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. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Expected value of the sample mean is the population mean
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

2. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Distribution with only two possible outcomes
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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

3. Beta distribution
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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!)

4. Homoskedastic only F - stat
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 an individual asset
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

5. Maximum likelihood method
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Choose parameters that maximize the likelihood of what observations occurring
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

6. T distribution
SSR
Distribution with only two possible outcomes
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

7. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
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
(a^2)(variance(x)) + (b^2)(variance(y))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

8. Statistical (or empirical) model
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Yi = B0 + B1Xi + ui
Sample mean +/ - t*(stddev(s)/sqrt(n))

9. Gamma distribution
SSR
We accept a hypothesis that should have been rejected
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
Sample mean will near the population mean as the sample size increases

10. Difference between population and sample variance
Variance = (1/m) summation(u<n - i>^2)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Population denominator = n - Sample denominator = n - 1

11. i.i.d.
Does not depend on a prior event or information
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Independently and Identically Distributed

12. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
(a^2)(variance(x)) + (b^2)(variance(y))
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

13. GEV
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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

14. Non - parametric vs parametric calculation of VaR
When the sample size is large - the uncertainty about the value of the sample is very small
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Variance(x) + Variance(Y) + 2*covariance(XY)

15. Covariance
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
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
E(XY) - E(X)E(Y)
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!)

16. Unbiased
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Mean of sampling distribution is the population mean
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

17. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
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
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
Variance(x)

18. Direction of OVB
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

19. Standard normal distribution
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
When one regressor is a perfect linear function of the other regressors
Transformed to a unit variable - Mean = 0 Variance = 1
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

20. Monte Carlo Simulations
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
P - value
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

21. Time series data
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Returns over time for an individual asset

22. 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
E(mean) = mean
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

23. Continuous representation of the GBM
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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)
Average return across assets on a given day
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

24. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Only requires two parameters = mean and variance
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

25. Confidence ellipse
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
Does not depend on a prior event or information
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

26. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Variance(y)/n = variance of sample Y
When the sample size is large - the uncertainty about the value of the sample is very small

27. Skewness
When one regressor is a perfect linear function of the other regressors
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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
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

28. Type II Error
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
We accept a hypothesis that should have been rejected
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

29. Bernouli Distribution
Returns over time for an individual asset
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Only requires two parameters = mean and variance
Distribution with only two possible outcomes

30. LAD
Variance(X) + Variance(Y) - 2*covariance(XY)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Least absolute deviations estimator - used when extreme outliers are not uncommon
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

31. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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
Variance(x) + Variance(Y) + 2*covariance(XY)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

32. Antithetic variable technique
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

33. Homoskedastic
When the sample size is large - the uncertainty about the value of the sample is very small
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

34. Consistent
Summation((xi - mean)^k)/n
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
(a^2)(variance(x)
When the sample size is large - the uncertainty about the value of the sample is very small

35. Mean(expected value)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Confidence level
Z = (Y - meany)/(stddev(y)/sqrt(n))

36. Deterministic Simulation
Summation((xi - mean)^k)/n
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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
Variance(y)/n = variance of sample Y

37. SER
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Var(X) + Var(Y)
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
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

38. Test for unbiasedness
Variance(y)/n = variance of sample Y
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Use historical simulation approach but use the EWMA weighting system
E(mean) = mean

39. POT
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
Peaks over threshold - Collects dataset in excess of some threshold
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Independently and Identically Distributed

40. Joint probability functions
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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)
Probability that the random variables take on certain values simultaneously

41. Discrete random variable
E(mean) = mean
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Does not depend on a prior event or information

42. Confidence interval for sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
Based on a dataset
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)

43. Unconditional vs conditional distributions
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Low Frequency - High Severity events

44. Sample correlation
Sample mean will near the population mean as the sample size increases
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Rxy = Sxy/(Sx*Sy)

45. Variance of aX + bY
Transformed to a unit variable - Mean = 0 Variance = 1
Variance(x)
(a^2)(variance(x)) + (b^2)(variance(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

46. Conditional probability functions
Variance(X) + Variance(Y) - 2*covariance(XY)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

47. Marginal unconditional probability function
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Variance(x)
Choose parameters that maximize the likelihood of what observations occurring
Does not depend on a prior event or information

48. Logistic distribution
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Has heavy tails

49. Efficiency
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
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
Among all unbiased estimators - estimator with the smallest variance is efficient

50. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Special type of pooled data in which the cross sectional unit is surveyed over time
Does not depend on a prior event or information
More than one random variable