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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. GEV
Yi = B0 + B1Xi + ui
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
Price/return tends to run towards a long - run level
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

2. Continuous random variable
Only requires two parameters = mean and variance
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Concerned with a single random variable (ex. Roll of a die)

3. Unconditional vs conditional distributions
Independently and Identically Distributed
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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

4. Cross - sectional
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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
Average return across assets on a given day
Sample mean will near the population mean as the sample size increases

5. Sample mean
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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
Expected value of the sample mean is the population mean
Choose parameters that maximize the likelihood of what observations occurring

6. Sample variance
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Sample variance = (1/(k - 1))Summation(Yi - mean)^2
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
Expected value of the sample mean is the population mean

7. Discrete representation of the GBM
If variance of the conditional distribution of u(i) is not constant
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
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

8. BLUE
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Distribution with only two possible outcomes
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Confidence level

9. Exponential distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Time to wait until an event takes place - F(x) = lambda e^( - lambdax) - Lambda = 1/beta
Normal - Student's T - Chi - square - F distribution
E(mean) = mean

10. Covariance calculations using weight sums (lambda)
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Independently and Identically Distributed

11. Central Limit Theorem
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
Nonlinearity
For n>30 - sample mean is approximately normal
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail

12. Economical(elegant)
Transformed to a unit variable - Mean = 0 Variance = 1
Only requires two parameters = mean and variance
Low Frequency - High Severity events
Rxy = Sxy/(Sx*Sy)

13. Variance(discrete)
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
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

14. Block maxima
Variance reverts to a long run level
Among all unbiased estimators - estimator with the smallest variance is efficient
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
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

15. P - value
Population denominator = n - Sample denominator = n - 1
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)
Normal - Student's T - Chi - square - F distribution
P(Z>t)

16. Sample covariance
E(XY) - E(X)E(Y)
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Choose parameters that maximize the likelihood of what observations occurring
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

17. Single variable (univariate) probability
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
When one regressor is a perfect linear function of the other regressors
Concerned with a single random variable (ex. Roll of a die)
P(Z>t)

18. Historical std dev
For n>30 - sample mean is approximately normal
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)

19. Exact significance level
P - value
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
Independently and Identically Distributed
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

20. T distribution
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
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Random walk (usually acceptable) - Constant volatility (unlikely)
i = ln(Si/Si - 1)

21. Two requirements of OVB
Probability that the random variables take on certain values simultaneously
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
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!)
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

22. SER
Normal - Student's T - Chi - square - F distribution
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one

23. Variance of sample mean
Variance(y)/n = variance of sample Y
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
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Confidence level

24. Variance of X - Y assuming dependence
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Variance(X) + Variance(Y) - 2*covariance(XY)

25. Pooled data
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Mean of sampling distribution is the population mean
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
Returns over time for a combination of assets (combination of time series and cross - sectional data)

26. Non - parametric vs parametric calculation of VaR
Average return across assets on a given day
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Probability that the random variables take on certain values simultaneously
Choose parameters that maximize the likelihood of what observations occurring

27. Mean reversion in variance
Variance reverts to a long run level
Normal - Student's T - Chi - square - F distribution
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Model dependent - Options with the same underlying assets may trade at different volatilities

28. Chi - squared distribution
Among all unbiased estimators - estimator with the smallest variance is efficient
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
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

29. Joint probability functions
Probability that the random variables take on certain values simultaneously
Variance(X) + Variance(Y) - 2*covariance(XY)
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

30. Law of Large Numbers
Sample mean will near the population mean as the sample size increases
Price/return tends to run towards a long - run level
SSR
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

31. Continuous representation of the GBM
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Has heavy tails
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^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)

32. Implications of homoscedasticity
SSR
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
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
Special type of pooled data in which the cross sectional unit is surveyed over time

33. Confidence interval (from t)
Probability that the random variables take on certain values simultaneously
Sample mean +/ - t*(stddev(s)/sqrt(n))
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
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

34. Variance - covariance approach for VaR of a portfolio
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
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
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

35. Type I error
We reject a hypothesis that is actually true
If variance of the conditional distribution of u(i) is not constant
Var(X) + Var(Y)
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d

36. Variance of X+Y assuming dependence
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Variance(x) + Variance(Y) + 2*covariance(XY)
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)
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

37. Cholesky factorization (decomposition)
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
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

38. Overall F - statistic
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Z = (Y - meany)/(stddev(y)/sqrt(n))

39. Standard error for Monte Carlo replications
Use historical simulation approach but use the EWMA weighting system
Nonlinearity
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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

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

41. Hybrid method for conditional volatility
Confidence set for two coefficients - two dimensional analog for the confidence interval
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
Use historical simulation approach but use the EWMA weighting system
Combine to form distribution with leptokurtosis (heavy tails)

42. Binomial distribution equations for mean variance and std dev
Returns over time for an individual asset
E(XY) - E(X)E(Y)
Mean = np - Variance = npq - Std dev = sqrt(npq)
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

43. Multivariate probability
Probability that the random variables take on certain values simultaneously
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
More than one random variable
Peaks over threshold - Collects dataset in excess of some threshold

44. Econometrics
Variance = (1/m) summation(u<n - i>^2)
Application of mathematical statistics to economic data to lend empirical support to models
We accept a hypothesis that should have been rejected
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption

45. Key properties of linear regression
Variance = (1/m) summation(u<n - i>^2)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Regression can be non - linear in variables but must be linear in parameters

46. Mean reversion
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)
We accept a hypothesis that should have been rejected

47. F distribution
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
When the sample size is large - the uncertainty about the value of the sample is very small
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!)
Variance(x) + Variance(Y) + 2*covariance(XY)

48. Gamma distribution
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
Rxy = Sxy/(Sx*Sy)
Combine to form distribution with leptokurtosis (heavy tails)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

49. POT
Peaks over threshold - Collects dataset in excess of some threshold
Distribution with only two possible outcomes
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
Mean = np - Variance = npq - Std dev = sqrt(npq)

50. Consistent
When the sample size is large - the uncertainty about the value of the sample is very small
Statement of the error or precision of an estimate
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)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)