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. Sample correlation
Rxy = Sxy/(Sx*Sy)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance = (1/m) summation(u<n - i>^2)
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

2. Binomial distribution
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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
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!)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

3. What does the OLS minimize?
Variance(x)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
SSR
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

4. Empirical frequency
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Based on a dataset
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

5. Covariance
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Independently and Identically Distributed
Variance reverts to a long run level
E(XY) - E(X)E(Y)

6. 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)
Var(X) + Var(Y)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
We reject a hypothesis that is actually true

7. Discrete representation of the GBM
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Independently and Identically Distributed
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

8. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Yi = B0 + B1Xi + ui
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

9. Maximum likelihood method
Transformed to a unit variable - Mean = 0 Variance = 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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Choose parameters that maximize the likelihood of what observations occurring

10. Sample variance
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Nonlinearity
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

11. Type II Error
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Mean = np - Variance = npq - Std dev = sqrt(npq)
We accept a hypothesis that should have been rejected
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

12. SER
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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

13. Variance of aX
Confidence set for two coefficients - two dimensional analog for the confidence interval
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
(a^2)(variance(x)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

14. Exponential distribution
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

15. Central Limit Theorem(CLT)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Sampling distribution of sample means tend to be normal
Based on a dataset
Sample mean +/ - t*(stddev(s)/sqrt(n))

16. Marginal unconditional probability function
Does not depend on a prior event or information
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

17. Variance(discrete)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Rxy = Sxy/(Sx*Sy)

18. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Returns over time for a combination of assets (combination of time series and cross - sectional data)

19. Least squares estimator(m)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
Statement of the error or precision of an estimate
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

20. Mean reversion in asset dynamics
Contains variables not explicit in model - Accounts for randomness
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
P(Z>t)
Price/return tends to run towards a long - run level

21. Cholesky factorization (decomposition)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Var(X) + Var(Y)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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

22. Time series data
Transformed to a unit variable - Mean = 0 Variance = 1
E(mean) = mean
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Returns over time for an individual asset

23. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Yi = B0 + B1Xi + ui
Distribution with only two possible outcomes

24. Extreme Value Theory
Normal - Student's T - Chi - square - F distribution
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Among all unbiased estimators - estimator with the smallest variance is efficient
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

25. Importance sampling technique
Attempts to sample along more important paths
Model dependent - Options with the same underlying assets may trade at different volatilities
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Returns over time for an individual asset

26. Mean reversion in variance
Variance reverts to a long run level
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
When the sample size is large - the uncertainty about the value of the sample is very small

27. Variance of sample mean
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Variance(y)/n = variance of sample Y

28. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Population denominator = n - Sample denominator = n - 1
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Mean = np - Variance = npq - Std dev = sqrt(npq)

29. Chi - squared distribution
Application of mathematical statistics to economic data to lend empirical support to 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
Sample mean +/ - t*(stddev(s)/sqrt(n))
P(X=x - Y=y) = P(X=x) * P(Y=y)

30. Multivariate Density Estimation (MDE)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)

31. R^2
P - value
Statement of the error or precision of an estimate
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

32. Mean(expected value)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Does not depend on a prior event or information
Mean = np - Variance = npq - Std dev = sqrt(npq)

33. Conditional probability functions
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Combine to form distribution with leptokurtosis (heavy tails)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

34. Hazard rate of exponentially distributed random variable
Price/return tends to run towards a long - run level
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Variance = (1/m) summation(u<n - i>^2)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

35. Standard normal distribution
Special type of pooled data in which the cross sectional unit is surveyed over time
Transformed to a unit variable - Mean = 0 Variance = 1
(a^2)(variance(x)
Variance(y)/n = variance of sample Y

36. Standard variable for non - normal distributions
Distribution with only two possible outcomes
Z = (Y - meany)/(stddev(y)/sqrt(n))
SSR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

37. Single variable (univariate) probability
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Concerned with a single random variable (ex. Roll of a die)
Has heavy tails
Based on an equation - P(A) = # of A/total outcomes

38. Implied standard deviation for options
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

39. Statistical (or empirical) model
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Yi = B0 + B1Xi + ui
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
Does not depend on a prior event or information

40. Consistent
Based on a dataset
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
When the sample size is large - the uncertainty about the value of the sample is very small

41. LFHS
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
Low Frequency - High Severity events
Expected value of the sample mean is the population mean
Returns over time for an individual asset

42. Priori (classical) probability
Has heavy tails
Yi = B0 + B1Xi + ui
Based on an equation - P(A) = # of A/total outcomes
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

43. ESS
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Choose parameters that maximize the likelihood of what observations occurring
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

44. Variance of X+b
More than one random variable
Z = (Y - meany)/(stddev(y)/sqrt(n))
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Variance(x)

45. Homoskedastic
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
Special type of pooled data in which the cross sectional unit is surveyed over time

46. Key properties of linear regression
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
Regression can be non - linear in variables but must be linear in parameters

47. Variance of sampling distribution of means when n<N
Average return across assets on a given day
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance(sample y) = (variance(y)/n)*(N - n/N - 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

48. Variance - covariance approach for VaR of a portfolio
Variance = (1/m) summation(u<n - i>^2)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

49. Continuously compounded return equation
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Concerned with a single random variable (ex. Roll of a die)
i = ln(Si/Si - 1)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

50. Test for unbiasedness
When one regressor is a perfect linear function of the other regressors
E(mean) = mean
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Z = (Y - meany)/(stddev(y)/sqrt(n))