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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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1. Panel data (longitudinal or micropanel)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Special type of pooled data in which the cross sectional unit is surveyed over time
When one regressor is a perfect linear function of the other regressors
Independently and Identically Distributed

2. Skewness
E(XY) - E(X)E(Y)
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
Combine to form distribution with leptokurtosis (heavy tails)
P - value

3. Square root rule
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

4. Non - parametric vs parametric calculation of VaR
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
If variance of the conditional distribution of u(i) is not constant
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

5. LAD
We reject a hypothesis that is actually true
Normal - Student's T - Chi - square - F distribution
Least absolute deviations estimator - used when extreme outliers are not uncommon
Based on a dataset

6. Unstable return distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Var(X) + Var(Y)
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

7. Joint probability functions
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
More than one random variable
Variance(x) + Variance(Y) + 2*covariance(XY)
Probability that the random variables take on certain values simultaneously

8. P - value
P(Z>t)
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
Special type of pooled data in which the cross sectional unit is surveyed over time
We reject a hypothesis that is actually true

9. Consistent
Based on an equation - P(A) = # of A/total outcomes
If variance of the conditional distribution of u(i) is not constant
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
When the sample size is large - the uncertainty about the value of the sample is very small

10. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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)
Variance(x)

11. Difference between population and sample variance
Least absolute deviations estimator - used when extreme outliers are not uncommon
We accept a hypothesis that should have been rejected
Population denominator = n - Sample denominator = n - 1
Returns over time for an individual asset

12. Bootstrap method
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
When the sample size is large - the uncertainty about the value of the sample is very small
Attempts to sample along more important paths
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

13. Discrete representation of the GBM
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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

14. Hazard rate of exponentially distributed random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Nonlinearity
When one regressor is a perfect linear function of the other regressors
Peaks over threshold - Collects dataset in excess of some threshold

15. Poisson distribution equations for mean variance and std deviation
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors
Z = (Y - meany)/(stddev(y)/sqrt(n))
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

16. 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)
Model dependent - Options with the same underlying assets may trade at different volatilities
Variance(X) + Variance(Y) - 2*covariance(XY)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

17. Variance of X - Y assuming dependence
Variance reverts to a long run level
Does not depend on a prior event or information
Variance(X) + Variance(Y) - 2*covariance(XY)
Sampling distribution of sample means tend to be normal

18. Stochastic error term
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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
Contains variables not explicit in model - Accounts for randomness
Easy to manipulate

19. Persistence
Mean of sampling distribution is the population mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
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)

20. Variance of X+Y assuming dependence
Confidence level
Rxy = Sxy/(Sx*Sy)
Independently and Identically Distributed
Variance(x) + Variance(Y) + 2*covariance(XY)

21. EWMA
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Choose parameters that maximize the likelihood of what observations occurring

22. Simplified standard (un - weighted) variance
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Variance = (1/m) summation(u<n - i>^2)
Choose parameters that maximize the likelihood of what observations occurring

23. GEV
P(Z>t)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

24. Four sampling distributions

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25. F distribution
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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
SSR
i = ln(Si/Si - 1)

26. Overall F - statistic
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

27. Two ways to calculate historical volatility
We accept a hypothesis that should have been rejected
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Random walk (usually acceptable) - Constant volatility (unlikely)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

28. Logistic distribution
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Has heavy tails
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

29. R^2
Returns over time for an individual asset
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

30. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Least absolute deviations estimator - used when extreme outliers are not uncommon
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

31. Poisson Distribution
Application of mathematical statistics to economic data to lend empirical support to models
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

32. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Population denominator = n - Sample denominator = n - 1
Normal - Student's T - Chi - square - F distribution
Attempts to sample along more important paths

33. Bernouli Distribution
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
For n>30 - sample mean is approximately normal
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Distribution with only two possible outcomes

34. Block maxima
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

35. What does the OLS minimize?
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
SSR
Variance(X) + Variance(Y) - 2*covariance(XY)

36. Multivariate Density Estimation (MDE)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

37. Two assumptions of square root rule
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Random walk (usually acceptable) - Constant volatility (unlikely)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Probability that the random variables take on certain values simultaneously

38. Implications of homoscedasticity
Yi = B0 + B1Xi + ui
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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
Expected value of the sample mean is the population mean

39. Sample variance
Nonlinearity
Statement of the error or precision of an estimate
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Variance reverts to a long run level

40. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
Only requires two parameters = mean and variance
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 = np - Variance = npq - Std dev = sqrt(npq)

41. Gamma distribution
Contains variables not explicit in model - Accounts for randomness
Population denominator = n - Sample denominator = n - 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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

42. GARCH
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
When one regressor is a perfect linear function of the other regressors
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)
Var(X) + Var(Y)

43. 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
Based on an equation - P(A) = # of A/total outcomes
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Rxy = Sxy/(Sx*Sy)

44. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Doesn't imply added variable is significant - doesn't imply regressors are a true cause of the dependent variable - doesn't imply there's no OVB - doesn't imply you have the most appropriate set of regressors
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

45. Result of combination of two normal with same means
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Combine to form distribution with leptokurtosis (heavy tails)
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)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

46. Confidence interval for sample mean
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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!)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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

47. Time series data
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Independently and Identically Distributed
Returns over time for an individual asset

48. Central Limit Theorem
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
For n>30 - sample mean is approximately normal
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Statement of the error or precision of an estimate

49. ESS
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
P(Z>t)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Expected value of the sample mean is the population mean

50. Law of Large Numbers
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!)
SSR
Sample mean will near the population mean as the sample size increases
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)