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The efficient frontier enables managers to reduce that number of possible portfolios considered because the portfolios on the efficient frontier:

A) have lower risk levels for every level of expected return than all other possible portfolios.

B) have higher risk levels for every level of expected return than all other possible portfolios.

C) have higher expected returns for every level of risk than all other possible portfolios.

If we are selecting portfolios from a large number of stocks, say the S&P 500, rather than just two stocks, the number of possible combinations is extremely large. We can restrict our search for possible portfolio combinations by focusing on those portfolios on the efficient frontier. We know they dominate all the other possible choices because they offer higher return for the same level of risk.
The minimum-variance frontier consists of portfolios that have lower risk levels for every level of expected return than all other possible portfolios.

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An analyst has gathered the following data:

Portfolio	Weight S&P(%)	Weight EAFE(%)	PORT σ (%)	E(Rp)(%)
A	100	0	10	10
B	70	30	6
C	30	70	11
D	0	100	15	20

Which portfolio represents the minimum variance portfolio?

A) Portfolio C.

B) Portfolio B.

C) Portfolio A.

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Minimum variance portfolio among the choices presented is portfolio B (70% S&P, 30% EAFE).

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For a U.S. investor with extreme risk aversion, is there a benefit to international diversification?

A) Yes, since a 100% weighting in international stocks results in a doubling of the expected return with only a 50% increase in risk.

B) Yes, since a 30% weighting in the EAFE index results in an increased return and decreased standard deviation than 100% investment in the S&P index.

C) Yes, since a 70% weighting in the EAFE index results in a much higher expected return with a minimal increase in portfolio standard deviation than 100% investment in the S&P index.

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To answer this question, it is necessary to complete the table.
ERportB = (0.70)(10) + (0.30)(20) = 13
ERportC = (0.30)(10) + (0.70)(20) = 17

Portfolio	Weight S&P(%)	Weight EAFE(%)	PORTσ(%)	E(Rp)(%)
A	100	0	10	10
B	70	30	6	13
C	30	70	11	17
D	0	100	15	20

For portfolio B, the addition of EAFE is return enhancing and risk reducing, so even in the presence of extreme risk aversion there is a benefit. By choosing portfolio B, E(r) increases to 13% and portfolio risk decreases to 6%. For portfolios C and D, returns are increasing but so is the risk level. Both of these risk-return trade-offs may have some merit, but we cannot be sure in the presence of extreme risk aversion.

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Assume the annual Treasury bill (T-bill) yields 4%. Which portfolio is the most desirable (i.e., highest Sharpe ratio)?

A) Portfolio B.

B) Portfolio A.

C) Portfolio C.

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Sharpe (Portfolio A) = (10 – 4) / 10 = 0.60

Sharpe (Portfolio B) = (13 – 4) / 6 = 1.5

Sharpe (Portfolio C) = (17 – 4) / 11 = 1.18

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The efficient frontier consists of portfolios that have:

A) the minimum standard deviation for any given level of expected return.

B) the maximum expected return for any given standard deviation.

C) capital allocation lines with slopes greater than 1.0.

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The efficient frontier consists of (efficient) portfolios that have the maximum expected return for any given standard deviation. The efficient frontier starts at the global minimum-variance portfolio and continues above it on the minimum variance frontier. The minimum-variance frontier is the expected return-standard deviation combinations of the set of portfolios that have the minimum variance for every given level of expected return. Efficient portfolios can have capital allocation line (CAL) slopes less than 1.0. These slopes, however, will all be less than that of the CAL of the market portfolio (the capital market line).

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The portfolio on the minimum-variance frontier that has the smallest standard deviation is the:

A) market portfolio.

B) global minimum-variance portfolio.

C) optimal efficient portfolio.

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The global minimum-variance portfolio is the portfolio on the minimum-variance frontier that has the smallest standard deviation (or variance). It is the portfolio at the tip of the bullet. The market portfolio, in which each asset is held in proportion to its market value, cannot have the smallest standard deviation of the portfolios on the minimum variance frontier.

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When solving for the minimum-variance frontier for many assets, the constraint is:

A) portfolio weights must sum to one.

B) weighted-average covariances must sum to zero.

C) weighted-average expected asset returns must sum to expected portfolio return.

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This is the second step in determining the minimum-variance frontier. For every expected return between the smallest and largest expected return, determine the single portfolio with the smallest variance. We assume that the portfolio weights add up to one (this is the constraint on the portfolio weights). This step requires expected returns, variances, and covariances to calculate expected return and variance of the portfolios.

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The efficient frontier is useful for portfolio management because:

A) portfolios on the efficient frontier are useful as factor portfolios.

B) portfolios on the efficient frontier are optimal: the correlation between each efficient portfolio, and the market portfolio is negative.

C) it significantly reduces the number of portfolios a manager must consider.

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If we are selecting portfolios from a large number of stocks, say the S&P 500, rather than just two stocks, the number of possible combinations is extremely large. We can restrict our search for possible portfolio combinations by focusing on those portfolios on the efficient frontier. We know they dominate all the other possible choices because they offer higher return

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Chris McDonald, CFA, is a portfolio manager for InvesTrack, a firm that seeks to closely track a selected index or indexes with each of its funds. McDonald is analyzing the returns of several of InvesTrack’s managed funds. The primary fund, Marketrack, or the MT portfolio, tracks a combination of a major stock index, bond index, real estate index, and a precious metals index. The stock index in the MT portfolio closely follows the S&P 500. The weights on each of the indexes in the MT target portfolio are approximately the same as the weights that the analysts at InvesTrack have estimated for these assets in the overall economy. McDonald believes that the MT portfolio is more likely to lie on the efficient frontier than a portfolio of only stocks. In a recent discussion with his assistants, Joseph Kreager and Maria Ito, McDonald said the low correlations between classes such as precious metals and real estate in the portfolio will improve the diversification of the portfolio. Kreager proposes that the ultimate goal should be to combine assets to achieve the minimum variance portfolio on the efficient frontier.
McDonald proposes that the returns of the MT portfolio can serve as a better representation of a market portfolio than an index like the Dow Jones Industrial Average or the S&P 500, which many analysts and portfolio managers use as a market proxy. For example, he asserts that betas estimated using the MT portfolio will be a more realistic representation of systematic risk, and this will make the betas more reliable in decisions concerning the effects of diversification. Furthermore, he suggests that the capital asset line (CAL) based upon the MT portfolio should be steeper than that based upon the S&P 500 alone. Kreager claims that that the MT portfolio will only have steeper CAL if the average returns of the indexes other than the stock index in the MT tracking portfolio are higher than the S&P 500. Ito responds that MT portfolio CAL will be higher than the S&P 500 CAL only if the standard deviation of the returns of the other indexes in the MT tracking portfolio are lower than the S&P 500.
Recently a customer holding a position in TTX stock wanted to explore the purchase of shares in a real estate investment trust (REIT). McDonald ran a regression of the return of the stock on the return of the MT portfolio, and he also ran a regression of the REIT’s return on the return of MT portfolio. Using monthly returns over three years, the results of the market model regressions are:

(Return of the TTX stock)t = −0.006 + 1.28 × (Return of MT portfolio)t + εt

(Return of the REIT) t = 0.014 + 0.60 × (Return of MT portfolio) t + ηt

The annualized standard deviations of the monthly returns for each of these investments are σTTXstock = 38, σREIT = 24, and σMT = 16. McDonald asks Kreager to compute the variance covariance matrix based upon these results. He also asks Kreager to compute the standard deviation of the unexplained risk for each of the assets.
After performing the regressions, Kreager investigates the property of beta drift. Using a monthly time-series, he finds that the betas of both the TTX stock and the REIT both follow an AR(1) process:

βt+1 = 0.1 + 0.9 × βt-1

Using this AR(1) process, Kreager tries to determine if the covariance between the two assets will increase or decrease in the next time period. He assumes the variance of the MT portfolio will remain the same.
After viewing the statistics, Ito gathers information on the S&P 500 and finds that its average return is 12%, and the standard deviation is 20%. The current risk-free rate is 5%. She wants to investigate whether McDonald’s assertion that the MT portfolio CAL is steeper than the S&P 500 CAL is true. In Kreager and Ito's responses to McDonald’s proposition that the CAL of the MT portfolio should be steeper than that of the S&P 500:

A) only one is correct.

B) both are correct.

C) both are incorrect.

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Kreager asserts that the CAL will be steeper if the average returns on the non-stock indexes are greater than the S&P 500. The fact is that the slope, which is also called the Sharpe Ratio, also depends upon the standard deviation of the MT portfolio. Without further information, it is impossible to know if Kreager is correct, but his statement is clearly not correct taken in isolation.
Ito asserts that the CAL will be steeper if the standard deviations of the non-stock indexes are less than the S&P 500. The fact is that the slope, which is also called the Sharpe Ratio, also depends upon the return of the MT portfolio. Without further information, it is impossible to know if Ito is correct, but her statement is clearly not correct taken in isolation.

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In response to Kreager’s assertion that the goal is to try to achieve the minimum variance portfolio on the efficient frontier, McDonald should:

A) disagree under any circumstances.

B) agree.

C) agree only if it can be achieved with long positions in assets.

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Any portfolio on the efficient frontier with a return greater than the minimum variance portfolio can be combined with the risk-free asset to create a portfolio that has a superior risk-return tradeoff when compared with the minimum variance portfolio. Thus, achieving the minimum variance portfolio would not be a worthwhile goal.

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With the given information, Ito finds that the CAL of the S&P 500 is equal to the CAL of the MT portfolio if the return of the MT portfolio equals:

A) 8.6%.

B) 10.6%.

C) 11.4%.

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The CAL of the S&P 500 is 0.35 = (12 − 5) / 20. To find the return that gives this slope for the CAL, Ito would solve for R in the expression 0.35 = (R − 5) / 16. This gives 5.6 = R − 5, R = 10.6.

luckygiftvn

An investor holds a single stock, Amgen, in her portfolio. She would like to add one additional stock to her portfolio. Which stock should she add to achieve the most diversification benefits?

Correlation Matrix
Fund	Amgen	WW	XX	ZZ
Amgen	1.0
WW	0.5	1.0
XX	0.1	-0.2	1.0
YY	0.3	0.4	0.8
ZZ	0.0	0.8	0.9	1.0

A) Stock WW.

B) Stock XX.

C) Stock ZZ.

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As the correlation between assets decreases, the benefits of diversification increase. Of the three stocks, ZZ has the lowest correlation with Amgen.

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It can be determined from the figure below that ρ2 is:

A) between 0.2 and 1.0.

B) between 0.0 and 0.2.

C) between -1.0 and 0.2.

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The diversification benefits are greater if the correlation between the returns of the assets in the portfolio is lower. If the correlation equals +1, the minimum variance frontier is a straight line and there is no benefit to diversification (ρ3). If the correlation equals = -1, the minimum variance frontier is two line segments (ρ1). Therefore ρ2 must be less than 0.2 and greater than –1.0. It could be equal to zero, but we can’t tell for sure given the information in the problem.

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Matton, CFA, has been asked to invest $100,000, choosing one or more of the following three stocks. All stocks have the same expected return and standard deviation. The correlation matrix for the three stocks is given below:

Stock Correlations
	X	Y	Z
X	1.00	0.15	0.70
Y	0.15	1.00	0.51
Z	0.70	0.51	1.00

Which of the three stocks, X, Y, and Z, should be included in the portfolio?

A) X, Y, and Z.

B) Any investment in the three stocks will result in the exact same expected return and risk.

C) X and Y only.

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Diversification benefits occur whenever a stock is added that is not perfectly positively correlated with other stocks in the portfolio. Since none of the stocks are perfectly positively correlated with the other stocks, it would be beneficial to purchase all three rather than just one or two stocks