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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. Unstable return distribution
Contains variables not explicit in model - Accounts for randomness
Parameters (mean - volatility - etc) vary over time due to variability in market conditions
Low Frequency - High Severity events
For n>30 - sample mean is approximately normal

2. Sample mean
Weights are not a function of time - but based on the nature of the historic period (more similar to historic stake - greater the weight)
Expected value of the sample mean is the population mean
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
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

3. Weibul distribution
Rxy = Sxy/(Sx*Sy)
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
Can Use alpha and beta weights to solve for the long - run average variance - VL = w/(1 - alpha - beta)

4. 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
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Confidence set for two coefficients - two dimensional analog for the confidence interval
Mean of sampling distribution is the population mean

5. GPD
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
We accept a hypothesis that should have been rejected
Variance(X) + Variance(Y) - 2*covariance(XY)
Has heavy tails

6. Bernouli Distribution
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
Sample mean will near the population mean as the sample size increases
Distribution with only two possible outcomes
Sample mean +/ - t*(stddev(s)/sqrt(n))

7. Type I error
Based on a dataset
We reject a hypothesis that is actually true
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!)
Reverse engineer the implied std dev from the market price - Cmarket = f(implied standard deviation)

8. Standard error
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
Variance = summation(alpha weight)(u<n - i>^2) - alpha weights must sum to one
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Coefficent of determination - fraction of variance explained by independent variables - R^2 = ESS/TSS = 1 - (SSR/TSS)

9. T distribution
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
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
Sample mean +/ - t*(stddev(s)/sqrt(n))
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)

10. Implications of homoscedasticity
i = ln(Si/Si - 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
Generalized Extreme Value Distribution - Uses a tail index - smaller index means fatter tails
E(XY) - E(X)E(Y)

11. P - value
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
P(Z>t)
Average return across assets on a given day
Transformed to a unit variable - Mean = 0 Variance = 1

12. Poisson distribution equations for mean variance and std deviation
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
Historical simulation with replacement - Vector is chosen at random from historic period for each simulated period
Use historical simulation approach but use the EWMA weighting system
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

13. Variance of X+Y
Easy to manipulate
Var(X) + Var(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)
Regression can be non - linear in variables but must be linear in parameters

14. Single variable (univariate) probability
Concerned with a single random variable (ex. Roll of a die)
Based on an equation - P(A) = # of A/total outcomes
Non - parametric directly uses a historical dataset - Parametric imposes a specific distribution assumption
Two parameters: alpha(center) and beta(shape) - - Popular for modeling recovery rates

15. Homoskedastic
Best Linear Unbiased Estimator - Sample mean for samples that are i.i.d.
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
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
Sample mean +/ - t*(stddev(s)/sqrt(n))

16. What does the OLS minimize?
Var(X) + Var(Y)
Apply today's weight for yesterday's returns "what would happen if we held this portfolio in the past"
SSR
Compute series of periodic returns - Choose a weighting scheme to translate a series into a single metric

17. 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)
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)
Sampling distribution of sample means tend to be normal
Variance(y)/n = variance of sample Y

18. Unbiased
Has heavy tails
Generalized Pareto Distribution - Models distribution of POT - Empirical distributions are rarely sufficient for this model
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Mean of sampling distribution is the population mean

19. Econometrics
Statement of the error or precision of an estimate
Unconditional is the same regardless of market or economic conditions (unrealistic) - Conditional depends on the economy - market - or other state
Application of mathematical statistics to economic data to lend empirical support to models
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

20. Reliability
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
Among all unbiased estimators - estimator with the smallest variance is efficient
Based on an equation - P(A) = # of A/total outcomes
Statement of the error or precision of an estimate

21. Stochastic error term
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Contains variables not explicit in model - Accounts for randomness
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

22. Cholesky factorization (decomposition)
Standard error of error term - SER = sqrt(SSR/(n - k - 1)) - K is the # of slope coefficients
P(Z>t)
Application of mathematical statistics to economic data to lend empirical support to models
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

23. LFHS
Low Frequency - High Severity events
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
(a^2)(variance(x)) + (b^2)(variance(y))
Variance(x) + Variance(Y) + 2*covariance(XY)

24. Empirical frequency
Normal - Student's T - Chi - square - F distribution
Based on a dataset
X - t(Sx/sqrt(n))<meanx<x + t(Sx/sqrt(n)) - Random interval since it will vary by the sample
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha

25. Variance of X+Y assuming dependence
E(XY) - E(X)E(Y)
Variance(x) + Variance(Y) + 2*covariance(XY)
Variance(X) + Variance(Y) - 2*covariance(XY)
Use historical simulation approach but use the EWMA weighting system

26. Standard error for Monte Carlo replications
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
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
Omitted variable is correlated with regressor - Omitted variable is a determinant of the dependent variable
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)

27. Panel data (longitudinal or micropanel)
Sampling distribution of sample means tend to be normal
SE(predicted std dev) = std dev * sqrt(1/2T) - Ten times more precision needs 100 times more replications
Special type of pooled data in which the cross sectional unit is surveyed over time
Mean of sampling distribution is the population mean

28. Inverse transform method
Expected value of the sample mean is the population mean
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
[1/(n - 1)]*summation((Xi - X)(Yi - Y))

29. Block maxima
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Sample mean will near the population mean as the sample size increases
We accept a hypothesis that should have been rejected
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

30. SER
T = (x - meanx)/(stddev(x)/sqrt(n)) - Symmetrical - mean = 0 - Variance = k/k - 2 - Slightly heavy tail (kurtosis>3)
Standard deviation of the sampling distribution SE = std dev(y)/sqrt(n)
Expected value of the sample mean is the population mean
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

31. Difference between population and sample variance
Population denominator = n - Sample denominator = n - 1
Dataset is parsed into blocks with greater length than the periodicity - Observations must be i.i.d.
Has heavy tails
Assumes a value among a finite set including x1 - x2 - etc - P(X=xk) = f(xk)

32. 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
(a^2)(variance(x)) + (b^2)(variance(y))
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)

33. Shortcomings of implied volatility
Model dependent - Options with the same underlying assets may trade at different volatilities
Probability that the random variables take on certain values simultaneously
For n>30 - sample mean is approximately normal
Changes the sign of the random samples - appropriate when distribution is symmetric - creates twice as many replications

34. Regime - switching volatility model
Statement of the error or precision of an estimate
Discrete: E(Y) = Summation(xi*pi) - Continuous: E(X) = integral(x*f(x)dx)
When a distribution switches from high to low volatility - but never in between - Will exhibit fat tails of unaccounted for
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

35. 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)
Combine to form distribution with leptokurtosis (heavy tails)
Explained sum of squares - Summation[(predicted yi - meany)^2] - Squared distance between the predicted y and the mean of y
Based on an equation - P(A) = # of A/total outcomes

36. EWMA
Normal - Student's T - Chi - square - F distribution
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
Mean = np - Variance = npq - Std dev = sqrt(npq)
Exponentially Weighted Moving Average - Weights decline in constant proportion given by lambda

37. Binomial distribution
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!)
Mean of sampling distribution is the population mean
Conditional mean is time - varying - Conditional volatility is time - varying (more likely)
Variance(y)/n = variance of sample Y

38. Standard variable for non - normal distributions
P - value
Variance(x) + Variance(Y) + 2*covariance(XY)
Z = (Y - meany)/(stddev(y)/sqrt(n))
Regression can be non - linear in variables but must be linear in parameters

39. Kurtosis
(a^2)(variance(x)
Generalized exponential distribution - Exponential is a Weibull distribution with alpha = 1.0 - F(x) = 1 - e^ - (x/beta)^alpha
E[(Y - meany)^2] = E(Y^2) - [E(Y)]^2
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment

40. Sample correlation
Rxy = Sxy/(Sx*Sy)
Regression can be non - linear in variables but must be linear in parameters
E[variance(n+t)] = VL + ((alpha + beta)^t)*(variance(n) - VL)
Easy to manipulate

41. Tractable
Easy to manipulate
Does not depend on a prior event or information
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
Among all unbiased estimators - estimator with the smallest variance is efficient

42. Variance - covariance approach for VaR of a portfolio
Change in S = S<t - 1>(meanchange in time + stddev E * sqrt(change in time))
Sampling distribution of sample means tend to be normal
Z = (Y - meany)/(stddev(y)/sqrt(n))
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events

43. Variance of X+b
Variance(x)
P - value
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Mean = lambda - Variance = lambda - Std dev = sqrt(lambda)

44. Efficiency
Among all unbiased estimators - estimator with the smallest variance is efficient
Sample mean will near the population mean as the sample size increases
Variance reverts to a long run level
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

45. Direction of OVB
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
P(Z>t)
Make parametric assumptions about covariances of each position and extend them to entire portfolio - Problem: correlations change during stressful market events
Depends on whether X and mean are positively or negatively correlated - Beta1 = beta1 + correlation(x -mean)*(stddev(mean)/stddev(x))

46. WLS
Measures degree of "peakedness" - Value of 3 indicates normal distribution - Sigma^4 = E[(X - mean)^4]/sigma^4 - Function of fourth moment
Easy to manipulate
Attempts to increase accuracy by reducing sample variance instead of increasing sample size
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE

47. Monte Carlo Simulations
Weighted least squares estimator - Weights the squares to account for heteroskedasticity and is BLUE
Mean = np - Variance = npq - Std dev = sqrt(npq)
Generation of a distribution of returns by use of random numbers - Return path decided by algorithm - Correlation must be modeled
Translates a random number into a cumulative standard normal distribution - EXCEL: NORMSINV(RAND())

48. Variance of aX + bY
Price/return tends to run towards a long - run level
Simplest approach to extending horizon - J - period VaR = sqrt(J) * 1 - period VaR - Only applies under i.i.d
Nonlinearity
(a^2)(variance(x)) + (b^2)(variance(y))

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

50. Homoskedastic only F - stat
Depends upon lambda - which indicates the rate of occurrence of the random events (binomial) over a time interval - (lambda^k)/(k!) * e^( - lambda)
Variance(sample y) = (variance(y)/n)*(N - n/N - 1)
F = [(SSR<restricted> - SSR<unrestricted>)/q]/(SSR<unrestricted>/(n - k<unrestricted> - 1)
Regression can be non - linear in variables but must be linear in parameters