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Test your basic knowledge |
FRM Foundations Of Risk Management Quantitative Methods
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Instructions:
Answer 50 questions in 15 minutes.
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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))