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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. Overall F - statistic
(a^2)(variance(x)) + (b^2)(variance(y))
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Based on an equation - P(A) = # of A/total outcomes

2. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Var(X) + Var(Y)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

3. Multivariate probability
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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
More than one random variable
Variance(y)/n = variance of sample Y

4. Regime - switching volatility model
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
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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

5. LAD
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
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Least absolute deviations estimator - used when extreme outliers are not uncommon

6. Panel data (longitudinal or micropanel)
Concerned with a single random variable (ex. Roll of a die)
Special type of pooled data in which the cross sectional unit is surveyed over time
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Based on a dataset

7. Binomial distribution equations for mean variance and std dev
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Mean = np - Variance = npq - Std dev = sqrt(npq)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Sampling distribution of sample means tend to be normal

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

9. Implications of homoscedasticity
Based on a dataset
Mean = np - Variance = npq - Std dev = sqrt(npq)
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
P - value

10. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Mean of sampling distribution is the population mean
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Low Frequency - High Severity events

11. Control variates technique
Distribution with only two possible outcomes
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

12. POT
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Peaks over threshold - Collects dataset in excess of some threshold
Sample mean +/ - t*(stddev(s)/sqrt(n))
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

13. Type I error
Special type of pooled data in which the cross sectional unit is surveyed over time
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
We reject a hypothesis that is actually true
Peaks over threshold - Collects dataset in excess of some threshold

14. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

15. SER
Sample mean will near the population mean as the sample size increases
Mean of sampling distribution is the population mean
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

16. Variance of aX
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
(a^2)(variance(x)
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

17. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Combine to form distribution with leptokurtosis (heavy tails)

18. Chi - squared distribution
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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

19. Monte Carlo Simulations
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Use historical simulation approach but use the EWMA weighting system
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Sample mean +/ - t*(stddev(s)/sqrt(n))

20. Direction of OVB
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Independently and Identically Distributed

21. R^2
Choose parameters that maximize the likelihood of what observations occurring
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

22. Binomial distribution
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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!)

23. Mean reversion
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Contains variables not explicit in model - Accounts for randomness

24. Weibul 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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Returns over time for an individual asset
Variance(X) + Variance(Y) - 2*covariance(XY)

25. Standard error
P(Z>t)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Regression can be non - linear in variables but must be linear in parameters
Z = (Y - meany)/(stddev(y)/sqrt(n))

26. 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
Population denominator = n - Sample denominator = n - 1
P(Z>t)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

27. Standard normal distribution
Least absolute deviations estimator - used when extreme outliers are not uncommon
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
Transformed to a unit variable - Mean = 0 Variance = 1

28. Beta distribution
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Yi = B0 + B1Xi + ui

29. Discrete representation of the GBM
Attempts to sample along more important paths
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)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

30. Two assumptions of square root rule
Regression can be non - linear in variables but must be linear in parameters
Based on a dataset
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Random walk (usually acceptable) - Constant volatility (unlikely)

31. LFHS
Low Frequency - High Severity events
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Price/return tends to run towards a long - run level

32. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Does not depend on a prior event or information

33. Pooled data
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Expected value of the sample mean is the population mean
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

34. Joint probability functions
Probability that the random variables take on certain values simultaneously
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

35. Two drawbacks of moving average series
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Variance(x) + Variance(Y) + 2*covariance(XY)
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

36. Confidence ellipse
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Low Frequency - High Severity events
Confidence set for two coefficients - two dimensional analog for the confidence interval
P - value

37. Biggest (and only real) drawback of GARCH mode
Based on an equation - P(A) = # of A/total outcomes
Rxy = Sxy/(Sx*Sy)
Sample mean will near the population mean as the sample size increases
Nonlinearity

38. WLS
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Among all unbiased estimators - estimator with the smallest variance is efficient
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

39. Kurtosis
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
E(XY) - E(X)E(Y)

40. Central Limit Theorem(CLT)
Yi = B0 + B1Xi + ui
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
Sampling distribution of sample means tend to be normal
Average return across assets on a given day

41. Perfect multicollinearity
Peaks over threshold - Collects dataset in excess of some threshold
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
When one regressor is a perfect linear function of the other regressors

42. Simulation models
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Variance(X) + Variance(Y) - 2*covariance(XY)
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
E(XY) - E(X)E(Y)

43. Variance of X+Y
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Var(X) + Var(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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

44. Variance of sample mean
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
SSR
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Variance(y)/n = variance of sample Y

45. Implied standard deviation for options
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance(x) + Variance(Y) + 2*covariance(XY)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

46. Economical(elegant)
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
Only requires two parameters = mean and variance
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

47. Sample mean
i = ln(Si/Si - 1)
Expected value of the sample mean is the population mean
Use historical simulation approach but use the EWMA weighting system
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

48. F distribution
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
(a^2)(variance(x)) + (b^2)(variance(y))
Model dependent - Options with the same underlying assets may trade at different volatilities
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

49. Empirical frequency
SSR
Summation((xi - mean)^k)/n
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
Based on a dataset

50. Sample correlation
Rxy = Sxy/(Sx*Sy)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Regression can be non - linear in variables but must be linear in parameters
Least absolute deviations estimator - used when extreme outliers are not uncommon