**6.12 CHAPTER 6: EXERCISES**

In the following exercises an * denotes greater difficulty.

1. Use CAPM Tutor's default dataset (CAPM.DAT) to answer the following.

a) Compare the variance of the optimal portfolio for a 15% expected return I.) with a risk-free asset market open, and ii.) without a risk-free asset market. Explain why the presence of a risk-free asset lowers the variance of the optimal portfolio.

b) Describe the shape of the CML line and explain why it has this shape.

2. Using CAPM Tutor and SP20.DAT, assume that short selling is permitted and answer the following.

a) From the financial press record the current 3-month Treasury bill rate. Using CAPM Tutor, open the risk-free market and change the default risk-free rate to the current risk-free rate that you have obtained. When entering this rate, be careful that you first convert it from its annualized form into a daily rate. You must convert this into a daily rate because SP20.DAT is the variance-covariance matrix of daily returns.

Click on the point of tangency between the minimum-variance frontier and the capital market line (the point of tangency is a green dot in default colors).

Record the expected return, portfolio standard deviation, portfolio beta.

b) Why is the portfolio beta in a) equal to 1?

c) Suppose you have invested 50% of your wealth at* r**f* and 50% of your wealth in the tangency portfolio. What is expected return and portfolio standard deviation for this portfolio? (To answer this, use CAPM Tutor to click on the midpoint lying on the CML between *r**f* and the tangency portfolio. When Alpha0 equals 0.5 you have identified the required portfolio.)

d) For a target return equal to 0.003, with the risk-free asset market open, record the portfolio standard deviation for this portfolio.

e) The portfolio identified in c) lies above the portfolio identified in a). If you were investing $1 million in this portfolio, how much is invested in risky assets, and how much is invested in the risk-free asset? (Hint: Again use the field "Alpha0" to answer this part.)

Suppose you could not borrow at the risk-free rate that you entered in part a), but you could borrow at 100 basis points above the annualized risk-free rate. __Be sure to convert the 100 basis points to a daily rate before__ answering the following parts:

e) With the risk-free asset market open, change the default risk-free rate to your borrowing rate expressed as a daily rate. Click on the point of tangency between the minimum-variance frontier and the capital market line (the point of tangency is a green dot in default colors). Record the expected return, portfolio standard deviation, and portfolio beta.

f) Compare your answer in e) with the answer you obtained in a).

g) Draw the shape of the CML, assuming that your borrowing and lending rates are as described above. You should mark clearly the tangency point(s).

h) For a target return equal to .003 with the risk-free asset market open at the borrowing rate of *r**f* plus 100 basis points (annualized), record the portfolio standard deviation for this portfolio.

i) The portfolio identified in h) lies above the portfolio identified in part e). If you were investing $1 million in this portfolio how much is invested in risky assets and how much is invested in the risk-free asset? (Hint again use the field Alpha0 to answer this part.)

j) In terms of portfolio risk for the given target return of .003 contrast the risk of your target portfolio when borrowing at the risk-free rate to the risk of your target portfolio when borrowing at 100 basis points over the risk-free rate (annualized). By relating each target portfolio to its respective position on the CML, explain why the risk varies as you increase the borrowing rate.

3. Using CAPM Tutor and SP20.DAT, answer the following assuming short selling is permitted and that these 20 securities represent the set of all available risky assets.

To answer questions 3a) and 3b) rescale the return/standard deviation graph to the return scale of +/- 0.015 and the standard deviation scale from 0 to 0.1.

a) Define what the SML is.

b) With the risk-free asset market open, and changing the default risk-free rate to the current risk-free rate expressed as a daily return, click on the SML button. Why does everthing lie on the SML?

4. Using CAPM Tutor with the default dataset (CAPM.DAT) and with a risk-free asset;

a) Vary the target return for 14%, 15%, and 16%, and record the portfolio weights and portfolio risk.

b) Why don't the weights change as you change the target return ?

c) How does portfolio risk change?

d) Turn off the riskfree asset in CAPM Tutor, and again vary the target return for 14%, 15%, and 16%, and record portfolio weights and portfolio risk.

e) Why do the weights now change as you vary the target return?

f) Compare the portfolio risk with and without a risk-free asset (i.e., compare your answers to parts a) and d).) Why does one set of portfolios dominate the other set of portfolios?

5. Using the default dataset in CAPM Tutor (CAPM.DAT), where there is a risk-free security, identify the tangency portfolio. What is this tangency portfolio and why is it called the market portfolio?

6. Using the default dataset in CAPM Tutor, fix the target return to equal 14% and include the risk-free security (i.e., click on yes for risk-free).

a) Record the optimal portfolio weights, the portfolio risk and the market price of risk.

b) What is the market price of risk, and how is computed?

c) Increase *r**f* to 0.122 and repeat a).

d) Decrease *r**f* to 0.118 and repeat a).

Contrast the solutions to a), b), and c) in terms of portfolio weights, portfolio risk, and the market price of risk.

e) Why do portfolio weights change?

f) Describe the relationship you observe between how the market price of risk changes and how the target portfolio risk changes. Why does this relationship occur?

7.* If you are given a variance-covariance matrix of returns, a set of expected returns, and a risk-free interest rate, how would you calculate the portfolio weights corresponding to the market portfolio ?

8. Suppose you hold only the market portfolio. Consider adding a small amount of the first stock to your portfolio. Explain how the resulting change in variance is related to the beta of the stock.

9a) What is a zero-beta portfolio?

9b) Why can the risk-free interest rate be replaced by a zero-beta portfolio?

10. Using the default dataset and a target return equal to 15%, compute the market price of risk.

a) When there is a risk-free security and with *r**f* = 0.12.

b) When there is not a risk-free security but with a zero-beta portfolio.

c) Why is your answer to part b "steeper" than your answer to part a)?

11. Using the default dataset (CAPM.DAT), turn on zero-beta in CAPM Tutor and vary the target return to be 14%, 15%, and 16%.

a) The zero-beta portfolio is sensitive to these changes in the target return. Why?

b) How does the slope of the market price of risk and the portfolio risk vary?

12. Explain what is meant by the statement "the CAPM is essentially untestable."

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