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THE ECONOMICS OF FINANCIAL MARKETS R. E. BAILEY
Solution Guide to Exercises for
Chapter 6 The capital asset pricing model
- The following information is provided for a stock market:
μj βj Asset 1 6.6% 0. Asset 2 9.8% 1. Asset 3 12.2% 1.
Notation: μj = expected rate of return on asset j; βj = beta-coefficient for asset j, j = 1, 2 , 3.
(a) In the context of the Capital Asset Pricing Model (CAPM), define the ‘beta-coefficient’, βj , corresponding to asset j. Discuss how assets’ beta-coefficients should be interpreted and explain how their values can be obtained in practice. Answer : The beta-coefficient can be defined in any of the following equivalent ways:
βj = σjM σ^2 M
ρjM σj σM σ M^2
= ρjM σj σM
where σjM is the covariance between the rate of return on asset j and the market rate of return, σM is the standard deviation of the market rate of return, σj is the standard deviation of the rate of return on asset j, and ρjM 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:
rj = r 0 + (rM − r 0 )βj + εj , j = 1, 2 ,... , n,
where εj is an unobserved random error. It is assumed that E[εj | rM ] = 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. Answer : The CAPM predicts that: μj = r 0 + (μM − r 0 )βj , where μj is the expected rate of return on asset j, μM is the expected rate of return on the market portfolio, and r 0 is the risk-free rate of return The SML treats μj as a function of βj 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:
0 .066 = r 0 + 0.4(μM − r 0 ), and 0 .098 = r 0 + 1.2(μM − r 0 ).
6
μj
βj
r 0
μM
SM L
Figure 1: The Security Market Line, SML
Hence it must follow that: μM = 0. 09 and r 0 = 0. 05. Thus, in this example the market price of risk is 4%. Hence the SML is:
μj = 0.05 + 0. 04 βj.
(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? Answer : 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:
μj = ω + (μM − ω)βj ,
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.] Answer : The CAPM predicts that the expected rate of return on the fourth asset is:
0 .082 = 0.05 + 0. 04 × 0. 8.
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.
- The following information is provided for a stock market:
σj ρjM Security A 50% 0. Security B 60% −0. Market Portfolio 20% 1.
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.
- 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: μj − r 0 = (μM − r 0 )βj , where βj = ρjM
σj σM
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, βE = σE /σM and the prediction becomes: μE − r 0 σE
μM − r 0 σM
which is the equation of the CML. The Characteristic Line, treats μj − r 0 as a function of μM − r 0 , with slope βj. This is useful for estimating βj. The Security Market Line treats μj − r 0 as a function of βj , with slope μM − 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 (‘homo- geneous’ or ‘unanimous’ beliefs) about asset returns. (f) What is the role of these assumptions? The mean variance assumption implies that for each investor: μj − r 0 βj σZ
μZ − r 0 σZ or μj − r 0 = (μZ − r 0 )βj , j = 1, 2 ,... , n,
where Z is the efficient portfolio comprising risky assets only. Note that, without further assumptions, μj , βj 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 μj and βj 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 in- vestors 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.)
