P1.T2. Quantitative Analysis

Practice questions for Quantitative Analysis: Econometrics, MCS, Volatility, Probability Distributions and VaR (Intro)

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1. Variance and Covariance Calculation Clarification

Hi David, Thanks! I work in Excel every day so being able to look at the numbers was a big help. What I was describing in the first part can be summed up as: Pr*(X-µ)^2 The second equation can be described as: Pr*X^2-(sum(Pr*X))^2. sum(Pr*X) = µ What you were showing in the second example was with samples it may be difficult to assign a true distribution, so instead for a sample mean, you...
Hi David, Thanks! I work in Excel every day so being able to look at the numbers was a big help. What I was describing in the first part can be summed up as: Pr*(X-µ)^2 The second equation can be described as: Pr*X^2-(sum(Pr*X))^2. sum(Pr*X) = µ What you were showing in the second example was with samples it may be difficult to assign a true distribution, so instead for a sample mean, you...
Hi David, Thanks! I work in Excel every day so being able to look at the numbers was a big help. What I was describing in the first part can be summed up as: Pr*(X-µ)^2 The second equation can be described as: Pr*X^2-(sum(Pr*X))^2. sum(Pr*X) = µ What you were showing in the second example was...
Hi David, Thanks! I work in Excel every day so being able to look at the numbers was a big help. What I was describing in the first part can be summed up as: Pr*(X-µ)^2 The second equation can...
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2. Uses of the Probability Density Function versus the Cumulative Distribution Function

a discrete distribution has a pmf (probability mass function) instead of a prob density function (pdf) which is its continuous analog. An easy example of pmf/CDF is a fair six-sided die: the CDF is F(X) = X/6; i.e., the probability of rolling a three or less is 3/6 = 50% the pmf is the derivative: if F(X) = 1/6*x, then f(X) = F'(X) = 1/6; ie the pmf of a fair die is f(x) = 1/6 if f(x) = ax +...
a discrete distribution has a pmf (probability mass function) instead of a prob density function (pdf) which is its continuous analog. An easy example of pmf/CDF is a fair six-sided die: the CDF is F(X) = X/6; i.e., the probability of rolling a three or less is 3/6 = 50% the pmf is the derivative: if F(X) = 1/6*x, then f(X) = F'(X) = 1/6; ie the pmf of a fair die is f(x) = 1/6 if f(x) = ax +...
a discrete distribution has a pmf (probability mass function) instead of a prob density function (pdf) which is its continuous analog. An easy example of pmf/CDF is a fair six-sided die: the CDF is F(X) = X/6; i.e., the probability of rolling a three or less is 3/6 = 50% the pmf is the...
a discrete distribution has a pmf (probability mass function) instead of a prob density function (pdf) which is its continuous analog. An easy example of pmf/CDF is a fair six-sided die: the CDF...
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3. Test-Questions as collection?

T. Flockert, Thanks for appreciating these quant questions... 1. They are from the 2008 season: I wrote them all. See http://www.bionicturtle.com/premium/quizzes/category/frm_product/ for inventory of practice questions. Near the bottom @ http://www.bionicturtle.com/premium/quiz/2008_quantitative/ (in the flash quiz that has printout ability) 2. Yes, that is the goal but, beyond the...
T. Flockert, Thanks for appreciating these quant questions... 1. They are from the 2008 season: I wrote them all. See http://www.bionicturtle.com/premium/quizzes/category/frm_product/ for inventory of practice questions. Near the bottom @ http://www.bionicturtle.com/premium/quiz/2008_quantitative/ (in the flash quiz that has printout ability) 2. Yes, that is the goal but, beyond the...
T. Flockert, Thanks for appreciating these quant questions... 1. They are from the 2008 season: I wrote them all. See http://www.bionicturtle.com/premium/quizzes/category/frm_product/ for inventory of practice questions. Near the bottom @...
T. Flockert, Thanks for appreciating these quant questions... 1. They are from the 2008 season: I wrote them all. See http://www.bionicturtle.com/premium/quizzes/category/frm_product/ for...
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5. Skewness and kurtosis

Hi Mike, The mu^3 is really a mistake (legacy notation actually). The correct numerator for skewness is the third moment E[(Y-mu)^3] and the correct numerator for kurtosis is the fourth moment E[(Y-mu)^3]. In general, the r-th moment is E[(Y-mean)^r]. Apologies, I've got to get rid of the mu^3 and mu^4 Thanks, David
Hi Mike, The mu^3 is really a mistake (legacy notation actually). The correct numerator for skewness is the third moment E[(Y-mu)^3] and the correct numerator for kurtosis is the fourth moment E[(Y-mu)^3]. In general, the r-th moment is E[(Y-mean)^r]. Apologies, I've got to get rid of the mu^3 and mu^4 Thanks, David
Hi Mike, The mu^3 is really a mistake (legacy notation actually). The correct numerator for skewness is the third moment E[(Y-mu)^3] and the correct numerator for kurtosis is the fourth moment E[(Y-mu)^3]. In general, the r-th moment is E[(Y-mean)^r]. Apologies, I've got to get rid of the...
Hi Mike, The mu^3 is really a mistake (legacy notation actually). The correct numerator for skewness is the third moment E[(Y-mu)^3] and the correct numerator for kurtosis is the fourth moment...
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6. Regression Analysis

Hi Eva, (1) Yes, interchangeable indeed! Gujarati has them as perfect synonyms (I sometimes connote serial correlation with time series as special case of regression versus autocorrelation for a generic regression, but I may have no basis) (2) No, both univariate/multivariate OLS regression assume constant variance (homoskedastic). For both, heteroskedasticity is an assumptional...
Hi Eva, (1) Yes, interchangeable indeed! Gujarati has them as perfect synonyms (I sometimes connote serial correlation with time series as special case of regression versus autocorrelation for a generic regression, but I may have no basis) (2) No, both univariate/multivariate OLS regression assume constant variance (homoskedastic). For both, heteroskedasticity is an assumptional...
Hi Eva, (1) Yes, interchangeable indeed! Gujarati has them as perfect synonyms (I sometimes connote serial correlation with time series as special case of regression versus autocorrelation for a generic regression, but I may have no basis) (2) No, both univariate/multivariate OLS...
Hi Eva, (1) Yes, interchangeable indeed! Gujarati has them as perfect synonyms (I sometimes connote serial correlation with time series as special case of regression versus autocorrelation for...
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7. Random Variable question

Hi David, Thanks, it actually FRM 2008 Practice Exam 1, question number 4.
Hi David, Thanks, it actually FRM 2008 Practice Exam 1, question number 4.
Hi David, Thanks, it actually FRM 2008 Practice Exam 1, question number 4.
Hi David, Thanks, it actually FRM 2008 Practice Exam 1, question number 4.
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8. question on: 208.3.C and 202.5

Hi Chris, I think you are correct on both, can you see the source question thread @ i.e., you've identified two errors. I apologize they are not yet fixed in the PDF (like all errors, we will revise the PDFs, but I felt it more helpful currently to prioritize the 2 fresh mock exams). Thanks,
Hi Chris, I think you are correct on both, can you see the source question thread @ i.e., you've identified two errors. I apologize they are not yet fixed in the PDF (like all errors, we will revise the PDFs, but I felt it more helpful currently to prioritize the 2 fresh mock exams). Thanks,
Hi Chris, I think you are correct on both, can you see the source question thread @ i.e., you've identified two errors. I apologize they are not yet fixed in the PDF (like all errors, we will revise the PDFs, but I felt it more helpful currently to prioritize the 2 fresh mock exams). Thanks,
Hi Chris, I think you are correct on both, can you see the source question thread @ i.e., you've identified two errors. I apologize they are not yet fixed in the PDF (like all errors, we will...
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9. Question about problem (21.5 at p. 6 from Quantitative Analysis:Hull, Ch.21, 2010 Practice Questions)

Hi, David, I got a question about problem in 21.5 at p.6 from Quantitative Analysis:Hull, Ch.21, 2010 Practice Questions. I am a bit confused about how you obtain the CI. Since the general formula for CI is x_bar +- (z *SE) In your answer, how did you calculate the SE ? Also, what did the NORMSINV mean in word? Thank you so much.
Hi, David, I got a question about problem in 21.5 at p.6 from Quantitative Analysis:Hull, Ch.21, 2010 Practice Questions. I am a bit confused about how you obtain the CI. Since the general formula for CI is x_bar +- (z *SE) In your answer, how did you calculate the SE ? Also, what did the NORMSINV mean in word? Thank you so much.
Hi, David, I got a question about problem in 21.5 at p.6 from Quantitative Analysis:Hull, Ch.21, 2010 Practice Questions. I am a bit confused about how you obtain the CI. Since the general formula for CI is x_bar +- (z *SE) In your answer, how did you calculate the SE ? Also, what did...
Hi, David, I got a question about problem in 21.5 at p.6 from Quantitative Analysis:Hull, Ch.21, 2010 Practice Questions. I am a bit confused about how you obtain the CI. Since the general...
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10. Question 9: Key operational process

Question: We observe that 5 errors per day are made with respect to a key operational process. What are the odds that tomorrow (a full day) we will observe at least two (2) errors? A. 86.72% B. 4.25% C. 91.26% D. 95.96% Answer: D Explanation: This calls for a Poisson(5) distribution. The odds are 1 - [P(X=0) + P(X=1)] because it's the odds that we don't observe zero or one errors....
Question: We observe that 5 errors per day are made with respect to a key operational process. What are the odds that tomorrow (a full day) we will observe at least two (2) errors? A. 86.72% B. 4.25% C. 91.26% D. 95.96% Answer: D Explanation: This calls for a Poisson(5) distribution. The odds are 1 - [P(X=0) + P(X=1)] because it's the odds that we don't observe zero or one errors....
Question: We observe that 5 errors per day are made with respect to a key operational process. What are the odds that tomorrow (a full day) we will observe at least two (2) errors? A. 86.72% B. 4.25% C. 91.26% D. 95.96% Answer: D Explanation: This calls for a Poisson(5) distribution....
Question: We observe that 5 errors per day are made with respect to a key operational process. What are the odds that tomorrow (a full day) we will observe at least two (2) errors? A. 86.72% ...
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11. Question 99: Estimating correlation

Question: When estimating correlation, what is the main challenge of in extending the GARCH model used for volatility to the multivariate GARCH model for correlations? A. Correlation not necessarily mean reverting B. Almost impossible to parameterize persistence C. Accuracy requires many lagged factors (long time series) D. Number of parameters increases exponentially Answer: D ...
Question: When estimating correlation, what is the main challenge of in extending the GARCH model used for volatility to the multivariate GARCH model for correlations? A. Correlation not necessarily mean reverting B. Almost impossible to parameterize persistence C. Accuracy requires many lagged factors (long time series) D. Number of parameters increases exponentially Answer: D ...
Question: When estimating correlation, what is the main challenge of in extending the GARCH model used for volatility to the multivariate GARCH model for correlations? A. Correlation not necessarily mean reverting B. Almost impossible to parameterize persistence C. Accuracy requires many...
Question: When estimating correlation, what is the main challenge of in extending the GARCH model used for volatility to the multivariate GARCH model for correlations? A. Correlation not...
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12. Question 98: Squared return

Question: If the average lambda under the RiskMetrics approach is 0.94 under daily intervals, what weight is effectively assigned to the squared return on day n-2 (not yesterday, but the day before yesterday)? A. 0.06 B. 0.0036 C. 0.0564 D. 0.94 Answer: C Explanation: The most recent weight is (1-lambda) = 6%. Throughout the series, each weight is a constant proportion of its...
Question: If the average lambda under the RiskMetrics approach is 0.94 under daily intervals, what weight is effectively assigned to the squared return on day n-2 (not yesterday, but the day before yesterday)? A. 0.06 B. 0.0036 C. 0.0564 D. 0.94 Answer: C Explanation: The most recent weight is (1-lambda) = 6%. Throughout the series, each weight is a constant proportion of its...
Question: If the average lambda under the RiskMetrics approach is 0.94 under daily intervals, what weight is effectively assigned to the squared return on day n-2 (not yesterday, but the day before yesterday)? A. 0.06 B. 0.0036 C. 0.0564 D. 0.94 Answer: C Explanation: The most recent...
Question: If the average lambda under the RiskMetrics approach is 0.94 under daily intervals, what weight is effectively assigned to the squared return on day n-2 (not yesterday, but the day...
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13. Question 97: Mean-reverting

Question: Here is the GARCH(1,1) specification: variance = omega + (alpha)(lagged, squared return)+(beta)(lagged variance). In the first series, alpha = 0.1 and beta = 0.7. In the second series, alpha = 0.2 and beta = 0.9. Which series is mean-reverting? A. First B. Second C. Both D. Neither Answer: A Explanation: Alpha and beta are here the weights assigned, respectively, to the...
Question: Here is the GARCH(1,1) specification: variance = omega + (alpha)(lagged, squared return)+(beta)(lagged variance). In the first series, alpha = 0.1 and beta = 0.7. In the second series, alpha = 0.2 and beta = 0.9. Which series is mean-reverting? A. First B. Second C. Both D. Neither Answer: A Explanation: Alpha and beta are here the weights assigned, respectively, to the...
Question: Here is the GARCH(1,1) specification: variance = omega + (alpha)(lagged, squared return)+(beta)(lagged variance). In the first series, alpha = 0.1 and beta = 0.7. In the second series, alpha = 0.2 and beta = 0.9. Which series is mean-reverting? A. First B. Second C. Both D....
Question: Here is the GARCH(1,1) specification: variance = omega + (alpha)(lagged, squared return)+(beta)(lagged variance). In the first series, alpha = 0.1 and beta = 0.7. In the second series,...
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14. Question 96: GARCH (1,1) and EWMA

Question: Which are advantages of the GARCH(1,1) approach over the EWMA approach? I. More weight on recent information, II. Mean reversion, III. Persistence A. I only B. II only C. II and III D. I, II, and III Answer: B Explanation: GARCH(1,1) incorporates reversion to the mean but EWMA does not. Both models, unlike the moving average, assign greater weight to more recent...
Question: Which are advantages of the GARCH(1,1) approach over the EWMA approach? I. More weight on recent information, II. Mean reversion, III. Persistence A. I only B. II only C. II and III D. I, II, and III Answer: B Explanation: GARCH(1,1) incorporates reversion to the mean but EWMA does not. Both models, unlike the moving average, assign greater weight to more recent...
Question: Which are advantages of the GARCH(1,1) approach over the EWMA approach? I. More weight on recent information, II. Mean reversion, III. Persistence A. I only B. II only C. II and III D. I, II, and III Answer: B Explanation: GARCH(1,1) incorporates reversion to the mean but...
Question: Which are advantages of the GARCH(1,1) approach over the EWMA approach? I. More weight on recent information, II. Mean reversion, III. Persistence A. I only B. II only C. II and...
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15. Question 95: Moving average

Question: The three prior daily returns for a stock are +1% , +2%, and +3% (day n-1 to day n-3, respectively). Apply Jorion's moving average (MA) over the three day window [i.e., MA(3)] to estimate current volatility. A. 0.02 B. 0.004 C. 0.0216 D. 0.024 Answer: C Explanation: Square each return (under MA the order does not matter) to produce this series: 0.0001, 0.0004, and 0.0009....
Question: The three prior daily returns for a stock are +1% , +2%, and +3% (day n-1 to day n-3, respectively). Apply Jorion's moving average (MA) over the three day window [i.e., MA(3)] to estimate current volatility. A. 0.02 B. 0.004 C. 0.0216 D. 0.024 Answer: C Explanation: Square each return (under MA the order does not matter) to produce this series: 0.0001, 0.0004, and 0.0009....
Question: The three prior daily returns for a stock are +1% , +2%, and +3% (day n-1 to day n-3, respectively). Apply Jorion's moving average (MA) over the three day window [i.e., MA(3)] to estimate current volatility. A. 0.02 B. 0.004 C. 0.0216 D. 0.024 Answer: C Explanation: Square...
Question: The three prior daily returns for a stock are +1% , +2%, and +3% (day n-1 to day n-3, respectively). Apply Jorion's moving average (MA) over the three day window [i.e., MA(3)] to...
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16. Question 94: Distribution

Question: An analysis of typical financial asset returns produces a distribution that has fatter tails than implied by a normal distribution. Which explanation is LEAST LIKELY? A. Non-normal distribution B. Sample too small; larger will converge to normal C. mean is time-varying D. volatility is time-varying Answer: B Explanation: Although the sample may be too small, the best...
Question: An analysis of typical financial asset returns produces a distribution that has fatter tails than implied by a normal distribution. Which explanation is LEAST LIKELY? A. Non-normal distribution B. Sample too small; larger will converge to normal C. mean is time-varying D. volatility is time-varying Answer: B Explanation: Although the sample may be too small, the best...
Question: An analysis of typical financial asset returns produces a distribution that has fatter tails than implied by a normal distribution. Which explanation is LEAST LIKELY? A. Non-normal distribution B. Sample too small; larger will converge to normal C. mean is time-varying D....
Question: An analysis of typical financial asset returns produces a distribution that has fatter tails than implied by a normal distribution. Which explanation is LEAST LIKELY? A. Non-normal...
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17. Question 93: P value

Hi Mike, It's tricky, strictly speaking, there is a difference (but i think an indirect relationship if you will) between the p-value and the significance level (the significance level = probably of committing a Type I error; i.e., mistakenly rejecting a true null). The trick part is that the p-value is the "exact significance level" The link above, in case you didn't drill down, is the...
Hi Mike, It's tricky, strictly speaking, there is a difference (but i think an indirect relationship if you will) between the p-value and the significance level (the significance level = probably of committing a Type I error; i.e., mistakenly rejecting a true null). The trick part is that the p-value is the "exact significance level" The link above, in case you didn't drill down, is the...
Hi Mike, It's tricky, strictly speaking, there is a difference (but i think an indirect relationship if you will) between the p-value and the significance level (the significance level = probably of committing a Type I error; i.e., mistakenly rejecting a true null). The trick part is that the...
Hi Mike, It's tricky, strictly speaking, there is a difference (but i think an indirect relationship if you will) between the p-value and the significance level (the significance level =...
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18. Question 92: Two-tailed test

Question: When working, our machine produces ball bearings that are 10 centimeters in diameter. But not perfectly, as the (population) standard deviation is known to be 0.4 centimeters. We recently took a sample of 64 ball bearings. The sample mean diameter was 10.1 centimeters. Is the machine broken at, respectively, 1% and 5% significance levels using a two-tailed test? A. No (at 1%) and...
Question: When working, our machine produces ball bearings that are 10 centimeters in diameter. But not perfectly, as the (population) standard deviation is known to be 0.4 centimeters. We recently took a sample of 64 ball bearings. The sample mean diameter was 10.1 centimeters. Is the machine broken at, respectively, 1% and 5% significance levels using a two-tailed test? A. No (at 1%) and...
Question: When working, our machine produces ball bearings that are 10 centimeters in diameter. But not perfectly, as the (population) standard deviation is known to be 0.4 centimeters. We recently took a sample of 64 ball bearings. The sample mean diameter was 10.1 centimeters. Is the machine...
Question: When working, our machine produces ball bearings that are 10 centimeters in diameter. But not perfectly, as the (population) standard deviation is known to be 0.4 centimeters. We...
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19. Question 91: Confidence interval

Question: A sample of 25 firms gives a mean PE ratio of 19. We happen to know the population standard deviation is 10. What is the 95% confidence interval? A. 15.1 and 22.9 B. 15.8 and 22.3 C. 16.4 and 21.5 D. 17.5 and 22.5 Answer: A Explanation: The standard error = 10/SQRT(25) = 2.0. The confidence interval = 19 +/- (1.96)(2). Note we know the population variance, so we can use...
Question: A sample of 25 firms gives a mean PE ratio of 19. We happen to know the population standard deviation is 10. What is the 95% confidence interval? A. 15.1 and 22.9 B. 15.8 and 22.3 C. 16.4 and 21.5 D. 17.5 and 22.5 Answer: A Explanation: The standard error = 10/SQRT(25) = 2.0. The confidence interval = 19 +/- (1.96)(2). Note we know the population variance, so we can use...
Question: A sample of 25 firms gives a mean PE ratio of 19. We happen to know the population standard deviation is 10. What is the 95% confidence interval? A. 15.1 and 22.9 B. 15.8 and 22.3 C. 16.4 and 21.5 D. 17.5 and 22.5 Answer: A Explanation: The standard error = 10/SQRT(25) =...
Question: A sample of 25 firms gives a mean PE ratio of 19. We happen to know the population standard deviation is 10. What is the 95% confidence interval? A. 15.1 and 22.9 B. 15.8 and 22.3 ...
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20. Question 90: Outcome

Question: Which of the following is the least likely outcome? A. Reject null but make Type I error B. Reject null with no error C. Fail to reject null but Type II error D. Fail to reject null but make Type I error Answer: D Explanation: A Type I error is to mistakenly reject a true null, so (D) is not only unlikely, (D) is IMPOSSIBLE. If we do not reject the error, either our...
Question: Which of the following is the least likely outcome? A. Reject null but make Type I error B. Reject null with no error C. Fail to reject null but Type II error D. Fail to reject null but make Type I error Answer: D Explanation: A Type I error is to mistakenly reject a true null, so (D) is not only unlikely, (D) is IMPOSSIBLE. If we do not reject the error, either our...
Question: Which of the following is the least likely outcome? A. Reject null but make Type I error B. Reject null with no error C. Fail to reject null but Type II error D. Fail to reject null but make Type I error Answer: D Explanation: A Type I error is to mistakenly reject a true...
Question: Which of the following is the least likely outcome? A. Reject null but make Type I error B. Reject null with no error C. Fail to reject null but Type II error D. Fail to reject...
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