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. Persistence
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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

2. Gamma distribution
Choose parameters that maximize the likelihood of what observations occurring
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
Population denominator = n - Sample denominator = n - 1
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. i.i.d.
Independently and Identically Distributed
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Mean = np - Variance = npq - Std dev = sqrt(npq)
If variance of the conditional distribution of u(i) is not constant

4. LAD
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Least absolute deviations estimator - used when extreme outliers are not uncommon
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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

5. Sample covariance
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

6. Unstable return distribution
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Var(X) + Var(Y)

7. Test for unbiasedness
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
E(mean) = mean
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

8. Continuous random variable
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
If variance of the conditional distribution of u(i) is not constant
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

9. Shortcomings of implied volatility
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
Application of mathematical statistics to economic data to lend empirical support to models
Model dependent - Options with the same underlying assets may trade at different volatilities

10. Continuously compounded return equation
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
When the sample size is large - the uncertainty about the value of the sample is very small
i = ln(Si/Si - 1)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

11. Cholesky factorization (decomposition)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Has heavy tails
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

12. Critical z values
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
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
We reject a hypothesis that is actually true
95% = 1.65 99% = 2.33 For one - tailed tests

13. Bernouli Distribution
Distribution with only two possible outcomes
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Mean = np - Variance = npq - Std dev = sqrt(npq)
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

14. Unbiased
Statement of the error or precision of an estimate
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Returns over time for an individual asset
Mean of sampling distribution is the population mean

15. Conditional probability functions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Transformed to a unit variable - Mean = 0 Variance = 1
(a^2)(variance(x)) + (b^2)(variance(y))
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

16. Potential reasons for fat tails in return distributions
Price/return tends to run towards a long - run level
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
We reject a hypothesis that is actually true

17. K - th moment
Summation((xi - mean)^k)/n
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!)
Low Frequency - High Severity events
If variance of the conditional distribution of u(i) is not constant

18. Econometrics
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.
Application of mathematical statistics to economic data to lend empirical support to models
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

19. Test for statistical independence
Least absolute deviations estimator - used when extreme outliers are not uncommon
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
P(X=x - Y=y) = P(X=x) * P(Y=y)
95% = 1.65 99% = 2.33 For one - tailed tests

20. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
For n>30 - sample mean is approximately normal
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

21. Poisson Distribution
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

22. SER
Variance(x)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance(x) + Variance(Y) + 2*covariance(XY)
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

23. Statistical (or empirical) model
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
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
Nonlinearity
Yi = B0 + B1Xi + ui

24. Reliability
When one regressor is a perfect linear function of the other regressors
Does not depend on a prior event or information
Statement of the error or precision of an estimate
Random walk (usually acceptable) - Constant volatility (unlikely)

25. Joint probability functions
Transformed to a unit variable - Mean = 0 Variance = 1
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
Variance = (1/m) summation(u<n - i>^2)

26. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Confidence set for two coefficients - two dimensional analog for the confidence interval

27. Variance of X+Y assuming dependence
Yi = B0 + B1Xi + ui
Variance(x) + Variance(Y) + 2*covariance(XY)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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

28. Efficiency
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
We accept a hypothesis that should have been rejected
Among all unbiased estimators - estimator with the smallest variance is efficient
Variance(x)

29. Hazard rate of exponentially distributed random variable
E(mean) = mean
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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
Var(X) + Var(Y)

30. Discrete random variable
Normal - Student's T - Chi - square - F distribution
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Summation((xi - mean)^k)/n

31. Direction of OVB
Population denominator = n - Sample denominator = n - 1
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Nonlinearity

32. SER
Expected value of the sample mean is the population mean
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
For n>30 - sample mean is approximately normal
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

33. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Variance(x)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

34. Variance of weighted scheme
Nonlinearity
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Statement of the error or precision of an estimate
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

35. Law of Large Numbers
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
i = ln(Si/Si - 1)
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 mean will near the population mean as the sample size increases

36. Cross - sectional
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
Nonlinearity
Average return across assets on a given day
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

37. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Combine to form distribution with leptokurtosis (heavy tails)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

38. Maximum likelihood method
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
i = ln(Si/Si - 1)
Choose parameters that maximize the likelihood of what observations occurring

39. Multivariate probability
Sample mean +/ - t*(stddev(s)/sqrt(n))
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
More than one random variable
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

40. Type II Error
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
We accept a hypothesis that should have been rejected
Average return across assets on a given day

41. Variance of X - Y assuming dependence
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Price/return tends to run towards a long - run level
Variance(X) + Variance(Y) - 2*covariance(XY)

42. Tractable
Easy to manipulate
Variance(y)/n = variance of sample Y
Summation((xi - mean)^k)/n
Nonlinearity

43. Biggest (and only real) drawback of GARCH mode
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Rxy = Sxy/(Sx*Sy)
Mean of sampling distribution is the population mean
Nonlinearity

44. R^2
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Probability that the random variables take on certain values simultaneously
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

45. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Statement of the error or precision of an estimate

46. Four sampling distributions

47. Beta distribution
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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)

48. BLUE
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

49. Antithetic variable technique
When one regressor is a perfect linear function of the other regressors
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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

50. Regime - switching volatility model
Probability that the random variables take on certain values simultaneously
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications