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. Chi - squared distribution
Combine to form distribution with leptokurtosis (heavy tails)
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(y)/n = variance of sample Y
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

2. Type I error
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
We reject a hypothesis that is actually true
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

3. Covariance
E(XY) - E(X)E(Y)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

4. Importance sampling technique
Choose parameters that maximize the likelihood of what observations occurring
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Attempts to sample along more important paths

5. Simulation models
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Mean of sampling distribution is the population mean
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

6. Sample covariance
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

7. Sample variance
For n>30 - sample mean is approximately normal
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
E(mean) = mean

8. Standard variable for non - normal distributions
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Z = (Y - meany)/(stddev(y)/sqrt(n))
Least absolute deviations estimator - used when extreme outliers are not uncommon
Mean = np - Variance = npq - Std dev = sqrt(npq)

9. Reliability
Based on a dataset
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Statement of the error or precision of an estimate
Among all unbiased estimators - estimator with the smallest variance is efficient

10. Confidence interval for sample mean
P(X=x - Y=y) = P(X=x) * P(Y=y)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Price/return tends to run towards a long - run level
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

11. Mean reversion in asset dynamics
Variance(y)/n = variance of sample Y
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
We accept a hypothesis that should have been rejected
Price/return tends to run towards a long - run level

12. BLUE
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
Rxy = Sxy/(Sx*Sy)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Sample mean +/ - t*(stddev(s)/sqrt(n))

13. Normal distribution
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
Confidence level
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

14. Empirical frequency
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Based on a dataset
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

15. Exponential distribution
Special type of pooled data in which the cross sectional unit is surveyed over time
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

16. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

17. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Nonlinearity
Population denominator = n - Sample denominator = n - 1

18. Discrete representation of the GBM
When one regressor is a perfect linear function of the other regressors
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Low Frequency - High Severity events
Expected value of the sample mean is the population mean

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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Confidence level

20. Poisson Distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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!)
Var(X) + Var(Y)
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

21. Potential reasons for fat tails in return distributions
Variance reverts to a long run level
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
When one regressor is a perfect linear function of the other regressors

22. Weibul distribution
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 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
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)

23. Skewness
Population denominator = n - Sample denominator = n - 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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

24. Variance of sample mean
Mean of sampling distribution is the population mean
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
Transformed to a unit variable - Mean = 0 Variance = 1
Variance(y)/n = variance of sample Y

25. Priori (classical) probability
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Rxy = Sxy/(Sx*Sy)
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

26. Variance of aX + bY
Variance(x)
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))
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

27. Tractable
Easy to manipulate
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Does not depend on a prior event or information

28. Variance of X - Y assuming dependence
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Population denominator = n - Sample denominator = n - 1
Variance(X) + Variance(Y) - 2*covariance(XY)

29. Shortcomings of implied volatility
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Model dependent - Options with the same underlying assets may trade at different volatilities
Easy to manipulate
When the sample size is large - the uncertainty about the value of the sample is very small

30. Kurtosis
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
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Contains variables not explicit in model - Accounts for randomness
Has heavy tails

31. Law of Large Numbers
Variance(X) + Variance(Y) - 2*covariance(XY)
P - value
Sample mean will near the population mean as the sample size increases
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

32. Simulating for VaR
(a^2)(variance(x)) + (b^2)(variance(y))
Easy to manipulate
(a^2)(variance(x)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

33. Standard error for Monte Carlo replications
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Mean = np - Variance = npq - Std dev = sqrt(npq)

34. Test for unbiasedness
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Average return across assets on a given day
E(mean) = mean
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

35. F distribution
Only requires two parameters = mean and variance
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Least absolute deviations estimator - used when extreme outliers are not uncommon
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

36. Continuous representation of the GBM
Variance = (1/m) summation(u<n - i>^2)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

37. i.i.d.
Independently and Identically Distributed
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

38. Hazard rate of exponentially distributed random variable
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Average return across assets on a given day

39. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Low Frequency - High Severity events

40. Simplified standard (un - weighted) variance
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
Variance = (1/m) summation(u<n - i>^2)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
P - value

41. Confidence ellipse
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Confidence set for two coefficients - two dimensional analog for the confidence interval
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Mean = np - Variance = npq - Std dev = sqrt(npq)

42. Binomial distribution equations for mean variance and std dev
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!)
Based on an equation - P(A) = # of A/total outcomes
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Mean = np - Variance = npq - Std dev = sqrt(npq)

43. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

44. Standard normal distribution
Variance(y)/n = variance of sample Y
Transformed to a unit variable - Mean = 0 Variance = 1
P(X=x - Y=y) = P(X=x) * P(Y=y)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

45. WLS
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Among all unbiased estimators - estimator with the smallest variance is efficient
Low Frequency - High Severity events
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

46. Efficiency
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Among all unbiased estimators - estimator with the smallest variance is efficient
Application of mathematical statistics to economic data to lend empirical support to models
Variance(y)/n = variance of sample Y

47. Control variates technique
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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
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

48. Pooled data
Summation((xi - mean)^k)/n
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Variance(x)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

49. LFHS
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
Low Frequency - High Severity events
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
Easy to manipulate

50. Block maxima
Returns over time for an individual asset
If variance of the conditional distribution of u(i) is not constant
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
P(Z>t)