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CHAPTER 8

AN INTRODUCTION TO ASSET PRICING MODELS

Answers to Questions

1. Explain why the set of points between the risk-free asset and a portfolio on the Markowitz efficient frontier is a straight line.It can be shown that the expected return function is a weighted average of the individual returns. In

addition, it is shown that combining any portfolio with the risk-free asset, that the standard deviation of the combination is only a function of the weight for the risky asset portfolio. Therefore, since both the expected return and the variance are simple weighted averages, the combination will lie along a straight line.

2. Draw a graph that shows what happens to the Markowitz efficient frontier when you combine a risk-free asset with alternative risky asset portfolios on the Markowitz efficient frontier. Explain this graph.

Expected Rate of Return * F M * P * * B RFR *A

E

Expected Risk ( of return)

The existence of a risk-free asset excludes the E-A segment of the efficient frontier because any point below A is dominated by the RFR. In fact, the entire efficient frontier below M is dominated by points on the RFR-M Line (combinations obtained by investing a part of the portfolio in the risk-free asset and the remainder in M), e.g., the point P dominates the previously efficient B because it has lower risk for the same level of return. As shown, M is at the point where the ray from RFR is tangent to the efficient frontier. The new efficient frontier thus becomes RFR-M-F.

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3. Draw and explain why the line from the RFR that is tangent to the efficient frontier defines the dominant set of portfolio possibilities.

Expected Rate of Return M C

B

RFR A

E

Expected Risk ( of return)This figure indicates what happens as a risk-free asset is combined with risky portfolios higher and higher on the efficient frontier. In each case, as you combine with the higher return portfolio, the new line will dominate all portfolios below this line. This program continues until you combine with the portfolio at the point of tangency and this line becomes dominant over all prior lines. It is not possible to do any better because there are no further risky asset portfolios at a higher point.

4. Discuss what risky assets are in portfolio M and why they are in it.The “M” or “market” portfolio contains all risky assets available. If a risky asset, be it an obscure bond or a rare stamp, was not included in the market portfolio, then there would be no demand for this asset, and consequently, its price would fall. Notably, the price decline would continue to the point where the return would make the asset desirable such that it would be part of the M portfolio - e.g., if the bonds of ABC Corporation were selling for 100 and had a coupon of 8 percent, the investor’s return would be 8 percent; however, if there was no demand for ABC bonds the price would fall, say to 80, at which point the 10 percent (80/800) return might make it a desirable investment. Conversely, if the demand for ABC bonds was greater than supply, prices would be bid up to the point where the return would be in equilibrium. In either case, ABC bonds would be included in the market portfolio.

5.Disscuss leverage and its effect on the CML.Leverage indicates the ability to borrow funds and invest these added funds in the market portfolio of risky assets. The idea is to increase the risk of the portfolio (because of the leverage), and also the expected return from the portfolio. It is shown that if you can borrow at the RFR then the set of leveraged portfolios is simply a linear extension of the set of portfolios along the line from the RFR to the market portfolio. Therefore, the full CML becomes a line from the RFR to the M portfolio and continuing upward

.

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6. Disscuss and justify a measure of diversification for a portfolio in terms of capital market theory.You can measure how well diversified a portfolio is by computing the extent of correlation between the portfolio in question and a completely diversified portfolio - i.e., the market portfolio. The idea is that, if a portfolio is completely diversified and, therefore, has only systematic risk, it should be perfectly correlated with another portfolio that only has systematic risk.

7. What changes would you expect in the standard deviation for a portfolio of between 4 & 10 stocks, between 10 & 20 stocks and between 50 & 100 stocks?Standard deviation would be expected to decrease with an increase in stocks in the portfolio because an increase in number will increase the probability of having more inversely correlated stocks. There will be a major decline from 4 to 10 stocks, a continued decline from 10 to 20 but at a slower rate. Finally, from 50 to 100 stocks, there is a further decline but at a very slow rate because almost all unsystematic risk is eliminated by about 18 stocks.

8.Discuss why the investment and financing decisions are separate when you have a CML.Given the existence of the CML, everyone should invest in the same risky asset portfolio, the market portfolio. The only difference among individual investors should be in the financing decision they make, which depends upon their risk preference. Specifically, investors initially make investment decisions to invest in the market portfolio, M. Subsequently, based upon their risk preferences, they make financing decisions as to whether to borrow or lend to attain the preferred point on the CML.

9. Given the CML, discuss and justify the relevant measure of risk for an individual security.Recall that the relevant risk variable for an individual security in a portfolio is its average covariance with all other risky assets in the portfolio. Given the CML, however, there is only one relevant portfolio and this portfolio is the market portfolio that contains all risky assets. Therefore, the relevant risk measure for an individual risky asset is its covariance with all other assets, namely the market portfolio

10. Capital market theory divides the variance of returns for a security into systematic variance and unsystematic or unique variance. Describe each of these terms.Systematic risk refers to that portion of total variability of returns caused by factors affecting the

prices of all securities, e.g., economic, political and sociological changes -factors that are uncontrollable, external, and broad in their effect on all securities.

Unsystematic risk refers to factors that are internal and “unique” to the industry or company, e.g., management capability, consumer preferences, labor strikes, etc. Notably, it is not possible to get rid of the overall systematic risk, but it is possible to eliminate the “unique” risk for an individual asset in a diversified portfolio.

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11. The capital asset pricing model (CAPM) contends that there is systematic & unsystematic risk of individual security. What is the relevant risk variable and why is it relevant?In a capital asset pricing model (CAPM) world the relevant risk variable is the security’s systematic risk - its covariance of return with all other risky assets in the market. This risk cannot be eliminated. The unsystematic risk is not relevant because it can be eliminated through diversification - for instance, when you hold a large number of securities, the poor management capability, etc., of some companies will be offset by the above average capability of others.

12. How does SML differ from CML?For plotting, the SML the vertical axis measures the rate of return while the horizontal axis measures normalized systematic risk (the security’s covariance of return with the market portfolio divided by the variance of the market portfolio). By definition, the beta (normalized systematic risk) for the market portfolio is 1.0 and is zero for the risk-free asset. It differs from the CML where the measure of risk is the standard deviation of return (referred to as total risk).

13. Identify and briefly discuss three criticisms of beta as used in the capital asset pricing model (CAPM).CFA Examination I (1993)

Any three of the following are criticisms of beta as used in CAPM.

1. Theory does not measure up to practice. In theory, a security with a zero beta should give a return exactly equal to the risk-free rate. But actual results do not come out that way, implying that the market values something besides a beta measure of risk.

2. Beta is a fickle short-term performer. Some short-term studies have shown risk and return to be negatively related. For example, Black, Jensen and Scholes found that from April 1957 through December 1965, securities with higher risk produced lower returns than less risky securities. This result suggests that (1) in some short periods, investors may be penalized for taking on more risk, (2) in the long run, investors are not rewarded enough for high risk and are overcompensated for buying securities with low risk, and (3) in all periods, some unsystematic risk is being valued by the market.

3. Estimated betas are unstable. Major changes in a company affecting the character of the stock or some unforeseen event not reflected in past returns may decisively affect the security’s future returns.4. Beta is easily rolled over. Richard Roll has demonstrated that by changing the market index against which betas are measured, one can obtain quite different measures of the risk level of individual stocks and portfolios. As a result, one would make different predictions about the expected returns, and by changing indexes, one could change the risk-adjusted performance ranking of a manager.

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14. Briefly explain whether investors should expect a higher return from holding portfolio A versus portfolio B under capital asset pricing theory (CAPM).Assume that both portfolios are fully diversified.

Portfolio A Portfolio BSystematic risk (beta) 1 1Specific risk for each individual security High Low

CFA Examination I (1993)

Under CAPM, the only risk that investors should be compensated for bearing is the risk that cannot be diversified away (systematic risk). Because systematic risk (measured by beta) is equal to one for both portfolios, an investor would expect the same return for Portfolio A and Portfolio B.Since both portfolios are fully diversified, it doesn’t matter if the specified risk for each individual security is high or low. The specific risk has been diversified away for both portfolios.

15. You have been recently been appointed chief investment officer of a major charitable foundation. Its large endowment fund is currently invested in a broadly diversified portfolio of stocks (60%) and bonds (40%).The foundation’s board of trustees is a group of prominent individuals whose knowledge of investment theory and practice is superficial. You decide a discussion of basic investment principle would be helpful.15(a) Explain the concepts of specific risk, variance, covariance, standard deviation and beta as they relate to investment management.

CFA Examination II (1994)15(a). The concepts are explained as follows:The Foundation’s portfolio currently holds a number of securities from two asset classes. Each of the individual securities has its own risk (and return) characteristics, described as specific risk. By including a sufficiently large number of holdings, the specific risk of the individual holdings offset each other, diversifying away much of the overall specific risk and leaving mostly no diversifiable or market-related risk.Systematic risk is market-related risk that cannot be diversified away. Because systematic risk cannot be diversified away, investors are rewarded for assuming this risk.

The variance of an individual security is the sum of the probability-weighted average of the squared differences between the security’s expected return and its possible returns. The standard deviation is the square root of the variance. Both variance and standard deviation measure total risk, including both systematic and specific risk. Assuming the rates of return are normally distributed, the likelihood for a range of rates may be expressed using standard deviations. For example, 68 percent of returns may be expressed using standard deviations. Thus, 68 percent of returns can be expected to fall within + or -1 standard deviation of the mean, and 95 percent within 2 standard deviations of the mean.Covariance measures the extent to which two securities tend to move, or not move, together. The level of covariance is heavily influenced by the degree of correlation between the securities (the correlation coefficient) as well as by each security’s standard deviation. As long as the correlation coefficient is less than 1, the portfolio standard deviation is less than the weighted average of the

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individual securities’ standard deviations. The lower the correlation, the lower the covariance and the greater the diversification benefits (negative correlations provide more diversification benefits than positive correlations).The capital asset pricing model (CAPM) asserts that investors will hold only fully diversified portfolios. Hence, total risk as measured by the standard deviation is not relevant because it includes specific risk (which can be diversified away).

Under the CAPM, beta measures the systematic risk of an individual security or portfolio. Beta is the slope of the characteristic line that relates a security’s returns to the returns of the market portfolio. By definition, the market itself has a beta of 1.0. The beta of a portfolio is the weighted average of the betas of each security contained in the portfolio. Portfolios with betas greater than 1.0 have systematic risk higher than that of the market; portfolios with betas less than 1.0 have lower systematic risk. By adding securities with betas that are higher (lower), the systematic risk (beta) of the portfolio can be increased (decreased) as desired.

You believe that the addition of other asset classes to endowment portfolio would improve the portfolio by reducing risk and enhancing return. You are aware that depressed conditions in U.S. real estate markets are providing opportunities for property acquisition at levels of expected return that are unusually high by historical standards.You believe that an investment in U.S. real estate would be both appropriate and timely and have decided to recommend a 20% position be established with funds taken equally from stocks & bonds.Preliminary discussions revealed that several trustees believe real estate is too risky to include in the portfolio.The board chairman,however,has scheduled a special meeting for further discussion of the matter and has asked you to provide back ground information that will clarify the risk issue.To assist you,the following expectational data have been developed:

CORRELATION MATRIXAsset class Return Standard

deviationUS Stocks

US Bonds

US Real Estate US T-Bills

US Stocks 12% 21% 1.00US Bonds 8% 10.5% 0.14 1.00US Real Estate 12% 9% -0.04 -0.03 1.00US T-Bills 4% 0 -0.05 -0.03 0.25 1.00

15(b) Explain the effect on both portfolio risk and return that would result from the addition of U.S. real estate. Include in your answer two reasons for any change you expect in portfolio risk.Without performing the calculations, one can see that the portfolio return would increase because:(1) Real estate has an expected return equal to that of stocks. (2) Its expected return is higher than

the return on bonds.The addition of real estate would result in a reduction of risk because: (1) The standard deviation of real estate is less than that of both stocks and bonds. (2) The covariance of real estate with both stocks and bonds is negative.The addition of an asset class that is not perfectly correlated with existing assets will reduce variance. The fact that real estate has a negative covariance with the existing asset classes will reduce risk even more.

15(c) Your understanding of capital market theory causes you to doubt the validity of the expected return and risk for U.S. real estate .Justify your skepticism.

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Capital market theory holds that efficient markets prevent mispricing of assets and that expected return is proportionate to the level of risk taken. In this instance, real estate is expected to provide the same return as stocks and a higher return than bonds. Yet, it is expected to provide this return at a lower level of risk than both bonds and stocks. If these expectations were realistic, investors would sell the other asset classes and buy real estate, pushing down its return until it was proportionate to the level of risk.

Appraised values differ from transaction prices, reducing the accuracy of return and volatility measures for real estate. Capital market theory was developed and applied to the stock market, which is a very liquid market with relatively small transaction costs. In contrast to the stock market, real estate markets are very thin and lack liquidity.

16. In the empirical testing of the CAPM, what are two major concerns? Why are they important?First, the stability of beta: It is important to know whether it is possible to use past betas as estimates of future betas. Second, is there a relationship between beta and rates of return? This would indicate whether the CAPM is a relevant pricing model that can explain rates of return on risky assets

17. Briefly discuss why it is important for beta coefficients to be stationary over time.Given that beta is the principal risk measure, stable betas make it easier to forecast future beta measures of systematic risk - i.e., can betas measured from past data be used in making investment decisions.

18. Discuss the empirical results relative to beta stability for individual stocks and portfolios of stocks.The results of the stability of beta studies indicate that betas for individual stocks are generally not stable, but portfolios of stocks have stable betas.

19. In the tests of the relationship between systematic risk (beta) and return, what are you looking for? Is there a positive linear relationship between the systematic risk of risky assets and the rates of return on these assets? Are the coefficients positive and significant? Is the intercept close to the risk-free rate of return?

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20. Draw an ideal SML .Based on the early empirical results, what did the actual risk return relationship look like relative to the ideal relationship implied by the CAPM?

E(R) Theoretical SML

Empirical SML RFR

RM

1.0 Risk (Beta)In the empirical line, low risk securities did better than expected, while high risk securities did not do as well as predicted.

21. According to the CAPM, what assets are included in the market portfolio and what are the relative weightings? In empirical studies of the CAPM, what are the typical proxies used for the market portfolio?The “market” portfolio contains all risky assets available. If a risky asset, be it an obscure bond or rare stamp, was not included in the market portfolio, then there would be no demand for this asset and, consequently, its price would fall. Notably, the price decline would continue to the point where the return would make the asset desirable such that it would be part of the “market” portfolio. The weights for all risky assets are equal to their relative market value.

22. Assuming that the empirical proxy for the market portfolio is not a good proxy, what factors related to the CAPM will be affected?According to Roll, a mistakenly specified proxy for the market portfolio can have two effects. First, the beta computed for alternative portfolios would be wrong because the market portfolio is inappropriate. Second, the SML derived would be wrong because it goes from the RFR through the improperly specified market portfolio. In general, when comparing the performance of a portfolio manager to the “benchmark” portfolio, these errors will tend to overestimate the performance of portfolio managers because the proxy market portfolio employed is probably not as efficient as the true market portfolio, so the slope of the SML will be underestimated 23. Some studies related to the efficient market hypothesis generated results that implied additional factors beyond beta should be considered to estimate expected returns. What are these other variables and why should they be considered?Studies of the efficient markets hypothesis suggest that additional factors affecting estimates of expected returns include firm size, the price-earnings ratio, and financial leverage. These variables have been shown to have predictive ability with respect to security returns.

24. According to Fema-French study, discuss what variables you should consider when selecting a cross section of stocks.Fama and French found that size, leverage, earnings-price ratios, and book value to market value of equity all have a significant impact on univariate tests on average return. In multivariate tests, size and book to market equity value are the major explanatory factors.

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Problems and solution

Problem 1: Assume that you expect the economy’s rate of inflation to be 3 percent, giving an RFR of 6 percent and a market return (Rm) of 12 percent.a. Draw the SML under these assumptions.b. Subsequently, you expect the rate of inflation to increase from 3 percent to 6 percent. What

effect would this have on the RFR and the Rm? Draw another SML on the graph from part a.c. Draw an SML on the same graph to reflect an RFR of 9 percent and an Rm of 17 percent.

How does this SML differ from that derived in part b? Explain what has transpired.Solution:

1. Rate of SMLc

Return SMLb

E(Rmc) .17 SMLc

E(Rmb) .15

E(Rmc) .12

RFRc=RFRb .09

RFRa .06

1.0 Systematic Risk (Beta)

In (b), a change in risk-free rate, with other things being equal, would result in a new SMLb, which would intercept with the vertical axis at the new risk-free rate (.09) and would be parallel in the original SMLa.

In (c), this indicates that not only did the risk-free rate change from .06 to .09, but the market risk premium per unit of risk [E(Rm) - Rf] also changed from .06 (.12 - .06) to .08 (.17 - .09). Therefore, the new SMLc will have an intercept at .09 and a different slope so it will no longer be parallel to SMLa.

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You expect an RFR of 10 percent and the market return (R m) of 14 percent. Compute the expected (required) return for the following stocks, and plot them on an SML graph.

Stock Beta E(Ri)

UND

0.851.25-0.20

Solution: E (Ri) = RFR + i(RM - RFR)

= .10 + i(.14 - .10)

= .10 + .04i

Stock Beta (Required Return) E(Ri) = .10 + .04 iU 85 .10 + .04(.85) = .10 + .034 = .134N 1.25 .10 + .04(1.25)= .10 + .05 = .150D -.20 .10 + .04(-.20) = .10 - .008 = .092

Problem 3: You ask a stockbroker what the firm’s research department expects for the three stocks in problem 2. The broker responds with the following information:

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Stock Current price Expected price Expected DividendUND

224837

245140

0.752.001.25

Solution: Stock Current

PriceExpected

PriceExpected Dividend

Estimated Return

U22 24 0.75

N 48 51 2.00

D 37 40 1.25

Stock Beta Required Estimated Evaluation U .85 .134 .1250 OvervaluedN 1.25 .150 .1042 OvervaluedD -.20 .092 .1149 Undervalued

If you believe the appropriateness of these estimated returns, you would buy stocks D and sell stocks U and N.

E(R)

N 14% U

*U’ *D’

* N’

D -0.5 -0.2 0.5 .085 1.0 1.25

Problem 11: Based on five years of monthly data, you derive the following information for the co mpanies listed.

Company δt (Intercept) σt rm

IntelFord

Anheuser Busch

0.220.100.17

12.10%14.607.60

0.720.330.55

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MerckS & P 500

0.050.00

10.205.50

0.601.00

a. compute the beta coefficient for each stock.b. Assuming a risk-free rate of 8 percent and an expected return for the market portfolio of 15

percent and an expected return for the market portfolio of 15 percent, compute the expected (required) return for all the stocks and plot them on the SML.

c. Plot the following estimated returns for the next year on the SML and indicate which stocks are undervalued or overvalued.

Intel – 20 percent. Ford -15 percent. Anheuser Busch – 19 percent. Merck – 10 percent.

Solution: a.

Then COVi,m = (ri,m)(i)( m)

For Intel:

COV i,m = (.72)(.1210)(.0550) = .00479

For Ford:

COV i,m = (.33)(.1460)(.0550) = .00265

For Anheuser Busch:

COV i,m = (.55)(.0760)(.0550) = .00230

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For Merck:

COV i,m = (.60)(.1020)(.0550) = .00337

11(b). E(Ri) = RFR + Bi(RM - RFR)

= .08 + Bi(.15 - .08) = .08 + .07Bi

Stock Beta E(Ri) = .08 + .07Bi

Intel 1.597 .08 + .1118 = .1918Ford .883 .08 + .0618 = .1418 Anheuser Busch .767 .08 + .0537 = .1337Merck 1.123 .08 + .0786 = .1586

11(c). .20 *Intel *AB

RM = .15 *Ford

.10 *Merck RFR=.08

1.0 Beta

Problem 12: Calculate the expected (required) return for each of the following stocks when the risk-free rate is 0.08 and you expect the market return to be 0.14.

Stocks BetaABCDE

1.721.140.760.440.03

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F -0.79 Solution:

E(Ri) = RFR + i (RM - RFR)

= .068 + i (.14 - .08)

= .08 + .06i

12(a). E(RA) = .08 + .06(1.72) = .08 + .1050 = .1850 = 18.50%

12(b). E(RB) = .08 + .06(1.14) = .08 + .0684 = .1484 = 14.84%

12(c). E(RC) = .08 + .06(0.76) = .08 + .0456 = .1256 = 12.56%

12(d). E(RD) = .08 + .06(0.44) = .08 + .0264 = .1064 = 10.64%

12(e). E(RE) = .08 + .06(0.03) = .08 + .0018 = .0818 = 8.18%

12(f). E(RF) = .08 + .06(-0.79) = .08 - .0474 = .0326 = 3.26%

Problem 13:

The following are the historic returns for the Chelle Computer Company:

Year Chelle Computer General Index

12345

379

-11811

151314-912

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6 4 9Based on this information, compute the following:

a. the correlation coefficient between Chelle Computer and the General Index.b. The standard deviation for the company and index.c. The beta for the Chelle Computer Company.

Solution:

. Anita General (R1 - E(R1) xYear (R1) Index (RM) R1 - E(R1) RM - E(RM) RM - E(RM)

1 37 15 27.33 6 163.98 2 9 13 -.67 4 -2.683 -11 14 -20.67 5 -103.354 8 -9 -1.67 -18 30.065 11 12 1.33 3 3.996 4 9 -5.67 0 0 .00

= 58 = 54 = 92.00

E(R1) = 9.67 E(M) = 9

13(a). The correlation coefficient can be computed as follows:

13(b). The standard deviations are: 14.21% for Anita Computer and 8.27% for index, respectively.

13(c). Beta for Anita Computer is computed as follows:

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Problem 14: CFA Examination Level IIThe following information describes the expected return and risk relationship for

the stocks of two of WAH’s competitors.

Expected Return Standard Deviation Beta Stock XStock Y

Market IndexRisk-free rate

12.0%9.010.05.0

20%1512

1.30.71.0

Using only the data shown in the preceding table:a. draw and label a graph showing the security market line and position stocks X and Y

relative to it.[ 5 minutes ]b. compute the alphas both for stock X and for stock Y. Show your work. [4 minutes ]c. assume that the risk-free rate increases to 7 percent with the other data in the preceding

matrix remaining unchanged. Select the stock providing the higher expected risk-adjusted return and justify your selection. Show your calculations. [6 minutes]

Solution:

14. CFA Examination II (1995)

14(a). The security market line (SML) shows the required return for a given level of systematic risk. The SML is described by a line drawn from the risk-free rate: expected return is 5 percent, where beta equals 0 through the market return; expected return is 10 percent, where beta equal 1.0.

15% Security Market Line (SML)

12% *Stock Y 10% *Market 9% *Stock Y

5%

.5 .7 1.0 1.3 1.5 2.0 Beta()

14(b). The expected risk-return relationship of individual securities may deviate from that suggested by the SML, and that difference is the asset’s alpha. Alpha is the difference between the expected (estimated) rate of return for a stock and its required rate of return based on its systematic risk Alpha is computed as

ALPHA () = E(ri) - [rf + (E(rM) - rf)]

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where

E(ri) = expected return on Security irf = risk-free ratei = beta for Security iE(rM) = expected return on the market

Calculation of alphas:

Stock X: = 12% - [5% + 1.3% (10% - 5%)] = 0.5%Stock Y: = 9% - [5% + 0.7%(10% - 5%)] = 0.5%

In this instance, the alphas are equal and both are positive, so one does not dominate the other.

Another approach is to calculate a required return for each stock and then subtract that

required return from a given expected return. The formula for required return (k) is

k = rf + i (rM - rf ).

Calculations of required returns:

Stock X: k = 5% + 1.3(10% - 5%) = 11.5% = 12% - 11.5% = 0.5%

Stock Y: k = 5% + 0.7(10% - 5%) = 8.5% = 9% - 8.5% = 0.5%

14(c). By increasing the risk-free rate from 5 percent to 7 percent and leaving all other factors

unchanged, the slope of the SML flattens and the expected return per unit of incremental

risk becomes less. Using the formula for alpha, the alpha of Stock X increases to 1.1 percent

and the alpha of Stock Y falls to -0.1 percent. In this situation, the expected return (12.0

percent) of Stock X exceeds its required return (10.9 percent) based on the CAPM.

Therefore, Stock X’s alpha (1.1 percent) is positive. For Stock Y, its expected return (9.0

percent) is below its required return (9.1 percent) based on the CAPM. Therefore, Stock Y’s

alpha (-0.1 percent) is negative. Stock X is preferable to Stock Y under these circumstances.

Calculations of revised alphas:

Stock X = 12% - [7% + 1.3 (10% - 7%] = 12% - 10.95% = 1.1%

Stock Y = 9% - [7% + 0.7(10% - 7%)] = 9% - 9.1% = -00.1%

Problem 15: CFA Examination Level II

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An analyst expects a risk-free return of 4.5 percent, a market return of 14.5 percent, and the returns for stocks A and B that are shown in the following table.

STOCK INFORMATION

Stock Beta Analyst’s Estimated ReturnAB

1.20.8

16 %14 %

a. Show on the graph provided in the answer book:(1) where stock A and B would plot on the security market line (SML) if they were

fairly valued using the capital asset pricing model (CAPM).(2) Where stock A and B actually plot on the same graph according to the returns

estimated by the analyst and shown in the table [6 minutes]b. state whether stock A and B are undervalued or overvalued if the analyst uses the SML for

strategic investment decisions.[4 minutes]Solution: CFA Examination II (1998) 15(a). Security Market Line

i. Fair-value plot. The following template shows, using the CAPM, the expected return, ER, of Stock A and Stock B on the SML. The points are consistent with the following equations:

ER on stock = Risk-free rate + Beta x (Market return – Risk-free rate)

ER for A = 4.5% + 1.2(14.5% - 4.5%) = 16.5%

ER for B = 4.5% + 0.8(14.5% - 4.5%) = 12.5%

ii. Analyst estimate plot. Using the analyst’s estimates, Stock A plots below the SML and Stock B, above the SML.

*Stock A14.5% *Stock B

4.5%

0.8 1.2

15(b). Over vs. UndervalueStock A is overvalued because it should provide a 16.5% return according to the CAPM whereas the analyst has estimated only a 16.0% return.

Stock B is undervalued because it should provide a 12.5% return according to the CAPM

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whereas the analyst has estimated a 14% return.

Problem 16: Given the following results, indicate what will happen to the beta for stock E, relative to the market proxy, compared to the beta relative to the time market portfolio:

Yearly Rates Of Return

YearStock E

(Percent)Market Proxy

(Percent)True Market

( Percent)12345

1020-14-2015

814-10-1812

611-7-1210

Discuss the reason for the differences in measured beta. Does the suggested relationship appear reasonable? Why or why not?

Solution: Rproxy = 1.2; Rtrue = 1.6

The beta for using the proxy is given by Cov(i,proxy)/Var(proxy). Given the data, proxy = 256.7/205.2 = 1.251

true = 187.6/109.3 = 1.716.

The proxy is not mean-variance efficient, as it is dominated by the true market portfolio.

Problem 17: Draw the impled SMLs for the following two sets of conditions:

a. RFR = 0.07; Rm (S + P 500) = 0.16b. Rt = 0.09; Rm (True) = 0.18Under which set of conditions would it be more difficult for a portfolio manager to be superior?

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Solution:

R

.18 .16

.09 .07

1.0 Beta

It would be more difficult to show superior performance relative to the true market index.

Problem 17: Using the graph and equations from problem 17, which of the following portfolios

would be superior?

a. Ra = 11 %; β = 0.09b. Rb = 14 %; β = 1.00c. Rc = 12 %; β = - 0.40d. Rd = 20 %; β = 1.10Does it matter which SML you use?

Solution: SMLS&P = 0.07 + x(0.16 – 0.07)

SMLTrue = 0.09 + x(0.18 – 0.09)

18(a). Ra = 0.11, a = 0.09Using the S&P proxy:

E(Ra) = 0.07 + 0.09x(0.09)= 0.07 + 0.0081= 0.0781

Using the true market:E(Ra) = 0.09 + 0.09x(0.09)

= 0.09 + 0.0081= 0.0981

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A would be superior in either case.

18(b). Rb = .14, b = 1.00Using the S&P proxy:E(Rb) = 0.07 + 1.0x0.09

= 0.16

Using the true market:E(Rb) = 0.09 + 1.0x0.09

= 0.18

Inferior performance in both cases.

18(c). Rc = 0.12 c = -0.4Using the S&P proxy:E(Rc) = 0.07 – 0.40x0.09 = 0.07 – 0.036 = 0.034

Using the true market:E(Rc) = 0.09 – 0.40x0.09 = 0.09 –0.036 = 0.054

Superior performance in both cases. 18(d). Rd = 0.20 d = 1.10

Using the market proxy: E(Rd) = 0.07 + 1.1x0.09

= 0.07 + 0.99= 0.169

Using the true market:E(Rd) = 0.09 + 1.1x0.09

= 0.09 +0.099= 0.189

Superior performance in both cases.

Problem 17: Draw the security market line for each of the following conditions:

a. (1) RFR = 0.08 Rm (proxy) = 0.12

(2) Rz = 0.06 Rm (true) = 0.15b. rader Tire has the following results for the last six periods. Calculate and compare the betas using each index.

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Period

Return of Rader

(Percent )

Proxy Specific Index

(Percent )

True General Index

(Percent )

1

2

3

4

5

6

29

12

-12

.17

20

-5

12

10

-9

14

25

-10

15

13

-8

18

28

0

If the current period return for the market is 12 percent and for Rader is 11 percent , are superior

results being obtained for either index beta?

Solution:

R

0.08 0.06

1.0 Beta19(b). = Cov i,m/(m)2

From a spreadsheet program, we find

Cov i,m = 187.4m

2 = 190.4Using the proxy:p = 187.4/190.4 = .984

Using the true index:t = 176.4/168 = 1.05

19(c). Using the proxy:E(RR) = 0.08 + 0.984x(0.12 - 0.08)

= 0.08 + 0.0394 = .1194

Using the true market:E(RR) = 0.06 + 1.05x(0.12 – 0.06)

= 0.06 + 0.063 = 0.123

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Rader’s performance would be inferior compared to either.

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