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FRM Foundations Of Risk Management Quantitative Methods

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Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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1. Mean reversion in variance
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
Variance reverts to a long run level
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

2. Tractable
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Peaks over threshold - Collects dataset in excess of some threshold
Easy to manipulate
Z = (Y - meany)/(stddev(y)/sqrt(n))

3. Cross - sectional
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Average return across assets on a given day
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

4. Biggest (and only real) drawback of GARCH mode
Sample mean +/ - t*(stddev(s)/sqrt(n))
Does not depend on a prior event or information
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
Nonlinearity

5. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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

6. Monte Carlo Simulations
Least absolute deviations estimator - used when extreme outliers are not uncommon
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
When one regressor is a perfect linear function of the other regressors
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

7. Variance of X - Y assuming dependence
Does not depend on a prior event or information
Variance(X) + Variance(Y) - 2*covariance(XY)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Normal - Student's T - Chi - square - F distribution

8. Extending the HS approach for computing value of a portfolio
Regression can be non - linear in variables but must be linear in parameters
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

9. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
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
Low Frequency - High Severity 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

10. Variance of X+b
Random walk (usually acceptable) - Constant volatility (unlikely)
For n>30 - sample mean is approximately normal
Variance(x)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

11. Test for statistical independence
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
P(X=x - Y=y) = P(X=x) * P(Y=y)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Easy to manipulate

12. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
P(Z>t)

13. 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)
Normal - Student's T - Chi - square - F distribution
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Model dependent - Options with the same underlying assets may trade at different volatilities

14. Poisson distribution equations for mean variance and std deviation
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

15. Simulation models
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
Application of mathematical statistics to economic data to lend empirical support to models

16. Continuous representation of the GBM
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Sample mean +/ - t*(stddev(s)/sqrt(n))
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)

17. Importance sampling technique
Attempts to sample along more important paths
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
When the sample size is large - the uncertainty about the value of the sample is very small

18. Difference between population and sample variance
SSR
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Population denominator = n - Sample denominator = n - 1
Average return across assets on a given day

19. Confidence ellipse
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Peaks over threshold - Collects dataset in excess of some threshold
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!)

20. Hybrid method for conditional volatility
Use historical simulation approach but use the EWMA weighting system
E(XY) - E(X)E(Y)
Based on an equation - P(A) = # of A/total outcomes
Independently and Identically Distributed

21. Variance of weighted scheme
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Probability that the random variables take on certain values simultaneously

22. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
(a^2)(variance(x)) + (b^2)(variance(y))
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
When one regressor is a perfect linear function of the other regressors

23. LFHS
Yi = B0 + B1Xi + ui
Low Frequency - High Severity events
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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)

24. Central Limit Theorem
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
For n>30 - sample mean is approximately normal
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

25. What does the OLS minimize?
E(XY) - E(X)E(Y)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Yi = B0 + B1Xi + ui
SSR

26. SER
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance(X) + Variance(Y) - 2*covariance(XY)
Peaks over threshold - Collects dataset in excess of some threshold
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

27. Logistic distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Has heavy tails
When one regressor is a perfect linear function of the other regressors

28. Binomial distribution equations for mean variance and std dev
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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 = np - Variance = npq - Std dev = sqrt(npq)

29. Binomial distribution
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!)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
More than one random variable

30. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Use historical simulation approach but use the EWMA weighting system
Low Frequency - High Severity events

31. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Only requires two parameters = mean and variance
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

32. Unconditional vs conditional distributions
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Regression can be non - linear in variables but must be linear in parameters
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

33. Regime - switching volatility model
Population denominator = n - Sample denominator = n - 1
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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)

34. Deterministic Simulation
Random walk (usually acceptable) - Constant volatility (unlikely)
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
Population denominator = n - Sample denominator = n - 1
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

35. Control variates technique
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Combine to form distribution with leptokurtosis (heavy tails)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

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

37. LAD
We reject a hypothesis that is actually true
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Peaks over threshold - Collects dataset in excess of some threshold

38. Type I error
Confidence set for two coefficients - two dimensional analog for the confidence interval
Application of mathematical statistics to economic data to lend empirical support to models
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
We reject a hypothesis that is actually true

39. GPD
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Does not depend on a prior event or information
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

40. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
When the sample size is large - the uncertainty about the value of the sample is very small
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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

41. Variance of sampling distribution of means when n<N
Summation((xi - mean)^k)/n
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

42. POT
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Peaks over threshold - Collects dataset in excess of some threshold
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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

43. Simulating for VaR
Based on an equation - P(A) = # of A/total outcomes
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form 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
Variance(x)

44. Limitations of R^2 (what an increase doesn't necessarily imply)
Mean of sampling distribution is the population mean
F = ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

45. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Z = (Y - meany)/(stddev(y)/sqrt(n))
95% = 1.65 99% = 2.33 For one - tailed tests

46. Multivariate probability
More than one random variable
Mean = np - Variance = npq - Std dev = sqrt(npq)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Special type of pooled data in which the cross sectional unit is surveyed over time

47. Continuous random variable
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Sample mean will near the population mean as the sample size increases
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Confidence set for two coefficients - two dimensional analog for the confidence interval

48. Test for unbiasedness
Low Frequency - High Severity events
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
E(mean) = mean
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

49. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Based on an equation - P(A) = # of A/total outcomes
P(Z>t)
Returns over time for an individual asset

50. Sample mean
Least absolute deviations estimator - used when extreme outliers are not uncommon
Variance(y)/n = variance of sample Y
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Expected value of the sample mean is the population mean

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