Reading 66: Portfolio Concepts-LOS g 习题精选 respect, center, interpret, returns, 2011

土豆妮

Session 18: Portfolio Management: Capital Market Theory and the Portfolio Management Process
Reading 66: Portfolio Concepts

LOS g: Explain the market model, and state and interpret the market model’s predictions with respect to asset returns, variances, and covariances.

 

 

The single-factor market model predicts that the covariance between two assets (asset i and asset j) is equal to:

A) the beta of i times the beta of j.

B) the beta of i times the beta of j times the variance of the market portfolio.

C) the beta of i times the beta of j divided by the standard deviation of the market portfolio.

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One of the predictions of the single-factor market model is that Cov(R_i,R_j) = b_ib_js_M². In other words, the covariance between two assets is related to the betas of the two assets and the variance of the market portfolio.

土豆妮

The single-factor market model predicts that the systematic portion of the variance of an asset’s return is equal to the:

A) square of the asset's beta times the variance of the market portfolio.

B) covariance between the asset's returns and the market returns.

C) asset's beta.

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One of the predictions of the single-factor market model is that Var(R_i) = b_i²s_M² + s_ei². In other words, there are two components to the variance of the returns on asset i: a systematic component related to the asset’s beta (b_i²s_M²) and an unsystematic component related to firm-specific surprises (s_ei²).

土豆妮

Joseph Capital Management is considering implementing a mean-variance optimization model as part of their portfolio management process, however, the firm’s investment committee is unsure whether the model should use historical estimates or market model estimates for the inputs to the model. Joseph’s Senior Portfolio Manager, Travis Palmer, puts together a memo to the committee contrasting the two methods of calculating inputs. The memo includes the following points:

Point 1:	Using the historical estimate is far simpler and involves fewer computations than the market model method.
Point 2:	The use of market model estimates implicitly assumes that the market itself is mean-variance efficient.
Point 3:	Both the use of market model estimates and historical estimates rely on historical data to some degree.
Point 4:	One of the problems with using market model estimates for estimating returns is that the market model implicitly assumes the market index is representative of the entire market.

After reviewing Palmer’s memo, Joseph’s investment committee would be CORRECT to:

A) agree with Points 2 and 3, but disagree with Point 1.

B) agree with Point 3, but disagree with Points 2 and 4.

C) agree with Points 1 and 4, but disagree with Point 3.

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The committee should disagree with Point 1. The use of historical estimates involves computing the covariance of between each stock in a portfolio with every other stock in the portfolio, while the use of the market model only relies on computing the covariance of each stock with the market index, resulting in fewer computations.

The committee should agree with Points 2, 3, and 4. The market model regresses historical returns of a stock/portfolio with the corresponding returns of a market index and implicitly assumes that historical relationships are reflective of future relationships. The market model also implicitly assumes that the market itself is mean-variance efficient and that the index used for market returns is representative of the entire market.

土豆妮

Which of the following is NOT an assumption necessary to derive the single-factor market model? The:

A) market portfolio is the tangency portfolio.

B) firm-specific surprises are uncorrelated across assets.

C) expected value of firm-specific surprises is zero.

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The result that the market portfolio is the tangency portfolio is a prediction of the CAPM model, not the market model. The market model assumes that there are two sources of risk, unanticipated macroeconomic events and firm-specific events. We use the return on the market portfolio as a proxy for the macroeconomic factor and assume all stocks have varying degrees of sensitivity to this macro factor. In addition, each stock’s returns are uniquely affected by firm-specific events uncorrelated across stocks and with the macro events. The remaining choices are the assumptions necessary to derive the single-factor market model.

土豆妮

Carl Dursham recently earned the CFA designation and has just been hired by Quad Cities Consultants, which is a money management firm for private, high net worth clients. Quad Cities Consultants has just assigned Dursham his first client. The client’s name is Sally Litner. Litner has just received a multi-million dollar inheritance consisting of certificates of deposit that are about to mature. She is only 30 years old and recognizes that she should probably invest in assets like stocks that have a higher risk and return. Litner is a high school mathematics teacher and has an aptitude for formulas and equations, but she has never applied it to investments. Litner feels that Dursham will probably do a good job for her, but she wants him to explain to her how he will approach creating her portfolio.

When Dursham and Litner first meet, Litner says that she has heard of a stock that has done very well and is expected to continue to experience dramatic increases in the future. The name of the stock is IntMarket Corporation, which is a company that facilitates commerce on the Internet, and its recent return and standard deviation are 24% and 60% respectively. She asks Dursham if he thinks she should invest 100% of her portfolio in IntMarket Corporation. Dursham looks up the beta of IntMarket and finds that it is 1.6. He says that IntMarket Corporation might be a good first position, and he says that a good second position might be Granite Bank. The return and standard deviation of the bank stock is 12% and 30% respectively. Its beta is 0.9. The covariance of the bank stock with IntMarket Corporation is 576.

Dursham explains how diversification can lower risk and computes the statistics for portfolios that have various weights in IntMarket and Granite Bank. Litner is intrigued by Dursham’s demonstration concerning the effects of diversification. She asks about the effect of adding a third asset to the portfolio. To help illustrate the benefits of diversification further, Dursham chooses Capital Commodities Mutual fund, which invests in assets related to the production of raw materials and other commodities. The recent return and standard deviation of Capital Commodities has been 8% and 18% respectfully. The correlation of Capital Commodities with the other two stocks is effectively zero. Dursham computes the return and standard deviation of a portfolio consisting of 50% IntMarket, 30% Granite Bank, and 20% Capital Commodities.

Dursham takes time to explain the principle and assumptions behind mean-variance analysis and why it is important. He says the four underlying principals are i) investors are risk averse, ii) necessary statistics of returns can be calculated, iii) the returns have a normal distribution, and iv) the tax rate is fixed at some positive rate like 28%. During the discussion, Litner says she thinks the three stocks IntMarket Corporation, Granite Bank, and Capital Commodities may be all she needs in her portfolio. She asks Dursham to choose the weights for those three stocks that will minimize the variance and let that be her portfolio. If they desire a higher return, she adds using terms she has just learned, they can just leverage up that portfolio.

If the recent return of the market was 14%, and the risk-free rate is 3%, using the market model what was the alpha of IntMarket Corporation?

A) +1.6%.

B) +4.4%.

C) +1.4%.

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When using the market model, alpha is the difference between the realized return and that predicted by the product of the beta and the market return. The risk-free rate is not a part of the computation. The recent return of IntMarket was 24%. The predicted return based upon a beta equal to 1.6 and a market return of 14% is the product of these values: 22.4%. Thus the alpha is 24% ? 22.4% = 1.6%. (Study Session 18, LOS 64.g)

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Of Dursham’s list of the assumptions underlying mean-variance analysis, which of the following is the least likely to be one of the generally accepted assumptions?

A) The tax rate is fixed at some positive rate like 28%.

B) Necessary statistics of returns can be calculated.

C) The returns have a normal distribution.

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The assumption should be that there are no taxes and that there are no transactions costs. (Study Session 18, LOS 64.g)

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A portfolio invested 50% in IntMarket and 50% in Granite Bank would have an expected return:

A) lower than that of Granite Bank and a higher standard deviation than that of Granite Bank.

B) greater than that of Granite Bank and a higher standard deviation than that of Granite Bank.

C) greater than that of Granite Bank and a lower standard deviation than that of Granite Bank.

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The average will obviously be higher than that of Granite Bank. The average is 18% = (0.5 × 24%) + (0.5 × 12%). The variance of the 50/50 portfolio is 1413 = (0.5 × 0.5 × 60 × 60) + (0.5 × 0.5 × 30 × 30) + (2 × 0.5 × 0.5 × 576); the standard deviation is about 37.6%, which is greater than the 30% standard deviation of Granite Bank. (Study Session 18, LOS 64.a)

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The portfolio that Dursham recommends using the two stocks and the mutual fund would have a standard deviation that is closest in value to:

A) 36.0%.

B) 36.7%.

C) 34.2%.

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Since the return of Capital Commodities is uncorrelated with the returns of the two stocks, the variance of the portfolio is 1166.8 = (0.5 × 0.5 × 60 × 60) + (0.3 × 0.3 × 30 × 30) + (0.2 × 0.2 × 18 × 18) + (2 × 0.5 × 0.3 × 576) The standard deviation is √34.2%. (Study Session 18, LOS 64.a)

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When compared to all other possible portfolios, the portfolio that has the smallest variance, which Litner requests, would have a Sharpe ratio that:

A) may or may not be the highest of all possible portfolios; there is no general rule.

B) is the highest of all possible portfolios.

C) could not be the highest of all possible portfolios.

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Minimizing the variance does not produce the portfolio with the highest Sharpe ratio. A point along the efficient frontier above the minimum variance portfolio will have both a higher return and standard deviation, but it will have a higher Sharpe ratio. (Study Session 18, LOS 64.b)

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The portfolio that Litner requests, the one that has the smallest variance of all possible portfolios, would best be described as the:

A) global minimum variance portfolio.

B) market portfolio.

C) efficient variance portfolio.

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This is the definition of the global minimum variance portfolio. (Study Session 18, LOS 64.b)

土豆妮

Michael Carr and Karen Bocock are analysts for the Portfolio Optimization Group. Carr and Bocock are discussing the firm’s mean variance optimization model for equity holdings and the pros and cons of using market model estimates or historical estimates as inputs to the model.

• Carr states, “One of the main concerns I have about the model is that whether we are using market model estimates or historical estimates, we are implicitly assuming that the historical relationship between the stock and the market is indicative of the future.”
• Bocock replies, “One of the main advantages to using the market model estimates is the fact that there are fewer parameters to estimate.”

With regard to their statements about methods for computing the inputs for a mean-optimization model:

A) both are correct.

B) both are incorrect.

C) only one is correct.

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Carr’s statement is correct. Using historical estimates and market model estimates both involve the implicit assumption that the historical relationship between a stock and the market is indicative of the future relationship. The historical estimate method uses direct historical means, variances, and correlations as inputs to the model. The market model method regresses historical returns against returns for the market and assumes that returns for each asset are correlated with returns to the market. Since both methods use some form of historical data, both assume that history is indicative of the future.

Bocock is also correct. The historical estimate method requires a large number of estimates, especially for computing the covariances between every stock in a portfolio. The market model estimate method simplifies the process significantly (resulting in fewer parameters) since all stock returns are assumed to be correlated with the market.

土豆妮

The single-factor market model assumes there are how many sources of risk in asset returns?

A) Two.

B) One.

C) Three.

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The market model assumes that there are two sources of risk in asset returns, unanticipated macroeconomic events and firm-specific events.

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Bill Tanner is a new associate at Global Western Investments. Tanner approaches his supervisor, Eric Simms, with some questions about risk. Specifically, Tanner lacks a complete understanding of many portfolio concepts, including the following:

• How the presence of a risk-free asset will affect the efficient frontier.
• The difference between total risk, systematic risk, and unsystematic risk.
• Market and Macroeconomic models.

Tanner is concerned with providing the best investment advice possible for his clients. He seeks advice from some of his former Midwestern college friends who now happen to be CFA charterholders. One of his old roommates suggests that he look into using the market model or a multifactor model based on the arbitrage pricing theory (APT).

Tanner researches alternative pricing models and starts to become confused as all the equations look similar. He writes down the following notes from memory:

• The intercept for the market model is derived from the APT.
• The intercept for the APT is the risk free rate.
• The intercept for a macroeconomic factor model is the expected return on the stock when there are no surprises to the factors.

Simms makes the predictions for Tanner shown in Exhibit 1.

Exhibit 1: Simm’s Predictions for Tanner

Beta for Stock B	1.10
Beta for Stock C	1.50
Correlation between Stock A and the S& 500	0.50
Standard deviation for Stock A	28%
Standard deviation for the S& 500	20%
1-year Treasure bill rate	5%
Expected return on the S& 500	12%

Tanner uses the market model predictions (and the S& 500 as a proxy for the market portfolio) to calculate the covariance of Stock B and C at 0.33. Using the market model, he also determines that the systematic component of the variance for Stock B is equal to 0.048.

Next, he heads out to meet a friend, Del Torres, for lunch. Torres excitedly tells Tanner about his latest work with tracking and factor portfolios. Torres says he has developed a tracking portfolio to aid in speculating on oil prices and is working on a factor portfolio with a specific set of factor sensitivities to the Russell 2000.

Which of the following is the most appropriate response to Tanner’s question about the presence of a risk-free asset and the Markowitz efficient frontier? The presence of a risk-free asset changes the characteristics of the Markowitz efficient frontier by:

A) reducing the total risk and the systematic risk of the market portfolio.

B) converting the Markowitz efficient frontier from a curve into a linear risk/return relationship.

C) allowing risk averse investors to include in their portfolios an asset that is negatively correlated with stocks, thereby reducing the risk related to investing in equities.

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The presence of a risk-free asset changes the characteristics of the Markowitz efficient frontier by converting the Markowitz efficient frontier from a curve into a straight line called the capital market line (CML). (Study Session 18, LOS 64.d)

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Which of the following statements best describes the concept of systematic risk? Systematic risk:

A) is approximately equal to total risk divided by unsystematic risk.

B) as measured by the standard deviation is the only risk rewarded by the market.

C) remains even for a well-diversified portfolio.

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Systematic risk remains even if a portfolio is well diversified. (Study Session 18, LOS 64.g)

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Are Tanner’s notes on the intercepts for the pricing models correct?

A) No, because the intercept for the APT is the stock’s alpha.

B) No, because the intercept for the market model is the return on the stock when the return on the market is zero.

C) No, because the intercept for the market model is the risk-free rate.

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Tanner is incorrect with regard to the market model. The intercept is equal to the return when the market return is zero. Tanner’s other two comments on intercepts are correct. (Study Session 18, LOS 64.g)

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The beta of Stock A is closest to:

A) 0.36.

B) 0.70.

C) 0.50.

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(Study Session 18, LOS 64.h)

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According to the predictions of the market model, did Tanner correctly calculate the covariance of Stock B and C and Stock B’s systematic component of variance?

Covariance

Systematic component

A) No Yes

B) Yes Yes

C) Yes No

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Tanner incorrectly calculated the covariance and correctly calculated the systematic variance component.

According to the market model, the covariance between any two stocks is calculated as the product of their betas and the variance of the market portfolio. Here, the S& 500 is a proxy for the market portfolio.

Here, Cov_B,C = 1.10(1.50)(0.2)² = 0.066. Tanner incorrectly used the standard deviation of the market.

The variance of the returns on asset i consists of two components: a systematic component related to the asset’s beta, , and an unsystematic component related to firm-specific events, .

For Stock B, the systematic component = 1.10²(0.2)² = 0.048 (Study Session 18, LOS 64.a)

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Did Torres correctly describe tracking and factor portfolios?

Tracking

Factor

A) Yes No

B) No Yes

C) No No

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Torres reversed the concepts and is thus incorrect on both counts. A factor portfolio is a portfolio with a factor sensitivity of 1 to a particular factor and zero to all other factors. It represents a pure bet on one factor, and can be used for speculation or hedging purposes. A tracking portfolio is a portfolio with a specific set of factor sensitivities. Tracking portfolios are often designed to replicate the factor exposures of a benchmark index like the Russell 2000. (Study Session 18, LOS 64.m)