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. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Statement of the error or precision of an estimate
Normal - Student's T - Chi - square - F distribution
Variance(X) + Variance(Y) - 2*covariance(XY)

2. Unbiased
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
(a^2)(variance(x)
Application of mathematical statistics to economic data to lend empirical support to models
Mean of sampling distribution is the population mean

3. Consistent
Application of mathematical statistics to economic data to lend empirical support to models
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
When the sample size is large - the uncertainty about the value of the sample is very small
We reject a hypothesis that is actually true

4. Joint probability functions
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Among all unbiased estimators - estimator with the smallest variance is efficient
Probability that the random variables take on certain values simultaneously
E(XY) - E(X)E(Y)

5. Key properties of linear regression
Among all unbiased estimators - estimator with the smallest variance is efficient
Rxy = Sxy/(Sx*Sy)
Regression can be non - linear in variables but must be linear in parameters
Has heavy tails

6. Perfect multicollinearity
Random walk (usually acceptable) - Constant volatility (unlikely)
When one regressor is a perfect linear function of the other regressors
Only requires two parameters = mean and variance
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

7. Variance of aX
(a^2)(variance(x)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

8. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Price/return tends to run towards a long - run level

9. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Contains variables not explicit in model - Accounts for randomness
Independently and Identically Distributed
Variance reverts to a long run level

10. Maximum likelihood method
Use historical simulation approach but use the EWMA weighting system
Distribution with only two possible outcomes
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
Choose parameters that maximize the likelihood of what observations occurring

11. Multivariate Density Estimation (MDE)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

12. Variance of X - Y assuming dependence
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Variance(X) + Variance(Y) - 2*covariance(XY)

13. Standard variable for non - normal distributions
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Z = (Y - meany)/(stddev(y)/sqrt(n))

14. Unstable return distribution
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

15. Overall F - statistic
SSR
E(mean) = mean
Mean of sampling distribution is the population mean
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

16. Hybrid method for conditional volatility
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Based on an equation - P(A) = # of A/total outcomes
Use historical simulation approach but use the EWMA weighting system

17. Multivariate probability
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Sample mean +/ - t*(stddev(s)/sqrt(n))
More than one random variable

18. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

19. Confidence ellipse
Summation((xi - mean)^k)/n
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Confidence set for two coefficients - two dimensional analog for the confidence interval

20. Bernouli Distribution
Distribution with only two possible outcomes
Returns over time for an individual asset
Random walk (usually acceptable) - Constant volatility (unlikely)
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

21. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Distribution with only two possible outcomes
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

22. Mean reversion
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Based on a dataset

23. Monte Carlo Simulations
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Attempts to sample along more important paths
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

24. R^2
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Easy to manipulate
Only requires two parameters = mean and variance

25. Homoskedastic
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
Attempts to sample along more important paths
Variance(x) + Variance(Y) + 2*covariance(XY)
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

26. Conditional probability functions
Average return across assets on a given day
Based on a dataset
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

27. LFHS
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Low Frequency - High Severity events

28. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Easy to manipulate
Least absolute deviations estimator - used when extreme outliers are not uncommon
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

29. Result of combination of two normal with same means
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Combine to form distribution with leptokurtosis (heavy tails)
Summation((xi - mean)^k)/n
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

30. Variance - covariance approach for VaR of a portfolio
E(mean) = mean
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Distribution with only two possible outcomes
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

31. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
Variance(x)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

32. Reliability
Statement of the error or precision of an estimate
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Returns over time for an individual asset
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

33. Persistence
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
We reject a hypothesis that is actually true
Sampling distribution of sample means tend to be normal
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

34. ESS
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
SSR
P - value

35. SER
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Average return across assets on a given day
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

36. Variance of X+Y
Has heavy tails
We reject a hypothesis that is actually true
Var(X) + Var(Y)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

37. Two ways to calculate historical volatility
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
P(Z>t)
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

38. Econometrics
Mean of sampling distribution is the population mean
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Application of mathematical statistics to economic data to lend empirical support to models

39. GEV
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Has heavy tails

40. Logistic distribution
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Has heavy tails
Variance(x) + Variance(Y) + 2*covariance(XY)

41. i.i.d.
Regression can be non - linear in variables but must be linear in parameters
SSR
Does not depend on a prior event or information
Independently and Identically Distributed

42. Statistical (or empirical) model
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
P - value
Yi = B0 + B1Xi + ui

43. 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

44. Exponential distribution
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Summation((xi - mean)^k)/n
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

45. Least squares estimator(m)
P(Z>t)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Low Frequency - High Severity events

46. Two requirements of OVB
Concerned with a single random variable (ex. Roll of a die)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of 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

47. Cross - sectional
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Average return across assets on a given day
Has heavy tails

48. Deterministic Simulation
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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
Variance = (1/m) summation(u<n - i>^2)
Flexible and postulate stochastic process or resample historical data - Full valuation on target date - More prone to model risk - Slow and loses precision due to sampling variation

49. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Only requires two parameters = mean and variance
When one regressor is a perfect linear function of the other regressors
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

50. Economical(elegant)
Only requires two parameters = mean and variance
Variance reverts to a long run level
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)