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FRM Foundations Of Risk Management Quantitative Methods

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1. Critical z values
95% = 1.65 99% = 2.33 For one - tailed tests
Transformed to a unit variable - Mean = 0 Variance = 1
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

2. Sample correlation
Rxy = Sxy/(Sx*Sy)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
95% = 1.65 99% = 2.33 For one - tailed tests

3. Poisson distribution equations for mean variance and std deviation
Confidence level
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Use historical simulation approach but use the EWMA weighting system

4. Sample covariance
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

5. Block maxima
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Least absolute deviations estimator - used when extreme outliers are not uncommon
For n>30 - sample mean is approximately normal

6. WLS
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!)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

7. Mean reversion
Regression can be non - linear in variables but must be linear in parameters
Mean of sampling distribution is the population mean
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

8. Deterministic Simulation
Only requires two parameters = mean and variance
Based on a dataset
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)
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

9. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
(a^2)(variance(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)
Sample mean will near the population mean as the sample size increases

10. Regime - switching volatility model
Sample mean +/ - t*(stddev(s)/sqrt(n))
Variance(X) + Variance(Y) - 2*covariance(XY)
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

11. Beta distribution
Normal - Student's T - Chi - square - F distribution
Sampling distribution of sample means tend to be normal
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

12. Bootstrap method
When one regressor is a perfect linear function of the other regressors
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Normal - Student's T - Chi - square - F distribution
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

13. P - value
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
Nonlinearity
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
P(Z>t)

14. Exponential distribution
Z = (Y - meany)/(stddev(y)/sqrt(n))
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

15. Sample mean
Expected value of the sample mean is the population mean
Easy to manipulate
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

16. Discrete representation of the GBM
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
SSR

17. Confidence ellipse
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Variance reverts to a long run level
Confidence set for two coefficients - two dimensional analog for the confidence interval

18. Monte Carlo Simulations
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

19. Law of Large Numbers
Distribution with only two possible outcomes
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Sample mean will near the population mean as the sample size increases
More than one random variable

20. Multivariate probability
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
More than one random variable
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

21. Consistent
Z = (Y - meany)/(stddev(y)/sqrt(n))
When the sample size is large - the uncertainty about the value of the sample is very small
Based on a dataset
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

22. Unbiased
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
Mean of sampling distribution is the population mean
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Does not depend on a prior event or information

23. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

24. Weibul distribution
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Var(X) + Var(Y)
(a^2)(variance(x)) + (b^2)(variance(y))
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

25. Cross - sectional
We reject a hypothesis that is actually true
Variance(x) + Variance(Y) + 2*covariance(XY)
Average return across assets on a given day
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

26. SER
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

27. Persistence
Probability that the random variables take on certain values simultaneously
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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

28. Simulating for VaR
95% = 1.65 99% = 2.33 For one - tailed tests
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Mean = np - Variance = npq - Std dev = sqrt(npq)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

29. Confidence interval for sample mean
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
For n>30 - sample mean is approximately normal

30. Continuously compounded return equation
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
i = ln(Si/Si - 1)
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

31. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Population denominator = n - Sample denominator = n - 1
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

32. GPD
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Population denominator = n - Sample denominator = n - 1
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Only requires two parameters = mean and variance

33. SER
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 mean will near the population mean as the sample size increases
Variance(X) + Variance(Y) - 2*covariance(XY)
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

34. Result of combination of two normal with same means
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)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

35. Normal distribution
Does not depend on a prior event or information
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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
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

36. Extending the HS approach for computing value of a portfolio

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37. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

38. Econometrics
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Transformed to a unit variable - Mean = 0 Variance = 1
Application of mathematical statistics to economic data to lend empirical support to models
Population denominator = n - Sample denominator = n - 1

39. T distribution
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Independently and Identically Distributed
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

40. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 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

41. Empirical frequency
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Based on a dataset
Combine to form distribution with leptokurtosis (heavy tails)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

42. Biggest (and only real) drawback of GARCH mode
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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)
Nonlinearity
Probability that the random variables take on certain values simultaneously

43. Variance of X+Y
Var(X) + Var(Y)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
When one regressor is a perfect linear function of the other regressors
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

44. ESS
Mean = np - Variance = npq - Std dev = sqrt(npq)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Contains variables not explicit in model - Accounts for randomness
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

45. Statistical (or empirical) model
E(XY) - E(X)E(Y)
Yi = B0 + B1Xi + ui
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Average return across assets on a given day

46. Bernouli Distribution
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)
Distribution with only two possible outcomes
Variance = (1/m) summation(u<n - i>^2)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

47. GARCH
Price/return tends to run towards a long - run 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)
SSR
P(Z>t)

48. Marginal unconditional probability function
Does not depend on a prior event or information
Based on an equation - P(A) = # of A/total outcomes
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Returns over time for an individual asset

49. Continuous representation of the GBM
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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)

50. Pooled data
Random walk (usually acceptable) - Constant volatility (unlikely)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

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