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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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1. Importance sampling technique
95% = 1.65 99% = 2.33 For one - tailed tests
Population denominator = n - Sample denominator = n - 1
Attempts to sample along more important paths
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

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

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3. Limitations of R^2 (what an increase doesn't necessarily imply)

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4. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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
Z = (Y - meany)/(stddev(y)/sqrt(n))
Summation((xi - mean)^k)/n

5. Four sampling distributions

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6. Poisson Distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Nonlinearity
Probability that the random variables take on certain values simultaneously
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

7. Joint probability functions
i = ln(Si/Si - 1)
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

8. Bernouli Distribution
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Distribution with only two possible outcomes
We accept a hypothesis that should have been rejected
Contains variables not explicit in model - Accounts for randomness

9. Simulation models
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
Returns over time for an individual asset
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

10. Heteroskedastic
Regression can be non - linear in variables but must be linear in parameters
Based on a dataset
If variance of the conditional distribution of u(i) is not constant
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

11. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Independently and Identically Distributed
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

12. Adjusted R^2
Sample mean +/ - t*(stddev(s)/sqrt(n))
Independently and Identically Distributed
Regression can be non - linear in variables but must be linear in parameters
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

13. Conditional probability functions
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Transformed to a unit variable - Mean = 0 Variance = 1

14. Chi - squared distribution
Combine to form distribution with leptokurtosis (heavy tails)
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)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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

15. Significance =1
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
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Confidence level
When one regressor is a perfect linear function of the other regressors

16. Two assumptions of square root rule
Nonlinearity
If variance of the conditional distribution of u(i) is not constant
Mean of sampling distribution is the population mean
Random walk (usually acceptable) - Constant volatility (unlikely)

17. i.i.d.
Regression can be non - linear in variables but must be linear in parameters
Independently and Identically Distributed
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

18. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
Yi = B0 + B1Xi + ui
Normal - Student's T - Chi - square - F distribution
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

19. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

20. Variance of X+b
Based on a dataset
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance(x)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

21. Single variable (univariate) probability
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Concerned with a single random variable (ex. Roll of a die)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

22. Sample variance
Probability that the random variables take on certain values simultaneously
Does not depend on a prior event or information
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

23. Inverse transform method
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*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)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

24. Variance of sampling distribution of means when n<N
More than one random variable
Variance = (1/m) summation(u<n - i>^2)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

25. Test for statistical independence
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
P(X=x - Y=y) = P(X=x) * P(Y=y)
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!)

26. Standard variable for non - normal distributions
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
Z = (Y - meany)/(stddev(y)/sqrt(n))
P - value
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

27. Covariance calculations using weight sums (lambda)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Contains variables not explicit in model - Accounts for randomness
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

28. Variance(discrete)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Variance reverts to a long run level
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
i = ln(Si/Si - 1)

29. ESS
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)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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

30. Critical z values
We reject a hypothesis that is actually true
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
95% = 1.65 99% = 2.33 For one - tailed tests
Low Frequency - High Severity events

31. Extreme Value Theory
Transformed to a unit variable - Mean = 0 Variance = 1
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Sample mean will near the population mean as the sample size increases
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

32. BLUE
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

33. EWMA
E(mean) = mean
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

34. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Concerned with a single random variable (ex. Roll of a die)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Nonlinearity

35. Homoskedastic
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
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
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!)

36. Biggest (and only real) drawback of GARCH mode
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
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

37. Continuous representation of the GBM
Average return across assets on a given day
Based on an equation - P(A) = # of A/total outcomes
Confidence level
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)

38. Exact significance level
P - value
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
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
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)

39. Binomial distribution equations for mean variance and std dev
Z = (Y - meany)/(stddev(y)/sqrt(n))
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Mean = np - Variance = npq - Std dev = sqrt(npq)

40. Historical std dev
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Expected value of the sample mean is the population mean

41. Variance of aX + bY
Z = (Y - meany)/(stddev(y)/sqrt(n))
Among all unbiased estimators - estimator with the smallest variance is efficient
Concerned with a single random variable (ex. Roll of a die)
(a^2)(variance(x)) + (b^2)(variance(y))

42. Stochastic error term
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
Mean of sampling distribution is the population mean
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Contains variables not explicit in model - Accounts for randomness

43. Variance of sample mean
Variance(y)/n = variance of sample Y
We accept a hypothesis that should have been rejected
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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

44. Confidence interval (from t)
Mean = np - Variance = npq - Std dev = sqrt(npq)
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))
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

45. LAD
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Model dependent - Options with the same underlying assets may trade at different volatilities
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Least absolute deviations estimator - used when extreme outliers are not uncommon

46. Tractable
Model dependent - Options with the same underlying assets may trade at different volatilities
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Easy to manipulate
Variance(X) + Variance(Y) - 2*covariance(XY)

47. Discrete random variable
Sample mean +/ - t*(stddev(s)/sqrt(n))
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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)

48. Variance of weighted scheme
Variance = (1/m) summation(u<n - i>^2)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

49. Perfect multicollinearity
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
When one regressor is a perfect linear function of the other regressors
Low Frequency - High Severity events
Yi = B0 + B1Xi + ui

50. Gamma distribution
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
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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