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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. Type II Error
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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
Summation((xi - mean)^k)/n
We accept a hypothesis that should have been rejected

2. Joint probability functions
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
For n>30 - sample mean is approximately normal
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)
Probability that the random variables take on certain values simultaneously

3. Direction of OVB
Variance reverts to a long run level
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Attempts to sample along more important paths

4. Simulation models
We reject a hypothesis that is actually true
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

5. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance(x)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

6. Overall F - statistic
If variance of the conditional distribution of u(i) is not constant
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Choose parameters that maximize the likelihood of what observations occurring
(a^2)(variance(x)) + (b^2)(variance(y))

7. What does the OLS minimize?
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
For n>30 - sample mean is approximately normal
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
SSR

8. 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
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
We accept a hypothesis that should have been rejected

9. Antithetic variable technique
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Easy to manipulate
Expected value of the sample mean is the population mean
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

10. Kurtosis
Peaks over threshold - Collects dataset in excess of some threshold
Statement of the error or precision of an estimate
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Rxy = Sxy/(Sx*Sy)

11. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Statement of the error or precision of an estimate
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

12. Conditional probability functions
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
If variance of the conditional distribution of u(i) is not constant
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

13. Cholesky factorization (decomposition)
Nonlinearity
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
We accept a hypothesis that should have been rejected
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

14. Tractable
Distribution with only two possible outcomes
Easy to manipulate
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Z = (Y - meany)/(stddev(y)/sqrt(n))

15. Covariance
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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!)
E(XY) - E(X)E(Y)
Variance(y)/n = variance of sample Y

16. SER
Only requires two parameters = mean and variance
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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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

17. Block maxima
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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)
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

18. Implications of homoscedasticity
We reject a hypothesis that is actually true
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Does not depend on a prior event or information

19. Extending the HS approach for computing value of a portfolio

20. Adjusted R^2
If variance of the conditional distribution of u(i) is not constant
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Normal - Student's T - Chi - square - F distribution
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

21. Statistical (or empirical) model
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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
Yi = B0 + B1Xi + ui
E(XY) - E(X)E(Y)

22. Variance of X+Y
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Distribution with only two possible outcomes
Var(X) + Var(Y)
Returns over time for an individual asset

23. WLS
Population denominator = n - Sample denominator = n - 1
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

24. Type I error
Variance(X) + Variance(Y) - 2*covariance(XY)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Combine to form distribution with leptokurtosis (heavy tails)
We reject a hypothesis that is actually true

25. Two ways to calculate historical volatility
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Variance(x) + Variance(Y) + 2*covariance(XY)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Returns over time for a combination of assets (combination of time series and cross - sectional data)

26. Mean reversion in asset dynamics
When the sample size is large - the uncertainty about the value of the sample is very small
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Price/return tends to run towards a long - run level
Regression can be non - linear in variables but must be linear in parameters

27. Efficiency
When the sample size is large - the uncertainty about the value of the sample is very small
Among all unbiased estimators - estimator with the smallest variance is efficient
(a^2)(variance(x)) + (b^2)(variance(y))
Variance(X) + Variance(Y) - 2*covariance(XY)

28. Biggest (and only real) drawback of GARCH mode
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Attempts to sample along more important paths
Nonlinearity

29. Standard variable for non - normal distributions
P(Z>t)
Price/return tends to run towards a long - run level
Confidence level
Z = (Y - meany)/(stddev(y)/sqrt(n))

30. Consistent
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
When the sample size is large - the uncertainty about the value of the sample is very small
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

31. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
i = ln(Si/Si - 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
Combine to form distribution with leptokurtosis (heavy tails)

32. Sample correlation
We accept a hypothesis that should have been rejected
Yi = B0 + B1Xi + ui
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!)
Rxy = Sxy/(Sx*Sy)

33. Regime - switching volatility model
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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
We accept a hypothesis that should have been rejected

34. Variance of aX + bY
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
(a^2)(variance(x)) + (b^2)(variance(y))
(a^2)(variance(x)

35. Discrete representation of the GBM
For n>30 - sample mean is approximately normal
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Based on an equation - P(A) = # of A/total outcomes
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

36. Variance of sample mean
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Variance(y)/n = variance of sample Y
We reject a hypothesis that is actually true

37. Two drawbacks of moving average series
Var(X) + Var(Y)
If variance of the conditional distribution of u(i) is not constant
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

38. Beta distribution
Sample mean +/ - t*(stddev(s)/sqrt(n))
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Variance = (1/m) summation(u<n - i>^2)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

39. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Special type of pooled data in which the cross sectional unit is surveyed over time
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

40. Persistence
Based on an equation - P(A) = # of A/total outcomes
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
Expected value of the sample mean is the population mean
Sampling distribution of sample means tend to be normal

41. Confidence ellipse
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

42. Econometrics
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)
Based on a dataset
We reject a hypothesis that is actually true

43. Mean reversion in variance
Variance reverts to a long run level
If variance of the conditional distribution of u(i) is not constant
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

44. ESS
More than one random variable
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Based on a dataset
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

45. R^2
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Independently and Identically Distributed
Only requires two parameters = mean and variance
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

46. Extreme Value Theory
Sample mean will near the population mean as the sample size increases
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

47. Variance - covariance approach for VaR of a portfolio
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

48. 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)
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)
Distribution with only two possible outcomes
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

49. EWMA
Choose parameters that maximize the likelihood of what observations occurring
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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
Only requires two parameters = mean and variance

50. Variance of X+b
Var(X) + Var(Y)
Independently and Identically Distributed
Variance(x)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))