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FRM Foundations Of Risk Management Quantitative Methods

Subjects : business-skills, certifications, frm

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. BLUE
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Low Frequency - High Severity events
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

2. GPD
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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
Var(X) + Var(Y)

3. Result of combination of two normal with same means
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Least absolute deviations estimator - used when extreme outliers are not uncommon
Combine to form distribution with leptokurtosis (heavy tails)

4. Confidence interval (from t)
Variance reverts to a long run level
Easy to manipulate
Sample mean +/ - t*(stddev(s)/sqrt(n))
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model

5. Reliability
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Statement of the error or precision of an estimate
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

6. GARCH
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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)

7. Persistence
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
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
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
Choose parameters that maximize the likelihood of what observations occurring

8. Adjusted R^2
Variance = (1/m) summation(u<n - i>^2)
Transformed to a unit variable - Mean = 0 Variance = 1
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
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

9. Historical std dev
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Variance(y)/n = variance of sample Y
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

10. Multivariate Density Estimation (MDE)
Choose parameters that maximize the likelihood of what observations occurring
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 variance = (1/(k - 1))Summation(Yi - mean)^2
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)

11. Importance sampling technique
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Attempts to sample along more important paths
Application of mathematical statistics to economic data to lend empirical support to models

12. Biggest (and only real) drawback of GARCH mode
Nonlinearity
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

13. Tractable
Easy to manipulate
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps

14. Logistic distribution
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Statement of the error or precision of an estimate
Has heavy tails
Yi = B0 + B1Xi + ui

15. Mean reversion in asset dynamics
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
P - value
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
Price/return tends to run towards a long - run level

16. Mean reversion in variance
Variance reverts to a long run level
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
(a^2)(variance(x)
Based on a dataset

17. Perfect multicollinearity
Use historical simulation approach but use the EWMA weighting system
95% = 1.65 99% = 2.33 For one - tailed tests
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
When one regressor is a perfect linear function of the other regressors

18. Exponential distribution
For n>30 - sample mean is approximately normal
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Based on a dataset
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

19. LFHS
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Low Frequency - High Severity events
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

20. Test for unbiasedness
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
E(mean) = mean
(a^2)(variance(x)) + (b^2)(variance(y))
Low Frequency - High Severity events

21. Homoskedastic only F - stat
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
E(XY) - E(X)E(Y)
Independently and Identically Distributed

22. Cross - sectional
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Average return across assets on a given day
Regression can be non - linear in variables but must be linear in parameters
Statement of the error or precision of an estimate

23. Lognormal
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
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

24. Significance =1
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
We reject a hypothesis that is actually true
Statement of the error or precision of an estimate
Confidence level

25. Econometrics
Model dependent - Options with the same underlying assets may trade at different volatilities
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Application of mathematical statistics to economic data to lend empirical support to models
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

26. Discrete representation of the GBM
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
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
P(X=x - Y=y) = P(X=x) * P(Y=y)

27. Chi - squared distribution
Peaks over threshold - Collects dataset in excess of some threshold
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
P(Z>t)
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

28. Simplified standard (un - weighted) variance
Population denominator = n - Sample denominator = n - 1
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance = (1/m) summation(u<n - i>^2)
Summation((xi - mean)^k)/n

29. Economical(elegant)
Only requires two parameters = mean and variance
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Random walk (usually acceptable) - Constant volatility (unlikely)
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

30. Pooled data
Variance reverts to a long run level
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

31. Sample correlation
Price/return tends to run towards a long - run level
Attempts to sample along more important paths
We accept a hypothesis that should have been rejected
Rxy = Sxy/(Sx*Sy)

32. P - value
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
P(Z>t)
Based on a dataset
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

33. Conditional probability functions
Rxy = Sxy/(Sx*Sy)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)

34. Unstable return distribution
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

35. Time series data
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
E(XY) - E(X)E(Y)
Returns over time for an individual asset

36. Least squares estimator(m)
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Sample mean +/ - t*(stddev(s)/sqrt(n))
95% = 1.65 99% = 2.33 For one - tailed tests

37. Variance of aX
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
(a^2)(variance(x)
Z = (Y - meany)/(stddev(y)/sqrt(n))

38. Marginal unconditional probability function
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
Does not depend on a prior event or information
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
For n>30 - sample mean is approximately normal

39. 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
E(mean) = mean
Least absolute deviations estimator - used when extreme outliers are not uncommon
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

40. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

41. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

42. Mean reversion
Based on an equation - P(A) = # of A/total outcomes
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
i = ln(Si/Si - 1)
Sampling distribution of sample means tend to be normal

43. Variance of X+b
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Variance(x)

44. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Confidence set for two coefficients - two dimensional analog for the confidence interval
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
Z = (Y - meany)/(stddev(y)/sqrt(n))

45. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

46. Variance of aX + bY
Choose parameters that maximize the likelihood of what observations occurring
Returns over time for an individual asset
Sampling distribution of sample means tend to be normal
(a^2)(variance(x)) + (b^2)(variance(y))

47. 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)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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

48. Sample mean
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Expected value of the sample mean is the population mean
Contains variables not explicit in model - Accounts for randomness

49. 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
Use historical simulation approach but use the EWMA weighting system
P(Z>t)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

50. Multivariate probability
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
More than one random variable
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample