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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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Match each statement with the correct term.
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1. Kurtosis
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
More than one random variable

2. Implications of homoscedasticity
Confidence level
Among all unbiased estimators - estimator with the smallest variance is efficient
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
Use historical simulation approach but use the EWMA weighting system

3. i.i.d.
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
Independently and Identically Distributed
Nonlinearity
Distribution with only two possible outcomes

4. Deterministic Simulation
When one regressor is a perfect linear function of the other regressors
E(mean) = mean
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

5. Cholesky factorization (decomposition)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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
Sample mean +/ - t*(stddev(s)/sqrt(n))
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

6. Historical std dev
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Mean of sampling distribution is the population mean
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

7. Simulation models
Based on a dataset
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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

8. Binomial distribution equations for mean variance and std dev
Regression can be non - linear in variables but must be linear in parameters
Mean = np - Variance = npq - Std dev = sqrt(npq)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Does not depend on a prior event or information

9. Standard variable for non - normal distributions
Only requires two parameters = mean and variance
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
Z = (Y - meany)/(stddev(y)/sqrt(n))
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

10. Two ways to calculate historical volatility
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Rxy = Sxy/(Sx*Sy)

11. Two drawbacks of moving average series
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Sample mean +/ - t*(stddev(s)/sqrt(n))
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Statement of the error or precision of an estimate

12. Inverse transform method
Variance(y)/n = variance of sample Y
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
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

13. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

14. Continuously compounded return equation
i = ln(Si/Si - 1)
Independently and Identically Distributed
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

15. Priori (classical) probability
Based on an equation - P(A) = # of A/total outcomes
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Transformed to a unit variable - Mean = 0 Variance = 1
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

16. Efficiency
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
Among all unbiased estimators - estimator with the smallest variance is efficient
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
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

17. Biggest (and only real) drawback of GARCH mode
Nonlinearity
Does not depend on a prior event or information
Independently and Identically Distributed
Peaks over threshold - Collects dataset in excess of some threshold

18. Discrete representation of the GBM
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
(a^2)(variance(x)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

19. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
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
Among all unbiased estimators - estimator with the smallest variance is efficient
Low Frequency - High Severity events

20. Bernouli Distribution
Rxy = Sxy/(Sx*Sy)
Distribution with only two possible outcomes
Concerned with a single random variable (ex. Roll of a die)
Sample mean +/ - t*(stddev(s)/sqrt(n))

21. 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)
E(mean) = mean
Among all unbiased estimators - estimator with the smallest variance is efficient
Rxy = Sxy/(Sx*Sy)

22. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

23. Test for unbiasedness
E(mean) = mean
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

24. Hazard rate of exponentially distributed random variable
Use historical simulation approach but use the EWMA weighting system
Sampling distribution of sample means tend to be normal
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

25. Maximum likelihood method
Choose parameters that maximize the likelihood of what observations occurring
For n>30 - sample mean is approximately normal
Variance(x)
Rxy = Sxy/(Sx*Sy)

26. Beta distribution
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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

27. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Sample mean +/ - t*(stddev(s)/sqrt(n))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

28. Sample covariance
Sample mean +/ - t*(stddev(s)/sqrt(n))
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Variance(x) + Variance(Y) + 2*covariance(XY)

29. Variance of sample mean
We reject a hypothesis that is actually true
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Mean of sampling distribution is the population mean
Variance(y)/n = variance of sample Y

30. GPD
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
E(mean) = mean
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

31. Hybrid method for conditional volatility
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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))

32. POT
Summation((xi - mean)^k)/n
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Peaks over threshold - Collects dataset in excess of some threshold
Average return across assets on a given day

33. Type II Error
Regression can be non - linear in variables but must be linear in parameters
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
We accept a hypothesis that should have been rejected
Returns over time for an individual asset

34. Sample correlation
Based on an equation - P(A) = # of A/total outcomes
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
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
Rxy = Sxy/(Sx*Sy)

35. Sample variance
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Variance reverts to a long run level
If variance of the conditional distribution of u(i) is not constant

36. Variance of aX + bY
E(XY) - E(X)E(Y)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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
(a^2)(variance(x)) + (b^2)(variance(y))

37. P - value
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(Z>t)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

38. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Easy to manipulate
Mean of sampling distribution is the population mean

39. Two assumptions of square root rule
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Random walk (usually acceptable) - Constant volatility (unlikely)

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

41. Time series data
Least absolute deviations estimator - used when extreme outliers are not uncommon
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Returns over time for an individual asset
Mean = np - Variance = npq - Std dev = sqrt(npq)

42. Importance sampling technique
Attempts to sample along more important paths
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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!)
E(mean) = mean

43. Cross - sectional
Variance(x) + Variance(Y) + 2*covariance(XY)
Average return across assets on a given day
For n>30 - sample mean is approximately normal
Sampling distribution of sample means tend to be normal

44. Skewness
Normal - Student's T - Chi - square - F distribution
Contains variables not explicit in model - Accounts for randomness
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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

45. Logistic distribution
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Has heavy tails

46. Antithetic variable technique
Choose parameters that maximize the likelihood of what observations occurring
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Returns over time for a combination of assets (combination of time series and cross - sectional data)

47. Result of combination of two normal with same means
When the sample size is large - the uncertainty about the value of the sample is very small
Confidence set for two coefficients - two dimensional analog for the confidence interval
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

48. BLUE
Expected value of the sample mean is the population mean
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

49. 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
Normal - Student's T - Chi - square - F distribution
Nonlinearity
Expected value of the sample mean is the population mean

50. SER
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.