CFA Level 1 - Quantitative Methods Session 2 - Reading 9

CFA Level 1 - Quantitative Methods Session 2 - Reading 9 (Notes, Practice Questions, Sample Questions) 1. Which of the following statements about the normal probability distribution is most accurate? A)Five percent of the normal curve probability is more than two standard deviations from the mean. B)Sixty-eight percent of the area under the normal curve falls between the mean and 1 standard deviation above the mean. C)The normal curve is asymmetrical about its mean Explanation: A — The normal curve is symmetrical about its mean with 34% of the area under the normal curve falling between the mean and one standard deviation above the mean. Ninety-five percent of the normal curve is within two standard deviations of the mean, so five percent of the normal curve falls outside two standard deviations from the mean 2. Which of the following is least likely a probability distribution? A)Roll an irregular die: p(1) = p(2) = p(3) = p(4) = 0.2 and p(5) = p(6) = 0.1. B)Flip a coin: P(H) = P(T) = 0.5. C)Zeta Corp.: P(dividend increases) = 0.60, P(dividend decreases) = 0.30 Explanation: C — All the probabilities must be listed. In the case of Zeta Corp. the probabilities do not sum to one 3. Which of the following statements about probability distributions is least accurate? A)A discrete random variable is a variable that can assume only certain clearly separated values resulting from a count of some set of items. B)The skewness of a normal distribution is zero. C)A binomial probability distribution is an example of a continuous probability distribution Explanation: C — The binomial probability distribution is an example of a discrete probability distribution. There are only two possible outcomes of each trial and the outcomes are mutually exclusive. For example, in a coin toss the outcome is either heads or tails. The other responses are both correct definitions 4. Which of the following statements about probability distributions is least accurate? A)In a binomial distribution each observation has only two possible outcomes that are mutually exclusive. B)A probability distribution includes a listing of all the possible outcomes of an experiment. C)A probability distribution is, by definition, normally distributed Explanation: C — Probabilities must be zero or positive, but a probability distribution is not necessarily normally distributed. Binomial distributions are either successes or failures 5. A probability distribution is least likely to: A)contain all the possible outcomes. B)give the probability that the distribution is realistic. C)have only non-negative probabilities Explanation: B — The probability distribution may or may not reflect reality. But the probability distribution must list all possible outcomes, and probabilities can only have non-negative values 6. A dealer in a casino has rolled a five on a single die three times in a row. What is the probability of her rolling another five on the next roll, assuming it is a fair die? A)0.001. B)0.167. C)0.200. Explanation: B — The probability of a value being rolled is 1/6 regardless of the previous value rolled 7. The number of ships in the harbor is an example of what kind of variable? A)Indiscrete. B)Continuous. C)Discrete Explanation: C — A discrete variable is one that is represented by finite units 8. Which of the following is a discrete random variable? A)The number of advancing stocks in the DJIA in a day. B)The realized return on a corporate bond. C)The amount of time between two successive stock trades Explanation: A — Since the DJIA consists of only 30 stocks, the answer associated with it would be a discrete random variable. Random variables measuring time, rates of return and weight will be continuous 9. Which of the following statements about probability distributions is most accurate? A)A discrete uniform random variable has varying probabilities for each outcome that total to one. B)A binomial distribution counts the number of successes that occur in a fixed number of independent trials that have mutually exclusive (i.e. yes or no) outcomes. C)A continuous uniform distribution has a lower limit but no upper limit Explanation: B — Binomial probability distributions give the result of a single outcome and are used to study discrete random variables where you want to know the probability that an exact event will happen. A continuous uniform distribution has both an upper and a lower limit. A discrete uniform random variable has equal probabilities for each outcome 10. Assume a discrete distribution for the number of possible sunny days in Provo, Utah during the week of April 20 through April 26. For this discrete distribution, p(x) = 0 when x cannot occur, or p(x) > 0 if it can. Based on this information, what is the probability of it being sunny on 5 days and on 10 days during the week, respectively? A)A positive value; infinite. B)A positive value; zero. C)Zero; infinite Explanation: B — The probability of it being sunny on 5 days during the week has some positive value, but the probability of having sunshine 10 days within a week of 7 days is zero because this cannot occur 11. Which of the following is least likely to be an example of a discrete random variable? A)The rate of return on a real estate investment. B)The number of days of sunshine in the month of May 2006 in a particular city. C)Quoted stock prices on the NASDAQ Explanation: A — The rate of return on a real estate investment, or any other investment, is an example of a continuous random variable because the possible outcomes of rates of return are infinite (e.g., 10.0%, 10.01%, 10.001%, etc.). Both of the other choices are measurable (countable) 12. A random variable that has a countable number of possible values is called a: A)probability distribution. B)continuous random variable. C)discrete random variable. Explanation: C — A discrete random variable is one for which the number of possible outcomes are countable, and for each possible outcome, there is a measurable and positive probability. A continuous random variable is one for which the number of outcomes is not countable 13. A probability function: A)is often referred to as the "cdf." B)specifies the probability that the random variable takes on a specific value. C)only applies to continuous distributions Explanation: B — This is true by definition 14. Which of the following could least likely be a probability function? A)X1,2,3,4) p(x) = 0.2. B)X1,2,3,4) p(x) = (x × x) / 30. C)X1,2,3,4) p(x) = x / 10 Explanation: A — In a probability function, the sum of the probabilities for all of the outcomes must equal one. Only one of the probability functions in these answers fails to sum to one 15. If a smooth curve is to represent a probability density function, what two requirements must be satisfied? The area under the curve must be: A)one and the curve must not fall below the horizontal axis. B)one and the curve must not rise above the horizontal axis. C)zero and the curve must not fall below the horizontal axis Explanation: A — If a smooth curve is to represent a probability density function, the total area under the curve must be one (probability of all outcomes equals 1) and the curve must not fall below the horizontal axis (no outcome can have a negative chance of occurring) 16. In a continuous probability density function, the probability that any single value of a random variable occurs is equal to what? A)One. B)Zero. C)1/N Explanation: B — Since there are infinite potential outcomes in a continuous pdf, the probability of any single value of a random variable occurring is 1/infinity = 0 17. A cumulative distribution function for a random variable X is given as follows: x F(x) 5 0.14 10 0.25 15 0.86 20 1.00 The probability of an outcome less than or equal to 10 is: A)14%. B)25%. C)39%. Explanation: B — A cumulative distribution function (cdf) gives the probability of an outcome for a random variable less than or equal to a specific value. For the random variable X, the cdf for the outcome 10 is 0.25, which means there is a 25% probability that X will take a value less than or equal to 10 18. Which of the following qualifies as a cumulative distribution function? A)F(1) = 0, F(2) = 0.25, F(3) = 0.50, F(4) = 1. B)F(1) = 0, F(2) = 0.5, F(3) = 0.5, F(4) = 0. C)F(1) = 0.5, F(2) = 0.25, F(3) = 0.25 Explanation: A — Because a cumulative probability function defines the probability that a random variable takes a value equal to or less than a given number, for successively larger numbers, the cumulative probability values must stay the same or increase 19. A random variable X is continuous and bounded between zero and five, X0 ≤ X ≤ 5). The cumulative distribution function (cdf) for X is F(x) = x / 5. Calculate P(2 ≤ X ≤ 4). A)1.00. B)0.40. C)0.50. Explanation: B — For a continuous distribution, P(a ≤ X ≤b) = F(b) − F(a). Here, F(4) = 0.8 and F(2) = 0.4. Note also that this is a uniform distribution over 0 ≤ x ≤ 5 so Prob(2 < x < 4) = (4 − 2) / 5 = 40% 20. Which of the following random variables would be most likely to follow a discrete uniform distribution? A)The outcome of a roll of a standard, six-sided die where X equals the number facing up on the die. B)The number of heads on the flip of two coins. C)The outcome of the roll of two standard, six-sided dice where X is the sum of the numbers facing up Explanation: A — The discrete uniform distribution is characterized by an equal probability for each outcome. A single die roll is an often-used example of a uniform distribution. In combining two random variables, such as coin flip or die roll outcomes, the sum will not be uniformly distributed 21. The number of days a particular stock increases in a given five-day period is uniformly distributed between zero and five inclusive. In a given five-day trading week, what is the probability that the stock will increase exactly three days? A)0.333. B)0.167. C)0.600 Explanation: B — If the possible outcomes are X0,1,2,3,4,5), then the probability of each of the six outcomes is 1 / 6 = 0.167 22. The Night Raiders, an expansion team in the National Indoor Football League, is having a challenging first season with a current win loss record of 0 and 4. However, the team recently signed four new defensive players and one of the team sponsors (who also happens to hold a CFA charter) calculates the probability of the team winning a game at 0.40. Assuming that whether the team wins a game is independent of whether it wins any other game, the probability that the team will win 6 out of the next 10 games is closest to: A)0.417. B)0.112. C)0.350. Explanation: B — Use the formula for a binomial random variable to calculate the answer to this question. We will define "success" as the team winning a game. The formula is: p(x) = P(X = x) = [number of ways to choose x from n] × px × (1 p)n-x, where [number of ways to choose x from n] = n! / [(n - x)! × x!]. Here, p(x) = P(X = 6) = [10! / (10 − 6)! × 6!] × 0.406 × (1 − 0.40)10-6 = 210.0 × 0.00410 × 0.12960 = 0.11159, or approximately 0.112. To calculate factorial using your financial calculator: On the TI, factorial is [2nd] ¡→ [x!]. On the HP, factorial is [g] → [n!]. To compute 10! on the TI, enter [10] → [2nd] → [x!] = 3,628,800. On the HP, use [10] → [ENTER] → [g] → [n!] 23. Assume 30% of the CFA candidates have a degree in economics. A random sample of three CFA candidates is selected. What is the probability that none of them has a degree in economics? A)0.343. B)0.027. C)0.900 Explanation: A — 3! / (0!3!) (0.3)0 (0.7)3 = 0.343 24. A casual laborer has a 70% chance of finding work on each day that she reports to the day labor marketplace. What is the probability that she will work three days out of five? A)0.6045. B)0.3192. C)0.3087 Explanation: C — P(3) = 5! / [(5 – 3)! × 3!] × (0.73) × (0.32) = 0.3087 = 5 →2nd→ nCr → 3 × 0.343 × 0.09 25. Which of the following could be the set of all possible outcomes for a random variable that follows a binomial distribution? A)(-1, 0, 1). B)(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11). C)(1, 2). Explanation: B — This reflects a basic property of binomial outcomes. They take on whole number values that must start at zero up to the upper limit n. The upper limit in this case is 11 26. Which of the following is NOT an assumption of the binomial distribution? A)The trials are independent. B)The expected value is a whole number. C)Random variable X is discrete. Explanation: B — The expected value is n × p. A simple example shows us that the expected value does not have to be a whole number: n = 5, p = 0.5, n × p = 2.5. The other conditions are necessary for the binomial distribution 27. For a certain class of junk bonds, the probability of default in a given year is 0.2. Whether one bond defaults is independent of whether another bond defaults. For a portfolio of five of these junk bonds, what is the probability that zero or one bond of the five defaults in the year ahead? A)0.0819. B)0.4096. C)0.7373. Explanation: C — The outcome follows a binomial distribution where n = 5 and p = 0.2. In this case p(0) = 0.85 = 0.3277 and p(1) = 5 × 0.84 × 0.2 = 0.4096, so P(X=0 or X=1) = 0.3277 + 0.4096 28. A stock priced at $10 has a 60% probability of moving up and a 40% probability of moving down. If it moves up, it increases by a factor of 1.06. If it moves down, it decreases by a factor of 1/1.06. What is the expected stock price after two successive periods? A)$10.27. B)$10.03. C)$11.24 Explanation: A — If the stock moves up twice, it will be worth $10 × 1.06 × 1.06 = $11.24. The probability of this occurring is 0.60 × 0.60 = 0.36. If the stock moves down twice, it will be worth $10 × (1/1.06) × (1/1.06) = $8.90. The probability of this occurring is 0.40 × 0.40 = 0.16. If the stock moves up once and down once, it will be worth $10 × 1.06 × (1/1.06) = $10.00. This can occur if either the stock goes up then down or down then up. The probability of this occurring is 0.60 × 0.40 + 0.40 × 0.60 = 0.48. Multiplying the potential stock prices by the probability of them occurring provides the expected stock price: ($11.24 × 0.36) + ($8.90 × 0.16) + ($10.00 × 0.48) = $10.27 29. A stock priced at $20 has an 80% probability of moving up and a 20% probability of moving down. If it moves up, it increases by a factor of 1.05. If it moves down, it decreases by a factor of 1/1.05. What is the expected stock price after two successive periods? A)$21.24. B)$20.05. C)$22.05 Explanation: A — If the stock moves up twice, it will be worth $20 × 1.05 × 1.05 = $22.05. The probability of this occurring is 0.80 × 0.80 = 0.64. If the stock moves down twice, it will be worth $20 × (1/1.05) × (1/1.05) = $18.14. The probability of this occurring is 0.20 × 0.20 = 0.04. If the stock moves up once and down once, it will be worth $20 × 1.05 × (1/1.05) = $20.00. This can occur if either the stock goes up then down or down then up. The probability of this occurring is 0.80 × 0.20 + 0.20 × 0.80 = 0.32. Multiplying the potential stock prices by the probability of them occurring provides the expected stock price: ($22.05 × 0.64) + ($18.14 × 0.04) + ($20.00 × 0.32) = $21.24 30. A stock priced at $100 has a 70% probability of moving up and a 30% probability of moving down. If it moves up, it increases by a factor of 1.02. If it moves down, it decreases by a factor of 1/1.02. What is the probability that the stock will be $100 after two successive periods? A)21%. B)9%. C)42% Explanation: C — For the stock to be $100 after two periods, it must move up once and move down once: $100 × 1.02 × (1/1.02) = $100. This can happen in one of two ways: 1) the stock moves up during period one and down during period two; or 2) the stock moves down during period one and up during period two. The probability of either event is 0.70 × 0.30 = 0.21. The combined probability of either event is 2(0.21) = 0.42 or 42% 31. Tracking error for a portfolio is best described as the: A)sample mean minus population mean. B)portfolio return minus a benchmark return. C)standard deviation of differences between an index return and portfolio return Explanation: B — Tracking error is the difference between the total return on a portfolio and the total return on the benchmark used to measure the portfolio’s performance. The difference between a sample statistic and a population parameter is sampling error. The standard deviation of the difference between a portfolio return and an index (or any chosen benchmark return) is more often referred to as tracking risk 32. A portfolio begins the year with a value of $100,000 and ends the year with a value of $95,000. The manager’s performance is measured against an index that declined by 7% on a total return basis during the year. The tracking error of this portfolio is closest to: A)−2%. B)−5%. C)2%. Explanation: C — Tracking error is the portfolio total return minus the benchmark total return. The portfolio return is ($95,000 − $100,000) / $100,000 = −5%. Tracking error = −5% − (−7%) = +2% 33. A discount brokerage firm states that the time between a customer order for a trade and the execution of the order is uniformly distributed between three minutes and fifteen minutes. If a customer orders a trade at 11:54 A.M., what is the probability that the order is executed after noon? A)0.750. B)0.500. C)0.250 Explanation: A — The limits of the uniform distribution are three and 15. Since the problem concerns time, it is continuous. Noon is six minutes after 11:54 A.M. The probability the order is executed after noon is (15 − 6) / (15 − 3) = 0.75. 34. The probability density function of a continuous uniform distribution is best described by a: A)line segment with a curvilinear slope. B)line segment with a 45-degree slope. C)horizontal line segment Explanation: C — By definition, for a continuous uniform distribution, the probability density function is a horizontal line segment over a range of values such that the area under the segment (total probability of an outcome in the range) equals one 35. Consider a random variable X that follows a continuous uniform distribution: 7 ≤ X ≤ 20. Which of the following statements is least accurate? A)F(21) = 0.00. B)F(12 ≤ X ≤ 16) = 0.307. C)F(10) = 0.23 Explanation: A — F(21) = 1.00 The probability density function for a continuous uniform distribution is calculated as follows: F(X) = (X – a) / (b – a), where a and b are the upper and lower endpoints, respectively. (If the given X is greater than the upper limit, the probability is 1.0.) Shortcut: If you know the properties of this function, you do not need to do any calculations to check the other choices. The other choices are true. F(10) = (10 – 7) / (20 – 7) = 3 / 13 = 0.23 F(12 ≤ X ≤ 16) = F(16) – F(12) = [(16 – 7) / (20 – 7)] − [(12 – 7) / (20 – 7)] = 0.692 − 0.385 = 0.307 36. A random variable follows a continuous uniform distribution over 27 to 89. What is the probability of an outcome between 34 and 38? A)0.0645. B)0.0546. C)0.0719 Explanation: A — P(34 ≤ X ≤ 38) = (38 − 34) / (89 − 27) = 0.0645 37. A multivariate normal distribution that includes three random variables can be completely described by the means and variances of each of the random variables and the: A)correlation coefficient of the three random variables. B)conditional probabilities among the three random variables. C)correlations between each pair of random variables. Explanation: C — A multivariate normal distribution that includes three random variables can be completely described by the means and variances of each of the random variables and the correlations between each pair of random variables. Correlation measures the strength of the linear relationship between two random variables (thus, "the correlation coefficient of the three random variables" is inaccurate) 38. Which of the following would least likely be categorized as a multivariate distribution? A)The return of a stock and the return of the DJIA. B)The days a stock traded and the days it did not trade. C)The returns of the stocks in the DJIA Explanation: B — The number of days a stock traded and did not trade describes only one random variable. Both of the other cases involve two or more random variables 39. Joan Biggs, CFA, acquires a large database of past returns on a variety of assets. Biggs then draws random samples of sets of returns from the database and analyzes the resulting distributions. Biggs is engaging in: A)Monte Carlo simulation. B)historical simulation. C)discrete analysis Explanation: B — This is a typical example of historical simulation 40. Many analysts prefer to use Monte Carlo simulation rather than historical simulation because: A)computers can manipulate theoretical data much more quickly than historical data. B)past distributions cannot address changes in correlations or events that have not happened before. C)it is much easier to generate the required variables Explanation: B — While the past is often a good predictor of the future, simulations based on past distributions are limited to reflecting changes and events that actually occurred. Monte Carlo simulation can be used to model based on parameters that are not limited to past experience 41. The difference between a Monte Carlo simulation and a historical simulation is that a historical simulation uses randomly selected variables from past distributions, while a Monte Carlo simulation: A)projects variables based on a priori principles. B)uses randomly selected variables from future distributions. C)uses a computer to generate random variables Explanation: C — A Monte Carlo simulation uses a computer to generate random variables from specified distributions 42. A drawback of historical simulation is it: A)depends on the accuracy of the random number generator. B)may not accurately reflect possible outcomes. C)may not account for very rare events Explanation: C — There are two major problems with historical simulation. The first is that it cannot account for events that do not occur in the sample. If a security began trading after 1987, for example, there would be no evidence of its behavior in a market crash. The other drawback is that the analyst cannot change the parameters of the distribution to examine how small changes might affect the asset’s behavior