Reading 51: An Introduction to Asset Pricing Models LOS d习题 2010, including, security, capital

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LOS d, (Part 1): Explain the capital asset pricing model, including the security market line (SML) and beta.

 

Beta is least accurately described as:

A) a measure of the sensitivity of a security’s return to the market return.

B) a standardized measure of the total risk of a security.

C) the covariance of a security’s returns with the market return, divided by the variance of market returns.

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Beta is a standardized measure of the systematic risk of a security. β = Cov_r,mkt / σ²_mkt. Beta is multiplied by the market risk premium in the CAPM: E(R_i) = RFR + β[E(R_mkt) – RFR].

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The security market line (SML) will resemble a band, rather than a line, if some of its underlying assumptions are lifted. About which of the following assumptions is this least accurate?

A) No transactions costs.

B) Heterogeneous investor expectations.

C) Unequal borrowing and lending rates.

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If investors have heterogeneous expectations, or if transaction costs are not assumed to be zero, the SML becomes a band rather than a line. Differences in borrowing and lending rates can be assumed to be appropriate and used in the construction of the capital market line (CML). The result is a CML that is bent around the market portfolio. The portion of the CML connecting the risk-free asset and market portfolio will have a steeper slope than the portion of the CML extending beyond the market portfolio.

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Which of the following assumptions associated with the capital asset pricing model (CAPM), when relaxed, will be least likely to result in turning the security market line (SML) into a band rather than a line?

A) A single holding period.

B) No transaction costs.

C) Equal borrowing and lending rates.

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If the assumption of equal borrowing and lending rates is relaxed then the CAPM cannot be derived since there is no unique market portfolio. In effect, the CML will become kinked.

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LOS d, (Part 2): Describe the effects of relaxing the capital asset pricing model's underlying assumptions.

 

A basic assumption of the capital asset pricing model (CAPM) is that there are no transaction costs. If this assumption is relaxed, which of the following would be the least likely to occur?

A) All securities will plot very close to the security market line.

B) Each investor can have a unique view of a security market line.

C) Diversification benefits will be realized up to the point that they offset transactions costs.

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If the assumption of “no transaction cost” is relaxed, then investors will correct mispricing only up to the point where transaction costs begin to offset potential excess return. As a result, all securities will plot within a band around the SML. It also would impact diversification, since at some point the transaction cost will offset the benefits of diversification.

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21st Century Investments manages a portfolio, Z, that has zero correlation with the market index and examines the prospects for AMI Enterprises, a manufacturer of laptop batteries. 21^st Century Investments derives the following market forecasts:

• Expected return on portfolio Z -   8%
• Expected return on the market index -  14%
• Risk-free rate -  5%
• AMI beta -  1.50

Using the zero-beta form of the capital asset pricing model (CAPM), the equilibrium expected return for AMI is closest to:

A) 17.0%.

B) 18.5%.

C) 14.0%.

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The zero beta form of the CAPM replaces the risk-free rate with the return on a zero beta portfolio. Portfolio Z has zero correlation with the market portfolio. Therefore, the beta for portfolio Z also equals zero. Recall the formula for beta:

where cov_im is the covariance between any asset i and the market index m, σ_i is the standard deviation of returns for asset i, σ_m is the standard deviation of returns for the market index, and ρ_im is the correlation between asset i and the market index. Therefore, the beta will equal zero if the correlation equals zero.

The equation for the zero-beta CAPM is:

E(R) = E(R_z) + β[E(R_m) – E(R_z)] = 0.08 + 1.50[0.14 – 0.08] = 0.17 = 17%

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Andrew Howell uses the security market line (SML) to make investment decisions. His firm incurs 2% transaction costs on all purchases. How does the existence of the 2% transaction cost change the intercept and slope of the SML for stock purchases faced by Howell’s firm?

Intercept

Slope

A) Increase by 2% No change

B) No change Increase by 2%

C) No change No change

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The existence of transaction costs causes the SML to change from a line to a thick band. The width of the band equals the transactions cost (2% above the line for purchases, and 2% below the line for sales). The question asks about purchases, so the intercept will increase by 2%. There is no change in the slope if the 2% transaction applies to all stocks as stated in the question.

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Which of the following statements about asset pricing models is most accurate?

A) According to the Capital Asset Pricing Model (CAPM), the expected rate of return of a portfolio with a beta of 1.0 is the market expected return.

B) Assuming assets are not perfectly positively correlated, the systematic risk of a portfolio decreases as more assets are added.

C) Adding the risk-free asset to a portfolio will reduce return and total risk.

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Diversification reduces unsystematic, or unique risk. With the risk-free asset and a portfolio of risky assets, the equation for the expected standard deviation is linear: w_As_A .  A combination of the risk free asset and a portfolio always gives more return for a given level of risk.  Risk tends to be reduced, but assuming that assets are not perfectly positively correlated, an investor can achieve the benefits of diversification by adding just one security (Markowitz). Studies have shown that approximately 18-30 stocks are needed for proper diversification. The main point is that the number of stocks required is small and is significantly less than all securities (and significantly less than 1,000 securities).

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Given the following data, what is the correlation coefficient between the two stocks and the Beta of stock A?

• standard deviation of returns of Stock A is 10.04%
• standard deviation of returns of Stock B is 2.05%
• standard deviation of the market is 3.01%
• covariance between the two stocks is 0.00109
• covariance between the market and stock A is 0.002

Correlation Coefficient

Beta (stock A)

A) 0.6556 2.20

B) 0.5296 0.06

C) 0.5296 2.20

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correlation coefficient = 0.00109 / (0.0205)(0.1004) = 0.5296.

beta of stock A = covariance between stock and the market / variance of the market

Beta = 0.002 / 0.0301² = 2.2

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Which of the following statements about the capital market line (CML) is least accurate?

A) Investors choose a portfolio on the CML by varying their weightings of the risk-free asset and the market portfolio.

B) The CML will not be a linear relationship if investors' borrowing and lending rates are not equal.

C) The market portfolio lies on the CML and has only unsystematic risk.

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The first part of this statement is true - the market portfolio does lie on the CML. However, the market portfolio is well diversified and thus has no unsystematic risk. The risk that remains is market risk, or nondiversifiable, or systematic risk.

The CML measures standard deviation (or total risk) against returns. The CML will “kink” if the borrowing rate and lending rate are not equal. Investors choose a portfolio on the CML by lending or borrowing at the risk-free rate to vary the weighting of their investments in the risk-free asset and the market portfolio.

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If the standard deviation of the market’s returns is 5.8%, the standard deviation of a stock’s returns is 8.2%, and the covariance of the market’s returns with the stock’s returns is 0.003, what is the beta of the stock?

A) 0.05.

B) 0.89.

C) 1.07.

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The formula for beta is: (Cov_stock,market)/(Var_market), or (0.003)/(0.058)² = 0.89.