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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
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
Independently and Identically Distributed
Peaks over threshold - Collects dataset in excess of some threshold

2. Importance sampling technique
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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
Attempts to sample along more important paths
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

3. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Least absolute deviations estimator - used when extreme outliers are not uncommon
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

4. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

5. ESS
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
95% = 1.65 99% = 2.33 For one - tailed tests
Yi = B0 + B1Xi + ui

6. What does the OLS minimize?
Variance(X) + Variance(Y) - 2*covariance(XY)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
SSR

7. Exact significance level
More than one random variable
P - value
Has heavy tails
(a^2)(variance(x)) + (b^2)(variance(y))

8. Heteroskedastic
Transformed to a unit variable - Mean = 0 Variance = 1
Concerned with a single random variable (ex. Roll of a die)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
If variance of the conditional distribution of u(i) is not constant

9. Conditional probability functions
95% = 1.65 99% = 2.33 For one - tailed tests
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
P(Z>t)

10. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Sample mean will near the population mean as the sample size increases
Based on an equation - P(A) = # of A/total outcomes
P(Z>t)

11. Two drawbacks of moving average series
Nonlinearity
Confidence level
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

12. Central Limit Theorem
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
For n>30 - sample mean is approximately normal
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

13. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

14. Variance of X+b
Variance(x)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

15. Potential reasons for fat tails in return distributions
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

16. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Combine to form distribution with leptokurtosis (heavy tails)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Variance(x)

17. Panel data (longitudinal or micropanel)
(a^2)(variance(x)) + (b^2)(variance(y))
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

18. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
E(mean) = mean
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Contains variables not explicit in model - Accounts for randomness

19. Hazard rate of exponentially distributed random variable
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
(a^2)(variance(x)) + (b^2)(variance(y))

20. Square root rule
Sampling distribution of sample means tend to be normal
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)
Expected value of the sample mean is the population mean

21. Continuously compounded return equation
i = ln(Si/Si - 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!)
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

22. Covariance calculations using weight sums (lambda)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Does not depend on a prior event or information
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

23. Unbiased
Var(X) + Var(Y)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Based on a dataset
Mean of sampling distribution is the population mean

24. Biggest (and only real) drawback of GARCH mode
Nonlinearity
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Contains variables not explicit in model - Accounts for randomness

25. Variance of X+Y
(a^2)(variance(x)
Mean of sampling distribution is the population mean
When one regressor is a perfect linear function of the other regressors
Var(X) + Var(Y)

26. Beta distribution
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)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
P(Z>t)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

27. Tractable
Low Frequency - High Severity events
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Easy to manipulate
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

28. Mean reversion in asset dynamics
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
95% = 1.65 99% = 2.33 For one - tailed tests
Price/return tends to run towards a long - run level
When the sample size is large - the uncertainty about the value of the sample is very small

29. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Variance(X) + Variance(Y) - 2*covariance(XY)
(a^2)(variance(x)) + (b^2)(variance(y))

30. Standard normal distribution
P(X=x - Y=y) = P(X=x) * P(Y=y)
Transformed to a unit variable - Mean = 0 Variance = 1
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Low Frequency - High Severity events

31. Variance of X - Y assuming dependence
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Confidence set for two coefficients - two dimensional analog for the confidence interval
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Variance(X) + Variance(Y) - 2*covariance(XY)

32. Stochastic error term
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
E(XY) - E(X)E(Y)
Contains variables not explicit in model - Accounts for randomness
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

33. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Mean = np - Variance = npq - Std dev = sqrt(npq)
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
Independently and Identically Distributed

34. Marginal unconditional probability function
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Does not depend on a prior event or information
For n>30 - sample mean is approximately normal
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

35. Non - parametric vs parametric calculation of VaR
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

36. Kurtosis
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Variance reverts to a long run level
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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

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

38. Bernouli Distribution
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
Distribution with only two possible outcomes
If variance of the conditional distribution of u(i) is not constant

39. Critical z values
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
95% = 1.65 99% = 2.33 For one - tailed tests
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

40. Law of Large Numbers
Use historical simulation approach but use the EWMA weighting system
Sample mean will near the population mean as the sample size increases
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Distribution with only two possible outcomes

41. POT
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Easy to manipulate
Peaks over threshold - Collects dataset in excess of some threshold
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

42. Time series data
Returns over time for an individual asset
When one regressor is a perfect linear function of the other regressors
If variance of the conditional distribution of u(i) is not constant
Variance(X) + Variance(Y) - 2*covariance(XY)

43. K - th moment
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
i = ln(Si/Si - 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
Summation((xi - mean)^k)/n

44. Economical(elegant)
Variance = (1/m) summation(u<n - i>^2)
Variance(x) + Variance(Y) + 2*covariance(XY)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Only requires two parameters = mean and variance

45. Gamma distribution
(a^2)(variance(x)
Peaks over threshold - Collects dataset in excess of some threshold
Low Frequency - High Severity events
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

46. Deterministic Simulation
Combine to form distribution with leptokurtosis (heavy tails)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Sample mean +/ - t*(stddev(s)/sqrt(n))
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

47. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Rxy = Sxy/(Sx*Sy)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Combine to form distribution with leptokurtosis (heavy tails)

48. Unstable return distribution
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Mean of sampling distribution is the population mean

49. Test for unbiasedness
Summation((xi - mean)^k)/n
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
E(mean) = mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

50. Bootstrap method
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
For n>30 - sample mean is approximately normal