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. Deterministic Simulation
Combine to form distribution with leptokurtosis (heavy tails)
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

2. Sample variance
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Confidence level
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

3. BLUE
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
P(Z>t)

4. Chi - squared distribution
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
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
E(XY) - E(X)E(Y)
95% = 1.65 99% = 2.33 For one - tailed tests

5. P - value
P(Z>t)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
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

6. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
Mean of sampling distribution is the population mean
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Sample mean will near the population mean as the sample size increases

7. Continuously compounded return equation
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
i = ln(Si/Si - 1)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

8. Mean(expected value)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Expected value of the sample mean is the population mean
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

9. Standard normal distribution
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Transformed to a unit variable - Mean = 0 Variance = 1
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

10. Mean reversion
For n>30 - sample mean is approximately normal
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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

11. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Based on a dataset
Regression can be non - linear in variables but must be linear in parameters

12. Exponential distribution
Variance(y)/n = variance of sample Y
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Variance = (1/m) summation(u<n - i>^2)

13. Joint probability functions
Statement of the error or precision of an estimate
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Combine to form distribution with leptokurtosis (heavy tails)
Probability that the random variables take on certain values simultaneously

14. Exact significance level
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Summation((xi - mean)^k)/n
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
P - value

15. 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)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Price/return tends to run towards a long - run level

16. Variance of X+Y
Var(X) + Var(Y)
Variance reverts to a long run level
Low Frequency - High Severity events
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

17. Hybrid method for conditional volatility
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
Confidence level
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Use historical simulation approach but use the EWMA weighting system

18. Economical(elegant)
i = ln(Si/Si - 1)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Price/return tends to run towards a long - run level
Only requires two parameters = mean and variance

19. GEV
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Confidence set for two coefficients - two dimensional analog for the confidence interval
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

20. Heteroskedastic
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
Model dependent - Options with the same underlying assets may trade at different volatilities
If variance of the conditional distribution of u(i) is not constant

21. Standard variable for non - normal distributions
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
Combine to form distribution with leptokurtosis (heavy tails)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

22. Hazard rate of exponentially distributed random variable
We reject a hypothesis that is actually true
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 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
P(X=x - Y=y) = P(X=x) * P(Y=y)

23. 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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Model dependent - Options with the same underlying assets may trade at different volatilities

24. Logistic distribution
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!)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Has heavy tails
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

25. Homoskedastic
Statement of the error or precision of an estimate
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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

26. Multivariate probability
More than one random variable
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!)
E(XY) - E(X)E(Y)
Special type of pooled data in which the cross sectional unit is surveyed over time

27. 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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

28. Shortcomings of implied volatility
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
Model dependent - Options with the same underlying assets may trade at different volatilities

29. Critical z values
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
95% = 1.65 99% = 2.33 For one - tailed tests
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

30. F distribution
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Does not depend on a prior event or information
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

31. Biggest (and only real) drawback of GARCH mode
Based on an equation - P(A) = # of A/total outcomes
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Nonlinearity
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

32. Square root rule
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
We reject a hypothesis that is actually true
Variance(X) + Variance(Y) - 2*covariance(XY)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

33. Historical std dev
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Concerned with a single random variable (ex. Roll of a die)
P(X=x - Y=y) = P(X=x) * P(Y=y)

34. Extending the HS approach for computing value of a portfolio
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Least absolute deviations estimator - used when extreme outliers are not uncommon
Has heavy tails
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

35. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Sample mean +/ - t*(stddev(s)/sqrt(n))
Use historical simulation approach but use the EWMA weighting system
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

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

37. Binomial 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
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!)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

38. Two requirements of OVB
Regression can be non - linear in variables but must be linear in parameters
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)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Expected value of the sample mean is the population mean

39. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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)
P(X=x - Y=y) = P(X=x) * P(Y=y)

40. Marginal unconditional probability function
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Based on an equation - P(A) = # of A/total outcomes
Does not depend on a prior event or information
(a^2)(variance(x)) + (b^2)(variance(y))

41. Continuous random variable
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Contains variables not explicit in model - Accounts for randomness
Only requires two parameters = mean and variance
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

42. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

43. POT
Peaks over threshold - Collects dataset in excess of some threshold
Expected value of the sample mean is the population mean
Special type of pooled data in which the cross sectional unit is surveyed over time
(a^2)(variance(x)

44. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Has heavy tails
P(X=x - Y=y) = P(X=x) * P(Y=y)
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

45. Cholesky factorization (decomposition)
Based on a dataset
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
Low Frequency - High Severity events
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

46. Difference between population and sample variance
Attempts to sample along more important paths
Population denominator = n - Sample denominator = n - 1
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Model dependent - Options with the same underlying assets may trade at different volatilities

47. Potential reasons for fat tails in return distributions
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Use historical simulation approach but use the EWMA weighting system
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

48. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

49. Implied standard deviation for options
Combine to form distribution with leptokurtosis (heavy tails)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Only requires two parameters = mean and variance

50. What does the OLS minimize?
SSR
Only requires two parameters = mean and variance
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
P(Z>t)

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