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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. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Application of mathematical statistics to economic data to lend empirical support to models
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
Based on a dataset

2. Panel data (longitudinal or micropanel)
Combine to form distribution with leptokurtosis (heavy tails)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Special type of pooled data in which the cross sectional unit is surveyed over time
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

3. SER
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Rxy = Sxy/(Sx*Sy)
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
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

4. Weibul distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Variance(x) + Variance(Y) + 2*covariance(XY)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
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

5. R^2
Transformed to a unit variable - Mean = 0 Variance = 1
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Probability that the random variables take on certain values simultaneously
Random walk (usually acceptable) - Constant volatility (unlikely)

6. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Only requires two parameters = mean and variance
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

7. Importance sampling technique
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Attempts to sample along more important paths
Population denominator = n - Sample denominator = n - 1
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

8. Sample mean
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Expected value of the sample mean is the population mean
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Application of mathematical statistics to economic data to lend empirical support to models

9. Antithetic variable technique
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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
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
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

10. Variance of X+Y
Attempts to sample along more important paths
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Var(X) + Var(Y)
Yi = B0 + B1Xi + ui

11. Multivariate Density Estimation (MDE)
Combine to form distribution with leptokurtosis (heavy tails)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Variance(X) + Variance(Y) - 2*covariance(XY)
Among all unbiased estimators - estimator with the smallest variance is efficient

12. Econometrics
For n>30 - sample mean is approximately normal
Application of mathematical statistics to economic data to lend empirical support to models
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

13. Binomial 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!)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Application of mathematical statistics to economic data to lend empirical support to models
Sample mean will near the population mean as the sample size increases

14. Variance of X+Y assuming dependence
Among all unbiased estimators - estimator with the smallest variance is efficient
i = ln(Si/Si - 1)
Variance(x) + Variance(Y) + 2*covariance(XY)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

15. Simulation models
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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
P(Z>t)

16. Variance of X+b
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Variance(x)

17. Confidence interval for sample mean
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

18. Confidence ellipse
Special type of pooled data in which the cross sectional unit is surveyed over time
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
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

19. Homoskedastic
Confidence set for two coefficients - two dimensional analog for the confidence interval
Has heavy tails
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

20. Empirical frequency
Based on a dataset
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
Variance(X) + Variance(Y) - 2*covariance(XY)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

21. Deterministic Simulation
Sample mean will near the population mean as the sample size increases
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
For n>30 - sample mean is approximately normal
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

22. Persistence
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
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
Average return across assets on a given day
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

23. Type II Error
We accept a hypothesis that should have been rejected
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Variance = (1/m) summation(u<n - i>^2)
Low Frequency - High Severity events

24. Variance of aX
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
(a^2)(variance(x)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Special type of pooled data in which the cross sectional unit is surveyed over time

25. GEV
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

26. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

27. Test for statistical independence
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Attempts to sample along more important paths
P(X=x - Y=y) = P(X=x) * P(Y=y)
P(Z>t)

28. Two drawbacks of moving average series
Only requires two parameters = mean and variance
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

29. Result of combination of two normal with same means
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
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
Combine to form distribution with leptokurtosis (heavy tails)
Returns over time for a combination of assets (combination of time series and cross - sectional data)

30. Adjusted R^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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Combine to form distribution with leptokurtosis (heavy tails)
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

31. Block maxima
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Sample mean +/ - t*(stddev(s)/sqrt(n))
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

32. Standard normal distribution
Average return across assets on a given day
Transformed to a unit variable - Mean = 0 Variance = 1
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
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

33. Unconditional vs conditional distributions
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Confidence level

34. Hybrid method for conditional volatility
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
Use historical simulation approach but use the EWMA weighting system
Confidence set for two coefficients - two dimensional analog for the confidence interval
Among all unbiased estimators - estimator with the smallest variance is efficient

35. Key properties of linear regression
Contains variables not explicit in model - Accounts for randomness
Population denominator = n - Sample denominator = n - 1
(a^2)(variance(x)) + (b^2)(variance(y))
Regression can be non - linear in variables but must be linear in parameters

36. Variance of aX + bY
Low Frequency - High Severity events
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
(a^2)(variance(x)) + (b^2)(variance(y))
Confidence set for two coefficients - two dimensional analog for the confidence interval

37. Monte Carlo Simulations
(a^2)(variance(x)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Average return across assets on a given day

38. Kurtosis
Yi = B0 + B1Xi + ui
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

39. Control variates technique
Expected value of the sample mean is the population mean
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
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

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

41. Cross - sectional
Average return across assets on a given day
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

42. Economical(elegant)
Model dependent - Options with the same underlying assets may trade at different volatilities
Only requires two parameters = mean and variance
Based on an equation - P(A) = # of A/total outcomes
Concerned with a single random variable (ex. Roll of a die)

43. Exact significance level
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
Among all unbiased estimators - estimator with the smallest variance is efficient
P - value
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

44. Mean reversion in asset dynamics
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Price/return tends to run towards a long - run level
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Variance(y)/n = variance of sample Y

45. Shortcomings of implied volatility
(a^2)(variance(x)
Model dependent - Options with the same underlying assets may trade at different volatilities
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

46. Perfect multicollinearity
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
Special type of pooled data in which the cross sectional unit is surveyed over time
When one regressor is a perfect linear function of the other regressors
Peaks over threshold - Collects dataset in excess of some threshold

47. Reliability
Statement of the error or precision of an estimate
Use historical simulation approach but use the EWMA weighting system
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)
Based on a dataset

48. Logistic distribution
Has heavy tails
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Variance(x)
When one regressor is a perfect linear function of the other regressors

49. Sample correlation
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
We reject a hypothesis that is actually true
Rxy = Sxy/(Sx*Sy)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

50. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
(a^2)(variance(x)) + (b^2)(variance(y))