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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)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Variance(x) + Variance(Y) + 2*covariance(XY)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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

2. Poisson Distribution
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Nonlinearity
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

3. P - value
Distribution with only two possible outcomes
P(Z>t)
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
Average return across assets on a given day

4. Mean reversion in variance
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
We accept a hypothesis that should have been rejected
Yi = B0 + B1Xi + ui
Variance reverts to a long run level

5. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

6. Shortcomings of implied volatility
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Variance = (1/m) summation(u<n - i>^2)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Model dependent - Options with the same underlying assets may trade at different volatilities

7. Persistence
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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
When one regressor is a perfect linear function of the other regressors
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

8. Cross - sectional
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Average return across assets on a given day
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Contains variables not explicit in model - Accounts for randomness

9. 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!)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
If variance of the conditional distribution of u(i) is not constant

10. Variance of aX + bY
P(Z>t)
We reject a hypothesis that is actually true
(a^2)(variance(x)) + (b^2)(variance(y))
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

11. Weibul distribution
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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

12. Adjusted R^2
Mean of sampling distribution is the population mean
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Probability that the random variables take on certain values simultaneously
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. SER
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
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

14. Variance - covariance approach for VaR of a portfolio
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
Independently and Identically Distributed
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Mean = np - Variance = npq - Std dev = sqrt(npq)

15. Panel data (longitudinal or micropanel)
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
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
Special type of pooled data in which the cross sectional unit is surveyed over time
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

16. Reliability
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Normal - Student's T - Chi - square - F distribution
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
Statement of the error or precision of an estimate

17. F distribution
i = ln(Si/Si - 1)
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
Combine to form distribution with leptokurtosis (heavy tails)
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

18. Result of combination of two normal with same means
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Combine to form distribution with leptokurtosis (heavy tails)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Variance(X) + Variance(Y) - 2*covariance(XY)

19. Bernouli Distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Distribution with only two possible outcomes
Easy to manipulate
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

20. What does the OLS minimize?
SSR
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
More than one random variable
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

21. POT
Peaks over threshold - Collects dataset in excess of some threshold
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
P(X=x - Y=y) = P(X=x) * P(Y=y)
Mean of sampling distribution is the population mean

22. Mean reversion
Sample mean +/ - t*(stddev(s)/sqrt(n))
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

23. Poisson distribution equations for mean variance and std deviation
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
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!)

24. i.i.d.
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
When one regressor is a perfect linear function of the other regressors
Independently and Identically Distributed
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

25. Consistent
Mean = np - Variance = npq - Std dev = sqrt(npq)
When the sample size is large - the uncertainty about the value of the sample is very small
Only requires two parameters = mean and variance
Summation((xi - mean)^k)/n

26. Continuously compounded return equation
i = ln(Si/Si - 1)
Peaks over threshold - Collects dataset in excess of some threshold
Choose parameters that maximize the likelihood of what observations occurring
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

27. Two ways to calculate historical volatility
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

28. Perfect multicollinearity
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
When one regressor is a perfect linear function of the other regressors
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

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

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30. Central Limit Theorem
Probability that the random variables take on certain values simultaneously
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
For n>30 - sample mean is approximately normal
E(XY) - E(X)E(Y)

31. Standard variable for non - normal distributions
Sample mean +/ - t*(stddev(s)/sqrt(n))
Yi = B0 + B1Xi + ui
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!)
Z = (Y - meany)/(stddev(y)/sqrt(n))

32. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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!)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

33. Sample covariance
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
Variance = (1/m) summation(u<n - i>^2)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
(a^2)(variance(x)

34. Gamma distribution
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
P - value

35. Central Limit Theorem(CLT)
Regression can be non - linear in variables but must be linear in parameters
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Sampling distribution of sample means tend to be normal

36. Multivariate probability
Average return across assets on a given day
More than one random variable
Summation((xi - mean)^k)/n
Variance(x) + Variance(Y) + 2*covariance(XY)

37. Difference between population and sample variance
More than one random variable
Use historical simulation approach but use the EWMA weighting system
Population denominator = n - Sample denominator = n - 1
Sampling distribution of sample means tend to be normal

38. Time series data
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Returns over time for an individual asset
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

39. Joint probability functions
Probability that the random variables take on certain values simultaneously
Based on a dataset
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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

40. Continuous representation of the GBM
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Peaks over threshold - Collects dataset in excess of some threshold
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)
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

41. Beta distribution
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
We accept a hypothesis that should have been rejected
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

42. Key properties of linear regression
Only requires two parameters = mean and variance
Regression can be non - linear in variables but must be linear in parameters
Combine to form distribution with leptokurtosis (heavy tails)
(a^2)(variance(x)

43. Variance of X+Y assuming dependence
P(X=x - Y=y) = P(X=x) * P(Y=y)
Variance(x) + Variance(Y) + 2*covariance(XY)
We accept a hypothesis that should have been rejected
Least absolute deviations estimator - used when extreme outliers are not uncommon

44. Implied standard deviation for options
95% = 1.65 99% = 2.33 For one - tailed tests
Normal - Student's T - Chi - square - F distribution
Average return across assets on a given day
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

45. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
We accept a hypothesis that should have been rejected
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Choose parameters that maximize the likelihood of what observations occurring

46. Standard normal distribution
Rxy = Sxy/(Sx*Sy)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
We accept a hypothesis that should have been rejected
Transformed to a unit variable - Mean = 0 Variance = 1

47. Variance(discrete)
Peaks over threshold - Collects dataset in excess of some threshold
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

48. Historical std dev
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Use historical simulation approach but use the EWMA weighting system
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

49. Economical(elegant)
Does not depend on a prior event or information
Variance(x)
Only requires two parameters = mean and variance
Combine to form distribution with leptokurtosis (heavy tails)

50. Chi - squared distribution
Average return across assets on a given day
We accept a hypothesis that should have been rejected
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients