Portfolio Management【Reading 60】Sample 2012

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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.

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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.

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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.

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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.

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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%

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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%

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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%.

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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)

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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%.

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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)

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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.

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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)

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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.

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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)

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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.

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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)

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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.

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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)

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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.

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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)

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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%.

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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)

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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.

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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)

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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.

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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)

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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.

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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)

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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.

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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)