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周教授CFA金融课程:2019/2020年CFA一二三级系列课程

Portfolio Management【Reading 60】Sample

Mean-variance analysis assumes that investor preferences depend on all of the following EXCEPT:
A)
correlations among asset returns.
B)
skewness of the distribution of asset returns.
C)
expected asset returns.



Mean-variance analysis assumes that investors only need to know expected returns, variances, and covariances in order create optimal portfolios. The skewness of the distribution of expected returns can be ignored.

One of the assumptions of mean-variance analysis is that all investors are risk-averse, which means they:
A)
are not willing to make risky investments.
B)
prefer less risk to more for any given level of volatility.
C)
prefer less risk to more for any given level of expected return.



In mean-variance analysis we assume that all investors are risk averse, which means they prefer less risk to more for any given level of expected return (NOT for any given level of volatility.) It does NOT mean that they are unwilling to take on any risk.

TOP

What are the expected return and expected standard deviation for the two-asset portfolio described as:

Expected Return/Correlation

Variance

Weight


E(R1) = 10%

Var(1) = 9%

w1 = 30%


E(R2) = 15%

Var(2) = 25%

w2 = 70%


r1,2 = 0.4

E(Rport)σport
A)
10.5%15.58%
B)
13.5%39.47%
C)
11.5%3.95%



E(Rport) = w1E(R1) + w2E(R2) = (0.3)(10.0) + (0.7)(15.0) = 13.5%
σport = [(w1)2(σ1)2 + (w2)2(σ2)2 + 2w1w2σ1σ2ρ1,2]1/2
= [(0.3)2(0.09) + (0.7)2(0.25) + 2(0.3)(0.7)(0.3)(0.5)(0.4)]1/2 = 39.47%

TOP

Allen Marko, CFA, is analyzing the diversification benefits available from investing in three equity funds. He is basing his analysis on monthly returns for the three funds and an appropriate market index over the past twenty years. He feels that there is no reason that the past performance should not carry forward into the future. Treasury bills currently pay 5%.

Table 1: Expected Returns, Variances, and Covariance for Funds A, B, & C

Equity Fund A

Equity Fund B

Equity Fund C


Average Return

12%

9%

8%


Variance

0.0256

0.0196

0.0172


Correlation of A & B is 0.50
Correlation of A & C is 0.38
Correlation of B & C is 0.85

Marko has also obtained information about a fourth fund, Fund D. He does not have any information regarding the covariance of Fund D with Funds A, B, and C. The average return and variance for fund D are 10% and 0.018, respectively. The beta of Fund D is 0.714. Based on this data, what is the expected return of a portfolio that is made up of 60% of Fund A, 30% of Fund B, and 10% of Fund C?
A)
10.2%.
B)
10.7%.
C)
11.4%.



Expected return for the portfolio = (0.6)(0.12) + (0.3)(0.09) +(0.1)(0.08)= 0.107 or 10.7%. (Study Session 18, LOS 60.a)

Which of the following is closest to the standard deviation of a portfolio that is made up of 60% of Fund A, 30% of Fund B, and 10% of Fund C?
A)
14.840%.
B)
2.205%.
C)
13.062%.



Standard deviation of a three asset portfolio:
σportfolio = [(0.6)2(0.0256) + (0.3)2(0.0196) + (0.1)2(0.0172) + 2(0.60)(0.30)(0.50)(0.16)(0.14) + 2(0.60)(0.10)(0.38)(0.16)(0.13)+ 2(0.3)(0.1)(0.85)(0.14)(0.13)]0.5

= [0.017062]1/2 = 0.13062 or 13.062%.


(Study Session 18, LOS 60.a)


With respect to the relative efficiencies of the Funds, which of the following is most accurate?
A)
Fund B is inefficient relative to Fund D.
B)
No determination is possible.
C)
Fund B and D are both inefficient.



To be inefficient, the return must be lower while the variance is higher. The only case where that relationship exists is with respect to Fund B and D. (Study Session 18, LOS 60.b)

If Marko had to choose to form a portfolio using only T-bills and one of the four funds, which should he choose?
A)
Fund A.
B)
Fund B.
C)
Fund D.



The easiest way to approach this question is to calculate the Sharpe ratio for each fund and choose the one with the highest ratio. The highest Sharpe ratio reflects the highest excess return for a given level of risk.
The Sharpe ratios are as follows:

Fund A = (12 − 5) / 16.00 = 0.44

Fund B = (9 − 5) / 14.00 = 0.29

Fund D = (10 − 5) / 13.42 = 0.37


Fund A has the highest Sharpe ratio and therefore would be the best one to combine with T-bills.
An alternative way to answer the question can be seen by combining Fund A with T-bills in a portfolio to get an average/expected return equal to each of the other portfolios and computing the variance for each of those portfolios. Then compare the variance of the portfolio composed of A and the T-bills to the corresponding variance of the other asset.
To find the appropriate weights for the portfolio to earn the return of Fund B, solve for W in the following equation: 9% = W × 12% + (1 − W) × 5%. The solution is W = 0.5714.
0.5714 in Fund A and 0.429 in T-bills has a variance equal to (0.5714)(0.5714)(0.0256) = 0.00836.
Applying the same procedure to Fund D gives W = 0.80
0.80 in Fund D and 0.20 in T-bills has a variance equal to (0.80)(0.80)(0.018) = 0.01152.
Thus, a CAL formed with Fund A can dominate the CAL of each of the other three portfolios. (Study Session 18, LOS 60.d)


Which of the following statements regarding the graph of return vs. risk for all possible portfolio combinations consisting of Funds A, B, and C is least accurate?
A)
Combinations of Fund A, B, and C will dominate all other combinations of portfolios that have a lower return for the same level of risk.
B)
If the objective of the portfolio manager is to minimize risk the optimal portfolio must lie on the curved line below the minimum-variance portfolio.
C)
If the objective of the portfolio manager is to maximize return the optimal portfolio must lie on the curved line above the minimum-variance portfolio.



The curved line below the minimum-variance portfolio represents all portfolio combinations that are dominated by other portfolio combinations. Based on the efficient frontier created by these two funds higher returns at the same level of risk can be achieved above the minimum-variance portfolio. (Study Session 18, LOS 60.b)

The beta of Fund A is 1.2, the expected return of T-bills is 5% and the standard deviation for the market is 13%. What is the covariance between the market portfolio and Fund A?
A)
0.020.
B)
0.081.
C)
0.156.


The beta for fund A is equal to the covariance of fund A and the market divided by the variance of the market. Therefore, 1.2 = COV(A,Market) / (0.13)2   
Solving for COV(A,Market) = (1.2)(0.13)2 = 0.0203. (Study Session 18, LOS 60.a)

TOP

Sandy Wilson is a research analyst for WWW Equities Investments. She has just finished collecting the information on Table 1 to answer questions posed by her supervisor, Jackie Lewis. For example, using the Capital Market Line (CML), Lewis wants to know the market price of risk. Also, given all the attention paid to index funds in recent years, Lewis asked Wilson to see if any one of the securities would prove a better investment than the S&P 500. If not, can she compose a portfolio from stocks A, B, and C that is more efficient than the S&P 500?
Lewis wants Wilson to explore whether the results on Table 1 are congruent with the Capital Asset Pricing Model (CAPM). Using a regression analysis where the S&P 500 represents the market portfolio, she computes the beta of Stock A, and finds that it equals one. Using this, she will derive the betas of the other stocks and compare them to betas estimated with other techniques. As she performs her calculations, she reviews reasons why her results might not be congruent with the CAPM. Lewis asserts that the S&P 500 may not be a good proxy for “the market portfolio” needed for CAPM calculations.
Table 1

Expected Return and Risk for Selected Investments

Investment

Expected Return

Standard Deviation


Stock A

12%

30%


Stock B

15%

35%


Stock C

11%

40%


S&P 500

12%

22%


Treasury Bills

3%

0%


Correlation Coefficient for Stocks A and B equals 0.4.
Correlation Coefficient for Stocks A and C equals -0.5.
Correlation Coefficient for Stocks B and C equals 0.1.
Assuming that the S&P 500 is the market portfolio and her estimates are accurate, what is the price of risk based on the slope of the Capital Market Line (CML)?
A)
0.409.
B)
0.545.
C)
0.250.



The market price of risk, or return per unit of standard deviation risk, is determined as follows: (0.12 − 0.03) / 0.22 = (0.09 / 0.22) = 0.409. (Study Session 18, LOS 60.d)

What is the expected return and standard deviation of a portfolio that consists of 40% of stock A and 60% of stock B?
A)
Expected Return: 13.8%, Standard Deviation: 29.5%.
B)
Expected Return: 13.8%, Standard Deviation: 28.0%.
C)
Expected Return: 13.8%, Standard Deviation: 33.0%.



E(RP) = 0.4(0.12) + 0.6(0.15) = 0.048 + 0.09 = 0.138 or 13.8%The portfolio standard deviation is:
[(0.4)2(0.3)2 + (0.6)2(0.35)2 + 2(0.4)(0.6)(0.3)(0.35)(0.4)]0.5 = [0.0144 + 0.0441 + 0.02016]0.5 = 0.2805
(Study Session 18, LOS 60.a)


Wilson uses the computed beta of stock A, the covariance of stock A and B, and their standard deviations to compute stock B’s beta. Given stock B’s expected return, the results are:
A)
not congruent with the CAPM, which does not support Lewis’ assertion concerning the S&P 500 as a proxy for the market.
B)
congruent with the CAPM, which does not support Lewis’ assertion concerning the S&P 500 as a proxy for the market.
C)
not congruent with the CAPM, which supports Lewis’ assertion concerning the S&P 500 as a proxy for the market.



The provided standard deviations and covariance and the beta of stock A can be entered into the following relationship:
covariance(A,B)=(beta of A) × (beta of B) × (Variance of market) gives us
(0.3 × 0.35 × 0.40) = 0.042 = 1 × (beta of B) × (0.22 × 0.22)
beta of B = 0.042 / 0.0484 = 0.868.
expected return of B = risk free rate + (beta of B) × (Market risk premium),
expected return of B = 0.03 + (0.868) × (0.12 − 0.03) = 0.108 < 0.15, which is the expected return she computed from her analysis. One explanation for this is that the S&P 500 is not a good proxy for the market portfolio. (Study Session 18, LOS 60.a,g)


Based upon the given information, can Wilson compose a portfolio with any one of the three stocks and Treasury bills that is more efficient than the S&P 500?
A)
No, the S&P 500 is more efficient than any of the individual stocks.
B)
Yes, stock B.
C)
Yes, stock A.



To investigate this, Wilson can first rule out stocks A and C. Both of them have an expected return that is less than or equal to the S&P 500, but their standard deviations are higher. Wilson must perform some calculations to see if stock B is more efficient than the S&P 500. Wilson would first determine the portfolio weights that can make the expected return of the stock B and T-bill portfolio equal to the S&P 500 portfolio. By setting up 0.12 = w × 0.15 + (1 − w) × 0.03 and solving for w, Wilson finds that a (0.75 / 0.25) stock B/T-bill portfolio has the same expected return of 0.12. The standard deviation of that portfolio is (0.75 × 35%) = 26.25% > 24% which is the standard deviation of the S&P 500. Thus, the portfolio using Stock B and Treasury bills is not more efficient than the S&P 500. (Study Session 18, LOS 60.b)

With regard to the capital allocation line (CAL), moving along the CAL above the point of the tangency portfolio represents:
A)
borrowing at the risk-free rate to be invested in more than 100% of the tangency portfolio.
B)
buying T-bills to reduce risk yet still maximize efficiency by being on the CAL.
C)
increasing risk exposure by being above the efficient frontier.



Moving along the CAL above the tangency portfolio represents borrowing at the risk free rate (shorting T-bills) to invest in more than your original capital in the tangency portfolio. The CAL becomes the efficient frontier when the risk free asset is available to invest in. (Study Session 18, LOS 60.d)

Which of the following is least likely an assumption of the Capital Asset Pricing Model (CAPM)?
A)
Capital markets are perfectly competitive and all assets are marketable.
B)
The distribution of investors' forecasts of a given asset’s return is normal.
C)
Investors can borrow and lend at the risk-free rate.



The CAPM assumes that investors have the same forecast of a given asset’s return. Thus, according to the required assumption, the distribution will not be normal because the variance of the forecasts is zero. (Study Session 18, LOS 60.e)

TOP

Given the following information, what is the expected return on the portfolio of the two funds?

The Washington Fund

The Jefferson Fund


Expected Return

30%

36%


Variance

0.0576

0.1024


Investment

$2,000,000

$6,000,000


Correlation

0.40

A)
33.0%.
B)
34.5%.
C)
31.5%.



First calculate the portfolio weights on each fund:

WWash = $2 million/$8 million = 0.25
WJeff = $6 million/$8 million = 0.75

The expected portfolio return is the weighted average of the funds' expected returns:

E(RP) = (0.25)(30%) + (0.75)(36%) = 34.5%.

TOP

Which of the following statements is least accurate regarding modern portfolio theory?
A)
The capital market line is developed under the assumption that investors can borrow or lend at the risk-free rate.
B)
All portfolios on the capital allocation line are perfectly negatively correlated.
C)
For a portfolio made up of the risk-free asset and a risky asset, the standard deviation is the weighted proportion of the standard deviation of the risky asset.



All portfolios on the capital allocation line are perfectly positively correlated. Both remaining statements are each true.

TOP

Joe Janikowski owns a portfolio consisting of 2 stocks. Janikowski has compiled the following information:

Stock

Topper Manufacturing

Base Construction

Expected Return (percent

12

11

Standard Deviation (percent)

10

15

Portfolio Weighting (percent)

75

25

Correlation

0.22

The expected return for the portfolio is:
A)
11.50%.
B)
12.00%.
C)
11.75%.



Expected return is computed by weighting each stock as a percentage of the entire portfolio, and then multiplying each stock by the expected return. The expected return is: ((0.75 × 12) + (0.25 × 11)) = 11.75.

The standard deviation of the portfolio is closest to:
A)
0.0909.
B)
0.0839.
C)
0.0070.



The formula for the standard deviation of a two-stock portfolio is: the square root of [((0.75)² × (0.10)²) + ((0.25)² × (0.15)²) + (2 × (0.75) × (0.25) × (0.22) × (0.15) × (0.10))] = 0.0909.

TOP

Andy Green, CFA, and Sue Hutchinson, CFA, are considering adding alternative investments to the portfolio they manage for a private client. They have found that it is recommended that a large, well-diversified portfolio like the one that they manage should include a 5 to 10% allocation in alternative investments such as commodities, distressed companies, emerging markets, etc.. After much discussion, Green and Hutchinson have decided that they will not choose individual assets themselves. Instead of choosing individual alternative investments, they will add a hedge fund to the portfolio. They decide to divide up their research by having each of them take a different focus. In their research of hedge funds, Green focuses on hedge funds that have the highest returns. Hutchinson focuses on finding hedge funds that can allow the client’s portfolio to lower risk while, with the use of leverage, maintain the same level of return.
After completing their research into finding appropriate hedge funds, Green proposes two hedge funds: the New Horizon Emerging Market Fund, which takes long-term positions in emerging markets, and the Hi Rise Real Estate Fund, which holds a highly leveraged real estate portfolio. Hutchinson proposes two hedge funds: the Quality Commodity Fund, which takes conservative long-term positions in commodities, and the Beta Naught Fund, which manages an equity long/short portfolio that has the goal of targeting the portfolio’s market risk to zero. The Beta Naught Fund engages in short-term pair trading to capture additional returns while keeping the beta of the fund equal to zero. The table below lists the statistics for the client’s portfolio without any alternative investments and for the four hedge funds based upon recent data. The expected return, standard deviation and beta of the client portfolio and the hedge funds are expected to have the same values in the near future. Green uses the market model to estimate covariances between portfolios with their respective betas and the variance of the market return. The variance of the market return is 324(%2).


Current Client Portfolio

New Horizon

Hi Rise Real Estate

Quality Commodity

Beta Naught

Average

10%

20%

10%

6%

4%

Std. Dev.

16%

50%

16%

16%

25%

Beta

0.8

0.9

0.4

-0.2

0

Green and Hutchinson have decided to sell off 10% of the current client portfolio and replace it with one of the four hedge funds. They have agreed to select the hedge fund that will provide the highest Sharpe Ratio when 10% of the client’s portfolio is allocated to that hedge fund.
As an alternative to investing 10% in one hedge fund, Green and Hutchinson have discussed investing 5% in the Beta Naught Fund and 5% in one of the other three hedge funds. This new 50/50 hedge fund portfolio would then serve as the 10% allocation in alternative investments for the client’s portfolio. Green and Hutchinson divided up their research into return enhancement and diversification benefits. Based upon the stated goals of their research, which of the two approaches is more likely to lead to an appropriate choice? The focus of:
A)
neither manager is appropriate and will not achieve a meaningful result.
B)
Green’s research.
C)
Hutchinson’s research.



Simply increasing return may not be appropriate if the risk level increases more than the return increases. Focusing on assets that help diversify the existing portfolio is more appropriate because any reduction in return can be offset by an increase in leverage. (Study Session 18, LOS 60.a, b)

Of the proposed hedge funds, which is most likely to introduce active risk into the client’s portfolio?
A)
Hi Rise Real Estate Fund.
B)
New Horizon Emerging Market Fund.
C)
The Beta Naught Fund.


The Beta Naught Fund is the only one that takes short-term positions. (Study Session 18, LOS 60.a)

Which of the following is closest to the expected return of the client’s portfolio if 10% of the portfolio is invested in the New Horizon Emerging Market Fund?
A)
10.2%.
B)
11.0%.
C)
11.8%.



11% = (0.9 × 10%) + (0.1 × 20%) (Study Session 18, LOS 60.a)

Which of the following is closest to the expected standard deviation of the client’s portfolio if 10% of the portfolio is invested in the Quality Commodity Fund?
A)
9.6%.
B)
14.2%.
C)
16.0%.



The market model offers a simple way to estimate the covariance between two assets, using the beta of each asset and the variance of the market return. Here, covariance is -51.84 = 0.8 × (-0.2) × 324. The variance of the new client portfolio is 200.59 = (0.9 × 0.9 × 16 × 16) + (0.1 × 0.1 × 16 × 16) + (2 × 0.9 × 0.1 × (-51.84)). The square root of the variance of the new client portfolio is approximately 14.2%. (Study Session 18, LOS 60.a,g)

Which of the following is closest to the expected return of a portfolio that consists of 90% of the original client’s portfolio, 5% of the Hi Rise Real Estate Fund and 5% in the Beta Naught Fund?
A)
9.0%.
B)
9.7%.
C)
10.4%.



9.7% = (0.9 × 10%) + (0.05 × 10%) + (0.05 × 4%) (Study Session 18, LOS 60.a)

There was a discussion of allocating 5% each in Beta Naught and one of the other funds. When combined with Beta Naught in a 50/50 portfolio, which of the other three funds will produce a portfolio that has the lowest standard deviation?
A)
Either Hi Rise or Quality Commodity.
B)
New Horizon only.
C)
Quality Commodity only.



Since the beta of Beta Naught is zero, its covariance with any of the other funds is zero. Thus, the lowest standard deviation will be achieved with the fund with the lowest standard deviation. Since Hi Rise and Quality Commodity have the same standard deviation, which is less than New Horizon, either of them would produce the same result. (Study Session 18, LOS 60.a)

TOP

Which of the portfolios represented in the table below are NOT efficient?

Portfolio

A

B

C

D

E

F

G

H

(Rp)

10%

12.5%

15%

16%

17%

18%

18%

20%

sp

23%

21%

25%

29%

29%

32%

35%

45%

A)
B, D, and F.
B)
A, D, and G.
C)
B, E, and F.



Relative to any other portfolio, an inefficient portfolio has greater risk at the same return (portfolio G), less return at the same level of risk (portfolio D), or less return and more risk (portfolio A).

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