Chapter 7, Portiolio Theory and Other Asset Pricing Models Video Solutions, Financial Management: Theory and Practice

Problem 1

The standard deviation of stock returns for Stock A is $40 \%$. The standard deviation of the market return is $20 \%$. If the correlation between Stock $A$ and the market is $0.70,$ what is Stock A's beta?

Zachary Zhao

Numerade Educator

01:20

Problem 2

An analyst has modeled the stock of Crisp Trucking using a two-factor APT model. The risk-free rate is $6 \%$, the expected return on the first factor $\left(\mathrm{r}_{1}\right)$ is $12 \%$ and the expected return on the second factor $\left(\mathrm{r}_{2}\right)$ is $8 \% .$ If $\mathrm{b}_{\mathrm{i} 1}=0.7$ and $\mathrm{b}_{\mathrm{i} 2}=0.9$ what is Crisp's required return?

James Kiss

Numerade Educator

01:03

Problem 3

An analyst has modeled the stock of a company using a Fama-French three-factor model. The risk-free rate is $5 \%$, the required market return is $10 \%$, the risk premium for small stocks $\left(\mathrm{r}_{\mathrm{SMB}}\right)$ is $3.2 \%,$ and the risk premium for value stocks $\left(\mathrm{r}_{\mathrm{HML}}\right)$ is $4.8 \% .$ If $a_{i}=0, b_{i}=1.2, c_{i}=-0.4,$ and $d_{i}=1.3,$ what is the stock's required return?

Breanna Ollech

Numerade Educator

01:21

Problem 4

Stock A has an expected return of $12 \%$ and a standard deviation of $40 \% .$ Stock B has an expected return of $18 \%$ and a standard deviation of $60 \% .$ The correlation coefficient between Stocks A and B is 0.2. What are the expected return and standard deviation of a portfolio invested $30 \%$ in Stock $\mathrm{A}$ and $70 \%$ in Stock $\mathrm{B}$ ?

Christopher Stanley

Numerade Educator

08:51

Problem 5

The beta coefficient of an asset can be expressed as a function of the asset's correlation with the market as follows:
$$\mathrm{b}_{\mathrm{i}}=\frac{\rho_{\mathrm{i}, \mathrm{M}} \sigma_{\mathrm{i}}}{\sigma_{\mathrm{M}}}$$
a. Substitute this expression for beta into the Security Market Line (SML), Equation $7-9 .$ This results in an alternative form of the SML.
b. Compare your answer to part a with the Capital Market Line (CML), Equation 7-6. What similarities are observed? What conclusions can be drawn?

Chris Trentman

Numerade Educator

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Problem 6

Suppose you are given the following information. The beta of company i, bis, is 1.1 the risk-free rate, $r_{R F^{\prime}}$ is $7 \%,$ and the expected market premium, $r_{M}-r_{R F^{\prime}}$ is $6.5 \%$ (Assume that $\left.a_{i}=0.0 .\right)$
a. Use the Security Market Line (SML) of CAPM to find the required return for this company.
b. Because your company is smaller than average and more successful than average (that is, it has a low book-to-market ratio), you think the Fama-French three-factor model might be more appropriate than the CAPM. You estimate the additional coefficients from the Fama-I rench three-factor model: The coefficient for the size effect, $c_{i}$, is 0.7 , and the coefficient for the book-to-market effect, $\mathrm{d}_{\mathrm{i}}$ is $-0.3 .$ If the expected value of the size factor is $5 \%$ and the expected value of the book-to-market factor is $4 \%,$ what is the required return using the Fama-French three-factor model?

Shu Naito

Numerade Educator

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Problem 7

You are given the following set of data:
a. Use a spreadsheet (or a calculator with a linear regression function) to determine Stock X's beta coefficient.
b. Determine the arithmetic average rates of return for Stock $X$ and the NYSE over the period given. Calculate the standard deviations of returns for both Stock X and the NYSE.
c. Assuming (1) that the situation during Years 1 to 7 is expected to hold true in the future (that is, $\hat{\mathrm{r}}_{\mathrm{X}}=\overline{\mathrm{r}}_{\mathrm{X}^{\prime}} \hat{\mathrm{r}}_{\mathrm{M}}=\overline{\mathrm{r}}_{\mathrm{M}^{\prime}}$ and both $\sigma_{\mathrm{X}}$ and $\mathrm{b}_{\mathrm{X}}$ in the future will equal
their past values), and ( 2 ) that Stock $X$ is in equilibrium (that is, it plots on the Security Market Line), what is the risk-free rate?
d. Plot the Security Market Line.
e. Suppose you hold a large, well-diversified portfolio and are considering adding to the portfolio either Stock $X$ or another stock, Stock $Y$, that has the same beta as Stock X but a higher standard deviation of returns. Stocks X and Y have the same expected returns; that is, $\hat{r}_{X}=\hat{r}_{Y}=10.6 \% .$ Which stock should you choose?

Jason Gerber

Numerade Educator

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Problem 8

You are given the following set of data:
a. Construct a scatter diagram showing the relationship between returns on Stock $Y$ and the market. Use a spreadsheet or a calculator with a linear regression function to estimate beta.
b. Give a verbal interpretation of what the regression line and the beta coefficient show about Stock $Y^{\prime}$ s volatility and relative risk as compared with those of other stocks.
c. Suppose the scatter of points had been more spread out, but the regression line was exactly where your present graph shows it. How would this affect
(1) the firm's risk if the stock is held in a one-asset portfolio and (2) the actual risk premium on the stock if the CAPM holds exactly?
d. Suppose the regression line had been downward sloping and the beta coefficient had been negative. What would this imply about (1) Stock Y's relative risk, (2) its correlation with the market, and (3) its probable risk premium?

Shu Naito

Numerade Educator

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Problem 9

Start with the partial model in the file $F M 12 C h 07$ P09 Build a Model.xls from the textbook's Web site. Following is information for the required returns and standard deviations of returns for $\mathrm{A}, \mathrm{B},$ and $\mathrm{C}$ The correlation coefficients for each pair are shown below in a matrix, with each cell in the matrix showing the correlation between the stock in the same row and column. For example, $\rho_{\mathrm{AB}}=0.1571$ is in the row for A and the column for B. Notice that the diagonal values are equal to $1,$ because a variable is always perfectly positively correlated with itself.
a. Suppose a portfulio has $30 \%$ invested in $A, 50 \%$ in $B,$ and $20 \%$ in $C .$ What are the expected return and standard deviation of the portfolio?
b. The partial model lists 66 different cumbinations of portfolio weights. For each combination of weights, find the required return and standard deviation.
c. The partial model provides a scatter diagram showing the required returns and standard deviations calculated above. This provides a visual indicator of the feasible set. If you would like a return of $10.50 \%$, what is the smallest standard deviation that you must accept?

Jason Gerber

Numerade Educator