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Investments

Zvi Bodie, Alex Kane, Alan J. Marcus

Chapter 6

Capital Allocation to Risky Assets - all with Video Answers

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Chapter Questions

Problem 1

Which of the following choices best completes the following statement? Explain. An investor with a higher degree of risk aversion, compared to one with a lower degree, will demand investment portfolios
a. with higher risk premiums.
b. that are riskier (with higher standard deviations).
c. with lower Sharpe ratios.
d. with lower trading costs.

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Problem 2

Which of the following statements are true? Explain.
a. A lower allocation to the risky portfolio reduces the Sharpe (reward-to-volatility) ratio.
b. The higher the borrowing rate, the lower the Sharpe ratios of levered portfolios.
c. With a fixed risk-free rate, doubling the expected return and standand deviation of the risky portfolio will double the Sharpe ratio.
d. Holding constant the risk premium of the risky portfolio, a higher risk-free rate will increase the Sharpe ratio of investments with a positive allocation to the risky asset.

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01:04

Problem 3

What do you think would happen to the equilibrium expected return on stocks if investors perceived higher volatility in the equity market? Relate your answer to Equation 6.7.

Achintya Suden
Achintya Suden
Numerade Educator

Problem 4

Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $$\$ 70,000$$ or $$\$ 200,000$$ with equal probabilities of 5 . The alternative risk free investment in T-bills pays $2 \%$ per year.
a. If you require a risk premium of $8 \%$, how much will you be willing to pay for the portfolio?
b. Suppose that the portfolio can be purchased for the amount you found in (a). What will be the expected rate of return on the portfolio?
c. Now suppose that you require a risk premium of $12 \%$. What price are you willing to pay?
d. Comparing your answers to (a) and (c), what do you conclude about the relationship between the required risk premium on a portfolio and the price at which the portfolio will sell?

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Problem 5

Consider a portfolio that offers an expected rate of return of $7 \%$ and a standard deviation of 18\%. T-bills offer a risk-free $2 \%$ rate of return. What is the maximum level of risk aversion for which the risky portfolio is still preferred to T-bills?

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04:06

Problem 6

Draw the indifference curve in the expected return-standard deviation plane corresponding to a utility level of .02 for an investor with a risk aversion coefficient of 3.

Jasper Baltz
Jasper Baltz
Numerade Educator

Problem 7

Now draw the indifference curve corresponding to a utility level of .02 for an investor with risk aversion coefficient $A=4$. Comparing your answer to Problem 6, what do you conclude?

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Problem 8

Draw an indifference curve for a risk-neutral investor providing utility level .02 .

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Problem 9

What must be true about the sign of the risk aversion coefficient, $A$, for a risk lover? Draw the indifference curve for a utility level of, 02 for a risk lover.

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Problem 10

Calculate the expected return and variance of portfolios invested in T-bills and the S\&P 500 index with weights as follows:
$$
\begin{array}{ll}
W_{\text {nili }} & W_{\text {nsor }} \\
\hline 0 & 1.0 \\
0.2 & 0.8 \\
0.4 & 0.6 \\
0.6 & 0.4 \\
0.8 & 0.2 \\
1.0 & 0
\end{array}
$$

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Problem 11

Calculate the utility levels of each portfolio of Problem 10 for an investor with $A=2$. What do you conclude?

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02:57

Problem 12

Repeat Problem 11 for an investor with $A=3$. What do you conclude?

Eric Mockensturm
Eric Mockensturm
Numerade Educator

Problem 13

Your client chooses to invest $70 \%$ of a portfolio in your fund and $30 \%$ in an essentially risk-free moncy market fund. What are the expected value and standard deviation of the rate of return on his portfolio?

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08:19

Problem 14

Suppose that your risky portfolio includes the following investments in the given proportions:
$$
\begin{array}{ll}
\text { Stock A } & 25 \% \\
\text { Stock B } & 32 \% \\
\text { Stock C } & 43 \%
\end{array}
$$
What are the investment proportions of each asset in your client's overall portfolio, including the position in T-hills?

Chris Trentman
Chris Trentman
Numerade Educator

Problem 15

What is the reward-to-volatility (Sharpe) ratio (S) of your risky portfolio? Your client's?

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Problem 16

Draw the CAL. of your portfolio on an expected return-standard deviation diagram. What is the slope of the CAL? Show the position of your client on your fund's CAL.

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Problem 17

Suppose that your client decides to invest in your portfolio a proportion $y$ of the total investment budget so that the overall portfolio will have an expected rate of return of $10 \%$.
a. What is the proportion $y$ ?
b. What are your client's investment proportions in your three stocks and the T-bill fund?
c. What is the standard deviation of the rate of return on your client's portfolio?

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Problem 18

Suppose that your client prefers to invest in your fund a proportion $y$ that maximizes the expected return on the complete portfolio subject to the constraint that the complete portfolio's standard deviation will not exceed $12 \%$.
a. What is the investment proportion, $y$ ?
b. What is the expected rate of return on the complete portfolio?

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Problem 19

Your client's degree of risk aversion is $A=3.5$.
a. What proportion, $y$, of the total investment should be invested in your fund?
b. What are the expected value and standard deviation of the rate of return on your client's optimized portfolio?

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01:31

Problem 20

Look at the data in Table 6.7 on the average excess return of the U.S. equity market and the standard deviation of that excess return. Suppose that the U.S. Market is your risky portfolio.
a. If your risk-aversion coefficient is $A=4$ and you believe that the entire 1927-2021 period is representative of future expected performance, what fraction of your portfolio should be allocated to T-bills and what fraction to equity?
b. What if you believe that the 1975-1998 period is representative?
c. What do you conclude upon comparing your answers to $(a)$ and $(b)$ ?

Breanna Ollech
Breanna Ollech
Numerade Educator

Problem 21

Consider the following information about a risky portfolio that you manage and a risk-free asset: $E\left(r_P\right)=8 \%, \sigma_P=15 \%, r_f=2 \%$.
a. Your client wants to invest a proportion of her total investment budget in your risky fund to provide an expected rate of return on her overall or complete portfolio equal to $5 \%$. What proportion should she invest in the risky portfolio, $P$, and what proportion in the risk-free asset?
b. What will be the standard deviation of the rate of return on her portfolio?
c. Another client wants the highest return possible subject to the constraint that you limit his standard deviation to be no more than $12 \%$. Which client is more risk averse?

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Problem 22

Investment Management Inc. (IMI) uses the capital market line to make asset allocation recommendations. IMI derives the following forecasts:
- Expected return on the market portfolio: $12 \%$
- Standard deviation on the market portfolio: $20 \%$
- Risk-free rate: $5 \%$
Samuel Johnson seeks IMI's advice for a portfolio asset allocation. Johnson wants the standard deviation of the portfolio to equal half that of the market portfolio. Using the capital market line, what expected return can IMI provide subject to Johnson's risk constraint?

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Problem 23

Draw a diagram of your client's CML, accounting for the higher borrowing rate. Superimpose on it two sets of indifference curves, one for a client who will choose to borrow and one for a client who will invest in both the index fund and a money market fund.

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Problem 24

What is the range of risk aversion for which a client will neither borrow nor lend, that is, for which $y=1$ ?

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Problem 25

Solve Problems 23 and 24 for a client who uses your fund rather than an index fund.

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04:10

Problem 26

What is the largest percentage fee that a client who currently is lending $(y<1)$ will be willing to pay to imest in your fund? What about a clicnt who is borrowing $(y>1)$ ?

Shivani Sharma
Shivani Sharma
Numerade Educator

Problem 27

Draw the CML and your funds' CAL on an expected return-standard deviation diagram.
a. What is the slope of the CML?
b. Characterize in one short paragraph the advantage of your fund over the passive fund.

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Problem 28

Your client ponders whether to switch the $70 \%$ that is invested in your fund to the passive portfolio.
a. Explain to your client the disadvantage of the switch.
b. Show him the maximum fee you could charge (as a percentage of the investment in your fund, dedacted at the end of the year) that would leave him at least as well off investing in your fund as in the passive one.

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Problem 29

Consider again the client in Problem 19 with $A=3.5$.
a. If he chose to invest in the passive portfolio, what proportion, $y$, would he select?
b. Is the fee (percentage of the investment in your fund, deducted at the end of the year) that you can charge to make the client indifferent between your fund and the passive strategy affected by his capital allocation decision (i.e., his choice of $y$ )?

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