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. Standard error
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Use historical simulation approach but use the EWMA weighting system
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

2. Unstable return distribution
We reject a hypothesis that is actually true
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

3. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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
Random walk (usually acceptable) - Constant volatility (unlikely)
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

4. Cholesky factorization (decomposition)
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
When the sample size is large - the uncertainty about the value of the sample is very small
Population denominator = n - Sample denominator = n - 1
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

5. Implied standard deviation for options
Based on an equation - P(A) = # of A/total outcomes
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

6. Mean reversion in asset dynamics
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Price/return tends to run towards a long - run level

7. Statistical (or empirical) model
Sample mean +/ - t*(stddev(s)/sqrt(n))
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Normal - Student's T - Chi - square - F distribution
Yi = B0 + B1Xi + ui

8. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
i = ln(Si/Si - 1)
P(Z>t)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

9. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Sample mean +/ - t*(stddev(s)/sqrt(n))
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

10. Least squares estimator(m)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Variance(X) + Variance(Y) - 2*covariance(XY)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

11. Hybrid method for conditional volatility
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!)
Use historical simulation approach but use the EWMA weighting system
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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

12. Four sampling distributions
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
E(XY) - E(X)E(Y)
Normal - Student's T - Chi - square - F distribution

13. Test for unbiasedness
E(mean) = mean
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Easy to manipulate

14. F distribution
i = ln(Si/Si - 1)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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 approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

15. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Model dependent - Options with the same underlying assets may trade at different volatilities
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
Concerned with a single random variable (ex. Roll of a die)

16. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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
Does not depend on a prior event or information
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

17. Lognormal
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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

18. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
If variance of the conditional distribution of u(i) is not constant
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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

19. Direction of OVB
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Combine to form distribution with leptokurtosis (heavy tails)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

20. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Combine to form distribution with leptokurtosis (heavy tails)
When one regressor is a perfect linear function of the other regressors

21. Tractable
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Random walk (usually acceptable) - Constant volatility (unlikely)
Choose parameters that maximize the likelihood of what observations occurring
Easy to manipulate

22. Deterministic Simulation
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
Sample mean +/ - t*(stddev(s)/sqrt(n))
Average return across assets on a given day

23. Variance of aX + bY
Mean = np - Variance = npq - Std dev = sqrt(npq)
Sampling distribution of sample means tend to be normal
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
(a^2)(variance(x)) + (b^2)(variance(y))

24. Difference between population and sample variance
Only requires two parameters = mean and variance
Rxy = Sxy/(Sx*Sy)
Population denominator = n - Sample denominator = n - 1
P - value

25. Bootstrap method
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
If variance of the conditional distribution of u(i) is not constant
Average return across assets on a given day

26. Discrete random variable
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
E(mean) = mean
Variance(x) + Variance(Y) + 2*covariance(XY)

27. Unbiased
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
Mean of sampling distribution is the population mean

28. Time series data
Variance reverts to a long run level
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Returns over time for an individual asset
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

29. Importance sampling technique
Returns over time for an individual asset
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Attempts to sample along more important paths

30. Variance(discrete)
Based on a dataset
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
Contains variables not explicit in model - Accounts for randomness

31. Standard variable for non - normal distributions
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Based on a dataset
Does not depend on a prior event or information
Z = (Y - meany)/(stddev(y)/sqrt(n))

32. WLS
For n>30 - sample mean is approximately normal
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
Transformed to a unit variable - Mean = 0 Variance = 1
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

33. Hazard rate of exponentially distributed random variable
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
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

34. Overall F - statistic
Confidence level
F = ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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

35. Type II Error
E(mean) = mean
Returns over time for an individual asset
We accept a hypothesis that should have been rejected
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

36. LFHS
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Low Frequency - High Severity events
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
E(mean) = mean

37. 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
P(X=x - Y=y) = P(X=x) * P(Y=y)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

38. Square root rule
Independently and Identically Distributed
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Var(X) + Var(Y)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

39. SER
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
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
Population denominator = n - Sample denominator = n - 1
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!)

40. Marginal unconditional probability function
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
P - value
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Does not depend on a prior event or information

41. Reliability
Statement of the error or precision of an estimate
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Sampling distribution of sample means tend to be normal
Among all unbiased estimators - estimator with the smallest variance is efficient

42. Confidence interval (from t)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Use historical simulation approach but use the EWMA weighting system
Sample mean +/ - t*(stddev(s)/sqrt(n))

43. Pooled data
Easy to manipulate
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Independently and Identically Distributed
Normal - Student's T - Chi - square - F distribution

44. Biggest (and only real) drawback of GARCH mode
Nonlinearity
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
Contains variables not explicit in model - Accounts for randomness
More than one random variable

45. Variance of sample mean
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Variance(y)/n = variance of sample Y

46. Variance of weighted scheme
(a^2)(variance(x)) + (b^2)(variance(y))
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

47. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Nonlinearity
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
Contains variables not explicit in model - Accounts for randomness

48. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

49. Variance of X+Y
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Var(X) + Var(Y)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

50. 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)
Variance(y)/n = variance of sample Y
Random walk (usually acceptable) - Constant volatility (unlikely)
Sampling distribution of sample means tend to be normal

Test your basic knowledge |

FRM Foundations Of Risk Management Quantitative Methods

Link to This Test

Related Subjects