiple sources of systematic risk. We know the CAPM is not a perfect model and that ultimately, it will be far from the last word on security pricing. Still, the model’s logic is compelling, and more sophisticated models of security pricing all rely on the key distinction between systematic versus diversifiable risk. The CAPM therefore provides a useful framework for thinking rigorously about the relationship between security risk and return. Multifactor models: Models of security returns positing that returns respond to several systematic factors. CAPM alone is not enough to explain how security returns behave. One way to improve is to consider multiple sources of systematic risk. We know the CAPM is not a perfect model and that ultimately, it will be far from the last word on security pricing. Still, the model’s logic is compelling, and more sophisticated models of security pricing all rely on the key distinction between systematic versus diversifiable risk. The CAPM therefore provides a useful framework for thinking rigorously about the relationship between security risk and return.

55. 55 Summary Asset pricing models tell us how to figure a risky asset’s expected return. The CAPM is such a model. It says that there is only one source of systematic risk – the market. In the CAPM world, ? is the measure of risk for individual securities. CAPM says that the riskier the security, the higher the required/expected return for holding that asset. SML depicts the relationship between ? and expected return. CAPM can be used in a number of financial applications. Newer asset pricing models use more than one systematic risk factor.

56. Homework 4 Chapter 6: 20, 21. Chapter 7: 1,8 abcd, 9-15,17,20,21, CFA 6,7,8. 56