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

2. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Concerned with a single random variable (ex. Roll of a die)

3. Persistence
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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
Choose parameters that maximize the likelihood of what observations occurring
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

4. Square root rule
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

5. Type II Error
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
We accept a hypothesis that should have been rejected
Application of mathematical statistics to economic data to lend empirical support to models
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

6. Joint probability functions
Probability that the random variables take on certain values simultaneously
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
95% = 1.65 99% = 2.33 For one - tailed tests
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)

7. Panel data (longitudinal or micropanel)
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
Special type of pooled data in which the cross sectional unit is surveyed over time
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 = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

8. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
We accept a hypothesis that should have been rejected
We reject a hypothesis that is actually true
Statement of the error or precision of an estimate

9. Binomial distribution
Variance(x)
Peaks over threshold - Collects dataset in excess of some threshold
Variance reverts to a long run level
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!)

10. Unconditional vs conditional distributions
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
E(mean) = mean
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

11. Importance sampling technique
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Attempts to sample along more important paths

12. Exact significance level
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
P - value
More than one random variable

13. Marginal unconditional probability function
Z = (Y - meany)/(stddev(y)/sqrt(n))
Does not depend on a prior event or information
Has heavy tails
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

14. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
We accept a hypothesis that should have been rejected
For n>30 - sample mean is approximately normal
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

15. Chi - squared distribution
Summation((xi - mean)^k)/n
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
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

16. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
Regression can be non - linear in variables but must be linear in parameters
Transformed to a unit variable - Mean = 0 Variance = 1
Sample mean +/ - t*(stddev(s)/sqrt(n))

17. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Sampling distribution of sample means tend to be normal
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
Independently and Identically Distributed

18. Kurtosis
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Normal - Student's T - Chi - square - F distribution
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

19. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Random walk (usually acceptable) - Constant volatility (unlikely)
(a^2)(variance(x)
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

20. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
P(Z>t)
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)

21. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Has heavy tails
Returns over time for an individual asset
Average return across assets on a given day

22. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
Nonlinearity
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

23. Variance of sampling distribution of means when n<N
Model dependent - Options with the same underlying assets may trade at different volatilities
Low Frequency - High Severity events
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Application of mathematical statistics to economic data to lend empirical support to models

24. Significance =1
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Confidence level

25. Potential reasons for fat tails in return distributions
Returns over time for an individual asset
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Special type of pooled data in which the cross sectional unit is surveyed over time
Nonlinearity

26. Two assumptions of square root rule
Statement of the error or precision of an estimate
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Random walk (usually acceptable) - Constant volatility (unlikely)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

27. Priori (classical) probability
Easy to manipulate
Based on an equation - P(A) = # of A/total outcomes
Dataset is parsed into blocks with greater length than the periodicity - Observations must be 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)

28. Test for statistical independence
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

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

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

30. Confidence interval for sample mean
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

31. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

32. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Sampling distribution of sample means tend to be normal
Special type of pooled data in which the cross sectional unit is surveyed over time
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

33. Sample mean
Application of mathematical statistics to economic data to lend empirical support to models
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Expected value of the sample mean is the population mean
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

34. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Sampling distribution of sample means tend to be normal
Normal - Student's T - Chi - square - F distribution
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

35. Least squares estimator(m)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

36. Hazard rate of exponentially distributed random variable
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Does not depend on a prior event or information

37. Normal distribution
i = ln(Si/Si - 1)
Attempts to sample along more important paths
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
Sample mean +/ - t*(stddev(s)/sqrt(n))

38. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
If variance of the conditional distribution of u(i) is not constant
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

39. Mean reversion in asset dynamics
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Price/return tends to run towards a long - run level
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

40. Variance of X+b
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)
We reject a hypothesis that is actually true
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

41. F distribution
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Variance = (1/m) summation(u<n - i>^2)
Easy to manipulate
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

42. 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

43. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
We reject a hypothesis that is actually true
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Combine to form distribution with leptokurtosis (heavy tails)

44. Implications of homoscedasticity
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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

45. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Nonlinearity
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

46. Continuous random variable
Mean of sampling distribution is the population mean
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

47. Heteroskedastic
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
If variance of the conditional distribution of u(i) is not constant
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Peaks over threshold - Collects dataset in excess of some threshold

48. Confidence interval (from t)
Independently and Identically Distributed
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Attempts to sample along more important paths
Sample mean +/ - t*(stddev(s)/sqrt(n))

49. Simulating for VaR
Only requires two parameters = mean and variance
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

50. Homoskedastic only F - stat
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
Returns over time for an individual asset
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)