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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Regime - switching volatility model
Probability that the random variables take on certain values simultaneously
95% = 1.65 99% = 2.33 For one - tailed tests
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Among all unbiased estimators - estimator with the smallest variance is efficient

2. LAD
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Least absolute deviations estimator - used when extreme outliers are not uncommon
Independently and Identically Distributed
Normal - Student's T - Chi - square - F distribution

3. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Combine to form distribution with leptokurtosis (heavy tails)
Transformed to a unit variable - Mean = 0 Variance = 1

4. Continuous representation of the GBM
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
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)
For n>30 - sample mean is approximately normal
E(mean) = mean

5. P - value
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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!)
P(Z>t)
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

6. Standard variable for non - normal distributions
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Var(X) + Var(Y)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

7. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Summation((xi - mean)^k)/n
E(mean) = mean
Combine to form distribution with leptokurtosis (heavy tails)

8. Implied standard deviation for options
If variance of the conditional distribution of u(i) is not constant
P(Z>t)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

9. T distribution
Special type of pooled data in which the cross sectional unit is surveyed over time
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)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

10. Reliability
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Statement of the error or precision of an estimate
Var(X) + Var(Y)

11. Confidence ellipse
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Sample mean will near the population mean as the sample size increases
Confidence set for two coefficients - two dimensional analog for the confidence interval

12. What does the OLS minimize?
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Special type of pooled data in which the cross sectional unit is surveyed over time
Contains variables not explicit in model - Accounts for randomness
SSR

13. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
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
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

14. Variance(discrete)
Model dependent - Options with the same underlying assets may trade at different volatilities
Sample mean +/ - t*(stddev(s)/sqrt(n))
We reject a hypothesis that is actually true
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

15. Sample covariance
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
P(Z>t)
Variance reverts to a long run level

16. Sample 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)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Use historical simulation approach but use the EWMA weighting system
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

17. R^2
Z = (Y - meany)/(stddev(y)/sqrt(n))
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
Among all unbiased estimators - estimator with the smallest variance is efficient
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

18. Confidence interval for sample mean
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
(a^2)(variance(x)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Expected value of the sample mean is the population mean

19. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
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
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

20. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
P - value
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

21. Control variates technique
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Has heavy tails

22. Adjusted R^2
Var(X) + Var(Y)
Average return across assets on a given day
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
Normal - Student's T - Chi - square - F distribution

23. Economical(elegant)
Normal - Student's T - Chi - square - F distribution
E(XY) - E(X)E(Y)
Only requires two parameters = mean and variance
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

24. SER
95% = 1.65 99% = 2.33 For one - tailed tests
Mean of sampling distribution is the population mean
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Concerned with a single random variable (ex. Roll of a die)

25. WLS
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Nonlinearity
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

26. Bootstrap method
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
Regression can be non - linear in variables but must be linear in parameters
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

27. Confidence interval (from t)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Sample mean +/ - t*(stddev(s)/sqrt(n))
Variance = (1/m) summation(u<n - i>^2)

28. Logistic distribution
Normal - Student's T - Chi - square - F distribution
When the sample size is large - the uncertainty about the value of the sample is very small
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Has heavy tails

29. Mean(expected value)
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
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

30. Hazard rate of exponentially distributed random variable
Sample mean +/ - t*(stddev(s)/sqrt(n))
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

31. Homoskedastic only F - stat
Average return across assets on a given day
Use historical simulation approach but use the EWMA weighting system
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

32. Difference between population and sample variance
Variance(x)
Population denominator = n - Sample denominator = n - 1
i = ln(Si/Si - 1)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

33. Consistent
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
When the sample size is large - the uncertainty about the value of the sample is very small
Least absolute deviations estimator - used when extreme outliers are not uncommon
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

34. Historical std dev
Variance = (1/m) summation(u<n - i>^2)
Expected value of the sample mean is the population mean
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

35. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Low Frequency - High Severity events

36. Maximum likelihood method
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
For n>30 - sample mean is approximately normal
More than one random variable
Choose parameters that maximize the likelihood of what observations occurring

37. Standard error for Monte Carlo replications
Choose parameters that maximize the likelihood of what observations occurring
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
SSR

38. Test for unbiasedness
E(mean) = mean
Concerned with a single random variable (ex. Roll of a die)
Choose parameters that maximize the likelihood of what observations occurring
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

39. Multivariate Density Estimation (MDE)
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Choose parameters that maximize the likelihood of what observations occurring
Rxy = Sxy/(Sx*Sy)

40. Tractable
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Distribution with only two possible outcomes
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Easy to manipulate

41. Unbiased
Mean of sampling distribution is the population mean
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Population denominator = n - Sample denominator = n - 1
95% = 1.65 99% = 2.33 For one - tailed tests

42. Multivariate probability
More than one random variable
Based on an equation - P(A) = # of A/total outcomes
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
P(X=x - Y=y) = P(X=x) * P(Y=y)

43. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
Sample mean will near the population mean as the sample size increases
Var(X) + Var(Y)
Regression can be non - linear in variables but must be linear in parameters

44. Critical z values
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Sample mean +/ - t*(stddev(s)/sqrt(n))
95% = 1.65 99% = 2.33 For one - tailed tests

45. Two drawbacks of moving average series
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

46. Mean reversion in asset dynamics
SSR
Normal - Student's T - Chi - square - F distribution
Summation((xi - mean)^k)/n
Price/return tends to run towards a long - run level

47. Poisson distribution equations for mean variance and std deviation
Variance(X) + Variance(Y) - 2*covariance(XY)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

48. Non - parametric vs parametric calculation of VaR
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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

49. Persistence
Sample mean +/ - t*(stddev(s)/sqrt(n))
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Concerned with a single random variable (ex. Roll of a die)
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

50. Variance of aX
Returns over time for an individual asset
Sample mean +/ - t*(stddev(s)/sqrt(n))
(a^2)(variance(x)
Among all unbiased estimators - estimator with the smallest variance is efficient