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. Sample correlation
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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
Rxy = Sxy/(Sx*Sy)
E(XY) - E(X)E(Y)

2. Sample variance
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Independently and Identically Distributed
P - value
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

3. Discrete random variable
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)
Rxy = Sxy/(Sx*Sy)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

4. Sample mean
Least absolute deviations estimator - used when extreme outliers are not uncommon
Peaks over threshold - Collects dataset in excess of some threshold
Expected value of the sample mean is the population mean
Confidence level

5. Deterministic Simulation
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
Sampling distribution of sample means tend to be normal
Variance(X) + Variance(Y) - 2*covariance(XY)
Confidence set for two coefficients - two dimensional analog for the confidence interval

6. Variance of aX
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
(a^2)(variance(x)
Confidence level
Contains variables not explicit in model - Accounts for randomness

7. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

8. Conditional probability functions
Easy to manipulate
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

9. Multivariate Density Estimation (MDE)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Nonlinearity

10. Law of Large Numbers
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
i = ln(Si/Si - 1)
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)
Sample mean will near the population mean as the sample size increases

11. Direction of OVB
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)
If variance of the conditional distribution of u(i) is not constant
95% = 1.65 99% = 2.33 For one - tailed tests

12. Mean reversion in asset dynamics
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Price/return tends to run towards a long - run level
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

13. Simplified standard (un - weighted) variance
Regression can be non - linear in variables but must be linear in parameters
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
More than one random variable
Variance = (1/m) summation(u<n - i>^2)

14. Variance of weighted scheme
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
95% = 1.65 99% = 2.33 For one - tailed tests
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

15. Historical std dev
Variance reverts to a long run level
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Special type of pooled data in which the cross sectional unit is surveyed over time
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)

16. Unbiased
Mean of sampling distribution is the population mean
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Confidence set for two coefficients - two dimensional analog for the confidence interval

17. Joint probability functions
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Probability that the random variables take on certain values simultaneously
When the sample size is large - the uncertainty about the value of the sample is very small
Special type of pooled data in which the cross sectional unit is surveyed over time

18. Simulation models
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

19. Empirical frequency
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
E(XY) - E(X)E(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
Based on a dataset

20. Adjusted R^2
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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
E(XY) - E(X)E(Y)
Has heavy tails

21. Key properties of linear regression
Peaks over threshold - Collects dataset in excess of some threshold
Regression can be non - linear in variables but must be linear in parameters
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!)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

22. Mean(expected value)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
We accept a hypothesis that should have been rejected
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

23. Result of combination of two normal with same means
95% = 1.65 99% = 2.33 For one - tailed tests
Has heavy tails
Combine to form distribution with leptokurtosis (heavy tails)
Transformed to a unit variable - Mean = 0 Variance = 1

24. Variance of X - Y assuming dependence
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
P - value
Variance(X) + Variance(Y) - 2*covariance(XY)
Least absolute deviations estimator - used when extreme outliers are not uncommon

25. Monte Carlo Simulations
Population denominator = n - Sample denominator = n - 1
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Variance(sample y) = (variance(y)/n)*(N - n/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

26. Continuously compounded return equation
Special type of pooled data in which the cross sectional unit is surveyed over time
i = ln(Si/Si - 1)
Yi = B0 + B1Xi + ui
(a^2)(variance(x)

27. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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)
P(Z>t)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

28. Variance of sample mean
Variance(y)/n = variance of sample Y
Contains variables not explicit in model - Accounts for randomness
We accept a hypothesis that should have been rejected
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

29. Potential reasons for fat tails in return distributions
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Mean of sampling distribution is the population mean
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

30. Central Limit Theorem
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
For n>30 - sample mean is approximately normal
Variance reverts to a long run level

31. Extending the HS approach for computing value of a portfolio

32. 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
Average return across assets on a given day
Rxy = Sxy/(Sx*Sy)
95% = 1.65 99% = 2.33 For one - tailed tests

33. SER
Independently and Identically Distributed
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

34. Perfect multicollinearity
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
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
When one regressor is a perfect linear function of the other regressors

35. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
P - value
Average return across assets on a given day
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

36. Two requirements of OVB
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Mean = np - Variance = npq - Std dev = sqrt(npq)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Variance(X) + Variance(Y) - 2*covariance(XY)

37. Variance - covariance approach for VaR of a portfolio
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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)

38. Multivariate probability
Concerned with a single random variable (ex. Roll of a die)
Model dependent - Options with the same underlying assets may trade at different volatilities
Variance(X) + Variance(Y) - 2*covariance(XY)
More than one random variable

39. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Population denominator = n - Sample denominator = n - 1
Combine to form distribution with leptokurtosis (heavy tails)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

40. Importance sampling technique
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Attempts to sample along more important paths
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

41. Simulating for VaR
95% = 1.65 99% = 2.33 For one - tailed tests
When one regressor is a perfect linear function of the other 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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

42. Efficiency
Has heavy tails
Among all unbiased estimators - estimator with the smallest variance is efficient
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Model dependent - Options with the same underlying assets may trade at different volatilities

43. BLUE
(a^2)(variance(x)
Variance(x)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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

44. Poisson distribution equations for mean variance and std deviation
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

45. Marginal unconditional probability function
Does not depend on a prior event or information
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
95% = 1.65 99% = 2.33 For one - tailed tests
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

46. Variance of sampling distribution of means when n<N
Rxy = Sxy/(Sx*Sy)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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
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

47. WLS
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Z = (Y - meany)/(stddev(y)/sqrt(n))
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

48. Single variable (univariate) probability
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 reverts to a long run level
Concerned with a single random variable (ex. Roll of a die)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

49. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
We accept a hypothesis that should have been rejected
More than one random variable
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

50. Tractable
Variance(x) + Variance(Y) + 2*covariance(XY)
Based on a dataset
Easy to manipulate
E(mean) = mean