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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. Expected future variance rate (t periods forward)
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)

2. Bernouli Distribution
Distribution with only two possible outcomes
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Population denominator = n - Sample denominator = n - 1

3. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
P(X=x - Y=y) = P(X=x) * P(Y=y)
Normal - Student's T - Chi - square - F distribution
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

4. Mean reversion in variance
Variance reverts to a long run level
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
Regression can be non - linear in variables but must be linear in parameters
Based on a dataset

5. Shortcomings of implied volatility
Mean = np - Variance = npq - Std dev = sqrt(npq)
Model dependent - Options with the same underlying assets may trade at different volatilities
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
(a^2)(variance(x)

6. Variance of sampling distribution of means when n<N
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
More than one random variable
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

7. POT
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
When one regressor is a perfect linear function of the other regressors
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
Peaks over threshold - Collects dataset in excess of some threshold

8. i.i.d.
Independently and Identically Distributed
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
For n>30 - sample mean is approximately normal

9. WLS
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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
P(Z>t)
Z = (Y - meany)/(stddev(y)/sqrt(n))

10. Standard error for Monte Carlo 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
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

11. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Normal - Student's T - Chi - square - F distribution
Sample mean +/ - t*(stddev(s)/sqrt(n))
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

12. Variance(discrete)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

13. Panel data (longitudinal or micropanel)
Variance(x)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Special type of pooled data in which the cross sectional unit is surveyed over time
Based on an equation - P(A) = # of A/total outcomes

14. Confidence interval for sample mean
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Variance reverts to a long run level
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Variance(x)

15. Mean reversion
Mean = np - Variance = npq - Std dev = sqrt(npq)
Normal - Student's T - Chi - square - F distribution
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
When the sample size is large - the uncertainty about the value of the sample is very small

16. Kurtosis
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Confidence level
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Returns over time for a combination of assets (combination of time series and cross - sectional data)

17. F distribution
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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
Independently and Identically Distributed

18. Standard error
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

19. Type II Error
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
We accept a hypothesis that should have been rejected
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
For n>30 - sample mean is approximately normal

20. Marginal unconditional probability function
Does not depend on a prior event or information
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Attempts to sample along more important paths

21. Homoskedastic
Var(X) + Var(Y)
Application of mathematical statistics to economic data to lend empirical support to models
Variance(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

22. Test for statistical independence
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
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
(a^2)(variance(x)) + (b^2)(variance(y))
P(X=x - Y=y) = P(X=x) * P(Y=y)

23. Block maxima
If variance of the conditional distribution of u(i) is not constant
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

24. Test for unbiasedness
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
Sample mean +/ - t*(stddev(s)/sqrt(n))
E(mean) = mean
P(X=x - Y=y) = P(X=x) * P(Y=y)

25. Homoskedastic only F - stat
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

26. Binomial distribution equations for mean variance and std dev
Mean = np - Variance = npq - Std dev = sqrt(npq)
Model dependent - Options with the same underlying assets may trade at different volatilities
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Variance reverts to a long run level

27. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

28. Historical std dev
Expected value of the sample mean is the population mean
E(XY) - E(X)E(Y)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

29. Exponential distribution
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Yi = B0 + B1Xi + ui
For n>30 - sample mean is approximately normal
Expected value of the sample mean is the population mean

30. Chi - squared distribution
Choose parameters that maximize the likelihood of what observations occurring
i = ln(Si/Si - 1)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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

31. Extreme Value Theory
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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
Price/return tends to run towards a long - run level

32. Critical z values
Returns over time for a combination of assets (combination of time series and cross - sectional data)
95% = 1.65 99% = 2.33 For one - tailed tests
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Least absolute deviations estimator - used when extreme outliers are not uncommon

33. K - th moment
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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!)
Summation((xi - mean)^k)/n

34. Lognormal
(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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Rxy = Sxy/(Sx*Sy)

35. Least squares estimator(m)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

36. Poisson Distribution
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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

37. Variance of X+Y assuming dependence
If variance of the conditional distribution of u(i) is not constant
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
i = ln(Si/Si - 1)
Variance(x) + Variance(Y) + 2*covariance(XY)

38. Implied standard deviation for options
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
P(Z>t)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

39. Economical(elegant)
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 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
Only requires two parameters = mean and variance
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

40. P - value
P(Z>t)
Variance(X) + Variance(Y) - 2*covariance(XY)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
When the sample size is large - the uncertainty about the value of the sample is very small

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

42. Variance of weighted scheme
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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

43. Potential reasons for fat tails in return distributions
Distribution with only two possible outcomes
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

44. Consistent
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Model dependent - Options with the same underlying assets may trade at different volatilities
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
When the sample size is large - the uncertainty about the value of the sample is very small

45. Mean(expected value)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Variance(y)/n = variance of sample Y
Concerned with a single random variable (ex. Roll of a die)

46. Gamma distribution
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

47. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

48. Antithetic variable technique
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Confidence level
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

49. Discrete random variable
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

50. ESS
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Returns over time for a combination of assets (combination of time series and cross - sectional data)