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. Significance =1
Attempts to sample along more important paths
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Nonlinearity
Confidence level

2. Test for statistical independence
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
(a^2)(variance(x)
P(X=x - Y=y) = P(X=x) * P(Y=y)
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

3. GPD
Use historical simulation approach but use the EWMA weighting system
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Average return across assets on a given day
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

4. Importance sampling technique
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Attempts to sample along more important paths
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Confidence set for two coefficients - two dimensional analog for the confidence interval

5. Confidence interval for sample mean
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

6. Homoskedastic only F - stat
Sample mean will near the population mean as the sample size increases
Least absolute deviations estimator - used when extreme outliers are not uncommon
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Only requires two parameters = mean and variance

7. Two drawbacks of moving average series
Population denominator = n - Sample denominator = n - 1
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Normal - Student's T - Chi - square - F distribution

8. Bernouli Distribution
Mean of sampling distribution is the population mean
Distribution with only two possible outcomes
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

9. Mean(expected value)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
When one regressor is a perfect linear function of the other regressors

10. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over 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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

11. Four sampling distributions

12. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

13. Variance of X+Y assuming dependence
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Variance(x) + Variance(Y) + 2*covariance(XY)
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

14. Skewness
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
E(mean) = mean
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Variance(X) + Variance(Y) - 2*covariance(XY)

15. P - value
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
P(X=x - Y=y) = P(X=x) * P(Y=y)
Does not depend on a prior event or information
P(Z>t)

16. Marginal unconditional probability function
Mean = np - Variance = npq - Std dev = sqrt(npq)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Sample mean will near the population mean as the sample size increases
Does not depend on a prior event or information

17. Exact significance level
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Z = (Y - meany)/(stddev(y)/sqrt(n))
P - value
Contains variables not explicit in model - Accounts for randomness

18. Simulation models
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
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
95% = 1.65 99% = 2.33 For one - tailed tests

19. 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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

20. Key properties of linear regression
Z = (Y - meany)/(stddev(y)/sqrt(n))
Sampling distribution of sample means tend to be normal
If variance of the conditional distribution of u(i) is not constant
Regression can be non - linear in variables but must be linear in parameters

21. K - th moment
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
Summation((xi - mean)^k)/n
Combine to form distribution with leptokurtosis (heavy tails)

22. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
Returns over time for an individual asset
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Expected value of the sample mean is the population mean

23. i.i.d.
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Contains variables not explicit in model - Accounts for randomness
Independently and Identically Distributed
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

24. Unconditional vs conditional distributions
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
Price/return tends to run towards a long - run level
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

25. Bootstrap method
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 deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

26. Sample covariance
Regression can be non - linear in variables but must be linear in parameters
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

27. Adjusted R^2
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
More than one random variable
Z = (Y - meany)/(stddev(y)/sqrt(n))

28. WLS
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

29. Logistic distribution
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Has heavy tails
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Statement of the error or precision of an estimate

30. Potential reasons for fat tails in return distributions
More than one random variable
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

31. Priori (classical) probability
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
95% = 1.65 99% = 2.33 For one - tailed tests
Based on an equation - P(A) = # of A/total outcomes

32. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

33. Sample correlation
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
Population denominator = n - Sample denominator = n - 1
Rxy = Sxy/(Sx*Sy)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

34. Type II Error
Price/return tends to run towards a long - run level
We accept a hypothesis that should have been rejected
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Peaks over threshold - Collects dataset in excess of some threshold

35. Deterministic Simulation
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
Easy to manipulate
Summation((xi - mean)^k)/n

36. Single variable (univariate) probability
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Concerned with a single random variable (ex. Roll of a die)

37. F distribution
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

38. Two requirements of OVB
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

39. Perfect multicollinearity
Attempts to sample along more important paths
Based on a dataset
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
When one regressor is a perfect linear function of the other regressors

40. Poisson distribution equations for mean variance and std deviation
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Attempts to sample along more important paths
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Price/return tends to run towards a long - run level

41. Gamma distribution
Variance(y)/n = variance of sample Y
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
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

42. Pooled data
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)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Special type of pooled data in which the cross sectional unit is surveyed over time
Returns over time for a combination of assets (combination of time series and cross - sectional data)

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

44. Central Limit Theorem(CLT)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
Sampling distribution of sample means tend to be normal

45. Exponential distribution
Peaks over threshold - Collects dataset in excess of some threshold
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
When one regressor is a perfect linear function of the other regressors

46. Direction of OVB
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
P(X=x - Y=y) = P(X=x) * P(Y=y)

47. Variance - covariance approach for VaR of a portfolio
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance(x)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

48. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Nonlinearity
Statement of the error or precision of an estimate

49. GEV
If variance of the conditional distribution of u(i) is not constant
Variance = (1/m) summation(u<n - i>^2)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

50. ESS
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Sample mean will near the population mean as the sample size increases
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y