Test your basic knowledge |

FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

Instructions:

Answer 50 questions in 15 minutes.
If you are not ready to take this test, you can study here.
Match each statement with the correct term.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. SER
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Create covariance matrix - Covariance matrix (R) is decomposed into lower - triangle matrix (L) and upper - triangle matrix (U) - are mirrors of each other - R=LU - solve for all matrix elements - LU is the result and is used to simulate vendor varia
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

2. Standard normal distribution
Variance(x)
Transformed to a unit variable - Mean = 0 Variance = 1
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

3. Variance of weighted scheme
In EWMA - the lambda parameter - In GARCH(1 -1) - sum of alpha and beta - Higher persistence implies slow decay toward the long - run average variance
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Regression can be non - linear in variables but must be linear in parameters

4. Gamma distribution
Sampling distribution of sample means tend to be normal
F(x) = (1/(beta tao(alpha)) e^( - x/beta) * (x/beta)^(alpha - 1) - Alpha = 1 - becomes exponential - Alpha = k/2 beta = 2 - becomes chi - squared
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Variance reverts to a long run level

5. Variance of X+Y assuming dependence
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Variance(x) + Variance(Y) + 2*covariance(XY)
Peaks over threshold - Collects dataset in excess of some threshold
Choose parameters that maximize the likelihood of what observations occurring

6. Standard error
Create covariance matrix - Covariance matrix (R) is decomposed into lower - triangle matrix (L) and upper - triangle matrix (U) - are mirrors of each other - R=LU - solve for all matrix elements - LU is the result and is used to simulate vendor varia
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

7. Variance(discrete)
Confidence level
Contains variables not explicit in model - Accounts for randomness
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

8. Standard variable for non - normal distributions
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Least absolute deviations estimator - used when extreme outliers are not uncommon
Z = (Y - meany)/(stddev(y)/sqrt(n))
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

9. Reliability
Statement of the error or precision of an estimate
Contains variables not explicit in model - Accounts for randomness
F(x) = (1/stddev(x)sqrt(2pi))e^ - (x - mean)^2/(2variance) - skew = 0 - Parsimony = only requires mean and variance - Summation stability = combination of two normal distributions is a normal distribution - Kurtosis = 3
Variance(y)/n = variance of sample Y

10. Expected future variance rate (t periods forward)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Confidence set for two coefficients - two dimensional analog for the confidence interval
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Average return across assets on a given day

11. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
We reject a hypothesis that is actually true
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Among all unbiased estimators - estimator with the smallest variance is efficient

12. Exact significance level
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
P - value
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

13. Variance of sampling distribution of means when n<N
Mean = np - Variance = npq - Std dev = sqrt(npq)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

14. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Observe sample variance and compare it to hypothetical population variance (sample variance/population variance)(n - 1) = chi - squared - Non - negative and skewed right - approaches zero as n increases - mean = k where k = degrees of freedom - Varia
Generalized Auto Regressive Conditional Heteroscedasticity model - GARCH(1 -1) is the weighted sum of a long term variance (weight=gamma) - the most recent squared return (weight=alpha) and the most recent variance (weight=beta)

15. Empirical frequency
Sample mean +/ - t*(stddev(s)/sqrt(n))
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Var(X) + Var(Y)
Based on a dataset

16. Consistent
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Has heavy tails
Contains variables not explicit in model - Accounts for randomness
When the sample size is large - the uncertainty about the value of the sample is very small

17. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
In EWMA - the lambda parameter - In GARCH(1 -1) - sum of alpha and beta - Higher persistence implies slow decay toward the long - run average variance
E(XY) - E(X)E(Y)
Random walk (usually acceptable) - Constant volatility (unlikely)

18. Variance of sample mean
Contains variables not explicit in model - Accounts for randomness
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Only requires two parameters = mean and variance
Variance(y)/n = variance of sample Y

19. Extending the HS approach for computing value of a portfolio

Warning: Invalid argument supplied for foreach() in /var/www/html/basicversity.com/show_quiz.php on line 183

20. Binomial distribution
Normal - Student's T - Chi - square - F distribution
Based on a dataset
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Sum of n i.i.d. Bernouli variables - Probability of k successes: (combination n over k)(p^k)(1 - p)^(n - k) - (n over k) = (n!)/((n - k)!k!)

21. Logistic distribution
Has heavy tails
Variance(y)/n = variance of sample Y
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Create covariance matrix - Covariance matrix (R) is decomposed into lower - triangle matrix (L) and upper - triangle matrix (U) - are mirrors of each other - R=LU - solve for all matrix elements - LU is the result and is used to simulate vendor varia

22. Discrete random variable
Contains variables not explicit in model - Accounts for randomness
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

23. Poisson Distribution
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

24. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Has heavy tails
When asset return(r) is normally distributed - the continuously compounded future asset price level is lognormal - Reverse is true - if a variable is lognormal - its natural log is normal
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

25. Least squares estimator(m)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

26. ESS
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
When the sample size is large - the uncertainty about the value of the sample is very small
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Combine to form distribution with leptokurtosis (heavy tails)

27. Tractable
Based on a dataset
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Easy to manipulate

28. Shortcomings of implied volatility
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Model dependent - Options with the same underlying assets may trade at different volatilities
Confidence set for two coefficients - two dimensional analog for the confidence interval
Sampling distribution of sample means tend to be normal

29. Monte Carlo Simulations
E(mean) = mean
F(x) = (1/stddev(x)sqrt(2pi))e^ - (x - mean)^2/(2variance) - skew = 0 - Parsimony = only requires mean and variance - Summation stability = combination of two normal distributions is a normal distribution - Kurtosis = 3
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Expected value of the sample mean is the population mean

30. Statistical (or empirical) model
Random walk (usually acceptable) - Constant volatility (unlikely)
Yi = B0 + B1Xi + ui
Regression can be non - linear in variables but must be linear in parameters
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

31. Bernouli Distribution
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
(a^2)(variance(x)) + (b^2)(variance(y))
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Distribution with only two possible outcomes

32. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
For n>30 - sample mean is approximately normal
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Regression can be non - linear in variables but must be linear in parameters

33. GARCH
Generalized Auto Regressive Conditional Heteroscedasticity model - GARCH(1 -1) is the weighted sum of a long term variance (weight=gamma) - the most recent squared return (weight=alpha) and the most recent variance (weight=beta)
Use historical simulation approach but use the EWMA weighting system
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

34. Unstable return distribution
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

35. Square root rule
Returns over time for an individual asset
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
95% = 1.65 99% = 2.33 For one - tailed tests

36. Key properties of linear regression
Mean of sampling distribution is the population mean
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Regression can be non - linear in variables but must be linear in parameters
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

37. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

38. Biggest (and only real) drawback of GARCH mode
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
E(XY) - E(X)E(Y)
Nonlinearity

39. Sample mean
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Expected value of the sample mean is the population mean
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors

40. Lognormal
When one regressor is a perfect linear function of the other regressors
When asset return(r) is normally distributed - the continuously compounded future asset price level is lognormal - Reverse is true - if a variable is lognormal - its natural log is normal
Summation((xi - mean)^k)/n
F(x) = (1/(beta tao(alpha)) e^( - x/beta) * (x/beta)^(alpha - 1) - Alpha = 1 - becomes exponential - Alpha = k/2 beta = 2 - becomes chi - squared

41. Critical z values
Only requires two parameters = mean and variance
95% = 1.65 99% = 2.33 For one - tailed tests
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Var(X) + Var(Y)

42. Difference between population and sample variance
Expected value of the sample mean is the population mean
i = ln(Si/Si - 1)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Population denominator = n - Sample denominator = n - 1

43. K - th moment
Summation((xi - mean)^k)/n
Peaks over threshold - Collects dataset in excess of some threshold
(a^2)(variance(x)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

44. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Peaks over threshold - Collects dataset in excess of some threshold
Sample mean will near the population mean as the sample size increases
Statement of the error or precision of an estimate

45. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Attempts to sample along more important paths
Mean = np - Variance = npq - Std dev = sqrt(npq)

46. Two assumptions of square root rule
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Random walk (usually acceptable) - Constant volatility (unlikely)

47. Mean reversion in variance
When the sample size is large - the uncertainty about the value of the sample is very small
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance reverts to a long run level

48. Implied standard deviation for options
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
95% = 1.65 99% = 2.33 For one - tailed tests

49. F distribution
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance ratio distribution F = (variance(x)/variance(y)) - Greater sample variance is numerator - Nonnegative and skewed right - Approaches normal as df increases - Square of t - distribution has a F distribution with 1 -k df - M*F(m -n) = Chi - s
Confidence level

50. Significance =1
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Concerned with a single random variable (ex. Roll of a die)
Confidence level

Sorry!:) No result found.

Can you answer 50 questions in 15 minutes?

Major Subjects
Tests & Exams AP CLEP DSST GRE SAT GMAT	Certifications CISSP go to https://www.isc2.org/ PMP ITIL RHCE MCTS More...	IT Skills Android Programming Data Modeling Objective C Programming Basic Python Programming Adobe Illustrator More...	Business Skills Advertising Techniques Business Accounting Basics Business Strategy Human Resource Management Marketing Basics More...
Soft Skills Body Language People Skills Public Speaking Persuasion Job Hunting And Resumes More...	Vocabulary GRE Vocab SAT Vocab TOEFL Essential Vocab Basic English Words For All Global Words You Should Know Business English More...	Languages AP German Vocab AP Latin Vocab SAT Subject Test: French Italian Survival Norwegian Survival More...	Engineering Audio Engineering Computer Science Engineering Aerospace Engineering Chemical Engineering Structural Engineering More...
Health Sciences Basic Nursing Skills Health Science Language Fundamentals Veterinary Technology Medical Language Cardiology Clinical Surgery More...	English Grammar Fundamentals Literary And Rhetorical Vocab Elements Of Style Vocab Introduction To English Major Complete Advanced Sentences Literature Homonyms More...	Math Algebra Formulas Basic Arithmetic: Measurements Metric Conversions Geometric Properties Important Math Facts Number Sense Vocab Business Math More...	Other Major Subjects Science Economics History Law Performing-arts Cooking Logic & Reasoning Trivia

Test your basic knowledge |

FRM Foundations Of Risk Management Quantitative Methods

Sorry!:) No result found.

Can you answer 50 questions in 15 minutes?

Let me suggest you:

Browse all subjects

Browse all tests

Most popular tests

Major Subjects

Tests & Exams

Certifications

IT Skills

Business Skills

Soft Skills

Vocabulary

Languages

Engineering

Health Sciences

English

Math

Other Major Subjects

Browse all subjects

Browse all tests

Most popular tests