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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. Consistent
When the sample size is large - the uncertainty about the value of the sample is very small
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Special type of pooled data in which the cross sectional unit is surveyed over time
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

2. Statistical (or empirical) model
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Population denominator = n - Sample denominator = n - 1
Yi = B0 + B1Xi + ui
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

3. P - value
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
P(Z>t)

4. Central Limit Theorem
For n>30 - sample mean is approximately normal
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
i = ln(Si/Si - 1)

5. Variance of aX
(a^2)(variance(x)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
E(mean) = mean
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

6. Tractable
For n>30 - sample mean is approximately normal
Easy to manipulate
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
i = ln(Si/Si - 1)

7. Skewness
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

8. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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 = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

9. Type I error
We reject a hypothesis that is actually true
When one regressor is a perfect linear function of the other regressors
Mean = np - Variance = npq - Std dev = sqrt(npq)
Probability that the random variables take on certain values simultaneously

10. Variance of X+Y
Sampling distribution of sample means tend to be normal
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Low Frequency - High Severity events
Var(X) + Var(Y)

11. F distribution
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Application of mathematical statistics to economic data to lend empirical support to models
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

12. Direction of OVB
Sample mean will near the population mean as the sample size increases
When one regressor is a perfect linear function of the other regressors
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

13. Binomial distribution
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Variance reverts to a long run level
Normal - Student's T - Chi - square - F 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!)

14. Efficiency
Variance(x)
Among all unbiased estimators - estimator with the smallest variance is efficient
Combine to form distribution with leptokurtosis (heavy tails)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

15. Two requirements of OVB
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Confidence set for two coefficients - two dimensional analog for the confidence interval
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

16. Econometrics
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
We reject a hypothesis that is actually true
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Application of mathematical statistics to economic data to lend empirical support to models

17. LFHS
Low Frequency - High Severity events
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Average return across assets on a given day
Z = (Y - meany)/(stddev(y)/sqrt(n))

18. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Variance reverts to a long run level
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Expected value of the sample mean is the population mean

19. Confidence ellipse
i = ln(Si/Si - 1)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Only requires two parameters = mean and variance
Confidence set for two coefficients - two dimensional analog for the confidence interval

20. Type II Error
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
We accept a hypothesis that should have been rejected
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

21. Shortcomings of implied volatility
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
Model dependent - Options with the same underlying assets may trade at different volatilities
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

22. 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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Independently and Identically Distributed

23. Panel data (longitudinal or micropanel)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Use historical simulation approach but use the EWMA weighting system
Special type of pooled data in which the cross sectional unit is surveyed over time
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

24. Continuous random variable
Concerned with a single random variable (ex. Roll of a die)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Based on a dataset

25. SER
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

26. Discrete representation of the GBM
Yi = B0 + B1Xi + ui
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"

27. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
E(XY) - E(X)E(Y)
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
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

28. EWMA
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
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

29. LAD
Least absolute deviations estimator - used when extreme outliers are not uncommon
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Variance(y)/n = variance of sample Y

30. Weibul distribution
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

31. Poisson Distribution
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Among all unbiased estimators - estimator with the smallest variance is efficient
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
P(X=x - Y=y) = P(X=x) * P(Y=y)

32. Variance of X+b
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
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(x)

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

34. Hazard rate of exponentially distributed random variable
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Confidence set for two coefficients - two dimensional analog for the confidence interval
We accept a hypothesis that should have been rejected
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!)

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

36. Standard variable for non - normal distributions
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

37. Cholesky factorization (decomposition)
Price/return tends to run towards a long - run level
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Sampling distribution of sample means tend to be normal
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

38. Beta distribution
Peaks over threshold - Collects dataset in excess of some threshold
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

39. Two ways to calculate historical volatility
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

40. 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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Nonlinearity
(a^2)(variance(x)

41. Poisson distribution equations for mean variance and std deviation
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Probability that the random variables take on certain values simultaneously
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

42. Lognormal
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

43. Conditional probability functions
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Average return across assets on a given day
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

44. Antithetic variable technique
Use historical simulation approach but use the EWMA weighting system
Independently and Identically Distributed
Based on an equation - P(A) = # of A/total outcomes
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

45. Inverse transform method
95% = 1.65 99% = 2.33 For one - tailed tests
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

46. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Sample mean +/ - t*(stddev(s)/sqrt(n))
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

47. Time series data
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!)
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
Returns over time for an individual asset
Var(X) + Var(Y)

48. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Attempts to sample along more important paths
(a^2)(variance(x)) + (b^2)(variance(y))

49. Test for unbiasedness
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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
E(mean) = mean

50. Simulating for VaR
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Only requires two parameters = mean and variance
Sample mean +/ - t*(stddev(s)/sqrt(n))
Statement of the error or precision of an estimate