# P1.T2. Quantitative Analysis

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

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1. ### P1.T2.307. Skew and Kurtosis (Miller)

Hi @puneet_ For this bond (a Bernoulli) we have Pr[X=0] = 95% and P[X=1] = 1- 95% = 5%. The mean, µ = (95%*0)+(5%*1) = 0.050. Skew is the third central moment divided by (ie, standardized by) σ^3, just as kurtosis is the fourth central moment divided by σ^4. With respect to skew, the third central moment = (1.0 outcome - 0.05 mean µ )^3*5% probability of 1.0 + (0 outcome - 0.05 mean µ )^3*95%...
Hi @puneet_ For this bond (a Bernoulli) we have Pr[X=0] = 95% and P[X=1] = 1- 95% = 5%. The mean, µ = (95%*0)+(5%*1) = 0.050. Skew is the third central moment divided by (ie, standardized by) σ^3, just as kurtosis is the fourth central moment divided by σ^4. With respect to skew, the third central moment = (1.0 outcome - 0.05 mean µ )^3*5% probability of 1.0 + (0 outcome - 0.05 mean µ )^3*95%...
Hi @puneet_ For this bond (a Bernoulli) we have Pr[X=0] = 95% and P[X=1] = 1- 95% = 5%. The mean, µ = (95%*0)+(5%*1) = 0.050. Skew is the third central moment divided by (ie, standardized by) σ^3, just as kurtosis is the fourth central moment divided by σ^4. With respect to skew, the third...
Hi @puneet_ For this bond (a Bernoulli) we have Pr[X=0] = 95% and P[X=1] = 1- 95% = 5%. The mean, µ = (95%*0)+(5%*1) = 0.050. Skew is the third central moment divided by (ie, standardized by) σ^3,...
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2. ### L1.T2.70 Standard error

Oh okay, sorry @bpdulog thank you for the heads-up! (you already copied Nicole so I won't again ....)
Oh okay, sorry @bpdulog thank you for the heads-up! (you already copied Nicole so I won't again ....)
Oh okay, sorry @bpdulog thank you for the heads-up! (you already copied Nicole so I won't again ....)
Oh okay, sorry @bpdulog thank you for the heads-up! (you already copied Nicole so I won't again ....)
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9
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3. ### L1.T2.69 Sampling distribution

Hi @bpdulog the variance is (technically) the second moment about the mean (or "around the mean"); aka, the second central moment. See As the standard deviation is the square root of variance, we can say the standard deviation is a function of the second (central) moment. The question just means to query this idea: the standard deviation of a sampling distribution is called a standard error....
Hi @bpdulog the variance is (technically) the second moment about the mean (or "around the mean"); aka, the second central moment. See As the standard deviation is the square root of variance, we can say the standard deviation is a function of the second (central) moment. The question just means to query this idea: the standard deviation of a sampling distribution is called a standard error....
Hi @bpdulog the variance is (technically) the second moment about the mean (or "around the mean"); aka, the second central moment. See As the standard deviation is the square root of variance, we can say the standard deviation is a function of the second (central) moment. The question just...
Hi @bpdulog the variance is (technically) the second moment about the mean (or "around the mean"); aka, the second central moment. See As the standard deviation is the square root of variance, we...
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4. ### L1.T2.71 Central limit theorem (CLT)

Hi @bpdulog, I see this as, we are asked to calculate the probability of more than 120 loans will default but we are not given the mean value. But we have pd and n as you know we can calculate the mean=n*p=5000*2%=100 that is mean or we can say probability of default is 100. Thank you
Hi @bpdulog, I see this as, we are asked to calculate the probability of more than 120 loans will default but we are not given the mean value. But we have pd and n as you know we can calculate the mean=n*p=5000*2%=100 that is mean or we can say probability of default is 100. Thank you
Hi @bpdulog, I see this as, we are asked to calculate the probability of more than 120 loans will default but we are not given the mean value. But we have pd and n as you know we can calculate the mean=n*p=5000*2%=100 that is mean or we can say probability of default is 100. Thank you
Hi @bpdulog, I see this as, we are asked to calculate the probability of more than 120 loans will default but we are not given the mean value. But we have pd and n as you know we can calculate...
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162
5. ### P1.T2.404. Basic Statistics

Hi @theproman23 Yes, for observations i = {1 ....n), the sample variance is [Σ (Xi - µ)^2] /(N-1), where the numerator is the sum of squared differences (from the sample mean, µ, which i am here using to denote sample mean although it should be x-bar). Sample standard deviation is the square root of the sample variance. This question is testing logic against an understanding of these...
Hi @theproman23 Yes, for observations i = {1 ....n), the sample variance is [Σ (Xi - µ)^2] /(N-1), where the numerator is the sum of squared differences (from the sample mean, µ, which i am here using to denote sample mean although it should be x-bar). Sample standard deviation is the square root of the sample variance. This question is testing logic against an understanding of these...
Hi @theproman23 Yes, for observations i = {1 ....n), the sample variance is [Σ (Xi - µ)^2] /(N-1), where the numerator is the sum of squared differences (from the sample mean, µ, which i am here using to denote sample mean although it should be x-bar). Sample standard deviation is the square...
Hi @theproman23 Yes, for observations i = {1 ....n), the sample variance is [Σ (Xi - µ)^2] /(N-1), where the numerator is the sum of squared differences (from the sample mean, µ, which i am here...
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6. ### P1.T2.405. Distributions I

HI @theproman23 There is no sample so there is no standard error; question 405.1 is just asking about the properties of the given distribution. To contrast, let me ask a question that does invoke the standard error (which, in this case, is the standard deviation of a sample mean not a population). Here is the alternate question just for contrast: Assume a population with mean earnings of $2.5... HI @theproman23 There is no sample so there is no standard error; question 405.1 is just asking about the properties of the given distribution. To contrast, let me ask a question that does invoke the standard error (which, in this case, is the standard deviation of a sample mean not a population). Here is the alternate question just for contrast: Assume a population with mean earnings of$2.5...
HI @theproman23 There is no sample so there is no standard error; question 405.1 is just asking about the properties of the given distribution. To contrast, let me ask a question that does invoke the standard error (which, in this case, is the standard deviation of a sample mean not a...
HI @theproman23 There is no sample so there is no standard error; question 405.1 is just asking about the properties of the given distribution. To contrast, let me ask a question that does invoke...
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7. ### P1.T2.406. Distributions II

Hi @fjc120 F-distiribution is on page 60 of P1.T2. Miler (see below). It's also in the Miller reading, although personally I do not find Miller's explanation awesomely helpful. See also ; i.e., we just take the ratio of the two sample variances, and this F-ratio (aka, variance ratio) is used to test the null hypothesis that the (underlying population) variances are equal. If the population...
Hi @fjc120 F-distiribution is on page 60 of P1.T2. Miler (see below). It's also in the Miller reading, although personally I do not find Miller's explanation awesomely helpful. See also ; i.e., we just take the ratio of the two sample variances, and this F-ratio (aka, variance ratio) is used to test the null hypothesis that the (underlying population) variances are equal. If the population...
Hi @fjc120 F-distiribution is on page 60 of P1.T2. Miler (see below). It's also in the Miller reading, although personally I do not find Miller's explanation awesomely helpful. See also ; i.e., we just take the ratio of the two sample variances, and this F-ratio (aka, variance ratio) is used to...
Hi @fjc120 F-distiribution is on page 60 of P1.T2. Miler (see below). It's also in the Miller reading, although personally I do not find Miller's explanation awesomely helpful. See also ; i.e.,...
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8. ### L1.T2.62 Expectation & variance of variable

Hi @bpdulog (b) is a constant, given the domain it must be 0.10, such that the function is: f(x) = 0.1*x over [0, 1, 2, 3, 4]. So f(1) = 0.1, f(2) = 0.2, f(3) = 0.3 and f(4) = 0.4; these sum to 1.0 per the definition of a probability (the requirement is how we solved for constant b). Separately, we can ask, what is the mean of this function? The answer to that question is the sum of X*f(x),...
Hi @bpdulog (b) is a constant, given the domain it must be 0.10, such that the function is: f(x) = 0.1*x over [0, 1, 2, 3, 4]. So f(1) = 0.1, f(2) = 0.2, f(3) = 0.3 and f(4) = 0.4; these sum to 1.0 per the definition of a probability (the requirement is how we solved for constant b). Separately, we can ask, what is the mean of this function? The answer to that question is the sum of X*f(x),...
Hi @bpdulog (b) is a constant, given the domain it must be 0.10, such that the function is: f(x) = 0.1*x over [0, 1, 2, 3, 4]. So f(1) = 0.1, f(2) = 0.2, f(3) = 0.3 and f(4) = 0.4; these sum to 1.0 per the definition of a probability (the requirement is how we solved for constant b). Separately,...
Hi @bpdulog (b) is a constant, given the domain it must be 0.10, such that the function is: f(x) = 0.1*x over [0, 1, 2, 3, 4]. So f(1) = 0.1, f(2) = 0.2, f(3) = 0.3 and f(4) = 0.4; these sum to...
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9. ### P1.T2.503. One-factor model (Hull)

Hi @bpdulog I apologize for the confusion, I inadvertently posted from the wrong Hull text such that I posted the Factor model (which applies to questions 503.2 and 503.3, while 503.1 refers to correlated normal random variables). They are essentially similar but there is a difference. Superficially, the factor model generates a vector (i.e., a single column) of random standard normals U(1),...
Hi @bpdulog I apologize for the confusion, I inadvertently posted from the wrong Hull text such that I posted the Factor model (which applies to questions 503.2 and 503.3, while 503.1 refers to correlated normal random variables). They are essentially similar but there is a difference. Superficially, the factor model generates a vector (i.e., a single column) of random standard normals U(1),...
Hi @bpdulog I apologize for the confusion, I inadvertently posted from the wrong Hull text such that I posted the Factor model (which applies to questions 503.2 and 503.3, while 503.1 refers to correlated normal random variables). They are essentially similar but there is a difference....
Hi @bpdulog I apologize for the confusion, I inadvertently posted from the wrong Hull text such that I posted the Factor model (which applies to questions 503.2 and 503.3, while 503.1 refers to...
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10. ### L1.T2.86 OLS Regression

I was barking up wrong tree as I assumed I had to include the sample size of 10 in my calculations. Thanks
I was barking up wrong tree as I assumed I had to include the sample size of 10 in my calculations. Thanks
I was barking up wrong tree as I assumed I had to include the sample size of 10 in my calculations. Thanks
I was barking up wrong tree as I assumed I had to include the sample size of 10 in my calculations. Thanks
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11. ### L1.T2.83 Regression function

Please see An estimator is basically a statistic as opposed to a parameter (although if we want to get technical, here is ). As such, to my knowledge, a tell-tale sign of an estimator would be that it varies (even if slightly) with each different sample. In the case of a SRF, each time we regress based on a different draw of observations (from the same population), the slope coefficient...
Please see An estimator is basically a statistic as opposed to a parameter (although if we want to get technical, here is ). As such, to my knowledge, a tell-tale sign of an estimator would be that it varies (even if slightly) with each different sample. In the case of a SRF, each time we regress based on a different draw of observations (from the same population), the slope coefficient...
Please see An estimator is basically a statistic as opposed to a parameter (although if we want to get technical, here is ). As such, to my knowledge, a tell-tale sign of an estimator would be that it varies (even if slightly) with each different sample. In the case of a SRF, each time we...
Please see An estimator is basically a statistic as opposed to a parameter (although if we want to get technical, here is ). As such, to my knowledge, a tell-tale sign of an estimator would be...
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12. ### P1.T2.304. Covariance (Miller)

Hi @bpdulog You are referring to Covariance Method 1 (see above in the answer). The E[X] = -3.0%*30% + 1.0%*50% + 5.0%*20% = -0.900% + 0.500% + 1.000% [see fourth column above, under Exp Value X] = 0.600% is the expected value of X because it's just a weighted average. Then see under Covariance Method 1, the first row is given by (-3.0% - 0.6000%)*(-2.0% - 1.000%)*30% = 0.0324%; i.e., (Xi -...
Hi @bpdulog You are referring to Covariance Method 1 (see above in the answer). The E[X] = -3.0%*30% + 1.0%*50% + 5.0%*20% = -0.900% + 0.500% + 1.000% [see fourth column above, under Exp Value X] = 0.600% is the expected value of X because it's just a weighted average. Then see under Covariance Method 1, the first row is given by (-3.0% - 0.6000%)*(-2.0% - 1.000%)*30% = 0.0324%; i.e., (Xi -...
Hi @bpdulog You are referring to Covariance Method 1 (see above in the answer). The E[X] = -3.0%*30% + 1.0%*50% + 5.0%*20% = -0.900% + 0.500% + 1.000% [see fourth column above, under Exp Value X] = 0.600% is the expected value of X because it's just a weighted average. Then see under Covariance...
Hi @bpdulog You are referring to Covariance Method 1 (see above in the answer). The E[X] = -3.0%*30% + 1.0%*50% + 5.0%*20% = -0.900% + 0.500% + 1.000% [see fourth column above, under Exp Value X]...
Fran ... 2
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13. ### P1.T2.303 Mean and variance of continuous probability density functions (pdf)

Hi @bpdulog Because several candidates were asking, last month we asked Bill May, SVP at GARP, the following question. My question and his response: Response from Bill May (March 8, 2016):
Hi @bpdulog Because several candidates were asking, last month we asked Bill May, SVP at GARP, the following question. My question and his response: Response from Bill May (March 8, 2016):
Hi @bpdulog Because several candidates were asking, last month we asked Bill May, SVP at GARP, the following question. My question and his response: Response from Bill May (March 8, 2016):
Hi @bpdulog Because several candidates were asking, last month we asked Bill May, SVP at GARP, the following question. My question and his response: Response from Bill May (March 8, 2016):
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14. ### P1.T2.203. Skew and kurtosis (Stock & Watson)

Hi @bpdulog of course, many distributions can have a mean of zero (e.g., student's t). The normal is but one of many.
Hi @bpdulog of course, many distributions can have a mean of zero (e.g., student's t). The normal is but one of many.
Hi @bpdulog of course, many distributions can have a mean of zero (e.g., student's t). The normal is but one of many.
Hi @bpdulog of course, many distributions can have a mean of zero (e.g., student's t). The normal is but one of many.
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15. ### L1.T2.110 Rachev's distributions

Your question raised a good point. Wasn't a bad question at all
Your question raised a good point. Wasn't a bad question at all
Your question raised a good point. Wasn't a bad question at all
Your question raised a good point. Wasn't a bad question at all
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16. ### L1.T2.73 Chi-square distribution

Thank you @ami44 ! Terrific answers, I agree wholeheartedly with your replies to both 73.2 and 73.4!
Thank you @ami44 ! Terrific answers, I agree wholeheartedly with your replies to both 73.2 and 73.4!
Thank you @ami44 ! Terrific answers, I agree wholeheartedly with your replies to both 73.2 and 73.4!
Thank you @ami44 ! Terrific answers, I agree wholeheartedly with your replies to both 73.2 and 73.4!
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17. ### L1.T2.80 Confidence intervals

Thank you @Deepak Chitnis ! It's true, [USER=38486]@ : for the test of the any individual slope coefficient in a regression, we want a student's t where (n - variables) = df, including the dependent variable. This is because it's actually the estimated coefficients that determine the df: the intercept plus the (partial) slope coefficients. In the case of y = α+β*x, as per Q80.3 above, df = n-2...
Thank you @Deepak Chitnis ! It's true, [USER=38486]@ : for the test of the any individual slope coefficient in a regression, we want a student's t where (n - variables) = df, including the dependent variable. This is because it's actually the estimated coefficients that determine the df: the intercept plus the (partial) slope coefficients. In the case of y = α+β*x, as per Q80.3 above, df = n-2...
Thank you @Deepak Chitnis ! It's true, [USER=38486]@ : for the test of the any individual slope coefficient in a regression, we want a student's t where (n - variables) = df, including the dependent variable. This is because it's actually the estimated coefficients that determine the df: the...
Thank you @Deepak Chitnis ! It's true, [USER=38486]@ : for the test of the any individual slope coefficient in a regression, we want a student's t where (n - variables) = df, including the...
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18. ### L1.T2.76 Critical t-values

Thank you [USER=38486]@ I suppose another way to look at this is from the perspective that the student's t distribution converges toward the normal distribution with variance = df/(df-2), see Therefore, as sample size or df increase (where df = sample - 1, in the case of a sample mean), the variance is decreasing. Indeed, the way that I think of this is, I visualize the student's t lookup...
Thank you [USER=38486]@ I suppose another way to look at this is from the perspective that the student's t distribution converges toward the normal distribution with variance = df/(df-2), see Therefore, as sample size or df increase (where df = sample - 1, in the case of a sample mean), the variance is decreasing. Indeed, the way that I think of this is, I visualize the student's t lookup...
Thank you [USER=38486]@ I suppose another way to look at this is from the perspective that the student's t distribution converges toward the normal distribution with variance = df/(df-2), see Therefore, as sample size or df increase (where df = sample - 1, in the case of a sample mean), the...
Thank you [USER=38486]@ I suppose another way to look at this is from the perspective that the student's t distribution converges toward the normal distribution with variance = df/(df-2), see ...
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19. ### P1.T2.305. Minimum variance hedge (Miller)

Thanks David! I did just read that update...fortunately calc is slowly coming back to me with each example. Based on Bill's comments it would seem some other prep providers I have had access to may be light in this area. Thanks for the deep dive on the great, albeit challenging, questions!
Thanks David! I did just read that update...fortunately calc is slowly coming back to me with each example. Based on Bill's comments it would seem some other prep providers I have had access to may be light in this area. Thanks for the deep dive on the great, albeit challenging, questions!
Thanks David! I did just read that update...fortunately calc is slowly coming back to me with each example. Based on Bill's comments it would seem some other prep providers I have had access to may be light in this area. Thanks for the deep dive on the great, albeit challenging, questions!
Thanks David! I did just read that update...fortunately calc is slowly coming back to me with each example. Based on Bill's comments it would seem some other prep providers I have had access to...
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20. ### P1.T2.308. Coskewness and cokurtosis

Thank you [USER=38486]@ Fixed above.
Thank you [USER=38486]@ Fixed above.
Thank you [USER=38486]@ Fixed above.
Thank you [USER=38486]@ Fixed above.
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