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. Non - parametric vs parametric calculation of VaR
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

2. Confidence interval for sample mean
Z = (Y - meany)/(stddev(y)/sqrt(n))
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Confidence set for two coefficients - two dimensional analog for the confidence interval
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

3. Multivariate probability
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Regression can be non - linear in variables but must be linear in parameters
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
More than one random variable

4. Two requirements of OVB
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Price/return tends to run towards a long - run level
P(Z>t)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

5. LAD
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
Attempts to sample along more important paths
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)

6. i.i.d.
Independently and Identically Distributed
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Regression can be non - linear in variables but must be linear in parameters
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

7. Expected future variance rate (t periods forward)
Variance reverts to a long run level
P(X=x - Y=y) = P(X=x) * P(Y=y)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
When one regressor is a perfect linear function of the other regressors

8. POT
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
We accept a hypothesis that should have been rejected
Sample mean +/ - t*(stddev(s)/sqrt(n))
Peaks over threshold - Collects dataset in excess of some threshold

9. Confidence interval (from t)
Regression can be non - linear in variables but must be linear in parameters
Confidence set for two coefficients - two dimensional analog for the confidence interval
Returns over time for an individual asset
Sample mean +/ - t*(stddev(s)/sqrt(n))

10. Continuous representation of the GBM
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)
E(XY) - E(X)E(Y)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

11. Simplified standard (un - weighted) variance
For n>30 - sample mean is approximately normal
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance(X) + Variance(Y) - 2*covariance(XY)
Variance = (1/m) summation(u<n - i>^2)

12. Homoskedastic only F - stat
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Sample mean will near the population mean as the sample size increases

13. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Distribution with only two possible outcomes
Based on an equation - P(A) = # of A/total outcomes
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

14. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
P - value
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

15. Exponential 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)
Concerned with a single random variable (ex. Roll of a die)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

16. Time series data
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Sampling distribution of sample means tend to be normal
Returns over time for an individual asset
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

17. WLS
Z = (Y - meany)/(stddev(y)/sqrt(n))
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
When one regressor is a perfect linear function of the other regressors
Expected value of the sample mean is the population mean

18. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Use historical simulation approach but use the EWMA weighting system
When the sample size is large - the uncertainty about the value of the sample is very small
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

19. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
E(XY) - E(X)E(Y)
Population denominator = n - Sample denominator = n - 1
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

20. Continuous random variable
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Confidence level
Special type of pooled data in which the cross sectional unit is surveyed over time
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

21. Standard error
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
More than one random variable
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

22. Variance of sample mean
Sample mean +/ - t*(stddev(s)/sqrt(n))
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance(y)/n = variance of sample Y
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

23. Heteroskedastic
Concerned with a single random variable (ex. Roll of a die)
Easy to manipulate
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
If variance of the conditional distribution of u(i) is not constant

24. Type I error
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Var(X) + Var(Y)
Combine to form distribution with leptokurtosis (heavy tails)
We reject a hypothesis that is actually true

25. Sample variance
Attempts to sample along more important paths
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
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

26. P - value
P(Z>t)
Summation((xi - mean)^k)/n
Easy to manipulate
For n>30 - sample mean is approximately normal

27. Central Limit Theorem(CLT)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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
Sampling distribution of sample means tend to be normal

28. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)

29. Implied standard deviation for options
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
We accept a hypothesis that should have been rejected
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
When the sample size is large - the uncertainty about the value of the sample is very small

30. Mean reversion
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Var(X) + Var(Y)
E(XY) - E(X)E(Y)

31. Multivariate Density Estimation (MDE)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Choose parameters that maximize the likelihood of what observations occurring
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

32. Persistence
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
P - value
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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. Hybrid method for conditional volatility
P - value
Average return across assets on a given day
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Use historical simulation approach but use the EWMA weighting system

34. Conditional probability functions
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
Statement of the error or precision of an estimate
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Combine to form distribution with leptokurtosis (heavy tails)

35. Skewness
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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(X=x - Y=y) = P(X=x) * P(Y=y)
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

36. Panel data (longitudinal or micropanel)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Special type of pooled data in which the cross sectional unit is surveyed over time
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

37. Priori (classical) probability
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
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
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

38. Reliability
Statement of the error or precision of an estimate
Peaks over threshold - Collects dataset in excess of some threshold
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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)

39. Kurtosis
Contains variables not explicit in model - Accounts for randomness
Use historical simulation approach but use the EWMA weighting system
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

40. 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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Based on a dataset
Sample mean will near the population mean as the sample size increases

41. Economical(elegant)
Only requires two parameters = mean and variance
Statement of the error or precision of an estimate
Based on an equation - P(A) = # of A/total outcomes
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

42. Least squares estimator(m)
Mean of sampling distribution is the population mean
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
When one regressor is a perfect linear function of the other regressors

43. Unbiased
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Mean of sampling distribution is the population mean
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

44. Homoskedastic
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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
When the sample size is large - the uncertainty about the value of the sample is very small
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

45. LFHS
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Variance = (1/m) summation(u<n - i>^2)
Low Frequency - High Severity events
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

46. Poisson Distribution
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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)

47. Chi - squared distribution
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

48. Bernouli Distribution
Does not depend on a prior event or information
Distribution with only two possible outcomes
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
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

49. R^2
Population denominator = n - Sample denominator = n - 1
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
When the sample size is large - the uncertainty about the value of the sample is very small
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

50. Lognormal
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption