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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Extreme Value Theory
Use historical simulation approach but use the EWMA weighting system
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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

2. Variance of aX
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
(a^2)(variance(x)

3. Central Limit Theorem
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
For n>30 - sample mean is approximately normal
Confidence 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

4. Biggest (and only real) drawback of GARCH mode
P - value
Nonlinearity
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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

5. Variance of X+Y assuming dependence
Variance = (1/m) summation(u<n - i>^2)
When the sample size is large - the uncertainty about the value of the sample is very small
Contains variables not explicit in model - Accounts for randomness
Variance(x) + Variance(Y) + 2*covariance(XY)

6. Potential reasons for fat tails in return distributions
Least absolute deviations estimator - used when extreme outliers are not uncommon
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)
Population denominator = n - Sample denominator = n - 1
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

7. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Expected value of the sample mean is the population mean
Transformed to a unit variable - Mean = 0 Variance = 1
Rxy = Sxy/(Sx*Sy)

8. What does the OLS minimize?
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
SSR
We reject a hypothesis that is actually true

9. T distribution
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Average return across assets on a given day

10. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
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

11. Sample correlation
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Use historical simulation approach but use the EWMA weighting system
Rxy = Sxy/(Sx*Sy)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

12. Variance of aX + bY
Yi = B0 + B1Xi + ui
(a^2)(variance(x)) + (b^2)(variance(y))
Independently and Identically Distributed
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

13. Time series data
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Variance reverts to a long run level
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Returns over time for an individual asset

14. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

15. Importance sampling technique
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Attempts to sample along more important paths

16. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Choose parameters that maximize the likelihood of what observations occurring

17. Econometrics
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
(a^2)(variance(x)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Application of mathematical statistics to economic data to lend empirical support to models

18. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
Has heavy tails
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

19. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Easy to manipulate
P - value
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

20. Confidence interval for sample mean
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
(a^2)(variance(x)) + (b^2)(variance(y))
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Summation((xi - mean)^k)/n

21. Marginal unconditional probability function
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
Does not depend on a prior event or information
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

22. Monte Carlo Simulations
Regression can be non - linear in variables but must be linear in parameters
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

23. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
i = ln(Si/Si - 1)
Variance(x)

24. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Yi = B0 + B1Xi + ui

25. Economical(elegant)
Choose parameters that maximize the likelihood of what observations occurring
Contains variables not explicit in model - Accounts for randomness
Has heavy tails
Only requires two parameters = mean and variance

26. Regime - switching volatility model
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Rxy = Sxy/(Sx*Sy)

27. Confidence ellipse
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Confidence set for two coefficients - two dimensional analog for the confidence interval
Contains variables not explicit in model - Accounts for randomness
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

28. Standard variable for non - normal distributions
SSR
Z = (Y - meany)/(stddev(y)/sqrt(n))
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

29. Two requirements of OVB
Sample mean will near the population mean as the sample size increases
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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)

30. WLS
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
Model dependent - Options with the same underlying assets may trade at different volatilities
Confidence level
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

31. Variance - covariance approach for VaR of a portfolio
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

32. Covariance
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
E(XY) - E(X)E(Y)
When the sample size is large - the uncertainty about the value of the sample is very small
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

33. 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)
If variance of the conditional distribution of u(i) is not constant
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

34. 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)
E(mean) = mean
Variance(x)
Variance(x) + Variance(Y) + 2*covariance(XY)

35. Persistence
Special type of pooled data in which the cross sectional unit is surveyed over time
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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
E(XY) - E(X)E(Y)

36. Consistent
For n>30 - sample mean is approximately normal
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
When the sample size is large - the uncertainty about the value of the sample is very small

37. Block maxima
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.
Var(X) + Var(Y)
Yi = B0 + B1Xi + ui

38. Overall F - statistic
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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 = ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Use historical simulation approach but use the EWMA weighting system

39. Result of combination of two normal with same means
More than one random variable
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)
Combine to form distribution with leptokurtosis (heavy tails)
Model dependent - Options with the same underlying assets may trade at different volatilities

40. K - th moment
Summation((xi - mean)^k)/n
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
i = ln(Si/Si - 1)
F = ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

41. Unbiased
Mean of sampling distribution is the population mean
Confidence set for two coefficients - two dimensional analog for the confidence interval
Summation((xi - mean)^k)/n
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

42. Variance of X - Y assuming dependence
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
Combine to form distribution with leptokurtosis (heavy tails)
Variance(X) + Variance(Y) - 2*covariance(XY)
Confidence level

43. F distribution
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
When the sample size is large - the uncertainty about the value of the sample is very small
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

44. Multivariate probability
Contains variables not explicit in model - Accounts for randomness
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
More than one random variable
Does not depend on a prior event or information

45. Mean reversion
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

46. Cholesky factorization (decomposition)
Population denominator = n - Sample denominator = n - 1
Statement of the error or precision of an estimate
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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
Low Frequency - High Severity events
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

48. Tractable
Easy to manipulate
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Special type of pooled data in which the cross sectional unit is surveyed over time
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

49. Difference between population and sample variance
Mean of sampling distribution is the population mean
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Population denominator = n - Sample denominator = n - 1
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

50. R^2
Population denominator = n - Sample denominator = n - 1
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Distribution with only two possible outcomes
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

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FRM Foundations Of Risk Management Quantitative Methods

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