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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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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. Cross - sectional
Based on an equation - P(A) = # of A/total outcomes
Statement of the error or precision of an estimate
Average return across assets on a given day
Probability that the random variables take on certain values simultaneously

2. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails

3. Gamma distribution
Z = (Y - meany)/(stddev(y)/sqrt(n))
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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
(a^2)(variance(x)

4. Bernouli Distribution
Var(X) + Var(Y)
Population denominator = n - Sample denominator = n - 1
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Distribution with only two possible outcomes

5. Central Limit Theorem(CLT)
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
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
If variance of the conditional distribution of u(i) is not constant
Sampling distribution of sample means tend to be normal

6. WLS
Has heavy tails
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
P - value

7. Normal distribution
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Peaks over threshold - Collects dataset in excess of some threshold
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!)

8. POT
When one regressor is a perfect linear function of the other regressors
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Peaks over threshold - Collects dataset in excess of some threshold
P - value

9. R^2
Sample mean +/ - t*(stddev(s)/sqrt(n))
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)
Price/return tends to run towards a long - run level
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

10. Unbiased
Combine to form distribution with leptokurtosis (heavy tails)
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
Mean of sampling distribution is the population mean
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

11. Critical z values
i = ln(Si/Si - 1)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
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)
95% = 1.65 99% = 2.33 For one - tailed tests

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

13. Sample covariance
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
SSR

14. Econometrics
Application of mathematical statistics to economic data to lend empirical support to models
Returns over time for an individual asset
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

15. Chi - squared distribution
Based on an equation - P(A) = # of A/total outcomes
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
Has heavy tails
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

16. Perfect multicollinearity
When one regressor is a perfect linear function of the other regressors
Special type of pooled data in which the cross sectional unit is surveyed over time
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
We accept a hypothesis that should have been rejected

17. Sample mean
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Expected value of the sample mean is the population mean

18. Mean reversion
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
SSR
Attempts to sample along more important paths
We accept a hypothesis that should have been rejected

19. Two assumptions of square root rule
Variance(x)
Random walk (usually acceptable) - Constant volatility (unlikely)
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
Use historical simulation approach but use the EWMA weighting system

20. Historical std dev
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
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

21. Unstable return distribution
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
Does not depend on a prior event or information
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
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

22. GPD
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Mean of sampling distribution is the population mean
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Random walk (usually acceptable) - Constant volatility (unlikely)

23. Variance of X - Y assuming dependence
Only requires two parameters = mean and variance
Variance(X) + Variance(Y) - 2*covariance(XY)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients

24. Empirical frequency
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Based on a dataset
Model dependent - Options with the same underlying assets may trade at different volatilities
Does not depend on a prior event or information

25. Key properties of linear regression
Regression can be non - linear in variables but must be linear in parameters
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
When one regressor is a perfect linear function of the other regressors
P - value

26. Simulation models
Has heavy tails
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
i = ln(Si/Si - 1)
Among all unbiased estimators - estimator with the smallest variance is efficient

27. Tractable
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
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!)
Easy to manipulate
Choose parameters that maximize the likelihood of what observations occurring

28. What does the OLS minimize?
SSR
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

29. Square root rule
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Generate sequence of variables from which price is computed - Calculate value of asset with these prices - Repeat to form distribution
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

30. Sample correlation
Sample mean will near the population mean as the sample size increases
Rxy = Sxy/(Sx*Sy)
Easy to manipulate
Probability that the random variables take on certain values simultaneously

31. Heteroskedastic
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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
If variance of the conditional distribution of u(i) is not constant
Use historical simulation approach but use the EWMA weighting system

32. 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)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
(a^2)(variance(x)
Does not depend on a prior event or information

33. Variance of X+b
P(Z>t)
When the sample size is large - the uncertainty about the value of the sample is very small
Contains variables not explicit in model - Accounts for randomness
Variance(x)

34. Exponential distribution
For n>30 - sample mean is approximately normal
Standard deviation of the sampling distribution SE = std dev(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)
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta

35. Beta distribution
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Summation((xi - mean)^k)/n

36. Pooled data
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Yi = B0 + B1Xi + ui
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Summation((xi - mean)^k)/n

37. Result of combination of two normal with same means
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Combine to form distribution with leptokurtosis (heavy tails)
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

38. Time series data
Model dependent - Options with the same underlying assets may trade at different volatilities
Population denominator = n - Sample denominator = n - 1
For n>30 - sample mean is approximately normal
Returns over time for an individual asset

39. Standard variable for non - normal distributions
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Normal - Student's T - Chi - square - F distribution
Z = (Y - meany)/(stddev(y)/sqrt(n))
Special type of pooled data in which the cross sectional unit is surveyed over time

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

41. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
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
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

42. Overall F - statistic
Choose parameters that maximize the likelihood of what observations occurring
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

43. SER
[1/(n - 1)]*summation((Xi - X)(Yi - 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
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Variance(x) + Variance(Y) + 2*covariance(XY)

44. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Mean of sampling distribution is the population mean
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
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

45. LFHS
E(XY) - E(X)E(Y)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Low Frequency - High Severity events
Distribution with only two possible outcomes

46. Unconditional vs conditional distributions
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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

47. Statistical (or empirical) model
Mean = np - Variance = npq - Std dev = sqrt(npq)
Yi = B0 + B1Xi + ui
Rxy = Sxy/(Sx*Sy)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

48. EWMA
Variance = (1/m) summation(u<n - i>^2)
i = ln(Si/Si - 1)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
When the sample size is large - the uncertainty about the value of the sample is very small

49. Skewness
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
Peaks over threshold - Collects dataset in excess of some threshold
When the sample size is large - the uncertainty about the value of the sample is very small
Does not depend on a prior event or information

50. Variance - covariance approach for VaR of a portfolio
Based on an equation - P(A) = # of A/total outcomes
E(XY) - E(X)E(Y)
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