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. K - th moment
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
Summation((xi - mean)^k)/n
Yi = B0 + B1Xi + ui

2. Central Limit Theorem
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
For n>30 - sample mean is approximately 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
We reject a hypothesis that is actually true

3. i.i.d.
Sampling distribution of sample means tend to be normal
Independently and Identically Distributed
For n>30 - sample mean is approximately normal
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

4. Key properties of linear regression
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Regression can be non - linear in variables but must be linear in parameters
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Mean of sampling distribution is the population mean

5. Joint probability functions
Probability that the random variables take on certain values simultaneously
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
P - value
Among all unbiased estimators - estimator with the smallest variance is efficient

6. Discrete random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

7. Variance - covariance approach for VaR of a portfolio
Independently and Identically Distributed
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
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

8. Exact significance level
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
P - value
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

9. Gamma distribution
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
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 variance = (1/(k - 1))Summation(Yi - mean)^2
Contains variables not explicit in model - Accounts for randomness

10. Logistic distribution
Has heavy tails
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Variance reverts to a long run level
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

11. What does the OLS minimize?
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Attempts to sample along more important paths
SSR
(a^2)(variance(x)) + (b^2)(variance(y))

12. Mean(expected value)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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
When the sample size is large - the uncertainty about the value of the sample is very small

13. Pooled data
Combine to form distribution with leptokurtosis (heavy tails)
For n>30 - sample mean is approximately normal
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

14. Type II Error
Random walk (usually acceptable) - Constant volatility (unlikely)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
We accept a hypothesis that should have been rejected
Choose parameters that maximize the likelihood of what observations occurring

15. Marginal unconditional probability function
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Sample mean will near the population mean as the sample size increases
Does not depend on a prior event or information

16. Historical std dev
Rxy = Sxy/(Sx*Sy)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Variance = (1/m) summation(u<n - i>^2)

17. Efficiency
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Among all unbiased estimators - estimator with the smallest variance is efficient
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
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

18. 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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Random walk (usually acceptable) - Constant volatility (unlikely)
P(Z>t)

19. Limitations of R^2 (what an increase doesn't necessarily imply)

Warning: Invalid argument supplied for foreach() in /var/www/html/basicversity.com/show_quiz.php on line 183

20. Normal distribution
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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
Variance = (1/m) summation(u<n - i>^2)
When the sample size is large - the uncertainty about the value of the sample is very small

21. Mean reversion in asset dynamics
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)
Price/return tends to run towards a long - run level
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

22. Four sampling distributions

Warning: Invalid argument supplied for foreach() in /var/www/html/basicversity.com/show_quiz.php on line 183

23. Potential reasons for fat tails in return distributions
Rxy = Sxy/(Sx*Sy)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Independently and Identically Distributed
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

24. Beta distribution
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Variance(sample y) = (variance(y)/n)*(N - n/N - 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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

25. Mean reversion in variance
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Variance reverts to a long run level
Variance(X) + Variance(Y) - 2*covariance(XY)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

26. Cholesky factorization (decomposition)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
95% = 1.65 99% = 2.33 For one - tailed tests
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
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

27. Multivariate Density Estimation (MDE)
Regression can be non - linear in variables but must be linear in parameters
Rxy = Sxy/(Sx*Sy)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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

28. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
Concerned with a single random variable (ex. Roll of a die)
Has heavy tails
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

29. Adjusted 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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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
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

30. Sample correlation
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Rxy = Sxy/(Sx*Sy)
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

31. Two requirements of OVB
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
Z = (Y - meany)/(stddev(y)/sqrt(n))

32. Standard normal distribution
Mean = np - Variance = npq - Std dev = sqrt(npq)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Transformed to a unit variable - Mean = 0 Variance = 1

33. Shortcomings of implied volatility
Variance(x)
Model dependent - Options with the same underlying assets may trade at different volatilities
Variance(X) + Variance(Y) - 2*covariance(XY)
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

34. SER
(a^2)(variance(x)) + (b^2)(variance(y))
Statement of the error or precision of an estimate
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
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

35. Variance(discrete)
E(XY) - E(X)E(Y)
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
Only requires two parameters = mean and variance
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

36. Continuously compounded return equation
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Population denominator = n - Sample denominator = n - 1
i = ln(Si/Si - 1)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

37. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Variance = (1/m) summation(u<n - i>^2)
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

38. Priori (classical) probability
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Statement of the error or precision of an estimate
Based on an equation - P(A) = # of A/total outcomes
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

39. Control variates technique
Application of mathematical statistics to economic data to lend empirical support to models
E(mean) = mean
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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. Test for statistical independence
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Random walk (usually acceptable) - Constant volatility (unlikely)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Variance = (1/m) summation(u<n - i>^2)

41. Least squares estimator(m)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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

42. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
Concerned with a single random variable (ex. Roll of a die)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

43. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

44. Deterministic Simulation
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under 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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

45. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
Var(X) + Var(Y)
Confidence level
Probability that the random variables take on certain values simultaneously

46. Sample mean
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Expected value of the sample mean is the population mean
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

47. Test for unbiasedness
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
We reject a hypothesis that is actually true
E(mean) = mean
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

48. Stochastic error term
Random walk (usually acceptable) - Constant volatility (unlikely)
P - value
Mean = np - Variance = npq - Std dev = sqrt(npq)
Contains variables not explicit in model - Accounts for randomness

49. Type I error
Nonlinearity
We reject a hypothesis that is actually true
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance(x)

50. Conditional probability functions
Only requires two parameters = mean and variance
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Distribution with only two possible outcomes
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)