Reading 7: Statistical Concepts and Market Returns-LOS e 习题 2011, including, interpret, measures, average

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Session 2: Quantitative Methods: Basic Concepts
Reading 7: Statistical Concepts and Market Returns

LOS e: Define, calculate, and interpret measures of central tendency, including the population mean, sample mean, arithmetic mean, weighted average or mean (including a portfolio return viewed as a weighted mean), geometric mean, harmonic mean, median, and mode.

 

 

Consider the following statements about the geometric and arithmetic means as measures of central tendency. Which statement is least accurate?

A) The difference between the geometric mean and the arithmetic mean increases with an increase in variability between period-to-period observations.

B) The geometric mean may be used to estimate the average return over a one-period time horizon because it is the average of one-period returns.

C) The geometric mean calculates the rate of return that would have to be earned each year to match the actual, cumulative investment performance.

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The arithmetic mean may be used to estimate the average return over a one-period time horizon because it is the average of one-period returns. Both remaining statements are true.

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A stock had the following returns over the last five years: 15%, 2%, 9%, 44%, 23%. What is the respective geometric mean and arithmetic mean for this stock?

A) 17.76%; 23.0%.

B) 17.76%; 18.6%.

C) 0.18%; 18.6%.

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Geometric mean = [(1.15)(1.02)(1.09)(1.44)(1.23)]^1/5 ? 1 = 1.17760 = 17.76%.

Arithmetic mean = (15 + 2 + 9 + 44 + 23) / 5 = 18.6%.

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Trina Romel, mutual fund manager, is taking over a poor-performing fund from a colleague. Romel wants to calculate the return on the portfolio. Over the last five years, the fund’s annual percentage returns were: 25, 15, 12, -8, and –14. Determine if the geometric return of the fund will be less than or greater than the arithmetic return and calculate the fund’s geometric return:

Geometric Return

Geometric compared to Arithmetic

 

A) 4.96%   greater than

B) 4.96%   less than

C) 12.86%   greater than

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The geometric return is calculated as follows:

[(1 + 0.25)(1 + 0.15)(1 + 0.12)(1 - 0.08)(1 – 0.14)]^1/5 – 1,

or [1.25 × 1.15 × 1.12 × 0.92 × 0.86]^0.2 – 1 = 0.4960, or 4.96%.

The geometric return will always be less than or equal to the arithmetic return. In this case the arithmetic return was 6%.

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An investor has a $12,000 portfolio consisting of $7,000 in stock A with an expected return of 20% and $5,000 in stock B with an expected return of 10%. What is the investor’s expected return on the portfolio?

A) 12.2%.

B) 15.0%.

C) 15.8%.

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Find the weighted mean where the weights equal the proportion of $12,000. (7,000 / 12,000)(0.20) + (5,000 / 12,000)(0.10) = 15.8%.

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Michael Philizaire is studying for the Level I CFA examination. During his review of measures of central tendency, he decides to calculate the geometric average of the appreciation/deprecation of his home over the last five years. Using comparable sales and market data he obtains from a local real estate appraiser, Philizaire calculates the year-to-year percentage change in the value of his home as follows: 20, 15, 0, -5, -5. The geometric return is closest to:

A) 11.60%.

B) 4.49%.

C) 0.00%.

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The geometric return is calculated as follows:

[(1 + 0.20) × (1 + 0.15) × (1 + 0.0) (1 ? 0.05) (1 ? 0.05)]^1/5 – 1,

or [1.20 × 1.15 × 1.0 × 0.95 × 0.95]^0.2 – 1 = 0.449, or 4.49%.

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The owner of a company has recently decided to raise the salary of one employee, who was already making the highest salary in the company, by 40%. Which of the following value(s) is (are) expected to be affected by this raise?

A) mean and median only.

B) median only.

C) mean only.

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Mean is affected because it is the sum of all values / number of observations.  Median is not affected as it the midpoint between the top half of values and the bottom half of values.

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An investor has a portfolio with 10% cash, 30% bonds, and 60% stock. Last year, the cash returns was 2.0%, the bonds’ return was 9.5%, and the stocks’ return was –32.5%. What was the return on the investor’s portfolio?

A) –33.33%.

B) –7.00%.

C) –16.45%.

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Find the weighted mean. (0.10)(0.02) + (0.30)(0.095) + (0.60)(–0.325) = –16.45%.

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Which measure of central tendency can be used for both numerical and categorical variables?

A) Mean.

B) Median.

C) Mode.

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The mode is the only choice that makes sense since you cannot take an average or median of categorical data such as bond ratings (AAA, AA, A, etc.) but the mode is simply the most frequently occurring number or category.

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For the last four years, the returns for XYZ Corporation’s stock have been 10.4%, 8.1%, 3.2%, and 15.0%. The equivalent compound annual rate is:

A) 9.1%.

B) 9.2%.

C) 8.9%.

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(1.104 × 1.081 × 1.032 × 1.15)^0.25 ? 1 = 9.1%

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What is the compound annual growth rate for stock A which has annual returns of 5.60%, 22.67%, and -5.23%?

A) 7.08%.

B) 6.00%.

C) 8.72%.

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Compound annual growth rate is the geometric mean. (1.056 × 1.2267 × 0.9477)^1/3 – 1 = 7.08%