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FRM Foundations Of Risk Management Quantitative Methods

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1. LFHS
Low Frequency - High Severity events
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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!)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

2. Hazard rate of exponentially distributed random variable
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

3. Confidence interval for sample mean
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)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

4. Variance of X+Y
Yi = B0 + B1Xi + ui
Var(X) + Var(Y)
Summation((xi - mean)^k)/n
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

5. Antithetic variable technique
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

6. Extending the HS approach for computing value of a portfolio
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Sample mean +/ - t*(stddev(s)/sqrt(n))
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

7. SER
Special type of pooled data in which the cross sectional unit is surveyed over time
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

8. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
Model dependent - Options with the same underlying assets may trade at different volatilities
Low Frequency - High Severity events
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

9. Key properties of linear regression
Price/return tends to run towards a long - run level
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Among all unbiased estimators - estimator with the smallest variance is efficient
Regression can be non - linear in variables but must be linear in parameters

10. Time series data
Returns over time for an individual asset
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Rxy = Sxy/(Sx*Sy)
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. Cholesky factorization (decomposition)
Variance(x) + Variance(Y) + 2*covariance(XY)
Least absolute deviations estimator - used when extreme outliers are not uncommon
More than one random variable
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

12. Multivariate Density Estimation (MDE)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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
Probability that the random variables take on certain values simultaneously
Variance(x) + Variance(Y) + 2*covariance(XY)

13. Efficiency
Mean of sampling distribution is the population mean
Nonlinearity
Among all unbiased estimators - estimator with the smallest variance is efficient
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

14. K - th moment
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Summation((xi - mean)^k)/n
Based on a dataset
P - value

15. Panel data (longitudinal or micropanel)
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
Independently and Identically Distributed
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Special type of pooled data in which the cross sectional unit is surveyed over time

16. Biggest (and only real) drawback of GARCH mode
Nonlinearity
Does not depend on a prior event or information
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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)

17. Weibul distribution
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Mean = np - Variance = npq - Std dev = sqrt(npq)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
(a^2)(variance(x)

18. Continuous representation of the GBM
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
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)
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

19. Test for statistical independence
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
P(X=x - Y=y) = P(X=x) * P(Y=y)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

20. Regime - switching volatility model
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Independently and Identically Distributed

21. Exponential distribution
Among all unbiased estimators - estimator with the smallest variance is efficient
We accept a hypothesis that should have been rejected
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

22. Control variates technique
Summation((xi - mean)^k)/n
Sample mean +/ - t*(stddev(s)/sqrt(n))
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Population denominator = n - Sample denominator = n - 1

23. Implied standard deviation for options
Random walk (usually acceptable) - Constant volatility (unlikely)
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)
Summation((xi - mean)^k)/n

24. Variance of sample mean
Variance(y)/n = variance of sample Y
F = ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

25. Beta distribution
Sample mean +/ - t*(stddev(s)/sqrt(n))
Normal - Student's T - Chi - square - F distribution
95% = 1.65 99% = 2.33 For one - tailed tests
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

26. Block maxima
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Low Frequency - High Severity events
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

27. Implications of homoscedasticity
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
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
SSR
Has heavy tails

28. Central Limit Theorem(CLT)
Var(X) + Var(Y)
Sampling distribution of sample means tend to be normal
Based on a dataset
Sample mean +/ - t*(stddev(s)/sqrt(n))

29. Sample covariance
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Peaks over threshold - Collects dataset in excess of some threshold
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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

30. Result of combination of two normal with same means
Special type of pooled data in which the cross sectional unit is surveyed over time
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Combine to form distribution with leptokurtosis (heavy tails)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

31. Continuously compounded return equation
i = ln(Si/Si - 1)
E(XY) - E(X)E(Y)
Variance(X) + Variance(Y) - 2*covariance(XY)
Variance(x)

32. Standard variable for non - normal distributions
Independently and Identically Distributed
Z = (Y - meany)/(stddev(y)/sqrt(n))
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
Regression can be non - linear in variables but must be linear in parameters

33. Sample variance
Price/return tends to run towards a long - run level
P(X=x - Y=y) = P(X=x) * P(Y=y)
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

34. Central Limit Theorem
For n>30 - sample mean is approximately normal
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Special type of pooled data in which the cross sectional unit is surveyed over time
Distribution with only two possible outcomes

35. What does the OLS minimize?
SSR
Variance reverts to a long run level
Among all unbiased estimators - estimator with the smallest variance is efficient
Only requires two parameters = mean and variance

36. Variance(discrete)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Application of mathematical statistics to economic data to lend empirical support to models

37. Logistic distribution
Has heavy tails
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)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

38. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Contains variables not explicit in model - Accounts for randomness
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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!)

39. Four sampling distributions
Normal - Student's T - Chi - square - F distribution
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
(a^2)(variance(x)) + (b^2)(variance(y))

40. Reliability
F = ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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!)
Statement of the error or precision of an estimate
Sampling distribution of sample means tend to be normal

41. Confidence ellipse
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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)
Population denominator = n - Sample denominator = n - 1

42. Exact significance level
Random walk (usually acceptable) - Constant volatility (unlikely)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Expected value of the sample mean is the population mean
P - value

43. Significance =1
Confidence level
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
SSR
Choose parameters that maximize the likelihood of what observations occurring

44. Type I error
We reject a hypothesis that is actually true
Population denominator = n - Sample denominator = n - 1
Confidence set for two coefficients - two dimensional analog for the confidence interval
95% = 1.65 99% = 2.33 For one - tailed tests

45. SER
Normal - Student's T - Chi - square - F distribution
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 error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Sample mean +/ - t*(stddev(s)/sqrt(n))

46. GEV
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Use historical simulation approach but use the EWMA weighting system
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

47. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Probability that the random variables take on certain values simultaneously
Peaks over threshold - Collects dataset in excess of some threshold
Average return across assets on a given day

48. Variance of sampling distribution of means when n<N
When the sample size is large - the uncertainty about the value of the sample is very small
Expected value of the sample mean is the population mean
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

49. P - value
Special type of pooled data in which the cross sectional unit is surveyed over time
Combine to form distribution with leptokurtosis (heavy tails)
Random walk (usually acceptable) - Constant volatility (unlikely)
P(Z>t)

50. Two drawbacks of moving average series
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Concerned with a single random variable (ex. Roll of a die)
Variance(y)/n = variance of sample Y

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