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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. Variance of aX
Low Frequency - High Severity events
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Probability that the random variables take on certain values simultaneously
(a^2)(variance(x)

2. Mean reversion
P - value
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

3. Standard normal distribution
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Transformed to a unit variable - Mean = 0 Variance = 1
Expected value of the sample mean is the population mean

4. Reliability
95% = 1.65 99% = 2.33 For one - tailed tests
Statement of the error or precision of an estimate
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

5. Hybrid method for conditional volatility
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Use historical simulation approach but use the EWMA weighting system
Variance = (1/m) summation(u<n - i>^2)
SSR

6. Bootstrap method
Based on an equation - P(A) = # of A/total outcomes
Only requires two parameters = mean and variance
Mean = np - Variance = npq - Std dev = sqrt(npq)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

7. Law of Large Numbers
E(XY) - E(X)E(Y)
Peaks over threshold - Collects dataset in excess of some threshold
Least absolute deviations estimator - used when extreme outliers are not uncommon
Sample mean will near the population mean as the sample size increases

8. Covariance
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
Sample mean will near the population mean as the sample size increases
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
E(XY) - E(X)E(Y)

9. Shortcomings of implied volatility
Probability that the random variables take on certain values simultaneously
Does not depend on a prior event or information
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Model dependent - Options with the same underlying assets may trade at different volatilities

10. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Easy to manipulate

11. WLS
E(mean) = mean
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Model dependent - Options with the same underlying assets may trade at different volatilities

12. Biggest (and only real) drawback of GARCH mode
Var(X) + Var(Y)
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
Nonlinearity
Normal - Student's T - Chi - square - F distribution

13. Two assumptions of square root rule
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Expected value of the sample mean is the population mean
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Random walk (usually acceptable) - Constant volatility (unlikely)

14. Poisson distribution equations for mean variance and std deviation
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Based on an equation - P(A) = # of A/total outcomes
Variance reverts to a long run level

15. Key properties of linear regression
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Regression can be non - linear in variables but must be linear in parameters
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 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)

16. Marginal unconditional probability function
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Does not depend on a prior event or information
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

17. Cross - sectional
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Choose parameters that maximize the likelihood of what observations occurring
Average return across assets on a given day

18. Continuously compounded return equation
Choose parameters that maximize the likelihood of what observations occurring
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)
Summation((xi - mean)^k)/n
i = ln(Si/Si - 1)

19. Statistical (or empirical) model
Based on a dataset
Random walk (usually acceptable) - Constant volatility (unlikely)
Yi = B0 + B1Xi + ui
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

20. Binomial 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!)
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Variance = (1/m) summation(u<n - i>^2)

21. Maximum likelihood method
Choose parameters that maximize the likelihood of what observations occurring
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

22. Critical z values
Z = (Y - meany)/(stddev(y)/sqrt(n))
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
95% = 1.65 99% = 2.33 For one - tailed tests
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

23. Two ways to calculate historical volatility
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
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance(X) + Variance(Y) - 2*covariance(XY)

24. Variance of X+b
Variance(x)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Probability that the random variables take on certain values simultaneously
Summation((xi - mean)^k)/n

25. Type I error
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Least absolute deviations estimator - used when extreme outliers are not uncommon
Variance(X) + Variance(Y) - 2*covariance(XY)
We reject a hypothesis that is actually true

26. Covariance calculations using weight sums (lambda)
(a^2)(variance(x)) + (b^2)(variance(y))
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Choose parameters that maximize the likelihood of what observations occurring
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

27. Inverse transform method
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Mean of sampling distribution is the population mean
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
E(mean) = mean

28. Mean(expected value)
P(Z>t)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Among all unbiased estimators - estimator with the smallest variance is efficient
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

29. ESS
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Mean of sampling distribution is the population mean

30. R^2
P(Z>t)
SSR
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
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

31. Pooled data
Sampling distribution of sample means tend to be normal
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Returns over time for an individual asset
Contains variables not explicit in model - Accounts for randomness

32. Confidence interval (from t)
Statement of the error or precision of an estimate
Sample mean +/ - t*(stddev(s)/sqrt(n))
We reject a hypothesis that is actually true
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

33. P - value
When the sample size is large - the uncertainty about the value of the sample is very small
Price/return tends to run towards a long - run level
Mean = np - Variance = npq - Std dev = sqrt(npq)
P(Z>t)

34. Unstable return distribution
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Sampling distribution of sample means tend to be normal

35. Heteroskedastic
If variance of the conditional distribution of u(i) is not constant
(a^2)(variance(x)
Combine to form distribution with leptokurtosis (heavy tails)
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!)

36. Variance of X - Y assuming dependence
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)
SSR
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Variance(X) + Variance(Y) - 2*covariance(XY)

37. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
More than one random variable
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

38. Importance sampling technique
Independently and Identically Distributed
Confidence level
Attempts to sample along more important paths
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)

39. Unconditional vs conditional distributions
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
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

40. Homoskedastic
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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Special type of pooled data in which the cross sectional unit is surveyed over time

41. GEV
(a^2)(variance(x)) + (b^2)(variance(y))
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

42. Limitations of R^2 (what an increase doesn't necessarily imply)

43. 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)
Statement of the error or precision of an estimate
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
Variance reverts to a long run level

44. 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
Probability that the random variables take on certain values simultaneously
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!)
For n>30 - sample mean is approximately normal

45. EWMA
Concerned with a single random variable (ex. Roll of a die)
Sample mean will near the population mean as the sample size increases
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Random walk (usually acceptable) - Constant volatility (unlikely)

46. Sample variance
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
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
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)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

47. Significance =1
Confidence level
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
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)
Does not depend on a prior event or information

48. Economical(elegant)
Only requires two parameters = mean and variance
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

49. Implied standard deviation for options
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Variance(x) + Variance(Y) + 2*covariance(XY)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

50. Gamma distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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