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. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
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
Use historical simulation approach but use the EWMA weighting system
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

2. Marginal unconditional probability function
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Distribution with only two possible outcomes
Does not depend on a prior event or information
Confidence set for two coefficients - two dimensional analog for the confidence interval

3. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Refers to whether distribution is symmetrical - Sigma^3 = E[(x - mean)^3]/sigma^3 - Positive skew = mean>median>mode - Negative skew = mean<median<mode - if zero - all are equal - Function of the third moment
P - value

4. Central Limit Theorem(CLT)
Yi = B0 + B1Xi + ui
Sampling distribution of sample means tend to be normal
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Low Frequency - High Severity events

5. Continuously compounded return equation
Variance(x) + Variance(Y) + 2*covariance(XY)
i = ln(Si/Si - 1)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
E(XY) - E(X)E(Y)

6. Variance of X - Y assuming dependence
Variance(y)/n = variance of sample Y
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance(X) + Variance(Y) - 2*covariance(XY)
Transformed to a unit variable - Mean = 0 Variance = 1

7. Cholesky factorization (decomposition)
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
Summation((xi - mean)^k)/n
Regression can be non - linear in variables but must be linear in parameters
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

8. Sample variance
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Sample mean will near the population mean as the sample size increases
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

9. GEV
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Variance(x) + Variance(Y) + 2*covariance(XY)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

10. Perfect multicollinearity
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
When one regressor is a perfect linear function of the other regressors
Choose parameters that maximize the likelihood of what observations occurring

11. Chi - squared distribution
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
Probability that the random variables take on certain values simultaneously
Mean = np - Variance = npq - Std dev = sqrt(npq)
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

12. Discrete representation of the GBM
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

13. Covariance
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
E(XY) - E(X)E(Y)

14. ESS
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 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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

15. Gamma distribution
Low Frequency - High Severity events
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
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

16. GARCH
Mean of sampling distribution is the population mean
Contains variables not explicit in model - Accounts for randomness
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
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)

17. Covariance calculations using weight sums (lambda)
Variance of conditional distribution of u(i) is constant - T - stat for slope of regression T = (b1 - beta)/SE(b1) - beta is a specified value for hypothesis test
Adjusted R^2 does not increase from addition of new independent variables -Adjusted R^2 = 1 - (n - 1)/(n - k - 1) * (SSR/TSS) = 1 - su^2/sy^2
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

18. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
OLS estimators are unbiased - consistent - and normal regardless of homo or heterskedasticity - OLS estimates are efficient - Can use homoscedasticity - only variance formula - OLS is BLUE
Sample mean +/ - t*(stddev(s)/sqrt(n))

19. Binomial distribution
Price/return tends to run towards a long - run level
Sample mean will near the population mean as the sample size increases
Confidence level
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!)

20. Sample correlation
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Confidence set for two coefficients - two dimensional analog for the confidence interval
Rxy = Sxy/(Sx*Sy)
Instead of independent samples - systematically fills space left by previous numbers in the series - Std error shrinks at 1/k instead of 1/sqrt(k) but accuracy determination is hard since variables are not independent

21. Time series data
Returns over time for an individual asset
Standard error of the regression - SER = sqrt(SSR/(n - 2)) = sqrt((ei^2)/(n - 2)) SSR - Sum of squared residuals - Summation[(Yi - predicted Yi)^2] - Summation of each squared deviation between the actual Y and the predicted Y - Directly related
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

22. Variance of sample mean
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Variance(y)/n = variance of sample Y
Mean of sampling distribution is the population mean

23. F distribution
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
Variance of conditional distribution of u(i) is constant - T - stat for slope of regression T = (b1 - beta)/SE(b1) - beta is a specified value for hypothesis test

24. Deterministic Simulation
Price/return tends to run towards a long - run level
Mean = np - Variance = npq - Std dev = sqrt(npq)
Application of mathematical statistics to economic data to lend empirical support to models
Instead of independent samples - systematically fills space left by previous numbers in the series - Std error shrinks at 1/k instead of 1/sqrt(k) but accuracy determination is hard since variables are not independent

25. POT
Independently and Identically Distributed
Adjusted R^2 does not increase from addition of new independent variables -Adjusted R^2 = 1 - (n - 1)/(n - k - 1) * (SSR/TSS) = 1 - su^2/sy^2
Z = (Y - meany)/(stddev(y)/sqrt(n))
Peaks over threshold - Collects dataset in excess of some threshold

26. Heteroskedastic
Yi = B0 + B1Xi + ui
If variance of the conditional distribution of u(i) is not constant
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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

27. SER
Distribution with only two possible outcomes
Variance(x)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

28. SER
Confidence set for two coefficients - two dimensional analog for the confidence interval
Attempts to sample along more important paths
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Standard error of the regression - SER = sqrt(SSR/(n - 2)) = sqrt((ei^2)/(n - 2)) SSR - Sum of squared residuals - Summation[(Yi - predicted Yi)^2] - Summation of each squared deviation between the actual Y and the predicted Y - Directly related

29. Discrete random variable
Attempts to sample along more important paths
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance(x) + Variance(Y) + 2*covariance(XY)

30. Variance of aX + bY
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
(a^2)(variance(x)) + (b^2)(variance(y))
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

31. Variance of aX
(a^2)(variance(x)
Concerned with a single random variable (ex. Roll of a die)
Variance reverts to a long run level
Only requires two parameters = mean and variance

32. Biggest (and only real) drawback of GARCH mode
Regression can be non - linear in variables but must be linear in parameters
Yi = B0 + B1Xi + ui
Nonlinearity
If variance of the conditional distribution of u(i) is not constant

33. Poisson distribution equations for mean variance and std deviation
Based on an equation - P(A) = # of A/total outcomes
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Normal - Student's T - Chi - square - F distribution

34. 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

35. Two drawbacks of moving average series
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
E(XY) - E(X)E(Y)

36. Adjusted R^2
If variance of the conditional distribution of u(i) is not constant
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Adjusted R^2 does not increase from addition of new independent variables -Adjusted R^2 = 1 - (n - 1)/(n - k - 1) * (SSR/TSS) = 1 - su^2/sy^2
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

37. Econometrics
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Application of mathematical statistics to economic data to lend empirical support to models
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
Does not depend on a prior event or information

38. EWMA
Variance of conditional distribution of u(i) is constant - T - stat for slope of regression T = (b1 - beta)/SE(b1) - beta is a specified value for hypothesis test
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Confidence level

39. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
When the sample size is large - the uncertainty about the value of the sample is very small
Expected value of the sample mean is the population mean
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

40. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Normal - Student's T - Chi - square - F distribution
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
(a^2)(variance(x)

41. Bernouli Distribution
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Low Frequency - High Severity events
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Distribution with only two possible outcomes

42. Two requirements of OVB
Normal - Student's T - Chi - square - F distribution
OLS estimators are unbiased - consistent - and normal regardless of homo or heterskedasticity - OLS estimates are efficient - Can use homoscedasticity - only variance formula - OLS is BLUE
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Transformed to a unit variable - Mean = 0 Variance = 1

43. Skewness
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
Refers to whether distribution is symmetrical - Sigma^3 = E[(x - mean)^3]/sigma^3 - Positive skew = mean>median>mode - Negative skew = mean<median<mode - if zero - all are equal - Function of the third moment
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Instead of independent samples - systematically fills space left by previous numbers in the series - Std error shrinks at 1/k instead of 1/sqrt(k) but accuracy determination is hard since variables are not independent

44. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Rxy = Sxy/(Sx*Sy)

45. Regime - switching volatility model
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Statement of the error or precision of an estimate

46. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Application of mathematical statistics to economic data to lend empirical support to models
Nonlinearity

47. Binomial distribution equations for mean variance and std dev
Attempts to sample along more important paths
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

48. Confidence interval (from t)
Special type of pooled data in which the cross sectional unit is surveyed over time
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Nonlinearity
Sample mean +/ - t*(stddev(s)/sqrt(n))

49. What does the OLS minimize?
SSR
Confidence set for two coefficients - two dimensional analog for the confidence interval
Z = (Y - meany)/(stddev(y)/sqrt(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

50. Continuous random variable
Probability that the random variables take on certain values simultaneously
(a^2)(variance(x)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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