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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. Importance sampling technique
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
Attempts to sample along more important paths
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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

2. Standard error
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
Variance = (1/m) summation(u<n - i>^2)
Use historical simulation approach but use the EWMA weighting system
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

3. Empirical frequency
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Based on a dataset
95% = 1.65 99% = 2.33 For one - tailed tests
Expected value of the sample mean is the population mean

4. Binomial distribution equations for mean variance and std dev
Easy to manipulate
Application of mathematical statistics to economic data to lend empirical support to models
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Mean = np - Variance = npq - Std dev = sqrt(npq)

5. Key properties of linear regression
Confidence level
Z = (Y - meany)/(stddev(y)/sqrt(n))
Regression can be non - linear in variables but must be linear in parameters
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

6. Direction of OVB
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Mean = np - Variance = npq - Std dev = sqrt(npq)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

7. Inverse transform method
Use historical simulation approach but use the EWMA weighting system
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Special type of pooled data in which the cross sectional unit is surveyed over time
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

8. Logistic distribution
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Variance(x)
Has heavy tails
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

9. GPD
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

10. Covariance
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
E(XY) - E(X)E(Y)
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
Model dependent - Options with the same underlying assets may trade at different volatilities

11. Standard variable for non - normal distributions
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Z = (Y - meany)/(stddev(y)/sqrt(n))
Has heavy tails

12. Poisson distribution equations for mean variance and std deviation
(a^2)(variance(x)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Independently and Identically Distributed
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

13. Hybrid method for conditional volatility
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Nonlinearity
Application of mathematical statistics to economic data to lend empirical support to models
Use historical simulation approach but use the EWMA weighting system

14. i.i.d.
Regression can be non - linear in variables but must be linear in parameters
Independently and Identically Distributed
Transformed to a unit variable - Mean = 0 Variance = 1
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

15. Tractable
Sample mean will near the population mean as the sample size increases
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Easy to manipulate
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

16. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

17. Exponential distribution
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Variance(y)/n = variance of sample Y
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

18. Law of Large Numbers
Statement of the error or precision of an estimate
Sample mean will near the population mean as the sample size increases
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

19. Perfect multicollinearity
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
When one regressor is a perfect linear function of the other regressors
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
We accept a hypothesis that should have been rejected

20. Mean(expected value)
Random walk (usually acceptable) - Constant volatility (unlikely)
Var(X) + Var(Y)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

21. Confidence interval (from t)
Variance(x)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

22. WLS
Mean of sampling distribution is the population mean
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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. Two assumptions of square root rule
Random walk (usually acceptable) - Constant volatility (unlikely)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Concerned with a single random variable (ex. Roll of a die)
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

24. Variance of weighted scheme
Transformed to a unit variable - Mean = 0 Variance = 1
Low Frequency - High Severity events
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

25. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Attempts to sample along more important paths
Random walk (usually acceptable) - Constant volatility (unlikely)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

26. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
Variance(x) + Variance(Y) + 2*covariance(XY)

27. Variance of aX + bY
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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
Model dependent - Options with the same underlying assets may trade at different volatilities
(a^2)(variance(x)) + (b^2)(variance(y))

28. 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
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
Least absolute deviations estimator - used when extreme outliers are not uncommon
When one regressor is a perfect linear function of the other regressors

29. Variance of X+b
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Variance(x)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

30. T distribution
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

31. Continuous representation of the GBM
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Returns over time for an individual asset
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)
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

32. Mean reversion in variance
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
Variance reverts to a long run level
Rxy = Sxy/(Sx*Sy)
Z = (Y - meany)/(stddev(y)/sqrt(n))

33. Normal distribution
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
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
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!)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

34. Overall F - statistic
Combine to form distribution with leptokurtosis (heavy tails)
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
Summation((xi - mean)^k)/n
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

35. Significance =1
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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)
Variance(y)/n = variance of sample Y

36. Potential reasons for fat tails in return distributions
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

37. POT
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Z = (Y - meany)/(stddev(y)/sqrt(n))
Low Frequency - High Severity events
Peaks over threshold - Collects dataset in excess of some threshold

38. Maximum likelihood method
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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
Choose parameters that maximize the likelihood of what observations occurring
P(X=x - Y=y) = P(X=x) * P(Y=y)

39. Multivariate probability
More than one random variable
SSR
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

40. Variance of sample mean
Probability that the random variables take on certain values simultaneously
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Variance(y)/n = variance of sample Y
SSR

41. Type I error
Variance(x) + Variance(Y) + 2*covariance(XY)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Returns over time for an individual asset
We reject a hypothesis that is actually true

42. Panel data (longitudinal or micropanel)
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Special type of pooled data in which the cross sectional unit is surveyed over time
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

43. R^2
Contains variables not explicit in model - Accounts for randomness
Based on an equation - P(A) = # of A/total outcomes
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Price/return tends to run towards a long - run level

44. Continuously compounded return equation
i = ln(Si/Si - 1)
When one regressor is a perfect linear function of the other regressors
Variance(x) + Variance(Y) + 2*covariance(XY)
Variance = (1/m) summation(u<n - i>^2)

45. Stochastic error term
Confidence level
Independently and Identically Distributed
Contains variables not explicit in model - Accounts for randomness
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

46. ESS
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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
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)

47. Joint probability functions
Sample mean will near the population mean as the sample size increases
Sample mean +/ - t*(stddev(s)/sqrt(n))
i = ln(Si/Si - 1)
Probability that the random variables take on certain values simultaneously

48. Sample correlation
Variance(y)/n = variance of sample Y
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Rxy = Sxy/(Sx*Sy)

49. Two ways to calculate historical volatility
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Average return across assets on a given day
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
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

50. Homoskedastic
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
95% = 1.65 99% = 2.33 For one - tailed tests
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Transformed to a unit variable - Mean = 0 Variance = 1