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. Efficiency
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Among all unbiased estimators - estimator with the smallest variance is efficient
Yi = B0 + B1Xi + ui
Variance(x) + Variance(Y) + 2*covariance(XY)

2. i.i.d.
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
Independently and Identically Distributed
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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

3. Binomial distribution
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Rxy = Sxy/(Sx*Sy)
When one regressor is a perfect linear function of the other regressors
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. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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

5. Two requirements of OVB
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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)
For n>30 - sample mean is approximately normal

6. Four sampling distributions

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

7. Significance =1
Returns over time for a combination of assets (combination of time series and cross - sectional data)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Price/return tends to run towards a long - run level
Confidence level

8. Econometrics
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(x) + Variance(Y) + 2*covariance(XY)
Application of mathematical statistics to economic data to lend empirical support to models
Concerned with a single random variable (ex. Roll of a die)

9. 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)
Var(X) + Var(Y)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Low Frequency - High Severity events

10. Exponential distribution
Normal - Student's T - Chi - square - F distribution
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)
SSR
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

11. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
(a^2)(variance(x)
When the sample size is large - the uncertainty about the value of the sample is very small

12. Kurtosis
P - value
Normal - Student's T - Chi - square - F distribution
Model dependent - Options with the same underlying assets may trade at different volatilities
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

13. Implications of homoscedasticity
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
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
Concerned with a single random variable (ex. Roll of a die)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

14. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Use historical simulation approach but use the EWMA weighting system
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Has heavy tails

15. Difference between population and sample variance
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Population denominator = n - Sample denominator = n - 1

16. Type I error
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
We reject a hypothesis that is actually true
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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

17. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Variance = (1/m) summation(u<n - i>^2)
Sampling distribution of sample means tend to be normal
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

18. Variance of X+b
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Based on an equation - P(A) = # of A/total outcomes
Variance(x)

19. P - value
P(Z>t)
Does not depend on a prior event or information
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Independently and Identically Distributed

20. Type II Error
Price/return tends to run towards a long - run level
Least absolute deviations estimator - used when extreme outliers are not uncommon
We accept a hypothesis that should have been rejected
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

21. Extreme Value Theory
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)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Variance(y)/n = variance of sample Y

22. Reliability
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Statement of the error or precision of an estimate
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

23. Central Limit Theorem
For n>30 - sample mean is approximately normal
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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

24. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of 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
Statement of the error or precision of an estimate
Only requires two parameters = mean and variance

25. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Statement of the error or precision of an estimate
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

26. T distribution
Returns over time for an individual asset
We reject a hypothesis that is actually true
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Concerned with a single random variable (ex. Roll of a die)

27. Non - parametric vs parametric calculation of VaR
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Mean of sampling distribution is the population mean
Summation((xi - mean)^k)/n

28. Marginal unconditional probability function
Rxy = Sxy/(Sx*Sy)
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
Does not depend on a prior event or information
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

29. LFHS
Probability that the random variables take on certain values simultaneously
Low Frequency - High Severity events
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

30. Lognormal
Confidence level
Mean = np - Variance = npq - Std dev = sqrt(npq)
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

31. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Easy to manipulate
When one regressor is a perfect linear function of the other regressors
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

32. Homoskedastic only F - stat
Sampling distribution of sample means tend to be normal
Nonlinearity
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

33. Standard error
When one regressor is a perfect linear function of the other regressors
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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
Probability that the random variables take on certain values simultaneously

34. Variance of X+Y assuming dependence
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
When the sample size is large - the uncertainty about the value of the sample is very small
Confidence set for two coefficients - two dimensional analog for the confidence interval
Variance(x) + Variance(Y) + 2*covariance(XY)

35. SER
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
95% = 1.65 99% = 2.33 For one - tailed tests
Independently and Identically Distributed
Based on an equation - P(A) = # of A/total outcomes

36. Inverse transform method
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Probability that the random variables take on certain values simultaneously

37. Mean(expected value)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Confidence level
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

38. Covariance calculations using weight sums (lambda)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

39. Continuous representation of the GBM
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)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Expected value of the sample mean is the population mean

40. 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
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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)

41. Confidence interval (from t)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Random walk (usually acceptable) - Constant volatility (unlikely)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Sample mean +/ - t*(stddev(s)/sqrt(n))

42. Gamma distribution
95% = 1.65 99% = 2.33 For one - tailed tests
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
E(XY) - E(X)E(Y)

43. Hazard rate of exponentially distributed random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
More than one random variable
Normal - Student's T - Chi - square - F distribution

44. Simplified standard (un - weighted) variance
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Variance = (1/m) summation(u<n - i>^2)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

45. Confidence ellipse
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Confidence set for two coefficients - two dimensional analog for the confidence interval

46. Test for unbiasedness
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
E(mean) = mean
Var(X) + Var(Y)

47. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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

48. Covariance
More than one random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
E(XY) - E(X)E(Y)
Variance(x)

49. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Regression can be non - linear in variables but must be linear in parameters
P(X=x - Y=y) = P(X=x) * P(Y=y)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

50. Empirical frequency
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
i = ln(Si/Si - 1)
Easy to manipulate
Based on a dataset