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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. i.i.d.
Independently and Identically Distributed
Choose parameters that maximize the likelihood of what observations occurring
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Application of mathematical statistics to economic data to lend empirical support to models

2. Mean reversion
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
Yi = B0 + B1Xi + ui
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
E(XY) - E(X)E(Y)

3. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Variance reverts to a long run level

4. Stochastic error term
Variance(y)/n = variance of sample Y
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Contains variables not explicit in model - Accounts for randomness
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

5. Square root rule
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Low Frequency - High Severity events

6. Discrete random variable
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

7. P - value
Z = (Y - meany)/(stddev(y)/sqrt(n))
P(Z>t)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

8. Control variates technique
Attempts to sample along more important paths
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
95% = 1.65 99% = 2.33 For one - tailed tests
Variance(x) + Variance(Y) + 2*covariance(XY)

9. Central Limit Theorem
For n>30 - sample mean is approximately normal
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Yi = B0 + B1Xi + ui

10. Regime - switching volatility model
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Only requires two parameters = mean and variance
Based on an equation - P(A) = # of A/total outcomes

11. Law of Large Numbers
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Sample mean will near the population mean as the sample size increases
We reject a hypothesis that is actually true
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

12. Statistical (or empirical) model
Yi = B0 + B1Xi + ui
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable

13. Direction of OVB
Z = (Y - meany)/(stddev(y)/sqrt(n))
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

14. Hazard rate of exponentially distributed random 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
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Sample mean +/ - t*(stddev(s)/sqrt(n))

15. Poisson Distribution
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
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

16. GPD
Expected value of the sample mean is the population mean
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
i = ln(Si/Si - 1)

17. Test for unbiasedness
E(mean) = mean
Based on an equation - P(A) = # of A/total outcomes
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.

18. Adjusted R^2
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance(y)/n = variance of sample Y
For n>30 - sample mean is approximately normal
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

19. Potential reasons for fat tails in return distributions
Regression can be non - linear in variables but must be linear in parameters
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

20. Simulation models
Nonlinearity
Contains variables not explicit in model - Accounts for randomness
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

21. Logistic distribution
Only requires two parameters = mean and variance
Has heavy tails
Var(X) + Var(Y)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)

22. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
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

23. F distribution
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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
Sample mean +/ - t*(stddev(s)/sqrt(n))
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

24. Poisson distribution equations for mean variance and std deviation
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
E(mean) = mean

25. POT
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Peaks over threshold - Collects dataset in excess of some threshold
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Expected value of the sample mean is the population mean

26. Maximum likelihood method
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
Choose parameters that maximize the likelihood of what observations occurring
Normal - Student's T - Chi - square - F distribution
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

27. Hybrid method for conditional volatility
Based on a dataset
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
(a^2)(variance(x)
Use historical simulation approach but use the EWMA weighting system

28. Standard variable for non - normal distributions
We reject a hypothesis that is actually true
Transformed to a unit variable - Mean = 0 Variance = 1
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Z = (Y - meany)/(stddev(y)/sqrt(n))

29. Time series data
Rxy = Sxy/(Sx*Sy)
Returns over time for an individual asset
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
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

30. Mean(expected value)
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Population denominator = n - Sample denominator = n - 1
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)

31. Sample correlation
i = ln(Si/Si - 1)
Yi = B0 + B1Xi + ui
Rxy = Sxy/(Sx*Sy)
Variance(X) + Variance(Y) - 2*covariance(XY)

32. Inverse transform method
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Contains variables not explicit in model - Accounts for randomness
Least absolute deviations estimator - used when extreme outliers are not uncommon

33. Variance of X+Y assuming dependence
Variance(x) + Variance(Y) + 2*covariance(XY)
When the sample size is large - the uncertainty about the value of the sample is very small
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

34. Discrete representation of the GBM
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Independently and Identically Distributed
E(XY) - E(X)E(Y)
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))

35. Efficiency
When one regressor is a perfect linear function of the other regressors
Combine to form distribution with leptokurtosis (heavy tails)
Among all unbiased estimators - estimator with the smallest variance is efficient
Summation((xi - mean)^k)/n

36. Covariance
E(XY) - E(X)E(Y)
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
i = ln(Si/Si - 1)

37. Variance of X+b
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Variance(x)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Rxy = Sxy/(Sx*Sy)

38. Block maxima
Model dependent - Options with the same underlying assets may trade at different volatilities
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
P(Z>t)

39. Weibul distribution
Variance(y)/n = variance of sample Y
Based on a dataset
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Among all unbiased estimators - estimator with the smallest variance is efficient

40. Joint probability functions
Probability that the random variables take on certain values simultaneously
Independently and Identically Distributed
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

41. Unconditional vs conditional distributions
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
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

42. Persistence
(a^2)(variance(x)
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
Regression can be non - linear in variables but must be linear in parameters
P - value

43. Chi - squared distribution
P(Z>t)
P(X=x - Y=y) = P(X=x) * P(Y=y)
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
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx

44. Test for statistical independence
P(X=x - Y=y) = P(X=x) * P(Y=y)
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)
We accept a hypothesis that should have been rejected
Confidence set for two coefficients - two dimensional analog for the confidence interval

45. Sample covariance
[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
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y

46. Exact significance level
P - value
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!)
Confidence set for two coefficients - two dimensional analog for the confidence interval
Expected value of the sample mean is the population mean

47. SER
Application of mathematical statistics to economic data to lend empirical support to models
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)

48. Variance(discrete)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Summation((xi - mean)^k)/n
Attempts to sample along more important paths

49. Non - parametric vs parametric calculation of VaR
When one regressor is a perfect linear function of the other regressors
Yi = B0 + B1Xi + ui
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Parameters (mean - volatility - etc) vary over time due to variability in market conditions

50. Variance of sample mean
Independently and Identically Distributed
Rxy = Sxy/(Sx*Sy)
Variance(y)/n = variance of sample Y
Parameters (mean - volatility - etc) vary over time due to variability in market conditions