Solution Guide to Exercises for Chapter 6 The capital asset pricing model

Transcription

1 THE ECONOMICS OF FINANCIAL MARKETS R. E. BAILEY Solution Guide to Exercises for Chapter 6 The capital asset pricing model 1. The following information is provided for a stock market: µ j β j Asset 1 6.6% 0.4 Asset 2 9.8% 1.2 Asset % 1.8 Notation: µ j = expected rate of return on asset j; β j = beta-coefficient for asset j, j = 1, 2, 3. (a) In the context of the Capital Asset Pricing Model (CAPM), define the beta-coefficient, β j, corresponding to asset j. Discuss how assets beta-coefficients should be interpreted and explain how their values can be obtained in practice. The beta-coefficient can be defined in any of the following equivalent ways: β j = M σ 2 M = ρ jm σ 2 M = ρ jm, where M is the covariance between the rate of return on asset j and the market rate of return, is the standard deviation of the market rate of return, is the standard deviation of the rate of return on asset j, and ρ jm is the correlation coefficient between the rate of return on asset j and the market rate of return. An asset s beta-coefficient is a measure of the relationship between its rate of return and the market rate of return. It can be interpreted as a measure of the asset s risk, relative to the market as a whole. An asset s beta-coefficient is formally the slope co-efficient on the excess rate of return on the market in a regression of the excess rate of return on asset j on the excess rate of return on the market: r j = r 0 + (r M r 0 )β j + ε j, j = 1, 2,..., n, where ε j is an unobserved random error. It is assumed that E[ε j r M ] = 0, that is, the expected value of the error, conditional upon the rate of return on the market portfolio, is zero. Typically (almost always) beta-coefficients are estimated from data on past rates of return (in the regression described above). (b) Assuming that a risk-free asset is available, explain and interpret the Security Market Line (SML) in the context of the CAPM. Construct the SML from the given information and interpret the values of its coefficients. The CAPM predicts that: µ j = r 0 + (µ M r 0 )β j, where µ j is the expected rate of return on asset j, µ M is the expected rate of return on the market portfolio, and r 0 is the risk-free rate of return The SML treats µ j as a function of β j and shows how the expected rate of return on each asset differs according to its beta-coefficient. The slope of the SML is then a measure of the market price of risk. See figure 1. The data in the question must satisfy: = r (µ M r 0 ), and = r (µ M r 0 ).

2 µ j SML µ M r 0 1 β j Figure 1: The Security Market Line, SML Hence it must follow that: µ M = 0.09 and r 0 = Thus, in this example the market price of risk is 4%. Hence the SML is: µ j = β j. (Check that the data for asset 3 also satisfy the SML.) (c) Now suppose a risk-free asset is not available, although the other assumptions of the CAPM remain valid. How should the SML be constructed and interpreted in this case? The formal analysis is the same as for the previous part, except that now the intercept of the SML is interpreted as the expected rate of return on a zero beta portfolio (i.e., a portfolio for which the beta-coefficient is zero). Formally: µ j = ω + (µ M ω)β j, where ω denotes the expected rate of return on a zero beta portfolio. Essentially, the only difference is that the risk-free rate of return is replaced with ω. (Answers should include a brief interpretation of the ω in terms of the Black version of the CAPM check your lecture notes on this.) (d) You are informed that a fourth asset, with β 4 = 0.8, is available. Recent observations reveal that its average rate of return is 7.0%. What inferences, if any, would you draw from this information? [Your answer may be in the context of either (b) or (c), above.] The CAPM predicts that the expected rate of return on the fourth asset is: = But the observed average rate is 7.0% < 8.2%. Hence, the fourth asset is overpriced. This evidence could be indicative either that the market is in disequilibrium or that the CAPM is not a good representation of the market. 2. The following information is provided for a stock market: ρ jm Security A 50% 0.6 Security B 60% 0.2 Market Portfolio 20% 1.0

3 Notation: = standard deviation of the rate of return on asset j = A and j = B; ρ jm = correlation coefficient between the return on asset j and the return on the market portfolio. The mean rate of return on the market portfolio is 8% and the risk-free rate of return is 5%. (a) In the Capital Asset Pricing Model, explain what is meant by the Security Market Line, SML. Calculate the SML from the given information. The Securty Market Line, SML, expresses the relationship between the expected rate of return µ j on assets, or portfolios of assets, and their beta coefficients, β j. Each β j is defined by β j = cov(r j, r M )/var(r M ). In words, the beta coefficient for asset j is the covariance between its rate of return and the market rate of return divided by the variance of the market rate of return. The market rate of return expresses the rate of return on a portfolio in which every asset is represented in proportion to its capital value in the entire market. Thus the beta coefficients are measures of the linear relationship between the rate of return on assets and the rate of return on the market as a whole. Formally, β j can be interpreted as a regression coefficient for the rate of return on asset j as a function of the rate of return on the market portfolio, both being interpreted as rates in excess of a risk-free rate, r 0. The SML in the CAPM can be represented by: µ j = r 0 + (µ M r 0 )β j that is as a straight line with intercept r 0 and slope µ M r 0. From the given information: µ j = ( )β j (1) = β j (2) (b) In the Capital Asset Pricing Model, explain what is meant by the beta coefficient, β j, for a security. Calculate the beta coefficients for the two securities from the given information. As noted above, the β j are defined by: β j = cov(r j, r M )/var(r M ). (See above for an explanation.) From the definition of variances and covariances, it follows that β j = ρ jm / where is the standard deviation of the rate of return on asset j and ρ jm is the correlation coefficient between the rate of return on asset j and the rate of return on the market portfolio. β A = σ A ρ AM = 0.2 β B = σ B ρ BM = = 1.5 (3) = 0.6 (4) (c) You are told that the mean rates of return for securities A and B are 7.5% and 4.6% respectively. What would you infer from this information in the context of the Capital Asset Pricing Model? From the SML, it follows that the predicted rate of return on asset A equals = Given that the observed rate is it follows that asset A is overpriced it is predicted to yield more than is observed. From the SML, it follows that the predicted rate of return on asset B equals = Given that the observed rate is it follows that asset B is underpriced it is predicted to yield less than is observed.

4 Consequently, the evidence suggests that either (i) the markets are in disequilibrium (and offer profitable investment opportunities) or (ii) perhaps the CAPM is not a very good model for these asset markets, or both. 3. What are the main predictions of the Capital Asset Pricing Model (CAPM)? Discuss the role and significance of the assumptions needed to obtain the predictions. Guidance: This is a typical final examination question for which there are many correct answers of varying standards (as well as even more bad answers). What follows are some pointers about how you should set about answering a question like this: (a) Read the question carefully and try to answer it, not just write about the CAPM. This question focuses on the predictions of the CAPM and the underlying assumptions that generate the predictions. (b) Begin by defining the most important terms in the question. Then define the concepts you need. In answering this question, obviously you will concentrate on the CAPM. Describe, briefly, the CAPM in terms of its origins in mean-variance analysis. That is, the CAPM is a model of market equilibrium in which investors choose their portfolios according to a mean-variance criterion and in which they all agree about the means and variances (i.e. homogeneous beliefs). (c) Now you are ready to state the main predictions. These can be summarised according to the three lines : the Capital Market Line, the Characteristic Line and the Security Market Line. Your answer should contain a brief statement of each of these. (Refer to chapter 6 of EFM. Then put EFM aside and then try to write a short paragraph on each.) It would make your answer coherent to tie the predictions together in terms of the equation: µ j r 0 = (µ M r 0 )β j, where β j = ρ jm. In your answer be sure to define what the symbols mean! The Capital Market Line (CML) is such that j denotes an efficient portfolio. The rate of return on any efficient porfolio, say E, is perfectly correlated with the market return. Hence, β E = σ E / and the prediction becomes: µ E r 0 = µ M r 0, σ E which is the equation of the CML. The Characteristic Line, treats µ j r 0 as a function of µ M r 0, with slope β j. This is useful for estimating β j. The Security Market Line treats µ j r 0 as a function of β j, with slope µ M r 0. This is useful for testing the cross-section patterns of asset returns. (d) Next move on to describing the assumptions. While it is not wrong to just give a long list of assumptions, the examiners will be more impressed if you can group the assumptions into catagories and offer some appraisal of their role. (Check chapter 6 of EFM. Then put EFM aside and write a few paragraphs describing the assumptions.) (e) The crucial assumptions are (i) that there is market equilibrium in the sense of a balance between the demand and supply to hold assets, (ii) that all investors choose portfolios according to a mean-variance criterion, and (iii) that they have the same beliefs ( homogeneous or unanimous beliefs) about asset returns. (f) What is the role of these assumptions? The mean variance assumption implies that for each investor: µ j r 0 β j σ Z = µ Z r 0 σ Z or µ j r 0 = (µ Z r 0 )β j, j = 1, 2,..., n,

5 where Z is the efficient portfolio comprising risky assets only. Note that, without further assumptions, µ j, β j and Z could differ from one investor to another. In your answer you should now describe briefly (check EFM, chapter 6, if necessary) why in market equilibrium the portfolio Z can be understood as the market portfolio. Also, the assumption of homogeneous beliefs implies that µ j and β j are the same for each investor. (g) Finally, you could conclude your answer by briefly mentioning the extensions of the CAPM, for example to allow for cases when it is unreasonable to assume that all investors can borrow or lend at a risk-free rate, or to encompass intertemporal planning (Consumption CAPM). Now put EFM and your notes aside and try writing an answer yourself. This will benefit you much more than trying to memorise someone else s answer because in an examination you will almost certainly not be asked to answer this question, instead one based on the same material. While it may help you to memorise definitions, concepts and analysis, learn how to answer questions not to memorise answers! (Rote learning is not rewarded.) *****