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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. Bootstrap method
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Variance = (1/m) summation(u<n - i>^2)

2. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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

3. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Regression can be non - linear in variables but must be linear in parameters
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Least absolute deviations estimator - used when extreme outliers are not uncommon

4. Variance of sample mean
Variance(X) + Variance(Y) - 2*covariance(XY)
Variance(y)/n = variance of sample Y
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

5. Chi - squared distribution
Independently and Identically Distributed
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
Sample mean +/ - t*(stddev(s)/sqrt(n))
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

6. Unstable return distribution
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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)

7. Joint probability functions
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.
Model dependent - Options with the same underlying assets may trade at different volatilities
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

8. Key properties of linear regression
Choose parameters that maximize the likelihood of what observations occurring
Mean of sampling distribution is the population mean
Regression can be non - linear in variables but must be linear in parameters
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

9. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Distribution with only two possible outcomes
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
If variance of the conditional distribution of u(i) is not constant

10. Type I error
Special type of pooled data in which the cross sectional unit is surveyed over time
Nonlinearity
We reject a hypothesis that is actually true
Population denominator = n - Sample denominator = n - 1

11. SER
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
We reject a hypothesis that is actually true

12. Stochastic error term
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Contains variables not explicit in model - Accounts for randomness
Only requires two parameters = mean and variance
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

13. What does the OLS minimize?
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
SSR
Variance(X) + Variance(Y) - 2*covariance(XY)

14. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Mean = np - Variance = npq - Std dev = sqrt(npq)
Expected value of the sample mean is the population mean
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

15. Biggest (and only real) drawback of GARCH mode
Nonlinearity
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

16. Shortcomings of implied volatility
Contains variables not explicit in model - Accounts for randomness
Model dependent - Options with the same underlying assets may trade at different volatilities
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

17. Consistent
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
When the sample size is large - the uncertainty about the value of the sample is very small
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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

18. Two requirements of OVB
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
E(XY) - E(X)E(Y)
P(Z>t)

19. Four sampling distributions

20. Variance of X - Y assuming dependence
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance(X) + Variance(Y) - 2*covariance(XY)
Combine to form distribution with leptokurtosis (heavy tails)

21. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

22. Beta distribution
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Transformed to a unit variable - Mean = 0 Variance = 1
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Peaks over threshold - Collects dataset in excess of some threshold

23. Panel data (longitudinal or micropanel)
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
Model dependent - Options with the same underlying assets may trade at different volatilities
Special type of pooled data in which the cross sectional unit is surveyed over time
Distribution with only two possible outcomes

24. Unbiased
Price/return tends to run towards a long - run level
Mean of sampling distribution is the population mean
Mean = np - Variance = npq - Std dev = sqrt(npq)
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

25. Exponential distribution
P(Z>t)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

26. Cross - sectional
Average return across assets on a given day
Choose parameters that maximize the likelihood of what observations occurring
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
Distribution with only two possible outcomes

27. Square root rule
E(XY) - E(X)E(Y)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Statement of the error or precision of an estimate

28. Variance of aX
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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
(a^2)(variance(x)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

29. Confidence interval for sample mean
Special type of pooled data in which the cross sectional unit is surveyed over time
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Variance(x)

30. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
More than one random variable
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

31. Two drawbacks of moving average series
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Does not depend on a prior event or information

32. Economical(elegant)
Random walk (usually acceptable) - Constant volatility (unlikely)
Only requires two parameters = mean and variance
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
We accept a hypothesis that should have been rejected

33. Priori (classical) probability
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Based on an equation - P(A) = # of A/total outcomes
Mean of sampling distribution is the population mean
Contains variables not explicit in model - Accounts for randomness

34. Poisson Distribution
Sampling distribution of sample means tend to be normal
(a^2)(variance(x)) + (b^2)(variance(y))
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

35. Variance of X+b
Mean of sampling distribution is the population mean
Variance(x)
Based on an equation - P(A) = # of A/total outcomes
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

36. Poisson distribution equations for mean variance and std deviation
Price/return tends to run towards a long - run level
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Does not depend on a prior event or information

37. Bernouli Distribution
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
Distribution with only two possible outcomes
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
For n>30 - sample mean is approximately normal

38. Time series data
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Returns over time for an individual asset
Expected value of the sample mean is the population mean

39. Standard error
SSR
(a^2)(variance(x)) + (b^2)(variance(y))
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Summation((xi - mean)^k)/n

40. Variance of sampling distribution of means when n<N
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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

41. Continuous random variable
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

42. Homoskedastic only F - stat
When one regressor is a perfect linear function of the other regressors
Var(X) + Var(Y)
Population denominator = n - Sample denominator = n - 1
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

43. WLS
Mean = np - Variance = npq - Std dev = sqrt(npq)
Sampling distribution of sample means tend to be normal
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

44. 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)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Z = (Y - meany)/(stddev(y)/sqrt(n))

45. LFHS
Low Frequency - High Severity events
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Contains variables not explicit in model - Accounts for randomness
Application of mathematical statistics to economic data to lend empirical support to models

46. EWMA
Statement of the error or precision of an estimate
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!)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

47. Gamma distribution
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
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
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Returns over time for a combination of assets (combination of time series and cross - sectional data)

48. Tractable
Least absolute deviations estimator - used when extreme outliers are not uncommon
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Easy to manipulate
Independently and Identically Distributed

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

50. P - value
Mean = np - Variance = npq - Std dev = sqrt(npq)
When the sample size is large - the uncertainty about the value of the sample is very small
Price/return tends to run towards a long - run level
P(Z>t)