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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. Variance of aX
E(mean) = mean
Peaks over threshold - Collects dataset in excess of some threshold
Normal - Student's T - Chi - square - F distribution
(a^2)(variance(x)

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

3. Single variable (univariate) probability
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Concerned with a single random variable (ex. Roll of a die)

4. Kurtosis
Transformed to a unit variable - Mean = 0 Variance = 1
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

5. Multivariate probability
SSR
More than one random variable
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

6. Cholesky factorization (decomposition)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Special type of pooled data in which the cross sectional unit is surveyed over time
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)

7. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
Confidence level
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Sampling distribution of sample means tend to be normal

8. Potential reasons for fat tails in return distributions
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Random walk (usually acceptable) - Constant volatility (unlikely)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

9. Lognormal
P - value
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

10. Bootstrap method
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Easy to manipulate
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

11. Difference between population and sample variance
Based on an equation - P(A) = # of A/total outcomes
Mean = np - Variance = npq - Std dev = sqrt(npq)
Population denominator = n - Sample denominator = n - 1
Contains variables not explicit in model - Accounts for randomness

12. Antithetic variable technique
Regression can be non - linear in variables but must be linear in parameters
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
P - value
Returns over time for an individual asset

13. Time series data
Expected value of the sample mean is the population mean
Returns over time for an individual asset
Z = (Y - meany)/(stddev(y)/sqrt(n))
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

14. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Mean = np - Variance = npq - Std dev = sqrt(npq)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
For n>30 - sample mean is approximately normal

15. Skewness
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Normal - Student's T - Chi - square - F distribution

16. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Only requires two parameters = mean and variance
Variance(y)/n = variance of sample Y
Model dependent - Options with the same underlying assets may trade at different volatilities

17. F distribution
If variance of the conditional distribution of u(i) is not constant
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

18. Two assumptions of square root rule
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Expected value of the sample mean is the population mean
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
Random walk (usually acceptable) - Constant volatility (unlikely)

19. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
E(mean) = mean
SSR

20. Efficiency
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)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Among all unbiased estimators - estimator with the smallest variance is efficient

21. Two ways to calculate historical volatility
Probability that the random variables take on certain values simultaneously
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Sampling distribution of sample means tend to be normal

22. Binomial distribution
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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!)
(a^2)(variance(x)
Variance = (1/m) summation(u<n - i>^2)

23. Confidence interval (from t)
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
Sample mean +/ - t*(stddev(s)/sqrt(n))
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
We reject a hypothesis that is actually true

24. Significance =1
Variance(x) + Variance(Y) + 2*covariance(XY)
Confidence level
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

25. Mean(expected value)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

26. Square root rule
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Only requires two parameters = mean and variance
Price/return tends to run towards a long - run level
When the sample size is large - the uncertainty about the value of the sample is very small

27. GPD
Peaks over threshold - Collects dataset in excess of some threshold
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

28. SER
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Expected value of the sample mean is the population mean
E(mean) = mean

29. Two requirements of OVB
Random walk (usually acceptable) - Constant volatility (unlikely)
Based on an equation - P(A) = # of A/total outcomes
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

30. Homoskedastic only F - stat
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Attempts to sample along more important paths
E(mean) = mean
Variance reverts to a long run level

31. Gamma distribution
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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

32. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Contains variables not explicit in model - Accounts for randomness
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

33. Importance sampling technique
Variance(x)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Attempts to sample along more important paths

34. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
For n>30 - sample mean is approximately normal
Confidence level

35. Law of Large Numbers
Does not depend on a prior event or information
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
For n>30 - sample mean is approximately normal
Sample mean will near the population mean as the sample size increases

36. Unstable return distribution
Based on a dataset
Least absolute deviations estimator - used when extreme outliers are not uncommon
Mean = np - Variance = npq - Std dev = sqrt(npq)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

37. Pooled data
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Regression can be non - linear in variables but must be linear in parameters
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)

38. LFHS
Variance reverts to a long run level
Low Frequency - High Severity events
(a^2)(variance(x)
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

39. Variance of aX + bY
Variance(x) + Variance(Y) + 2*covariance(XY)
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))
P(Z>t)

40. Chi - squared distribution
Mean of sampling distribution is the population mean
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
Sampling distribution of sample means tend to be 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

41. Type II Error
We accept a hypothesis that should have been rejected
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
i = ln(Si/Si - 1)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

42. Exact significance level
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
P - value
Sample mean will near the population mean as the sample size increases
P(Z>t)

43. Hazard rate of exponentially distributed random variable
Normal - Student's T - Chi - square - F distribution
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Only requires two parameters = mean and variance
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

44. Homoskedastic
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Application of mathematical statistics to economic data to lend empirical support to models
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
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

45. Variance of X+b
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
Variance(x)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

46. Central Limit Theorem(CLT)
Sampling distribution of sample means tend to be normal
SSR
Mean = np - Variance = npq - Std dev = sqrt(npq)
Does not depend on a prior event or information

47. Multivariate Density Estimation (MDE)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

48. Standard error for Monte Carlo replications
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
95% = 1.65 99% = 2.33 For one - tailed tests
Easy to manipulate
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

49. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Low Frequency - High Severity events
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

50. Control variates technique
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)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
SSR