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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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Match each statement with the correct term.
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1. Empirical frequency
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Based on a dataset
Confidence set for two coefficients - two dimensional analog for the confidence interval
More than one random variable

2. Covariance
E(mean) = mean
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
E(XY) - E(X)E(Y)
Contains variables not explicit in model - Accounts for randomness

3. Exact significance level
Concerned with a single random variable (ex. Roll of a die)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Peaks over threshold - Collects dataset in excess of some threshold
P - value

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

5. Continuous representation of the GBM
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
E(XY) - E(X)E(Y)
Based on an equation - P(A) = # of A/total outcomes
dS<t> = (mean<t>)(S<t>)dt + stddev(t)S<t>dt- GBM - Geometric Brownian Motion - Represented as drift + shock - Drift = mean * change in time - Shock = std dev E sqrt(change in time)

6. K - th moment
Summation((xi - mean)^k)/n
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

7. R^2
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Mean of sampling distribution is the population mean
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

8. Econometrics
Mean = np - Variance = npq - Std dev = sqrt(npq)
Application of mathematical statistics to economic data to lend empirical support to models
Variance reverts to a long run level
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

9. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
Sample mean +/ - t*(stddev(s)/sqrt(n))
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Z = (Y - meany)/(stddev(y)/sqrt(n))

10. Discrete representation of the GBM
Peaks over threshold - Collects dataset in excess of some threshold
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

11. Panel data (longitudinal or micropanel)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Special type of pooled data in which the cross sectional unit is surveyed over time
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)

12. Homoskedastic only F - stat
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Sample mean +/ - t*(stddev(s)/sqrt(n))
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

13. Central Limit Theorem
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
For n>30 - sample mean is approximately normal
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

14. Binomial distribution equations for mean variance and std dev
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Regression can be non - linear in variables but must be linear in parameters

15. Type I error
We reject a hypothesis that is actually true
Based on an equation - P(A) = # of A/total outcomes
Probability that the random variables take on certain values simultaneously
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

16. LFHS
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Based on a dataset
We accept a hypothesis that should have been rejected
Low Frequency - High Severity events

17. Four sampling distributions

18. Implied standard deviation for options
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Model dependent - Options with the same underlying assets may trade at different volatilities
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

19. Confidence interval for sample 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
Easy to manipulate
SSR
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample

20. Consistent
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
When the sample size is large - the uncertainty about the value of the sample is very small
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window

21. Joint probability functions
Peaks over threshold - Collects dataset in excess of some threshold
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Probability that the random variables take on certain values simultaneously

22. Block maxima
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Probability that the random variables take on certain values simultaneously
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

23. Heteroskedastic
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
If variance of the conditional distribution of u(i) is not constant
P(Z>t)
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

24. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Variance = (1/m) summation(u<n - i>^2)
Easy to manipulate
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

25. P - value
P(Z>t)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Attempts to sample along more important paths
Based on a dataset

26. T distribution
[1/(n - 1)]*summation((Xi - X)(Yi - 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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

27. Multivariate Density Estimation (MDE)
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Sample mean will near the population mean as the sample size increases
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
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

28. Deterministic Simulation
Statement of the error or precision of an estimate
Rxy = Sxy/(Sx*Sy)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
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

29. Historical std dev
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Based on a dataset

30. Direction of OVB
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Regression can be non - linear in variables but must be linear in parameters
i = ln(Si/Si - 1)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

31. Hybrid method for conditional volatility
P(X=x - Y=y) = P(X=x) * P(Y=y)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Population denominator = n - Sample denominator = n - 1
Use historical simulation approach but use the EWMA weighting system

32. Antithetic variable technique
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
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
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Use historical simulation approach but use the EWMA weighting system

33. Gamma distribution
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
Concerned with a single random variable (ex. Roll of a die)
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
Var(X) + Var(Y)

34. Two drawbacks of moving average series
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 = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Price/return tends to run towards a long - run level

35. Continuously compounded return equation
Price/return tends to run towards a long - run level
i = ln(Si/Si - 1)
When one regressor is a perfect linear function of the other regressors
Summation((xi - mean)^k)/n

36. SER
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
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

37. Non - parametric vs parametric calculation of VaR
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Use historical simulation approach but use the EWMA weighting system
Sample mean will near the population mean as the sample size increases
Confidence set for two coefficients - two dimensional analog for the confidence interval

38. Standard variable for non - normal distributions
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 of sampling distribution is the population mean
Z = (Y - meany)/(stddev(y)/sqrt(n))
Combine to form distribution with leptokurtosis (heavy tails)

39. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Expected value of the sample mean is the population mean
For n>30 - sample mean is approximately normal
Attempts to sample along more important paths

40. Chi - squared distribution
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
Easy to manipulate
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

41. Mean reversion in variance
Variance reverts to a long run level
Low Frequency - High Severity events
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Based on an equation - P(A) = # of A/total outcomes

42. GPD
Variance reverts to a long run level
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)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Contains variables not explicit in model - Accounts for randomness

43. Two requirements of OVB
Based on an equation - P(A) = # of A/total outcomes
Variance = (1/m) summation(u<n - i>^2)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

44. Importance sampling technique
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
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
Attempts to sample along more important paths

45. Standard error for Monte Carlo replications
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
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Use historical simulation approach but use the EWMA weighting system

46. Adjusted R^2
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

47. Variance of X - Y assuming dependence
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(X) + Variance(Y) - 2*covariance(XY)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

48. What does the OLS minimize?
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
(a^2)(variance(x)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
SSR

49. Extreme Value Theory
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Peaks over threshold - Collects dataset in excess of some threshold
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Low Frequency - High Severity events

50. Sample variance
Rxy = Sxy/(Sx*Sy)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
We reject a hypothesis that is actually true
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))