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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. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Sampling distribution of sample means tend to be normal
Price/return tends to run towards a long - run level
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

2. Maximum likelihood method
SSR
We reject a hypothesis that is actually true
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Choose parameters that maximize the likelihood of what observations occurring

3. i.i.d.
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Independently and Identically Distributed
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

4. Confidence ellipse
Use historical simulation approach but use the EWMA weighting system
We accept a hypothesis that should have been rejected
Mean = np - Variance = npq - Std dev = sqrt(npq)
Confidence set for two coefficients - two dimensional analog for the confidence interval

5. Variance of aX
We reject a hypothesis that is actually true
Transformed to a unit variable - Mean = 0 Variance = 1
(a^2)(variance(x)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

6. Historical std dev
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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
Mean = np - Variance = npq - Std dev = sqrt(npq)

7. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/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
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

8. Multivariate probability
We reject a hypothesis that is actually true
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
More than one random variable
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

9. Result of combination of two normal with same means
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Combine to form distribution with leptokurtosis (heavy tails)

10. Homoskedastic only F - stat
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)
P - value
Nonlinearity

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

12. Covariance calculations using weight sums (lambda)
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Nonlinearity
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

13. Unbiased
When one regressor is a perfect linear function of the other regressors
Mean of sampling distribution is the population mean
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

14. F distribution
Z = (Y - meany)/(stddev(y)/sqrt(n))
Statement of the error or precision of an estimate
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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

15. ESS
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
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(y)/n = variance of sample Y
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

16. Sample variance
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
E(XY) - E(X)E(Y)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Least absolute deviations estimator - used when extreme outliers are not uncommon

17. Biggest (and only real) drawback of GARCH mode
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
P - value
Nonlinearity
Independently and Identically Distributed

18. Joint probability functions
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
Sample mean will near the population mean as the sample size increases
Probability that the random variables take on certain values simultaneously
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

19. Central Limit Theorem
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
E(mean) = mean
For n>30 - sample mean is approximately normal
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

20. Kurtosis
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Nonlinearity
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

21. SER
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)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
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

22. T distribution
Variance(X) + Variance(Y) - 2*covariance(XY)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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
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. Variance of X+Y assuming dependence
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Variance(x) + Variance(Y) + 2*covariance(XY)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

24. GEV
(a^2)(variance(x)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

25. Panel data (longitudinal or micropanel)
Average return across assets on a given day
E(XY) - E(X)E(Y)
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

26. Antithetic variable technique
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

27. Key properties of linear regression
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
Regression can be non - linear in variables but must be linear in parameters
Yi = B0 + B1Xi + ui
For n>30 - sample mean is approximately normal

28. Persistence
Sample mean will near the population mean as the sample size increases
P(X=x - Y=y) = P(X=x) * P(Y=y)
Easy to manipulate
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

29. Inverse transform method
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Z = (Y - meany)/(stddev(y)/sqrt(n))

30. Heteroskedastic
Average return across assets on a given day
P - value
If variance of the conditional distribution of u(i) is not constant
Summation((xi - mean)^k)/n

31. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
Confidence level
Sample mean +/ - t*(stddev(s)/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

32. Statistical (or empirical) model
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Variance(y)/n = variance of sample Y
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

33. Mean(expected value)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
More than one random variable
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
(a^2)(variance(x)) + (b^2)(variance(y))

34. What does the OLS minimize?
Low Frequency - High Severity events
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
SSR

35. Adjusted R^2
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
We accept a hypothesis that should have been rejected
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
Distribution with only two possible outcomes

36. Perfect multicollinearity
Transformed to a unit variable - Mean = 0 Variance = 1
Has heavy tails
When one regressor is a perfect linear function of the other regressors
Independently and Identically Distributed

37. Variance of sample mean
Sample mean will near the population mean as the sample size increases
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Variance(y)/n = variance of sample Y
SSR

38. Mean reversion in variance
Easy to manipulate
Variance reverts to a long run level
Regression can be non - linear in variables but must be linear in parameters
Sample mean +/ - t*(stddev(s)/sqrt(n))

39. Cholesky factorization (decomposition)
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
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

40. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Only requires two parameters = mean and variance
P(Z>t)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

41. Logistic distribution
Has heavy tails
Contains variables not explicit in model - Accounts for randomness
Var(X) + Var(Y)
Application of mathematical statistics to economic data to lend empirical support to models

42. Type I error
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

43. P - value
Random walk (usually acceptable) - Constant volatility (unlikely)
P(Z>t)
Concerned with a single random variable (ex. Roll of a die)
Based on an equation - P(A) = # of A/total outcomes

44. Lognormal
Expected value of the sample mean is the population mean
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

45. Importance sampling technique
Z = (Y - meany)/(stddev(y)/sqrt(n))
Attempts to sample along more important paths
Summation((xi - mean)^k)/n
For n>30 - sample mean is approximately normal

46. Skewness
Returns over time for a combination of assets (combination of time series and cross - sectional data)
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
Special type of pooled data in which the cross sectional unit is surveyed over time
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

47. Standard normal distribution
Sample mean +/ - t*(stddev(s)/sqrt(n))
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!)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Transformed to a unit variable - Mean = 0 Variance = 1

48. Overall F - statistic
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Combine to form distribution with leptokurtosis (heavy tails)
P(Z>t)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

49. Two ways to calculate historical volatility
Choose parameters that maximize the likelihood of what observations occurring
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
P(Z>t)
Variance(X) + Variance(Y) - 2*covariance(XY)

50. Marginal unconditional probability function
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Attempts to sample along more important paths
Variance(x)
Does not depend on a prior event or information