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. Confidence ellipse
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Returns over time for an individual asset
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

2. Regime - switching volatility model
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
dS<t> = (mean<t>)(S<t>)dt + stddev(t)S<t>dt- GBM - Geometric Brownian Motion - Represented as drift + shock - Drift = mean * change in time - Shock = std dev E sqrt(change in time)

3. Implied standard deviation for options
Statement of the error or precision of an estimate
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

4. Importance sampling technique
Contains variables not explicit in model - Accounts for randomness
Mean of sampling distribution is the population mean
Attempts to sample along more important paths
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

5. 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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Summation((xi - mean)^k)/n
95% = 1.65 99% = 2.33 For one - tailed tests

6. 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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
When the sample size is large - the uncertainty about the value of the sample is very small
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

7. GPD
Among all unbiased estimators - estimator with the smallest variance is efficient
Price/return tends to run towards a long - run level
Special type of pooled data in which the cross sectional unit is surveyed over time
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

8. Adjusted R^2
Random walk (usually acceptable) - Constant volatility (unlikely)
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
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

9. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Confidence set for two coefficients - two dimensional analog for the confidence interval
(a^2)(variance(x)) + (b^2)(variance(y))
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

10. Variance of aX + bY
When the sample size is large - the uncertainty about the value of the sample is very small
P(Z>t)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
(a^2)(variance(x)) + (b^2)(variance(y))

11. Bernouli Distribution
dS<t> = (mean<t>)(S<t>)dt + stddev(t)S<t>dt- GBM - Geometric Brownian Motion - Represented as drift + shock - Drift = mean * change in time - Shock = std dev E sqrt(change in time)
Distribution with only two possible outcomes
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

12. Reliability
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Statement of the error or precision of an estimate
(a^2)(variance(x)) + (b^2)(variance(y))
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

13. Type II Error
Variance(x)
Nonlinearity
Confidence level
We accept a hypothesis that should have been rejected

14. Logistic distribution
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Has heavy tails
Combine to form distribution with leptokurtosis (heavy tails)
Low Frequency - High Severity events

15. ESS
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

16. Marginal unconditional probability function
Variance(x) + Variance(Y) + 2*covariance(XY)
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
Low Frequency - High Severity events
Does not depend on a prior event or information

17. F distribution
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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
dS<t> = (mean<t>)(S<t>)dt + stddev(t)S<t>dt- GBM - Geometric Brownian Motion - Represented as drift + shock - Drift = mean * change in time - Shock = std dev E sqrt(change in time)
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

18. T distribution
Easy to manipulate
Yi = B0 + B1Xi + ui
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

19. Block maxima
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Concerned with a single random variable (ex. Roll of a die)
Variance(y)/n = variance of sample Y
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

20. Covariance calculations using weight sums (lambda)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
We accept a hypothesis that should have been rejected
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

21. Continuous representation of the GBM
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
dS<t> = (mean<t>)(S<t>)dt + stddev(t)S<t>dt- GBM - Geometric Brownian Motion - Represented as drift + shock - Drift = mean * change in time - Shock = std dev E sqrt(change in time)
Rxy = Sxy/(Sx*Sy)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

22. Deterministic Simulation
Variance(x) + Variance(Y) + 2*covariance(XY)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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 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

23. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
95% = 1.65 99% = 2.33 For one - tailed tests
Random walk (usually acceptable) - Constant volatility (unlikely)
Among all unbiased estimators - estimator with the smallest variance is efficient

24. Hazard rate of exponentially distributed random variable
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)
Based on an equation - P(A) = # of A/total outcomes
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

25. GEV
Variance reverts to a long run level
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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

26. Time series data
Normal - Student's T - Chi - square - F distribution
Model dependent - Options with the same underlying assets may trade at different volatilities
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Returns over time for an individual asset

27. Two requirements of OVB
Normal - Student's T - Chi - square - F distribution
Concerned with a single random variable (ex. Roll of a die)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

28. Cholesky factorization (decomposition)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
dS<t> = (mean<t>)(S<t>)dt + stddev(t)S<t>dt- GBM - Geometric Brownian Motion - Represented as drift + shock - Drift = mean * change in time - Shock = std dev E sqrt(change in time)
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

29. Mean reversion in variance
Least absolute deviations estimator - used when extreme outliers are not uncommon
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance reverts to a long run level
We accept a hypothesis that should have been rejected

30. Homoskedastic
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 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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
(a^2)(variance(x)

31. Mean reversion in asset dynamics
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Price/return tends to run towards a long - run level

32. Two assumptions of square root rule
Probability that the random variables take on certain values simultaneously
Sampling distribution of sample means tend to be normal
Random walk (usually acceptable) - Constant volatility (unlikely)
Combine to form distribution with leptokurtosis (heavy tails)

33. Simplified standard (un - weighted) variance
Distribution with only two possible outcomes
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Variance = (1/m) summation(u<n - i>^2)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

34. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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

35. Tractable
Application of mathematical statistics to economic data to lend empirical support to models
Easy to manipulate
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Based on an equation - P(A) = # of A/total outcomes

36. Four sampling distributions
P(X=x - Y=y) = P(X=x) * P(Y=y)
(a^2)(variance(x)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Normal - Student's T - Chi - square - F distribution

37. Standard error
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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Low Frequency - High Severity events

38. SER
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
We accept a hypothesis that should have been rejected
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

39. Significance =1
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Easy to manipulate
Use historical simulation approach but use the EWMA weighting system
Confidence level

40. Antithetic variable technique
Easy to manipulate
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

41. Extending the HS approach for computing value of a portfolio
Var(X) + Var(Y)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

42. Potential reasons for fat tails in return distributions
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
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

43. BLUE
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
F = ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Sampling distribution of sample means tend to be normal

44. Monte Carlo Simulations
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
(a^2)(variance(x)
Average return across assets on a given day

45. P - value
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
When one regressor is a perfect linear function of the other regressors
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
P(Z>t)

46. 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)
Transformed to a unit variable - Mean = 0 Variance = 1
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

47. K - th moment
Use historical simulation approach but use the EWMA weighting system
Summation((xi - mean)^k)/n
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance = (1/m) summation(u<n - i>^2)

48. Difference between population and sample variance
Variance reverts to a long run level
E(mean) = mean
Only requires two parameters = mean and variance
Population denominator = n - Sample denominator = n - 1

49. Implications of homoscedasticity
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Peaks over threshold - Collects dataset in excess of some threshold
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
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

50. Control variates technique
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

Test your basic knowledge |

FRM Foundations Of Risk Management Quantitative Methods

13 Skills Types | 550 Pages | Only $9.95 | PDF / EPub, Kindle Ready

Link to This Test

Related Subjects