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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. Sample variance
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^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!)
Choose parameters that maximize the likelihood of what observations occurring
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

2. Heteroskedastic
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
If variance of the conditional distribution of u(i) is not constant
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled

3. Result of combination of two normal with same means
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Combine to form distribution with leptokurtosis (heavy tails)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

4. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
Sampling distribution of sample means tend to be normal
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

5. Standard variable for non - normal distributions
Z = (Y - meany)/(stddev(y)/sqrt(n))
Price/return tends to run towards a long - run level
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
When one regressor is a perfect linear function of the other regressors

6. Sample correlation
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Rxy = Sxy/(Sx*Sy)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

7. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

8. Sample covariance
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Variance(x) + Variance(Y) + 2*covariance(XY)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Returns over time for an individual asset

9. Econometrics
Transformed to a unit variable - Mean = 0 Variance = 1
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Application of mathematical statistics to economic data to lend empirical support to models
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)

10. Implied standard deviation for options
Nonlinearity
Var(X) + Var(Y)
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)

11. Variance(discrete)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

12. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance(x)
Variance(X) + Variance(Y) - 2*covariance(XY)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

13. R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
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
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Application of mathematical statistics to economic data to lend empirical support to models

14. Priori (classical) probability
Peaks over threshold - Collects dataset in excess of some threshold
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Low Frequency - High Severity events
Based on an equation - P(A) = # of A/total outcomes

15. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Variance(x)

16. i.i.d.
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
P(Z>t)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Independently and Identically Distributed

17. GPD
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Statement of the error or precision of an estimate
Among all unbiased estimators - estimator with the smallest variance is efficient
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

18. Variance of X - Y assuming dependence
We reject a hypothesis that is actually true
Independently and Identically Distributed
Probability that the random variables take on certain values simultaneously
Variance(X) + Variance(Y) - 2*covariance(XY)

19. Bootstrap method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

20. Kurtosis
Summation((xi - mean)^k)/n
P - value
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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

21. P - value
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
P(Z>t)

22. Beta distribution
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Concerned with a single random variable (ex. Roll of a die)
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

23. Conditional probability functions
E(XY) - E(X)E(Y)
(a^2)(variance(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)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

24. Standard error
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Population denominator = n - Sample denominator = n - 1
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

25. Antithetic variable technique
Nonlinearity
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

26. Logistic distribution
Has heavy tails
Sample mean +/ - t*(stddev(s)/sqrt(n))
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

27. What does the OLS minimize?
SSR
Easy to manipulate
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
More than one random variable

28. SER
Does not depend on a prior event or information
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Variance(X) + Variance(Y) - 2*covariance(XY)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

29. Panel data (longitudinal or micropanel)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Only requires two parameters = mean and variance
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
Special type of pooled data in which the cross sectional unit is surveyed over time

30. Difference between population and sample variance
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Yi = B0 + B1Xi + ui
Population denominator = n - Sample denominator = n - 1

31. Variance of sample mean
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Combine to form distribution with leptokurtosis (heavy tails)
Variance(y)/n = variance of sample Y
Price/return tends to run towards a long - run level

32. Reliability
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Statement of the error or precision of an estimate
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

33. Economical(elegant)
Price/return tends to run towards a long - run level
Mean = np - Variance = npq - Std dev = sqrt(npq)
P - value
Only requires two parameters = mean and variance

34. Type II Error
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
We accept a hypothesis that should have been rejected
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

35. Time series data
Returns over time for an individual asset
Independently and Identically Distributed
Z = (Y - meany)/(stddev(y)/sqrt(n))
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)

36. Joint probability functions
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Variance = (1/m) summation(u<n - i>^2)
Probability that the random variables take on certain values simultaneously

37. 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!)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

38. Discrete representation of the GBM
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Variance reverts to a long run level
SSR
Low Frequency - High Severity events

39. Test for unbiasedness
E(mean) = mean
We reject a hypothesis that is actually true
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
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

40. Block maxima
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
P(Z>t)

41. Variance of aX + bY
Returns over time for an individual asset
Variance(x) + Variance(Y) + 2*covariance(XY)
Based on an equation - P(A) = # of A/total outcomes
(a^2)(variance(x)) + (b^2)(variance(y))

42. Variance of X+b
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
P(X=x - Y=y) = P(X=x) * P(Y=y)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Variance(x)

43. Sample mean
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Expected value of the sample mean is the population mean
Based on a dataset
Variance = (1/m) summation(u<n - i>^2)

44. Continuous random variable
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
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
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

45. Potential reasons for fat tails in return distributions
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
When the sample size is large - the uncertainty about the value of the sample is very small
Var(X) + Var(Y)

46. Homoskedastic only F - stat
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Special type of pooled data in which the cross sectional unit is surveyed over time

47. Maximum likelihood method
Yi = B0 + B1Xi + ui
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
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

48. Variance - covariance approach for VaR of a portfolio
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
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
Sample mean +/ - t*(stddev(s)/sqrt(n))
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

49. Mean reversion in asset dynamics
Price/return tends to run towards a long - run level
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
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
E(XY) - E(X)E(Y)

50. ESS
Variance(x) + Variance(Y) + 2*covariance(XY)
Does not depend on a prior event or information
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y