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. Critical z values
Application of mathematical statistics to economic data to lend empirical support to models
95% = 1.65 99% = 2.33 For one - tailed tests
(a^2)(variance(x)
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

2. Continuously compounded return equation
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
i = ln(Si/Si - 1)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

3. Central Limit Theorem
For n>30 - sample mean is approximately normal
Variance = (1/m) summation(u<n - i>^2)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
(a^2)(variance(x)

4. Variance of X+b
Yi = B0 + B1Xi + ui
Variance(x)
Price/return tends to run towards a long - run level
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

5. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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
Least absolute deviations estimator - used when extreme outliers are not uncommon

6. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Population denominator = n - Sample denominator = n - 1
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

7. Continuous random variable
SSR
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

8. Block maxima
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
E(XY) - E(X)E(Y)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

9. 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

10. Simulating for VaR
Based on a dataset
Only requires two parameters = mean and variance
If variance of the conditional distribution of u(i) is not constant
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

11. Maximum likelihood method
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Low Frequency - High Severity events
Choose parameters that maximize the likelihood of what observations occurring
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

12. Poisson distribution equations for mean variance and std deviation
Sample mean +/ - t*(stddev(s)/sqrt(n))
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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
Model dependent - Options with the same underlying assets may trade at different volatilities

13. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
95% = 1.65 99% = 2.33 For one - tailed tests
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
Transformed to a unit variable - Mean = 0 Variance = 1

14. Two ways to calculate historical volatility
Expected value of the sample mean is the population mean
Z = (Y - meany)/(stddev(y)/sqrt(n))
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

15. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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

16. Test for statistical independence
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
P(X=x - Y=y) = P(X=x) * P(Y=y)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
i = ln(Si/Si - 1)

17. Implications of homoscedasticity
Nonlinearity
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Among all unbiased estimators - estimator with the smallest variance is efficient

18. Standard variable for non - normal distributions
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Regression can be non - linear in variables but must be linear in parameters
Z = (Y - meany)/(stddev(y)/sqrt(n))

19. Tractable
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Easy to manipulate
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

20. Logistic distribution
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
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

21. Unbiased
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!)
Mean of sampling distribution is the population mean
Variance(y)/n = variance of sample Y
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

22. Kurtosis
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Z = (Y - meany)/(stddev(y)/sqrt(n))
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 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

23. Discrete random variable
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Change in S = S<t - 1>(meanchange in time + stddev 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

24. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
Rxy = Sxy/(Sx*Sy)
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

25. What does the OLS minimize?
SSR
More than one random variable
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
i = ln(Si/Si - 1)

26. Variance of X+Y assuming dependence
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Variance(x) + Variance(Y) + 2*covariance(XY)
Distribution with only two possible outcomes
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

27. Economical(elegant)
Variance = (1/m) summation(u<n - i>^2)
Only requires two parameters = mean and variance
(a^2)(variance(x)
If variance of the conditional distribution of u(i) is not constant

28. Lognormal
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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

29. Persistence
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Yi = B0 + B1Xi + ui
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
We reject a hypothesis that is actually true

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

31. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
We accept a hypothesis that should have been rejected
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

32. Chi - squared distribution
P(X=x - Y=y) = P(X=x) * P(Y=y)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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
P(Z>t)

33. GEV
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

34. Adjusted R^2
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Returns over time for an individual asset
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
Statement of the error or precision of an estimate

35. Deterministic Simulation
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

36. Variance - covariance approach for VaR of a portfolio
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Attempts to sample along more important paths

37. Covariance
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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
E(XY) - E(X)E(Y)
Does not depend on a prior event or information

38. Mean(expected value)
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

39. Multivariate probability
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Concerned with a single random variable (ex. Roll of a die)
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
More than one random variable

40. Cholesky factorization (decomposition)
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 accept a hypothesis that should have been rejected
95% = 1.65 99% = 2.33 For one - tailed tests
Variance(X) + Variance(Y) - 2*covariance(XY)

41. Poisson Distribution
We accept a hypothesis that should have been rejected
Mean of sampling distribution is the population mean
E(mean) = mean
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

42. Variance of aX + bY
Variance(x)
(a^2)(variance(x)) + (b^2)(variance(y))
Normal - Student's T - Chi - square - F distribution
Sampling distribution of sample means tend to be normal

43. Implied standard deviation for options
Confidence set for two coefficients - two dimensional analog for the confidence interval
We accept a hypothesis that should have been rejected
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)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

44. Simulation models
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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
(a^2)(variance(x)) + (b^2)(variance(y))

45. Binomial distribution equations for mean variance and std dev
Least absolute deviations estimator - used when extreme outliers are not uncommon
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Mean = np - Variance = npq - Std dev = sqrt(npq)
Probability that the random variables take on certain values simultaneously

46. Joint probability functions
When the sample size is large - the uncertainty about the value of the sample is very small
Probability that the random variables take on certain values simultaneously
We reject a hypothesis that is actually true
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

47. Two requirements of OVB
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
P - value
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

48. Simplified standard (un - weighted) variance
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Price/return tends to run towards a long - run level
Variance = (1/m) summation(u<n - i>^2)

49. Skewness
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

50. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
Transformed to a unit variable - Mean = 0 Variance = 1
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Variance reverts to a long run level