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. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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)

2. Poisson Distribution
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
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Model dependent - Options with the same underlying assets may trade at different volatilities

3. Weibul distribution
Variance(x)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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!)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

4. Monte Carlo Simulations
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

5. Bootstrap 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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Confidence level
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

6. Overall F - statistic
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

7. Test for statistical independence
When the sample size is large - the uncertainty about the value of the sample is very small
Mean of sampling distribution is the population mean
P(X=x - Y=y) = P(X=x) * P(Y=y)
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

8. Gamma distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

9. Variance of sampling distribution of means when n<N
Nonlinearity
P - value
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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

10. Panel data (longitudinal or micropanel)
Variance(y)/n = variance of sample Y
Contains variables not explicit in model - Accounts for randomness
Special type of pooled data in which the cross sectional unit is surveyed over time
Rxy = Sxy/(Sx*Sy)

11. Multivariate Density Estimation (MDE)
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(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)
Variance reverts to a long run level

12. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Probability that the random variables take on certain values simultaneously
Sample mean will near the population mean as the sample size increases
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

13. 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

14. Central Limit Theorem
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
For n>30 - sample mean is approximately normal
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

15. Efficiency
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Attempts to sample along more important paths
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Among all unbiased estimators - estimator with the smallest variance is efficient

16. Type II Error
We accept a hypothesis that should have been rejected
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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

17. Skewness
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
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

18. Direction of OVB
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

19. Variance of X+Y
Population denominator = n - Sample denominator = n - 1
Var(X) + Var(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
Sample mean +/ - t*(stddev(s)/sqrt(n))

20. Two requirements of OVB
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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

21. Four sampling distributions

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

22. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Variance(x) + Variance(Y) + 2*covariance(XY)
Variance(x)
We accept a hypothesis that should have been rejected

23. GARCH
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)
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
Expected value of the sample mean is the population mean
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

24. Shortcomings of implied volatility
If variance of the conditional distribution of u(i) is not constant
Mean = np - Variance = npq - Std dev = sqrt(npq)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Model dependent - Options with the same underlying assets may trade at different volatilities

25. Standard variable for non - normal distributions
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Choose parameters that maximize the likelihood of what observations occurring
Z = (Y - meany)/(stddev(y)/sqrt(n))

26. Cross - sectional
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
Average return across assets on a given day
Variance(X) + Variance(Y) - 2*covariance(XY)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

27. Test for unbiasedness
Concerned with a single random variable (ex. Roll of a die)
E(mean) = mean
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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. Confidence interval for sample mean
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Normal - Student's T - Chi - square - F distribution
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Variance reverts to a long run level

29. Mean reversion
Special type of pooled data in which the cross sectional unit is surveyed over time
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Nonlinearity
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

30. Hybrid method for conditional volatility
Concerned with a single random variable (ex. Roll of a die)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Use historical simulation approach but use the EWMA weighting system
More than one random variable

31. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Special type of pooled data in which the cross sectional unit is surveyed over time
Z = (Y - meany)/(stddev(y)/sqrt(n))
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

32. Priori (classical) probability
Normal - Student's T - Chi - square - F distribution
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Based on an equation - P(A) = # of A/total outcomes
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

33. Hazard rate of exponentially distributed random variable
Sample mean +/ - t*(stddev(s)/sqrt(n))
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)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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

34. Sample mean
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth 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
Expected value of the sample mean is the population mean
Yi = B0 + B1Xi + ui

35. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Among all unbiased estimators - estimator with the smallest variance is efficient
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

36. Reliability
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
SSR
Statement of the error or precision of an estimate

37. K - th moment
Z = (Y - meany)/(stddev(y)/sqrt(n))
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
(a^2)(variance(x)
Summation((xi - mean)^k)/n

38. Beta distribution
(a^2)(variance(x)) + (b^2)(variance(y))
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
We accept a hypothesis that should have been rejected
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

39. Least squares estimator(m)
Mean of sampling distribution is the population mean
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

40. Heteroskedastic
Variance(y)/n = variance of sample Y
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
If variance of the conditional distribution of u(i) is not constant
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

41. Normal distribution
Rxy = Sxy/(Sx*Sy)
Yi = B0 + B1Xi + ui
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
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

42. Non - parametric vs parametric calculation of VaR
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Distribution with only two possible outcomes
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

43. What does the OLS minimize?
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
SSR
P(Z>t)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

44. Variance of X+Y assuming dependence
Var(X) + Var(Y)
Variance(x) + Variance(Y) + 2*covariance(XY)
When the sample size is large - the uncertainty about the value of the sample is very small
Among all unbiased estimators - estimator with the smallest variance is efficient

45. Economical(elegant)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Only requires two parameters = mean and variance
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

46. BLUE
Peaks over threshold - Collects dataset in excess of some threshold
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Contains variables not explicit in model - Accounts for randomness

47. Single variable (univariate) probability
Summation((xi - mean)^k)/n
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Concerned with a single random variable (ex. Roll of a die)

48. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
For n>30 - sample mean is approximately normal
Only requires two parameters = mean and variance

49. Variance - covariance approach for VaR of a portfolio
Only requires two parameters = mean and variance
Rxy = Sxy/(Sx*Sy)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Z = (Y - meany)/(stddev(y)/sqrt(n))

50. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
E(XY) - E(X)E(Y)
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)