CFA Program — Quantitative Methods

Question 1

A hypothesis test is conducted with a null hypothesis that a population mean equals 100. The test yields a p-value of 0.03. At a 5% significance level, the appropriate conclusion is:

• A) Fail to reject the null hypothesis, since 0.03 < 0.05
• B) Reject the null hypothesis, since the p-value is less than the significance level
• C) The test is inconclusive without the sample size
• D) Accept the alternative hypothesis with certainty

Question 2

An investment has a 30% chance of returning 20% and a 70% chance of returning 5%. What is the expected return?

• A) 8.5%
• B) 9.5%
• C) 10.5%
• D) 12.5%

Question 3

An investment offers to pay $10,000 in 5 years. If the discount rate is 6% annually, what is the approximate present value?

• A) $6,650
• B) $7,130
• C) $7,470
• D) $8,000

Question 4

An investor deposits $2,000 at the end of each year for 4 years into an account earning 5% annually. What is the approximate future value at the end of year 4?

• A) $8,000
• B) $8,320
• C) $8,620
• D) $9,000

Question 5

A data set has a mean of 50 and a standard deviation of 8. Using Chebyshev's inequality, at least what proportion of observations must lie within 2 standard deviations of the mean?

• A) 50%
• B) 68%
• C) 75%
• D) 95%

Question 6

Which measure of central tendency is most appropriate for describing the typical compound rate of growth of an investment over multiple periods?

• A) Arithmetic mean
• B) Geometric mean
• C) Median
• D) Mode

Question 7

A sample of five values is: 12, 18, 15, 21, 9. The arithmetic mean is closest to:

• A) 18.0
• B) 15.0
• C) 16.5
• D) 14.0

Question 8

A researcher rejects a null hypothesis when it is actually true. This is an example of:

• A) A confidence interval error
• B) A sampling frame error
• C) A Type I error
• D) A Type II error

Question 9

A random variable is normally distributed with a mean of 100 and a standard deviation of 15. Approximately what percentage of observations fall between 85 and 115?

• A) 95%
• B) 50%
• C) 99.7%
• D) 68%

Question 10

An investor needs $50,000 in 8 years and can earn 6% annually. The amount that must be invested today is closest to:

• A) $31,371
• B) $35,000
• C) $34,000
• D) $28,000

Question 11

An investor deposits $15,000 today into an account earning 7% annually, compounded annually. The value of the account after 5 years is closest to:

• A) $19,800
• B) $21,038
• C) $20,250
• D) $22,500

Question 12

Two events, A and B, are described as mutually exclusive. What does this imply about P(A and B)?

• A) P(A and B) = 0, since the events cannot occur simultaneously.
• B) P(A and B) = P(A) x P(B), always.
• C) P(A and B) = 1, since one of them must occur.
• D) P(A and B) cannot be determined from this information alone.

Question 13

A sample of six values is: 22, 28, 25, 31, 19, 24. The arithmetic mean is closest to:

• A) 23.0
• B) 27.5
• C) 24.8
• D) 26.0

Question 14

The risk-free rate is 2.5%, the expected market return is 8%, and a stock has a beta of 1.4. Using CAPM, the stock's expected return is closest to:

• A) 10.20%
• B) 8.00%
• C) 11.20%
• D) 7.70%

Question 15

In hypothesis testing, the "p-value" represents:

• A) A fixed value that is always equal to the significance level.
• B) The probability of obtaining a test result at least as extreme as the one observed, assuming the null hypothesis is true.
• C) The probability that the null hypothesis is true.
• D) The probability that the alternative hypothesis is false.

Question 16

An analyst wants to summarize the "typical" value in a data set that contains several extreme outliers. Which measure of central tendency is generally most appropriate in this situation?

• A) The median, since it is less sensitive to extreme values than the mean.
• B) The arithmetic mean, since it uses every data point.
• C) The geometric mean, which cannot be used with negative values.
• D) The range, which measures spread rather than central tendency.

Question 17

An investor deposits $6,000 today into an account earning 5% annually. The value after 10 years is closest to:

• A) $10,500
• B) $9,773
• C) $9,000
• D) $8,000

Question 18

Which of the following best describes "covariance" between two variables?

• A) A measure that is always between -1 and +1.
• B) The average value of a single variable.
• C) A measure exclusively used for categorical, non-numeric data.
• D) A measure of how two variables move together, with the sign indicating the direction of the relationship.

Question 19

A hypothesis test yields a p-value of 0.03. At a significance level of 5%, the analyst should:

• A) Fail to reject the null hypothesis
• B) Reject the null hypothesis
• C) Increase the sample size before concluding
• D) Accept the alternative hypothesis with certainty

Question 20

An investor deposits $10,000 today and $5,000 one year from now into an account earning 6% annually. The total value at the end of two years is closest to:

• A) $16,536
• B) $15,900
• C) $17,236
• D) $16,180

Question 21

A bond pays $80 annually and matures in 5 years with a face value of $1,000. If the required rate of return is 7%, its price is closest to:

• A) $1,041
• B) $959
• C) $1,000
• D) $918

Question 22

The effective annual rate (EAR) corresponding to a stated annual rate of 12% compounded monthly is closest to:

• A) 12.00%
• B) 12.36%
• C) 12.68%
• D) 12.55%

Question 23

A sample of 25 observations has a mean of 50 and a standard deviation of 10. The 95% confidence interval for the population mean is closest to:

• A) 46.08 to 53.92
• B) 45.00 to 55.00
• C) 30.00 to 70.00
• D) 47.20 to 52.80

Question 24

An investment has the following annual returns: +20%, −10%, +15%, −5%. The geometric mean return is closest to:

• A) 4.22%
• B) 5.00%
• C) 4.57%
• D) 3.97%

Question 25

An investor wants to know the minimum number of standard deviations from the mean, applicable to any distribution regardless of shape, within which at least 75% of observations must lie. According to Chebyshev's inequality, this is closest to:

• A) 2 standard deviations
• B) 1 standard deviation
• C) 1.5 standard deviations
• D) 3 standard deviations

Question 26

A distribution with positive skew (right-skewed) typically has which of the following relationships among its mean, median, and mode?

• A) Median > Mean > Mode
• B) Mean > Median > Mode
• C) Mode > Median > Mean
• D) Mean = Median = Mode

Question 27

A sample of five annual returns (%) is: 8, 12, 10, 14, 6. The sample standard deviation is closest to:

• A) 4.00%
• B) 2.53%
• C) 3.16%
• D) 2.83%

Question 28

An investment returns 20% in Year 1, -10% in Year 2, and 15% in Year 3. The geometric mean annual return is closest to:

• A) 8.33%
• B) 10.50%
• C) 7.20%
• D) 7.49%

Question 29

A stated annual interest rate of 8%, compounded quarterly, is closest to which effective annual rate?

• A) 8.24%
• B) 8.00%
• C) 8.16%
• D) 8.50%

Question 30

An investor deposits $8,000 today and $4,000 one year from now into an account earning 5% annually. The total value at the end of three years is closest to:

• A) $12,271
• B) $13,671
• C) $12,871
• D) $14,271

Question 31

An investor deposits $22,000 today into an account earning 6.5% annually, compounded annually. The value of the account after 6 years is closest to:

• A) $29,500
• B) $34,750
• C) $27,830
• D) $32,101

Question 32

A project requires an initial investment of $2,500 and returns $1,000 at the end of each of the next 3 years. The project's internal rate of return (IRR) is closest to:

• A) 9.7%
• B) 14.0%
• C) 6.5%
• D) 18.0%

Question 33

A project requires an initial investment of $5,000 and is expected to generate $1,800 at the end of each of the next 4 years. Using a discount rate of 7%, the project's NPV is closest to:

• A) $2,200
• B) $450
• C) $700
• D) $1,097

Question 34

A stated annual rate of 12%, compounded monthly, corresponds to an effective annual rate closest to:

• A) 12.00%
• B) 12.36%
• C) 12.68%
• D) 13.00%

Question 35

An investor deposits $4,000 at the end of each year for 8 years into an account earning 6% annually. The value of the account immediately after the final deposit is closest to:

• A) $32,000
• B) $39,590
• C) $42,800
• D) $36,200

Question 36

An investor needs $60,000 in 7 years and can earn 5.5% annually on invested funds. The amount that must be invested today is closest to:

• A) $41,246
• B) $45,000
• C) $38,500
• D) $50,000