FRM Foundations Of Risk Management Quantitative Methods

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. Joint probability functions
95% = 1.65 99% = 2.33 For one - tailed tests
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Probability that the random variables take on certain values simultaneously
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.


2. Central Limit Theorem
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
For n>30 - sample mean is approximately normal
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
P - value


3. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Average return across assets on a given day
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Application of mathematical statistics to economic data to lend empirical support to models


4. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
We reject a hypothesis that is actually true


5. Critical z values
Confidence level
95% = 1.65 99% = 2.33 For one - tailed tests
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Yi = B0 + B1Xi + ui


6. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
P(X=x - Y=y) = P(X=x) * P(Y=y)


7. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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
(a^2)(variance(x)) + (b^2)(variance(y))


8. Cross - sectional
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Average return across assets on a given day


9. 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)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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
Summation((xi - mean)^k)/n


10. Poisson Distribution
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)


11. Standard error
Easy to manipulate
SSR
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance(X) + Variance(Y) - 2*covariance(XY)


12. GPD
Variance = (1/m) summation(u<n - i>^2)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
95% = 1.65 99% = 2.33 For one - tailed tests


13. Importance sampling technique
P(Z>t)
Yi = B0 + B1Xi + ui
Attempts to sample along more important paths
Variance reverts to a long run level


14. Confidence interval for sample mean
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Peaks over threshold - Collects dataset in excess of some threshold
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample


15. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Price/return tends to run towards a long - run level
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption


16. Adjusted R^2
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance(x) + Variance(Y) + 2*covariance(XY)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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


17. Multivariate probability
If variance of the conditional distribution of u(i) is not constant
More than one random variable
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Normal - Student's T - Chi - square - F distribution


18. Standard error for Monte Carlo replications
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Statement of the error or precision of an estimate
More than one random variable
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications


19. R^2
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
E(mean) = mean
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)


20. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
If variance of the conditional distribution of u(i) is not constant
P(Z>t)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)


21. Mean(expected value)
Low Frequency - High Severity events
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)


22. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Var(X) + Var(Y)
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.


23. Deterministic Simulation
Among all unbiased estimators - estimator with the smallest variance is efficient
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance(x)


24. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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(X) + Variance(Y) - 2*covariance(XY)


25. Central Limit Theorem(CLT)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Variance(x)
Sampling distribution of sample means tend to be normal


26. Extreme Value Theory
We reject a hypothesis that is actually true
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Yi = B0 + B1Xi + ui


27. Sample variance
95% = 1.65 99% = 2.33 For one - tailed tests
Confidence set for two coefficients - two dimensional analog for the confidence interval
E(XY) - E(X)E(Y)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2


28. LAD
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!)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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)
Least absolute deviations estimator - used when extreme outliers are not uncommon


29. Economical(elegant)
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
Confidence level
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


30. Mean reversion in asset dynamics
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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
Price/return tends to run towards a long - run level
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one


31. Two assumptions of square root rule
Variance(X) + Variance(Y) - 2*covariance(XY)
i = ln(Si/Si - 1)
Random walk (usually acceptable) - Constant volatility (unlikely)
Sample mean will near the population mean as the sample size increases


32. Overall F - statistic
Variance = (1/m) summation(u<n - i>^2)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Average return across assets on a given day


33. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
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
Does not depend on a prior event or information
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)


34. Normal distribution
Variance reverts to a long run level
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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


35. WLS
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
When one regressor is a perfect linear function of the other regressors
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail


36. Conditional probability functions
Distribution with only two possible outcomes
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter 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)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications


37. Homoskedastic
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
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
SSR
Nonlinearity


38. Binomial distribution
Variance reverts to a long run level
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!)
When the sample size is large - the uncertainty about the value of the sample is very small
E(mean) = mean


39. Homoskedastic only F - stat
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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


40. Single variable (univariate) probability
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)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Concerned with a single random variable (ex. Roll of a die)
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


41. Antithetic variable technique
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))


42. SER
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
We accept a hypothesis that should have been rejected
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients


43. Two drawbacks of moving average series
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Yi = B0 + B1Xi + ui
Choose parameters that maximize the likelihood of what observations occurring


44. Logistic distribution
Has heavy tails
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Probability that the random variables take on certain values simultaneously


45. Unstable return distribution
Probability that the random variables take on certain values simultaneously
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Based on an equation - P(A) = # of A/total outcomes


46. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
We accept a hypothesis that should have been rejected
Attempts to increase accuracy by reducing sample variance instead of increasing sample size


47. Variance of X+b
Variance(x)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
95% = 1.65 99% = 2.33 For one - tailed tests
Variance(X) + Variance(Y) - 2*covariance(XY)


48. Persistence
Sample mean will near the population mean as the sample size increases
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric


49. Hazard rate of exponentially distributed random variable
Statement of the error or precision of an estimate
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Application of mathematical statistics to economic data to lend empirical support to models
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)


50. POT
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Peaks over threshold - Collects dataset in excess of some threshold