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. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
(a^2)(variance(x)
Sample mean +/ - t*(stddev(s)/sqrt(n))

2. Shortcomings of implied volatility
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Distribution with only two possible outcomes
Model dependent - Options with the same underlying assets may trade at different volatilities

3. Empirical frequency
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
We accept a hypothesis that should have been rejected
Based on a dataset

4. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Mean = np - Variance = npq - Std dev = sqrt(npq)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

5. F distribution
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
Confidence level
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Sample mean will near the population mean as the sample size increases

6. Antithetic variable technique
Statement of the error or precision of an estimate
Model dependent - Options with the same underlying assets may trade at different volatilities
For n>30 - sample mean is approximately normal
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

7. Sample correlation
Rxy = Sxy/(Sx*Sy)
Based on a dataset
Confidence level
Model dependent - Options with the same underlying assets may trade at different volatilities

8. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Concerned with a single random variable (ex. Roll of a die)

9. Normal distribution
Special type of pooled data in which the cross sectional unit is surveyed over time
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)
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. Variance of X - Y assuming dependence
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance(X) + Variance(Y) - 2*covariance(XY)
Statement of the error or precision of an estimate
Combine to form distribution with leptokurtosis (heavy tails)

11. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Nonlinearity
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
Z = (Y - meany)/(stddev(y)/sqrt(n))

12. 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

13. POT
Variance(y)/n = variance of sample Y
Peaks over threshold - Collects dataset in excess of some threshold
Independently and Identically Distributed
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

14. Two ways to calculate historical volatility
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

15. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Normal - Student's T - Chi - square - F distribution
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

16. Simulation models
Var(X) + Var(Y)
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be 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

17. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under 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
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

18. Poisson distribution equations for mean variance and std deviation
SSR
Model dependent - Options with the same underlying assets may trade at different volatilities
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

19. Type I error
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
When one regressor is a perfect linear function of the other regressors
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
We reject a hypothesis that is actually true

20. Multivariate probability
Var(X) + Var(Y)
Variance(X) + Variance(Y) - 2*covariance(XY)
More than one random variable
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

21. Continuously compounded return equation
Transformed to a unit variable - Mean = 0 Variance = 1
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
i = ln(Si/Si - 1)
Probability that the random variables take on certain values simultaneously

22. Confidence ellipse
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
Confidence set for two coefficients - two dimensional analog for the confidence interval
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Special type of pooled data in which the cross sectional unit is surveyed over time

23. SER
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

24. GARCH
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
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)
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
Low Frequency - High Severity events

25. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
Average return across assets on a given day
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

26. Continuous representation of the GBM
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
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
Transformed to a unit variable - Mean = 0 Variance = 1
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

28. Variance of X+Y assuming dependence
Based on an equation - P(A) = # of A/total outcomes
Statement of the error or precision of an estimate
Variance(x) + Variance(Y) + 2*covariance(XY)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

29. Unbiased
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Mean of sampling distribution is the population mean
Contains variables not explicit in model - Accounts for randomness

30. Time series data
Mean of sampling distribution is the population mean
Normal - Student's T - Chi - square - F distribution
Variance(X) + Variance(Y) - 2*covariance(XY)
Returns over time for an individual asset

31. Variance of sample mean
Variance(y)/n = variance of sample Y
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
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

32. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Concerned with a single random variable (ex. Roll of a die)

33. Deterministic Simulation
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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

34. Single variable (univariate) probability
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
Concerned with a single random variable (ex. Roll of a die)
Variance(x) + Variance(Y) + 2*covariance(XY)

35. R^2
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

36. LFHS
Application of mathematical statistics to economic data to lend empirical support to models
Price/return tends to run towards a long - run level
Low Frequency - High Severity events
Regression can be non - linear in variables but must be linear in parameters

37. Implied standard deviation for options
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Random walk (usually acceptable) - Constant volatility (unlikely)

38. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Independently and Identically Distributed
Random walk (usually acceptable) - Constant volatility (unlikely)
Normal - Student's T - Chi - square - F distribution

39. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

40. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
i = ln(Si/Si - 1)
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

41. P - value
Has heavy tails
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
P(Z>t)
E(mean) = mean

42. Standard error
Variance(x)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
When one regressor is a perfect linear function of the other regressors
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

43. Joint probability functions
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
Yi = B0 + B1Xi + ui
Probability that the random variables take on certain values simultaneously
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

44. Exact significance level
Low Frequency - High Severity events
P - value
Independently and Identically Distributed
Easy to manipulate

45. Econometrics
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Application of mathematical statistics to economic data to lend empirical support to models
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

46. Perfect multicollinearity
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
When one regressor is a perfect linear function of the other regressors

47. 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
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
Normal - Student's T - Chi - square - F distribution

48. Homoskedastic only F - stat
Variance = (1/m) summation(u<n - i>^2)
Among all unbiased estimators - estimator with the smallest variance is efficient
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

49. Cross - sectional
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Average return across assets on a given day
Expected value of the sample mean is the population mean
Special type of pooled data in which the cross sectional unit is surveyed over time

50. Four sampling distributions

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