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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. Block maxima
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
P - value
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.

2. Joint probability functions
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
Probability that the random variables take on certain values simultaneously
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
Sample mean will near the population mean as the sample size increases

3. Simplified standard (un - weighted) variance
Variance = (1/m) summation(u<n - i>^2)
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)
When the sample size is large - the uncertainty about the value of the sample is very small

4. Economical(elegant)
Only requires two parameters = mean and variance
Summation((xi - mean)^k)/n
Variance(x) + Variance(Y) + 2*covariance(XY)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

5. Cholesky factorization (decomposition)
Use historical simulation approach but use the EWMA weighting system
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)
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 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

6. BLUE
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
Choose parameters that maximize the likelihood of what observations occurring
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

7. P - value
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
P(Z>t)
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
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

8. GARCH
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)
Mean of sampling distribution is the population mean
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
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)

9. Homoskedastic only F - stat
Returns over time for a combination of assets (combination of time series and cross - sectional data)
Based on a dataset
Least absolute deviations estimator - used when extreme outliers are not uncommon
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)

10. Importance sampling technique
Sampling distribution of sample means tend to be normal
Attempts to sample along more important paths
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
Regression can be non - linear in variables but must be linear in parameters

11. Bootstrap method
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period

12. Statistical (or empirical) model
P(X=x - Y=y) = P(X=x) * P(Y=y)
Normal - Student's T - Chi - square - F distribution
Yi = B0 + B1Xi + ui
When the sample size is large - the uncertainty about the value of the sample is very small

13. Law of Large Numbers
Based on a dataset
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)
Sample mean will near the population mean as the sample size increases
SSR

14. 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
Variance(x)
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
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

15. Variance of aX + bY
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
(a^2)(variance(x)) + (b^2)(variance(y))
E(XY) - E(X)E(Y)
Variance reverts to a long run level

16. Historical std dev
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)
Simplest and most common way to estimate future volatility - Variance(t) = (1/N) Summation(r^2)
Variance reverts to a long run level
Probability that the random variables take on certain values simultaneously

17. What does the OLS minimize?
SSR
Average return across assets on a given day
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Peaks over threshold - Collects dataset in excess of some threshold

18. 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
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
We accept a hypothesis that should have been rejected

19. Hybrid method for conditional volatility
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
Use historical simulation approach but use the EWMA weighting system
Model dependent - Options with the same underlying assets may trade at different volatilities
Covariance = (lambda)(cov(n - 1)) + (1 - lambda)(xn - 1)(yn - 1)

20. Normal distribution
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Normal - Student's T - Chi - square - F 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
Yi = B0 + B1Xi + ui

21. Variance of aX
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
(a^2)(variance(x)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Statement of the error or precision of an estimate

22. Sample correlation
Application of mathematical statistics to economic data to lend empirical support to models
Probability of an outcome given another outcome P(Y|X) = P(X -Y)/P(X) - P(B|A) = P(A and B)/P(A)
Rxy = Sxy/(Sx*Sy)
More than one random variable

23. Single variable (univariate) probability
Among all unbiased estimators - estimator with the smallest variance is efficient
Rxy = Sxy/(Sx*Sy)
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
Concerned with a single random variable (ex. Roll of a die)

24. Variance of sampling distribution of means when n<N
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
Least absolute deviations estimator - used when extreme outliers are not uncommon
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
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state

25. Deterministic Simulation
[1/(n - 1)]*summation((Xi - X)(Yi - Y))
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
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
P(Z>t)

26. Variance of X - Y assuming dependence
Among all unbiased estimators - estimator with the smallest variance is efficient
Variance(X) + Variance(Y) - 2*covariance(XY)
95% = 1.65 99% = 2.33 For one - tailed tests
Price/return tends to run towards a long - run level

27. Variance of sample mean
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)
We reject a hypothesis that is actually true
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Variance(y)/n = variance of sample Y

28. Variance(discrete)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Summation(Yi - m)^2 = 1 - Minimizes the sum of squares gaps
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
We accept a hypothesis that should have been rejected

29. Direction of OVB
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Attempts to sample along more important paths
Infinite number of values within an interval - P(a<x<b) = interval from a to b of f(x)dx
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications

30. Stochastic error term
Contains variables not explicit in model - Accounts for randomness
Special type of pooled data in which the cross sectional unit is surveyed over time
Application of mathematical statistics to economic data to lend empirical support to models
Concerned with a single random variable (ex. Roll of a die)

31. Homoskedastic
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
For n>30 - sample mean is approximately normal
Application of mathematical statistics to economic data to lend empirical support to models
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

32. Continuous representation of the GBM
For n>30 - sample mean is approximately normal
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)
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
1/lambda is hazard rate of default intensity - Lambda = 1/beta - f(x) = lambda e^( - lambdax) -F(x) = 1 - e^( - lambda*x)

33. F distribution
Sampling distribution of sample means tend to be normal
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Nonlinearity
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

34. R^2
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

35. Panel data (longitudinal or micropanel)
Special type of pooled data in which the cross sectional unit is surveyed over time
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
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

36. Mean(expected value)
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
Sample variance = (1/(k - 1))Summation(Yi - mean)^2

37. Implied standard deviation for options
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

38. Covariance
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
E(XY) - E(X)E(Y)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

39. ESS
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Rxy = Sxy/(Sx*Sy)
Attempts to increase accuracy by reducing sample variance instead of increasing sample size

40. Antithetic variable technique
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Peaks over threshold - Collects dataset in excess of some threshold
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

41. Marginal unconditional probability function
Does not depend on a prior event or information
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
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)
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

42. Confidence interval (from t)
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Sample mean will near the population mean as the sample size increases
Price/return tends to run towards a long - run level
Sample mean +/ - t*(stddev(s)/sqrt(n))

43. Two assumptions of square root rule
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
Random walk (usually acceptable) - Constant volatility (unlikely)
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric
Based on an equation - P(A) = # of A/total outcomes

44. Potential reasons for fat tails in return distributions
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Sample mean will near the population mean as the sample size increases
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates
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!)

45. POT
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Peaks over threshold - Collects dataset in excess of some threshold
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
F = ½ ((t1^2)+(t2^2) - (correlation t1 t2))/(1 - 2correlation)

46. Extreme Value Theory
P(X=x - Y=y) = P(X=x) * P(Y=y)
EVT - Fits a separate distribution to the extreme loss tail - Only uses tail
Confidence set for two coefficients - two dimensional analog for the confidence interval
Transformed to a unit variable - Mean = 0 Variance = 1

47. WLS
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)
Regression can be non - linear in variables but must be linear in parameters
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Confidence set for two coefficients - two dimensional analog for the confidence interval

48. Two drawbacks of moving average series
Choose parameters that maximize the likelihood of what observations occurring
Ignores order of observations (no weight for most recent observations) - Has a ghosting feature where data points are dropped due to length of window
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2

49. K - th moment
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
We reject a hypothesis that is actually true
P(Z>t)
Summation((xi - mean)^k)/n

50. Hazard rate of exponentially distributed 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)
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
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)