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FRM Foundations Of Risk Management Quantitative Methods

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1. Covariance calculations using weight sums (lambda)
Mean = np - Variance = npq - Std dev = sqrt(npq)
(a^2)(variance(x)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Application of mathematical statistics to economic data to lend empirical support to models

2. LFHS
Low Frequency - High Severity events
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Mean of sampling distribution is the population mean
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

3. Homoskedastic only F - stat
When the sample size is large - the uncertainty about the value of the sample is very small
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

4. K - th moment
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
Summation((xi - mean)^k)/n
Contains variables not explicit in model - Accounts for randomness
Population denominator = n - Sample denominator = n - 1

5. WLS
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
When one regressor is a perfect linear function of the other regressors
Normal - Student's T - Chi - square - F distribution

6. Empirical frequency
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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
Based on a dataset
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

7. Type I error
Average return across assets on a given day
Probability that the random variables take on certain values simultaneously
We reject a hypothesis that is actually true
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

8. Variance - covariance approach for VaR of a portfolio
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
E(mean) = mean
Least absolute deviations estimator - used when extreme outliers are not uncommon

9. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Statement of the error or precision of an estimate
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

10. Monte Carlo Simulations
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Variance(y)/n = variance of sample Y
Based on an equation - P(A) = # of A/total outcomes
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

11. Result of combination of two normal with same means
Combine to form distribution with leptokurtosis (heavy tails)
95% = 1.65 99% = 2.33 For one - tailed tests
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Concerned with a single random variable (ex. Roll of a die)

12. Sample variance
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

13. Significance =1
Confidence level
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

14. Reliability
Statement of the error or precision of an estimate
Variance(x)
Attempts to sample along more important paths
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

15. Multivariate Density Estimation (MDE)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
F = ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

16. Antithetic variable technique
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Variance(x)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

17. Discrete random variable
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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

18. Extending the HS approach for computing value of a portfolio
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Sample mean +/ - t*(stddev(s)/sqrt(n))
95% = 1.65 99% = 2.33 For one - tailed tests
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

19. Kurtosis
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Transformed to a unit variable - Mean = 0 Variance = 1
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
E(XY) - E(X)E(Y)

20. Historical std dev
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

21. F distribution
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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
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

22. Bootstrap method
Rxy = Sxy/(Sx*Sy)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Based on a dataset

23. Time series data
When one regressor is a perfect linear function of the other regressors
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Returns over time for an individual asset
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

24. Exact significance level
If variance of the conditional distribution of u(i) is not constant
Distribution with only two possible outcomes
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
P - value

25. BLUE
Z = (Y - meany)/(stddev(y)/sqrt(n))
Returns over time for a combination of assets (combination of time series and cross - sectional data)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

26. Persistence
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
If variance of the conditional distribution of u(i) is not constant
Does not depend on a prior event or information

27. Importance sampling technique
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
F = ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Attempts to sample along more important paths

28. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
SSR
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
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

29. Difference between population and sample variance
P - value
Population denominator = n - Sample denominator = n - 1
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

30. Least squares estimator(m)
Change in S = S<t - 1>(meanchange in time + stddev 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!)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

31. Implied standard deviation for options
Transformed to a unit variable - Mean = 0 Variance = 1
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
(a^2)(variance(x)) + (b^2)(variance(y))

32. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
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
We reject a hypothesis that is actually true
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

33. ESS
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

34. Two ways to calculate historical volatility
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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

35. Normal 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
We accept a hypothesis that should have been rejected
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
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

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

37. Tractable
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Easy to manipulate
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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

38. Logistic distribution
P(Z>t)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Has heavy tails
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

39. Lognormal
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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

40. Hazard rate of exponentially distributed random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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

41. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
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

42. GPD
Among all unbiased estimators - estimator with the smallest variance is efficient
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
P(Z>t)

43. Implications of homoscedasticity
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Regression can be non - linear in variables but must be linear in parameters
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 an individual asset

44. Statistical (or empirical) model
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Yi = B0 + B1Xi + ui
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
Confidence set for two coefficients - two dimensional analog for the confidence interval

45. EWMA
Distribution with only two possible outcomes
E(XY) - E(X)E(Y)
Does not depend on a prior event or information
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

46. 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
Variance(sample y) = (variance(y)/n)*(N - n/N - 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
Combine to form distribution with leptokurtosis (heavy tails)

47. Test for unbiasedness
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Sampling distribution of sample means tend to be normal
E(mean) = mean
Model dependent - Options with the same underlying assets may trade at different volatilities

48. Continuous representation of the GBM
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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)
Has heavy tails
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

49. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Has heavy tails
Z = (Y - meany)/(stddev(y)/sqrt(n))
Average return across assets on a given day

50. Standard error
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

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FRM Foundations Of Risk Management Quantitative Methods

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