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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Standard normal distribution
Summation((xi - mean)^k)/n
Transformed to a unit variable - Mean = 0 Variance = 1
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

2. Discrete random variable
Choose parameters that maximize the likelihood of what observations occurring
Low Frequency - High Severity events
Variance(X) + Variance(Y) - 2*covariance(XY)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

3. Standard error
Low Frequency - High Severity events
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Average return across assets on a given day
Concerned with a single random variable (ex. Roll of a die)

4. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
Z = (Y - meany)/(stddev(y)/sqrt(n))

5. K - th moment
SSR
Population denominator = n - Sample denominator = n - 1
Summation((xi - mean)^k)/n
Among all unbiased estimators - estimator with the smallest variance is efficient

6. Mean(expected value)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Sampling distribution of sample means tend to be normal
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

7. Result of combination of two normal with same means
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Combine to form distribution with leptokurtosis (heavy tails)
Application of mathematical statistics to economic data to lend empirical support to models
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

8. Sample variance
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

9. Test for unbiasedness
When one regressor is a perfect linear function of the other regressors
SSR
E(mean) = mean
Only requires two parameters = mean and variance

10. Simulating for VaR
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Population denominator = n - Sample denominator = n - 1

11. Unbiased
Mean of sampling distribution is the population mean
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

12. LFHS
Low Frequency - High Severity events
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
For n>30 - sample mean is approximately normal

13. Pooled data
Nonlinearity
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)

14. Two ways to calculate historical volatility
Z = (Y - meany)/(stddev(y)/sqrt(n))
Least absolute deviations estimator - used when extreme outliers are not uncommon
Variance(X) + Variance(Y) - 2*covariance(XY)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

15. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Z = (Y - meany)/(stddev(y)/sqrt(n))
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
Confidence level

16. Variance - covariance approach for VaR of a portfolio
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
More than one random variable
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

17. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Regression can be non - linear in variables but must be linear in parameters
Model dependent - Options with the same underlying assets may trade at different volatilities

18. GEV
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Variance(x) + Variance(Y) + 2*covariance(XY)

19. 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
SSR
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Yi = B0 + B1Xi + ui

20. Heteroskedastic
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
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
If variance of the conditional distribution of u(i) is not constant

21. Tractable
Regression can be non - linear in variables but must be linear in parameters
Independently and Identically Distributed
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Easy to manipulate

22. Sample mean
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Expected value of the sample mean is the population mean
When one regressor is a perfect linear function of the other regressors

23. Significance =1
Peaks over threshold - Collects dataset in excess of some threshold
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Confidence level
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

24. Implications of homoscedasticity
Distribution with only two possible outcomes
When the sample size is large - the uncertainty about the value of the sample is very small
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
Easy to manipulate

25. Sample correlation
Z = (Y - meany)/(stddev(y)/sqrt(n))
Variance(X) + Variance(Y) - 2*covariance(XY)
Rxy = Sxy/(Sx*Sy)
Contains variables not explicit in model - Accounts for randomness

26. Least squares estimator(m)
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
E(mean) = mean
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

27. Standard variable for non - normal distributions
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Z = (Y - meany)/(stddev(y)/sqrt(n))
Summation((xi - mean)^k)/n

28. Discrete representation of the GBM
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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

29. GPD
Transformed to a unit variable - Mean = 0 Variance = 1
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Distribution with only two possible outcomes
Only requires two parameters = mean and variance

30. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Variance(x)

31. Covariance calculations using weight sums (lambda)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Based on an equation - P(A) = # of A/total outcomes
We accept a hypothesis that should have been rejected

32. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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
Probability that the random variables take on certain values simultaneously
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

33. Econometrics
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
Transformed to a unit variable - Mean = 0 Variance = 1
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Application of mathematical statistics to economic data to lend empirical support to models

34. Direction of OVB
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

35. Adjusted R^2
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 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
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

36. Extreme Value Theory
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Independently and Identically Distributed
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

37. What does the OLS minimize?
Based on a dataset
SSR
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
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

38. Priori (classical) probability
P(Z>t)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Among all unbiased estimators - estimator with the smallest variance is efficient
Based on an equation - P(A) = # of A/total outcomes

39. Bernouli Distribution
Distribution with only two possible outcomes
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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(x) = (1/(beta tao(alpha)) e^( - x/beta) * (x/beta)^(alpha - 1) - Alpha = 1 - becomes exponential - Alpha = k/2 beta = 2 - becomes chi - squared

40. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Independently and Identically Distributed
Yi = B0 + B1Xi + ui

41. Variance of aX + bY
(a^2)(variance(x)) + (b^2)(variance(y))
Sample mean +/ - t*(stddev(s)/sqrt(n))
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

42. Standard error for Monte Carlo replications
Combine to form distribution with leptokurtosis (heavy tails)
Variance(X) + Variance(Y) - 2*covariance(XY)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
We reject a hypothesis that is actually true

43. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Sampling distribution of sample means tend to be normal
Based on a dataset
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

44. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
More than one random variable
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
SSR

45. Continuous representation of the GBM
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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)
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
Variance = (1/m) summation(u<n - i>^2)

46. T distribution
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

47. Normal distribution
Var(X) + Var(Y)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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

48. Confidence interval (from t)
Among all unbiased estimators - estimator with the smallest variance is efficient
Sample mean +/ - t*(stddev(s)/sqrt(n))
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

49. Marginal unconditional probability function
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance = (1/m) summation(u<n - i>^2)
Does not depend on a prior event or information

50. Four sampling distributions

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