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. Cross - sectional
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Average return across assets on a given day
Variance(X) + Variance(Y) - 2*covariance(XY)

2. Weibul distribution
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
Normal - Student's T - Chi - square - F distribution

3. Central Limit Theorem
Easy to manipulate
For n>30 - sample mean is approximately normal
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

4. GPD
Random walk (usually acceptable) - Constant volatility (unlikely)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

5. Non - parametric vs parametric calculation of VaR
E(XY) - E(X)E(Y)
Variance(y)/n = variance of sample Y
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Regression can be non - linear in variables but must be linear in parameters

6. Logistic distribution
Model dependent - Options with the same underlying assets may trade at different volatilities
For n>30 - sample mean is approximately normal
Has heavy tails
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

7. Mean reversion in variance
Variance reverts to a long run level
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

8. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Sample mean will near the population mean as the sample size increases
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

9. Test for unbiasedness
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
E(mean) = mean

10. Cholesky factorization (decomposition)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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(X=x - Y=y) = P(X=x) * P(Y=y)
Statement of the error or precision of an estimate

11. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Use historical simulation approach but use the EWMA weighting system
Combine to form distribution with leptokurtosis (heavy tails)
E(XY) - E(X)E(Y)

12. Hazard rate of exponentially distributed random variable
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Variance(y)/n = variance of sample Y
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

13. Homoskedastic only F - stat
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Mean = np - Variance = npq - Std dev = sqrt(npq)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

14. F distribution
Dataset is parsed into blocks with greater length than the periodicity - Observations must be 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
When the sample size is large - the uncertainty about the value of the sample is very small
Contains variables not explicit in model - Accounts for randomness

15. Time series data
Has heavy tails
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Returns over time for an individual asset
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

16. Beta distribution
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Based on a dataset
When the sample size is large - the uncertainty about the value of the sample is very small
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

17. Variance(discrete)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Nonlinearity
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

18. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Variance(y)/n = variance of sample Y
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

19. Economical(elegant)
Only requires two parameters = mean and variance
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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

20. Unbiased
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Mean of sampling distribution is the population mean
Statement of the error or precision of an estimate
Has heavy tails

21. 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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Sample mean will near the population mean as the sample size increases
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

22. Kurtosis
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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

23. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
(a^2)(variance(x)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

24. Mean reversion in asset dynamics
Least absolute deviations estimator - used when extreme outliers are not uncommon
Price/return tends to run towards a long - run level
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

25. Antithetic variable technique
Random walk (usually acceptable) - Constant volatility (unlikely)
Variance(X) + Variance(Y) - 2*covariance(XY)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

26. Confidence interval for sample mean
Based on a dataset
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

27. Direction of OVB
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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

28. Type I error
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Variance(X) + Variance(Y) - 2*covariance(XY)
We reject a hypothesis that is actually true
Model dependent - Options with the same underlying assets may trade at different volatilities

29. Panel data (longitudinal or micropanel)
Sample mean will near the population mean as the sample size increases
Special type of pooled data in which the cross sectional unit is surveyed over time
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Price/return tends to run towards a long - run level

30. i.i.d.
Transformed to a unit variable - Mean = 0 Variance = 1
For n>30 - sample mean is approximately normal
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
Independently and Identically Distributed

31. Two ways to calculate historical volatility
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
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

32. Regime - switching volatility model
Distribution with only two possible outcomes
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

33. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Among all unbiased estimators - estimator with the smallest variance is efficient
E(mean) = mean
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

34. R^2
Population denominator = n - Sample denominator = n - 1
Contains variables not explicit in model - Accounts for randomness
Nonlinearity
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

35. Type II Error
Independently and Identically Distributed
We accept a hypothesis that should have been rejected
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Low Frequency - High Severity events

36. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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

37. Continuously compounded return equation
i = ln(Si/Si - 1)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

38. Lognormal
Based on a dataset
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent 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
Variance(x) + Variance(Y) + 2*covariance(XY)

39. Standard normal distribution
Low Frequency - High Severity events
Transformed to a unit variable - Mean = 0 Variance = 1
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

40. Unconditional vs conditional distributions
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

41. What does the OLS minimize?
Among all unbiased estimators - estimator with the smallest variance is efficient
Easy to manipulate
SSR
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

42. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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

43. Poisson distribution equations for mean variance and std deviation
Rxy = Sxy/(Sx*Sy)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

44. Importance sampling technique
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Based on a dataset
Var(X) + Var(Y)
Attempts to sample along more important paths

45. Reliability
Population denominator = n - Sample denominator = n - 1
F = ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Statement of the error or precision of an estimate

46. Persistence
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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

47. Critical z values
Population denominator = n - Sample denominator = n - 1
(a^2)(variance(x)
95% = 1.65 99% = 2.33 For one - tailed tests
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

48. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
SSR
Z = (Y - meany)/(stddev(y)/sqrt(n))
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

49. Gamma distribution
Yi = B0 + B1Xi + ui
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Rxy = Sxy/(Sx*Sy)
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

50. Binomial distribution
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Model dependent - Options with the same underlying assets may trade at different volatilities
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!)
Summation((xi - mean)^k)/n

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