标题: Quantitative Methods 【Reading 9】Sample [打印本页]
作者: cross-ied 时间: 2012-3-22 14:47 标题: [2012 L1] Quantitative Methods 【Session 3 - Reading 9】Sample
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. |
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B)
| Sixty-eight percent of the area under the normal curve falls between the mean and 1 standard deviation above the mean. |
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C)
| The normal curve is asymmetrical about its mean. |
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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.
作者: cross-ied 时间: 2012-3-22 14:48
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. |
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B)
| Flip a coin: P(H) = P(T) = 0.5. |
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C)
| Zeta Corp.: P(dividend increases) = 0.60, P(dividend decreases) = 0.30. |
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All the probabilities must be listed. In the case of Zeta Corp. the probabilities do not sum to one.
作者: cross-ied 时间: 2012-3-22 14:48
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. |
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B)
| The skewness of a normal distribution is zero. |
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C)
| A binomial probability distribution is an example of a continuous probability distribution. |
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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.
作者: cross-ied 时间: 2012-3-22 14:49
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. |
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B)
| A probability distribution includes a listing of all the possible outcomes of an experiment. |
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C)
| A probability distribution is, by definition, normally distributed. |
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Probabilities must be zero or positive, but a probability distribution is not necessarily normally distributed. Binomial distributions are either successes or failures.
作者: cross-ied 时间: 2012-3-22 14:49
A probability distribution is least likely to:A)
| contain all the possible outcomes. |
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B)
| give the probability that the distribution is realistic. |
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C)
| have only non-negative probabilities. |
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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.
作者: cross-ied 时间: 2012-3-22 14:50
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?
The probability of a value being rolled is 1/6 regardless of the previous value rolled.
作者: cross-ied 时间: 2012-3-22 14:50
The number of ships in the harbor is an example of what kind of variable?
A discrete variable is one that is represented by finite units.
作者: cross-ied 时间: 2012-3-22 14:50
Which of the following is a discrete random variable? A)
| The number of advancing stocks in the DJIA in a day. |
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B)
| The realized return on a corporate bond. |
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C)
| The amount of time between two successive stock trades. |
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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.
作者: cross-ied 时间: 2012-3-22 14:50
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. |
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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. |
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C)
| A continuous uniform distribution has a lower limit but no upper limit. |
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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.
作者: cross-ied 时间: 2012-3-22 14:51
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. |
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B)
| A positive value; zero. |
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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.
作者: cross-ied 时间: 2012-3-22 14:51
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. |
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B)
| The number of days of sunshine in the month of May 2006 in a particular city. |
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C)
| Quoted stock prices on the NASDAQ. |
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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).
作者: cross-ied 时间: 2012-3-22 14:52
A random variable that has a countable number of possible values is called a: A)
| probability distribution. |
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B)
| continuous random variable. |
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C)
| discrete random variable. |
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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.
作者: cross-ied 时间: 2012-3-22 14:52
A probability function: A)
| is often referred to as the "cdf." |
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B)
| specifies the probability that the random variable takes on a specific value. |
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C)
| only applies to continuous distributions. |
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This is true by definition.
作者: cross-ied 时间: 2012-3-22 14:52
Which of the following could least likely be a probability function?A)
| X 1,2,3,4) p(x) = 0.2. |
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B)
| X1,2,3,4) p(x) = (x × x) / 30. |
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C)
| X1,2,3,4) p(x) = x / 10. |
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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.
作者: cross-ied 时间: 2012-3-22 14:53
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. |
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B)
| one and the curve must not rise above the horizontal axis. |
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C)
| zero and the curve must not fall below the horizontal axis. |
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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).
作者: cross-ied 时间: 2012-3-22 14:54
In a continuous probability density function, the probability that any single value of a random variable occurs is equal to what?
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.
作者: cross-ied 时间: 2012-3-22 14:55
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 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.
作者: cross-ied 时间: 2012-3-22 14:56
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. |
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B)
| F(1) = 0, F(2) = 0.5, F(3) = 0.5, F(4) = 0. |
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C)
| F(1) = 0.5, F(2) = 0.25, F(3) = 0.25. |
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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.
作者: cross-ied 时间: 2012-3-22 14:56
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).
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%.
作者: cross-ied 时间: 2012-3-22 14:57
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. |
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B)
| The number of heads on the flip of two coins. |
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C)
| The outcome of the roll of two standard, six-sided dice where X is the sum of the numbers facing up. |
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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.
作者: cross-ied 时间: 2012-3-22 14:57
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?
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.
作者: cross-ied 时间: 2012-3-22 14:57
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:
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!].
作者: JustasS 时间: 2012-3-22 14:59
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?
3! / (0!3!) (0.3)0 (0.7)3 = 0.343
作者: JustasS 时间: 2012-3-22 15:00
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?
P(3) = 5! / [(5 – 3)! × 3!] × (0.73) × (0.32) = 0.3087 = 5 →2nd→ nCr → 3 × 0.343 × 0.09
作者: JustasS 时间: 2012-3-22 15:00
Which of the following could be the set of all possible outcomes for a random variable that follows a binomial distribution? |
B)
| (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11). |
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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.
作者: JustasS 时间: 2012-3-22 15:00
Which of the following is NOT an assumption of the binomial distribution?A)
| The trials are independent. |
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B)
| The expected value is a whole number. |
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C)
| Random variable X is discrete. |
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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.
作者: JustasS 时间: 2012-3-22 15:01
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?
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.
作者: JustasS 时间: 2012-3-22 15:01
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?
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.
作者: JustasS 时间: 2012-3-22 15:01
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?
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.
作者: JustasS 时间: 2012-3-22 15:02
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?
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%.
作者: JustasS 时间: 2012-3-22 15:03
Tracking error for a portfolio is best described as the: A)
| sample mean minus population mean. |
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B)
| portfolio return minus a benchmark return. |
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C)
| standard deviation of differences between an index return and portfolio return. |
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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.
作者: JustasS 时间: 2012-3-22 15:03
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:
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%.
作者: JustasS 时间: 2012-3-22 15:03
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?
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.
作者: JustasS 时间: 2012-3-22 15:04
The probability density function of a continuous uniform distribution is best described by a: A)
| line segment with a curvilinear slope. |
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B)
| line segment with a 45-degree slope. |
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C)
| horizontal line segment. |
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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.
作者: JustasS 时间: 2012-3-22 15:04
Consider a random variable X that follows a continuous uniform distribution: 7 ≤ X ≤ 20. Which of the following statements is least accurate? |
B)
| F(12 ≤ X ≤ 16) = 0.307. |
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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
作者: JustasS 时间: 2012-3-22 15:04
A random variable follows a continuous uniform distribution over 27 to 89. What is the probability of an outcome between 34 and 38?
P(34 ≤ X ≤ 38) = (38 − 34) / (89 − 27) = 0.0645
作者: JustasS 时间: 2012-3-22 15:04
If X has a normal distribution with μ = 100 and σ = 5, then there is approximately a 90% probability that:
100 +/- 1.65 (5) = 91.75 to 108.25 or P ( P(91.75 < X < 108.25).
作者: JustasS 时间: 2012-3-22 15:05
Which of the following statements about a normal distribution is least accurate? A)
| Approximately 34% of the observations fall within plus or minus one standard deviation of the mean. |
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B)
| The distribution is completely described by its mean and variance. |
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C)
| Kurtosis is equal to 3. |
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Approximately 68% of the observations fall within one standard deviation of the mean. Approximately 34% of the observations fall within the mean plus one standard deviation (or the mean minus one standard deviation).
作者: JustasS 时间: 2012-3-22 15:05
A normal distribution is completely described by its: A)
| mean, mode, and skewness. |
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By definition, a normal distribution is completely described by its mean and variance.
作者: JustasS 时间: 2012-3-22 15:05
In a normal distribution, the: A)
| mean is less than the mode. |
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B)
| mean is greater than the median. |
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C)
| median equals the mode. |
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In a normal distribution, the mean, median, and mode are all equal.
作者: JustasS 时间: 2012-3-22 15:06
The lower limit of a normal distribution is:
By definition, a true normal distribution has a positive probability density function from negative to positive infinity.
作者: JustasS 时间: 2012-3-22 15:06
A normal distribution can be completely described by its: A)
| skewness and kurtosis. |
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The normal distribution can be completely described by its mean and variance.
作者: JustasS 时间: 2012-3-22 15:06
A stock portfolio has had a historical average annual return of 12% and a standard deviation of 20%. The returns are normally distributed. The range –27.2 to 51.2% describes a:A)
| 95% confidence interval. |
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B)
| 68% confidence interval. |
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C)
| 99% confidence interval. |
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The upper limit of the range, 51.2%, is (51.2 − 12) = 39.2 / 20 = 1.96 standard deviations above the mean of 12. The lower limit of the range is (12 − (-27.2)) = 39.2 / 20 = 1.96 standard deviations below the mean of 12. A 95% confidence level is defined by a range 1.96 standard deviations above and below the mean.
作者: JustasS 时间: 2012-3-22 15:07
A stock portfolio's returns are normally distributed. It has had a mean annual return of 25% with a standard deviation of 40%. The probability of a return between -41% and 91% is closest to:
A 90% confidence level includes the range between plus and minus 1.65 standard deviations from the mean. (91 − 25) / 40 = 1.65 and (-41 − 25) / 40 = -1.65.
作者: JustasS 时间: 2012-3-22 15:07
For a normal distribution, what approximate percentage of the observations fall within ±3 standard deviation of the mean?
For normal distributions, approximately 99% of the observations fall within ±3 standard deviations of the mean.
作者: JustasS 时间: 2012-3-22 15:08
The mean return of a portfolio is 20% and its standard deviation is 4%. The returns are normally distributed. Which of the following statements about this distribution are least accurate? The probability of receiving a return: A)
| in excess of 16% is 0.16. |
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B)
| of less than 12% is 0.025. |
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C)
| between 12% and 28% is 0.95. |
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The probability of receiving a return greater than 16% is calculated by adding the probability of a return between 16% and 20% (given a mean of 20% and a standard deviation of 4%, this interval is the left tail of one standard deviation from the mean, which includes 34% of the observations.) to the area from 20% and higher (which starts at the mean and increases to infinity and includes 50% of the observations.) The probability of a return greater than 16% is 34 + 50 = 84%.Note: 0.16 is the probability of receiving a return less than 16%.
作者: JustasS 时间: 2012-3-22 15:08
A group of investors wants to be sure to always earn at least a 5% rate of return on their investments. They are looking at an investment that has a normally distributed probability distribution with an expected rate of return of 10% and a standard deviation of 5%. The probability of meeting or exceeding the investors' desired return in any given year is closest to:
The mean is 10% and the standard deviation is 5%. You want to know the probability of a return 5% or better. 10% - 5% = 5% , so 5% is one standard deviation less than the mean. Thirty-four percent of the observations are between the mean and one standard deviation on the down side. Fifty percent of the observations are greater than the mean. So the probability of a return 5% or higher is 34% + 50% = 84%.
作者: JustasS 时间: 2012-3-22 15:08
A client will move his investment account unless the portfolio manager earns at least a 10% rate of return on his account. The rate of return for the portfolio that the portfolio manager has chosen has a normal probability distribution with an expected return of 19% and a standard deviation of 4.5%. What is the probability that the portfolio manager will keep this account?
Since we are only concerned with values that are below a 10% return this is a 1 tailed test to the left of the mean on the normal curve. With μ = 19 and σ = 4.5, P(X ≥ 10) = P(X ≥ μ − 2σ) therefore looking up -2 on the cumulative Z table gives us a value of 0.0228, meaning that (1 − 0.0228) = 97.72% of the area under the normal curve is above a Z score of -2. Since the Z score of -2 corresponds with the lower level 10% rate of return of the portfolio this means that there is a 97.72% probability that the portfolio will earn at least a 10% rate of return.
作者: JustasS 时间: 2012-3-22 15:08
A portfolio manager is looking at an investment that has an expected annual return of 10% with a standard deviation of annual returns of 5%. Assuming the returns are approximately normally distributed, the probability that the return will exceed 20% in any given year is closest to:
Given that the standard deviation is 5%, a 20% return is two standard deviations above the expected return of 10%. Assuming a normal distribution, the probability of getting a result more than two standard deviations above the expected return is 1 − Prob(Z ≤ 2) = 1 − 0.9772 = 0.0228 or 2.28% (from the Z table).
作者: JustasS 时间: 2012-3-22 15:09
An investment has a mean return of 15% and a standard deviation of returns equal to 10%. If returns are normally distributed, which of the following statements is least accurate? The probability of obtaining a return: A)
| between 5% and 25% is 0.68. |
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B)
| greater than 25% is 0.32. |
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C)
| greater than 35% is 0.025. |
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Sixty-eight percent of all observations fall within +/- one standard deviation of the mean of a normal distribution. Given a mean of 15 and a standard deviation of 10, the probability of having an actual observation fall within one standard deviation, between 5 and 25, is 68%. The probability of an observation greater than 25 is half of the remaining 32%, or 16%. This is the same probability as an observation less than 5. Because 95% of all observations will fall within 20 of the mean, the probability of an actual observation being greater than 35 is half of the remaining 5%, or 2.5%.
作者: JustasS 时间: 2012-3-22 15:09
A food retailer has determined that the mean household income of her customers is $47,500 with a standard deviation of $12,500. She is trying to justify carrying a line of luxury food items that would appeal to households with incomes greater than $60,000. Based on her information and assuming that household incomes are normally distributed, what percentage of households in her customer base has incomes of $60,000 or more?
Z = ($60,000 – $47,500) / $12,500 = 1.0
From the table of areas under the normal curve, 84.13% of observations lie to the left of +1 standard deviation of the mean. So, 100% – 84.13% = 15.87% with incomes of $60,000 or more.
作者: JustasS 时间: 2012-3-22 15:09
The average annual rainfall amount in Yucutat, Alaska, is normally distributed with a mean of 150 inches and a standard deviation of 20 inches. The 90% confidence interval for the annual rainfall in Yucutat is closest to:
The 90% confidence interval is µ ± 1.65 standard deviations. 150 − 1.65(20) = 117 and 150 + 1.65(20) = 183.
作者: cyber21 时间: 2012-3-22 15:11
The standard normal distribution is most completely described as a: A)
| distribution that exhibits zero skewness and no excess kurtosis. |
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B)
| symmetrical distribution with a mean equal to its median. |
|
C)
| normal distribution with a mean of zero and a standard deviation of one. |
|
The standard normal distribution is defined as a normal distribution that has a mean of zero and a standard deviation of one. The other choices apply to any normal distribution.
作者: cyber21 时间: 2012-3-22 15:11
Monthly sales of hot water heaters are approximately normally distributed with a mean of 21 and a standard deviation of 5. What is the probabilility of selling 12 hot water heaters or less next month?
Z = (12 – 21) / 5 = -1.8
From the cumulative z-table, the probability of being more than 1.8 standard deviations below the mean, probability x < -1.8, is 3.59%.
作者: cyber21 时间: 2012-3-22 15:11
The owner of a bowling alley determined that the average weight for a bowling ball is 12 pounds with a standard deviation of 1.5 pounds. A ball denoted “heavy” should be one of the top 2% based on weight. Assuming the weights of bowling balls are normally distributed, at what weight (in pounds) should the “heavy” designation be used?
The first step is to determine the z-score that corresponds to the top 2%. Since we are only concerned with the top 2%, we only consider the right hand of the normal distribution. Looking on the cumulative table for 0.9800 (or close to it) we find a z-score of 2.05. To answer the question, we need to use the normal distribution given: 98 percentile = sample mean + (z-score)(standard deviation) = 12 + 2.05(1.5) = 15.08.
作者: cyber21 时间: 2012-3-22 15:12
Which of the following represents the mean, standard deviation, and variance of a standard normal distribution?
By definition, for the standard normal distribution, the mean, standard deviation, and variance are 0, 1, 1.
作者: cyber21 时间: 2012-3-22 15:12
Standardizing a normally distributed random variable requires the: A)
| mean, variance and skewness. |
|
B)
| natural logarithm of X. |
|
C)
| mean and the standard deviation. |
|
All that is necessary is to know the mean and the variance. Subtracting the mean from the random variable and dividing the difference by the standard deviation standardizes the variable.
作者: cyber21 时间: 2012-3-22 15:13
A grant writer for a local school district is trying to justify an application for funding an after-school program for low-income families. Census information for the school district shows an average household income of $26,200 with a standard deviation of $8,960. Assuming that the household income is normally distributed, what is the percentage of households in the school district with incomes of less than $12,000?
Z = ($12,000 – $26,200) / $8,960 = –1.58.
From the table of areas under the standard normal curve, 5.71% of observations are more than 1.58 standard deviations below the mean.
作者: cyber21 时间: 2012-3-22 15:13
Given a normally distributed population with a mean income of $40,000 and standard deviation of $7,500, what percentage of the population makes between $30,000 and $35,000?
The z-score for $30,000 = ($30,000 – $40,000) / $7,500 or –1.3333, which corresponds with 0.0918. The z-score for $35,000 = ($35,000 – $40,000) / $7,500 or –0.6667, which corresponds with 0.2514. The difference is 0.1596 or 15.96%.
作者: cyber21 时间: 2012-3-22 15:14
John Cupp, CFA, has several hundred clients. The values of the portfolios Cupp manages are approximately normally distributed with a mean of $800,000 and a standard deviation of $250,000. The probability of a randomly selected portfolio being in excess of $1,000,000 is:
Although the number of clients is discrete, since there are several hundred of them, we can treat them as continuous. The selected random value is standardized (its z-value is calculated) by subtracting the mean from the selected value and dividing by the standard deviation. This results in a z-value of (1,000,000 – 800,000) / 250,000 = 0.8. Looking up 0.8 in the z-value table yields 0.7881 as the probability that a random variable is to the left of the standardized value (i.e., less than $1,000,000). Accordingly, the probability of a random variable being to the right of the standardized value (i.e., greater than $1,000,000) is 1 – 0.7881 = 0.2119.
作者: cyber21 时间: 2012-3-22 15:14
If a stock's return is normally distributed with a mean of 16% and a standard deviation of 50%, what is the probability of a negative return in a given year?
The selected random value is standardized (its z-value is calculated) by subtracting the mean from the selected value and dividing by the standard deviation. This results in a z-value of (0 − 16) / 50 = -0.32. Changing the sign and looking up +0.32 in the z-value table yields 0.6255 as the probability that a random variable is to the right of the standardized value (i.e. more than zero). Accordingly, the probability of a random variable being to the left of the standardized value (i.e. less than zero) is 1 − 0.6255 = 0.3745.
作者: cyber21 时间: 2012-3-22 15:14
The average amount of snow that falls during January in Frostbite Falls is normally distributed with a mean of 35 inches and a standard deviation of 5 inches. The probability that the snowfall amount in January of next year will be between 40 inches and 26.75 inches is closest to:
To calculate this answer, we will use the properties of the standard normal distribution. First, we will calculate the Z-value for the upper and lower points and then we will determine the approximate probability covering that range. Note: This question is an example of why it is important to memorize the general properties of the normal distribution.
Z = (observation – population mean) / standard deviation - Z26.75 = (26.75 – 35) / 5 = -1.65. (1.65 standard deviations to the left of the mean)
- Z40 = (40 – 35) / 5 = 1.0 (1 standard deviation to the right of the mean)
Using the general approximations of the normal distribution: 68% of the observations fall within ± one standard deviation of the mean. So, 34% of the area falls between 0 and +1 standard deviation from the mean.
90% of the observations fall within ± 1.65 standard deviations of the mean. So, 45% of the area falls between 0 and +1.65 standard deviations from the mean.
Here, we have 34% to the right of the mean and 45% to the left of the mean, for a total of 79%.
作者: cyber21 时间: 2012-3-22 15:15
Three portfolios with normally distributed returns are available to an investor who wants to minimize the probability that the portfolio return will be less than 5%. The risk and return characteristics of these portfolios are shown in the following table:Portfolio | Expected return | Standard deviation |
Epps | 6% | 4% |
Flake | 7% | 9% |
Grant | 10% | 15% |
Based on Roy’s safety-first criterion, which portfolio should the investor select?
Roy’s safety-first ratios for the three portfolios:
Epps = (6 - 5) / 4 = 0.25
Flake = ( 7 - 5) / 9 = 0.222
Grant = (10 - 5) / 15 = 0.33
The portfolio with the largest safety-first ratio has the lowest probability of a return less than 5%. The investor should select the Grant portfolio.
作者: cyber21 时间: 2012-3-22 15:15
The mean and standard deviation of four portfolios are listed below in percentage terms. Using Roy's safety first criteria and a threshold of 3%, select the respective mean and standard deviation that corresponds to the optimal portfolio.
According to the safety-first criterion, the optimal portfolio is the one that has has the largest value for the SFRatio (mean − threshold) / Standard Deviation. A mean = 5 and Standard Deviation = 3 yields the largest SFRatio from the choices given: (5 − 3) / 3 = 0.67.
作者: cyber21 时间: 2012-3-22 15:16
The safety-first criterion rules focuses on:
The safety-first criterion focuses on shortfall risk which is the probability that a portfolio’s value or return will not fall below a given threshold level. The safety-first criterion usually dictate choosing a portfolio with the lowest probability of falling below the threshold level or return.
作者: cyber21 时间: 2012-3-22 15:16
If the threshold return is higher than the risk-free rate, what will be the relationship between Roy’s safety-first ratio (SF) and Sharpe’s ratio? A)
| The SF ratio will be lower. |
|
B)
| The SF ratio may be higher or lower depending on the standard deviation. |
|
C)
| The SF ratio will be higher. |
|
Since each ratio has the standard deviation of returns in the denominator, the difference depends upon the effect on the numerator. Since both the risk-free rate (in the Sharpe ratio) and the threshold rate (in the SF ratio) are subtracted from the expected return, a larger threshold rate would result in a smaller SF ratio value.
作者: cyber21 时间: 2012-3-22 15:17
The mean and standard deviation of three portfolios are listed below in percentage terms. Using Roy's safety-first criteria and a threshold of 4%, select the respective mean and standard deviation that corresponds to the optimal portfolio.
According to the safety-first criterion, the optimal portfolio is the one that has the largest value for the SFRatio (mean − threshold) / Standard Deviation. A mean = 19 and Standard Deviation = 28 yields the largest SFRatio from the choices given: (19 − 4) / 28 = 0.5357.
作者: cyber21 时间: 2012-3-22 15:17
The farthest point on the left side of the lognormal distribution: A)
| is skewed to the left. |
|
B)
| can be any negative number. |
|
|
The lognormal distribution is skewed to the right with a long right hand tail and is bounded on the left hand side of the curve by zero.
作者: cyber21 时间: 2012-3-22 15:18
Which of the following statements regarding the distribution of returns used for asset pricing models is most accurate? A)
| Normal distribution returns are used for asset pricing models because they will only allow the asset price to fall to zero. |
|
B)
| Lognormal distribution returns are used for asset pricing models because they will not result in an asset return of less than -100%. |
|
C)
| Lognormal distribution returns are used because this will allow for negative returns on the assets. |
|
Lognormal distribution returns are used for asset pricing models because this will not result in asset returns of less than 100% because the lowest the asset price can decrease to is zero which is the lowest value on the lognormal distribution. The normal distribution allows for asset prices less than zero which could result in a return of less than -100% which is impossible.
作者: cyber21 时间: 2012-3-22 15:18
If a random variable x is lognormally distributed then ln x is: A)
| abnormally distributed. |
|
|
|
For any random variable that is normally distributed its natural logarithm (ln) will be lognormally distributed. The opposite is also true: for any random variable that is lognormally distributed its natural logarithm (ln) will be normally distributed.
作者: cyber21 时间: 2012-3-22 15:19
If random variable Y follows a lognormal distribution then the natural log of Y must be: |
|
C)
| lognormally distributed. |
|
For any random variable that is lognormally distributed its natural logarithm (ln) will be normally distributed.
作者: cyber21 时间: 2012-3-22 15:19
Given Y is lognormally distributed, then ln Y is: A)
| a lognormal distribution. |
|
|
|
If Y is lognormally distributed, then ln Y is normally distributed.
作者: cyber21 时间: 2012-3-22 15:19
A stock increased in value last year. Which will be greater, its continuously compounded or its holding period return? A)
| Neither, they will be equal. |
|
B)
| Its continuously compounded return. |
|
C)
| Its holding period return. |
|
When a stock increases in value, the holding period return is always greater than the continuously compounded return that would be required to generate that holding period return. For example, if a stock increases from $1 to $1.10 in a year, the holding period return is 10%. The continuously compounded rate needed to increase a stock's value by 10% is Ln(1.10) = 9.53%.
作者: cyber21 时间: 2012-3-22 15:20
Over a period of one year, an investor’s portfolio has declined in value from 127,350 to 108,427. What is the continuously compounded rate of return?
The continuously compounded rate of return = ln( S1 / S0 ) = ln(108,427 / 127,350) = –16.09%.
作者: cyber21 时间: 2012-3-22 15:20
If a stock decreases in one period and then increases by an equal dollar amount in the next period, will the respective arithmetic average of the continuously compounded and holding period rates of return be positive, negative, or zero?
The holding period return will have an upward bias that will give a positive average. For example, a fall from 100 to 90 is 10%, and the rise from 90 to 100 is an increase of 11.1%. The continuously compounded return will have an arithmetic average of zero. Since we can sum continuously compounded rates for multiple periods, the continuously compounded rate for the two periods (0%), means the rates for the two periods must sum to zero, and their average must therefore be zero.
作者: cyber21 时间: 2012-3-22 15:20
Assume an investor purchases a stock for $50. One year later, the stock is worth $60. After one more year, the stock price has fallen to the original price of $50. Calculate the continuously compounded return for year 1 and year 2.
Given a holding period return of R, the continuously compounded rate of return is: ln(1 + R) = ln(Price1/Price0). Here, if the stock price increases to $60, r = ln(60/50) = 0.18232, or 18.23%.
Note: Calculator keystrokes are as follows. First, obtain the result of 60/50, or 1. On the TI BA II Plus, enter 1.20 and then click on LN. On the HP12C, 1.2 [ENTER] g [LN] (the LN appears in blue on the %T key).The return for year 2 is ln(50/60), or ln(0.833) = negative 18.23%.
作者: cyber21 时间: 2012-3-22 15:21
If a stock decreases from $90 to $80, the continuously compounded rate of return for the period is:
This is given by the natural logarithm of the new price divided by the old price; ln(80 / 90) = -0.1178.
作者: cyber21 时间: 2012-3-22 15:21
Given a holding period return of R, the continuously compounded rate of return is:
This is the formula for the continuously compounded rate of return.
作者: cyber21 时间: 2012-3-22 15:21
The continuously compounded rate of return that will generate a one-year holding period return of -6.5% is closest to:
Continuously compounded rate of return = ln(1 − 0.065) = -6.72%.
作者: cyber21 时间: 2012-3-22 15:21
Mei Tekei just celebrated her 22nd birthday. When she is 27, she will receive a $100,000 inheritance. Tekei needs funds for the down payment on a co-op in Manhattan and has found a bank that will give her the present value of her inheritance amount, assuming an 8.0% stated annual interest rate with continuous compounding. Will the proceeds from the bank be sufficient to cover her down payment of $65,000? A)
| Yes, Tekei will receive $68,058. |
|
B)
| Yes, Tekei will receive $67,028. |
|
C)
| No, Tekei will only receive $61,878. |
|
Because the rate is 8% compounded continuously, the effective annual rate is e0.08 - 1 = 8.33%. To find the present value of the inheritance, enter N=5, I/Y=8.33, PMT=0, FV=100,000 CPT PV = 67,028.
Alternatively, 100,000e-0.08(5) = 67,032.
作者: cyber21 时间: 2012-3-22 15:22
Monte Carlo simulation is necessary to: A)
| reduce sampling error. |
|
B)
| compute continuously compounded returns. |
|
C)
| approximate solutions to complex problems. |
|
This is the purpose of this type of simulation. The point is to construct distributions using complex combinations of hypothesized parameters.
作者: cyber21 时间: 2012-3-22 15:23
In which of the following cases would Monte Carlo simulation least likely be needed? Payoff of a:
The probability distribution of a roulette wheel would be easy to estimate using empirical or a priori methodology.
作者: cyber21 时间: 2012-3-22 15:23
A multivariate distribution:A)
| specifies the probabilities associated with groups of random variables. |
|
B)
| applies only to binomial distributions. |
|
C)
| gives multiple probabilities for the same outcome. |
|
This is the definition of a multivariate distribution.
作者: cyber21 时间: 2012-3-22 15:24
In addition to the usual parameters that describe a normal distribution, to completely describe 10 random variables, a multivariate normal distribution requires knowing the:
The number of correlations in a multivariate normal distribution of n variables is computed by the formula ((n) × (n-1)) / 2, in this case (10 × 9) / 2 = 45.
作者: cyber21 时间: 2012-3-22 15:24
A multivariate distribution is best defined as describing the behavior of: A)
| two or more independent random variables. |
|
B)
| a random variable with more than two possible outcomes. |
|
C)
| two or more dependent random variables. |
|
A multivariate distribution describes the relationships between two or more random variables, when the behavior of each random variable is dependent on the others in some way.
作者: cyber21 时间: 2012-3-22 15:25
In a multivariate normal distribution, a correlation tells the: A)
| relationship between the means and variances of the variables. |
|
B)
| overall relationship between all the variables. |
|
C)
| strength of the linear relationship between two of the variables. |
|
This is true by definition. The correlation only applies to two variables at a time.
作者: cyber21 时间: 2012-3-22 15:25
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. |
|
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).
作者: cyber21 时间: 2012-3-22 15:26
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. |
|
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.
作者: cyber21 时间: 2012-3-22 15:26
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. |
|
|
This is a typical example of historical simulation.
作者: cyber21 时间: 2012-3-22 15:27
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. |
|
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.
作者: cyber21 时间: 2012-3-22 15:27
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. |
|
A Monte Carlo simulation uses a computer to generate random variables from specified distributions.
作者: cyber21 时间: 2012-3-22 15:27
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. |
|
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.
作者: terpsichorefan 时间: 2013-3-19 22:50
thanks for sharing
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