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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. Hybrid method for conditional volatility
Distribution with only two possible outcomes
Variance reverts to a long run level
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Use historical simulation approach but use the EWMA weighting system

2. Central Limit Theorem
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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)
For n>30 - sample mean is approximately normal
Choose parameters that maximize the likelihood of what observations occurring

3. Variance of aX
Sample mean +/ - t*(stddev(s)/sqrt(n))
Independently and Identically Distributed
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!)
(a^2)(variance(x)

4. Expected future variance rate (t periods forward)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
P(X=x - Y=y) = P(X=x) * P(Y=y)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

5. Maximum likelihood method
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
Sample mean will near the population mean as the sample size increases
Choose parameters that maximize the likelihood of what observations occurring

6. Direction of OVB
Average return across assets on a given day
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(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
Confidence level

7. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Variance(y)/n = variance of sample Y
Easy to manipulate
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

8. Poisson Distribution
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Price/return tends to run towards a long - run level

9. Reliability
Model dependent - Options with the same underlying assets may trade at different volatilities
Statement of the error or precision of an estimate
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

10. Multivariate probability
Attempts to sample along more important paths
Statement of the error or precision of an estimate
More than one random variable
Population denominator = n - Sample denominator = n - 1

11. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance reverts to a long run level
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

12. Result of combination of two normal with same means
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Regression can be non - linear in variables but must be linear in parameters
Variance(x)
Combine to form distribution with leptokurtosis (heavy tails)

13. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Rxy = Sxy/(Sx*Sy)
P(Z>t)

14. Implied standard deviation for options
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Sampling distribution of sample means tend to be normal
Variance(y)/n = variance of sample Y

15. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Sampling distribution of sample means tend to be normal
Sample mean +/ - t*(stddev(s)/sqrt(n))
Attempts to sample along more important paths

16. Continuous representation of the GBM
Application of mathematical statistics to economic data to lend empirical support to models
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Least absolute deviations estimator - used when extreme outliers are not uncommon
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)

17. Standard error for Monte Carlo replications
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Combine to form distribution with leptokurtosis (heavy tails)

18. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Contains variables not explicit in model - Accounts for randomness
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Rxy = Sxy/(Sx*Sy)

19. Mean reversion in variance
Variance reverts to a long run level
P(X=x - Y=y) = P(X=x) * P(Y=y)
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

20. K - th moment
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
Variance(y)/n = variance of sample Y
Summation((xi - mean)^k)/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

21. BLUE
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

22. Consistent
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
When the sample size is large - the uncertainty about the value of the sample is very small
Independently and Identically Distributed
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

23. F distribution
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
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

24. GARCH
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
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)
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
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

25. R^2
P - value
SSR
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

26. Binomial distribution equations for mean variance and std dev
Least absolute deviations estimator - used when extreme outliers are not uncommon
Mean = np - Variance = npq - Std dev = sqrt(npq)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Probability that the random variables take on certain values simultaneously

27. Two requirements of OVB
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Normal - Student's T - Chi - square - F distribution
Peaks over threshold - Collects dataset in excess of some threshold
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

28. Unstable return distribution
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Z = (Y - meany)/(stddev(y)/sqrt(n))
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

29. GPD
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Special type of pooled data in which the cross sectional unit is surveyed over time
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

30. Central Limit Theorem(CLT)
Application of mathematical statistics to economic data to lend empirical support to models
P - value
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
Sampling distribution of sample means tend to be normal

31. Type II Error
We accept a hypothesis that should have been rejected
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Combine to form distribution with leptokurtosis (heavy tails)
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

32. Variance of X - Y assuming dependence
Variance = (1/m) summation(u<n - i>^2)
Variance(X) + Variance(Y) - 2*covariance(XY)
Returns over time for an individual asset
Population denominator = n - Sample denominator = n - 1

33. Weibul distribution
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
P(Z>t)
Use historical simulation approach but use the EWMA weighting system
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

34. Multivariate Density Estimation (MDE)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Choose parameters that maximize the likelihood of what observations occurring

35. Variance of weighted scheme
Random walk (usually acceptable) - Constant volatility (unlikely)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Distribution with only two possible outcomes
For n>30 - sample mean is approximately normal

36. Historical std dev
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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

37. Key properties of linear regression
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Expected value of the sample mean is the population mean
P - value
Regression can be non - linear in variables but must be linear in parameters

38. Mean(expected value)
Regression can be non - linear in variables but must be linear in parameters
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

39. Cholesky factorization (decomposition)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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
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

40. Statistical (or empirical) model
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Mean = np - Variance = npq - Std dev = sqrt(npq)
(a^2)(variance(x)) + (b^2)(variance(y))
Yi = B0 + B1Xi + ui

41. Sample mean
Based on a dataset
Expected value of the sample mean is the population mean
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

42. Homoskedastic only F - stat
Low Frequency - High Severity events
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Model dependent - Options with the same underlying assets may trade at different volatilities

43. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Transformed to a unit variable - Mean = 0 Variance = 1
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

44. SER
Nonlinearity
E(mean) = mean
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Least absolute deviations estimator - used when extreme outliers are not uncommon

45. Biggest (and only real) drawback of GARCH mode
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Nonlinearity
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. Variance of sampling distribution of means when n<N
Model dependent - Options with the same underlying assets may trade at different volatilities
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Var(X) + Var(Y)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

47. SER
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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
Concerned with a single random variable (ex. Roll of a die)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

48. Variance of X+Y assuming dependence
Independently and Identically Distributed
Mean of sampling distribution is the population mean
(a^2)(variance(x)) + (b^2)(variance(y))
Variance(x) + Variance(Y) + 2*covariance(XY)

49. Critical z values
Least absolute deviations estimator - used when extreme outliers are not uncommon
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
95% = 1.65 99% = 2.33 For one - tailed tests
Mean = np - Variance = npq - Std dev = sqrt(npq)

50. Variance of X+b
Variance(x)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)