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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. Adjusted R^2
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
Based on an equation - P(A) = # of A/total outcomes
Least absolute deviations estimator - used when extreme outliers are not uncommon
Special type of pooled data in which the cross sectional unit is surveyed over time

2. Maximum likelihood method
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Model dependent - Options with the same underlying assets may trade at different volatilities
Choose parameters that maximize the likelihood of what observations occurring
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

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

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4. Standard variable for non - normal distributions
Rxy = Sxy/(Sx*Sy)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Probability that the random variables take on certain values simultaneously
Z = (Y - meany)/(stddev(y)/sqrt(n))

5. Pooled data
Mean of sampling distribution is the population mean
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Sampling distribution of sample means tend to be normal

6. Chi - squared distribution
Transformed to a unit variable - Mean = 0 Variance = 1
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
Based on a dataset
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

7. Variance of sample mean
Variance(y)/n = variance of sample Y
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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!)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

8. Mean reversion
Choose parameters that maximize the likelihood of what observations occurring
Average return across assets on a given day
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

9. Variance(discrete)
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Variance(X) + Variance(Y) - 2*covariance(XY)
Rxy = Sxy/(Sx*Sy)

10. Discrete random variable
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)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Based on a dataset

11. Heteroskedastic
Sampling distribution of sample means tend to be normal
Variance reverts to a long run level
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
If variance of the conditional distribution of u(i) is not constant

12. GARCH
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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)

13. Variance of X - Y assuming dependence
Variance(X) + Variance(Y) - 2*covariance(XY)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

14. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
For n>30 - sample mean is approximately normal
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
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

15. Cholesky factorization (decomposition)
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
Sampling distribution of sample means tend to be normal
Use historical simulation approach but use the EWMA weighting system
E(XY) - E(X)E(Y)

16. Shortcomings of implied volatility
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))
Peaks over threshold - Collects dataset in excess of some threshold
Model dependent - Options with the same underlying assets may trade at different volatilities

17. Covariance
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(y)/n = variance of sample Y
Application of mathematical statistics to economic data to lend empirical support to models
E(XY) - E(X)E(Y)

18. Sample mean
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
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Attempts to sample along more important paths
Expected value of the sample mean is the population mean

19. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

20. Importance sampling technique
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
Attempts to sample along more important paths
i = ln(Si/Si - 1)
Variance = (1/m) summation(u<n - i>^2)

21. Least squares estimator(m)
Statement of the error or precision of an estimate
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Has heavy tails

22. Conditional probability functions
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
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
Variance(X) + Variance(Y) - 2*covariance(XY)

23. Tractable
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance(x)
Easy to manipulate
Combine to form distribution with leptokurtosis (heavy tails)

24. Stochastic error term
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Contains variables not explicit in model - Accounts for randomness
Expected value of the sample mean is the population mean
Variance(X) + Variance(Y) - 2*covariance(XY)

25. Simulating for VaR
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)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

26. Historical std dev
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

27. Unbiased
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Mean of sampling distribution is the population mean

28. Marginal unconditional probability function
Variance(X) + Variance(Y) - 2*covariance(XY)
Does not depend on a prior event or information
Price/return tends to run towards a long - run level
(a^2)(variance(x)

29. Covariance calculations using weight sums (lambda)
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Concerned with a single random variable (ex. Roll of a die)

30. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Combine to form distribution with leptokurtosis (heavy tails)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

31. Extreme Value Theory
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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))
E(mean) = mean

32. Non - parametric vs parametric calculation of VaR
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

33. Implications of homoscedasticity
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
(a^2)(variance(x)) + (b^2)(variance(y))
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

34. Central Limit Theorem
For n>30 - sample mean is approximately normal
Combine to form distribution with leptokurtosis (heavy tails)
Contains variables not explicit in model - Accounts for randomness
Mean of sampling distribution is the population mean

35. GPD
Independently and Identically Distributed
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Returns over time for a combination of assets (combination of time series and cross - sectional data)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

36. Variance of X+Y
Sample mean will near the population mean as the sample size increases
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Var(X) + Var(Y)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

37. Sample covariance
Use historical simulation approach but use the EWMA weighting system
Sample mean will near the population mean as the sample size increases
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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

38. Discrete representation of the GBM
Easy to manipulate
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Based on an equation - P(A) = # of A/total outcomes
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

39. Mean(expected value)
Concerned with a single random variable (ex. Roll of a die)
Variance(x) + Variance(Y) + 2*covariance(XY)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

40. Two assumptions of square root rule
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Attempts to sample along more important paths
Random walk (usually acceptable) - Constant volatility (unlikely)

41. Standard normal distribution
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Transformed to a unit variable - Mean = 0 Variance = 1
Mean = np - Variance = npq - Std dev = sqrt(npq)

42. GEV
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Price/return tends to run towards a long - run level
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

43. Binomial distribution equations for mean variance and std dev
Choose parameters that maximize the likelihood of what observations occurring
Z = (Y - meany)/(stddev(y)/sqrt(n))
Mean = np - Variance = npq - Std dev = sqrt(npq)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

44. Potential reasons for fat tails in return distributions
P(Z>t)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

45. Exact significance level
Probability that the random variables take on certain values simultaneously
Price/return tends to run towards a long - run level
P - value
Mean of sampling distribution is the population mean

46. Expected future variance rate (t periods forward)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
P(X=x - Y=y) = P(X=x) * P(Y=y)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

47. Variance of aX
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
(a^2)(variance(x)
Summation((xi - mean)^k)/n
We accept a hypothesis that should have been rejected

48. Normal distribution
P(X=x - Y=y) = P(X=x) * P(Y=y)
Normal - Student's T - Chi - square - F distribution
Does not depend on a prior event or information
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

49. R^2
Least absolute deviations estimator - used when extreme outliers are not uncommon
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
When the sample size is large - the uncertainty about the value of the sample is very small

50. EWMA
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
We reject a hypothesis that is actually true
Easy to manipulate
Expected value of the sample mean is the population mean