Reading 50: An Introduction to Portfolio Management LOS e习题 manager, standard, existing, face, 2010

honeycfa

LOS e: List the components of the portfolio standard deviation formula.

A portfolio manager adds a new stock that has the same standard deviation of returns as the existing portfolio but has a correlation coefficient with the existing portfolio that is less than +1. Adding this stock will have what effect on the standard deviation of the revised portfolio's returns? The standard deviation will:

A) decrease.

B) increase.

C) decrease only if the correlation is negative.

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If the correlation coefficient is less than 1, there are benefits to diversification. Thus, adding the stock will reduce the portfolio's standard deviation.

 

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Which of the following measures is NOT considered when calculating the risk (variance) of a two-asset portfolio?

A) Each asset’s standard deviation.

B) The beta of each asset.

C) Each asset weight in the portfolio.

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The formula for calculating the variance of a two-asset portfolio is:

σ_p² = W_A²σ_A² + W_B²σ_B² + 2W_AW_BCov(a,b)

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Assets A (with a variance of 0.25) and B (with a variance of 0.40) are perfectly positively correlated. If an investor creates a portfolio using only these two assets with 40% invested in A, the portfolio standard deviation is closest to:

A) 0.3742.

B) 0.3400.

C) 0.5795.

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The portfolio standard deviation = [(0.4)²(0.25) + (0.6)²(0.4) + 2(0.4)(0.6)1(0.25)^0.5(0.4)^0.5]^0.5 = 0.5795

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As the correlation between the returns of two assets becomes lower, the risk reduction potential becomes:

A) greater.

B) smaller.

C) decreased by the same level.

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Perfect positive correlation (r = +1) of the returns of two assets offers no risk reduction, whereas perfect negative correlation (r = -1) offers the greatest risk reduction.

honeycfa

Adding a stock to a portfolio will reduce the risk of the portfolio if the correlation coefficient is less than which of the following?

A) 0.00.

B) +1.00.

C) +0.50.

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Adding any stock that is not perfectly correlated with the portfolio (+1) will reduce the risk of the portfolio.

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An investor has a two-stock portfolio (Stocks A and B) with the following characteristics:

• σ_A = 55%
• σ_B = 85%
• Covariance_A,B = 0.9
• W_A = 70%
• W_B = 30%

The variance of the portfolio is closest to:

A) 0.39

B) 0.59

C) 0.54

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The formula for the variance of a 2-stock portfolio is:

s² = [W_A²σ_A² + W_B²σ_B² + 2W_AW_Bσ_Aσ_Br_A,B]

Since σ_Aσ_Br_A,B = Cov_A,B, then

s² = [(0.7² × 0.55²) + (0.3² × 0.85²) + (2 × 0.7 × 0.3 × 0.9)] = [0.14822 + 0.06502 + 0.378] = 0.59124, or approximately 0.59.

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An investor’s portfolio currently consists of 100% of stocks that have a mean return of 16.5% and an expected variance of 0.0324. The investor plans to diversify slightly by replacing 20% of her portfolio with U.S. Treasury bills that earn 4.75%. Assuming the investor diversifies, what are the expected return and expected standard deviation of the portfolio?

ERPortfolio

σPortfolio

A) 14.15% 14.40%

B) 14.15% 2.59%

C) 10.63% 2.59%

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Since Treasury bills (T-bills) are considered risk-free, we know that the standard deviation of this asset and the correlation between T-bills and the other stocks is 0. Thus, we can calculate the portfolio expected return and standard deviation.

Step 1: Calculate the expected return
Expected Return_Portfolio = (w_T-bills × ER_T-bills) + (w_Stocks × ER_Stocks)
= (0.20) × (0.0475) + (1.00-0.20) × (0.165) = 0.1415, or 14.15%.

Step 2: Calculate the expected standard deviation
When combining a risk-free asset and a risky asset (or portfolio or risky assets), the equation for the standard deviation, σ_1,2 = [(w₁²)(σ₁²) + (w₂²)(σ₂²) + 2w₁w₂ σ₁ σ₂ρ_1,2]^1/2, reduces to: σ_1,2 = [(w_Stocks)(σ_Stocks)] = 0.80 × 0.0324^1/2 = 0.14400, or 14.40%. (Remember to convert variance to standard deviation).

honeycfa

What is the variance of a two-stock portfolio if 15% is invested in stock A (variance of 0.0071) and 85% in stock B (variance of 0.0008) and the correlation coefficient between the stocks is –0.04?

A) 0.0020.

B) 0.0007.

C) 0.0026.

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The variance of the portfolio is found by:

[W₁² σ₁² + W₂² σ₂² + 2W₁W₂σ₁σ₂r_1,2], or [(0.15)²(0.0071) + (0.85)²(0.0008) + (2)(0.15)(0.85)(0.0843)(0.0283)(–0.04)] = 0.0007.

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Which of the following equations is least accurate?

A) Real Risk-Free Rate = [(1 + nominal risk-free rate) / (1 + expected inflation)] ? 1.

B) Required Returnnominal = [(1 + Risk Free Ratereal) × (1 + Expected Inflation) × (1 + Risk Premium)] ? 1.

C) Standard Deviation2-Stock Portfolio = [(w12 × σ12) + (w22 × σ22) + (2 × w1 × w2 σ1σ2 × ρ1,2)].

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This is the equation for the variance of a 2-stock portfolio. The standard deviation is the square root of the variance. The other equations are correct.

honeycfa

Stock A has a standard deviation of 4.1% and Stock B has a standard deviation of 5.8%. If the stocks are perfectly positively correlated, which portfolio weights minimize the portfolio’s standard deviation?

Stock A

Stock B

A) 100% 0%

B) 63% 37%

C) 0% 100%

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Because there is a perfectly positive correlation, there is no benefit to diversification. Therefore, the investor should put all his money into Stock A (with the lowest standard deviation) to minimize the risk (standard deviation) of the portfolio.