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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

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Answer 50 questions in 15 minutes.
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1. Cholesky factorization (decomposition)
Combine to form distribution with leptokurtosis (heavy tails)
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
SSR

2. GPD
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Variance reverts to a long run level
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Peaks over threshold - Collects dataset in excess of some threshold

3. Normal distribution
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
Choose parameters that maximize the likelihood of what observations occurring
Peaks over threshold - Collects dataset in excess of some threshold
Model dependent - Options with the same underlying assets may trade at different volatilities

4. Standard normal distribution
Transformed to a unit variable - Mean = 0 Variance = 1
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Combine to form distribution with leptokurtosis (heavy tails)

5. Variance of sampling distribution of means when n<N
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
95% = 1.65 99% = 2.33 For one - tailed tests
Application of mathematical statistics to economic data to lend empirical support to models

6. Two ways to calculate historical volatility
When one regressor is a perfect linear function of the other regressors
We accept a hypothesis that should have been rejected
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
More than one random variable

7. What does the OLS minimize?
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
SSR
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
Random walk (usually acceptable) - Constant volatility (unlikely)

8. Variance - covariance approach for VaR of a portfolio
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Normal - Student's T - Chi - square - F distribution
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

9. Non - parametric vs parametric calculation of VaR
P - value
Variance(X) + Variance(Y) - 2*covariance(XY)
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

10. Variance of X+Y
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Based on a dataset
Var(X) + Var(Y)
Mean of sampling distribution is the population mean

11. Historical std dev
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Contains variables not explicit in model - Accounts for randomness

12. Inverse transform method
P - value
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
If variance of the conditional distribution of u(i) is not constant
Yi = B0 + B1Xi + ui

13. Variance of X+b
Variance(x)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

14. Skewness
Z = (Y - meany)/(stddev(y)/sqrt(n))
Variance(X) + Variance(Y) - 2*covariance(XY)
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

15. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Mean = np - Variance = npq - Std dev = sqrt(npq)
Regression can be non - linear in variables but must be linear in parameters
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

16. Confidence interval (from t)
i = ln(Si/Si - 1)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Yi = B0 + B1Xi + ui
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

17. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Transformed to a unit variable - Mean = 0 Variance = 1
We accept a hypothesis that should have been rejected
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

18. Poisson distribution equations for mean variance and std deviation
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Concerned with a single random variable (ex. Roll of a die)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

19. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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)

20. Continuous random variable
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
P(Z>t)
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

21. T distribution
(a^2)(variance(x)
Variance reverts to a long run level
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

22. Type I error
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
Population denominator = n - Sample denominator = n - 1
We reject a hypothesis that is actually true
Does not depend on a prior event or information

23. Multivariate probability
More than one random variable
Variance(X) + Variance(Y) - 2*covariance(XY)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Has heavy tails

24. R^2
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

25. Homoskedastic only F - stat
Z = (Y - meany)/(stddev(y)/sqrt(n))
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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

26. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
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
Price/return tends to run towards a long - run level
Concerned with a single random variable (ex. Roll of a die)

27. Mean reversion
Average return across assets on a given day
P(X=x - Y=y) = P(X=x) * P(Y=y)
Nonlinearity
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

28. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
Model dependent - Options with the same underlying assets may trade at different volatilities
When one regressor is a perfect linear function of the other regressors
Rxy = Sxy/(Sx*Sy)

29. Central Limit Theorem
E(mean) = mean
Use historical simulation approach but use the EWMA weighting system
Transformed to a unit variable - Mean = 0 Variance = 1
For n>30 - sample mean is approximately normal

30. Discrete random variable
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!)
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)
Mean = np - Variance = npq - Std dev = sqrt(npq)

31. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

32. Standard variable for non - normal distributions
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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
Z = (Y - meany)/(stddev(y)/sqrt(n))

33. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
Combine to form distribution with leptokurtosis (heavy tails)
Contains variables not explicit in model - Accounts for randomness
Low Frequency - High Severity events

34. Multivariate Density Estimation (MDE)
SSR
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Returns over time for an individual asset

35. Tractable
Easy to manipulate
Standard deviation of the sampling distribution SE = std dev(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
Z = (Y - meany)/(stddev(y)/sqrt(n))

36. Heteroskedastic
Distribution with only two possible outcomes
Contains variables not explicit in model - Accounts for randomness
If variance of the conditional distribution of u(i) is not constant
Only requires two parameters = mean and variance

37. Consistent
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
When the sample size is large - the uncertainty about the value of the sample is very small
Sample mean +/ - t*(stddev(s)/sqrt(n))
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

38. Binomial distribution
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!)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Variance = (1/m) summation(u<n - i>^2)

39. Expected future variance rate (t periods forward)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Random walk (usually acceptable) - Constant volatility (unlikely)
Price/return tends to run towards a long - run level

40. Mean reversion in variance
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Normal - Student's T - Chi - square - F distribution
E(XY) - E(X)E(Y)
Variance reverts to a long run level

41. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Based on an equation - P(A) = # of A/total outcomes

42. Econometrics
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Independently and Identically Distributed
Application of mathematical statistics to economic data to lend empirical support to models
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

43. Unbiased
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Mean of sampling distribution is the population mean
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

44. Reliability
Statement of the error or precision of an estimate
Population denominator = n - Sample denominator = n - 1
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Among all unbiased estimators - estimator with the smallest variance is efficient

45. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Attempts to sample along more important paths
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Variance(y)/n = variance of sample Y

46. Chi - squared distribution
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
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

47. Least squares estimator(m)
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
Transformed to a unit variable - Mean = 0 Variance = 1
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

48. Gamma distribution
Statement of the error or precision of an estimate
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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!)
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

49. Time series data
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Returns over time for an individual asset
Based on an equation - P(A) = # of A/total outcomes
Among all unbiased estimators - estimator with the smallest variance is efficient

50. Adjusted R^2
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
P(X=x - Y=y) = P(X=x) * P(Y=y)
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
Low Frequency - High Severity events