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FRM Foundations Of Risk Management Quantitative Methods

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1. GARCH
Confidence level
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)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

2. SER
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Based on an equation - P(A) = # of A/total outcomes
Nonlinearity
Choose parameters that maximize the likelihood of what observations occurring

3. Sample mean
Use historical simulation approach but use the EWMA weighting system
Expected value of the sample mean is the population mean
E(XY) - E(X)E(Y)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

4. Control variates technique
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Rxy = Sxy/(Sx*Sy)
E(XY) - E(X)E(Y)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

5. Normal distribution
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
Attempts to sample along more important paths
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Peaks over threshold - Collects dataset in excess of some threshold

6. Efficiency
Combine to form distribution with leptokurtosis (heavy tails)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Among all unbiased estimators - estimator with the smallest variance is efficient
Variance(X) + Variance(Y) - 2*covariance(XY)

7. Simplified standard (un - weighted) variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Normal - Student's T - Chi - square - F distribution
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
Variance = (1/m) summation(u<n - i>^2)

8. Exact significance level
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
P - value

9. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Rxy = Sxy/(Sx*Sy)
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
Least absolute deviations estimator - used when extreme outliers are not uncommon

10. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

11. Bernouli Distribution
Contains variables not explicit in model - Accounts for randomness
We reject a hypothesis that is actually true
Distribution with only two possible outcomes
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

12. Variance of aX
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
(a^2)(variance(x)
P(Z>t)

13. Mean reversion in variance
Variance = (1/m) summation(u<n - i>^2)
Variance reverts to a long run level
Combine to form distribution with leptokurtosis (heavy tails)
SSR

14. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Least absolute deviations estimator - used when extreme outliers are not uncommon

15. Variance of weighted scheme
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Variance = (1/m) summation(u<n - i>^2)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Contains variables not explicit in model - Accounts for randomness

16. GPD
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
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Attempts to sample along more important paths

17. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Transformed to a unit variable - Mean = 0 Variance = 1
Regression can be non - linear in variables but must be linear in parameters
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

18. Least squares estimator(m)
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Application of mathematical statistics to economic data to lend empirical support to models
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

19. Difference between population and sample variance
Combine to form distribution with leptokurtosis (heavy tails)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Population denominator = n - Sample denominator = n - 1
Sampling distribution of sample means tend to be normal

20. Variance of sampling distribution of means when n<N
Nonlinearity
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
(a^2)(variance(x)

21. Confidence ellipse
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
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
Confidence set for two coefficients - two dimensional analog for the confidence interval
Variance(x)

22. Overall F - statistic
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Sample mean will near the population mean as the sample size increases
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

23. Standard normal distribution
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Transformed to a unit variable - Mean = 0 Variance = 1

24. Antithetic variable technique
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Based on a dataset
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

25. Discrete representation of the GBM
E(mean) = mean
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Low Frequency - High Severity events
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

26. Sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Mean = np - Variance = npq - Std dev = sqrt(npq)
Average return across assets on a given day
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

27. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Only requires two parameters = mean and variance
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

28. Skewness
Easy to manipulate
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

29. Four sampling distributions

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30. Exponential distribution
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Transformed to a unit variable - Mean = 0 Variance = 1
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

31. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
E(XY) - E(X)E(Y)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance = (1/m) summation(u<n - i>^2)

32. Implications of homoscedasticity
Independently and Identically Distributed
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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

33. Unstable return distribution
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
P(Z>t)
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

34. Empirical frequency
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Based on a dataset
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Confidence set for two coefficients - two dimensional analog for the confidence interval

35. Persistence
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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
Random walk (usually acceptable) - Constant volatility (unlikely)
Only requires two parameters = mean and variance

36. Cholesky factorization (decomposition)
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Only requires two parameters = mean and variance

37. Two ways to calculate historical volatility
Variance(y)/n = variance of sample Y
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
P(Z>t)

38. Regime - switching volatility model
Variance(X) + Variance(Y) - 2*covariance(XY)
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

39. Statistical (or empirical) model
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Attempts to sample along more important paths
Yi = B0 + B1Xi + ui

40. Central Limit Theorem
For n>30 - sample mean is approximately normal
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Average return across assets on a given day
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

41. What does the OLS minimize?
Contains variables not explicit in model - Accounts for randomness
SSR
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"

42. Chi - squared distribution
Yi = B0 + B1Xi + ui
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
Model dependent - Options with the same underlying assets may trade at different volatilities
Based on a dataset

43. Direction of OVB
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

44. Sample correlation
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Rxy = Sxy/(Sx*Sy)

45. P - value
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
P(Z>t)
Application of mathematical statistics to economic data to lend empirical support to models

46. Biggest (and only real) drawback of GARCH mode
Nonlinearity
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

47. Extreme Value Theory
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Transformed to a unit variable - Mean = 0 Variance = 1
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

48. i.i.d.
Independently and Identically Distributed
Probability that the random variables take on certain values simultaneously
Returns over time for a combination of assets (combination of time series and cross - sectional data)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

49. Discrete random variable
Z = (Y - meany)/(stddev(y)/sqrt(n))
Has heavy tails
Low Frequency - High Severity events
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

50. Shortcomings of implied volatility
Variance(X) + Variance(Y) - 2*covariance(XY)
Normal - Student's T - Chi - square - F distribution
Model dependent - Options with the same underlying assets may trade at different volatilities
Regression can be non - linear in variables but must be linear in parameters

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