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FRM Foundations Of Risk Management Quantitative Methods

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1. SER
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Average return across assets on a given day
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

2. Lognormal
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
Mean of sampling distribution is the population mean
Variance(x) + Variance(Y) + 2*covariance(XY)
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

3. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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)

4. Time series data
Returns over time for an individual asset
Contains variables not explicit in model - Accounts for randomness
Summation((xi - mean)^k)/n
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

5. Hybrid method for conditional volatility
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Choose parameters that maximize the likelihood of what observations occurring
Use historical simulation approach but use the EWMA weighting system
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

6. Covariance
P - value
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
E(XY) - E(X)E(Y)
When the sample size is large - the uncertainty about the value of the sample is very small

7. LFHS
Population denominator = n - Sample denominator = n - 1
Low Frequency - High Severity events
Average return across assets on a given day
When one regressor is a perfect linear function of the other regressors

8. Significance =1
Confidence level
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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

9. Implied standard deviation for options
Use historical simulation approach but use the EWMA weighting system
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Concerned with a single random variable (ex. Roll of a die)

10. Antithetic variable technique
Concerned with a single random variable (ex. Roll of a die)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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

11. Continuous random variable
Summation((xi - mean)^k)/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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

12. Variance of X - Y assuming dependence
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Variance(X) + Variance(Y) - 2*covariance(XY)

13. Persistence
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
P(Z>t)
95% = 1.65 99% = 2.33 For one - tailed tests
i = ln(Si/Si - 1)

14. Standard variable for non - normal distributions
Var(X) + Var(Y)
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))
Combine to form distribution with leptokurtosis (heavy tails)

15. Variance of X+Y
Var(X) + Var(Y)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
E(XY) - E(X)E(Y)
Sampling distribution of sample means tend to be normal

16. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Random walk (usually acceptable) - Constant volatility (unlikely)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

17. Implications of homoscedasticity
Peaks over threshold - Collects dataset in excess of some threshold
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
Price/return tends to run towards a long - run level
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

18. Extreme Value Theory
Use historical simulation approach but use the EWMA weighting system
Variance reverts to a long run level
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

19. Adjusted R^2
Variance(x) + Variance(Y) + 2*covariance(XY)
Only requires two parameters = mean and variance
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
Independently and Identically Distributed

20. Potential reasons for fat tails in return distributions
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)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
When one regressor is a perfect linear function of the other regressors

21. Discrete random variable
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Transformed to a unit variable - Mean = 0 Variance = 1

22. R^2
P - value
E(XY) - E(X)E(Y)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

23. Variance of sample mean
Variance(y)/n = variance of sample Y
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
E(mean) = mean

24. Exact significance level
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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
P - value

25. Shortcomings of implied volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance(y)/n = variance of sample Y
Model dependent - Options with the same underlying assets may trade at different volatilities
Use historical simulation approach but use the EWMA weighting system

26. Sample mean
Variance(x) + Variance(Y) + 2*covariance(XY)
We reject a hypothesis that is actually true
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
Expected value of the sample mean is the population mean

27. Variance - covariance approach for VaR of a portfolio
Based on an equation - P(A) = # of A/total outcomes
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

28. Importance sampling technique
Confidence set for two coefficients - two dimensional analog for the confidence interval
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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!)
Attempts to sample along more important paths

29. Statistical (or empirical) model
Z = (Y - meany)/(stddev(y)/sqrt(n))
Yi = B0 + B1Xi + ui
When the sample size is large - the uncertainty about the value of the sample is very small
Variance(x)

30. Confidence interval for sample mean
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!)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Mean of sampling distribution is the population mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

31. P - value
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Transformed to a unit variable - Mean = 0 Variance = 1
When the sample size is large - the uncertainty about the value of the sample is very small
P(Z>t)

32. 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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Variance(X) + Variance(Y) - 2*covariance(XY)
(a^2)(variance(x)) + (b^2)(variance(y))

33. Type II Error
We accept a hypothesis that should have been rejected
Variance(x) + Variance(Y) + 2*covariance(XY)
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

34. Consistent
When the sample size is large - the uncertainty about the value of the sample is very small
Mean of sampling distribution is the population mean
E(mean) = mean
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

35. Marginal unconditional probability function
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
(a^2)(variance(x)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Does not depend on a prior event or information

36. Binomial distribution
Transformed to a unit variable - Mean = 0 Variance = 1
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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!)
Least absolute deviations estimator - used when extreme outliers are not uncommon

37. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Based on an equation - P(A) = # of A/total outcomes
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Concerned with a single random variable (ex. Roll of a die)

38. Kurtosis
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

39. i.i.d.
Independently and Identically Distributed
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Among all unbiased estimators - estimator with the smallest variance is efficient

40. Cross - sectional
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Average return across assets on a given day
Least absolute deviations estimator - used when extreme outliers are not uncommon

41. Conditional probability functions
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
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
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

42. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
(a^2)(variance(x)) + (b^2)(variance(y))

43. Mean(expected value)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Summation((xi - mean)^k)/n

44. 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
Probability that the random variables take on certain values simultaneously
Based on an equation - P(A) = # of A/total outcomes
E(mean) = mean

45. Economical(elegant)
Variance reverts to a long run level
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Only requires two parameters = mean and variance
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

46. Empirical frequency
Based on a dataset
Variance(X) + Variance(Y) - 2*covariance(XY)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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

47. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

48. Square root rule
Summation((xi - mean)^k)/n
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Average return across assets on a given day
We reject a hypothesis that is actually true

49. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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

50. Skewness
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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
Sample mean +/ - t*(stddev(s)/sqrt(n))
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

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FRM Foundations Of Risk Management Quantitative Methods

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