CFA Level 1 - Quantitative Methods Session 2 - Reading 7

CFA Level 1 – Quantitative Methods, Session 2 – Reading 7

(Notes, Practice Questions, Sample Questions)

1. Fifty mutual funds are ranked according to performance. The five best performing funds are assigned the number 1, while the five worst performing funds are assigned the number 10. This is an example of a(n):

A)ordinal scale.

B)interval scale.

C)nominal scale.

[Explanation]: A — The ordinal scale of measurement categorizes and orders data with respect to some characteristic. In this example, the ordinal scale tells us that a fund ranked “1” performed better than a fund ranked “10,” but it does not tell us anything about the difference in performance

2. Which measure of scale has a true zero point as the origin?

A)Nominal scale.

B)Ordinal scale.

C)Ratio scale

[Explanation]: C — Ratio scales are the strongest level of measurement; they quantify differences in the size of data and have a true zero point as the origin

3. Which of the following statements regarding the terms population and sample is least accurate?

A)A sample includes all members of a specified group.

B)A sample’s characteristics are attributed to the population as a whole.

C)A descriptive measure of a sample is called a statistic

[Explanation]: A — A population includes all members of a specified group. A sample is a portion, or subset of the population of interest

4. Which of the following statements about statistical concepts is least accurate?

A)A frequency distribution is a tabular display of data summarized into a relatively small number of intervals.

B)A sample contains all members of a specified group, but a population contains only a subset.

C)A parameter is any descriptive measure of a population characteristic

[Explanation]: B — A population is defined as all members of a specified group, but a sample is a subset of a population

5. A summary measure that is computed to describe a population characteristic from a sample is called a:

A)census.

B)parameter.

C)statistic

[Explanation]: C — When sampling from a portion of the population, you compute a statistic to make inferences about the population

6. Which one of the following alternatives best describes the primary use of inferential statistics? Inferential statistics are used to:

A)summarize the important characteristics of a large data set based on statistical characteristics of a smaller sample.

B)make forecasts, estimates or judgments about a large set of data based on statistical characteristics of a smaller sample.

C)make forecasts based on large data sets

[Explanation]: B — Inferential statistics are used mainly to make forecasts, estimates or judgements about a large set of data based on statistical characteristics of a smaller set of data

7. Which one of the following alternatives best describes the primary use of descriptive statistics? Descriptive statistics are used to:

A)obtain data about the characteristics of any data set that can be used to assess the likelihood of the occurrence of future events.

B)arrive at estimates regarding a large set of data regarding the statistical characteristics of a smaller sample.

C)summarize important characteristics of large data sets

[Explanation]: C — Descriptive statistics are used mainly to summarize important characteristics of large data sets

8. What is the main difference between descriptive statistics and inferential statistics? Descriptive statistics are:

A)used to summarize data while inferential statistics are used to obtain precise information about a large data set.

B)used to make forecasts about the likelihood of upcoming events while inferential statistics are used to summarize any data set.

C)used to summarize a large data set while inferential statistics involves procedures used to make forecasts or judgments about a large data set by examining a smaller sample

[Explanation]: C — Descriptive statistics are used to summarize a large data set while inferential statistics are based on procedures used to make forecasts or judgments about a large data set by examining a smaller set of data

9. Which of the following statements regarding various statistical measures is least accurate?

A)The correlation coefficient is calculated by dividing the covariance of two random variables by the product of their standard deviations.

B)Variance equals the sum of the squared deviations from the mean times the probability that that each outcome will occur.

C)The coefficient of variation is calculated by dividing the mean by the standard deviation

[Explanation]: C — The coefficient of variation equals the standard deviation divided by the mean

10. Which of the following is an example of a parameter?

A)Population variance.

B)Sample standard deviation.

C)Sample mean

[Explanation]: A — A parameter is any descriptive measure of a population characteristic. The population variance describes a population while the sample standard deviation and sample mean are each descriptive measures of samples

11. A summary measure of a characteristic of an entire population is called a:

A)statistic.

B)parameter.

C)census

[Explanation]: B — A parameter measures a characteristic of the underlying population

12. Which of the following statements regarding frequency distributions is least accurate? Frequency distributions:

A)summarize data into a relatively small number of intervals.

B)organize data into overlapping groups.

C)work with all types of measurement scales

[Explanation]: B — Data in a frequency distribution must belong to only one group or interval. Intervals are mutually exclusive and non-overlapping

13. Which of the following best describes a frequency distribution? A frequency distribution is a grouping of:

A)selected data into intervals (classes) so that the number of observations in each of the non-overlapping intervals (classes) can be seen and tallied.

B)data into intervals (classes) so that the number of observations in each of the non-overlapping intervals (classes) can be seen and tallied.

C)independent intervals (classes) so that they can be seen and tallied.

[Explanation]: B — A frequency distribution is a tabular presentation of statistical data that aids the analysis of large data sets

14. How is the relative frequency of an interval computed?

A)Dividing the frequency of that interval by the sum of all frequencies.

B)Dividing the sum of the two interval limits by 2.

C)Subtracting the lower limit of the interval by the upper limit

[Explanation]: A — The relative frequency is the percentage of total observations falling within each interval. It is found by taking the frequency of the interval and dividing that number by the sum of all frequencies

15. In a frequency distribution histogram, the frequency of an interval is given by the:

A)height multiplied by the width of the corresponding bar.

B)width of the corresponding bar.

C)height of the corresponding bar

[Explanation]: A — In a histogram, intervals are placed on the horizontal axis, and frequencies are placed on the vertical axis. The frequency of a particular interval is given by the value on the vertical axis, or the height of the corresponding bar

16. Which of the following statements about histograms and frequency polygons is least accurate?

A)A frequency polygon is constructed by plotting the midpoint of each interval on the horizontal axis.

B)A histogram connects points with a straight line.

C)A histogram and a frequency polygon both plot the absolute frequency on the vertical axis.

[Explanation]: B — In constructing a frequency polygon, the midpoint of each interval is plotted on the horizontal axis and the frequency of each interval is plotted on the vertical axis. Points are then connected with straight lines. A histogram is a bar chart of data that has been grouped into a frequency distribution – because it is a bar chart, there are no individual points to connect

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

A)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.

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

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

[Explanation]: A — 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

18. 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)0.18%; 18.6%.

C)17.76%; 18.6%

[Explanation]: C — 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%

19. 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)12.86% greater than

C)4.96% less than

[Explanation]: C — 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%

20. 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%

[Explanation]: C — 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%

21. 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%

[Explanation]: B — 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%

22. 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

[Explanation]: C — 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

23. 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)–16.45%.

B)–33.33%.

C)–7.00%

[Explanation]: A — Find the weighted mean. (0.10)(0.02) + (0.30)(0.095) + (0.60)(–0.325) = –16.45%

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

A)Mean.

B)Median.

C)Mode.

[Explanation]: C — 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

25. 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)9.2%.

B)9.1%.

C)8.9%

[Explanation]: B — (1.104 × 1.081 × 1.032 × 1.15)0.25 − 1 = 9.1%

26. 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%

[Explanation]: A — Compound annual growth rate is the geometric mean. (1.056 × 1.2267 × 0.9477)1/3 – 1 = 7.08%

27. Find the mean, median, and mode, respectively, of the following data:

3, 3, 5, 8, 9, 13, 17

A)8; 8.28; 3.

B)8.28; 8; 3.

C)3; 8.28; 8

[Explanation]: B — Mean = (3 + 3 + 5 + 8 + 9 + 13 + 17) / 7 = 8.28; Median = middle of distribution = 8 (middle number); Mode = most frequent = 3

28. An investor has a $15,000 portfolio consisting of $10,000 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)7.9%.

C)16.7%

[Explanation]: C — Find the weighted mean where the weights equal the proportion of $15,000. [(10,000 / 15,000) × 0.20] + [(5,000 / 15,000 × 0.10] = 16.7%

29. An investor has a portfolio with 10% cash, 30% bonds, and 60% stock. If last year’s return on cash was 2.0%, the return on bonds was 9.5%, and the return on stock was 25%, what was the return on the investor’s portfolio?

A)36.50%.

B)18.05%.

C)22.30%

[Explanation]: B — Find the weighted mean of the returns. (0.10 × 0.02) + (0.30 × 0.095) + (0.60 × 0.25) = 18.05%

30. Which of the following statements about a normal distribution is least accurate?

A)Approximately 68% of the observations lie within +/- 1 standard deviation of the mean.

B)A normal distribution has excess kurtosis of three.

C)The mean and variance completely define a normal distribution

[Explanation]: B — Even though normal curves have different sizes, they all have identical shape characteristics. The kurtosis for all normal distributions is three; an excess kurtosis of three would indicate a leptokurtic distribution. Both remaining choices are true

31. An investor has the following assets:

$5,000 in bonds with an expected return of 8%.

$10,000 in equities with an expected return of 12%.

$5,000 in real estate with an expected return of 10%.

What is the portfolio’s expected return?

A)10.00%.

B)11.00%.

C)10.50%

[Explanation]: C — Expected return is the weighted average of the individual expected values. The expected return is: [(5,000) × (10.00) + (5,000) × (8.00) + (10,000) × (12.00)] / 20,000 = 10.50%

32. A portfolio is equally invested in Stock A, with an expected return of 6%, and Stock B, with an expected return of 10%, and a risk-free asset with a return of 5%. The expected return on the portfolio is:

A)7.0%.

B)8.0%.

C)7.4%

[Explanation]: A — (0.333)(0.06) + (0.333)(0.10) + 0.333(0.05) = 0.07

33. Which of the following statements about the arithmetic mean is least accurate?

A)The arithmetic mean of a frequency distribution is equal to the sum of the class frequency times the midpoint of the frequency class all divided by the number of observations.

B)If the distribution is skewed to the left then the mean will be greater than the median.

C)The arithmetic mean is the only measure of central tendency where the sum of the deviations of each observation from the mean is always zero

[Explanation]: B — If the distribution is skewed to the left, then the mean will be less than the median

34. Which of the following statements about the median is least accurate? It is:

A)equal to the 50th percentile.

B)more affected by extreme values than the mean.

C)equal to the mode in a normal distribution

[Explanation]: B — Median is less influenced by outliers since the median is computed as the “middle” observation. On the other hand, all of the data including outliers are used in computing the mean. Both remaining statements are true regarding the median.

35. What are the median and the third quintile of the following data points, respectively?

9.2%, 10.1%, 11.5%, 11.9%, 12.2%, 12.8%, 13.1%, 13.6%, 13.9%, 14.2%, 14.8%, 14.9%, 15.4%

A)13.1%; 13.6%.

B)13.1%; 13.7%.

C)12.8%; 13.6%

[Explanation]: B — The median is the midpoint of the data points. In this case there are 13 data points and the midpoint is the 7th term.

The formula for determining quantiles is: Ly = (n + 1)(y) / (100). Here, we are looking for the third quintile (60% of the observations lie below) and the formula is: (14)(60) / (100) = 8.4. The third quintile falls between 13.6% and 13.9%, the 8th and 9th numbers from the left. Since L is not a whole number, we interpolate as: 0.136 + (0.40)(0.139 − 0.136) = 0.1372, or 13.7%

36. In a positively skewed distribution, the:

A)median equals the mean.

B)mean is greater than the median.

C)mean is less than the median

[Explanation]: B — In a right-skewed distribution, there are large positive outliers. These outliers increase the mean of the distribution but have little effect on the median. Therefore, the mean is greater than the median

37. If a distribution is positively skewed, then generally:

A)mean < median < mode.

B)mean > median < mode.

C)mean > median > mode

[Explanation]: C — When a distribution is positively skewed the right side tail is longer than normal due to outliers. The mean will exceed the median, and the median will generally exceed the mode because large outliers falling to the far right side of the distribution can dramatically influence the mean

38. Twenty Level I CFA candidates in a study group took a practice exam and want to determine the distribution of their scores. When they grade their exams they discover that one of them skipped an ethics question and subsequently filled in the rest of his answers in the wrong places, leaving him with a much lower score than the rest of the group. If they include this candidate’s score, their distribution will most likely:

A)have a mode that is less than its median.

B)have a mean that is less than its median.

C)be positively skewed

[Explanation]: B — With the low outlier included, the distribution will be negatively skewed. For a negatively skewed distribution, the mean is less than the median, which is less than the mode

39. In a negatively skewed distribution, what is the order (from lowest value to highest) for the distribution’s mode, mean, and median values?

A)Median, mode, mean.

B)Mode, mean, median.

C)Mean, median, mode

[Explanation]: C — In a negatively skewed distribution, the mean is less than the median, which is less than the mode

40. In a positively skewed distribution, what is the order (from lowest value to highest) for the distribution’s mode, mean, and median values?

A)Mean, median, mode.

B)Mode, mean, median.

C)Mode, median, mean

[Explanation]: C — In a positively skewed distribution, the mode is less than the median, which is less than the mean