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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. Four sampling distributions

2. Confidence ellipse
Variance = (1/m) summation(u<n - i>^2)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Among all unbiased estimators - estimator with the smallest variance is efficient

3. What does the OLS minimize?
Sample mean +/ - t*(stddev(s)/sqrt(n))
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
SSR
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

4. Persistence
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
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)
Confidence level
Special type of pooled data in which the cross sectional unit is surveyed over time

5. Statistical (or empirical) model
(a^2)(variance(x)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
Yi = B0 + B1Xi + ui

6. 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))
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

7. Sample correlation
Rxy = Sxy/(Sx*Sy)
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
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!)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

8. POT
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!)
Var(X) + Var(Y)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Peaks over threshold - Collects dataset in excess of some threshold

9. EWMA
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

10. Two requirements of OVB
Contains variables not explicit in model - Accounts for randomness
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Sample mean +/ - t*(stddev(s)/sqrt(n))
Random walk (usually acceptable) - Constant volatility (unlikely)

11. Standard variable for non - normal distributions
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Z = (Y - meany)/(stddev(y)/sqrt(n))
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
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

12. Variance(discrete)
Transformed to a unit variable - Mean = 0 Variance = 1
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

13. Beta distribution
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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance(x) + Variance(Y) + 2*covariance(XY)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

14. F distribution
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
Population denominator = n - Sample denominator = n - 1
(a^2)(variance(x)) + (b^2)(variance(y))
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)

15. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
Mean = np - Variance = npq - Std dev = sqrt(npq)
Confidence set for two coefficients - two dimensional analog for the confidence interval
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

16. Mean reversion
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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
Sampling distribution of sample means tend to be normal

17. Confidence interval (from t)
Sample mean +/ - t*(stddev(s)/sqrt(n))
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
Confidence set for two coefficients - two dimensional analog for the confidence interval
Based on an equation - P(A) = # of A/total outcomes

18. Perfect multicollinearity
P(X=x - Y=y) = P(X=x) * P(Y=y)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
When one regressor is a perfect linear function of the other regressors
More than one random variable

19. Overall F - statistic
Variance = (1/m) summation(u<n - i>^2)
(a^2)(variance(x)
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 = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

20. Simplified standard (un - weighted) variance
Var(X) + Var(Y)
Variance = (1/m) summation(u<n - i>^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
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

21. Inverse transform method
We reject a hypothesis that is actually true
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Application of mathematical statistics to economic data to lend empirical support to models
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

22. Significance =1
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
Confidence level
Variance = (1/m) summation(u<n - i>^2)

23. 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!)
Probability that the random variables take on certain values simultaneously
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution

24. Antithetic variable technique
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
P - value
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

25. Poisson Distribution
Attempts to sample along more important paths
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

26. Cross - sectional
We reject a hypothesis that is actually true
Based on a dataset
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Average return across assets on a given day

27. Reliability
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
Statement of the error or precision of an estimate
(a^2)(variance(x)) + (b^2)(variance(y))
Confidence set for two coefficients - two dimensional analog for the confidence interval

28. Variance of X+b
Variance(x)
Only requires two parameters = mean and variance
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
Confidence set for two coefficients - two dimensional analog for the confidence interval

29. Law of Large Numbers
Application of mathematical statistics to economic data to lend empirical support to models
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Sample mean will near the population mean as the sample size increases

30. Variance of X - Y assuming dependence
E(XY) - E(X)E(Y)
Transformed to a unit variable - Mean = 0 Variance = 1
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Variance(X) + Variance(Y) - 2*covariance(XY)

31. Lognormal
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
We accept a hypothesis that should have been rejected

32. LFHS
Only requires two parameters = mean and variance
We reject a hypothesis that is actually true
Low Frequency - High Severity events
If variance of the conditional distribution of u(i) is not constant

33. Single variable (univariate) probability
Price/return tends to run towards a long - run level
Normal - Student's T - Chi - square - F distribution
Concerned with a single random variable (ex. Roll of a die)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for

34. SER
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

35. Adjusted R^2
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
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Concerned with a single random variable (ex. Roll of a die)

36. GEV
Normal - Student's T - Chi - square - F distribution
i = ln(Si/Si - 1)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

37. GPD
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
(a^2)(variance(x)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

38. Binomial distribution equations for mean variance and std dev
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance = (1/m) summation(u<n - i>^2)
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!)
Mean = np - Variance = npq - Std dev = sqrt(npq)

39. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

40. Extreme Value Theory
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
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

41. Bernouli Distribution
Low Frequency - High Severity events
Distribution with only two possible outcomes
Returns over time for an individual asset
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

42. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
(a^2)(variance(x)
Rxy = Sxy/(Sx*Sy)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

43. Difference between population and sample variance
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Application of mathematical statistics to economic data to lend empirical support to models
Population denominator = n - Sample denominator = n - 1
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

44. Variance of weighted scheme
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Variance(y)/n = variance of sample Y
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

45. Bootstrap method
Statement of the error or precision of an estimate
For n>30 - sample mean is approximately normal
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

46. Discrete random variable
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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!)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
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

47. Control variates technique
Variance(y)/n = variance of sample Y
Yi = B0 + B1Xi + ui
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

48. Maximum likelihood method
When the sample size is large - the uncertainty about the value of the sample is very small
Choose parameters that maximize the likelihood of what observations occurring
E(mean) = mean
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

49. Implications of homoscedasticity
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
Z = (Y - meany)/(stddev(y)/sqrt(n))
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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

50. Standard normal distribution
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Transformed to a unit variable - Mean = 0 Variance = 1
Distribution with only two possible outcomes
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled