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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. GEV
Concerned with a single random variable (ex. Roll of a die)
Low Frequency - High Severity events
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Nonlinearity

2. Empirical frequency
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Based on a dataset
Does not depend on a prior event or information
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

3. Central Limit Theorem
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Transformed to a unit variable - Mean = 0 Variance = 1
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
For n>30 - sample mean is approximately normal

4. Binomial distribution equations for mean variance and std dev
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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

5. Cholesky factorization (decomposition)
Expected value of the sample mean is the population mean
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Transformed to a unit variable - Mean = 0 Variance = 1

6. Conditional probability functions
95% = 1.65 99% = 2.33 For one - tailed tests
Sample mean +/ - t*(stddev(s)/sqrt(n))
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

7. Square root rule
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
Rxy = Sxy/(Sx*Sy)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Among all unbiased estimators - estimator with the smallest variance is efficient

8. Implications of homoscedasticity
95% = 1.65 99% = 2.33 For one - tailed tests
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Average return across assets on a given day

9. Statistical (or empirical) model
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Yi = B0 + B1Xi + ui
When one regressor is a perfect linear function of the other regressors

10. Time series data
Mean of sampling distribution is the population mean
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Rxy = Sxy/(Sx*Sy)
Returns over time for an individual asset

11. Single variable (univariate) probability
Mean = np - Variance = npq - Std dev = sqrt(npq)
i = ln(Si/Si - 1)
Concerned with a single random variable (ex. Roll of a die)
SSR

12. Continuous random variable
Variance reverts to a long run level
Among all unbiased estimators - estimator with the smallest variance is efficient
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

13. Bernouli Distribution
Returns over time for an individual asset
E(mean) = mean
Distribution with only two possible outcomes
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. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

15. Covariance calculations using weight sums (lambda)
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
Low Frequency - High Severity events
Population denominator = n - Sample denominator = n - 1
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

16. Lognormal
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
(a^2)(variance(x)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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

17. Difference between population and sample variance
When the sample size is large - the uncertainty about the value of the sample is very small
Population denominator = n - Sample denominator = n - 1
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Choose parameters that maximize the likelihood of what observations occurring

18. WLS
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Independently and Identically Distributed

19. Discrete random variable
Low Frequency - High Severity events
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

20. Standard error
Price/return tends to run towards a long - run level
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

21. GARCH
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Population denominator = n - Sample denominator = n - 1
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)

22. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

23. Test for unbiasedness
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
E(mean) = mean

24. 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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

25. Block maxima
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

26. Two ways to calculate historical volatility
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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 = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

27. Implied standard deviation for options
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Independently and Identically Distributed
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

28. Perfect multicollinearity
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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!)
When one regressor is a perfect linear function of the other regressors

29. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance(x) + Variance(Y) + 2*covariance(XY)
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)
Easy to manipulate

30. Law of Large Numbers
Use historical simulation approach but use the EWMA weighting system
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!)
Sample mean will near the population mean as the sample size increases
Variance(X) + Variance(Y) - 2*covariance(XY)

31. Multivariate probability
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
More than one random variable
Random walk (usually acceptable) - Constant volatility (unlikely)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

32. Joint probability functions
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Probability that the random variables take on certain values simultaneously
E(XY) - E(X)E(Y)
Easy to manipulate

33. Beta distribution
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

34. Standard error for Monte Carlo replications
Rxy = Sxy/(Sx*Sy)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Contains variables not explicit in model - Accounts for randomness

35. Covariance
E(XY) - E(X)E(Y)
Choose parameters that maximize the likelihood of what observations occurring
Expected value of the sample mean is the population mean
Sampling distribution of sample means tend to be normal

36. LFHS
Based on an equation - P(A) = # of A/total outcomes
Low Frequency - High Severity events
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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

37. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
For n>30 - sample mean is approximately normal
Mean of sampling distribution is the population mean

38. Key properties of linear regression
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Variance reverts to a long run level
Regression can be non - linear in variables but must be linear in parameters
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

39. Chi - squared distribution
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Variance(X) + Variance(Y) - 2*covariance(XY)

40. Homoskedastic
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
i = ln(Si/Si - 1)
Mean = np - Variance = npq - Std dev = sqrt(npq)

41. Result of combination of two normal with same means
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Combine to form distribution with leptokurtosis (heavy tails)
Yi = B0 + B1Xi + ui
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

42. Two assumptions of square root rule
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
When one regressor is a perfect linear function of the other regressors
Sample mean +/ - t*(stddev(s)/sqrt(n))
Random walk (usually acceptable) - Constant volatility (unlikely)

43. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

44. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
P(Z>t)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Variance(y)/n = variance of sample Y

45. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
(a^2)(variance(x)) + (b^2)(variance(y))
Sample mean +/ - t*(stddev(s)/sqrt(n))

46. Antithetic variable technique
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
P(X=x - Y=y) = P(X=x) * P(Y=y)

47. Variance of X+Y
Var(X) + Var(Y)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Normal - Student's T - Chi - square - F distribution
Variance(x) + Variance(Y) + 2*covariance(XY)

48. Mean(expected value)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Regression can be non - linear in variables but must be linear in parameters
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
When the sample size is large - the uncertainty about the value of the sample is very small

49. Two requirements of OVB
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
P - value
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

50. 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)
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!)
Var(X) + Var(Y)
When the sample size is large - the uncertainty about the value of the sample is very small