Risk and Return: Portfolio Theory and Asset Pricing Models

ANSWERS TO END-OF-CHAPTER QUESTIONS

5-1 a. A portfolio is made up of a group of individual assets held in combination. An asset that would be relatively risky if held in isolation may have little, or even no risk if held in a well-diversified portfolio.

The feasible, or attainable, set represents all portfolios that can be constructed from a given set of stocks. This set is only efficient for part of its combinations.

An efficient portfolio is that portfolio which provides the highest expected return for any degree of risk. Alternatively, the efficient portfolio is that which provides the lowest degree of risk for any expected return.

The efficient frontier is the set of efficient portfolios out of the full set of potential portfolios. On a graph, the efficient frontier constitutes the boundary line of the set of potential portfolios.

b. An indifference curve is the risk/return trade-off function for a particular investor and reflects that investor's attitude toward risk. The indifference curve specifies an investor's required rate of return for a given level of risk. The greater the slope of the indifference curve, the greater is the investor's risk aversion.

The optimal portfolio for an investor is the point at which the efficient set of portfolios--the efficient frontier--is just tangent to the investor's indifference curve. This point marks the highest level of satisfaction an investor can attain given the set of potential portfolios.

c. The Capital Asset Pricing Model (CAPM) is a general equilibrium market model developed to analyze the relationship between risk and required rates of return on assets when they are held in well-diversified portfolios. The SML is part of the CAPM.

The Capital Market Line (CML) specifies the efficient set of portfolios an investor can attain by combining a risk-free asset and the risky market portfolio M. The CML states that the expected return on any efficient portfolio is equal to the riskless rate plus a risk premium, and thus describes a linear relationship between expected return and risk.

d. The characteristic line for a particular stock is obtained by regressing the historical returns on that stock against the historical returns on the general stock market. The slope of the characteristic line is the stock's beta, which measures the amount by which the stock's expected return increases for a given increase in the expected return on the market.

The beta coefficient (b) is a measure of a stock's market risk. It measures the stock's volatility relative to an average stock, which has a beta of 1.0.

e. Arbitrage Pricing Theory (APT) is an approach to measuring the equilibrium risk/return relationship for a given stock as a function of multiple factors, rather than the single factor (the market return) used by the CAPM. The APT is based on complex mathematical and statistical theory, but can account for several factors (such as GNP and the level of inflation) in determining the required return for a particular stock.

The Fama-French 3-factor model has one factor for the excess market return (the market return minus the risk free rate), a second factor for size (defined as the return on a portfolio of small firms minus the return on a portfolio of big firms), and a third factor for the book-to-market effect (defined as the return on a portfolio of firms with a high book-to-market ratio minus the return on a portfolio of firms with a low book-to-market ratio).

Most people don’t behave rationally in all aspects of their personal lives, and behavioral finance assume that investors have the same types of psychological behaviors in their financial lives as in their personal lives.

5-2 Security A is less risky if held in a diversified portfolio because of its lower beta and negative correlation with other stocks. In a single-asset portfolio, Security A would be more risky because ?A > ?B and CVA > CVB.

SOLUTIONS TO END-OF-CHAPTER PROBLEMS

5-1 a. A plot of the approximate regression line is shown in the following figure:

Using Excel, the regression equation estimates are: Beta = 0.56; Intercept = 0.037; R2 = 0.96.

b. The arithmetic average return for Stock X is calculated as follows:

The arithmetic average rate of return on the market portfolio, determined similarly, is 12.1%.

For Stock X, the estimated standard deviation is 13.1 percent:

The standard deviation of returns for the market portfolio is similarly determined to be 22.6 percent. The results are summarized below:

Stock X Market Portfolio

Average return, 10.6% 12.1%

Standard deviation, ? 13.1 22.6

Several points should be noted: (1) ?M over this particular period is higher than the historic average ?M of about 15 percent, indicating that the stock market was relatively volatile during this period; (2) Stock X, with sX = 13.1%, has much less total risk than an average stock, with sAvg = 22.6%; and (3) this example demonstrates that it is possible for a very low-risk single stock to have less risk than a portfolio of average stocks, since ?X < ?M.

c. Since Stock X is in equilibrium and plots on the Security Market Line (SML), and given the further assumption that and --and this assumption often does not hold--then this equation must hold:

This equation can be solved for the risk-free rate, rRF, which is the only unknown:

d. The SML is plotted below. Data on the risk-free security (bRF = 0,

rRF = 8.6%) and Security X (bX = 0.56, = 10.6%) provide the two points through which the SML can be drawn. rM provides a third point.

e. In theory, you would be indifferent between the two stocks. Since they have the same beta, their relevant risks are identical, and in equilibrium they should provide the same returns. The two stocks would be represented by a single point on the SML. Stock Y, with the higher standard deviation, has more diversifiable risk, but this risk will be eliminated in a well-diversified portfolio, so the market will compensate the investor only for bearing market or relevant risk. In practice, it is possible that Stock Y would have a slightly higher required return, but this premium for diversifiable risk would be small.

5-2

a. The regression graph is shown above. Using a speadsheet, we find b = 0.62.

b. Because b = 0.62, Stock Y is about 62 percent as volatile as the market; thus, its relative risk is about 62 percent of that of an average firm.

c. 1. Total risk would be greater because the second term of the firm's risk equation, , would be greater.

2. CAPM assumes that company-specific risk will be eliminated in a portfolio, so the risk premium under the CAPM would not be affected.

d. 1. The stock's variance would not change, but the risk of the stock to an investor holding a diversified portfolio would be greatly reduced.

2. It would now have a negative correlation with rM.

3. Because of a relative scarcity of such stocks and the beneficial net effect on portfolios that include it, its "risk premium" is likely to be very low or even negative. Theoretically, it should be negative.

5-3 a.

b. CML: SML:

With some arranging, the similarities between the CML and SML are obvious. When in this form, both have the same market price of risk, or slope,(rM - rRF)/?M.

The measure of risk in the CML is ?p. Since the CML applies only to efficient portfolios, ?p not only represents the portfolio's total risk, but also its market risk. However, the SML applies to all portfolios and individual securities. Thus, the appropriate risk measure is not ?i, the total risk, but the market risk, which in this form of the SML is riM?i, and is less than for all assets except those which are perfectly positively correlated with the market, and hence have riM = +1.0.

5-4 a. Using the CAPM:

ri = rRF + (rM - rRF)bi = 7% + (1.1)(6.5%) = 14.15%

b. Using the 3-factor model:

ri = rRF + (rM – rrf)bi + (rSMB)ci + (rHML)di

= 7% + (1.1)(6.5%) + (5%)(0.7) + (4%)(-0.3) = 16.45%

MINI CASE

To begin, briefly review the Chapter 4 Mini Case. Then, extend your knowledge of risk and return by answering the following questions.

a. Suppose asset A has an expected return of 10 percent and a standard deviation of 20 percent. Asset B has an expected return of 16 percent and a standard deviation of 40 percent. If the correlation between A and B is 0.4, what are the expected return and standard deviation for a portfolio comprised of 30 percent asset A and 70 percent asset B?

Answer:

b. Plot the attainable portfolios for a correlation of 0.4. Now plot the attainable portfolios for correlations of +1.0 and -1.0.

Answer:

c. Suppose a risk-free asset has an expected return of 5 percent. By definition, its standard deviation is zero, and its correlation with any other asset is also zero. Using only asset A and the risk-free asset, plot the attainable portfolios.

Answer:

d. Construct a reasonable, but hypothetical, graph which shows risk, as measured by portfolio standard deviation, on the x axis and expected rate of return on the y axis. Now add an illustrative feasible (or attainable) set of portfolios, and show what portion of the feasible set is efficient. What makes a particular portfolio efficient? Don't worry about specific values when constructing the graph—merely illustrate how things look with "reasonable" data.

Answer:

The figure above shows the feasible set of portfolios. The points B, C, D, and E represent single securities (or portfolios containing only one security). All the other points in the shaded area, including its boundaries, represent portfolios of two or more securities. The shaded area is called the feasible, or attainable, set.

The boundary AB defines the efficient set of portfolios, which is also called the efficient frontier. Portfolios to the left of the efficient set are not possible because they lie outside the attainable set. Portfolios to the right of the boundary line (interior portfolios) are inefficient because some other portfolio would provide either a higher return with the same degree of risk or a lower level of risk for the same rate of return.

e. Now add a set of indifference curves to the graph created for part B. What do these curves represent? What is the optimal portfolio for this investor? Finally, add a second set of indifference curves which leads to the selection of a different optimal portfolio. Why do the two investors choose different portfolios?

Answer:

The figure above shows the indifference curves for two hypothetical investors, A and B. To determine the optimal portfolio for a particular investor, we must know the investor's attitude towards risk as reflected in his or her risk/return tradeoff function, or indifference curve. Curves Ia1, Ia2, and Ia3 represent the indifference curves for individual A, with the higher curve (Ia3) denoting a greater level of satisfaction (or utility). Thus, Ia3 is better than Ia2 for any level of risk.

The optimal portfolio is found at the tangency point between the efficient set of portfolios and one of the investor's indifference curves. This tangency point marks the highest level of satisfaction the investor can attain. The arrows point toward the optimal portfolios for both investors A and B.

The investors choose different optimal portfolios because their risk aversion is different. Investor A chooses the portfolio with the lower expected return, but the riskiness of that portfolio is also lower than investor's B optimal portfolio, because investor a is more risk averse.

f. What is the capital asset pricing model (CAPM)? What are the assumptions that underlie the model?

Answer: The Capital Asset Pricing Model (CAPM) is an equilibrium model which specifies the relationship between risk and required rates of return on assets when they are held in well-diversified portfolios. The CAPM requires an extensive set of assumptions:

All investors are single-period expected utility of terminal wealth maximizers, who choose among alternative portfolios on the basis of each portfolio's expected return and standard deviation.

All investors can borrow or lend an unlimited amount at a given risk-free rate of interest.

Investors have homogeneous expectations (that is, investors have identical estimates of the expected values, variances, and covariances of returns among all assets).

All assets are perfectly divisible and perfectly marketable at the going price, and there are no transactions costs.

There are no taxes.

All investors are price takers (that is, all investors assume that their own buying and selling activity will not affect stock prices).

The quantities of all assets are given and fixed.

g. Now add the risk-free asset. What impact does this have on the efficient frontier?

Answer: The risk-free asset by definition has zero risk, and hence ? = 0%, so it is plotted on the vertical axis. Now, given the possibility of investing in the risk-free asset, investors can create new portfolios that combine the risk-free asset with a portfolio of risky assets. This enables them to achieve any combination of risk and return that lies along any straight line connecting rRF with any portfolio in the feasible set of risky portfolios. However, the straight line connecting rRF with m, the point of tangency between the line and the portfolio's efficient set curve, is the one that all investors would choose. Since all portfolios on the line rRFmz are preferred to the other risky portfolio opportunities on the efficient frontier AB, the points on the line rRFmz now represent the best attainable combinations of risk and return. Any combination under the rRFmz line offers less return for the same amount of risk, or offers more risk for the same amount of return. Thus, everybody wants to hold portfolios which are located on the rRFmz line.

h. Write out the equation for the capital market line (CML) and draw it on the graph. Interpret the CML. Now add a set of indifference curves, and illustrate how an investor's optimal portfolio is some combination of the risky portfolio and the risk-free asset. What is the composition of the risky portfolio?

Answer: The line rRFmz in the figure above is called the capital market line (CML). It has an intercept of rRF and a slope of . Therefore the equation for the capital market line may be expressed as follows:

CML:

The CML tells us that the expected rate of return on any efficient portfolio (that is, any portfolio on the CML) is equal to the risk-free rate plus a risk premium, and the risk premium is equal to multiplied by the portfolio's standard deviation, . Thus, the CML specifies a linear relationship between expected return and risk, with the slope of the CML being equal to the expected return on the market portfolio of risky stocks, , minus the risk-free rate, rRF, which is called the market risk premium, all divided by the standard deviation of returns on the market portfolio, ?m.

The figure above shows a set of indifference curves (i1, i2, and i3), with i1 touching the CML. This point of tangency defines the optimal portfolio for this investor, and he or she will buy a combination of the market portfolio and the risk-free asset.

The risky portfolio, m, must contain every asset in exact proportion to that asset's fraction of the total market value of all assets; that is, if security g is x percent of the total market value of all securities, x percent of the market portfolio must consist of security g.

i. What is a characteristic line? How is this line used to estimate a stock’s beta coefficient? Write out and explain the formula that relates total risk, market risk, and diversifiable risk.

Answer: Betas are calculated as the slope of the characteristic line, which is the regression line formed by plotting returns on a given stock on the y axis against returns on the general stock market on the x axis. In practice, 5 years of monthly data, with 60 observations, would be used, and a computer would be used to obtain a least squares regression line.

The relationship between stock J's total risk, market risk, and diversifiable risk can be expressed as follows:

Here is the variance or total risk of stock j, is the variance of the market, bj is stock J's beta coefficient, and is the variance of stock J's regression error term. If stock J is held in isolation, then the investor must bear its total risk. However, when stock J is held as part of a well-diversified portfolio, the regression error term, is driven to zero; hence, only the market risk remains.

j. What are two potential tests that can be conducted to verify the CAPM? What are the results of such tests? What is roll’s critique of CAPM tests?

Answer: Since the CAPM was developed on the basis of a set of unrealistic assumptions, empirical tests should be used to verify the CAPM. The first test looks for stability in historical betas. If betas have been stable in the past for a particular stock, then its historical beta would probably be a good proxy for its ex-ante, or expected beta. Empirical work concludes that the betas of individual securities are not good estimators of their future risk, but that betas of portfolios of ten or more randomly selected stocks are reasonably stable, hence that past portfolio betas are good estimators of future portfolio volatility.

The second type of test is based on the slope of the SML. As we have seen, the CAPM states that a linear relationship exists between a security's required rate of return and its beta. Further, when the SML is graphed, the vertical axis intercept should be rRF, and the required rate of return for a stock (or portfolio) with beta = 1.0 should be rm, the required rate of return on the market. Various researchers have attempted to test the validity of the CAPM model by calculating betas and realized rates of return, plotting these values in graphs, and then observing whether or not (1) the intercept is equal to rRF, (2) the regression line is linear, and (3) the SML passes through the point b = 1.0, rm. Evidence shows a more-or-less linear relationship between realized returns and market risk, but the slope is less than predicted. Tests that attempt to assess the relative importance of market and company-specific risk do not yield definitive results, so the irrelevance of diversifiable risk specified in the CAPM model can be questioned.

Roll questioned whether it is even conceptually possible to test the CAPM. Roll showed that the linear relationship which prior researchers had observed in graphs resulted from the mathematical properties of the models being tested, hence that a finding of linearity proved nothing about the validity of the CAPM. Roll's work did not disprove the CAPM theory, but he did show that it is virtually impossible to prove that investors behave in accordance with the theory.

In general, evidence seems to support the CAPM model when it is applied to portfolios, but the evidence is less convincing when the CAPM is applied to individual stocks.

Nevertheless, the CAPM provides a rational way to think about risk and return as long as one recognizes the limitations of the CAPM when using it in practice.

k. Briefly explain the difference between the CAPM and the arbitrage pricing theory (APT).

Answer: The CAPM is a single-factor model, while the Arbitrage Pricing Theory (APT) can include any number of risk factors. It is likely that the required return is dependent on many fundamental factors such as the GNP growth, expected inflation, and changes in tax laws, and that different groups of stocks are affected differently by these factors. Thus, the apt seems to have a stronger theoretical footing than does the CAPM. However, the apt faces several major hurdles in implementation, the most severe being that the apt does not identify the relevant factors--a complex mathematical procedure called factor analysis must be used to identify the factors. To date, it appears that only three or four factors are required in the apt, but much more research is required before the apt is fully understood and presents a true challenge to the CAPM.

l. Suppose you are given the following information. The beta of company, bi, is 0.9, the risk free rate, rRF, is 6.8%, and the expected market premium, rM-rRF, is 6.3%. Because your company is larger than average and more successful than average (i.e., it has a lower book-to-market ratio), you think the Fama-French 3-factor model might be more appropriate than the CAPM. You estimate the additional coefficients from the Fama-French 3-factor model: the coefficient for the size effect, Ci, is -0.5, and the coefficient for the book-to-market effect, di, is –0.3. If the expected value of the size factor is 4% and the expected value of the book-to-market factor is 5%, what is the required return using the Fama-French 3-factor model? (assume that Ai = 0.0.) What is the required return using CAPM?

Answer: The Fama-French model:

ri = rRF + (rm - rRF)bi + (rsmb)ci + (rhmb)dj

ri = 6.8% + (6.3%)(0.9) + (4%)(-0.5) + (5%)(-0.3)

= 8.97%

The CAPM:

ri = rrf + (rm - rrf)bi

ri = 6.8% + (6.3%)(0.9)

= 12.47%