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. Chi - squared distribution
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Sampling distribution of sample means tend to be normal

2. Unstable return distribution
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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)

3. Test for statistical independence
Only requires two parameters = mean and variance
P(X=x - Y=y) = P(X=x) * P(Y=y)
Low Frequency - High Severity events
Has heavy tails

4. Sample correlation
Rxy = Sxy/(Sx*Sy)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
(a^2)(variance(x)) + (b^2)(variance(y))
Contains variables not explicit in model - Accounts for randomness

5. SER
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
P - value
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

6. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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

7. Empirical frequency
E(mean) = mean
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
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
Based on a dataset

8. ESS
(a^2)(variance(x)) + (b^2)(variance(y))
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
P - value
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

9. GPD
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
i = ln(Si/Si - 1)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

10. Discrete random variable
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Var(X) + Var(Y)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

11. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Based on an equation - P(A) = # of A/total outcomes
Z = (Y - meany)/(stddev(y)/sqrt(n))

12. Priori (classical) probability
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Based on an equation - P(A) = # of A/total outcomes
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

13. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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

14. Monte Carlo Simulations
Application of mathematical statistics to economic data to lend empirical support to models
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
i = ln(Si/Si - 1)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

15. Logistic distribution
Only requires two parameters = mean and variance
Has heavy tails
Peaks over threshold - Collects dataset in excess of some threshold
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

16. Covariance calculations using weight sums (lambda)
Normal - Student's T - Chi - square - F distribution
Price/return tends to run towards a long - run level
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

17. Test for unbiasedness
E(mean) = mean
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

18. Limitations of R^2 (what an increase doesn't necessarily imply)

19. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
Independently and Identically Distributed
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Yi = B0 + B1Xi + ui

20. R^2
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
E(mean) = mean
Sample mean +/ - t*(stddev(s)/sqrt(n))
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

21. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Model dependent - Options with the same underlying assets may trade at different volatilities
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

22. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Only requires two parameters = mean and variance
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

23. i.i.d.
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
Regression can be non - linear in variables but must be linear in parameters
Independently and Identically Distributed
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

24. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
P(X=x - Y=y) = P(X=x) * P(Y=y)
P - value

25. Confidence interval (from t)
Confidence level
Yi = B0 + B1Xi + ui
95% = 1.65 99% = 2.33 For one - tailed tests
Sample mean +/ - t*(stddev(s)/sqrt(n))

26. Regime - switching volatility model
Summation((xi - mean)^k)/n
We accept a hypothesis that should have been rejected
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Transformed to a unit variable - Mean = 0 Variance = 1

27. Cholesky factorization (decomposition)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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
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
Concerned with a single random variable (ex. Roll of a die)

28. Perfect multicollinearity
Independently and Identically Distributed
When one regressor is a perfect linear function of the other regressors
Average return across assets on a given day
Use historical simulation approach but use the EWMA weighting system

29. Normal distribution
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
Peaks over threshold - Collects dataset in excess of some threshold
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

30. Cross - sectional
Nonlinearity
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Average return across assets on a given day

31. GEV
Application of mathematical statistics to economic data to lend empirical support to models
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
(a^2)(variance(x)) + (b^2)(variance(y))
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

32. Joint probability functions
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Probability that the random variables take on certain values simultaneously
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

33. Confidence interval for sample mean
Peaks over threshold - Collects dataset in excess of some threshold
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Based on an equation - P(A) = # of A/total outcomes
Summation((xi - mean)^k)/n

34. Econometrics
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
E(mean) = mean
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
Application of mathematical statistics to economic data to lend empirical support to models

35. Gamma distribution
E(mean) = mean
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

36. EWMA
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Returns over time for an individual asset
E(mean) = mean
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

37. Binomial distribution
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Variance(y)/n = variance of sample Y
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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!)

38. Bootstrap method
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
Special type of pooled data in which the cross sectional unit is surveyed over time
Statement of the error or precision of an estimate
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

39. Four sampling distributions

40. Covariance
E(XY) - E(X)E(Y)
Average return across assets on a given day
Among all unbiased estimators - estimator with the smallest variance is efficient
Variance(x) + Variance(Y) + 2*covariance(XY)

41. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Does not depend on a prior event or information

42. K - th moment
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
Summation((xi - mean)^k)/n
Has heavy tails
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

43. Tractable
Easy to manipulate
Application of mathematical statistics to economic data to lend empirical support to models
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
P(Z>t)

44. Efficiency
Does not depend on a prior event or information
Among all unbiased estimators - estimator with the smallest variance is efficient
Population denominator = n - Sample denominator = n - 1
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

45. Lognormal
Least absolute deviations estimator - used when extreme outliers are not uncommon
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

46. Variance - covariance approach for VaR of a portfolio
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

47. Two drawbacks of moving average series
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
SSR
When the sample size is large - the uncertainty about the value of the sample is very small
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

48. Multivariate probability
When the sample size is large - the uncertainty about the value of the sample is very small
More than one random variable
Combine to form distribution with leptokurtosis (heavy tails)
SSR

49. Sample covariance
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Mean of sampling distribution is the population mean
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

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